id
string | text
string | source
string | created
timestamp[s] | added
string | metadata
dict |
|---|---|---|---|---|---|
1203.4057
|
Polarization and spin correlation parameters in proton knockout reactions from
s1/2-orbits at 1 GeV
O.V. Miklukho (1), A.Yu. Kisselev (1), D.A. Aksenov (1), G.M. Amalsky (1) ,
V.A. Andreev (1), S.V. Evstiukhin (1), A.E. Ezhilov (1), O.Ya. Fedorov (1),
G.E. Gavrilov (1), D.S. Ilyin (1), A.A. Izotov (1), L.M. Kochenda (1),
P.A. Kravtsov (1), M.P. Levchenko (1), D.A. Maysuzenko (1), V.A. Murzin (1),
T. Noro (2), D.V. Novinsky (1), V.A. Oreshkin (1), A.N. Prokofiev (1),
A.V. Shvedchikov (1), V.Yu. Trautman (1), S.I. Trush (1) and A.A. Zhdanov (1)
(1) St.Petersburg Nuclear Physics Institute, Gatchina, Russia
(2) Department of Physics, Kyushu University, Fukuoka, Japan
The polarization of the secondary protons (P1,2) in the (p,2p) reaction with
the S - shell protons of nuclei 4He, 6Li, 12C, 28Si, 40Ca was measured at 1
GeV unpolarized proton beam. The spin correlation parameters Ci,j for the 4He
and 12C targets also were for the first time obtained. The polarization
measurements were performed by means of a two - arm magnetic spectrometer,
each arm of which was equipped with multiwire - proportional chamber
polarimeter.
Comments : 17 pages, 5 figures, 6 tables.
Category : Nuclear Experiment (nucl-ex)
Abstract
The polarization of the secondary protons (P1,2) in the (p,2p) reaction with
the S - shell protons of nuclei 4He, 6Li, 12C, 28Si, 40Ca was measured at 1
GeV unpolarized proton beam. The spin correlation parameters Ci,j for the 4He
and 12C targets also were for the first time obtained. The polarization
measurements were performed by means of a two - arm magnetic spectrometer,
each arm of which was equipped with multiwire - proportional chamber
polarimeter. This experimental work was aimed to study a modification of the
proton - proton scattering matrix in the nuclear medium.
## 1 Introduction
In recent years the question of medium modifications of nucleons and mesons
masses and sizes, and of meson - nucleon coupling constants, and, as
consequence, of a nucleon - nucleon scattering matrix, has received a great
deal of attention [1-6]. These modifications have been motivated from a
variety of theoretical points of view, which include renormalization effects
due to strong relativistic nuclear fields, deconfinement of quarks, and
partial chiral symmetry restoration.
The present work is a part of the wide experimental program in the frame of
which medium - induced modifications of nucleon - nucleon scattering
amplitudes are studied at PNPI synchrocyclotron with 1 GeV of proton beam
energy [7-13]. A (p,2p) reaction with nuclei is considered as the proton -
proton scattering in nuclear matter. In the exclusive (p,2p) experiments the
two - arm magnetic spectrometer (MAP and NES spectrometers) is used, the shell
structure of the nuclei being clearly distinguished. To measure polarization
characteristics of the reaction, each arm of the spectrometer is equipped with
multiwire - proportional chamber polarimeter.
In joint PNPI - RCNP experiments in 2000 - 2002 years, the polarization of
both secondary protons P1,2 in the (p,2p) reactions with the 1S - shell
protons of nuclei 6Li, 12C and with the 2S - shell protons of the 40Ca nucleus
was measured at the nuclear proton momenta before the (p,2p) interaction close
to zero [10]. The polarization observed in the experiment (as well as the
analyzing power Ay in the RCNP (p,2p) experiment at the 392 MeV polarized
proton beam [14,15]) drastically differs from that calculated in the framework
of non - relativistic Plane Wave Impulse Approximation (PWIA) and of spin -
dependent Distorted Wave Impulse Approximation (DWIA)[16], based on free space
proton - proton interaction. This difference was found to be negative and
increases monotonously with the effective mean nuclear density $\bar{\rho}$
[14], which is determined by an absorption of initial and secondary protons in
nucleus matter. Note also that the observed small difference between the non -
relativistic PWIA and DWIA calculations pointed out on a small contribution
from the trivial depolarization of secondary protons due to the final state
interaction. All these facts strongly indicated a modification of proton -
proton scattering amplitudes due to a modification of main properties of
hadrons in nuclear medium. Relativistic calculations have been done to analyze
and explain the experimental data [10,12].
The result of the (p,2p) experiment with the 4He target, performed in 2004
year [11], broke the mentioned above monotonous dependence of a difference
between experimental obtained polarization and that calculated in the frame of
the PWIA on the effective mean nuclear density $\bar{\rho}$. This difference
for the 4He nucleus proved to be close to that for the 6Li nucleus in spite of
that the 4He nucleus, according to studies of the elastic nucleon - nucleus
scattering, has the largest mean nuclear density. On the other hand mentioned
above deviation from the PWIA keeps to be a monotonous function of the S -
shell proton binding energy Es for all the investigated nuclei. It is possible
that in light nuclei, where nuclear matter is strongly heterogeneous, the
value of $\bar{\rho}$ does not reflect good enough the scale of nuclear medium
influence on hadron properties and nucleon - nucleon interaction. The results
observed in the (p,2p) experiment with the light nuclei at least show that the
value of Es may also be a measure of the influence of nuclear medium on the pp
- interaction.
At present work polarization of the secondary protons P1,2 in the (p,2p)
reaction with the 1S - shell protons of the 28Si nucleus was measured in the
kinematics close to that of the elastic proton - proton scattering (momenta of
nuclear protons before interaction were close to zero). The goal of the
experiment was to define the relative deviation of experimental polarization
from that calculated in the PWIA is determined by the the 1S - proton binding
energy Es or by the the effective mean nuclear density $\bar{\rho}$. The mean
value of the Es for the 28Si nucleus (50 MeV) is essentially larger then that
for the 12C nucleus (35 MeV). In the same time the values of the $\bar{\rho}$
seen in the kinematics of the (p,2p) reactions for these nuclei are close to
each other due to a saturation of the nuclear matter.
This experimental program was extended to measure the spin correlation
parameters Ci,j in the (p,2p) reaction with the 1S - shell protons of the 4He
and 12C nuclei. The left index i (i = n, s,) and the right index j (j = n,
s,,) are correspond to the forward scattered proton analyzed by the MAP
polarimeter and the recoil proton analyzed by the NES polarimeter,
respectively. Here n is the unit vector perpendicular to the scattering plane
of the (p,2p) reaction. Unit vectors s, and s,,, which lie in the scattering
plane, are concerned to the internal coordinate systems of the MAP and NES
polarimeters.
The main interest was related to measuring the spin correlation parameter Cnn
since it’s value is the same in the center - of - mass and laboratory systems,
and dos’not distort by magnetic fields of the two - arm spectrometer due to
the proton anomalous magnetic moment [17]. Since the polarization P and the
spin correlation parameter Cnn depend differently on the scattering matrix
element [6], measurement of both these polarization observables in a (p,2p)
experiment with nuclei can give more comprehensive information about
modification hadron properties in nuclear medium.
## 2 Experimental method
The general layout of the experimental setup used to investigate (p,2p)
reaction with nuclei is presented in Fig. 1.
The experiment is performed at the non - symmetric scattering angles of the
final state protons in the coplanar quasi - free scattering geometry with a
complete reconstruction of the reaction kinematics. The measured secondary
proton momenta K1, K2 and scattering angles $\Theta_{1}$, $\Theta_{2}$ are
used together with the proton beam energy T0 to calculate nuclear proton
separation energy $\Delta$E and the residual nucleus momentum Kr for each
(p,2p) event. In impulse approximation, the Kr is equal to the momentum (K) of
nuclear proton before the interaction (Kr=-K).
External proton beam of the PNPI synchrocyclotron was focused onto the target
TS of two - arm spectrometer (the magnetic spectrometers MAP and NES). The
beam intensity was monitored by the scintillation telescope M1, M2, M3 and was
about of 5$\cdot$1010 protons/(s$\cdot$cm2).
The solid nuclear targets TS made from CH2 (for the setup calibration), 6Li,
12C, 28Si, 40Ca (Table 1) and the universal cryogenic target with the liquid
helium 4He (or with the liquid hydrogen for calibration) were used in the
experiment [11,18]. Cylindrical aluminium appendix of the cryogenic target had
the following dimensions: diameter - 65 mm, height - 70 mm, wall thickness -
0.1 mm. The diameter of the beam spot on the target was less than 15 mm.
The two - arm spectrometer was used for registration of the secondary protons
from the (p,2p) reaction in coincidence and for measurement of their momenta
and outgoing angles. The polarization of these protons $P_{1}$ and $P_{2}$,
and the spin correlation parameters Ci,j were measured by the polarimeters
located in the region of focal planes of spectrometers MAP and NES. The
polarimeter of spectrometer MAP (NES) consisted of proportional chambers
PC1$\div$PC4, PC1’, PC4’ (PC5$\div$PC8, PC5’, PC8’) and carbon analyzer A1
(A2).
The main parameters of the two - arm magnetic spectrometer and polarimeters
are listed in Table 2 and Table 3, respectively. The $\Delta$E resolution of
the spectrometer estimated on the elastic proton - proton scattering with the
22-mm-thick cylindrical CH2 target (see Table 1) was found to be about of 5
MeV (FWHM).
The track information from proportional chambers of both polarimeters was used
in the offline analysis to find the azimuthal $\phi_{1}$, $\phi_{2}$ and polar
$\theta_{1}$, $\theta_{2}$ angles of proton scattering from the analyzers A1,
A2 for each (p,2p) event. In the case of absence of the accidental coincidence
background (the case of the elastic proton - proton scattering) the
polarization parameters could be found as [19]
$P_{1,2}=\frac{2<\cos\phi_{1,2}>}{<A(\theta_{1,2},K_{1,2})>}\\\ ,$ (1)
$C_{nn}=\frac{4<\cos\phi_{1}\cos\phi_{2}>}{<A(\theta_{1},K_{1})><A(\theta_{2},K_{2})>}\\\
,$ (2)
$C_{s^{,}{s^{,,}}}=\frac{4<\sin\phi_{1}\sin\phi_{2}>}{<A(\theta_{1},K_{1})><A(\theta_{2},K_{2})>}\\\
,$ (3)
$C_{n{s^{,,}}}=\frac{4<\cos\phi_{1}\sin\phi_{2}>}{<A(\theta_{1},K_{1})><A(\theta_{2},K_{2})>}\\\
,$ (4)
$C_{s^{,}{n}}=\frac{4<\sin\phi_{1}\cos\phi_{2}>}{<A(\theta_{1},K_{1})><A(\theta_{2},K_{2})>}\\\
,$ (5)
where averaging was made over a set of events within the working angular range
of $\theta_{1,2}$ (see Table 3) for the MAP and NES polarimeters.
$A(\theta_{1},K_{1})$ and $A(\theta_{2},K_{2})$, which were averaged over the
same set of events, are the carbon analyzing power parameterized according to
[20] and [21] for the MAP and NES polarimeter, respectively.
At present work the polarization parameters were estimated by folding the
theoretical functional shape of the azimuthal angular distribution into
experimental one [13], using the CERNLIB MINUIT package [22] and likelihood
$\chi^{2}$ estimator [23]. This method permits to realize the control over
$\chi^{2}$ in the case the experimentally measured azimuthal distribution is
distorted due to the instrumental problems.
The Time-of-Flight (TOF), the time difference between the signals from the
scintillation counters S2 and S4 was measured. This measurement served to
control the accidental coincidence background. The events from four
neighboring proton beam bunches were recorded. Three of them contained the
background events only and were used in the offline analysis to estimate the
background polarization parameters and the background contribution at the main
bunch containing the correlated (p,2p) events.
The recoil spectrometer NES was installed at a fixed angle
$\Theta_{2}=53.22^{\circ}$. At a given value of the S - shell mean binding
energy of nucleus under investigation, the angular and momentum settings of
the MAP spectrometer and the momentum setting of the NES spectrometer were
chosen to get a kinematics of (p,2p) reaction close to that of the free
elastic proton - proton scattering. In this (p,2p) kinematics, momentum K of
the nuclear proton before the interaction is close to zero. At this condition
the counting rate of the S - shell proton knockout reaction should be maximal.
In Fig. 2 the proton separation energy spectrum for the (p,2p) reaction on the
28Si nucleus, obtained at present work, is presented. As seen from the figure,
even at the preferable condition for the S - shell proton knockout process,
the contribution from the scattering off the external shell protons is
dominant.
Figure 1: The experimental setup. TS is the target of two - arm spectrometer;
Q1$\div$Q4 are the magnetic quadrupoles; D1, D2 are the dipole magnets; C1, C2
are the collimators; S1$\div$S4 and M1$\div$M3 are the scintillation counters;
PC1$\div$PC4, PC1’, PC4’ (PC5$\div$PC8, PC5’, PC8’) and A1 (A2) are the
proportional chambers and carbon analyzer of the high - momentum (low -
momentum) polarimeter, respectively; PC1”$\div$PC4” are the proportional
chambers. Shown above is the kinematics for the (p,2p) reaction.
Measurements of the spin correlation parameters and even of the polarization
in the (p,2p) reaction with heavy nuclei became possible due the fast
proportional chamber readout system (CROS-3), developed and produced at PNPI
[13]. This electronics allowed to collect the correlation events without
distortion at a high rate of the accidental coincidence background.
TABLE 1: Solid target parameters
---
Target | Dimensions (mm) | Isotope concentration ($\%$)
| diameter x high |
CH2 | 22x70 |
| thick x wide x high |
6Li | 4.5x12x25 | 99.0
12C | 4.0x18x70 | 98.9
28Si | 6.0x25x70 | 99.9
40Ca | 4.0x10x13 | 97.0
TABLE 2: Parameters of the magnetic spectrometers
---
Spectrometer | NES | MAP
Maximum particle momentum | |
$(K/Z)^{max}$, GeV/c | 1.0 | 1.7
Axial trajectory radius $\rho$, m | 3.27 | 5.5
Deflection angle $\beta$, deg. | 37.2 | 24.0
Dispersion in focal plane $D_{f}$, $mm/\%$ | 24 | 22
Solid angle acceptance $\Omega$, sr | $3.1\cdot 10^{-3}$ | $4.0\cdot 10^{-4}$
Momentum acceptance $\Delta K/K$, $\%$ | 8.0 | 8.0
Energy resolution (FWHM), MeV | $\sim 2.0$ | $\sim 1.5$
TABLE 3: Polarimeter parameters
---
Polarimeter | NES | MAP
Carbon block thickness, mm | 79 | 199
Polar angular range, deg. | 5$\div$18 | 3$\div$16
Average analyzing power | $\geq$ 0.46 | $\geq$ 0.23
Figure 2: Proton separation energy spectrum for the reaction 28Si(p,2p)27Al.
## 3 Experimental results and discussion
The measured polarization in the (p,2p) reactions with the S - shell protons
of nuclei 4He, 6Li, 12C, 28Si, 40Ca is given in Table 4 (see Appendix). In
Fig. 3, the averaged values of the data with those obtained earlier in [10]
are plotted versus of the S-shell proton binding energy Es and the effective
mean nuclear density $\bar{\rho}$, normalized on the saturation nuclear
density $\rho_{0}\approx$ 0.18 fm-3 (see also Table 4). The points ($\circ$)
and ($\bullet$) in the figure correspond to the polarization P1 and P2 of the
forward scattered protons at angle $\Theta_{1}$ = 21${}^{\circ}\div$ 25∘ (with
the kinetic energy T1 = 745 $\div$ 735 MeV) and the recoil protons scattered
at the angle $\Theta_{2}\approx$ 53.2∘ (with the energy T2 = 205 $\div$ 255
MeV). The points at the Es = 0 are the polarizations P1 and P2
Figure 3: Polarizations P1 and P2 of the protons scattered at the angles
$\Theta_{1}$ ($\circ$) and $\Theta_{2}$ = 53.22∘ ($\bullet$) in the (p,2p)
reaction with the S - shell protons of nuclei at 1 GeV as a function of the
mean binding energy Es and the effective mean nuclear density, $\bar{\rho}$
[14], in units of the saturation density ($\rho_{0}$ = 0.18 fm-3). The points
at Es=0 correspond to the elastic proton - proton scattering ($\Theta_{1}$ =
26.0∘). The dashed curve and the solid curve are the results of calculation in
the PWIA and the DWIA, respectively, with the NN interaction in free space
[16]. The dotted curve is the DWIA result, in which the relativistic effect is
taken into account in a Schrödinger equivalent form [5].
in the elastic proton - proton scattering at the angles $\Theta_{1}$ = 26.0∘
and $\Theta_{1}$ = 53.2∘ ($\Theta_{cm}$ = 62.25∘). Note that these pp - data
were obtained by a renormalization of the polarimeter analyzing power
requiring that the measured polarization should match the value (P1,2 = 0.326)
given by the current phase - shift analysis SP07 [25]. The normalization
coefficient was about of 1.06 for both polarimeters. This correction of the
analyzing power was also done for the polarization data obtained in the (p,2p)
experiment with nuclei.
In Fig. 3 the experimental data are compared with the results of non -
relativistic PWIA (plane wave impulse approximation) and DWIA (distorted wave
impulse approximation) calculations employing an on - shell factorized
approximation. The dashed and solid curves, corresponding to PWIA and DWIA,
respectively, present the results of the calculations, which were obtained
using the computer code THREEDEE [16]. A global optical potential [26],
parametrized in the relativistic framework and converted to the Shrödinger -
equivalent form, was used to calculate the distorted waves of incident and
outgoing protons in the case of DWIA, and a conventional well - depth method
was used to construct bound - state wave function. To calculate free
observables in the density independent NN interaction, the THREEDEE code uses
the 1986 Arndt NN phase - shift analysis (SP86) [27]. The results of the
calculations presented in Fig. 3 were normalized on a ratio of the PWIA
predictions obtained with the current phase - shift analysis SP07 and old one
SP86. The value of ratio P(SP07)/P(SP86) was about of 1.025. Note here, that
the 4He polarization data were analyzed only in the framework of the PWIA.
Because the difference between P1 and P2 values in the DWIA calculations was
found to be small, no more than 0.02, only the P1 values obtained from DWIA
are plotted in Fig. 3. As seen from the figure, the difference between the
PWIA and DWIA results is quite small. This result suggests that the
distortion, in a conventional non - relativistic framework, does not play an
essential role in the polarization for the kinematic conditions employed in
the present work. The final energy prescription [24] was used for the PWIA and
DWIA calculation. We also found that the difference between the initial and
final prescriptions was small in these kinematic region. The strong positive
slope of the polarizations predicted by these calculations (see Fig. 3) is
caused by the kinematic effects of the binding energy of the struck proton.
The differences between the polarizations P1, P2 calculated in the PWIA and
those measured in the (p,2p) reaction with nuclei 40Ca, 6Li, 12C are
monotonically increasing functions of the effective mean density (see Fig. 3)
[14]. The relative polarization difference (Pexp - Pia)/Pia is shown in Fig.
4. This difference for the 28Si nucleus, for the forward proton polarization
P1 at least, confirms also that the depolarization effect is determined by the
effective mean nuclear density $\bar{\rho}$. Indeed, the values of the
relative differences for 12C and 28Si nuclei ($\circ$ \- points) are
practically equal to each other, these nuclei having the same value of the
$\bar{\rho}$ and strongly different the S - shell mean binding energy Es. This
observation provides further evidence that there exists a nuclear medium
effect.
Figure 4: A relative deviation of the polarization observed in the (p,2p)
reaction with S - shell protons of nuclei from that calculated in the PWIA.
The points ($\circ$) and ($\bullet$) correspond to the forward scattered
proton at the angle $\Theta_{1}$ = 21.0∘$\div$ 25.08∘ and the recoil proton
scattered at the angle $\Theta_{2}$ = 53.22∘, respectively (see Fig. 3).
As seen from Fig. 3-4, there is a systematic difference between the P1 and P2
values, though they have the same values in the case of elastic pp -
scattering. Possible origins of the difference between these values include
non - relativistic and relativistic distortions (though the former is excluded
if the present DWIA calculations are valid), contributions of multi - step
processes, and even nontrivial modification of nucleons in the nuclear field.
In Fig. 3 the experimental data are compared with a theoretical result for the
case when a relativistic effect, the distortion of the nucleon spinor, is
taken into account. The calculation was carried out in the Shröedinger
equivalent form [10] using the THREEDEE code [16]. More specifically, this
calculation consists of a non - relativistic DWIA calculation with a nucleon -
nucleon t - matrix, that is modified in the nuclear potential following a
procedure similar to that proposed by Horowits and Iqbal [5]. In this approach
a modified NN interaction in nuclear medium is assumed due to the effective
nucleon mass (smaller than the free mass) which affects the Dirac spinors used
in the calculations of the NN scattering matrix. A linear dependence of the
effective mass of nucleons on the nuclear density was assumed in the
calculations. As seen from the Fig. 3, this relativistic approach gives the
results (the dotted curve) close to the experimental values of the forward
scattered proton polarization P1 in the (p,2p) reactions with nuclei at the
transfered momenta q = 3.2$\div$3.7 fm-1 (see Table 4).
Another possible medium effect is the modifications of exchanged meson masses
and meson - nucleon coupling constants in the NN interaction. Krein et al.
have shown in the relativistic Love - Franey model (RLF) that these
modifications cause significant changes in the spin observables which include
suppression of $A_{y}$ [6]. A such type of modification was investigated in
[12] using our experimental data on polarization in the (p,2p) reaction with
the S - shell protons of the 12C nucleus obtained in a wide range of the
momentum transfer q [10]. Note that at present work we essentially improved a
statistic accuracy of the polarization measurements in the (p,2p) reaction
with the 12C nucleus at the q = 3.4 fm-1.
In the present work, the spin correlation parameters Ci,j in the (p,2p)
reactions with the S - shell protons of the 4He and 12C nuclei were for the
first time measured using an unpolarized 1 GeV proton beam.
Since the polarization P and the spin correlation parameter Cnn depend
differently on the scattering matrix elements [6], measurement of both these
polarization observables in a (p,2p) experiment with nuclei can give more
comprehensive information about modification of the hadron properties in
nuclear medium.
The results of the Ci,j measurement in the (p,2p) reaction with nuclei are
given in Fig. 5 (Table 5). In Table 5 the measured mean values of the Ci,j for
the accidental coincidence background, obtained in investigating the (p,2p)
reaction with nuclei 12C and 28Si, are also presented.
Figure 5: Spin correlation parameters Ci,j in the (p,2p) reaction at 1 GeV
with the S - shell protons of the 4He and 12C nuclei at the secondary proton
scattering angles $\Theta_{2}$ = 53.2∘, $\Theta_{1}$ = 24.2∘ and $\Theta_{2}$
= 53.2∘, $\Theta_{1}$ = 22.7∘, respectively. The points at Es = 0 correspond
to the free elastic proton-proton scattering ($\Theta_{1}$ = 26.0∘,
$\Theta_{cm}$ = 62.25∘), obtained in 2009 year experiment (Table 6). The
dashed curve and the dotted curve are the results of the PWIA calculation of
the Cnn and C${}_{s^{,}{s^{,,}}}$ spin correlation parameters.
In Fig. 5 the dashed and dotted curves are correspond to the PWIA calculations
for the Cnn and C${}_{s^{,}{s^{,,}}}$ spin correlation parameters. In these
calculations the current Arndt phase - shift analysis (SP07) was used [25].
The C${}_{s^{,}{s^{,,}}}$ parameter was found by taking into account it’s
distortion in the magnetic fields of the MAP and NES spectrometers due to an
anomalous proton magnetic moment [17]. The points at the mean binding energy
value of Es = 0 correspond to the free elastic proton - proton scattering (see
Table 6).
As seen from Fig. 5, the differences between the Cnn values measured in the
(p,2p) experiment with the nuclei and those calculated in the PWIA are within
the statistical error bars.
The measured value of the C${}_{s^{,}{s^{,,}}}$ parameter in the elastic
proton - proton scattering strongly differs from the SP07 prediction. This can
be related to a lack of the spin correlation parameter data from the elastic
pp - scattering experiments at 1 GeV in the base of the current phase - shift
analysis.
Due to the parity conservation in the free elastic proton - proton scattering
the the spin correlation parameters C${}_{ns{{}^{,,}}}$ and C${}_{s^{,}{n}}$
should be equal to zero. In Fig. 5, this is confirmed by the experimental data
at the Es=0. For a (p,2p) reaction with nuclei the parity in the pp -
interaction system can be violated since there exists a residual nucleus in
the knockout process. However in this case, according to the Pauli principle,
a relation C${}_{ns{{}^{,,}}}$ = -C${}_{s^{,}{n}}$ for the pp - interaction
system should be performed [28].
## 4 Acknowledgments
This work is partly supported by the Grant of President of the Russian
Federation for Scientific School, Grant-3383.2010.2.
The authors are grateful to PNPI 1 GeV proton accelerator staff for stable
beam operation. We thank members of PNPI HEP Radio-electronics Laboratory for
providing the CROS-3 proportional chamber readout system.
Also, the authors would like to express their gratitude to A.A. Vorobyov and
S.L. Belostotski for their support and fruitful discussions.
## 5 Appendix:
TABLE 4: Polarization of secondary protons $P_{1}$ and $P_{2}$ produced in the
(p,2p) reaction at 1 GeV with the S - shell protons of nucleus at lab. angle
$\Theta_{1}$ and $\Theta_{2}$
---
Nucleus | $\Theta_{1}$ | $\Theta_{2}$ | $T_{1}$ | $T_{2}$ | $q$ | $P_{1}$ | $P_{2}$ | $\bar{\rho}/\rho_{0}$
| deg. | deg. | MeV | MeV | fm-1 | | |
4He (1S) | 24.21 | 53.22 | 738 | 242 | | 0.336$\pm$0.005 | 0.274$\pm$0.005 |
$<$4He (1S)$>$ | 24.21 | 53.22 | 738 | 242 | 3.6 | 0.335$\pm$0.005 | 0.274$\pm$0.004 |
6Li (1S) | 24.0 | 53.22 | 738 | 241 | | 0.309$\pm$0.026 | 0.247$\pm$0.023 |
$<$6Li (1S)$>$ | 24.0 | 53.25 | 739 | 239 | 3.6 | 0.306$\pm$0.015 | 0.255$\pm$0.015 | 0.19
12C (1S) | 22.71 | 53.22 | 746 | 219 | | 0.329$\pm$0.009 | 0.227$\pm$0.008 |
$<$12C (1S)$>$ | 22.71 | 53.22 | 746 | 219 | 3.4 | 0.325$\pm$0.008 | 0.225$\pm$0.008 | 0.31
28Si (1S) | 21.0 | 53.22 | 746 | 204 | 3.2 | 0.383$\pm$0.037 | 0.183$\pm$0.036 | $\sim$0.30
40Ca (2S) | 25.05 | 53.22 | 733 | 256 | | 0.306$\pm$0.037 | 0.304$\pm$0.033 |
$<$40Ca (2S)$>$ | 25.08 | 53.15 | 734 | 255 | 3.7 | 0.306$\pm$0.024 | 0.286$\pm$0.023 | 0.07
TABLE 5: Spin correlation parameters Cij in the (p,2p) reaction at 1 GeV with
the 1S - shell protons of the 4He and 12C nuclei at lab. angles $\Theta_{1}$
and $\Theta_{2}$ = 53.22∘. The line ”Background” corresponds to the measured
mean values of the Cij for the accidental coincidence background, obtained in
investigating the (p,2p) reaction with nuclei 12C and 28Si
---
Nucleus | $\Theta_{1}$ | Cnn | C${}_{s^{,}}$${}_{s^{,,}}$ | C${}_{ns^{,,}}$ | C${}_{s^{,}}$n
| deg. | | | |
4He | 24.21 | 0.667$\pm$0.070 | 0.150$\pm$0.070 | -0.095$\pm$0.070 | 0.086$\pm$0.070
12C | 22.71 | 0.407$\pm$0.163 | 0.229$\pm$0.164 | -0.086$\pm$0.163 | -0.281$\pm$0.165
Background | | -0.003$\pm$0.020 | 0.005$\pm$0.020 | 0.019$\pm$0.020 | 0.004$\pm$0.020
TABLE 6: Spin correlation parameters Cij in the elastic proton - proton
scattering at 1 GeV at lab. angles $\Theta_{1}$ = 26.0∘ and $\Theta_{2}$ =
53.22∘ ($\Theta_{cm}$ = 62.25∘), obtained in 2007-2010 years. The current
phase - shift analysis predicts the Cnn value of 0.57. The statistical errors
in the measurements of the polarizations P1 and P2 are also given
---
Year | Cnn | C${}_{s^{,}}$${}_{s^{,,}}$ | Cn${}_{s^{,,}}$ | C${}_{s^{,}}$n | $\delta$P1 | $\delta$P2
2007 | 0.587$\pm$0.021 | 0.115$\pm$0.021 | 0.005$\pm$0.021 | 0.080$\pm$0.021 | 0.0015 | 0.0012
2008 | 0.584$\pm$0.014 | 0.195$\pm$0.014 | -0.004$\pm$0.014 | 0.008$\pm$0.014 | 0.0010 | 0.0009
2009 | 0.577$\pm$0.016 | 0.170$\pm$0.016 | -0.016$\pm$0.016 | -0.006$\pm$0.016 | 0.0015 | 0.0013
2010 | 0.455$\pm$0.052 | 0.162$\pm$0.052 | -0.050$\pm$0.052 | 0.009$\pm$0.052 | 0.0041 | 0.0036
## References
* [1] G.E. Braun and M. Rho, Phys.Lett. 66, 2720 (1991).
* [2] R.J. Furnstahl, D.K. Griegel and T.D.Cohen, Phys.Rev. C46, 1507 (1992).
* [3] T. Hatsuda, Nucl.Phys. A544, 27 (1992).
* [4] B.D. Serot and J.D. Walecka, in ”Advances in Nucl.Phys.”, edited by J.W. Negele and E. Vogt (Plenum Press, New York, 1986), Vol.16, p.116.
* [5] C.J. Horowitz and M.J. Iqbal, Phys.Rev. C33, 2059 (1986).
* [6] G. Krein, Th.A.J. Maris, B.B. Rodrigues and E.A. Veit, Phys.Rev. C51, 2646 (1995).
* [7] O.V. Miklukho et al., Phys.Atom.Nucl. 63 No.5, 824 (2000).
* [8] O.V. Miklukho et al., Nucl.Phys. A683, 145 (2001).
* [9] O.V. Miklukho et al., Czech.J.Phys. 52 Suppl.C, 293 (2002).
* [10] V.A. Andreev et al., Phys.Rev. C69, 024604 (2004).
* [11] O.V. Miklukho et al., Phys.Atom.Nucl. 69 No.3, 474 (2006).
* [12] G.C. Hillhouse and T. Noro, Phys.Rev. C74 , 064608 (2006).
* [13] O.V. Miklukho, A.Yu. Kisselev et al., Phys.Atom.Nucl. 73 No.6, 927 (2010).
* [14] K. Hatanaka et al., Phys.Rev.Lett. 78, 1014 (1997).
* [15] T. Noro et al., Nucl.Phys. A633-664, 517 (2000).
* [16] N.S. Chant and P.G. Roos, Phys.Rev. C27 No.3, 1060 (1983).
* [17] W.O. Lock and D.F. Measday, ”Intermediate-Energy Nuclear Physics” (Methuen, London, 1970; Atomizdat, Moscow, 1973).
* [18] L. Kotchenda et al., Preprint PNPI, No 2816, 19 P. (2009).
* [19] O.Ya. Fedorov, Preprint PNPI, No.2432, 22 P. (2001).
* [20] O.Ya. Fedorov, Preprint LNPI, No.484, 29 P. (1979).
* [21] G. Waters et al., Nucl.Instrum.Methods 153, 401 (1978).
* [22] F. James, MINUIT, CERN Program Library Long Writeup D506 , Geneva (1998).
* [23] S. Baker and R. Cousins, Nucl.Instrum.Methods 221, 437 (1984).
* [24] W.T.H. van Oers et al., Phys.Rev.C25 No.1, 390 (1982).
* [25] R.A. Arndt et al., arXiv:0706.2195v2 [nucl-th], 16 P. (2007) or http://gwdac.phys.gwn.edu.
* [26] E.D. Cooper, S. Hama, B.C. Clark and R.L. Mercer, Phys.Rev C47, 297 (1993).
* [27] J. Bystricky, F. Lehar, and P. Winternitz, J.Phys.(Paris) 39, 1 (1978).
* [28] H. Faissner, ”Polarisierte Nucleonen I: Polarisation durch streuung”, Springer, 167 P. (1959).
|
arxiv-papers
| 2012-03-19T09:09:04 |
2024-09-04T02:49:28.747699
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O. V. Miklukho, A. Yu. Kisselev, D. A. Aksenov, G. M. Amalsky, V. A.\n Andreev, S. V. Evstiukhin, A. E. Ezhilov, O. Ya. Fedorov, G. E. Gavrilov, D.\n S. Ilyin, A. A. Izotov, L. M. Kochenda, P. A. Kravtsov, M. P. Levchenko, D.\n A. Maysuzenko, V. A. Murzin, T. Noro, D. V. Novinsky, V. A. Oreshkin, A. N.\n Prokofiev, A. V. Shvedchikov, V. Yu. Trautman, S. I. Trush and A. A. Zhdanov",
"submitter": "Alexander Prokofiev",
"url": "https://arxiv.org/abs/1203.4057"
}
|
1203.4112
|
[labelstyle=]
Department of Mathematical Sciences
The PhD School of Science
Faculty of Science
University of Copenhagen
Denmark
On the classical and quantum momentum map
A thesis submitted in fulfillment of the requirements for the degree of Doctor
of Philosophy
Chiara Esposito
Advisor : Ryszard Nest
Discussed : 27 January 2012
##
To Giovanni and Isabel
## Acknowledgments
First of all, I would like to thank my supervisor Prof. Ryszard Nest for his
continuous encouragement and his enthusiasm: without our funny discussions,
our fights with pens, eraser and blackboard and his patience, this thesis
would have never been completed.
Many thanks to Prof. Alan Weinsten and his students for the hospitality and
for the beautiful experience I had at the University of California, Berkeley,
both scientific and personal. A special thanks to Benoit Jubin for his help
with my paper and all the funny “french conversations” and to Sobhan
Seyfaddini, for all the nice persian food he made me taste.
Many thanks to Prof. Eva Miranda for her nice hospitality at the Universitat
Politècnica de Catalunya, Barcelona, for her support with applications and all
the ideas and suggestions. And a special thanks to my dear friend Romero
Solha, who introduced me to Eva, for our crazy discussions about girls and
food. Thanks to Prof. Rui Loja Fernandes for his interesting comments on my
work and his suggestions.
Thanks to all my friends: Giulietto, for all the deep and silly conversations
and the projects we share; Rita, Diego, Ninfa and Sergio and all my
“neapolitan physics-family”. Thanks to my former advisor, zio Fedele, for his
help and support in any moment of my life. To Alexandre Albore for the fun I
had during these last months with his wrong proofs. To Hany and Bryndis,
because their tango classes have been one of the nicest moments of the week in
this last year.
And thanks to George, for all the intense moments we had in these three years,
for his patience in bearing me (it’s hard…), for the beautiful trips and the
lazy days, for always offering me a different point of view and for all the
laughs we have together.
##
### Abstract
In this thesis we study the classical and quantum momentum maps and the theory
of reduction. We focus on the notion of momentum map in Poisson geometry and
we discuss the classification of the momentum map in this framework.
Furthermore, we describe the so-called Poisson Reduction, a technique that
allows us to reduce the dimension of a manifold in presence of symmetries
implemented by Poisson actions.
Using techniques of deformation quantization and quantum groups, we introduce
the quantum momentum map as a deformation of the classical momentum map,
constructed in such a way that it factorizes the quantum action. As an
application we discuss some examples of quantum reduction.
### Resumé
I denne afhandling studerer vi den klassiske impulsafbildning og
kvanteimpulsafbildningen samt reduktionsteori. Vi fokuserer på
impulsafbildningen i Poisson-geometri og diskuterer klassifikation af
impulsafbildningen inden for den ramme. Endvidere beskriver vi den såkalte
Poisson-reduktion, som er en teknik, der tillader os at reducere dimensionen
af en mangfoldighed med symmetrier implementeret af Poisson-virkninger.
Ved at bruge metoder fra deformationskvantisering og kvantegrupper
introducerer vi kvanteimpulsafbildningen som en deformation af den klassiske
impulsafbildning, der er konstrueret så den faktoriserer kvantereduktionen.
Resultaterne anvendes til at undersøge eksempler på kvantereduktioner.
## Contents
toc
## Introduction
The modern notion of momentum map was first introduced by Kostant [29] and
Souriau [52], as a refinement of the idea developed by Lie [32] in 1890. The
momentum map provides a mathematical formalization of the notion of conserved
quantity associated to symmetries of a dynamical system. The definition of
momentum map only requires a canonical Lie algebra action and its existence is
guaranteed whenever the infinitesimal generators of the Lie algebra action are
Hamiltonian vector fields.
It is important to underline that the momentum map is a fundamental tool for
the study of the so-called symplectic reduction of certain manifolds [40].
This method is a synthesis of different techniques of reduction of the phase
space (generally a symplectic manifold) associated to a dynamical system, in
which the symmetries are divided out. In particular, the Marsden-Weinstein
reduction [40] gives a description of the symplectic leaves on the orbit
space, obtained by a Hamiltonian action on a symplectic manifold.
These theories can be further generalized to the Poisson geometry framework,
considering more general structures which reduce to the symplectic ones under
certain conditions. This generalization has been performed with several
approaches, see e.g. [15], [34] and [39]. Poisson geometry was introduced by
Lie in [32] as a geometrization of Poisson’s studies of classical mechanics
[46]. During the past 40 years, Poisson geometry has become an interesting and
active field of research, motivated by connections with many research fields
as mechanics of particles and continua (Arnold [1], Lichnerowicz [31],
Marsden–Weinstein [41]) and integrable systems (Gel’fand-Dickey [18] and
Kostant [30]).
Furthermore, the theory of Poisson Lie groups has been developed through the
work of Drinfel’d [10] and Semenov-Tian-Shansky [50], [49] on completely
integrable systems and quantum groups. These new structures can be naturally
used to define Poisson actions, i.e. actions of Poisson Lie groups on Poisson
manifolds, as a generalization of Lie group actions.
In this thesis we mainly focus on a generalization of the momentum map
provided by Lu [33], [34]. In particular, we give a detailed discussion about
existence and uniqueness of Lu’s momentum map. More precisely, we introduce a
weaker momentum map, called infinitesimal momentum map, and we study the
conditions under which the infinitesimal momentum map determines a momentum
map in the usual sense. We describe the theory of reconstruction of the
momentum map from the infinitesimal one in two explicit cases. Moreover, we
provide the conditions which ensure the uniqueness of the momentum map.
Furthermore, we exploit Lu’s momentum map to construct a theory of reduction
for Poisson actions. Indeed, the local description of Poisson manifolds given
by Weinstein in [55] and the properties of Lu’s momentum map provide an
explicit description of the infinitesimal generator of a Poisson action, which
allows us to define a Poisson reduced space.
A similar mathematical construction can be implemented for the study of
symmetries in quantum mechanics. The passage between classical and quantum
systems can be performed with the theory of deformation quantization [27],
[2], [12]. We use this approach to study the relation between symmetries in
classical and quantum mechanics. The key idea resides in defining a quantum
momentum map such that the classical conserved quantities can be regarded as a
classical limit of the quantum ones.
The problem of quantization of the momentum map and the reduction has been the
main topic of many works, see e.g. [13] and [35]. In [13], Fedosov uses the
theory of deformation quantization to define a quantum momentum map and, as a
consequence, a quantum reduction. The author proves that the quantum reduced
space is isomorphic to the algebra obtained by canonical deformation
quantization of the symplectic reduced manifold. In [35], Lu defines the
quantization of a Poisson action in terms of quantum group action and the
quantum momentum map as a map which induces the quantum group action.
Motivated by these two works, in the present thesis we discuss a quantization
procedure for the momentum map associated to a Poisson action, which uses
quantum group and deformation quantization techniques.
### Plan of Work
We give here a brief description of the contents of this thesis, underlining
the main new results.
Chapter 1 gives some background about momentum map in symplectic geometry.
After a short overview of Lie group actions and Hamiltonian systems, we
present the momentum map with some basic examples and we describe the Marsden-
Weinstein reduction. The chapter aims to introduce the language that we use
throughout this thesis.
Chapter 2 starts by recalling some elements of Poisson geometry. In
particular, we focus on Poisson Lie groups and Lie bialgebras and we discuss
some properties of Poisson manifolds. Within this framework, we introduce the
Poisson action and the momentum map, and, as a warm up, we see the way they
generalize the discussion in Chapter 1.
Afterwards we commence the study of the part of the momentum map, the
infinitesimal momentum map, given by a certain family of differential forms
associated to the momentum map and which has the advantage that it can be
generalized to the quantized case - see the definition 2.4.12.
In Theorem 2.4.13 we give explicit, computable conditions for the
infinitesimal momentum map to determine a global momentum map. In Section
2.4.3 we study concrete cases of this globalization question and we prove the
existence and uniqueness/nonuniqueness of a momentum map associated to a given
infinitesimal momentum map, for the particular case when the dual Poisson Lie
group is abelian, respectively the Heisenberg group.
In the following section we study the question of infinitesimal deformations
of a given momentum map. The main result is Theorem 2.4.19, which describes
explicitly the space tangent to the space of momentum maps at a given point.
In particular, modulo the trivial infinitesimal deformations (which are given
by vector fields on $M$ commuting with the Poisson bivector and invariant
under the group action), the tangent space is given by a subspace of the
cohomology group $H^{1}({\mathfrak{g}},C^{\infty}(M))$. As an application, in
the case of a compact and semisimple Poisson Lie group $G$ acting on a Poisson
manifold $M$, the only infinitesimal deformations of a momentum map are the
trivial ones.
The following section (Sec. 2.5) is devoted to the theory of Poisson
reduction. The main result of this section is the theorem 2.5.3, which gives a
generalization of the Marsden-Weinstein reduction to the general case of an
arbitrary Poisson Lie group action on a Poisson manifold - the results that
can be found in the literature are restricted to the case when the Poisson
manifold in question is in fact a symplectic manifold.
Chapter 3 is dedicated to the study of the quantum momentum map. As proved by
Kontsevich [27], any Poisson manifold admits a canonical quantization. We
start this chapter by giving a short introduction to Kontsevich’s theorem and
its corollary describing the deformation quantization theory for Poisson
manifolds. Afterwards we give some background information on Hopf algebras,
quantum groups and the quantization of Poisson Lie groups and Lie bialgebras.
The quantum action is naturally expressed in terms of the action of the Hopf
algebra on the deformation quantization of the smooth manifold.
Since we work with deformation quantization of the Lie bialgebra of our
Poisson Lie group, the natural context is quantization of the infinitesimal
momentum map. This is defined in Section 3.3.1. One can briefly describe it as
follows. Let $\mathcal{U}_{\hbar}$ denote the quantized Lie bialgebra. Then
the coproduct $\Delta_{\hbar}$ of $\mathcal{U}_{\hbar}$ extends to an odd
derivation on the tensor algebra $T(\mathcal{U}_{\hbar}[1])$. If
$\mathcal{A}_{\hbar}$ denotes the quantized algebra of functions on the
manifold, the action of $\mathcal{U}_{\hbar}$ on $\mathcal{A}_{\hbar}$ becomes
a morphism of complexes
$\Phi_{\hbar}:(T(\mathcal{U}_{\hbar}[1]),\delta)\rightarrow(C^{\bullet}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar}),b),$
and quantum momentum map is a factorization of this morphism through the
complex of formal differential forms
$\Omega^{\bullet}(\mathcal{A}_{\hbar}^{+})$, which reduces, modulo
$\hbar^{2}$, to the classical infinitesimal momentum map.
Section 3.3.2 is devoted to the discussion of some examples of quantum
momentum map and of quantum reduction. The interesting observation is that the
existence of the quantum momentum map in the above sense dictates the formulas
for the quantization of the Lie bialgebra. In fact, in the low dimensional
examples studied in this section, the quantization of the Lie bialgebra is
essentially uniquely determined by existence of universal formulas for the
quantum momentum map.
The corresponding quantum reduction seems to be much more subtle then in the
case of the undeformed action of the group. The problem is as follows. In the
case of undeformed group action, the quantum momentum map has the form
$X\rightarrow dH_{X}$, and the reduction has the form $(\mathcal{A}/\langle
H_{X},\;X\in{\mathfrak{g}}\rangle)^{\mathcal{U}({\mathfrak{g}})}$. In the
general case, the “Hamiltonian forms” become of the form $adb$ with, as the
examples show, some of the $a$’s (and $b$’s) invertible, hence the quotients
of this form vanish. We can construct the quantum reduction in our case, but
it seems, at the moment, somewhat ad hoc.
## Chapter 1 Momentum Map in Symplectic Geometry
In this chapter we explain how the symmetries of a Hamiltonian dynamical
system can be used to simplify the study of that system. The description of
symmetries is implemented via Lie group actions, discussed in the first part
of the chapter. The treatment of Lie group actions and Hamiltonian systems is
necessary to introduce the key concept of this chapter, the momentum map. This
is a mathematical construction introduced by Lie [32], Kostant [29] and
Souriau [52] that describes the conservation laws associated to the symmetries
of a Hamiltonian dynamical system.
As will be seen, the momentum map is the basic ingredient for the construction
of the symplectic reduction. This is a procedure that uses symmetries and
conserved quantities to reduce the dimensionality of a Hamiltonian system.
### 1.1 Lie Group Actions
This section presents a brief review of the theory of Lie group actions on a
manifold, with some examples. This provides the background necessary to
introduce the concepts of symmetries of a Hamiltonian system and momentum map.
For details the reader should consult e.g. [47] and [38].
###### Definition 1.1.1.
Let $M$ be a manifold and $G$ a Lie group. A left action of $G$ on $M$ is a
smooth mapping $\Phi:G\times M\rightarrow M$ such that
1. 1.
$\Phi(e,m)=m,\qquad e\in G,\quad\forall m\in M$,
2. 2.
$\Phi(g,\Phi(h,m))=\Phi(gh,m),\qquad\forall g,h\in G,m\in M$,
where $e$ denotes the identity of $G$.
We often use the notation $g\cdot m:=\Phi(g,m):=\Phi_{g}(m):=\Phi^{m}(g)$.
Similarly as for the left action, a right action is a smooth map $\Phi:M\times
G\rightarrow M$, such that $\Phi(m,e)=m$, for all $m\in M$, and
$\Phi(\Phi(m,g),h)=\Phi(m,gh)$, for all $g,h\in G$ and $m\in M$.
###### Example 1.1.2.
An example of left action of a Lie group $G$ on itself is given by the left
translation $L_{g}:G\rightarrow G:h\mapsto gh$. Similarly, the right
translation $R_{g}:G\rightarrow G$, $h\mapsto hg$ defines a right action. The
inner automorphism $AD_{g}\equiv I_{g}:G\rightarrow G$, given by
$I_{g}:=R_{g^{-1}}\circ L_{g}$, defines a left action of $G$ on itself called
conjugation.
###### Example 1.1.3.
The differential at the identity of the conjugation map defines a linear left
action of a Lie group $G$ on its Lie algebra $\mathfrak{g}$, called the
adjoint representation of $G$ on $\mathfrak{g}$, that is
$Ad_{g}:=T_{e}I_{g}:\mathfrak{g}\rightarrow\mathfrak{g}.$ (1.1)
If $Ad^{*}_{g}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}$ is the dual of
$Ad_{g}$, then the map
$\Phi:G\times\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}:(g,\nu)\mapsto
Ad^{*}_{g^{-1}}\nu$ (1.2)
defines also a linear left action of $G$ on $\mathfrak{g}^{*}$ called the
coadjoint representation of $G$ on $\mathfrak{g}^{*}$.
Given a (left) action $\Phi:G\times M\rightarrow M$, the infinitesimal
generator $\xi_{M}\in TM$ associated to $\xi\in\mathfrak{g}$ is the vector
field on $M$ defined by
$\xi_{M}(m):=\left.\frac{d}{dt}\right|_{t=0}\Phi_{\exp(-t\xi)}(m)=T_{e}\Phi^{m}\cdot\xi$
(1.3)
An infinitesimal generator is a complete vector field. Indeed, the flow of
$\xi_{M}$ equals $(t,m)\mapsto\exp t\xi\cdot m$. Moreover, the map
$\xi\in\mathfrak{g}\mapsto\xi_{M}\in TM$ is a Lie algebra homomorphism, that
is,
1. 1.
$(a\xi+b\eta)_{M}=a\xi_{M}+b\eta_{M}$
2. 2.
$[\xi,\eta]_{M}=[\xi_{M},\eta_{M}]$,
for all $a,b\in\mathbb{R}$ and $\xi,\eta\in\mathfrak{g}$. Motivated by these
properties, we introduce the following definition.
###### Definition 1.1.4.
Let $\mathfrak{g}$ be a Lie algebra and $M$ a smooth manifold. A right (left)
Lie algebra action of $\mathfrak{g}$ on $M$ is a Lie algebra homomorphism
$\xi\in\mathfrak{g}\mapsto\xi_{M}\in TM$ such that the mapping $(m,\xi)\in
M\times\mathfrak{g}\mapsto\xi_{M}(m)\in TM$ is smooth.
Given a Lie group action, we refer to the Lie algebra action induced by its
infinitesimal generators as the associated Lie algebra action. An example of
Lie algebra action is given by the adjoint representation of the algebra
$\mathfrak{g}$, defined by the map
$ad:\mathfrak{g}\rightarrow End(\mathfrak{g}):\xi\mapsto ad_{\xi}=[\xi,\cdot]$
(1.4)
Consider a Lie group $G$ acting on a manifold $M$. The isotropy subgroup or
stabilizer of an element $m\in M$ acted upon by the Lie group $G$ is the
closed subgroup
$G_{m}:=\left\\{g\in G|\,g\cdot m=m\right\\}\subset G$ (1.5)
whose Lie algebra $\mathfrak{g}_{m}$ equals
$\mathfrak{g}_{m}=\left\\{\xi\in\mathfrak{g}|\,\xi_{M}(m)=0\right\\}$ (1.6)
The orbit $\mathcal{O}_{m}$ of the element $m\in M$ under the group action
$\Phi$ is the set
$\mathcal{O}_{m}\equiv G\cdot m:=\left\\{g\cdot m|\,g\in G\right\\}.$ (1.7)
The notion of orbit can be used to characterize an equivalence relation on the
manifold $M$. Two elements $x,y\in M$ are equivalent, $x\sim y$, if and only
if they are in the same orbit, hence if there exists an element $g\in G$ such
that $\Phi_{g}(x)=y$. The orbit space is the space of these equivalence
classes and is denoted by $M/G$.
In the following we discuss the conditions which ensure that the orbit space
is a regular quotient manifold.
A group action on $M$ is said to be
1. -
transitive, if there is only one orbit,
2. -
free, if the isotropy of every element in $M$ consists only of the identity
element,
3. -
faithful, if $\xi_{M}=id_{M}$ implies that $g=e$
Let $X$ and $Y$ be two topological spaces with $Y$ first countable. A
continuous map $f:X\rightarrow Y$ is called proper if for any sequence
$\\{x_{n}\\}_{n\in\mathbb{N}}$ such that $f(x_{n})\rightarrow y$ there exist a
convergent subsequence $\\{x_{n_{k}}\\}$ such that $x_{n_{k}}\rightarrow x$
and $f(x)=y$. A map $f:X\rightarrow Y$ is proper if and only if it is closed
and $f^{-1}(y)$ is compact, for any $y\in Y$.
###### Definition 1.1.5.
Let $G$ be a Lie group acting on the manifold $M$ via the map $\Phi:G\times
M\rightarrow M$. We say that $\Phi$ is proper whenever the map $\Theta:G\times
M\rightarrow M\times M$ defined by $\Theta(g,m)=(m,\Phi(g,m))$ is proper.
The properness of the action is equivalent to the following condition: for any
two convergent sequences $\\{m_{n}\\}$ and $\\{g_{n}\cdot m_{n}\\}$ in $M$,
there exists a convergent subsequence $\\{g_{n_{k}}\\}$ in $G$. We say that
the action $\Phi$ is proper at the point $m\in M$ when for any two convergent
sequences $\\{m_{n}\\}$ and $\\{g_{n}\cdot m_{n}\\}$ in $M$ such that
$m_{n}\rightarrow m$ and $g_{n}\cdot m_{n}\rightarrow m$, there exists a
convergent subsequence $\\{g_{n_{k}}\\}$ in $G$.
The following proposition is crucial in the theory of symplectic reduction. A
proof can be found e.g. in [47] and [3].
###### Proposition 1.1.6.
Let $\Phi:G\times M\rightarrow M$ be a proper action of the Lie group $G$ on
the manifold $M$. Then
1. 1.
For any $m\in M$, the isotropy subgroup $G_{m}$ is compact.
2. 2.
The orbit space $M/G$ is a Hausdorff topological space
3. 3.
If the action is free, $M/G$ is a smooth manifold, and the canonical
projection $\mathfrak{p}:M\rightarrow M/G$ defines on $M$ the structure of a
smooth left principal $G$-bundle.
### 1.2 Hamiltonian systems
We start this section with a brief review of some basic notions about
symplectic manifolds and invariant Hamiltonian dynamics. This provides us all
the necessary background that we use in the next sections.
###### Definition 1.2.1.
A symplectic manifold is a pair $(M,\omega)$, where $M$ is a manifold and
$\omega\in\Omega^{2}(M)$ is a closed nondegenerate two form on $M$, that is,
$d\omega=0$ and, for every $m\in M$, the map $v\in
T_{m}M\mapsto\omega(m)(v,\cdot)\in T^{*}_{m}M$ is a linear isomorphism between
the tangent space $T_{m}M$ and the cotangent space $T^{*}_{m}M$.
Since $\omega$ is a differential two-form, hence skew-symmetric, the dimension
of $M$ is always even. A Hamiltonian dynamical system is a triple
$(M,\omega,H)$, where $(M,\omega)$ is a symplectic manifold and $H\in
C^{\infty}(M)$ is the Hamiltonian function of the system. By nondegeneracy of
the symplectic form $\omega$, to each Hamiltonian system one can associate a
Hamiltonian vector field $X_{H}\in TM$, defined by the identity
$i_{X_{H}}\omega=dH$ (1.8)
It’s clear that $X\in TM$ is a Hamiltonian vector field if and only if the one
form $i_{X}\omega$ is exact.
###### Definition 1.2.2.
Let $f,g\in C^{\infty}(M)$. The Poisson bracket of these functions is the
function $\left\\{f,g\right\\}\in C^{\infty}(M)$ defined by
$\left\\{f,g\right\\}(m)=\omega(m)\left(X_{f}(m),X_{g}(m)\right)=X_{g}[f](m)=-X_{f}[g](m).$
(1.9)
Given a symplectic manifold $(M,\omega)$, the set $C^{\infty}(M)$ can be
always equipped with a real Lie algebra structure relative to the Poisson
bracket:
###### Definition 1.2.3.
A Poisson manifold is a pair $(M,\left\\{\cdot,\cdot\right\\})$, where $M$ is
a smooth manifold and $\left\\{\cdot,\cdot\right\\}$ is a bilinear operation
on $C^{\infty}(M)$, such that the pair
$(C^{\infty}(M),\left\\{\cdot,\cdot\right\\})$ is a Lie algebra and
$\left\\{\cdot,\cdot\right\\}$ is a derivation in each argument. The pair
$(C^{\infty}(M),\left\\{\cdot,\cdot\right\\})$ is called Poisson algebra. The
functions in the center of the Lie algebra
$(C^{\infty}(M),\left\\{\cdot,\cdot\right\\})$ are called Casimir functions.
Since there is an isomorphism between derivations on $C^{\infty}(M)$ and
vector fields on $M$, it follows that each $H\in C^{\infty}(M)$ induces a
vector field on $M$ via the expression
$X_{H}=\left\\{\cdot,H\right\\},$ (1.10)
called the Hamiltonian vector field associated to the Hamiltonian function
$H$. The Hamiltonian equations $\dot{z}=X_{H}(z)$ can be equivalently written
in Poisson bracket form as
$\dot{f}=\left\\{f,H\right\\},$ (1.11)
for any $f\in C^{\infty}(M)$. The triple $(M,\left\\{\cdot,\cdot\right\\},H)$
is called a Poisson dynamical system. The Lie algebra map
$(C^{\infty}(M),\left\\{\cdot,\cdot\right\\})\rightarrow(TM,\left[\cdot,\cdot\right])$
that assigns to each function $f\in C^{\infty}(M)$ the associated Hamiltonian
vector field $X_{f}\in TM$ is a Lie algebra homomorphism:
$X_{\left\\{f,g\right\\}}=\left[X_{f},X_{g}\right]\qquad\forall f,g\in
C^{\infty}(M).$ (1.12)
Any Hamiltonian system on a symplectic manifold is a Poisson dynamical system
relative to the Poisson bracket induced by the symplectic structure. Given a
Poisson dynamical system $(M,\left\\{\cdot,\cdot\right\\},H)$, its conserved
quantities or integrals of motion are defined by the subalgebra
$C^{\infty}(M)^{G}$ of $(C^{\infty}(M),\left\\{\cdot,\cdot\right\\})$
consisting of the $G$-invariant functions on $M$, i.e $f\in C^{\infty}(M)$
such that $\left\\{f,H\right\\}=0$.
As mentioned above, the symmetries of a Hamiltonian system are encoded via Lie
group actions consistent with the structure of the given dynamical system.
This motivates the following definition:
###### Definition 1.2.4.
A canonical action is a map $\Phi:G\times M\rightarrow M$ such that
$\Phi_{g}^{*}\left\\{f,g\right\\}=\left\\{\Phi_{g}^{*}f,\Phi_{g}^{*}g\right\\}\qquad(\text{resp.}\quad\Phi_{g}^{*}\omega=\omega)$
(1.13)
We say that the Hamiltonian system $(M,\left\\{\cdot,\cdot\right\\},H)$ is
$G$-symmetric when the Lie group $G$ acts canonically on
$(M,\left\\{\cdot,\cdot\right\\})$ and the Hamiltonian function $H$ is
$G$-invariant, that is $\Phi^{*}_{g}(H)=H$.
The infinitesimal version of this concept is the canonical action of a Lie
algebra. An action of the Lie algebra $\mathfrak{g}$ on the Poisson
(respectively, symplectic) manifold $M$ is canonical if the vector fields
$\xi_{M}\in TM$ are infinitesimal Poisson automorphisms, that is, if $\pi$ is
the Poisson tensor we have that $L_{\xi_{M}}\pi=0$ (respectively,
$L_{\xi_{M}}\omega=0$). We say that the Hamiltonian system
$(M,\left\\{\cdot,\cdot\right\\},H)$ is $\mathfrak{g}$-symmetric if the Lie
algebra $\mathfrak{g}$ acts canonically on $(M,\left\\{\cdot,\cdot\right\\})$
and the Hamiltonian function $H$ is $\mathfrak{g}$-invariant.
### 1.3 Momentum map
In the previous sections we introduced the symmetries of a Hamiltonian systems
via Lie group actions. The conservation laws of this system can be described
with a mathematical construction called the momentum map. The definition of
momentum map only requires a canonical Lie algebra action and its existence is
guaranteed when the infinitesimal generators of the action are Hamiltonian
vector fields.
###### Definition 1.3.1.
Let $\mathfrak{g}$ be a Lie algebra acting canonically on the Poisson manifold
$(M,\left\\{\cdot,\cdot\right\\})$. Suppose that for any $\xi\in\mathfrak{g}$
the vector field $\xi_{M}$ is Hamiltonian, with Hamiltonian function
$\boldsymbol{\mu}^{\xi}\in C^{\infty}(M)$ such that
$\xi_{M}=X_{\boldsymbol{\mu}^{\xi}}.$ (1.14)
The map $\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$ defined by the
relation
$\boldsymbol{\mu}^{\xi}(m)=\left\langle\boldsymbol{\mu}(m),\xi\right\rangle$
(1.15)
for all $\xi\in\mathfrak{g}$ and $m\in M$, is called momentum map of the
$\mathfrak{g}$-action.
Notice that the momentum map is not uniquely determined; indeed,
$\boldsymbol{\mu}^{\xi}_{1}$ and $\boldsymbol{\mu}^{\xi}_{2}$ are momentum
maps for the same canonical action if and only if for any $\xi\in\mathfrak{g}$
$\boldsymbol{\mu}^{\xi}_{1}-\boldsymbol{\mu}^{\xi}_{2}$ (1.16)
is a Casimir function; if $M$ is symplectic and connected, then
$\boldsymbol{\mu}$ is determined up to a constant in $\mathfrak{g}^{*}$.
###### Example 1.3.2 (Linear momentum).
We consider the phase space $T^{*}\mathbb{R}^{3N}$ of a $N$-particle system.
The additive group $\mathbb{R}^{3}$ acts on it by applying spatial translation
on each factor:
$\mathbf{v}\cdot(\mathbf{q}_{i},\mathbf{p}^{i})=(\mathbf{q}_{i}+\mathbf{v},\mathbf{p}^{i})$,
with $i=1,\dots,N$. This action is canonical and has an associated momentum
map that coincides with the classical linear momentum
$\displaystyle\boldsymbol{\mu}:$ $\displaystyle
T^{*}\mathbb{R}^{3N}\rightarrow Lie(\mathbb{R}^{3})\simeq\mathbb{R}^{3}$
(1.17)
$\displaystyle(\mathbf{q},\mathbf{p})\mapsto\sum_{i=1}^{N}\mathbf{p}_{i}$
(1.18)
###### Example 1.3.3 (Angular momentum).
Let $SO(3)$ act on $\mathbb{R}^{3}$ and then, by lift, on
$T^{*}\mathbb{R}^{3}$, that is,
$A\cdot(\mathbf{q},\mathbf{p})=(A\mathbf{q},A\mathbf{p})$. This action is
canonical and has an associated momentum map
$\displaystyle\boldsymbol{\mu}:$ $\displaystyle
T^{*}\mathbb{R}^{3}\rightarrow\mathfrak{so(3)}^{*}\simeq\mathbb{R}^{3}$ (1.19)
$\displaystyle(\mathbf{q},\mathbf{p})\mapsto\mathbf{q}\times\mathbf{p},$
(1.20)
which is the classical angular momentum.
It can be shown that the momentum map satisfies Noether’s Theorem [45], as
stated in the following theorem.
###### Theorem 1.3.4 ([47]).
Let $G$ be a Lie group acting canonically on the Poisson manifold
$(M,\\{\cdot,\cdot\\})$. Assume that this action admits a momentum map
$\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$ and that $H\in C^{\infty}(M)$
is invariant under the action of $\Phi$. Then the momentum map is an integral
for the Hamiltonian vector field $X_{H}$ (i.e. if $F_{t}$ is the flow of
$X_{H}$ then $\boldsymbol{\mu}(F_{t}(x))=\boldsymbol{\mu}(x)$ for all $x$ and
$t$ where $F_{t}$ is defined).
Finally, we introduce the property of equivariance of the momentum map. Let
$(M,\\{\cdot,\cdot\\})$ be a Poisson manifold and $\mathfrak{g}$ act
canonically on it with a momentum map
$\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$. The map
$(\mathfrak{g},\left[\cdot,\cdot\right])\rightarrow(C^{\infty}(M),\\{\cdot,\cdot\\})$
defined by $\xi\mapsto\boldsymbol{\mu}^{\xi}$, $\xi\in\mathfrak{g}$ is a Lie
algebra homomorphism if and only if
$T_{z}\boldsymbol{\mu}\cdot\Phi_{\xi}(m)=ad^{*}_{\xi}\boldsymbol{\mu}(m)$
(1.21)
for any $\xi\in\mathfrak{g}$ and any $m\in M$. A momentum map that satisfies
this relation is called infinitesimally equivariant. When the Lie algebra
action is associated to the action of a Lie group $G$, we say that
$\boldsymbol{\mu}$ is $G$-equivariant if
$\boldsymbol{\mu}\circ\Phi_{g}=Ad_{g}^{*}\circ\boldsymbol{\mu}$ (1.22)
for all $g\in G$. A Lie algebra action with an infinitesimally equivariant
momentum map is called Hamiltonian action and a Lie group action with an
equivariant momentum map is called globally Hamiltonian.
Details about the problem of the existence of the momentum map can be found in
[47].
### 1.4 Symplectic Reduction
In this section we describe the simplest version of symplectic reduction that
constructs a symplectic manifold out of a given symmetric one, on which the
conservation laws and degeneracies associated to the symmetries have been
eliminated. Given a symmetric Hamiltonian dynamical system, the Marsden-
Weinstein reduced system is also a Hamiltonian system with reduced
dimensionality, as proved in the following theorem:
###### Theorem 1.4.1 (Marsden-Weinstein Reduction [40]).
Let $\Phi:G\times M\rightarrow M$ be a canonical action of the Lie group $G$
on the connected symplectic manifold $(M,\omega)$. Suppose that the action has
an associated equivariant momentum map
$\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$. Let $u\in\mathfrak{g}^{*}$ be
a regular value of $\boldsymbol{\mu}$ and assume that the isotropy group
$G_{u}$ under the $Ad^{*}$ action on $\mathfrak{g}^{*}$ acts freely and
properly on $\boldsymbol{\mu}^{-1}(u)$. Then:
1. 1.
the space $M_{u}:=\boldsymbol{\mu}^{-1}(u)/G_{u}$ is a regular quotient
manifold and there is a symplectic structure $\omega_{u}$ on $M_{u}$ uniquely
determined by $i^{*}_{u}\omega=\mathfrak{p}^{*}_{u}\omega_{u}$, where
$i_{u}:\boldsymbol{\mu}^{-1}(u)\hookrightarrow M$ is the natural inclusion and
$\mathfrak{p}$ is the natural projection of $\boldsymbol{\mu}^{-1}(u)$ onto
$M_{u}$. The pair $(M_{u},\omega_{u})$ is called the symplectic reduced space.
2. 2.
Let $H\in C^{\infty}(M)^{G}$ be a $G$-invariant Hamiltonian. The flow $F_{t}$
of the Hamiltonian vector field $X_{H}$ leaves the connected components of
$\boldsymbol{\mu}^{-1}(u)$ invariant and commutes with the $G$-action, so it
induces a flow $F_{t}^{u}$ on $M_{u}$ defined by
$\mathfrak{p}_{u}\circ F_{t}\circ i_{u}=F_{t}^{\nu}\circ\mathfrak{p}_{u}.$
(1.23)
3. 3.
The vector field generated by the flow $F_{t}^{u}$ on $(M_{u},\omega_{u})$ is
Hamiltonian with associated reduced Hamiltonian function $H_{u}\in
C^{\infty}(M_{u})$ defined by
$H_{u}\circ\mathfrak{p}_{u}=H\circ i_{u}.$ (1.24)
The vector fields $X_{H}$ and $X_{H_{u}}$ are $\mathfrak{p}_{u}$-related. The
triple $(M_{u},\omega_{u},H_{u})$ is called reduced Hamiltonian system.
4. 4.
Let $K\in C^{\infty}(M)^{G}$ be another $G$-invariant function. Then
$\\{H,K\\}$ is also $G$-invariant and
$\\{H,K\\}_{u}=\\{H_{u},K_{u}\\}_{M_{u}}$, where $\\{\cdot,\cdot\\}_{M_{u}}$
denotes the Poisson bracket associated to the symplectic form $\omega_{u}$ on
$M_{u}$.
The symplectic reduction can be rephrased in terms of algebra of functions. In
[51], Sniatycki and Weinstein proposed a different procedure that yields a
reduced Poisson algebra; they proved that this algebra coincides with the
Poisson algebra on the reduced space $M_{u}$. We briefly introduce the results
obtained in [51], which we use in the next chapters.
Consider the Hamiltonian function $\boldsymbol{\mu}^{\xi}\in C^{\infty}(M)$,
with $\xi\in\mathfrak{g}$ and let
$\boldsymbol{\mu}_{i}=\boldsymbol{\mu}^{e_{i}}$, $i=1,\dots,n$ be the
components of the Hamiltonian function on the basis $\\{e_{i}\\}$ of
$\mathfrak{g}$. Define the ideal $\mathcal{I}$ in $C^{\infty}(M)^{G}$
generated by the momenta $\boldsymbol{\mu}_{i}$ as
$\mathcal{I}=\\{f\in
C^{\infty}(M)^{G}|f=\sum_{i}g_{i}\boldsymbol{\mu}_{i},g_{i}\in
C^{\infty}(M)\\}.$ (1.25)
Then we have:
###### Lemma 1.4.2.
Let $\mathfrak{g}$ be a Lie algebra acting canonically on the Poisson manifold
$(M,\left\\{\cdot,\cdot\right\\})$ with $G$-equivariant momentum map
$\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$. The the ideal $\mathcal{I}$
is a Poisson subalgebra of $C^{\infty}(M)^{G}$.
The action of $G$ on $C^{\infty}(M)$ induces an action of $G$ on the quotient
algebra $C^{\infty}(M)^{G}$ such that the projection homomorphism
$\rho:C^{\infty}(M)\rightarrow C^{\infty}(M)/\mathcal{I}$ is $G$-equivariant.
In [51] the authors proved that the quotient
$\mathcal{R}=C^{\infty}(M)^{G}/\mathcal{I}$ naturally inherits a Poisson
algebra structure. More precisely, under the assumptions of Lemma 1.4.2, we
have
###### Lemma 1.4.3.
$\rho^{-1}(C^{\infty}(M)^{G})$ is the normalizer of $\mathcal{I}$ and it has
the structure of a Poisson subalgebra of $C^{\infty}(M)$.
###### Corollary 1.4.4.
$C^{\infty}(M)^{G}$ inherits the structure of a Poisson algebra such that
$\rho$ restricted to $\rho^{-1}(C^{\infty}(M)^{G})$ is a Poisson algebra
homomorphism.
The Poisson algebra $\mathcal{R}$ is called the reduced Poisson algebra of the
considered system. Finally, we have:
###### Theorem 1.4.5.
The Poisson algebra $\mathcal{R}$ is canonically isomorphic to the Poisson
algebra of the reduced phase space $C^{\infty}(M_{u})$ with Poisson structure
induced by $\omega_{u}$.
## Chapter 2 Momentum Map in Poisson Geometry
In this chapter we discuss the generalization of the theory of momentum map
and reduction to the Poisson geometry case. Similarly to the previous chapter,
in this formalism the description of symmetries is implemented via Poisson
actions. In order to introduce Poisson actions we give some background about
Poisson Lie groups and Lie bialgebras and we discuss a more complete
definition of Poisson manifolds with certain properties. We introduce the
generalization of momentum map given by Lu [33] and its related Hamiltonian
action. After this introductory part, we give a new definition of the momentum
map in terms of one-forms and we study its properties. As in the symplectic
case, the momentum map here plays a fundamental role in the construction of
the Poisson reduction theory.
### 2.1 Lie bialgebras
The first object under consideration is a generalization of the notion of Lie
algebra, called Lie bialgebra. In general, given a Lie algebra $\mathfrak{g}$,
its dual space $\mathfrak{g}^{*}$ is a vector space. In this section we see
that $\mathfrak{g}$ can be endowed with a structure which induces a Lie
algebra structure on its dual $\mathfrak{g}^{*}$. The corresponding Lie groups
carry a Poisson structure compatible with the group multiplication. For
details on this topic see e.g. [28]. As will be seen in next chapter, the
importance of these structures relies in the fact that they admit a standard
procedure of quantization.
Let $\mathfrak{g}$ be a finite dimensional Lie algebra and $\delta$ a linear
map from $\mathfrak{g}$ to $\mathfrak{g}\otimes\mathfrak{g}$ with transpose
${}^{t}\delta:\mathfrak{g^{*}}\otimes\mathfrak{g^{*}}\rightarrow\mathfrak{g^{*}}$.
Recall that a linear map on $\mathfrak{g^{*}}\otimes\mathfrak{g^{*}}$ can be
identified with a bilinear map on $\mathfrak{g}$.
###### Definition 2.1.1.
A Lie bialgebra is a Lie algebra $\mathfrak{g}$ with a linear map
$\delta:\mathfrak{g}\rightarrow\mathfrak{g}\wedge\mathfrak{g}$ such that
1. 1.
${}^{t}\delta:\mathfrak{g^{*}}\otimes\mathfrak{g^{*}}\rightarrow\mathfrak{g^{*}}$
defines a Lie bracket on $\mathfrak{g}^{*}$, and
2. 2.
$\delta$ is a 1-cocycle on $\mathfrak{g}$ relative to the adjoint
representation of $\mathfrak{g}$ on $\mathfrak{g}\otimes\mathfrak{g}$
Condition 2 means that the 2-cocycle
$ad_{\xi}(\delta(\eta))-ad_{\eta}(\delta(\xi))-\delta([\xi,\eta])=0$ (2.1)
for any $\xi,\eta\in\mathfrak{g}$.
In the following we will adopt the notation
$[x,y]_{*}={}^{t}\delta(x\otimes y),$ (2.2)
for any $x,y\in\mathfrak{g}^{*}$. Thus, by definition
$\langle[x,y]_{*},\xi\rangle=\langle\delta(\xi),x\otimes y\rangle$ (2.3)
for $\xi\in\mathfrak{g}$.
As discussed in the previous chapter any Lie algebra $\mathfrak{g}$ acts on
itself by the adjoint representation $ad:\xi\in\mathfrak{g}\mapsto ad_{\xi}\in
End\ \mathfrak{g}$, defined by $ad_{\xi}(\eta)=[\xi,\eta]$ (see eq. (1.4)).
We now introduce the definition of coadjoint representation of a Lie algebra
on the dual vector space. Let $\mathfrak{g}$ be a Lie algebra and let
$\mathfrak{g}^{*}$ be its dual vector space. For $\xi\in\mathfrak{g}$, we set
$ad_{\xi}^{*}=-{}^{t}(ad_{\xi}).$ (2.4)
Thus $ad_{\xi}^{*}$ is the endomorphism of $\mathfrak{g}^{*}$ satisfying
$\langle x,ad_{\xi}(y)\rangle=-\langle ad_{\xi}^{*}x,y\rangle.$ (2.5)
The map $\xi\in\mathfrak{g}\mapsto ad_{\xi}^{*}\in End\ \mathfrak{g}^{*}$ is a
representation of $\mathfrak{g}$ in $\mathfrak{g}^{*}$, that we call coadjoint
representation. Hence, eq. (2.1) can be written as
$\begin{split}\langle[x,y]_{\mathfrak{g}^{*}},[\xi,\eta]\rangle&+\langle[ad_{\xi}^{*}x,y]_{*},y\rangle+\langle[x,ad_{\xi}^{*}y],y\rangle\\\
&-\langle[ad_{\xi}^{*}x,y]_{*},\xi\rangle-\langle[x,ad_{\xi}^{*}y],\xi\rangle=0\end{split}$
(2.6)
It is important to stress that there is a symmetry between $\mathfrak{g}$,
with Lie bracket $[\cdot,\cdot],$ and $\mathfrak{g}^{*}$, with Lie bracket
$[\cdot,\cdot]_{*}$, defined by $\delta$. In fact, setting
$ad_{x}(y)=[x,y]_{\mathfrak{g}^{*}}$ (2.7)
and
$\langle ad_{x}y,\xi\rangle=-\langle y,ad_{x}^{*}\xi\rangle,$ (2.8)
the map $x\in\mathfrak{g}^{*}\mapsto ad_{x}^{*}\in End\ \mathfrak{g}$ is the
coadjoint representation of $\mathfrak{g}^{*}$ in the dual of
$\mathfrak{g}^{*}$, which is isomorphic to $\mathfrak{g}$. Hence, eq. (2.1) is
equivalent to
$\begin{split}\langle[x,y]_{\mathfrak{g}^{*}},[\xi,\eta]\rangle&+\langle
ad_{\xi}^{*}x,ad_{y}^{*}\eta\rangle-\langle
ad_{\xi}^{*}y,ad_{x}^{*}\eta\rangle\\\ &-\langle
ad_{\eta}^{*}x,ad_{y}^{*}\xi\rangle+\langle
ad_{\eta}^{*}y,ad_{x}^{*}\xi\rangle=0.\end{split}$ (2.9)
The symmetry between $\mathfrak{g}$ and $\mathfrak{g}^{*}$ then follows from
the fact that eq. (2.9) is equivalent to the condition on
${}^{t}[\cdot,\cdot]:\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}$
to be a 1-cocycle on $\mathfrak{g}^{*}$ with values on
$\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}$, where $\mathfrak{g}^{*}$ acts on
$\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}$ by the adjoint action.
###### Proposition 2.1.2.
If $(\mathfrak{g},\delta)$ is a Lie bialgebra, and $[\cdot,\cdot]$ is a Lie
bracket on $\mathfrak{g}$, then $(\mathfrak{g}^{*},{}^{t}\delta)$ is a Lie
bialgebra, where ${}^{t}[\cdot,\cdot]$ defines a Lie bracket on
$\mathfrak{g}^{*}$.
By definition, $(\mathfrak{g}^{*},{}^{t}[\cdot,\cdot])$ is the dual of the Lie
bialgebra $(\mathfrak{g},\delta)$. It is easy to see that the dual of
$(\mathfrak{g}^{*},{}^{t}[\cdot,\cdot])$ coincides with
$(\mathfrak{g},\delta)$.
###### Proposition 2.1.3.
Let $(\mathfrak{g},\delta)$ be a Lie bialgebra with dual
$(\mathfrak{g}^{*},{}^{t}[,])$. There exists a unique Lie algebra structure on
the vector space $\mathfrak{g}\oplus\mathfrak{g}^{*}$ such that
1. 1.
it restricts to the given brackets on $\mathfrak{g}$ and $\mathfrak{g}^{*}$
2. 2.
the scalar product $\langle\cdot,\cdot\rangle$ on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ is invariant.
It is given by
$\left[\xi+x,\eta+y\right]=\left[x,y\right]-ad^{*}_{\eta}x+ad^{*}_{\xi}y+\left[\xi,\eta\right]+ad^{*}_{x}\eta-
ad^{*}_{y}\xi.$ (2.10)
Moreover, the structure (2.10) is a Lie bracket on
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ if and only if $\mathfrak{g}$ is a Lie
bialgebra.
###### Definition 2.1.4.
The double $\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ of the Lie
bialgebra $\mathfrak{g}$ is defined by the vector space
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ together with the Lie bracket given by
(2.10).
Note that $\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ is also the
double of $\mathfrak{g}^{*}$·
#### 2.1.1 Classical Yang-Baxter equation and r-matrices
We now introduce a particular class of Lie bialgebra structures, given by a
coboundary of an element $r\in\mathfrak{g}\otimes\mathfrak{g}$, called
r-matrix. An $r$-matrix defines a cocycle $\delta$ as follows:
$\delta(x)=ad_{x}(r)=[x\otimes 1+1\otimes x,r].$ (2.11)
To each element $r$ in $\mathfrak{g}\otimes\mathfrak{g}$, we associate the map
$\underline{r}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}$ defined by
$\underline{r}(\xi)(\eta)=r(\xi,\eta),$ (2.12)
for $\xi,\eta\in\mathfrak{g}^{*}$. When $\delta$ is determined by $r$ we write
$[\xi,\eta]^{r}$ instead of $[\xi,\eta]_{*}$.
###### Proposition 2.1.5.
If $r$ is skew-symmetric, then
$[\xi,\eta]^{r}=ad_{\underline{r}\xi}^{*}\eta-ad_{\underline{r}\eta}^{*}\xi.$
(2.13)
In order to show when the $r$-matrix defines a Lie bialgebra, we introduce the
Schouten bracket of an element $r\in\mathfrak{g}\otimes\mathfrak{g}$ with
itself, denoted by $[r,r]$. It is the element of $\bigwedge^{3}\mathfrak{g}$
defined by
$[r,r](\xi,\eta,\varsigma)=-2\circlearrowleft\langle\varsigma,[\underline{r}\xi,\underline{r}\eta],$
(2.14)
where $\circlearrowleft$ denotes the summation over the circular permutation
of $\xi$, $\eta$ and $\varsigma$.
###### Proposition 2.1.6.
The $r$-matrix defines a Lie bracket on $\mathfrak{g}^{*}$ if and only if
$[r,r]$ is ad-invariant.
The condition for $[r,r]$ to be $ad$-invariant is sometimes called generalized
Yang-Baxter equation.
###### Definition 2.1.7.
Let $r$ be an element of $\mathfrak{g}\otimes\mathfrak{g}$, with symmetric
part $s$ and skew-symmetric part $a$. If $s$ and $[a,a]$ are ad-invariant,
then $r$ is called classical $r$-matrix. If $r$ is skew-symmetric ($r=a$) and
if $[r,r]=0$, then $r$ is called a triangular $r$-matrix.
Let us define the map
$\langle\underline{r,r}\rangle:\bigwedge^{2}\mathfrak{g}^{*}\rightarrow\mathfrak{g}$,
given by
$\langle\underline{r,r}\rangle(\xi,\eta)=[\underline{r}\xi,\underline{r}\eta]-\underline{r}[\xi,\eta]^{r}.$
(2.15)
Setting
$\langle
r,r\rangle(\xi,\eta,\varsigma)=\langle\varsigma,\langle\underline{r,r}\rangle(\xi,\eta)\rangle,$
(2.16)
the map $\langle\underline{r,r}\rangle$ is identified with an element $\langle
r,r\rangle\in\bigwedge^{2}\mathfrak{g}\otimes\mathfrak{g}$.
###### Theorem 2.1.8.
Let $\mathfrak{g}$ a finite dimensional Lie algebra.
1. 1.
Let $a$ be in $\mathfrak{g}\otimes\mathfrak{g}$ and skew-symmetric. Then
$\langle a,a\rangle$ is in $\bigwedge^{3}\mathfrak{g}$, and
$\langle a,a\rangle=-\frac{1}{2}[a,a],$ (2.17)
2. 2.
Let $s$ be in $\mathfrak{g}\otimes\mathfrak{g}$, symmetric and ad-invariant.
Then $\langle s,s\rangle$ is an ad-invariant element in
$\bigwedge^{3}\mathfrak{g}$, and
$\langle\underline{s,s}\rangle(\xi,\eta)=[\underline{s}\xi,\underline{s}\eta],$
(2.18)
3. 3.
For $r=s+a$, where $a$ is skew-symmetric and $s$ is symmetric and ad-
invariant, $\langle r,r\rangle$ is in $\bigwedge^{3}\mathfrak{g}$ and
$\langle r,r\rangle=\langle a,a\rangle+\langle s,s\rangle.$ (2.19)
From this theorem we obtain
###### Corollary 2.1.9.
Let $r=a+s$ where $a$ is skew-symmetric and $s$ is symmetric and ad-invariant.
Then $[a,a]$ is ad-invariant if $\langle r,r\rangle=0$.
Thus an element $r$ in $\mathfrak{g}\otimes\mathfrak{g}$ with ad-invariant
symmetric part, satisfying $\langle r,r\rangle=0$ is an $r$-matrix. The
condition $\langle r,r\rangle=0$ is called classical Yang-Baxter equation.
###### Definition 2.1.10.
An $r$-matrix satisfying the classical Yang-Baxter equation is called quasi-
triangular. If the symmetric part is invertible, then $r$ is called
factorisable.
#### 2.1.2 Tensor notation
Given $r\in\mathfrak{g}\otimes\mathfrak{g}$, we define $r_{12}$, $r_{13}$,
$r_{23}$ as elements in the third power tensor of the enveloping algebra of
$\mathfrak{g}$ (i.e. an associative algebra with unit such that $[x,y]=x\cdot
y-y\cdot x$),
$\begin{split}r_{12}&=r\otimes 1\\\ r_{23}&=1\otimes r.\end{split}$ (2.20)
If $r$ = $\sum_{i}u_{i}\otimes v_{i}$, it is clear that
$r_{13}=\sum_{i}u_{i}\otimes 1\otimes v_{i}$, where $1$ is the unit of the
enveloping algebra.
Let us define
$\begin{split}[r_{12},r_{13}]&=[\sum_{i}u_{i}\otimes v_{i}\otimes
1,\sum_{j}u_{j}\otimes 1\otimes v_{j}]=\sum_{i,j}[u_{i},u_{j}]\otimes
v_{i}\otimes v_{j},\\\ [r_{12},r_{23}]&=[\sum_{i}u_{i}\otimes v_{i}\otimes
1,\sum_{j}1\otimes u_{j}\otimes
v_{j}]=\sum_{i,j}u_{i}\otimes[v_{i},u_{j}]\otimes v_{j},\\\
[r_{13},r_{23}]&=[\sum_{i}u_{i}\otimes 1\otimes v_{i},\sum_{j}1\otimes
u_{j}\otimes v_{j}]=\sum_{i,j}u_{i}\otimes
u_{j}\otimes[v_{i},v_{j}].\end{split}$ (2.21)
in $\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}$. With this notation,
if the symmetric part of $r$ is ad-invariant, we have
$\langle r,r\rangle=[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]$ (2.22)
and
$\langle s,s\rangle=[s_{12},s_{13}]=[s_{12},s_{23}]=[s_{13},s_{23}].$ (2.23)
Hence, the classical Yang-Baxter equation reads
$[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0.$ (2.24)
###### Example 2.1.11.
Let $\mathfrak{g}$ be the 2-dimensional Lie algebra with basis $X$, $Y$ and
Lie bracket
$[X,Y]=X.$ (2.25)
Then $r=X\wedge Y=X\otimes Y-Y\otimes X$ is a skew-symmetric solution of the
classical Yang-Baxter equation. As a consequence, $\delta(X)=ad_{X}(r)=0$ and
$\delta(Y)=ad_{Y}(r)=-X\wedge Y$. In terms of dual basis $X^{*}$, $Y^{*}$ of
$\mathfrak{g}^{*}$, $[X^{*},Y^{*}]^{r}=-Y^{*}$.
###### Example 2.1.12.
Let $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$ and consider the Casimir
element
$t=\frac{1}{8}H\otimes H+\frac{1}{4}(X\otimes Y+Y\otimes X).$ (2.26)
We set
$t_{0}=\frac{1}{8}H\otimes H,\quad t_{\pm}=\frac{1}{4}X\otimes Y$ (2.27)
and we define
$r=t_{0}+2t_{\pm}=\frac{1}{8}(H\otimes H+4X\otimes Y).$ (2.28)
Then the symmetric part of $r$ is $t$ and the skew symmetric part is
$a=\frac{1}{4}X\wedge Y$, and $r$ is a factorisable $r$-matrix. Then
$\delta(H)=ad_{H}(a)=0$, $\delta(X)=ad_{X}(a)=\frac{1}{4}X\wedge H$ and
$\delta(Y)=ad_{Y}(a)=\frac{1}{4}Y\wedge H$. In terms of the dual basis
$H^{*}$, $X^{*}$, $Y^{*}$ of $\mathfrak{g}^{*}$,
$[H^{*},X^{*}]^{r}=\frac{1}{4}X^{*},\quad[H^{*},Y^{*}]^{r}=\frac{1}{4}Y^{*},\quad[X^{*},Y^{*}]^{r}=0.$
(2.29)
###### Example 2.1.13.
On $\mathfrak{sl}(2,\mathbb{C})$ we can consider also $r=X\otimes H-H\otimes
X$, which is a triangular $r$-matrix. Then $\delta(X)=0$, $\delta(Y)=2Y\wedge
X$ and $\delta(H)=X\wedge H$.
### 2.2 Poisson manifolds
As discussed in the previous chapter, it is always possible to define a
Poisson structure on a symplectic manifold, which give rise to Poisson
brackets on the space of smooth functions on the manifold. In the following,
we rephrase the definition of Poisson manifold given above in a more
convenient way [54] and we discuss the theory of symplectic foliation
[4],[55].
Let $\pi$ be a bivector on a manifold $M$, i.e. a skew-symmetric,
contravariant 2-tensor. At each point $m$, $\pi(m)$ can be viewed as a skew-
symmetric bilinear form on $T_{m}^{*}M$, or as the skew-symmetric linear map
$\pi^{\sharp}(m):T_{m}^{*}M\rightarrow T_{m}M$, such that
$\pi(m)(\alpha_{m},\beta_{m})=\pi^{\sharp}(\alpha_{m})(\beta_{m}),\quad\alpha_{m},\beta_{m}\in
T_{m}^{*}M.$ (2.30)
If $\alpha$, $\beta$ are 1-forms on $M$, we define $\pi(\alpha,\beta)$ to be
the function in $C^{\infty}(M)$ whose value at $m$ is
$\pi(m)(\alpha_{m},\beta_{m})$. Given $f,g\in C^{\infty}(M)$ we set
$\pi(m)(df,dg)=\\{f,g\\}(m).$ (2.31)
Note that $\pi^{\sharp}(df)$ is the Hamiltonian vector field defined in
(1.10). It is clear that the bracket induced by $\pi$ satisfies the Leibniz
rule.
###### Definition 2.2.1.
A Poisson manifold $(M,\pi)$ is a manifold $M$ with a Poisson bivector $\pi$
such that the bracket defined in eq. (2.31) satisfies the Jacobi identity.
###### Example 2.2.2.
If $M=\mathbb{R}^{2n}$, with coordinates $(q^{i},p_{i})$, $i=1,\cdots,n$ and
if
$\pi^{\sharp}(dq^{i})=-\frac{\partial}{\partial
q^{i}},\quad\pi^{\sharp}(dp_{i})=-\frac{\partial}{\partial p_{i}},$ (2.32)
then
$X_{f}=\frac{\partial f}{\partial p_{i}}\frac{\partial}{\partial
q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial}{\partial p_{i}}$
(2.33)
and
$\\{f,g\\}=\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
q^{i}}-\frac{\partial f}{\partial q^{i}}\frac{\partial g}{\partial p_{i}},$
(2.34)
is the standard Poisson bracket of functions on the phase space. The
corresponding bivector is $\pi=\frac{\partial}{\partial
p_{i}}\wedge\frac{\partial}{\partial q^{i}}$.
In local coordinates, a bivector $\pi$ is a Poisson bivector if and only if
$\pi^{hi}\partial_{h}\pi^{jk}+\pi^{hj}\partial_{h}\pi^{ki}+\pi^{hk}\partial_{h}\pi^{ij}=0.$
(2.35)
In terms of bivector field $\pi$ we have the following characterization of a
Poisson bivector:
###### Proposition 2.2.3.
The bivector field $\pi\in\wedge^{2}TM$ is a Poisson bivector if and only if
$[\pi,\pi]_{S}=0$, where $[\cdot,\cdot]_{S}$ is the Schouten-Nijenhuis
bracket.
###### Definition 2.2.4.
A mapping $\phi:(M_{1},\pi_{1})\rightarrow(M_{2},\pi_{2})$ between two Poisson
manifolds is called a Poisson mapping if $\forall f,g\in C^{\infty}(M_{2})$
one has
$\\{f\circ\phi,g\circ\phi\\}_{1}=\\{f,g\\}_{2}\circ\phi$ (2.36)
###### Definition 2.2.5.
Given two Poisson manifolds $(M_{1},\pi_{1})$ and $(M_{2},\pi_{2})$, the pair
$(M_{1}\times M_{2},\pi)$, is a Poisson manifold, called Poisson product, with
$\pi=\pi_{1}\oplus\pi_{2}$.
An interesting feature of Poisson manifolds is the existence of the
differential calculus of forms, which can be resumed in the following result:
###### Theorem 2.2.6 ([54]).
Let $(M,\pi)$ be a Poisson manifold. Then there exists a unique bilinear,
skew-symmetric operation
$[\cdot,\cdot]_{\pi}:\Omega^{1}(M)\times\Omega^{1}(M)\rightarrow\Omega^{1}(M)$
such that
$\begin{split}[df,dg]_{\pi}&=d[f,g]_{\pi},\qquad f,g\in C^{\infty}(M),\\\
[\alpha,f\beta]_{\pi}&=f[\alpha,\beta]_{\pi}+(\iota_{\pi^{\sharp}(\alpha)}f)\beta\qquad
f\in C^{\infty}(M),\alpha,\beta\in\Omega^{1}(M).\end{split}$ (2.37)
This operation is given by the general formula
$[\alpha,\beta]_{\pi}=\mathcal{L}_{\pi^{\sharp}(\alpha)}\beta-\mathcal{L}_{\pi^{\sharp}(\beta)}\alpha-d(\pi(\alpha,\beta))$
(2.38)
Furthermore, it provides $\Omega^{1}(M)$ with a Lie algebra structure such
that $\pi^{\sharp}:T^{*}M\rightarrow TM$ is a Lie algebra homomorphism.
#### 2.2.1 Symplectic Foliation
As mentioned above, every symplectic manifold admits a Poisson structure, but
the converse does not hold. In fact, any Poisson manifold can be seen as a
union of symplectic manifolds called symplectic leaves.
Locally, the symplectic foliation of $(M,\pi)$ can be described in terms of
coordinates. More precisely, the local structure of a Poisson manifold at
$O\in M$ is described by the Splitting Theorem [55]:
###### Theorem 2.2.7 (Weinstein).
On a Poisson manifold $(M,\pi)$, any point $O\in M$ has a coordinate
neighborhood with coordinates
$(q_{1},\dots,q_{k},p_{1},\dots,p_{k},\allowbreak y_{1},\dots,y_{l})$ centered
at $O$, such that
$\pi=\sum_{i}\frac{\partial}{\partial q_{i}}\wedge\frac{\partial}{\partial
p_{i}}+\frac{1}{2}\sum_{i,j}\phi_{ij}(y)\frac{\partial}{\partial
y_{i}}\wedge\frac{\partial}{\partial y_{j}}\quad\phi_{ij}(0)=0.$ (2.39)
The rank of $\pi$ at $O$ is $2k$. Since $\phi$ depends only on the $y_{i}$’s,
this theorem gives a decomposition of the neighborhood of $O$ as a product of
two Poisson manifolds: one with rank $2k$, and the other with rank 0 at $O$.
For any point $O$ of the Poisson manifold, if $(q,p,y)$ are normal coordinates
as in the previous theorem, then the symplectic leaf through $O$ is given
locally by the equation $y=0$. Hence, for any point $m\in M$, we have a
symplectic leaf through it. Locally, this leaf has canonical coordinates
$(q_{1},\dots,q_{k},p_{1},\dots,p_{k})$, where the bracket is given by
canonical symplectic relations. Notice that the symplectic leaf is well-
defined, but each choice of coordinates $(y_{1},\dots y_{l})$ in Theorem 2.2.7
gives rise to a different term
$\frac{1}{2}\sum_{i,j}\phi_{ij}(y)\frac{\partial}{\partial
y_{i}}\wedge\frac{\partial}{\partial y_{j}}$ (2.40)
called the transverse Poisson structure of dimension $l$. The transverse
structures are not uniquely defined, but they are all isomorphic. Locally, the
transverse structure is determined by the structure functions
$\pi_{ij}(y)=\\{y_{i},y_{j}\\}$ (which vanishes at $y=0$).
Given a Poisson manifold the Casimir functions are functions which are
constant on the symplectic leaves.
### 2.3 Poisson Lie groups
Poisson Lie groups are a particular class of Poisson manifolds. Namely, a
Poisson Lie group is a Poisson manifold that is also a Lie group and its
corresponding infinitesimal object is a Lie bialgebra. In particular we
discuss Poisson Lie groups defined by $r$-matrices and some basic examples.
Recall that, given a Lie group $G$, the left and right translations by an
element $g\in G$ are defined by
$\lambda_{g}(h)=gh,\quad\rho_{g}(h)=hg,$ (2.41)
for $h\in G$.
###### Definition 2.3.1.
A Poisson Lie group $(G,\pi_{G})$ is a Lie group equipped with a
multiplicative Poisson structure $\pi_{G}$.
In terms of the Poisson tensor $\pi_{G}$, the Poisson structure is
multiplicative if and only if
$\pi_{G}(gh)=\lambda_{g}\pi_{G}(h)+\rho_{h}\pi_{G}(g),\qquad\forall g,h\in G,$
(2.42)
This is equivalent to the following condition (see [28])
$\\{\varphi\circ\lambda_{g},\psi\circ\lambda_{g}\\}(h)+\\{\varphi\circ\rho_{h},\psi\circ\rho_{h}\\}(g)=\\{\varphi,\psi\\}(gh),$
(2.43)
for all functions $\varphi,\psi$ on $G$, and for all $g,h$ in $G$. This
condition means that the multiplication map $G\times G\rightarrow G$ is a
Poisson map. Note that a nonzero multiplicative Poisson structure is in
general neither left nor right invariant. In fact the left or right
translation by $g\in G$ preserves $\pi_{G}$ if and only if $\pi_{G}(g)=0$.
###### Example 2.3.2.
If $\pi_{G}=0$, it is obviously multiplicative, hence any Lie group $G$ with
the trivial Poisson structure is a Poisson Lie group. The direct product of
two Poisson Lie groups is again a Poisson Lie group.
A very important class of Poisson Lie groups arise from the $r$-matrix
formalism. Let $r\in\mathfrak{g}\wedge\mathfrak{g}$. Define a bivector
$\pi_{G}$ on $G$ by
$\pi_{G}(g)=\lambda_{g}r-\rho_{g}r\qquad\forall g\in G$ (2.44)
We see that this bivector is multiplicative. In particular we have:
###### Theorem 2.3.3 (Drinfeld).
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. Let
$r\in\mathfrak{g}\wedge\mathfrak{g}$. Define a bivector field on $G$ by eq.
(2.44). Then $(G,\pi_{G})$ is a Poisson Lie group if and only if
$\left[r,r\right]\in\mathfrak{g}\wedge\mathfrak{g}\wedge\mathfrak{g}$ is Ad-
invariant. In particular, when $r$ satisfies the Yang-Baxter equation, it
defines a multiplicative Poisson structure on $G$.
The Poisson structure $\pi$ vanishes at the identity $e\in G$, and its
linearization at $e$ is given by
$d_{e}\pi:\mathfrak{g}\rightarrow\mathfrak{g}\wedge\mathfrak{g}$. The map
$d_{e}\pi$ is a derivative and its dual map
$\left[\cdot,\cdot\right]_{*}:\mathfrak{g^{*}}\wedge\mathfrak{g^{*}}\rightarrow\mathfrak{g^{*}}$
is given by $\left[x,y\right]_{*}=d_{e}(\pi_{G}(\bar{x},\bar{y}))$, where
$x,y\in\mathfrak{g}^{*}$ and $\bar{x}$ and $\bar{y}$ can be any 1-forms on $G$
with $\bar{x}(e)=x$ and $\bar{y}(e)=y$.
When $\pi_{G}$ is a Poisson bivector, $\left[\cdot,\cdot\right]_{*}$ satisfies
the Jacobi identity, so it makes $\mathfrak{g^{*}}$ into a Lie algebra. The
Lie algebra $(\mathfrak{g^{*}},\left[\cdot,\cdot\right]_{*})$ is just the
linearization of the Poisson structure at $e$.
###### Theorem 2.3.4.
A multiplicative bivector field $\pi_{G}$ on a connected Lie group $G$ is
Poisson if and only if its linearization at $e$ defines a Lie bracket on
$\mathfrak{g^{*}}$.
The relation between Poisson Lie groups and Lie bialgebras is given by the
following theorem:
###### Theorem 2.3.5 (Drinfeld).
If $(G,\pi_{G})$ is a Poisson Lie group, then the linearization of $\pi_{G}$
at $e$ defines a Lie algebra structure on $\mathfrak{g}^{*}$ such that
$(\mathfrak{g},\mathfrak{g}^{*})$ form a Lie bialgebra over $\mathfrak{g}$,
called the tangent Lie bialgebra to $(G,\pi_{G})$. Conversely, if $G$ is
connected and simply connected, then every Lie bialgebra
$(\mathfrak{g},\mathfrak{g}^{*})$ over $\mathfrak{g}$ defines a unique
multiplicative Poisson structure $\pi_{G}$ on $G$ such that
$(\mathfrak{g},\mathfrak{g}^{*})$ is the tangent Lie bialgebra to the Poisson
Lie group $(G,\pi_{G})$.
It is important to emphasize that, given a connected Lie group $G$ with Lie
algebra $\mathfrak{g}$, for an arbitrary 1-cocycle $\delta$ on $\mathfrak{g}$,
there is no general way to integrate $\delta$ to the 1-cocycle $\epsilon$ on
$G$ such that $d_{e}\epsilon=\delta$. In the case of a Lie bialgebra
$(\mathfrak{g},\mathfrak{g}^{*},\delta)$ the integration can be reduced to
integrating Lie algebras to Lie groups and Lie algebra homomorphisms to Lie
groups homomorphisms.
#### 2.3.1 Examples
###### Quasi-triangular structure of $SL(2,\mathbb{R})$
Let $G=SL(2,\mathbb{R})$ be the group of real $2\times 2$ matrices with
determinant 1. We consider $r=\frac{1}{8}(H\otimes H+4X\otimes Y)$, as seen in
Example 2.1.12, as element in
$\mathfrak{sl}(2,\mathbb{R})\otimes\mathfrak{sl}(2,\mathbb{R})$, with skew-
symmetric part $r_{0}=\frac{1}{4}(X\otimes Y-Y\otimes X)$. Then, given a
generic element $g=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\in G$ and
using (2.44) for $r_{0}$ we get the quadratic Poisson brackets,
$\displaystyle\\{a,b\\}$ $\displaystyle=\frac{1}{4}ab$
$\displaystyle\\{a,c\\}$ $\displaystyle=\frac{1}{4}ac$
$\displaystyle\\{a,d\\}$ $\displaystyle=\frac{1}{2}bc$ (2.45)
$\displaystyle\\{b,c\\}$ $\displaystyle=0$ $\displaystyle\\{b,d\\}$
$\displaystyle=\frac{1}{4}bd$ $\displaystyle\\{c,d\\}$
$\displaystyle=\frac{1}{4}cd.$ (2.46)
It is easy to check that $ad-bc$ is a Casimir element for this Poisson
structure, which is indeed defined on $SL(2,\mathbb{C})$.
###### Triangular structure of $SL(2,\mathbb{R})$
Consider now the triangular $r$-matrix defined in Example 2.1.13,
$r=X\otimes H-H\otimes X=\left(\begin{matrix}0&-1&1&0\\\ 0&0&0&-1\\\
0&0&0&1\\\ 0&0&0&0\end{matrix}\right).$ (2.47)
We find
$\displaystyle\\{a,b\\}$ $\displaystyle=1-a^{2}$ $\displaystyle\\{a,c\\}$
$\displaystyle=c^{2}$ $\displaystyle\\{a,d\\}$ $\displaystyle=c(-a+d)$ (2.48)
$\displaystyle\\{b,c\\}$ $\displaystyle=c(a+d)$ $\displaystyle\\{b,d\\}$
$\displaystyle=d^{2}-1$ $\displaystyle\\{c,d\\}$ $\displaystyle=-c^{2}.$
(2.49)
Also in this case $ad-bc$ is a Casimir function.
###### Quasi-triangular structure of $SU(2)$
Let $G=SU(2)$ and $\mathfrak{g}=\mathfrak{su}(2)$. Let
$e_{1}=\frac{1}{2}\left(\begin{matrix}i&0\\\ 0&-i\end{matrix}\right)\quad
e_{2}=\frac{1}{2}\left(\begin{matrix}0&1\\\ -1&0\end{matrix}\right)\quad
e_{3}=\frac{1}{2}\left(\begin{matrix}0&i\\\ i&0\end{matrix}\right).$ (2.50)
Then ${e_{1},e_{2},e_{3}}$ is a basis for $\mathfrak{g}$, and
$[e_{1},e_{2}]=e_{3}$, $[e_{2},e_{3}]=e_{1}$, $[e_{3},e_{1}]=e_{2}$. Any
$r\in\mathfrak{g}\wedge\mathfrak{g}$ is such that $\left[r,r\right]$ is
$Ad_{G}$-invariant.
Let $r=2(e_{2}\wedge e_{3})$ and define the Poisson structure on $SU(2)$ by
$\pi(g)=2\left(\rho_{g}(e_{2}\wedge e_{3})-\lambda_{g}(e_{2}\wedge
e_{3})\right).$ (2.51)
Hence, given a generic element $g\in SU(2)$ as $g=\left(\begin{matrix}a&b\\\
c&d,\end{matrix}\right)$, the Poisson brackets are given by
$\displaystyle\left\\{a,b\right\\}$ $\displaystyle=iab$
$\displaystyle\left\\{a,c\right\\}$ $\displaystyle=iac$
$\displaystyle\left\\{a,d\right\\}$ $\displaystyle=2ibc$ (2.52)
$\displaystyle\left\\{b,c\right\\}$ $\displaystyle=0$
$\displaystyle\left\\{b,d\right\\}$ $\displaystyle=ibd$
$\displaystyle\left\\{c,d\right\\}$ $\displaystyle=icd.$ (2.53)
Finally, the Lie bracket defined by $r$ on $\mathfrak{su}(2)^{*}$ is given by
$[e_{1}^{*},e_{2}^{*}]=e_{2}^{*},\quad[e_{1}^{*},e_{3}^{*}]=e_{3}^{*},\quad[e_{2}^{*},e_{3}^{*}]=0.$
(2.54)
#### 2.3.2 The dual of a Poisson Lie group
In Section 2.1 we introduced the dual and the double of a Lie bialgebra, so
now we discuss the corresponding constructions at the level of the Lie group.
Given a Poisson Lie group $(G,\pi_{G})$, we consider its Lie bialgebra
$\mathfrak{g}$ whose 1-cocycle is $\delta=d_{e}\pi_{G}$. Let us denote the
dual Lie bialgebra by $(\mathfrak{g}^{*},\delta)$. By Theorem 2.3.5 we know
that there is a unique connected and simply connected Poisson Lie group
$(G^{*},\pi_{G^{*}})$, called the dual of $(G,\pi_{G})$, associated to the Lie
bialgebra $(\mathfrak{g}^{*},\delta)$. If $G$ is connected and simply
connected, then the dual of $G^{*}$ is $G$.
When $(G,\pi_{G})$ is a Poisson Lie group, its Lie bialgebra
$(\mathfrak{g},\delta)$ has a double $\mathfrak{d}$. The connected and simply
connected Lie group $\mathcal{D}$ with Lie algebra $\mathfrak{d}$ is called
the double of $(G,\pi_{G})$. Since $\mathfrak{d}$ is a factorisable Lie
bialgebra with $r$-matrix $r_{\mathfrak{d}}$, $\mathcal{D}$ is a factorisable
Poisson Lie group with Poisson structure
$\pi_{\mathcal{D}}(d)=\lambda_{d}r_{\mathfrak{d}}-\rho_{d}r_{\mathfrak{d}}$
(2.55)
where $d\in\mathcal{D}$.
###### Example 2.3.6.
Let $G=SU(2)$ be equipped with the Poisson structure given in Example 2.3.1.
Then $G^{*}$ can be identified with $SB(2,\mathbb{C})$. The double Lie algebra
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ is the Lie algebra
$\mathfrak{sl}(2,\mathbb{C})$ considered as a real Lie algebra. The
decomposition
$\mathfrak{sl}(2,\mathbb{C})=\mathfrak{su}(2)\oplus\mathfrak{sb}(2,\mathbb{C})$
is the well-known Gram-Schmidt decomposition.
###### Example 2.3.7.
Consider the Lie bialgebra $\mathfrak{g}=ax+b$ of Example 2.1.11. A matrix
representation of $\mathfrak{g}$ is the Lie algebra
$\mathfrak{gl}(2,\mathbb{R})$ via
$X=\left(\begin{matrix}1&0\\\ 0&0\end{matrix}\right),\quad
Y=\left(\begin{matrix}0&1\\\ 0&0\end{matrix}\right),\quad
X^{*}=\left(\begin{matrix}0&0\\\ 0&1\end{matrix}\right),\quad
Y^{*}=\left(\begin{matrix}0&0\\\ 1&0\end{matrix}\right)$ (2.56)
with metric
$\gamma(A,B)=tr(AJBJ),\quad\text{where}\quad J=\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right).$ (2.57)
The subgroups $G$ and $G^{*}$ of the Lie group $GL^{+}(2,\mathbb{R})$ of
matrices with determinant $>0$ are given by
$G=\left\\{\left(\begin{matrix}x&y\\\ 0&1\end{matrix}\right):x>0\right\\}\quad
G^{*}=\left\\{\left(\begin{matrix}1&0\\\ a&b\end{matrix}\right):b>0\right\\}.$
(2.58)
The Poisson bivector on $GL^{+}(2,\mathbb{R})$ in the coordinates
$\left(\begin{matrix}x&y\\\ a&b\end{matrix}\right)$ reads
$\pi=xy\partial_{x}\wedge\partial_{y}+ab\partial_{a}\wedge\partial_{b}+xb(\partial_{x}\wedge\partial_{b}+\partial_{a}\wedge\partial_{y}).$
(2.59)
It is degenerate at points with $xb=0$ and vanishes at $x=b=0$.
### 2.4 Poisson actions and Momentum maps
A Poisson action is a key concept for the generalization of the theory of
momentum map, since it generalizes the canonical action discussed in the
previous chapter.
Recall, from Definition 1.2.4, that a canonical action of a Lie group $G$ on a
Poisson manifold $M$ is defined as a group action which preserves the Poisson
structure. Instead, a Poisson action is an action of a Poisson Lie group on a
Poisson manifold satisfying a different property of compatibility between the
Poisson bivectors of both manifolds. When the Poisson structure is trivial we
recover the canonical actions.
In the following we always assume that $G$ is connected and simply connected,
such that Theorem 2.3.5 holds.
###### Definition 2.4.1.
The action of $(G,\pi_{G})$ on $(M,\pi)$ is called Poisson action if the map
$\Phi:G\times M\rightarrow M$ is Poisson, where $G\times M$ is a Poisson
product with structure $\pi_{G}\oplus\pi$
If $(G,\pi_{G})$ is a Poisson Lie group, the left and right actions of $G$ on
itself are Poisson actions. From the previous chapter, we have that, given an
action $\Phi$ of $G$ on $M$, the infinitesimal generator $\xi_{M}$ associated
to $\xi\in\mathfrak{g}$ is the vector field on $M$ defined by
$\xi_{M}(m):=\left.\frac{d}{dt}\right|_{t=0}\Phi_{\exp(-t\xi)}(m).$ (2.60)
This defines an action of $\mathfrak{g}$ on $M$ by
$\xi\in\mathfrak{g}\mapsto\xi_{M}$, in fact we have
$[\xi,\eta]_{M}=[\xi_{M},\eta_{M}],\quad\text{for}\quad\xi,\eta\in\mathfrak{g}.$
(2.61)
Note that if $G$ carries the zero Poisson structure $\pi_{G}=0$, the action is
Poisson if and only if it preserves $\pi$. In general, when $\pi_{G}\neq 0$,
the structure $\pi$ is not invariant with respect to the action.
###### Proposition 2.4.2.
Assume that $(G,\pi_{G})$ is a connected Poisson Lie group with associate
1-cocycle of $\mathfrak{g}$
$\delta=d_{e}\pi_{G}:\mathfrak{g}\rightarrow\wedge^{2}\mathfrak{g},$ (2.62)
and let $(M,\pi)$ be a Poisson manifold. The action $\Phi:G\times M\rightarrow
M$ is a Poisson action if and only if
$\mathcal{L}_{\xi_{M}}(\pi)=-(\delta(\xi))_{M}$ (2.63)
for any $\xi\in\mathfrak{g}$, where $\mathcal{L}$ denotes the Lie derivative.
###### Definition 2.4.3.
A Lie algebra action $\xi\mapsto\xi_{M}$ is called an infinitesimal Poisson
action of the Lie bialgebra $(\mathfrak{g},\delta)$ on $(M,\pi)$ if it
satisfies eq. (2.63).
In this formalism the definition of momentum map reads (Lu, [33], [34]):
###### Definition 2.4.4.
A momentum map for the Poisson action $\Phi:G\times M\rightarrow M$ is a map
$\boldsymbol{\mu}:M\rightarrow G^{*}$ such that
$\xi_{M}=\pi^{\sharp}(\boldsymbol{\mu}^{*}(\theta_{\xi}))$ (2.64)
where $\theta_{\xi}$ is the left invariant 1-form on $G^{*}$ defined by the
element $\xi\in\mathfrak{g}=(T_{e}G^{*})^{*}$ and $\boldsymbol{\mu}^{*}$ is
the cotangent lift $T^{*}G^{*}\rightarrow T^{*}M$.
If $G$ has trivial Poisson structure, then $G^{*}=\mathfrak{g}^{*}$, the
differential 1-form $\theta_{\xi}$ is the constant 1-form $\xi$ on
$\mathfrak{g}^{*}$, and
$\boldsymbol{\mu}^{*}(\theta_{\xi})=d(\boldsymbol{\mu}^{\xi}),\quad\text{where}\quad\boldsymbol{\mu}^{\xi}(m)=\langle\boldsymbol{\mu}(m),\xi\rangle.$
(2.65)
Thus, in this case, we recover the usual definition of a momentum map for a
Hamiltonian action $\boldsymbol{\mu}:M\rightarrow\mathfrak{g}^{*}$, that is
$\xi_{M}=\pi^{\sharp}(d(\boldsymbol{\mu}^{\xi})).$ (2.66)
In other words, $\xi_{M}$ is the Hamiltonian vector field with Hamiltonian
$\boldsymbol{\mu}^{\xi}\in C^{\infty}(M)$.
When $\pi_{G}$ is not trivial, $\theta_{\xi}$ is a Maurer-Cartan form, hence
$\boldsymbol{\mu}^{*}(\theta_{\xi})$ can not be written as a differential of a
Hamiltonian function.
#### 2.4.1 Dressing Transformations
One of the most important example of Poisson action is the dressing action of
$G$ on $G^{*}$. Consider a Poisson Lie group $(G,\pi_{G})$, its dual
$(G^{*},\pi_{G^{*}})$ and its double $\mathcal{D}$, with Lie algebras
$\mathfrak{g}$, $\mathfrak{g}^{*}$ and $\mathfrak{d}$, respectively.
###### Theorem 2.4.5.
Let $l(\xi)$ the vector field on $G^{*}$ defined by
$l(\xi)=\pi_{G^{*}}^{\sharp}(\theta_{\xi})$ (2.67)
for each $\xi\in\mathfrak{g}$. Here $\theta_{\xi}$ is the left invariant
1-form on $G^{*}$ defined by $\xi\in\mathfrak{g}=(T_{e}G^{*})^{*}$. Then
1. 1.
The map $\xi\mapsto l(\xi)$ is an action of $\mathfrak{g}$ on $G^{*}$, whose
linearization at $e$ is the coadjoint action of $\mathfrak{g}$ on
$\mathfrak{g}^{*}$.
2. 2.
The action $\xi\mapsto l(\xi)$ is an infinitesimal Poisson action of the Lie
bialgebra $\mathfrak{g}$ on the Poisson Lie group $G^{*}$.
A proof of this theorem can be found in [28].
The action defined in (2.67) is generally called left infinitesimal dressing
action of $\mathfrak{g}$ on $G^{*}$. In particular, when $G$ is a trivial
Poisson Lie group, its dual group $G^{*}$ is the Abelian group
$\mathfrak{g}^{*}$, and the left infinitesimal dressing action of
$\mathfrak{g}$ on $\mathfrak{g}^{*}$ is given by the linear vector fields
$l(\xi):\eta\in\mathfrak{g}^{*}\mapsto-ad^{*}_{\xi}\eta\in\mathfrak{g}^{*}$,
for each $\xi\in\mathfrak{g}$. It is easy to see that $\xi\mapsto l(\xi)$ is a
Lie algebra homomorphism from $\mathfrak{g}$ to the Lie algebra of linear
vector fields on $\mathfrak{g}^{*}$, in fact we find
$l([\xi,\eta])=l(\xi)l(\eta)-l(\eta)l(\xi)$.
Similarly, the right infinitesimal dressing action of $\mathfrak{g}$ on
$G^{*}$ is defined by
$r(\xi)=-\pi_{G^{*}}^{\sharp}(\theta_{\xi})$ (2.68)
where $\theta_{\xi}$ is the right invariant 1-form on $G^{*}$.
Let $l(\xi)$ (resp. $r(\xi)$) a left (resp. right) dressing vector field on
$G^{*}$. If all the dressing vector fields are complete, we can integrate the
$\mathfrak{g}$-action into a Poisson $G$-action on $G^{*}$ called the dressing
action and we say that the dressing actions consist of dressing
transformations.
###### Proposition 2.4.6.
The symplectic leaves of $G$ (resp. $G^{*}$) are the connected components of
the orbits of the right or left dressing action of $G^{*}$ (resp. $G$).
The momentum map for the dressing action of $G$ on $G^{*}$ is the opposite of
the identity map from $G^{*}$ to itself.
For the Poisson Lie group $G^{*}$ we identify $\mathfrak{g}$ with the space of
left invariant 1-forms on $G^{*}$. Then, this space is closed under the
bracket defined by $\pi_{G^{*}}$ and the induced bracket on $\mathfrak{g}$, by
the above identification, coincides with the original Lie bracket on
$\mathfrak{g}$ (see [56]). Given $\xi\in\mathfrak{g}$, we denote by
$\theta_{\xi}$ the left invariant form on $G^{*}$ whose value at identity is
$\xi$. The basic property of $\theta$’s is the Maurer-Cartan equation for
$G^{*}$:
$d\theta_{\xi}+\frac{1}{2}\theta\wedge\theta\circ\delta(\xi)=0$ (2.69)
In the following proposition we prove new relations satisfied by $\theta$:
###### Proposition 2.4.7.
Let $\theta_{\xi},\theta_{\eta}$ be two left invariant 1-forms on $G^{*}$,
such that $\theta_{\xi}(e)=\xi$, $\theta_{\eta}(e)=\eta$ then
$\theta_{[\xi,\eta]}=[\theta_{\xi},\theta_{\eta}]_{\pi_{G^{*}}}$ (2.70)
and
$\mathcal{L}_{X}\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})=x([\xi,\eta])+\pi_{G^{*}}(\theta_{ad^{*}_{x}\xi},\theta_{\eta})+\pi_{G^{*}}(\theta_{\xi},\theta_{ad^{*}_{x}\eta})$
(2.71)
###### Proof.
First, we prove that $[\theta_{\xi},\theta_{\eta}]_{\pi_{G^{*}}}$ is a left-
invariant 1-form. Let us consider and element $x\in\mathfrak{g}^{*}$ and the
correspondent left invariant vector field $X\in TG^{*}$. We contract $X$ with
the bracket $[\theta_{\xi},\theta_{\eta}]_{\pi_{G^{*}}}$ to show that we
obtain a constant. More precisely, we contract $X$ term by term with the
equation (2.38) and we get
$\begin{split}\iota_{X}d\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})&=(\mathcal{L}_{X}\pi_{G^{*}})(\theta_{\xi},\theta_{\eta})+\pi_{G^{*}}(\mathcal{L}_{X}\theta_{\xi},\theta_{\eta})+\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})\\\
\iota_{X}\mathcal{L}_{\pi^{\sharp}_{G^{*}}(\theta_{\xi})}\theta_{\eta}&=(\mathcal{L}_{X}\pi_{G^{*}})(\theta_{\eta},\theta_{\xi})-\pi_{G^{*}}(\mathcal{L}_{X}\theta_{\xi},\theta_{\eta})\\\
\iota_{X}\mathcal{L}_{\pi^{\sharp}_{G^{*}}(\theta_{\eta})}\theta_{\xi}&=(\mathcal{L}_{X}\pi_{G^{*}})(\theta_{\xi},\theta_{\eta})-\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})\end{split}$
(2.72)
Then, substituting the relations (2.72) in (2.38) we obtain
$\iota_{X}[\theta_{\xi},\theta_{\eta}]_{\pi_{G^{*}}}=(\mathcal{L}_{X}\pi_{G^{*}})(\theta_{\xi},\theta_{\eta}).$
(2.73)
Since $\mathcal{L}_{X}\pi_{G^{*}}(e)={}^{t}\delta(x)$, eq. (2.70) is proved.
Moreover, we have
$\begin{split}\mathcal{L}_{X}\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})&=(\mathcal{L}_{X}\pi_{G^{*}})(\theta_{\xi},\theta_{\eta})+\pi_{G^{*}}(\mathcal{L}_{X}\theta_{\xi},\theta_{\eta})+\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})\\\
&={}^{t}\delta(x)(\xi,\eta)+\pi_{G^{*}}(\theta_{ad^{*}_{x}\xi},\theta_{\eta})+\pi_{G^{*}}(\theta_{\xi},\theta_{ad^{*}_{x}\eta}),\end{split}$
(2.74)
since $\mathcal{L}_{X}\theta_{\xi}=\theta_{ad^{*}_{x}\xi}$. From
${}^{t}\delta(x)(\xi,\eta)=x([\xi,\eta])$, eq. (2.71) follows. ∎
For sake of completeness, we record an alternative way to define the dressing
action. Consider $g\in G$ and $u\in G^{*}$ and let $ug\in\mathcal{D}$ be their
product. Since $\mathfrak{d}=\mathfrak{g}\oplus\mathfrak{g}^{*}$, elements in
$\mathcal{D}$ close to the unit can be decomposed in a unique way as a product
of an element in $G$ and an element in $G^{*}$. Then, there exist elements
${}^{u}g\in G$ and $u^{g}\in G^{*}$ such that
$ug=^{u}gu^{g}.$ (2.75)
Hence, the action of $u\in G^{*}$ on $g\in G$ (resp. the action of $g\in G$ on
$u\in G^{*}$) is given by
$(u,g)\mapsto(ug)_{G}\quad(\text{resp.}\quad(u,g)\mapsto(ug)_{G^{*}}),$ (2.76)
where $(ug)_{G}$ (resp. $(ug)_{G^{*}}$) denotes the $G$-factor (resp.
$G^{*}$-factor) of $ug\in\mathcal{D}$ as $g^{\prime}u^{\prime}$, with
$g^{\prime}\in G$, $u^{\prime}\in G^{*}$. Accordingly, the product
$gu\in\mathcal{D}$ can be uniquely decomposed into ${}^{g}ug^{u}$, where
${}^{g}u\in G^{*}$ and $g^{u}\in G$. So, by definition,
$gu=^{g}ug^{u}.$ (2.77)
This defines locally a left action of $G$ on $G^{*}$ and a right action of
$G^{*}$ on $G$.
###### Definition 2.4.8.
A multiplicative Poisson tensor $\pi$ on $G$ is complete if each left (equiv.
right) dressing vector field is complete on $G$.
###### Proposition 2.4.9.
A Poisson Lie group is complete if and only if its dual Poisson Lie group is
complete.
Assume that $G$ is a complete Poisson Lie group. We denote respectively the
left (resp. right) dressing action of $G$ on its dual $G^{*}$ by $g\mapsto
l_{g}$ (resp. $g\mapsto r_{g}$).
###### Definition 2.4.10.
A momentum map $\boldsymbol{\mu}:M\rightarrow G^{*}$ for a left (resp. right)
Poisson action $\Phi$ is called G-equivariant if it is such with respect to
the left dressing action of $G$ on $G^{*}$, that is,
$\boldsymbol{\mu}\circ\Phi_{g}=\lambda_{g}\circ\boldsymbol{\mu}$ (resp.
$\boldsymbol{\mu}\circ\Phi_{g}=\rho_{g}\circ\boldsymbol{\mu}$)
A momentum map is $G$-equivariant if and only if it is a Poisson map, i.e.
$\boldsymbol{\mu}_{*}\pi=\pi_{G^{*}}$. Given this generalization of the
concept of equivariance introduced for Lie group actions, it is natural to
call Hamiltonian action a Poisson action induced by an equivariant momentum
map.
#### 2.4.2 Structure of the momentum map
In this section we introduce a weaker definition of momentum map in
infinitesimal terms. From Definition 2.4.4, it follows that one can associate
to a momentum map a 1-form $\alpha_{\xi}$. In the following, we discuss the
properties of these forms and, using the infinitesimal momentum map, we
analyze the conditions under which the momentum map is determined. Then, we
introduce the concept of gauge equivalence for the $\alpha$’s and we show the
relation of this equivalence class with a cohomological class in
$H^{1}(M,\mathfrak{g})$.
As a direct consequence of the properties of $\theta$’s stated in Section
2.4.1, we have the following proposition:
###### Proposition 2.4.11.
Given a Poisson action $\Phi:G\times M\rightarrow M$ with equivariant momentum
map $\boldsymbol{\mu}:M\rightarrow G^{*}$, the forms
$\alpha_{\xi}=\boldsymbol{\mu}^{*}(\theta_{\xi})$ satisfy the following
identities:
$\displaystyle\alpha_{[\xi,\eta]}$
$\displaystyle=[\alpha_{\xi},\alpha_{\eta}]_{\pi}$ (2.78) $\displaystyle
d\alpha_{\xi}$ $\displaystyle+\frac{1}{2}\alpha\wedge\alpha\circ\delta(\xi)=0$
(2.79)
###### Proof.
These identities are a direct consequence of the properties of
$\theta\in\Omega(G^{*})$ stated in the previous section. In particular, eq.
(2.79) follows from eq. (2.69) by simply recalling that the pullback and the
differential commute. Eq. (2.78) follows from eq. (2.70), using the
equivariance of the momentum map. ∎
In the following, we give the definition of infinitesimal momentum map that,
as will be seen in the next chapter, plays a fundamental role in the
quantization of the momentum map.
###### Definition 2.4.12.
Let $M$ be a Poisson manifold and $G$ a Poisson Lie group. An infinitesimal
momentum map is a morphism of Gerstenhaber algebras
$\alpha:(\wedge^{\bullet}\mathfrak{g},\delta,[\;,\;])\longrightarrow(\Omega^{\bullet}(M),d_{DR},[\;,\;]_{\pi}).$
(2.80)
The following theorem is crucial in the study of the conditions in which an
infinitesimal momentum map determines a momentum map in the usual sense.
###### Theorem 2.4.13.
Let $(M,\pi)$ be a Poisson manifold and
$\alpha:{\mathfrak{g}}\rightarrow\Omega^{1}(M)$ a linear map. Suppose that the
following relations
$\begin{split}\alpha_{[\xi,\eta]}&=[\alpha_{\xi},\alpha_{\eta}]_{\pi}\\\
d\alpha_{\xi}&=\alpha\wedge\alpha\circ\delta(\xi)\end{split}$ (2.81)
are satisfied. Then:
1. 1.
$\\{\alpha_{\xi}-\theta_{\xi},\ \xi\in{\mathfrak{g}}\\}$ generate an
involutive distribution $\mathcal{D}$ on $M\times G^{*}$.
2. 2.
Suppose, moreover, that $M$ is connected and simply connected. Then the leaves
$\mathcal{F}$ of $\mathcal{D}$ coincide with graphs of maps
$\boldsymbol{\mu}_{\mathcal{F}}:M\rightarrow G^{*}$ satisfying
$\alpha=\boldsymbol{\mu}^{*}_{\mathcal{F}}(\theta)$ and $G^{*}$ acts freely
transitively on the space of leaves by left multiplication on the second
factor.
3. 3.
Vector fields $\pi^{\sharp}(\alpha_{\xi})$ give a homomorphism
$\mathfrak{g}\rightarrow TM.$ (2.82)
Suppose that they integrate to the action of $G$ on $M$ (which is
automatically the case when $M$ is compact and $G$ simply connected). Then the
induced action of $G$ on $M$ is a Poisson action of the Poisson group $G$ and
$\boldsymbol{\mu}_{\mathcal{F}}$ is a momentum map for this action if and only
if the functions
$\phi(\xi,\eta)=\pi(\alpha_{\xi},\alpha_{\eta})-\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})$
(2.83)
satisfy
$\phi(\xi,\eta)|_{\mathcal{F}}=0$ (2.84)
for all $\xi,\eta\in\mathfrak{g}$.
###### Proof.
1. 1.
Using the eqs. (2.69)-(2.79), the $\mathfrak{g}$-valued form $\alpha-\theta$
on $M\times G^{*}$ satisfies
$d(\alpha-\theta)=(\alpha-\theta)\wedge(\alpha-\theta)$ (2.85)
and hence it defines a distribution on $M\times G^{*}$. Let $\mathcal{F}$ be
any of its leaves. Let $p_{i}$, $i=1,2$ denote the projection onto the first
(resp. second) factor in $M\times G^{*}$. Since the linear span of
$\theta_{\xi},\xi\in\mathfrak{g}$ at any point $u\in G^{*}$ coincides with
$T_{u}^{*}{G^{*}}$, the restriction of the projection $p_{1}:M\times
G\rightarrow M$ to $\mathcal{F}$ is an immersion. Moreover, since
$dim(M)=dim(\mathcal{F})$, $p_{1}$ is a covering map.
2. 2.
As we assumed that $M$ is simply connected, $p_{1}$ is a diffeomorphism and
$\boldsymbol{\mu}_{\mathcal{F}}=p_{2}\circ p_{1}^{-1}$ (2.86)
is a smooth map whose graph coincides with $\mathcal{F}$. It is immediate,
that $\alpha=\boldsymbol{\mu}_{\mathcal{F}}^{*}(\theta)$.
The statement about the action of $G^{*}$ on the space of leaves follows from
the fact that $\theta$’s are left invariant.
3. 3.
Suppose that the condition (2.84) is satisfied. Then
$\pi(\alpha_{\xi},\alpha_{\eta})=\boldsymbol{\mu}^{*}_{\mathcal{F}}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))$
(2.87)
and $Ker{\boldsymbol{\mu}_{\mathcal{F}}}_{*}$ coincides with the set of zero’s
of $\alpha_{\xi},\ {\xi}\in\mathfrak{g}$. Hence,
$\boldsymbol{\mu}_{\mathcal{F}}$ is a Poisson map and, in particular
${\boldsymbol{\mu}_{\mathcal{F}}}_{*}(\pi^{\sharp}(\alpha_{\xi}))=\pi_{G^{*}}^{\sharp}(\theta_{\xi}),$
(2.88)
i.e. it is a $G$-equivariant map.
∎
We are interested in understanding when the condition
$d\alpha_{\xi}+\frac{1}{2}\alpha\wedge\alpha\circ\delta(\xi)=0,$ (2.89)
is satisfied. We show that it can be solved explicitly, at least in the case
when $M$ is a Kähler manifold.
###### Definition 2.4.14.
Two solutions $\alpha$ and $\alpha^{\prime}$ of eq. (2.89) are said to be
gauge equivalent, if there exists a smooth function
$H:M\rightarrow\mathfrak{g}^{*}$ such that
$\alpha^{\prime}=\exp(adH)(\alpha)+\int_{0}^{1}dt\exp t(adH)(dH)$ (2.90)
###### Theorem 2.4.15.
Suppose that $M$ is is a Kähler manifold. The set of gauge equivalence classes
of $\alpha\in\Omega^{1}(M,\mathfrak{g}^{*})$ satisfying the equation
$d\alpha_{\xi}+\frac{1}{2}\alpha\wedge\alpha\circ\delta(\xi)=0$ (2.91)
is in bijective correspondence with the set of the cohomology classes $c\in
H^{1}(M,\mathfrak{g}^{*})$ satisfying
$[c,c]=0.$ (2.92)
###### Proof.
Since $M$ is is a Kähler manifold, $(\Omega^{\bullet}(M),d)$ is a formal CDGA
(commutative differential graded algebra) [20]. As a consequence,
$Hom({\mathfrak{g}^{*}},\Omega^{\bullet}(M)),d,[\cdot,\cdot])$ (2.93)
is a formal DGLA (some elements of DGLA’s will be given in the next chapter)
and, in particular, there exists a bijection between the equivalence classes
of Maurer Cartan elements of
$Hom({\mathfrak{g}^{*}},\Omega^{\bullet}(M),d,[\cdot,\cdot])$ and Maurer
Cartan elements of
$Hom({\mathfrak{g}^{*}},H_{DR}^{\bullet}(M),[\cdot,\cdot])$.
A Maurer-Cartan element in
$Hom({\mathfrak{g}^{*}},H_{DR}^{\bullet}(M),[\cdot,\cdot])$ is an element
$c\in H^{1}(M,\mathfrak{g}^{*})$ satisfying
$[c,c]=0,$ (2.94)
and the claim is proved. ∎
#### 2.4.3 Reconstruction problem
In this section we discuss the conditions under which the distribution
$\mathcal{D}$ defined in Theorem 2.4.13 admits a leaf satisfying eq. (2.84).
In particular, we analyze the case where the structure on $G^{*}$ is trivial
and the Heisenberg group case. In the following we keep the assumption that
$M$ is connected and simply connected.
###### Abelian case
Suppose that $G^{*}=\mathfrak{g}^{*}$ is abelian. Then, the forms
$\alpha_{\xi}$ satisfy $d\alpha_{\xi}=0$, hence $\alpha_{\xi}=dH_{\xi}$ (since
$H^{1}(M)=0$), for some $H_{\xi}\in C^{\infty}(M)$.
Let us denote by $ev_{\xi}$ the linear functions $\mathfrak{g}^{*}\ni
l\rightarrow z(\xi)$. Then $\theta_{\xi}=d(ev_{\xi})$ and the leaves of the
distribution $\mathcal{D}$ coincide with the level sets (on
$M\times\mathfrak{g}^{*}$) of the functions
$\\{H_{\xi}-ev_{\xi}\mid\xi\in\mathfrak{g}\\}.$ (2.95)
Furthermore, we have
$\phi(\xi,\eta)(m,z)=\\{H_{\xi},H_{\eta}\\}-z[[\xi,\eta]).$ (2.96)
In this case, the basic identity (2.38) reduces to
$d\\{H_{\xi},H_{\eta}\\}=dH_{[\xi,\eta]},$ (2.97)
hence
$\\{H_{\xi},H_{\eta}\\}-H_{[\xi,\eta]}=c(\xi,\eta),$ (2.98)
for some constants $c(\xi,\eta)$. By the Jacobi identity, the constants
$c(\xi,\eta)$ define a class $[c]\in H^{2}(\mathfrak{g},\mathbb{R})$. Suppose
that this class vanishes (for example if $\mathfrak{g}$ semisimple). Then,
there exists a $z_{0}\in\mathfrak{g}^{*}$ such that
$c(\xi,\eta)=z_{0}([\xi,\eta])$. Hence, given a leaf $\mathcal{F}$,
$\phi(\xi,\eta)|_{\mathcal{F}}=0$ (2.99)
if and only if $\mathcal{F}$ is given by
$H_{\xi}-ev_{\xi}-z_{0}(\xi)=0.$ (2.100)
In other words, the space of leaves of $\mathcal{D}$ which give a momentum map
coincides with the affine space modeled on
$\\{z\in\mathfrak{g}^{*}|z|_{[\mathfrak{g},\mathfrak{g}]}=0\\}$ (which again
vanishes when $\mathfrak{g}$ is semisimple). This proves the following
theorem.
###### Theorem 2.4.16.
Suppose that $G$ is a connected and simply connected Lie group with trivial
Poisson structure and $M$ is compact. Then an infinitesimal momentum map is a
map $\mathfrak{g}\ni\xi\rightarrow C^{\infty}(M)$ such that
$d\\{H_{\xi},H_{\eta}\\}=dH_{[\xi,\eta]},\ \forall\xi,\eta\in\mathfrak{g}.$
(2.101)
$c(\xi,\eta)=\\{H_{\xi},H_{\eta}\\}-H_{[\xi,\eta]}$ is a two cocycle $c$ on
$\mathfrak{g}$ with values in $\mathbb{R}$. The infinitesimal momentum map
$\alpha$ is generated by a momentum map $\boldsymbol{\mu}$ if this cocycle
vanishes and, in this case, $\boldsymbol{\mu}$ is unique.
###### Heisenberg group case
Suppose now that $G^{*}$ is the Heisenberg group. Let $x,y,z$ be a basis for
$\mathfrak{g}^{*}$, where $z$ is central and $[x,y]=z$. Let $\xi,\eta,\zeta$
be the dual basis of $\mathfrak{g}$. Recall that the cocycle $\delta$ on
$\mathfrak{g}$ dual to the Lie algebra structure on $\mathfrak{g}^{*}$ is
given by
$[l_{1},l_{2}](\xi)=(l_{1}\wedge l_{2})\delta(\xi).$ (2.102)
Hence, we have
$\delta(\xi)=\delta(\eta)=0\mbox{ and }\delta(\zeta)=\xi\wedge\eta.$ (2.103)
and
$\begin{split}d\alpha_{\xi}&=d\alpha_{\eta}=0\\\
d\alpha_{\zeta}&=\alpha_{\xi}\wedge\alpha_{\eta}.\end{split}$ (2.104)
There are essentially two possibilities for the Lie bialgebra structure on
$\mathfrak{g}^{*}$, which give the following two possibilities for the Lie
algebra structure on $\mathfrak{g}$. Either
$[\xi,\eta]=0,[\xi,\zeta]=\xi,[\eta,\zeta]=\eta$ (2.105)
or
$[\xi,\eta]=0,[\xi,\zeta]=\eta,[\eta,\zeta]=-\xi.$ (2.106)
The result below will turn out to be independent of the choice - the
computations will be done using the second choice, which corresponds to
$G=\mathbb{R}\ltimes\mathbb{R}^{2}$, with $\mathbb{R}$ acting by rotation on
$\mathbb{R}^{2}$. Below we use the notation
$\delta(\xi)=\sum_{i}\xi^{1}_{i}\wedge\xi^{2}_{i}.$ (2.107)
Applying the Cartan formula $\mathcal{L}=[\iota,d]$ and the identity
$[\alpha_{\xi},\alpha_{\eta}]_{\pi}=\alpha_{[\xi,\eta]}$ to the basic equation
(2.38), we get
$\sum_{i}\pi(\alpha_{\eta},\alpha_{\xi_{i}^{1}})\alpha_{\xi^{2}_{i}}-\sum_{i}\pi(\alpha_{\xi},\alpha_{\eta^{1}_{i}})\alpha_{\eta^{2}_{i}}=\alpha_{[\eta,\xi]}-d\pi(\alpha_{\eta},\alpha_{\xi}).$
(2.108)
In our case it gives the following equations
$\begin{split}d\pi(\alpha_{\xi},\alpha_{\eta})&=\alpha_{[\xi,\eta]}\\\
d\pi(\alpha_{\zeta},\alpha_{\eta})&=\alpha_{[\zeta,\eta]}+\pi(\alpha_{\eta},\alpha_{\xi})\alpha_{\eta}\\\
d\pi(\alpha_{\zeta},\alpha_{\xi})&=\alpha_{[\zeta,\xi]}-\pi(\alpha_{\xi},\alpha_{\eta})\alpha_{\xi}.\end{split}$
(2.109)
which are also satisfied after replacing $\alpha$ with $\theta$. Let
$\mathcal{I}$ denote the ideal generating our distribution $\mathcal{D}$.
Then, from above,
$\displaystyle d\phi(\xi,\eta)\in\mathcal{I}$ (2.110)
$\displaystyle\phi(\xi,\eta)|_{\mathcal{F}}=0\Longrightarrow
d\phi(\zeta,\eta)|_{\mathcal{F}},d\phi(\zeta,\xi)|_{\mathcal{F}}\in\mathcal{I}.$
(2.111)
Here, as before, $\mathcal{F}$ is a leaf of $\mathcal{D}$. Using the relation
(2.71), we get
$\begin{split}&\mathcal{L}_{z}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))=\mathcal{L}_{x}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))=\mathcal{L}_{y}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))=0\\\
&\mathcal{L}_{z}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\zeta}))=\mathcal{L}_{y}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\zeta}))=0\qquad\mathcal{L}_{x}^{*}(\pi_{G^{*}}(\theta_{\xi},\theta_{\zeta}))=1\\\
&\mathcal{L}_{z}^{*}(\pi_{G^{*}}(\theta_{\eta},\theta_{\zeta}))=\mathcal{L}_{x}^{*}(\pi_{G^{*}}(\theta_{\eta},\theta_{\zeta}))=0\qquad\mathcal{L}_{y}^{*}(\pi_{G^{*}}(\theta_{\eta},\theta_{\zeta}))=1\end{split}$
(2.112)
In particular, $\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})$ is invariant under
left translations. Since $\pi_{G^{*}}$ is zero at the identity, we get
$\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})=0$ (2.113)
By the first equations (2.110), $\phi(\xi,\eta)$ is leafwise constant, hence
so is $\pi(\alpha_{\xi},\alpha_{\eta})$. Hence we have
###### Lemma 2.4.17.
$\pi(\alpha_{\xi},\alpha_{\eta})=c$ is constant on $M$ and necessary condition
for existence of the momentum map is $c=0$.
Let us continue under the assumption that $c=0$. Then, given a leaf
$\mathcal{F}$, by eq. (2.110),
$\phi(\eta,\zeta)|_{\mathcal{F}}=c_{1}\quad\text{and}\quad\phi(\xi,\zeta)|_{\mathcal{F}}=c_{2}$
(2.114)
for some constants $c_{1}$ and $c_{2}$. Setting
$\mathcal{F}_{1}=id\times\exp(c_{1}x)\exp(c_{2}y)$ to $\mathcal{F}$, we get
$\phi(\eta,\zeta)|_{\mathcal{F}_{1}}=\phi(\xi,\zeta)|_{\mathcal{F}_{1}}=\phi(\xi,\eta)|_{\mathcal{F}_{1}}=0.$
(2.115)
###### Theorem 2.4.18.
Let $G$ be a Poisson Lie group acting on a Poisson manifold $M$ with an
infinitesimal momentum map $\alpha$ and such that $G^{*}$ is the Heisenberg
group. Let $\xi,\eta,\zeta$ denote the basis of $\mathfrak{g}$ dual to the
standard basis $x,y,z$ of $\mathfrak{g}^{*}$ (with $z$ central and $[x,y]=z$.
Then
$\pi(\alpha_{\xi},\alpha_{\eta})=c$ (2.116)
is constant on $M$. The form $\alpha$ lifts to a momentum map
$\boldsymbol{\mu}:M\rightarrow G^{*}$ if and only if $c=0$. When $c=0$ the set
of momentum maps with given $\alpha$ is one dimensional with free transitive
action of $\mathbb{R}$.
#### 2.4.4 Infinitesimal deformations of a momentum map
In the following we study the behavior of deformations of a momentum map,
close to the identity. Indeed, we consider a deformation of $\boldsymbol{\mu}$
given by the map $X:M\rightarrow\mathfrak{g}^{*}$ and we discuss the property
of the infinitesimal generator of the action induced by this deformed momentum
map.
###### Theorem 2.4.19.
Let $(M,\pi)$ be a Poisson manifold with a Poisson action of a Poisson Lie
group $(G,\pi_{G})$. Suppose that $[-\epsilon,\epsilon]\ni
t\rightarrow\boldsymbol{\mu}_{t}:M\rightarrow G^{*}$ is a differentiable path
of momentum maps for this action. Let $\exp:{\mathfrak{g}}^{*}\rightarrow
G^{*}$ denote the exponential map. We can assume that $\boldsymbol{\mu}_{t}$
is of the form $m\rightarrow\boldsymbol{\mu}(m)\exp(tX_{m})+o(\epsilon)$ for
some differentiable map $X:M\rightarrow{\mathfrak{g}}^{*}:m\mapsto X_{m}$.
Then, for all $\xi,\eta\in\mathfrak{g}$,
$\displaystyle\mathcal{L}_{\xi}X(\eta)-\mathcal{L}_{\eta}X(\xi)$
$\displaystyle=X([\xi,\eta])$ (2.117) $\displaystyle\\{X(\xi),\ \cdot\\}$
$\displaystyle=-\mathcal{L}_{ad^{*}_{X}\xi}.$ (2.118)
###### Proof.
Assuming that the deformed momentum maps can be written as
$\boldsymbol{\mu}_{t}(m)=\boldsymbol{\mu}(m)\exp(tX_{m})$ then we have
$\alpha^{t}_{\xi}=\boldsymbol{\mu}_{t}^{*}(\theta_{\xi})=\langle
d\boldsymbol{\mu}_{t},\theta_{\xi}\rangle$. We want to figure out its behavior
close to the identity so we calculate $\left.\frac{d}{dt}\right|_{t=0}\langle
d\boldsymbol{\mu}_{t},\theta_{\xi}\rangle$. First notice that
$d\boldsymbol{\mu}_{t}=(\rho_{\exp(tX)})_{*}d\boldsymbol{\mu}+(\lambda_{\boldsymbol{\mu}})_{*}d\exp(tX),$
(2.119)
so we get:
$\begin{split}\left.\frac{d}{dt}\right|_{t=0}\langle(\rho_{\exp(tX)})_{*}d\boldsymbol{\mu},\theta_{\xi}\rangle&=\left.\frac{d}{dt}\right|_{t=0}\langle
d\boldsymbol{\mu},(\rho_{\exp(tX)})^{*}\theta_{\xi}\rangle\\\ &=\langle
d\boldsymbol{\mu},\mathcal{L}_{X}\theta_{\xi}\rangle\\\ &=\langle
d\boldsymbol{\mu},\theta_{ad^{*}_{X}\xi}\rangle=\alpha_{ad^{*}_{X}\xi}\end{split}$
(2.120)
and
$\begin{split}\left.\frac{d}{dt}\right|_{t=0}\langle(\lambda_{\boldsymbol{\mu}})_{*}d\exp(tX),\theta_{\xi}\rangle&=\left.\frac{d}{dt}\right|_{t=0}\langle
d\exp(tX),(\lambda_{\boldsymbol{\mu}})^{*}\theta_{\xi}\rangle\\\
&=\left.\frac{d}{dt}\right|_{t=0}\langle
d\exp(tX),\theta_{\xi}\rangle\end{split}$ (2.121)
The differential of the exponential map $\exp:{\mathfrak{g}}^{*}\rightarrow
G^{*}$ is a map from the cotangent bundle of ${\mathfrak{g}}^{*}$ to the
cotangent bundle of $G^{*}$. It can be trivialized as
$d\exp:{\mathfrak{g}}^{*}\times{\mathfrak{g}}^{*}\rightarrow
G^{*}\times{\mathfrak{g}}^{*}$. Furthermore,
$(\exp^{-1},id):G^{*}\times{\mathfrak{g}}^{*}\rightarrow{\mathfrak{g}}^{*}\times{\mathfrak{g}}^{*}$
hence the map
${\mathfrak{g}}^{*}\times{\mathfrak{g}}^{*}\rightarrow{\mathfrak{g}}^{*}\times{\mathfrak{g}}^{*}$
is given by $tX+o(t^{2})$. We get
$\left.\frac{d}{dt}\right|_{t=0}\langle
d\exp(tX),\theta_{\xi}\rangle=\left.\frac{d}{dt}\right|_{t=0}\langle
d(tX+o(t)),\theta_{\xi}\rangle=d\langle X,\theta_{\xi}\rangle=d\langle
X,\xi\rangle$ (2.122)
and finally
$\beta_{\xi}=\left.\frac{d}{dt}\right|_{t=0}\alpha^{t}_{\xi}=\alpha_{ad^{*}_{X}\xi}+dX(\xi).$
(2.123)
Since $\pi^{\sharp}(\alpha^{t}_{\xi})=\mathcal{L}_{\xi}$ is independent of
$t$, we get the identity (2.118).
In order to prove the relation (2.117), recall that, since
$\boldsymbol{\mu}_{t}$ is a family of Poisson maps, one has
$\pi(\alpha^{t}_{\xi},\alpha^{t}_{\eta})(m)=\pi_{G^{*}}(\theta_{\xi},\theta_{\eta})(\boldsymbol{\mu}_{t}(m)).$
(2.124)
Applying $\left.\frac{d}{dt}\right|_{t=0}$ to both sides, we get
$\pi(\beta_{\xi},\alpha_{\eta})(m)+\pi(\alpha_{\xi},\beta_{\eta})(m)=\mathcal{L}_{X}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))(\boldsymbol{\mu}(m)).$
(2.125)
Substituting the expression of $\beta$’s (2.123) and using the following
identity
$\mathcal{L}_{X}(\pi_{G^{*}}(\theta_{\xi},\theta_{\eta}))(\boldsymbol{\mu}(m))=X[\xi,\eta]+\pi_{G^{*}}(\theta_{ad^{*}_{X}\xi},\theta_{\eta})+\pi_{G^{*}}(\theta_{\xi},\theta_{ad^{*}_{X}\eta})$
(2.126)
the claimed equality follows. ∎
We consider the case of a compact and semisimple Poisson Lie group $G$ to
obtain a uniqueness condition for the momentum map. From the relation (2.117)
we can conclude that there exists a function $\Phi$ such that
$\mathcal{L}_{\xi}\Phi=X(\xi).$ (2.127)
Using this expression we get
$\mathcal{L}_{ad^{*}_{X}\xi}f=\mathcal{L}_{\xi^{\prime}}\Phi\xi^{\prime\prime}(f)$,
where $\delta(\xi)=\xi^{\prime}\otimes\xi^{\prime\prime}$. Now observe that
$\xi\\{\Phi,f\\}=\mathcal{L}_{\xi}\pi(d\Phi,df)=(\mathcal{L}_{\xi}\pi)(d\Phi,df)+\\{\mathcal{L}_{\xi}\Phi,f\\}+\\{\Phi,\mathcal{L}_{\xi}f\\}$
(2.128)
hence
$\begin{split}\\{X(\xi),f\\}&=\\{\mathcal{L}_{\xi}\Phi,f\\}=\xi\\{\Phi,f\\}-(\mathcal{L}_{\xi}\pi)(d\Phi,df)-\\{\Phi,\mathcal{L}_{\xi}f\\}\\\
&=\xi\\{\Phi,f\\}-\delta(\xi)(\Phi,f)-\\{\Phi,\mathcal{L}_{\xi}f\\}\\\
&=\xi\\{\Phi,f\\}-\mathcal{L}_{\xi^{\prime}}\Phi\;\xi^{\prime\prime}(f)-\\{\Phi,\mathcal{L}_{\xi}f\\}.\end{split}$
(2.129)
Substituting these results in (2.118) we get
$\xi\\{\Phi,f\\}-\\{\Phi,\mathcal{L}_{\xi}f\\}=0.$ (2.130)
This means that there exists a vector field $H_{\Phi}$ associated with the
deformation of the momentum map which commutes with the action:
$[H_{\Phi},\mathcal{L}_{\xi}]=0.$ (2.131)
In other words, given the momentum map $\boldsymbol{\mu}:M\rightarrow G^{*}$,
if there exists an endomorphism on $M$ such that the associated vector field
commutes with the action, then we get another momentum map, as discussed in
the Theorem (2.4.19).
### 2.5 Poisson Reduction
Here we present the main result of this chapter. We show that, given a Poisson
action we can define a reduced manifold in terms of momentum map. A first
generalization of the Marsden-Weinstein reduction has been given by Lu in
[34], where it is shown that, given a Poisson Lie group acting on a symplectic
manifold $M$, the symplectic structure on $M$ induces a symplectic structure
on the leaves of $M/G$ generated by the momentum map.
Given a Poisson action $\Phi:G\times M\rightarrow M$ with momentum map
$\boldsymbol{\mu}:M\rightarrow G^{*}$, we define a $G$-invariant foliation
$\mathcal{F}$ of $M$. The leaves are not Poisson manifolds, but considering
the action of $G$ on the space of leaves, we prove that, for each leaf
$\mathcal{L}$, the Poisson structure on $M$ induces a Poisson structure on the
orbit space $\mathcal{L}/G_{\mathcal{L}}$, where $G_{\mathcal{L}}$ is the
isotropic group at any point of $\mathcal{L}$. This shows that we can reduce
$M$ to another Poisson manifold $\mathcal{L}/G_{\mathcal{L}}$ that we call the
Poisson reduced space.
#### 2.5.1 Poisson structure on $M/G$
In this section we prove that, given a Poisson Lie group $(G,\pi_{G})$ acting
on a Poisson manifold $(M,\pi)$, with equivariant momentum map
$\boldsymbol{\mu}:M\rightarrow G^{*}$, the orbit space inherits a Poisson
structure from $M$. From now on we further assume that the Poisson Lie group
$G$ is complete.
In [50] Semenov-Tian-Shansky showed that, given a Poisson action, if the orbit
space is a smooth manifold, it carries a Poisson structure such that the
natural projection $\text{pr}:M\rightarrow M/G$ is a Poisson mapping. More
precisely, given $f,h\in C^{\infty}(M)$ with the definitions
$\hat{f}(m,g):=f(g\cdot m),\quad\hat{h}(m,g):=h(g\cdot m),$ (2.132)
for any $f,h$ one finds
$\\{f,h\\}_{M}(g\cdot
m)=\\{\hat{f}(m,\cdot),\hat{h}(m,\cdot)\\}_{G}(g)+\\{\hat{f}(\cdot,g),\hat{h}(\cdot,g)\\}_{M}(m)$
(2.133)
Then $M/G$ inherits a Poisson structure from the Poisson structure on $M$:
$f,h\in C^{\infty}(M)^{G}\Longrightarrow\\{f,h\\}_{M}\in C^{\infty}(M)^{G}.$
(2.134)
Consider a Poisson action $\Phi:G\times M\rightarrow M$ with equivariant
momentum map $\boldsymbol{\mu}:M\rightarrow G^{*}$ and assume that the orbit
space $M/G$ is a smooth manifold. Recall that when the Poisson structure of
$G$ is trivial, the infinitesimal Poisson action $\xi_{M}$ is a Hamiltonian
vector field, but in general this does not hold. The first goal of this
section is to provide an explicit formulation for $\xi_{M}$, in terms of local
coordinates. We use the properties of the momentum map and dressing action to
obtain such a formulation.
We observe that the Poisson Lie group $G^{*}$ can be described locally in
terms of coordinates $(q,p,y)$ such that $\pi_{G^{*}}$ is given by the
Splitting Theorem 2.2.7. In particular, the transverse structure is determined
by the structure functions $\pi_{G^{*}}^{ij}(y)=\\{y_{i},y_{j}\\}$, which
vanishes on the symplectic leaves. As discussed in Section 2.4.1 the Poisson
Lie group $G$ acts on $G^{*}$ by dressing action and the dressing orbits are
the same as the symplectic leaves. Hence, the generic orbit $\mathcal{O}_{x}$
through $x\in G^{*}$ is a closed submanifold of $G^{*}$ and $y_{i}$ are
transversal coordinates such that $\\{y_{i},y_{j}\\}=0$. Note that
$\boldsymbol{\mu}$ is a submersion, hence it has open image. In particular,
the image of $M$ is an open neighborhood of a generic orbit of $G$ on $G^{*}$.
Define the functions $H_{i}$ as the pullbacks by $\boldsymbol{\mu}$ of the
transversal coordinates $y_{i}$ to the orbit on $G^{*}$:
$H_{i}:=y_{i}\circ\boldsymbol{\mu}.$ (2.135)
$H_{i}$ are defined locally in a $G$-invariant open neighborhood $U$ of the
preimage $N=\boldsymbol{\mu}^{-1}(\mathcal{O}_{x})$. We can assume that $x$ is
a regular value of $\boldsymbol{\mu}$, hence $N$ is a closed $G$-invariant
submanifold of $M$. Since $\\{y_{i},y_{j}\\}$ vanishes on the orbit
$\mathcal{O}_{x}$, $\\{H_{i},H_{j}\\}$ vanishes on the preimage $N$. The
1-forms $\alpha_{\xi}=\boldsymbol{\mu}^{*}(\theta_{\xi})$ are in the span of
the $dH_{i}$’s. Since the left invariant 1-form $\theta_{\xi}$ in local
coordinates can be expressed as a linear combination of $dy_{i}$, using the
definition (2.135) we have
$\alpha_{\xi}=\boldsymbol{\mu}^{*}(\theta_{\xi})=\sum_{i}c_{i}(\xi)dH_{i}$
(2.136)
for any $\xi\in\mathfrak{g}$. As a consequence, the infinitesimal generators
$\xi_{M}$ of the Hamiltonian action $\Phi$, induced by $\boldsymbol{\mu}$, can
be written as a linear combination of Hamiltonian vector fields:
$\xi_{M}=\pi^{\sharp}(\alpha_{\xi})=\sum_{i}c_{i}(\xi)\\{H_{j},\cdot\\}.$
(2.137)
We use this explicit formulation to prove that $M/G$ inherits a Poisson
structure from $M$:
###### Theorem 2.5.1.
Let $\Phi:G\times M\rightarrow M$ be a Poisson action with equivariant
momentum map $\boldsymbol{\mu}$. The algebra $C^{\infty}(M)^{G}$ of
$G$-invariant functions on $M$ is a Lie subalgebra of $C^{\infty}(M)$.
###### Proof.
Let $f,g\in C^{\infty}(M)^{G}$, then $\xi_{M}[f]=\xi_{M}[g]=0$ for any
$\xi\in\mathfrak{g}$. Applying the relation (2.137) we have that
$\sum_{i}c_{i}(\xi)\\{H_{i},f\\}=\sum_{i}c_{i}(\xi)\\{H_{i},g\\}=0$ (2.138)
that implies $\\{H_{i},f\\}=\\{H_{i},g\\}=0$ for any $i$. Then, using the
Jacobi identity we get $\\{H_{i},\\{f,g\\}\\}=0$. Since $G$ is connected we
proved that
$\xi_{M}\left[\\{f,g\\}\right]=0.$ (2.139)
Hence $\\{f,g\\}$ is $G$-invariant and the claim is proved. ∎
#### 2.5.2 Poisson structure on $\mathcal{L}/G_{\mathcal{L}}$
Consider the $\mathfrak{g}^{*}$-valued 1-forms $\alpha_{\xi}$ defined by
$\boldsymbol{\mu}$ by eq. (2.136). The distribution
$\\{\alpha_{\xi}|\xi\in\mathfrak{g}\\}$ defines a $G$-invariant foliation
$\mathcal{F}$ on $M_{reg}$, the open submanifold of regular values of
$\boldsymbol{\mu}$ of $M$ by
$T_{m}\mathcal{L}=\ker{\alpha_{\xi}}(m)=\bigcap_{i}\ker dH_{i}(m)$ (2.140)
for any leaf $\mathcal{L}$, which is of the form
$\mathcal{L}=\boldsymbol{\mu}^{-1}(x)$. The leaf $\mathcal{L}$ is not a
Poisson submanifold but we prove that, considering the action of $G$ on the
space of leaves, the quotient $\mathcal{L}/G_{\mathcal{L}}$ inherits a Poisson
structure by $M$, where
$G_{\mathcal{L}}=\\{g\in G|g\cdot\mathcal{L}=\mathcal{L}\\}$ (2.141)
is the stabilizer of the action of $G$ on $\mathcal{L}$.
In order to prove this statement we observe that, since $\pi_{G^{*}}$
restricted to $\mathcal{O}_{x}$ does not depend on the transversal coordinates
$y_{i}$’s, the Poisson structure $\pi$ on $M$ depends on the coordinates
$H_{i}$ defined in (2.135) only in the combination
$\partial_{x_{i}}\wedge\partial_{H_{i}}$. This is evident because the
differential $d\boldsymbol{\mu}$ between $TM|_{N}/TN$ and
$TG^{*}/T\mathcal{O}_{x}$ is a bijective map, so using the definition (2.135)
the claim is proved.
Now consider the ideal $\mathcal{I}$ generated by $H_{i}$. The coordinates
$H_{i}$ are locally defined but we can show that $\mathcal{I}$ is globally
defined. Considering a different neighborhood on the orbit of $G^{*}$ we have
transversal coordinates $y_{i}^{\prime}$ and their pullback to $M$ will be
$H_{i}^{\prime}=y_{i}^{\prime}\circ\boldsymbol{\mu}$. The coordinates
$H_{i}^{\prime}$ are defined in a different open neighborhood $V$ of $N$, but
we can see that the ideal $\mathcal{I}$ generated by $H_{i}$ coincides with
$\mathcal{I}^{\prime}$ generated by $H_{i}^{\prime}$ on the intersection of
$U$ and $V$, then it is globally defined. Moreover, since $\boldsymbol{\mu}$
is a Poisson map we have:
$\\{H_{i},H_{j}\\}=\\{y_{i}\circ\boldsymbol{\mu},y_{j}\circ\boldsymbol{\mu}\\}=\\{y_{i},y_{j}\\}\circ\boldsymbol{\mu}.$
(2.142)
Hence the ideal $\mathcal{I}$ is closed under Poisson brackets.
###### Lemma 2.5.2.
Suppose that $N/G$ is an embedded submanifold of the smooth manifold $M/G$,
then
$(C^{\infty}(M)/\mathcal{I})^{G}=(C^{\infty}(M)^{G}+\mathcal{I})/\mathcal{I}$
(2.143)
###### Proof.
Let $f$ be a smooth function on $M$ satisfying
$[f]\in(C^{\infty}(M)/\mathcal{I})^{G}$. If the equivalence class $[f]$ is
$G$-invariant, we have
$f(G\cdot m)=f(m)+i(m),$ (2.144)
where $i\in\mathcal{I}=\\{f\in C^{\infty}(M):f|_{N}=0\\}$. It is clear that
$f|_{N}$ is $G$-invariant and hence it defines a smooth function $\bar{f}\in
C^{\infty}(N/G)$. Since $N/G$ is a $k$-dimensional embedded submanifold of the
$n$-dimensional smooth manifold $M/G$, the inclusion map $\iota:N/G\rightarrow
M/G$ has local coordinates representation:
$(x_{1},\dots,x_{k})\mapsto(x_{1},\dots,x_{k},c_{k+1},\dots,c_{n})$ (2.145)
where $c_{i}$ are constants. Hence we can extend $\bar{f}$ to a smooth
function $\phi$ on $M/G$ by setting
$\bar{f}(x_{1},\dots,x_{k})=\phi(x_{1},\dots,x_{k},0,\dots,0)$. The pullback
$\tilde{f}$ of $\phi$ by $\text{pr}:M\rightarrow M/G$ is $G$-invariant and
satisfies
$\tilde{f}-f|_{N}=0,$ (2.146)
hence $\tilde{f}-f\in\mathcal{I}$. ∎
###### Theorem 2.5.3.
Let $\Phi:G\times M\rightarrow M$ be a Poisson action of $(G,\pi_{G})$ on a
Poisson manifold $(M,\pi)$ with equivariant momentum map
$\boldsymbol{\mu}:M\rightarrow G^{*}$. For each leaf, the orbit space
$\mathcal{L}/G_{\mathcal{L}}$ has a Poisson structure induced by $\pi$.
###### Proof.
First we prove that the Poisson bracket of $M$ induces a well defined Poisson
bracket on $(C^{\infty}(U)^{G}+\mathcal{I})/\mathcal{I}$. In fact, for any
$f+i\in C^{\infty}(U)^{G}/\mathcal{I}$ and $j\in\mathcal{I}$ the Poisson
bracket $\\{f+i,j\\}$ still belongs to the ideal $\mathcal{I}$. Since the
ideal $\mathcal{I}$ is closed under Poisson brackets, $\\{i,j\\}$ belongs to
$\mathcal{I}$. The function $j$, by definition on the ideal $\mathcal{I}$, can
be written as a linear combination of $H_{i}$, so
$\\{f,j\\}=\sum_{i}a_{i}\\{f,H_{i}\\}$. By the Theorem 2.5.1, we have
$\\{f,H_{i}\\}=0$, hence $\\{f+i,j\\}\in\mathcal{I}$ as stated. Finally, using
the isomorphism proved in the Lemma (2.5.2) and the identifications
$C^{\infty}(\mathcal{L}/G_{\mathcal{L}})\simeq
C^{\infty}(N/G)\simeq(C^{\infty}(U)/\mathcal{I})^{G}.$ (2.147)
the claim is proved. ∎
We refer to $\mathcal{L}/G_{\mathcal{L}}$ as the Poisson reduced space.
### 2.6 An example: $\mathbb{R}^{2}$ action
Here we want to discuss a concrete example for the Poisson reduction. Consider
the Lie bialgebra $\mathfrak{g}=\mathbb{R}^{2}$ with generators $\xi$ and
$\eta$ such that
$[\xi,\eta]=\eta$ (2.148)
and cobracket given by
$\delta(\xi)=0\quad\delta(\eta)=\xi\wedge\eta.$ (2.149)
The matrix representation of $\mathfrak{g}$ is the Lie algebra
$\mathfrak{gl}(2,\mathbb{R})$ and the subgroups $G$ and $G^{*}$ of
$GL(2,\mathbb{R})$ of matrices with positive determinant are given by
$G=\left\\{\left(\begin{matrix}1&0\\\
x&y\end{matrix}\right)\;:y>0\right\\}\qquad
G^{*}=\left\\{\left(\begin{matrix}a&b\\\
0&1\end{matrix}\right)\;:a>0\right\\}$ (2.150)
and we remark that the Poisson bivector on $G^{*}$ is
$\pi_{G^{*}}=ab\partial_{a}\wedge\partial_{b}.$ (2.151)
In this simple case it is clear that $\\{a,b\\}$ are global coordinates on
$G^{*}$. We analyze the orbits of the dressing action of $G$ on $G^{*}$ for
this example.
Remember that the dressing orbits $\mathcal{O}_{x}$ through a point $x\in
G^{*}$ are the same as the symplectic leaves, hence it is clear that they are
generated by the equation $b=0$. The symplectic foliation of the manifold
$G^{*}$ is now given by two open orbit, determined by the conditions $b>0$ and
$b<0$ respectively, and a closed orbit given by $b=0$ and $a\in\mathbb{R}$.
Consider a Poisson action of $G$ on a generic Poisson manifold $M$ induced by
the equivariant momentum map $\boldsymbol{\mu}:M\rightarrow G^{*}$. Its
pullback
$\boldsymbol{\mu}^{*}:C^{\infty}(G^{*})\longrightarrow C^{\infty}(M)$ (2.152)
maps the coordinates $a$ and $b$ on $G^{*}$ to
$\hat{a}(x)=a(\boldsymbol{\mu}(x))$ and $\hat{b}(x)=b(\boldsymbol{\mu}(x))$
resp. In order to simplify the notation we denote the coordinates on $M$ only
with $a$ and $b$. It is important to underline that we have no information on
the dimension of $M$, so $a$ and $b$ are just a couple of the possible
coordinates. Nevertheless, since $\boldsymbol{\mu}$ is a Poisson map, we have
$\\{a,b\\}=ab$ (2.153)
on $M$. The infinitesimal action of $\mathfrak{g}=\mathbb{R}^{2}$ on $M$ that
we consider can be written in terms of these coordinates $a,b$ as
$\Phi(\xi)=a\\{b,\cdot\\}\quad\Phi(\eta)=a\\{a^{-1},\cdot\\}.$ (2.154)
In the previous section we proved that the Poisson reduction is given
equivalently either as the Poisson algebra $C^{\infty}(N/G)$ on the quotient
$N/G$, with $N=\boldsymbol{\mu}^{-1}(\mathcal{O}_{x})$ or as
$(C^{\infty}(M)/\mathcal{I})^{G}$. In the following, we discuss 3 different
cases of dressing orbit.
###### Case 1: $b>0$.
Consider the dressing orbit $\mathcal{O}_{x}$ generated by the condition
$b>0$. Since $a$ and $b$ are both positive, we can put
$a=e^{p},\quad b=e^{q}$ (2.155)
and we have
$\\{p,q\\}=1$ (2.156)
since $\\{a,b\\}=ab$. For this reason we can claim that the preimage of the
dressing orbit can be split as $N=\mathbb{R}^{2}\times M_{1}$ and
$C^{\infty}(N)$ is given explicitly by the set of functions generated by
$b^{-1}$. The infinitesimal action is given by
$\Phi(\xi)=e^{p}\\{e^{q},\cdot\\}\qquad\Phi(\eta)=e^{p}\\{e^{-p},\cdot\\}$
(2.157)
which is just the action of $G$ on the plane. Hence the Poisson reduction in
this case is given by
$(C^{\infty}(M)[b^{-1}])^{G}.$ (2.158)
###### Case 2: $b<0$.
This case is similar, with the only difference that $b=-e^{q}$.
###### Case 3: $b=0$.
This case is slightly different. The orbit $\mathcal{O}_{x}$ is given by fixed
points on the line $b=0$, then we choose the point $a=1$. Clearly, in this
case we can not define $b=e^{p}$.
Consider the ideal $\mathcal{I}=\langle a-1,b\rangle$ of functions vanishing
on $N$. It is easy to check that it is $G$-invariant, hence the Poisson
reduction in this case is:
$(C^{\infty}(M)/\mathcal{I})^{G}.$ (2.159)
## Chapter 3 Quantum Momentum Map
The main goal of this chapter is the definition of the quantum momentum map as
a deformation of the classical momentum map, introduced in the previous
chapter. In the first part we introduce the reader to the theory of
deformation quantization of Poisson manifolds developed by M. Kontsevich [27].
Then we present some basic results about quantum groups and their connection
with Poisson Lie groups and Lie bialgebras [7]. Finally, we discuss the
quantization of the momentum map and analyze some examples of quantum
reduction.
### 3.1 Deformation quantization of Poisson manifolds
We start this section with a survey of deformation quantization sketching the
physical motivations which underlie such theory. In general, deformation
quantization establishes a correspondence between classical and quantum
mechanics.
In the Hamiltonian formalism of classical mechanics, physical observables are
represented by smooth functions on a certain space, called phase space. This
generally is a symplectic or Poisson manifold $M$. On the other hand, a
quantum system is usually described by a Hilbert space and the observables are
self-adjoint operators on it. However, a formal correspondence between the two
theories is still missing, despite the fact that many progresses in that
direction have been done.
The problem of finding a precise mathematical procedure to associate to a
classical observable (smooth function on $M$) a quantum analog, was first
approached by trying to construct a correspondence between the commutative
algebra $C^{\infty}(M)$ and the non-commutative algebra of operators. Starting
from the quantization of $\mathbb{R}^{2n}$, the first result was achieved by
Groenewold [21], which states that the Poisson algebra
$C^{\infty}(\mathbb{R}^{2n})$ can not be quantized in such a way that the
Poisson bracket of two classical observables is mapped into the Lie bracket of
the correspondent operators.
The idea of Bayen, Flato et al. [2],[16], [17] was a change of perspective:
instead of mapping functions to operators, the algebra of functions can be
deformed into a non-commutative one. In particular, they proved that on the
symplectic vector space $\mathbb{R}^{2n}$, there exists a standard deformation
quantization, or star product, known as the Moyal-Weyl product. The origins of
the Moyal-Weyl product can be found in the works of Weyl [57] and Wigner [58],
where they give an explicit correspondence between functions and operators,
and of Groenewold [21] and Moyal [43], where the product and the bracket of
operators defined by Weyl have been introduced. The existence of an
associative star product has been generalized to a symplectic manifold
admitting a flat connection $\nabla$ in [2]. The first proof of the existence
of star product for any symplectic manifold was given by De Wilde and Lecomte
[9] and few years later by Fedosov [12]. In subsequent works (e.g. [44], [23])
the equivalence classes of star products on symplectic manifolds and the
connection with de Rham cohomology has been studied. It came out that the
equivalence classes of star products and elements in
$H_{dR}^{2}(M)\llbracket\epsilon\rrbracket$ are in a one-to-one
correspondence.
The existence and classification of star product culminated with Kontsevich’s
Formality Theorem, that was first formalized in a conjecture in [26] and then
proved in [27]. Kontsevich showed that any finite dimensional Poisson manifold
$M$ admits a canonical deformation quantization and established a
correspondence between the set of isomorphism classes of deformations of
$C^{\infty}(M)$ and the set of equivalence classes of formal Poisson
structures on $M$.
#### 3.1.1 Classification of Star Products
We start discussing the problem of the existence of a formal deformation for
an arbitrary Poisson manifold. First, we recall the basic notion of formal
deformation of an algebra $A$ and then we explain the connection of
deformations with Poisson structures.
Kontsevich in [27] solved the problem of classifying star products on a given
Poisson manifold $M$ by proving that there is a one-to-one correspondence
between equivalence classes of star products and equivalence classes of
Poisson structures.
Let $k$ be a commutative ring and $A$ a $k$-algebra, associative and unital.
Denote by $k\llbracket\hbar\rrbracket$ the ring of formal power series in
$\hbar$ and by $A\llbracket\hbar\rrbracket$ the
$k\llbracket\hbar\rrbracket$-module of formal power series
$\sum_{n=0}^{\infty}\hbar^{n}a_{n}$ (3.1)
with coefficients in $A$. A formal deformation of the algebra $A$ is a formal
power series
$a\star b=ab+\sum_{k=1}^{\infty}\hbar^{k}P_{k}(a,b)$ (3.2)
where $P_{m}:A\times A\rightarrow A$ are $k$-bilinear maps such that
1. 1.
The product $\star$ is associative
2. 2.
$P_{k}(1,f)=P_{k}(f,1)=0$ for any $f\in A$
An isomorphism of two deformations $\star$, $\star^{\prime}$ is a formal power
series $T(a)=a+\sum_{m=0}^{\infty}t^{m}T_{m}(a)$ such that
$T(a\star b)=T(a)\star^{\prime}T(b)\quad\forall a,b\in A.$ (3.3)
Let $M$ be a smooth manifold. It has been proven in [2] that a deformation
quantization of $C^{\infty}(M)$, or a star product, is a deformation of
$\mathcal{A}=C^{\infty}(M)$ such that $P_{m}$ are bidifferential operators. An
isomorphism of two star products is an isomorphism of the corresponding
deformations such that the operators $T_{m}$ are differential.
Given a star product on a smooth manifold $M$, we can define a Poisson bracket
on $C^{\infty}(M)$ by setting
$\\{f,g\\}=P_{1}(f,g)-P_{1}(g,f).$ (3.4)
Recall from the previous chapter that we can associate a bivector $\pi$ to the
Poisson bracket, putting
$\\{f,g\\}_{\pi}=\pi(df,dg)$ (3.5)
From the associativity of $\star$ we obtain that the Poisson bracket (3.4) is
necessarily of the form $\\{f,g\\}_{\pi_{0}}$ for some bivector field
$\pi_{0}$.
From Proposition 2.2.3 we know that a bivector $\pi$ is a Poisson bivector if
and only if the Schouten bracket $\left[\pi,\pi\right]_{S}$ is zero. It is
easy to show that for any star product the bivector field $\pi_{0}$ in eq.
(3.5) is a Poisson structure. Moreover, we can define a formal Poisson
structure as a formal power series
$\pi_{\hbar}=\sum_{m=0}^{\infty}\hbar^{m}\pi_{m}$ such that
$\left[\pi_{\hbar},\pi_{\hbar}\right]_{S}=0$. For any Poisson structure $\pi$
it is possible to define a formal Poisson structure $\pi_{\hbar}=\hbar\pi$.
Two formal Poisson structures $\pi_{\hbar}$ and $\pi^{\prime}_{\hbar}$ are
equivalent if there is a formal power series
$X=\sum_{m=0}^{\infty}\hbar^{m}X_{m}$ such that
$\pi^{\prime}_{\hbar}=\exp(\mathcal{L}_{X})\pi_{\hbar}$. The connection
between formal Poisson structures and star products motivates the following
result:
###### Theorem 3.1.1 (Kontsevich, [27]).
There is a bijection, natural with respect to diffeomorphisms, between the set
of equivalence classes of formal Poisson structures on $M$ and the set of
isomorphism classes of deformation quantizations of $C^{\infty}(M)$.
In other words, this theorem states that classes of star products corresponds
to classes of deformations of the Poisson structure, i.e. any Poisson manifold
admits a deformation quantization. This result follows from a more general
one, called Formality theorem.
#### 3.1.2 Formality Theory
As mentioned in the previous section, a Poisson structure is completely
defined by the choice of a bivector field satisfying certain properties; on
the other hand a star product is specified by a family of bidifferential
operators. In order to work out the correspondence between these two objects,
we introduce the two differential graded Lie algebras they belong to:
multivector fields $\mathfrak{g}_{S}^{\bullet}(M)$ and multidifferential
operators $\mathfrak{g}_{G}^{\bullet}(C^{\infty}(M))$. In the Formality
theorem, Kontsevich constructed a $L_{\infty}$ quasi-isomorphism between these
differential graded Lie algebras.
###### Definition 3.1.2.
A graded Lie algebra (GLA) is a graded vector space
$\mathfrak{g}=\oplus_{i\in\mathbb{Z}}\mathfrak{g}^{i}$ endowed with a bilinear
operation
$[\cdot,\cdot]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ (3.6)
satisfying the following conditions:
1. 1.
homogeneity, $[a,b]\in\mathfrak{g}^{\alpha+\beta}$
2. 2.
skew-symmetry, $[a,b]=-(-1)^{\alpha\beta}[b,a]$
3. 3.
Jacoby identity, $[a,[b,c]]=[[a,b],c]+(-1)^{\alpha\beta}[b,[a,c]]$
for any $a\in\mathfrak{g}^{\alpha}$, $b\in\mathfrak{g}^{\beta}$ and
$c\in\mathfrak{g}^{\gamma}$.
As an example, any Lie algebra is a GLA concentrated in degree 0.
###### Definition 3.1.3.
A differential graded Lie algebra (DGLA) is a GLA $\mathfrak{g}$ together with
a differential $d:\mathfrak{g}\rightarrow\mathfrak{g}$, i.e. a linear operator
of degree 1 which satisfies the Leibniz rule
$d[a,b]=[da,b]+(-1)^{\alpha\beta}[a,db]\qquad a\in\mathfrak{g}^{\alpha},\quad
b\in\mathfrak{g}^{\beta}$ (3.7)
and $d^{2}=0$.
Given a DGLA we can define immediately the cohomology of $\mathfrak{g}$ as
$H^{i}(\mathfrak{g}):=Ker(d:\mathfrak{g}^{i}\rightarrow\mathfrak{g}^{i+1})/Im(d:\mathfrak{g}^{i-1}\rightarrow\mathfrak{g}^{i})$
(3.8)
The set $H:=\oplus_{i}H^{i}(\mathfrak{g})$ has a natural structure of graded
Lie algebra.
The morphism $f:\mathfrak{g}_{1}\rightarrow\mathfrak{g}_{2}$ of DGLA’s induces
a morphism $H(f):H_{1}\rightarrow H_{2}$ between cohomologies. Recall that a
quasi-isomorphism is a morphism of DGLA’s inducing isomorphisms in cohomology.
###### Definition 3.1.4.
A differential graded Lie algebra $\mathfrak{g}$ is formal if it is quasi-
isomorphic to its cohomology, regarded as a DGLA with zero differential and
the induced bracket.
#### Multivector fields and Multidifferential operators
In the following, we discuss two DGLA’s which will play a fundamental role in
deformation quantization (a useful presentation can be found in [6]). We start
with the DGLA of multidifferential operators, which is a subalgebra of the
Hochschild DGLA. In the following, we explain how this algebra is constructed.
The Hochschild complex of an associative unital algebra $A$ is the complex
$C(A,A)$ with vanishing components in degree $n<0$ and whose $n$-th component,
for $n\geq 0$ is the space
$\tilde{C}(A,A):=\sum_{n=-1}^{\infty}\tilde{C}^{n}(A,A)\qquad\tilde{C}^{n}(A,A)=Hom(A^{\otimes
n+1},A).$ (3.9)
If $A=C^{\infty}(M)$, we require that $\tilde{C}^{n}(A,A)$ consists of those
maps from $A^{\otimes n}$ to $A$ which are multi-differential. By definition,
the differential of a $n$-cochain $f$ is the $(n+1)$-cochain defined by
$\begin{split}(-1)^{n}(df)(a_{0},\dots,a_{n})&=a_{0}f(a_{1},\dots,a_{n})-\sum_{i=0}^{n-1}(-1)^{i}f(a_{0},\dots,a_{i}a_{i+1},\dots,a_{n})\\\
&+(-1)^{n-1}f(a_{0},\dots,a_{n-1})a_{n}\end{split}$ (3.10)
The Hochschild cohomology $H(A,A)$ of $A$ is the homology associated to the
Hochschild complex. The normalized Hochschild complex is
$C^{n}(A,A)=Hom(\bar{A}^{\otimes n},A)$ (3.11)
where $\bar{A}=A/k1$. Now we introduce a new structure on the Hochschild
complex, the Gerstenhaber bracket [19]. The Gerstenhaber product of
$f\in\tilde{C}^{n}(A,A)$ and $g\in\tilde{C}^{m}(A,A)$ is the $(n+m-1)$-cochain
defined by
$(f\circ
g)(a_{1},\dots,a_{n+m-1})=\sum_{j=0}^{n-1}(-1)^{(m-1)j}f(a_{1},\dots,a_{j},g(a_{j+1},\dots,a_{j+m}),\dots)$
(3.12)
that is not associative in general. As a consequence, we define the
Gerstenhaber bracket as follows:
$[D,E]_{G}=D\circ E-(-1)^{(n-1)(m-1)}E\circ D.$ (3.13)
We notice that the Hochschild differential can be expressed in terms of the
Gerstenhaber bracket and the multiplication $m$ of $A$ as
$d=[m,\cdot]_{G}:\tilde{C}^{\bullet}(A,A)\rightarrow\tilde{C}^{\bullet+1}(A,A)$
(3.14)
It follows that the Hochschild complex $\tilde{C}^{\bullet}(A,A)$ endowed with
the Gerstenhaber bracket is a DGLA [19] that we denote by
$\mathfrak{g}^{\bullet}_{G}(A)$. It is well known that the embedding of
$C^{\bullet}(A,A)$ into $\tilde{C}^{\bullet}(A,A)$ is a quasi-isomorphism [5].
The second DGLA we are interested in is given by the multivector fields on
$M$. By definition, a $k$-multivector field $X$ is a section of the $k$-th
exterior power $\wedge^{k}TM$ of the tangent space $TM$. We define the
Schouten-Nijenhuis bracket:
$[X,Y_{1}\wedge\dots\wedge Y_{k}]_{S}:=\sum_{i=1}^{k}(-1)^{i+1}[X,Y_{i}]\wedge
Y_{1}\wedge\dots\wedge\hat{Y}_{i}\wedge\dots\wedge Y_{k}.$ (3.15)
such that
1. 1.
$X\in\Gamma(M,T)$, $[X,\pi]_{S}=L_{X}\pi$,
2. 2.
for $f,g\in\Gamma(M,\wedge^{0}T)$, $[f,g]_{S}=0$,
3. 3.
the bracket $[\cdot,\cdot]_{S}$ turns $\Gamma(M,\wedge^{\bullet+1}T)$ into a
graded Lie algebra,
4. 4.
for any $\pi,\psi,\varphi\in\Gamma(M,\wedge^{\bullet}T)$,
$[\pi,\varphi\wedge\psi]_{S}=[\pi,\varphi]_{S}\wedge\psi+(-1)^{|\pi|(|\varphi|+1)}\varphi\wedge[\pi,\psi]_{S}.$
(3.16)
The space $\wedge^{k}TM$, endowed with the Schouten-Nijenhuis bracket and with
differential $d=0$, is a DGLA, which we denote by
$\mathfrak{g}_{S}^{\bullet}(M)=\Gamma(M,\wedge^{\bullet+1}T).$ (3.17)
#### Formality Theorem
As we mentioned above, Kontsevich’s main result is that
$\mathfrak{g}^{\bullet}_{G}(A)$ is a formal DGLA (see Def. 3.1.4). This result
relies on the existence of a previous result by Hochschild, Kostant and
Rosenberg [25] which establishes the existence of an isomorphism between the
cohomology of the algebra of multidifferential operators and the algebra of
multivector fields.
###### Theorem 3.1.5 (Hochschild-Kostant-Rosenberg [25]).
The formula
$D_{\pi}(a_{1},\dots,a_{n})=\langle\pi,da_{1}\dots da_{n}\rangle$ (3.18)
defines a quasi-isomorphism
$(\Gamma(T,\wedge^{\bullet}T),0)\rightarrow
C^{\bullet}(C^{\infty}(M),C^{\infty}(M))$ (3.19)
In particular, the cohomology groups
$H^{\bullet}(C^{\infty}(M),C^{\infty}(M))$ is isomorphic to
$\Gamma(T,\wedge^{\bullet}T),$ (3.20)
where the bracket induced by $[\cdot,\cdot]_{G}$ becomes the Schouten bracket
$[\cdot,\cdot]_{S}$.
The last tool we need is the notion of $L_{\infty}$-quasi isomorphism. Let
$L_{1}$ and $L_{2}$ be two DGLA. By definition, an
$\mathbf{L_{\infty}}$-morphism $f:L_{1}\rightarrow L_{2}$ is given by a
sequence of maps
$f_{n}:L_{1}^{\otimes n}\rightarrow L_{2},\quad n\geq 1,$ (3.21)
homogeneous of degree $1-n$ and such that the following conditions are
satisfied:
1. 1.
The morphism $f_{n}$ is graded antisymmetric, i.e. we have
$f_{n}(x_{1},\dots,x_{i},x_{i+1},\dots
x_{n})=-(-1)^{|x_{i}||x_{i+1}|}f_{n}(x_{1},\dots,x_{i+1},x_{i},\dots x_{n})$
(3.22)
for all homogeneous $x_{1},\dots,x_{n}$ of $L_{1}$.
2. 2.
We have $f_{1}\circ d=d\circ f_{1}$ i.e. the map $f_{1}$ is a morphism of
complexes.
3. 3.
$f_{1}$ is compatible with the brackets up to a homology given by $f_{2}$. In
particular, $f_{1}$ induces a morphism of graded Lie algebras from
$H^{\bullet}(L_{1})$ to $H^{\bullet}(L_{2})$.
4. 4.
More generally, for any homogeneous element $x_{1},\dots,x_{n}$ of
$\mathfrak{g}^{\bullet}$,
$\begin{split}\sum\pm&f_{q+1}([x_{i_{1}},\dots,x_{i_{p}}]_{p},x_{j_{1}},\dots,x_{j_{q}})=\\\
&\sum\pm\frac{1}{k!}[f_{n_{1}}(x_{i_{11}},\dots,x_{i_{1n_{1}}}),\dots,f_{n_{k}}(x_{i_{k1}},\dots,x_{i_{kn_{k}}})]\end{split}$
(3.23)
Roughly, an $L_{\infty}$-morphism is a map between DGLA which is compatible
with the brackets up to a given coherent system of higher homotopies.
An $\mathbf{L_{\infty}}$-quasi isomorphism is an $L_{\infty}$-morphism whose
first components is a quasi-isomorphism.
Kontsevitch’s Formality Theorem can be stated as follows:
###### Theorem 3.1.6 (Kontsevich [27]).
There exists natural $L_{\infty}$ quasi-isomorphism
$K:\mathfrak{g}_{S}^{\bullet}(M)\rightarrow\mathfrak{g}_{G}^{\bullet}(C^{\infty}(M))$
(3.24)
The component $K_{1}$ of $K$ coincides with the quasi-isomorphism defined in
the Hochschild-Kostant-Rosenberg Theorem 3.1.5.
Kontsevich’s formality map induces a one-to-one map from formal Poisson
structures on $M$ to star products on $C^{\infty}(M)$.
### 3.2 Quantization of a Poisson Lie group
The theory of Kontsevich provides a procedure to quantize a Poisson manifold;
we now introduce a theory of quantization for Poisson Lie groups and Lie
bialgebras. As defined in Section 2.3 a Poisson Lie group is a Poisson
manifold endowed with a Lie group structure. The quantization of these
structures can be done using the formalism of quantum groups; this allows us
to deform the Poisson manifold and group structures in a compatible way. More
precisely, given a Poisson Lie group or a Lie bialgebra, an associated Hopf
algebra can be defined and deformed to obtain the correspondent quantum group.
In the following we introduce the definitions of Hopf algebra and Hopf algebra
action and we explain how to quantize them. The interested reader can consult
the standard books about quantum groups e.g. [7] and [37] for details.
#### 3.2.1 Hopf algebras
An algebra with unit over a commutative ring $k$ is a $k$-module $A$ with a
multiplication, bilinear over $k$ and associative, and with the unit element
$1$ such that $a\cdot 1=1\cdot a=a$ for all $a\in A$. We reformulate this
definition in terms of commutative diagrams.
###### Definition 3.2.1.
An algebra over a commutative ring $k$ is a $k$-module $A$ equipped with
$k$-module maps $m^{A}:A\otimes_{k}A\rightarrow A$, the multiplication, and
$\iota^{A}:A\rightarrow A$, the unit, such that the following diagrams
commute:
$\begin{CD}A\otimes k@>{id\otimes\iota}>{}>A\otimes A\\\
@V{}V{\cong}V@V{}V{m}V\\\ A@>{id}>{}>A\end{CD}\qquad\qquad\begin{CD}k\otimes
A@>{\iota\otimes id}>{}>A\otimes A\\\ @V{}V{\cong}V@V{}V{m}V\\\
A@>{id}>{}>A\end{CD}$
$\begin{CD}A\otimes A\otimes A@>{m\otimes id}>{}>A\otimes A\\\ @V{}V{id\otimes
m}V@V{}V{m}V\\\ A\otimes A@>{m}>{}>A\end{CD}$
In terms of the traditional description of an algebra we have
$\iota(\lambda)=\lambda 1,\qquad m(a_{1}\otimes a_{2})=a_{1}\cdot a_{2}.$
(3.25)
The first two diagrams express the properties of the unit element and the
third the associativity of multiplication.
An algebra is commutative if the following diagram commutes
$\begin{CD}A\otimes A@>{\tau}>{}>A\otimes A\\\ @V{}V{m}V@V{}V{m}V\\\
A@>{id}>{}>A\end{CD}$
where $\tau:A\otimes A\rightarrow A\otimes A$ is the flip map
$\tau(a_{1}\otimes a_{2})=a_{2}\otimes a_{1}$. If we set $m_{op}=m\circ\tau$,
then $(A,\iota,m_{op})$ is the opposite algebra of $A$.
###### Definition 3.2.2.
A coalgebra over a commutative ring $k$ is a $k$-module $A$ equipped with
$k$-module maps $\Delta^{A}:A\rightarrow A\otimes A$, the coproduct, and
$\epsilon:A\rightarrow k$, the counit, such that the following diagrams
commute:
$\begin{CD}A\otimes k@<{id\otimes\epsilon}<{}<A\otimes A\\\
@A{}A{\cong}A@A{}A{\Delta}A\\\
A@<{id}<{}<A\end{CD}\qquad\qquad\begin{CD}k\otimes A@<{\epsilon\otimes
id}<{}<A\otimes A\\\ @A{}A{\cong}A@A{}A{\Delta}A\\\ A@<{id}<{}<A\end{CD}$
$\begin{CD}A\otimes A\otimes A@<{\Delta\otimes id}<{}<A\otimes A\\\
@A{}A{id\otimes\Delta}A@A{}A{\Delta}A\\\ A\otimes A@<{\Delta}<{}<A\end{CD}$
The commutativity of the third diagram is usually referred to as the
coassociativity of $A$. The coalgebra $A$ is called cocommutative if the
following diagram commutes:
$\begin{CD}A\otimes A@<{\tau}<{}<A\otimes A\\\ @A{}A{\Delta}A@A{}A{\Delta}A\\\
A@<{id}<{}<A\end{CD}$
If we set $\Delta^{op}=\tau\circ\Delta$, then $(A,\epsilon,\Delta^{op})$ is
the opposite coalgebra.
Given two coalgebras $A$ and $B$, a $k$-module map $\varphi:A\rightarrow B$ is
a coalgebra homomorphism if
$(\varphi\otimes\varphi)\circ\Delta^{A}=\Delta^{B}\circ\varphi,\quad\epsilon^{B}\circ\varphi=\epsilon^{A}.$
(3.26)
A Hopf algebra has compatible algebra and coalgebra structures and one extra
structure map.
###### Definition 3.2.3.
A Hopf algebra over a commutative ring $k$ is a $k$-module $A$ such that
1. 1.
$A$ is both an algebra and a coalgebra over $k$;
2. 2.
the coproduct $\Delta:A\rightarrow A\otimes A$ and the counit
$\epsilon:A\rightarrow k$ are algebra homomorphisms;
3. 3.
the product $m:A\otimes A$ and the unit $\iota:k\rightarrow A$ are coalgebra
homomorphisms;
4. 4.
$A$ is equipped with a bijective $k$-module map $S^{A}:A\rightarrow A$ called
the antipode, such that the following diagrams commute:
$\begin{CD}A\otimes A@>{S\otimes id}>{}>A\otimes A\\\
@A{}A{\Delta}A@V{}V{m}V\\\
A@>{\iota\circ\epsilon}>{}>A\end{CD}\qquad\begin{CD}A\otimes A@>{id\otimes
S}>{}>A\otimes A\\\ @A{}A{\Delta}A@V{}V{m}V\\\
A@>{\iota\circ\epsilon}>{}>A\end{CD}$
If $A$ and $B$ are Hopf algebras, a $k$-module map $\varphi:A\rightarrow B$ is
a Hopf algebra homomorphism if it is a homomorphism of both the algebra and
the coalgebra structures of $A$.
Let us consider two crucial examples.
###### Example 3.2.4.
Let $G$ be a compact topological group. Consider the space $C(G)$ of the
continuous functions on $G$ together with the following maps:
* -
$(f\cdot h)(g)=f(g)h(g)$
* -
$\Delta(f)(g_{1}\otimes g_{2})=f(g_{1}g_{2})$
* -
$\iota(x)=x1$ where $1(g)=1$ for any $g\in G$
* -
$\epsilon(f)=f(e)$ where $e$ is the unit element of $G$
* -
$S(f)(g)=f(g^{-1})$
where $g_{1},g_{2},g\in G$, $x\in k$ and $f,h\in C(G)$. The set $C(G)$
together with these maps is a Hopf algebra.
###### Example 3.2.5.
Let $\mathfrak{g}$ be a Lie algebra and $\mathcal{U}(\mathfrak{g})$ its
universal enveloping algebra, then $\mathcal{U}(\mathfrak{g})$ becomes a Hopf
algebra when
* -
the ordinary multiplication in $\mathcal{U}(\mathfrak{g})$
* -
$\Delta(x)=x\otimes 1+1\otimes x$
* -
$\iota(\alpha)=\alpha 1$
* -
$\epsilon(1)=1$ and zero on all the other elements
* -
$S(x)=-x$
where $x\in\mathfrak{g}$ is considered as a subset of
$\mathcal{U}(\mathfrak{g})$. To be precise, this defines $\Delta$, $\iota$,
$\epsilon$ and $S$ only on the subset $\mathfrak{g}$ of the universal
enveloping algebra, but these maps can be extended uniquely to all
$\mathcal{U}(\mathfrak{g})$ such that the Hopf algebra axioms are satisfied
everywhere.
These examples are in a sense dual each other. In general, suppose that
$(A,m,\Delta,\iota,\epsilon,S)$ is a Hopf algebra and $A^{*}$ is its dual
space; then using the structure maps of $A$ we define the structure maps
$(m^{*},\Delta^{*},\iota^{*},\epsilon^{*},S^{*})$ as follows:
* -
$\langle m^{*}(f\otimes g),x\rangle=\langle f\otimes g,\Delta(x)\rangle$
* -
$\langle\Delta^{*}(f),x\otimes y\rangle=\langle f,xy\rangle$
* -
$\langle\iota^{*}(\alpha),x\rangle=\alpha\cdot\epsilon(x)$
* -
$\epsilon^{*}(f)=\langle f,1\rangle$
* -
$\langle S^{*}(f),x\rangle=\langle f,S(x)\rangle$
where $f,g\in A^{*}$ an $x,y\in A$. The brackets $\langle\cdot,\cdot\rangle$
are the pairing between $A^{*}$ and $A$ and $\langle f\otimes g,x\otimes
y\rangle=\langle f,x\rangle\langle g,y\rangle$. It is easy to see that $A^{*}$
is also a Hopf algebra.
Since we are interested in the quantization of Poisson Lie groups, we need to
introduce the concept of Poisson Hopf algebra. Recall that a Poisson algebra
(1.2.3) is an algebra equipped with a bilinear map $\\{\cdot,\cdot\\}:A\otimes
A\rightarrow A$ such that $(A,\\{\cdot,\cdot\\})$ is a Lie algebra and
$\\{\cdot,\cdot\\}$ is a derivation.
###### Definition 3.2.6.
A Poisson algebra $(A,\\{\cdot,\cdot\\})$ is called a Poisson Hopf algebra if
it is also a Hopf algebra $(A,m,\Delta,\iota,\epsilon,S)$ over $k$ such that
both structure are compatible, i.e.
$\Delta\left(\\{a_{1},a_{2}\\}\right)=\\{\Delta(a_{1}),\Delta(a_{2})\\}_{A\otimes
A}$ (3.27)
for all $a_{1},a_{2}\in A$. Here the Poisson bracket
$\\{\cdot,\cdot\\}_{A\otimes A}$ is defined as
$\\{a\otimes a^{\prime},b\otimes b^{\prime}\\}_{A\otimes A}=\\{a,b\\}\otimes
a^{\prime}b^{\prime}+ab\otimes\\{a^{\prime},b^{\prime}\\}$ (3.28)
for all $a,a^{\prime},b,b^{\prime}\in A$.
Given a Poisson Lie group $(G,\pi)$, its Poisson algebra
$(C^{\infty}(G),\\{\cdot,\cdot\\})$ is a Poisson Hopf algebra with the Hopf
structure given in Example 3.2.4.
For the quantization of the Lie bialgebras, we need the dual version of the
above definition:
###### Definition 3.2.7.
A co-Poisson Hopf algebra is a co-commutative Hopf algebra
$(A,m,\Delta,\iota,\epsilon,S)$ together with a map
$\delta:A\rightarrow A\otimes A$ (3.29)
such that
1. 1.
$\tau\circ\delta=-\delta$ co-antisymmetry
2. 2.
$(1\otimes 1\otimes 1+(1\otimes\tau)(\tau\otimes 1))(1\otimes\delta)\delta=0$
co-Jacobi identity
3. 3.
$(\Delta\otimes 1)\delta=(1\otimes 1\otimes 1+\tau\otimes
1)(1\otimes\delta)\Delta$ co-Leibniz rule
4. 4.
$(m\otimes m)\circ\delta_{A\otimes A}=\delta\circ m$ m is co-Poisson
homomorphism
where $\delta_{A\otimes A}=(1\otimes\tau\otimes
1)(\delta\otimes\Delta+\Delta\otimes\delta)$ is the co-Poisson structure
naturally associated to the tensor product space
The universal enveloping algebra of a Lie biagebra $\mathfrak{g}$ is a co-
Poisson Hopf algebra with Hopf structure defined in Example 3.2.5.
#### 3.2.2 Quasi triangular Hopf algebras
In the following we discuss a particular class of Hopf algebras and, in
analogy with the classical case, the quantum Yang-Baxter equation.
As already mentioned, a Hopf algebra $H$ is cocommutative is
$\tau\circ\Delta=\Delta$. Here we consider Hopf algebras that are only
cocommutative up to conjugation by an element $R\in H\otimes H$. This element
$R$ is called the quasi triangular structure.
###### Definition 3.2.8.
A quasi triangular Hopf algebra is a pair $(H,R)$, where $H$ is a Hopf algebra
and $R\in H\otimes H$ is invertible and such that
$(\Delta\otimes id)R=R_{13}R_{23},\quad(id\otimes\Delta)R=R_{13}R_{12}$ (3.30)
For sake of completeness we record the following two lemmas, in analogy with
the classical case.
###### Lemma 3.2.9.
If $(H,R)$ is a quasitriangular bialgebra, then $R$ as an element of $H\otimes
H$ obeys
$(\epsilon\otimes id)R=(id\otimes\epsilon)R=1$ (3.31)
If $H$ is a Hopf algebra then one also has
$(S\otimes id)R=R^{-1},\quad(id\otimes S)R^{-1}=R$ (3.32)
and hence $(S\otimes S)R=R$.
###### Lemma 3.2.10.
Let $(H,R)$ be a quasitriangular bialgebra. Then
$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ (3.33)
is the quantum Yang-Baxter equation.
#### 3.2.3 Hopf algebra actions
In this section we introduce the notion of Hopf algebra action. This will be
used in the next sections to define a quantized action and a quantum momentum
map.
###### Definition 3.2.11.
For an algebra $A$, a (left) A-module is a $k$-space $M$ with a $k$-linear map
$\gamma:A\otimes M\rightarrow M$ such that $\gamma(m\otimes
id)=\gamma(id\otimes m)$ and $\gamma(u\otimes id)=$ scalar multiplication.
We have that $A$ acts on the $k$-space $M$ if $M$ is a left $A$-module. The
action is given by the map $\gamma$. The dual notion is the co-action of a
coalgebra:
###### Definition 3.2.12.
For a coalgebra $C$, a (right) C-comodule is a $k$-space $M$ with a $k$-linear
map $\rho:M\rightarrow M\otimes C$ such that
$(id\otimes\Delta)\rho=(\Delta\otimes id)\rho$ and $(id\otimes\epsilon)\rho=$
tensoring with 1.
We say that $\rho$ is a coaction of $C$ with $M$. Let us define the maps which
preserve the module and comodule structures on the corresponding spaces.
###### Definition 3.2.13.
Let $A$ be an algebra and $C$ a coalgebra.
1. 1.
Let $M$ and $N$ be (left) $A$-modules with structure maps $\gamma_{M}$ and
$\gamma_{N}$ respectively. A map $f:M\rightarrow N$ is called an A-module map
if $f\circ\gamma_{M}=\gamma_{N}\circ(id\otimes f)$.
2. 2.
Let $M$, $N$ be a (right) $C$-comodules, with structure maps $\rho_{M}$ and
$\rho_{N}$ respectively. A map $f:M\rightarrow N$ is called a C-comodule map
if $\rho_{N}\circ f=(f\otimes id)\circ\rho_{M}$.
Finally, the Hopf algebra actions are defined as follows:
###### Definition 3.2.14.
Let $H$ be a Hopf algebra. An algebra $A$ is a (left) H-module algebra if:
1. 1.
$A$ is a (left) $H$-module via $h\otimes a\mapsto\gamma(h)(a)$
2. 2.
$\gamma(ab)=m(\gamma\otimes\gamma)(\Delta h)(a\otimes b)$ for any $a$ and $b$
in $A$
3. 3.
$\gamma(h)1_{A}=\epsilon(h)1_{A}$
In this case we have a Hopf algebra action of $H$ on $A$ if the algebraic
structures of $A$ is compatible with this action. Similarly, a Hopf algebra
co-action is defined by:
###### Definition 3.2.15.
An algebra $A$ is a (right) H-comodule algebra if
1. 1.
$A$ is a (right) $H$-comodule via $\rho:A\rightarrow A\otimes H$ for any $a$
and $b$ in $A$
2. 2.
$\rho(ab)=m(\Delta a,\Delta b)$
3. 3.
$\rho(1_{A})=1_{A}\otimes 1_{A}$
#### 3.2.4 Quantization of Poisson structures
In this section we discuss the quantization of a Lie bialgebra $\mathfrak{g}$
as studied in [48] by Reshetikhin and in [11] by Etingof and Kazhdan. We will
see that the quantization of $\mathfrak{g}$ is provided by the quantum
universal enveloping algebra $\mathcal{U}_{\hbar}(\mathfrak{g})$. Recall that,
given a Poisson algebra $(A,\\{\cdot,\cdot\\})$, its quantization is given by
defining a star product as in eq. (3.2). Then, we describe the quantization of
a Poisson Hopf algebra, obtained as deformation of the Poisson algebra and the
Hopf algebra structures.
###### Definition 3.2.16 (Quantization of Hopf algebra).
A deformation of a Hopf algebra $(A,\iota,m,\epsilon,\Delta,S)$ over a field
$k$ is a Hopf algebra
$(A_{\hbar},\iota_{\hbar},m_{\hbar},\epsilon_{\hbar},\Delta_{\hbar},S_{\hbar})$
over the ring $k\llbracket\hbar\rrbracket$ of formal power series such that
1. 1.
$A_{\hbar}$ is isomorphic to $A\llbracket\hbar\rrbracket$ as a
$k\llbracket\hbar\rrbracket$-module;
2. 2.
$m_{\hbar}\equiv m$ (mod $\hbar$) and $\Delta_{\hbar}\equiv\Delta$ (mod
$\hbar$).
Next, let us define the quantization of a Poisson Hopf algebra:
###### Definition 3.2.17 (Quantization of Poisson Hopf algebra).
Let $A$ be a Poisson Hopf algebra over $k$. A quantization of $A$ is a Hopf
algebra deformation $A_{\hbar}$ such that
$\\{q(a),q(b)\\}=q\left(\frac{[a,b]_{\hbar}}{\hbar}\right)\quad\forall a,b\in
A$ (3.34)
where $q$ is the canonical quotient map $q:A_{\hbar}\rightarrow
A_{\hbar}/\hbar A_{\hbar}\cong A$ and $[\cdot,\cdot]_{\hbar}$ is the usual
commutator with respect to $m_{\hbar}$.
If $G$ is a Poisson Lie group, then the algebra of functions on $G$ is a Hopf
algebra and the two structure are compatible as in Definition 3.2.6. A
quantization of a Poisson Lie group $G$ is a quantization of this Poisson Hopf
algebra.
###### Definition 3.2.18 (Quantization of co-Poisson Hopf algebra).
Consider the co-Poisson Hopf algebra $(A,m,\Delta,\iota,\epsilon,S;\delta)$. A
quantization of $A$ is a non-commutative Hopf algebra
$(A_{\hbar},m_{\hbar},\Delta_{\hbar},\iota_{\hbar},\epsilon_{\hbar},S)$ over
$k\llbracket\hbar\rrbracket$ such that
1. 1.
$A_{\hbar}/\hbar A_{\hbar}\cong A$
2. 2.
$m\circ(q\otimes q)=q\circ m_{\hbar}$
3. 3.
$q\circ\iota_{\hbar}=\iota$
4. 4.
$\epsilon_{\hbar}\circ q=\epsilon$
5. 5.
$\delta(q(a))=q(\frac{1}{\hbar}(\Delta_{\hbar}(a)-\tau\circ\Delta_{\hbar}(a)))$
for all $a\in A$.
The product and coproduct need to be related by the following condition:
$\Delta_{\hbar}(a\star b)=\Delta_{\hbar}(a)\star\Delta_{\hbar}(b).$ (3.35)
Let us consider the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$
of the Lie bialgebra $\mathfrak{g}$. The Poisson structure on $G$ induces a
co-Poisson structure $\delta$ on its Lie bialgebra $\mathfrak{g}$, that can be
easily extended to $\mathcal{U}(\mathfrak{g})$. Using the Hopf structure
discussed in Example 3.2.5 we get that
$(\mathcal{U}(\mathfrak{g}),m,\Delta,\iota,\epsilon;\delta)$ is a co-Poisson
structure. From Definition 3.2.18, its quantization is a non-commutative Hopf
algebra
$(\mathcal{U}_{\hbar}(\mathfrak{g}),m_{\hbar},\Delta_{\hbar},\iota_{\hbar},\epsilon_{\hbar})$
with
$\left[\cdot,\cdot\right]_{\star}=\sum_{k=0}^{\infty}\hbar^{k+1}F_{k}(\cdot,\cdot)$
(3.36)
where $F_{0}(\cdot,\cdot)$ is the standard commutator, and
$\Delta_{\hbar}=\sum_{k}\hbar^{k}\Delta_{k}$ (3.37)
where $\Delta_{0}$ is the coproduct of $\mathcal{U}(\mathfrak{g})$ and the
antisymmetrization of $\Delta_{1}$ is given by the structure $\delta$.
###### Example 3.2.19 (Quantization of $\mathcal{U}(\mathfrak{sl})(2)$ [53]).
Let us consider the Lie algebra $\mathfrak{sl}(2)$ with basis ${H,E,F}$ and
commutation relations
$\left[H,E\right]=2E,\quad\left[H,F\right]=-2F\quad\left[E,F\right]=H.$ (3.38)
The co-Poisson structure of $\mathcal{U}(\mathfrak{sl}(2))$ is given by the
extension to the whole universal enveloping algebra of the map
$\displaystyle\delta(H)$ $\displaystyle=0$ (3.39) $\displaystyle\delta(E)$
$\displaystyle=\frac{1}{2}E\wedge H$ (3.40) $\displaystyle\delta(F)$
$\displaystyle=\frac{1}{2}F\wedge H$ (3.41)
The quantized space is the set of (formal) polynomials in $\hbar$ with
coefficients in $\mathcal{U}(\mathfrak{sl}(2))$. The coproduct
$\Delta_{\hbar}$ on this new space is determined by the following
requirements:
1. 1.
$\Delta_{\hbar}$ is co-associative
2. 2.
$\delta(q(a))=q(\frac{1}{\hbar}(\Delta_{\hbar}(a)-\tau\circ\Delta_{\hbar}(a)))$
3. 3.
In the classical limit $\hbar\rightarrow 0$ the coproduct $\Delta_{\hbar}$
reduces to the ordinary coproduct on $\mathcal{U}(\mathfrak{sl}(2))$
The coproduct $\Delta_{\hbar}$ has the general form
$\Delta_{\hbar}=\sum_{n=0}^{\infty}\frac{\hbar^{n}}{n!}\Delta_{n}.$ (3.42)
where the term $\Delta_{n}$ for an arbitrary $n$ and the coproduct of the
quantized universal enveloping algebra is
$\displaystyle\Delta_{\hbar}(H)$ $\displaystyle=H\otimes 1+1\otimes H$ (3.43)
$\displaystyle\Delta_{\hbar}(E)$ $\displaystyle=E\otimes 1+q^{-H}\otimes E$
(3.44) $\displaystyle\Delta_{\hbar}(F)$ $\displaystyle=F\otimes q^{H}+1\otimes
F.$ (3.45)
with $q=e^{\frac{\hbar}{4}}$. Now, imposing the condition (3.35) we get the
commutation relations
$\displaystyle\left[H,E\right]_{\star}$ $\displaystyle=2E$ (3.46)
$\displaystyle\left[H,F\right]_{\star}$ $\displaystyle=-2F$ (3.47)
$\displaystyle\left[E,F\right]_{\star}$ $\displaystyle=[H]_{q}$ (3.48)
where $[H]_{q}=\frac{q^{2H}-q^{-2H}}{q-q^{-1}}$. Finally, we have
$\displaystyle\epsilon_{\hbar}(E)$
$\displaystyle=\epsilon_{\hbar}(F)=\epsilon_{\hbar}(H)=0$ (3.49)
$\displaystyle\epsilon_{\hbar}(1)$ $\displaystyle=1$ (3.50)
and
$\displaystyle S_{\hbar}(E)$ $\displaystyle=-qE$ (3.51) $\displaystyle
S_{\hbar}(F)$ $\displaystyle=-q^{-1}F$ (3.52) $\displaystyle S_{\hbar}(H)$
$\displaystyle=-H.$ (3.53)
This Hopf algebra is called the quantum universal enveloping algebra of
$\mathfrak{sl}(2)$ and is denoted by $\mathcal{U}_{\hbar}(\mathfrak{sl}(2))$.
### 3.3 Quantum Momentum Map
The problem of quantizing the momentum map and the theory of reduction has
been the main topic of many works, e.g. [13] and [35]. In the following we
discuss two different methods that have been proposed to approach it.
The first one, due to Fedosov [13], uses deformation quantization. In this
approach, given a canonical action of a Lie group $G$ on a symplectic manifold
$M$, the quantum momentum map is defined as a Lie algebra homomorphism
$\boldsymbol{\mu}_{\hbar}$ from the Lie algebra $\mathfrak{g}$ into the
deformed algebra $C^{\infty}_{\hbar}(M)$. The corresponding quantum action is
given by the quantization of the Hamiltonian vector field induced by
$\boldsymbol{\mu}$. We notice that in this approach there is no quantization
of the group. Fedosov defined the quantum reduced space as
$C^{\infty}_{\hbar}(M)^{G}/\mathcal{I}_{\hbar},$ (3.54)
where $C^{\infty}_{\hbar}(M)^{G}$ is the set of the functions in
$C^{\infty}_{\hbar}(M)$ which are invariant under the quantized action. Here
$\mathcal{I}_{\hbar}$ is the ideal generated by the components
$\boldsymbol{\mu}^{i}_{\hbar}$ of the quantum momentum map
$\boldsymbol{\mu}_{\hbar}$. Furthermore he proved that, under the assumptions
of the Marsden-Weinstein Theorem, the quantum reduced algebra in eq. (3.54) is
isomorphic to the algebra obtained by canonical deformation quantization of
$C^{\infty}(M_{\xi})$.
A different approach has been developed in [35] by Lu. In this case the
quantization procedure is carried on via quantum group techniques. Here the
author considers a Hopf algebra $H$ with dual $H^{*}$ and assumes that $H$ is
the quantization of the Poisson Lie group $G$. Given an action
$\Phi:H^{*}\otimes V\rightarrow V$ of $H^{*}$ on the algebra $V$, then the
quantum momentum map is defined as an algebra homomorphism
$\boldsymbol{\mu}:H^{*}\rightarrow V$, provided that the action $\Phi$ can be
rewritten in terms of $\boldsymbol{\mu}$. We stress that this approach does
not guarantee that the quantum action $\Phi$ is the quantization of the given
Poisson action.
The idea we discuss in this section is the generalization of the deformation
quantization approach to the Poisson reduction case. We use quantum group
techniques to quantize Poisson Lie groups and the quantum momentum map is
basically defined as a linear map from the quantum group
$\mathcal{U}_{\hbar}(\mathfrak{g})$ to the deformed algebra
$C^{\infty}_{\hbar}(M)$. The induced quantum action will be a Hopf algebra
action, as defined in Definition 3.2.14. This will allows us to discuss some
examples of quantum momentum map and quantum reduction.
#### 3.3.1 Quantization of the momentum map
The main goal of this section is the quantization of the momentum map as
defined in (2.4.4). Basically, we consider a Poisson action of $G$ on $M$,
introduce the quantization of the structures using the techniques discussed in
the previous sections and we give a definition of the corresponding quantum
action in terms of Hopf algebra action. The quantum momentum map will be
defined as the map which factorizes such an action.
Let $C^{\infty}_{\hbar}(M)$ be a deformation quantization of $(M,\pi)$ and let
us denote
$\displaystyle m_{\star}:C^{\infty}_{\hbar}(M)\times
C^{\infty}_{\hbar}(M)\rightarrow C^{\infty}_{\hbar}(M)$ (3.55)
$\displaystyle[f,g]_{\star}=\sum_{n=0}^{\infty}P_{n}(f,g),\quad\forall f,g\in
C^{\infty}(M)$ (3.56)
the star-product and the deformed bracket in $C^{\infty}_{\hbar}(M)$.
As discussed in the previous section, $\mathcal{U}_{\hbar}(\mathfrak{g})$
denote the deformation quantization of the universal enveloping algebra of
$\mathfrak{g}$ and we denote with $m_{\hbar}$ and $\Delta_{\hbar}$ the
deformed product and coproduct on $\mathcal{U}_{\hbar}(\mathfrak{g})$, resp.
Given the quantization of all the structures, we define the quantum action as
follows:
###### Definition 3.3.1.
Given the infinitesimal generator $\Phi:\mathfrak{g}\rightarrow TM$ of a
Poisson action of $(G,\pi_{G})$ on $(M,\pi)$, the corresponding quantum action
is the linear map
$\Phi_{\hbar}:\mathcal{U}_{\hbar}(\mathfrak{g})\rightarrow
End\;C^{\infty}_{\hbar}(M):\xi\mapsto\Phi_{\hbar}(\xi)(f)$ (3.57)
continuous with respect to $C^{\infty}$-topology and such that it defines a
Hopf algebra action, i.e. such that
$\Phi_{\hbar}(\xi)(f\star
g)=m_{\star}(\Phi_{\hbar}\otimes\Phi_{\hbar}\circ\Delta_{\hbar}(\xi)(f\otimes
g)).$ (3.58)
and
$[\Phi_{\hbar}(\xi),\Phi_{\hbar}(\eta)](f)=\Phi_{\hbar}([\xi,\eta])(f)$ (3.59)
An example can be constructed when the deformation of
$\mathcal{U}(\mathfrak{g})$ comes from a twist
$\tau\in(\mathcal{U}(\mathfrak{g})\otimes\mathcal{U}(\mathfrak{g}))\llbracket\hbar\rrbracket$
such that $\Delta_{\hbar}=\tau\Delta\tau^{-1}$ is associative and the
corresponding associator is trivial [60]. In fact, one can define a
deformation quantization of $M$ setting, for any differential operator
$\mathcal{D}$ on $M$.
$\mathcal{D}(f\star g)=m((\mathcal{L}_{t}\Delta(\mathcal{D}))f\otimes g),$
(3.60)
where $\Delta$ is the usual coproduct on differential operators. The
$\star$-product defined by this formula is automatically consistent with the
action of deformed Hopf algebra $\mathcal{U}(\mathfrak{g})$.
In the following we define a quantum momentum map which, analogously to the
classical case, factorizes the quantum action (3.57). Let us recall from the
previous chapter the composition of Lie algebra homomorphisms which define the
momentum map:
$\displaystyle\mathfrak{g}$
$\displaystyle\longrightarrow\Omega^{1}(M)\longrightarrow TM$ (3.61)
$\displaystyle\xi$
$\displaystyle\longmapsto\alpha_{\xi}\longmapsto\pi^{\sharp}(\alpha_{\xi})$
(3.62)
Using the arguments of Section 2.4.2, it should be clear that a quantum
momentum map can be defined as a quantization of the infinitesimal momentum
map $\alpha$ (2.80). Recall that, in the classical construction, the map
$\pi^{\sharp}:\Omega^{1}(M)\rightarrow TM$ is defined by:
$\pi^{\sharp}:\Omega^{1}(M)\ni adb\longmapsto a\\{b,\cdot\\}\in TM$ (3.63)
where $a,b\in C^{\infty}(M)$. Hence, the classical construction can be
rephrased as follows:
$\displaystyle\mathfrak{g}$ $\displaystyle\longrightarrow C^{\infty}(M)\otimes
C^{\infty}(M)\longrightarrow End\;C^{\infty}(M)$ (3.64) $\displaystyle\xi$
$\displaystyle\longmapsto a_{\xi}^{i}\otimes
b_{\xi}^{i}\longmapsto\sum_{i}a_{\xi}^{i}\\{b_{\xi}^{i},f\\}$ (3.65)
This motivates the following definition of quantum momentum map.
###### Definition 3.3.2.
A quantum momentum map is defined to be a linear map
$\boldsymbol{\mu}_{\hbar}:\mathcal{U}_{\hbar}(\mathfrak{g})\rightarrow
C_{\hbar}^{\infty}(M)\otimes
C_{\hbar}^{\infty}(M):\xi\mapsto\sum_{i}a_{\xi}^{i}\otimes b_{\xi}^{i}.$
(3.66)
such that it is an algebra homomorphism and
$\Phi_{\hbar}(\xi)=\sum_{i}a_{\xi}^{i}\left[b_{\xi}^{i},\cdot\right]_{\star}$
(3.67)
is a quantized action.
To avoid cumbersome notation, we omitted the star notation $\star$ in the
star-product. Moreover, we denoted the functions $\hat{a}=a\mod\hbar$,
$\hat{b}=b\mod\hbar\in C_{\hbar}^{\infty}(M)$ simply by $a$ and $b$.
It is easy to see that, the classical action (3.64) can be recovered in the
limit $\hbar\rightarrow 0$ from eq. (3.67) and using eq. (3.4). In other
words, this definition gives the quantization of the construction (3.64) as
$\displaystyle\mathcal{U}_{\hbar}(\mathfrak{g})$ $\displaystyle\longrightarrow
C_{\hbar}^{\infty}(M)\otimes C_{\hbar}^{\infty}(M)\longrightarrow
End\;C^{\infty}_{\hbar}(M)$ (3.68) $\displaystyle\xi$
$\displaystyle\longmapsto a_{\xi}^{i}\otimes
b_{\xi}^{i}\longmapsto\frac{1}{\hbar}\sum_{i}a_{\xi}^{i}[b_{\xi}^{i},f]_{\star}$
(3.69)
On the other hand, introducing the space $\Omega^{1}(\mathcal{A}_{\hbar})$ of
differential forms on the algebra $\mathcal{A}_{\hbar}=C^{\infty}_{\hbar}(M)$
and identifying $\Omega^{1}(\mathcal{A}_{\hbar})$ with
$\mathcal{A}_{\hbar}\otimes\mathcal{A}_{\hbar}$, we have:
$\mathcal{U}_{\hbar}(\mathfrak{g})\longrightarrow\Omega^{1}(\mathcal{A}_{\hbar})\longrightarrow
End\;\mathcal{A}_{\hbar}\\\ $ (3.70)
We can define a non commutative product on the space of differential forms
$\Omega^{1}(\mathcal{A}_{\hbar})$, using the map
$\Omega^{1}(\mathcal{A}_{\hbar})\rightarrow
End\;\mathcal{A}_{\hbar}:adb\mapsto a[b,f]$. It associates $[b,[c,f]]$ to the
product of two closed forms $db\cdot dc$ and we have
$[b,[c,f]]=b[c,f]-[c,f]b=b[c,f]-[cb,f]+c[b,f].$ (3.71)
It is clear that the product $db\cdot dc$ on $\Omega^{1}(\mathcal{A}_{\hbar})$
has to be defined as follows:
$db\cdot dc=bdc-d(cb)+cdb.$ (3.72)
As introduced in (3.11) the space $End\;\mathcal{A}_{\hbar}$ defines the
Hochschild cochain of $\mathcal{A}_{\hbar}$ in itself
$C^{1}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar})$ with coboundary $b$. We
notice here that from the definition of the product (3.72), the map
$\Omega^{1}(\mathcal{A}_{\hbar})\longrightarrow End\;\mathcal{A}_{\hbar}$ is a
Lie algebra homomorphism only if the differential of the unit in
$\mathcal{A}_{\hbar}$ does not vanish in $\Omega^{1}(\mathcal{A}_{\hbar})$,
i.e. we work with the formal differential forms on the unitalization
$\mathcal{A}_{\hbar}^{+}$ of $\mathcal{A}_{\hbar}$.
These observations allow us to rewrite the definition of quantum momentum map
as follows:
###### Definition 3.3.3.
A quantum momentum map for the quantum action
$\Phi_{\hbar}:\mathcal{U}_{\hbar}(\mathfrak{g})\rightarrow
End\;\mathcal{A}_{\hbar}$ is a linear map
$\boldsymbol{\mu}_{\hbar}:\mathcal{U}_{\hbar}(\mathfrak{g})\longrightarrow\Omega^{1}(\mathcal{A}_{\hbar}):\xi\mapsto
a_{\xi}^{i}db_{\xi}^{i}$ (3.73)
such that it is an algebra homomorphism and
$\Phi_{\hbar}(\xi)(f)=\frac{1}{\hbar}\sum_{i}a_{\xi}^{i}[b_{\xi}^{i},f]_{\star},$
(3.74)
where $a_{\xi}^{i},b_{\xi}^{i}\in\mathcal{A}_{\hbar}$.
In Section (2.4.2) we rephrased the classical construction (3.61) in terms of
Gerstenhaber morphisms,
$\wedge^{\bullet}\mathfrak{g}\rightarrow\Omega^{\bullet}(M)\rightarrow\wedge^{\bullet}TM$
(3.75)
In order to generalize the quantum construction (3.68) in a similar way we
first notice that the map $\Omega^{1}(\mathcal{A}_{\hbar})\rightarrow
C^{1}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar})$ extends naturally to the map
$\Omega^{\bullet}(\mathcal{A}_{\hbar})\rightarrow
C^{\bullet}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar})$.
Consider the tensor algebra $T(\mathcal{U}_{\hbar}(\mathfrak{g})[1])$, where
the degree of $\xi_{1}\otimes\dots\otimes\xi_{n}$ is $n$. The coproduct on
$\mathcal{U}_{\hbar}(\mathfrak{g})$ extends naturally to
$T(\mathcal{U}_{\hbar}(\mathfrak{g})[1])$, simply putting
$\Delta_{\hbar}(\xi_{1}\otimes\xi_{2})=\Delta_{\hbar}(\xi_{1})\otimes\xi_{2}-\xi_{1}\otimes\Delta_{\hbar}(\xi_{2})$
(i. e. $\Delta$ is extended to an odd derivation of the tensor algebra). Then
$\Delta_{\hbar}^{2}=0$ and
$(T(\mathcal{U}_{\hbar}(\mathfrak{g})[1]),\Delta_{\hbar})$ is a complex. The
action $\mathcal{U}_{\hbar}(\mathfrak{g})\rightarrow
C^{1}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar})$ extends to the cochain map
$\displaystyle T(\mathcal{U}_{\hbar}(\mathfrak{g})[1])\longrightarrow
C^{\bullet}(\mathcal{A}_{\hbar},\mathcal{A}_{\hbar})$ (3.76)
These observations motivate the following rephrasing of the definition of
quantum momentum map
###### Definition 3.3.4.
A quantum momentum map is defined to be a linear map
$\boldsymbol{\mu}_{\hbar}:T(\mathcal{U}_{\hbar}(\mathfrak{g})[1])\rightarrow\Omega^{\bullet}(\mathcal{A}_{\hbar}):\xi_{1}\otimes\dots\otimes\xi_{n}\mapsto
a_{1}db_{1}\otimes\dots\otimes a_{n}db_{n}$ (3.77)
such that
$\Phi_{\hbar}(\xi_{1}\otimes\dots\otimes\xi_{n})(f_{1},\ldots,f_{n})=\frac{1}{\hbar^{n}}a_{1}[b_{1},f_{1}]\dots
a_{n}[b_{n},f_{n}]$ (3.78)
#### 3.3.2 Examples
In this section we apply the construction given above to some explicit
examples. We show that the existence of the quantum momentum map induces the
quantization of the Lie algebra. In fact, in the examples studied in this
section, the quantization of the Lie algebra is essentially uniquely
determined by existence of universal formulas for the quantum momentum map.
##### Two-dimensional case
Consider the Lie bialgebra $\mathfrak{g}=\mathbb{R}^{2}$ with generators
$\xi,\eta$ and a deformation quantization $C^{\infty}_{\hbar}(M)$ of a Poisson
manifold $M$. Assume that $\xi$ acts by
$\Phi_{\hbar}(\xi)=\frac{1}{\hbar}a[b,\cdot\;]$ (3.79)
for some $a,b\in C^{\infty}_{\hbar}(M)$. Let us impose that it is an Hopf
algebra action; then we have
$\Phi_{\hbar}(\xi)(fg)=\frac{1}{\hbar}a[b,fg]=\frac{1}{\hbar}a[b,f]g+\frac{1}{\hbar}[a,f][b,g]+\frac{1}{\hbar}fa[b,g].$
(3.80)
Suppose that $a$ is invertible, then $[a,f]=-a[a^{-1},f]a$ and, setting
$\Phi_{\hbar}(\eta)=\frac{1}{\hbar}a[a^{-1},\cdot]$ (3.81)
the coproduct which satisfies the condition (3.2.14) is given by
$\Delta_{\hbar}(\xi)=\xi\otimes 1-\hbar\;\eta\otimes\xi+1\otimes\xi.$ (3.82)
Similarly, for (3.81) we have
$\begin{split}\Phi_{\hbar}(\eta)(fg)&=\frac{1}{\hbar}a[a^{-1},fg]\\\
&=\frac{1}{\hbar}a[a^{-1},f]g-\frac{1}{\hbar}a[a^{-1},f]a[a^{-1},g]+\frac{1}{\hbar}fa[a^{-1},g].\end{split}$
(3.83)
hence
$\Delta_{\hbar}(\eta)=\eta\otimes 1-\hbar\;\eta\otimes\eta+1\otimes\eta.$
(3.84)
Finally we calculate the bracket of the generators to get the deformed algebra
structure of $\mathfrak{g}$:
$\begin{split}\left[\Phi_{\hbar}(\xi),\Phi_{\hbar}(\eta)\right]f&=\frac{1}{\hbar^{2}}(a[b,a[a^{-1},f]]-a[a^{-1},a[b,f]])\\\
&=a[b,a][a^{-1},f]+aa[b,[a^{-1},f]]-a[a^{-1},a][b,f]-aa[a^{-1},[b,f]]\\\
&=a[b,a][a^{-1},f]+a^{2}[[b,a^{-1}],f].\end{split}$ (3.85)
###### Remark 3.3.5.
One should notice that the equation (3.80) essentially says the following.
Given an element in the image of $\Phi_{\hbar}$ of the form
$\frac{1}{\hbar}a[b,\cdot]$, $a$ is invertible on the support of $b$ and,
assuming universality of our formulas, forces the image of $\Phi_{\hbar}$ also
to contain an element of the form $\frac{1}{\hbar}a[a^{-1},\cdot]$. In the
case when ${\mathfrak{g}}$ is two (or three dimensional), this essentially
forces the formulas for the deformed coproduct in the examples below.
We obtain different algebra structures that we discuss case by case
###### Case 1: $[a,b]=0$.
Under this assumption, from the relation (3.85) we obtain
$\left[\Phi_{\hbar}(\xi),\Phi_{\hbar}(\eta)\right]=0$ and imposing that
$\Phi_{\hbar}$ is a Lie algebra homomorphism we get
$[\xi,\eta]=0.$ (3.86)
Hence, the quantum group given by the universal enveloping algebra
$\mathcal{U}_{\hbar}(\mathbb{R}^{2})$ generated by the commuting elements
$\xi,\eta$ with coproduct
$\displaystyle\Delta_{\hbar}(\xi)$ $\displaystyle=\xi\otimes
1-\hbar\;\eta\otimes\xi+1\otimes\xi$ (3.87)
$\displaystyle\Delta_{\hbar}(\eta)$ $\displaystyle=\eta\otimes
1-\hbar\;\eta\otimes\eta+1\otimes\eta.$ (3.88)
is the deformation quantization of the abelian Lie bialgebra
$\mathfrak{g}=\mathbb{R}^{2}$, with cobracket
$\delta(\xi)=-\frac{1}{2}\eta\wedge\xi\qquad\delta(\eta)=0.$ (3.89)
The corresponding Poisson Lie group is $(\mathbb{R}^{2},\pi)$, where the
Poisson bivector is given by
$\pi=-\frac{1}{2}x\partial_{x}\wedge\partial_{y}$ (3.90)
Setting $a_{0}=a\mbox{ mod }\hbar$ and $b_{0}=b\mbox{ mod }\hbar$, the quantum
actions
$\Phi_{\hbar}(\xi)=\frac{1}{\hbar}a[b,\cdot]\qquad\Phi_{\hbar}(\eta)=\frac{1}{\hbar}a[a^{-1},\cdot]$
(3.91)
give the quantization of the Poisson action of $(\mathbb{R}^{2},\pi)$ on $M$
given by
$\Phi(\xi)=a_{0}\\{b_{0},\cdot\\}\qquad\Phi(\eta)=a_{0}\\{a^{-1}_{0},\cdot\\}.$
(3.92)
The Poisson reduction extends to the quantized version immediately. Given
$\lambda,\mu\in\mathcal{A}_{\hbar}$ with $\lambda\neq 0$ and considering the
ideal $\mathcal{I}_{\hbar}$ of functions generated by $a-\lambda$ and $b-\mu$,
the algebra
$(C^{\infty}_{\hbar}(M)/\mathcal{I}_{\hbar})^{\mathcal{U}_{\hbar}(\mathbb{R}^{2})}$
(3.93)
is a quantization of the Poisson algebra
$\\{a_{0}=\lambda,b_{0}=\mu\\}^{\mathbb{R}^{2}}.$ (3.94)
###### Case 2: $[a,b]=-\hbar$.
In this case we have hence $[b,a^{-1}]=a^{-2}\hbar$; using eq. (3.85) we
obtain
$\begin{split}\left[\Phi_{\hbar}(\xi),\Phi_{\hbar}(\eta)\right](f)&=Y(f)+a^{2}[a^{-2},f]\\\
&=Y(f)+2a[a^{-1},f]-a^{2}[a^{-1},[a^{-1},f]]\\\ &=3\hbar^{2}Y(f)-\hbar
Y^{2}(f).\end{split}$ (3.95)
Hence the quantum group $\mathcal{U}_{\hbar}(\mathfrak{g})$ has the following
structures
$\displaystyle[\xi,\eta]$ $\displaystyle=3\eta-\hbar\eta^{2}$ (3.96)
$\displaystyle\Delta_{\hbar}(\xi)=$ $\displaystyle=\xi\otimes
1-\hbar\eta\otimes\xi+1\otimes\xi$ (3.97) $\displaystyle\Delta_{\hbar}(\eta)$
$\displaystyle=\eta\otimes 1-\hbar\eta\otimes\eta+1\otimes\eta$ (3.98)
and defines a deformation quantization of the Lie bialgebra $\mathfrak{g}$
generated by $\xi$ and $\eta$ with
$\displaystyle[\xi,\eta]$ $\displaystyle=3\eta$ (3.99)
$\displaystyle\delta(\xi)$ $\displaystyle=-\frac{1}{2}\eta\wedge\xi$ (3.100)
$\displaystyle\delta(\eta)$ $\displaystyle=0$ (3.101)
The action of $\mathfrak{g}$ on $M$ is factorized by the momentum map
determined by the forms
$\boldsymbol{\mu}(\xi)=a_{0}db_{0}\quad\boldsymbol{\mu}(\eta)=d\log(a_{0})$
(3.102)
and is given by (3.92). Its quantization is given by (3.91) and it is
factorized by the quantum momentum map
$\boldsymbol{\mu}_{\hbar}(\xi)=adb\quad\boldsymbol{\mu}_{\hbar}(\eta)=d\log(a).$
(3.103)
The quantum reduction is given by
$(C^{\infty}_{\hbar}(M)[b^{-1}])^{\mathcal{U}_{\hbar}(\mathfrak{g})}$ (3.104)
and it is the quantization of the example discussed in Section 2.6.
In this case it is easy to check that $C^{\infty}_{\hbar}(M)[b^{-1}]$ is still
an algebra.
###### Case 3: $[a,b]=-\hbar ba$.
This example is the direct quantization of the Poisson reduction discussed in
Section 2.6. Similarly we discuss here the different cases that classically
give rise to different dressing orbits. First, if $b>0$ we can define
$b=e^{q}$ and $a=e^{p}$, then we recover the previous case, since we get
$[p,q]=\hbar$ (3.105)
i.e. the quantum plane. In this case, unfortunately
$C^{\infty}_{\hbar}(M)[b^{-1}]$ is not an algebra but we observe that we
easily get a well defined algebra by replacing it with
$C^{\infty}_{\hbar}(M)[[\hbar b^{-1}]]$. Hence the quantum reduction is given
by
$(C^{\infty}_{\hbar}(M)[[\hbar b^{-1}]])^{\mathcal{U}_{\hbar}(\mathfrak{g})}$
(3.106)
If $b=0$ we recover the result of the abelian case.
##### Three-dimensional case
The second example we discuss here is the Hopf algebra action of
$U_{\hbar}(\mathfrak{su}(2))$ on the deformed algebra $C^{\infty}_{\hbar}(M)$.
Consider $a,b,c\in C^{\infty}_{\hbar}(M)$ satisfying
$\displaystyle aba^{-1}$ $\displaystyle=e^{2\hbar}b$ (3.107) $\displaystyle
aca^{-1}$ $\displaystyle=e^{-2\hbar}c$ (3.108) $\displaystyle\left[b,c\right]$
$\displaystyle=\frac{\hbar^{2}}{e^{-\hbar}-e^{\hbar}}a^{-2}-(1-e^{2\hbar})cb$
(3.109)
and the generators $\xi,\eta,\zeta$ acting respectively by
$\displaystyle\Phi_{\hbar}(\xi)f$ $\displaystyle=\frac{1}{\hbar}a[b,f]$
(3.110) $\displaystyle\Phi_{\hbar}(\eta)f$
$\displaystyle=\frac{1}{\hbar}[c,f]a$ (3.111)
$\displaystyle\Phi_{\hbar}(\zeta)f$ $\displaystyle=afa^{-1}.$ (3.112)
Then by calculating the commutation relations of these generators and imposing
the Lie algebra homomorphism of $\Phi_{\hbar}$ we obtain that $\xi,\eta,\zeta$
satisfy the commutation relations:
$\displaystyle\zeta\xi\zeta^{-1}$ $\displaystyle=e^{2\hbar}\xi$ (3.113)
$\displaystyle\zeta\eta\zeta^{-1}$ $\displaystyle=e^{-2\hbar}\eta$ (3.114)
$\displaystyle\left[\xi,\eta\right]$
$\displaystyle=\frac{\zeta^{-1}-\zeta}{e^{-\hbar}-e^{\hbar}}$ (3.115)
Checking the condition (3.58) for any generator, we get
$\displaystyle\Phi_{\hbar}(\zeta)(fg)$
$\displaystyle=\Phi_{\hbar}(\zeta)(f)\Phi_{\hbar}(\zeta)(g)$ (3.116)
$\displaystyle\Phi_{\hbar}(\xi)(fg)$
$\displaystyle=\Phi_{\hbar}(\xi)(f)g+\Phi_{\hbar}(\zeta)(f)\Phi_{\hbar}(\xi)(g)$
(3.117) $\displaystyle\Phi_{\hbar}(\eta)(fg)$
$\displaystyle=f\Phi_{\hbar}(\eta)(g)+\Phi_{\hbar}(\eta)(f)\Phi_{\hbar}(\zeta)^{-1}(g)$
(3.118)
hence
$\displaystyle\Delta_{\hbar}(\zeta)$ $\displaystyle=\zeta\otimes\zeta$ (3.119)
$\displaystyle\Delta_{\hbar}(\xi)$ $\displaystyle=\xi\otimes
1+\zeta\otimes\xi$ (3.120) $\displaystyle\Delta_{\hbar}(\eta)$
$\displaystyle=1\otimes\eta+\eta\otimes\zeta^{-1}.$ (3.121)
Finally, we have that $\xi,\eta$ and $\zeta$ generate a Hopf algebra action of
$U_{\hbar}(\mathfrak{su}(2))$ on $C^{\infty}_{\hbar}(M)$.
## Conclusions
The work described in this thesis suggests a number of extensions and
directions for future work. Here we collect some possibilities.
### Momentum map and Reduction in Poisson geometry
The contributions of this thesis in the theory of momentum map and reduction
in Poisson geometry have been discussed in detail in Chapter 2. We give here a
summary of the main results.
We introduced a definition of momentum map in infinitesimal terms and we
proved the theory of reconstruction of momentum map from the infinitesimal one
(Sections 2.4.2 and 2.4.3). The reconstruction theory allowed us to prove the
existence of the momentum map in two explicit cases.
We studied the uniqueness of the momentum map, proving the Theorem 2.4.19 on
the infinitesimal deformations of a momentum map. We analyzed the uniqueness
of the momentum map in the case of a compact and semisimple Poisson Lie group
acting on a generic Poisson manifold. Finally, in Section 2.5 we introduced
the construction of a theory of Poisson reduction.
These results motivate the study of many open problems and we introduce and
briefly discuss in the following those we are interested in approaching:
1. –
The reconstruction problem has been discussed in the Section 2.4.3 only for
the abelian case and the Heisenberg group. An interesting question would be
the possibility of extend our result to an arbitrary two-step nilpotent group
$G^{*}$. Moreover, since the computation performed is a kind of spectral
sequence computation associated to the central series of a nilpotent Lie
algebra, it seems possible to extend the above result to arbitrary nilpotent
groups.
2. –
The connection of the Poisson reduction with the Lu’s point reduction defined
in [34] can be investigated. More precisely, the Poisson reduction can be
regarded as an orbit reduction, hence we aim to generalize the Reduction
diagram theorem, proved in [47] for canonical actions of Lie groups. This
would complete the analogy with the symplectic theory.
3. –
As suggested by Rui Loja Fernandes, Poisson reduction can be rephrased in
terms of Dirac structures [8]. Since $M$ is a Poisson manifold, it is known
that a Dirac structure on it can be defined by the graph of the map
$\pi^{\sharp}:T^{*}M\rightarrow TM$. We would like to prove that the reduced
space, as defined in Section 2.5, inherits an integrable Dirac structure from
$M$ and hence, a Poisson structure. Explicitly, we are interested to
demonstrate the following claim:
Consider the Poisson action $G\times M\rightarrow M$ with equivariant momentum
map $\boldsymbol{\mu}:M\rightarrow G^{*}$. Let $x\in G^{*}$ be a regular value
of $\boldsymbol{\mu}$ and assume that the action is proper and free on
$\boldsymbol{\mu}^{-1}(x)$. Then one has a natural isomorphism
$\boldsymbol{\mu}^{-1}(x)/G_{x}\simeq\boldsymbol{\mu}^{-1}(\mathcal{O}_{x})/G,$
(3.122)
where $\mathcal{O}_{x}\subset G^{*}$ denotes the dressing orbit of $G$ through
$x$ and $G_{x}$ denotes the isotropy group of $x$. This isomorphism is a
Poisson diffeomorphism for the unique Poisson structures on these quotients
which arise from the diagram where the inclusions are backward Dirac maps and
the projections are forward Dirac maps.
This new formulation of the Poisson reduction leads to a new question, i.e.
the relation with the theory of Dirac reduction.
### Quantum Momentum map and Quantum Reduction
The contribution of Chapter 3 is the study of a new definition of the quantum
momentum map associated to the quantized action. Some directions for the
future work are discussed below:
1. –
The first open question we would approach is a formal definition of the
quantum reduction. First we have to complete the example of the Hopf algebra
action of $U_{\hbar}(\mathfrak{su}(2))$ on $C^{\infty}_{\hbar}(M)$. Let
$H=a^{-2}-e^{\hbar}\frac{(1-e^{2\hbar})^{2}}{\hbar^{2}}cb$. Then we obtain
$\displaystyle a^{-1}Ha$ $\displaystyle=H$ (3.123) $\displaystyle[b,H]$
$\displaystyle=-(1-e^{2\hbar})Hb$ (3.124) $\displaystyle[c,H]$
$\displaystyle=c(1-e^{2\hbar})H$ (3.125)
In particular, the ideal $\mathcal{I}$ generated by $H$ in
$C^{\infty}_{\hbar}(M)$ is $\mathcal{U}_{\hbar}(\mathfrak{su}(2))$-invariant,
and
$(C^{\infty}_{\hbar}(M)/\mathcal{I})^{\mathcal{U}_{\hbar}(\mathfrak{su}(2))}$
(3.126)
is a deformation quantization of the Poisson reduction
$M/\\!/SU(2)$ (3.127)
corresponding to the symplectic leaf $a_{0}^{-2}-4b_{0}c_{0}=0$ in
$SU(2)^{*}=SB(2,{\mathbb{C}})$. Notice that the above leaf is not compact,
hence the action cannot integrate to the action of SU(2). It would be
interesting to investigate on the structure of the other leaves. This would
help us to understand how to define correctly the quantum reduction.
2. –
The procedure of quantization can applied to the Poisson actions lifted to
symplectic groupoids. It is known that a Poisson action often does not admit a
momentum map. For this reason Fernandes and Ponte in [14] define a
symplectization functor which turns this action into a Hamiltonian action.
Given a Poisson manifold there is a canonical symplectic groupoid
$\Sigma(M)\rightrightarrows M$ and the action lifted to symplectic groupoids
always admits a momentum map $\boldsymbol{\mu}:\Sigma(M)\rightarrow G^{*}$. It
would be interesting to analyze the quantization of this momentum map.
Similarly to the case of a Poisson Lie group, the quantization of the
symplectic groupoid is given by a quantum groupoid, using the theories studied
by Lu and Xu [36], [60].
3. –
Another significant future direction of research aims at comparing other
approaches to quantization different than the deformation one, in particular
we are interested in geometrical quantization [59]. This is motivated by an
example of the Gelfand-Cetlin system considered by Guillemin and Sternberg in
[22]. In this example geometric quantization is defined via higher cohomology
groups. More precisely, Gelfand-Cetlin system is an integrable system on
$\mathfrak{u(n)}^{*}$ obtained by Thimm’s method. This integrable system can
be viewed as an integrable system on the coadjoint orbits of
$\mathfrak{u(n)}^{*}$ and therefore as a collection of real polarization on
the symplectic leaves. There is a natural Hamiltonian action of a compact Lie
group in this setting: a toric action is associated to a natural choice of
action coordinates. In the case of geometric quantization of the Gelfand-
Cetlin system, Guillemin and Sternberg obtain results in representation
theory.
We would like to apply a similar scheme to other examples in Poisson Lie
groups which inherit real polarizations using a Thimm’s method [42], [24] and
compare the results obtained via deformation quantization.
## References
* [1] V.I. Arnol’d. Small denominators III. Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surveys, 1963.
* [2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Deformation theory and Quantization I-II. Annals of Physics, 111:61–110, 111–151, 1978.
* [3] N. Bourbaki. Lie groups and Lie algebras. Springer-Verlag, 1989.
* [4] A. Cannas de Silva and A. Weinstein. Geometrical models of Non Commutative Algebras. American Mathematical Society, 1999.
* [5] E. Cartan and S. Eilenberg. Homological algebra. Princeton University Press, 1956.
* [6] A. S. Cattaneo and D. Indelicato. Formality and star product. In Poisson geometry, deformation quantization and group representation, 2004. Cambridge University Press.
* [7] V. Chari. A guide to Quantum Groups. Cambridge University Press, 1994.
* [8] T. Courant. Dirac manifolds. Trans. A.M.S., 319:631–661, 1990.
* [9] M. De Wilde and P. B. A. Lecomte. Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys., 7:487–496, 1983.
* [10] V.G. Drinfel’d. Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang-Baxter equations. Soviet Math. Dokl., 27, 1983.
* [11] P. Etingof and D. Kazhdan. Quantization of Lie bialgebras I. Selecta Math., 2:1–41, 1996.
* [12] B.V. Fedosov. A simple Geometrical Construction of Deformation Quantization. Journal of Differential Geometry, 40:213–238, 1994.
* [13] B.V. Fedosov. Non abelian reduction in deformation quantization. Letters in Mathematical Physics, 43(2):137–154, 1998.
* [14] R. L. Fernandes and D. I. Ponte. Integrability of Poisson Lie group actions. Letters in Mathematical Physics, 90, 2009.
* [15] R. L. Fernandes, D. I. Ponte, and T. Ratiu. Momentum map in Poisson geometry. Americal Journal of Mathematics, 131(5), 2009.
* [16] M. Flato, A. Lichnerowicz, and D. Sternheimer. Déformations 1-différentiales des algèbres de Lie attachées à une variété symplectique ou de contact. Compositio Math., 31:47–82, 1975.
* [17] M. Flato, A. Lichnerowicz, and D. Sternheimer. Crochet de Moyal-Vey et quantification. C. R. Acad. Sci. Paris Sér. A-B, 283:61–110, 111–151, 1976.
* [18] I. M. Gel’fand and L. A. Dickey. A family of Hamiltonian structures connected with integrable nonlinear differential equations. Preprint 136, IPM AN SSSR, Moscow, 1978.
* [19] M. Gerstenhaber. The cohomology structure of an associative ring. Ann. Math., 78(2):267–288, 1963.
* [20] Ph. A. Griffiths, P. Deligne, J. W. Morgan, and D. Sullivan. Real homotopy theory of Kähler manifolds. Uspekhi Mat. Nauki, 32, 1997.
* [21] H. J. Groenewold. On the principles of elementary quantum mechanics. Physics, 12:405–460, 1946.
* [22] V. Guillemin and S. Sternberg. The Gel’fand-Cetlin system and quantization of the complex flag manifolds. J. Funct. Anal., 52:106–128, 1983.
* [23] S. Gutt and J. Rawnsley. Equivalence of star products on a symplectic manifold: an introduction to Deligne’s Cech cohomology classes. Journ. of Geom. and Phys, 29:347–392, 1999.
* [24] M. Hamilton and E. Miranda. Geometric quantization of integrable systems with hyperbolic singularities. Annales de l’Institut Fourier, 60:51–85, 2010.
* [25] G. Hochschild, B. Kostant, and A. Rosenberg. Differential forms on regular affine algebras. Trans. Amer. Math. Soc., 102(3):383–408, 1962.
* [26] M. Kontsevich. Formality conjecture. In J. Rawnsley D. Sternheimer and S. Gutt, editors, Deformation theory and symplectic geometry, pages 139–156, Ascona, 1996. Math. Phys. Stud.
* [27] M. Kontsevich. Deformation Quantization of Poisson Manifolds. Letters in Mathematical Physics, 66:157–216, 2003.
* [28] Y. Kosmann-Schwarzbach. Lie bialgebras, Poisson Lie groups and dressing transformations. In Y. Kosmann-Schwarzbach, B. Grammaticos, and K. M. Tamizhmani, editors, Integrability of Nonlinear Systems. Springer, 2004.
* [29] B. Kostant. Orbits, symplectic structures and representation theory. Proc. US-Japan Seminar on Diff. Geom., Kyoto. Nippon Hyronsha, Tokyo, 77, 1965.
* [30] B. Kostant. The solution to the generalized Toda lattice and representation theory. Adv. Math., 34, 1979.
* [31] A. Lichnerowicz. Les variétés de Poisson et leurs algébres de Lie associées. Journal of Differential Geometry, 12, 1977.
* [32] S. Lie. Theorie der Transformationgruppen. Zweiter Abschnitt. Teubner, Leipzig, 1890.
* [33] J-H. Lu. Multiplicative and Affine Poisson structure on Lie groups. PhD thesis, University of California (Berkeley), 1990.
* [34] J-H. Lu. Momentum mappings and reduction of Poisson actions. In Symplectic geometry, groupoids, and integrable systems, pages 209–226. Math. Sci. Res. Inst. Publ., Berkeley, California, 1991.
* [35] J-H. Lu. Moment Maps at the Quantum level. Communications in Mathematical Physics, pages 389–404, 1993.
* [36] J-H. Lu. Hopf Algebroids and quantum groupoids. International Journal of Mathematics, 7, 1995. arXiv:q-alg/9505024v1.
* [37] S. Majid. Foundations of Quantum Group Theory. Cambridge University Press, 1995.
* [38] J. Marsden and T. Ratiu. Introduction to Mechanics and Symmetry, volume 17. Springer-Verlag, 1999.
* [39] J. E. Marsden and T. Ratiu. Reduction of Poisson manifolds. Letters in Mathematical Physics, 11(2):161–169, 1986.
* [40] J. E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Rep. Math. Phys., 1974.
* [41] J. E. Marsden and A. Weinstein. The Hamiltonian structure of the Maxwell-Vlasov equations. Physica, 1982.
* [42] E. Miranda. From action-angle coordinates to geometric quantization: a round trip. Oberwolfach report, Geometric quantization in the non-compact setting, MFO Oberwolfach reports, 2011.
* [43] J. E. Moyal. Quantum mechanics as a statistical theory. Proc. Cambridge Phyl. Soc., 45:99–124, 1949.
* [44] R. Nest and B. Tsygan. Algebraic index theorem for families. Advances in Math., 113:151–205, 1995.
* [45] E. Noether. Invariant variation problems. Transport Theory and Statistical Physics, 1971.
* [46] S.-D. Poisson. Sur la variation des constantes arbitraires dans les questions de mécanique. J. Ecole Polytechnique, 1809.
* [47] T. Ratiu and J-P. Ortega. Momentum Maps and Hamiltonian Reduction, volume 222. Birkhäuser, 003.
* [48] N. Yu. Reshetikhin. Quantization of Lie bialgebras. Duke Math. J. Intern. Math. Res. Notices, 7, 1992.
* [49] M. A. Semenov-Tian-Shansky. What is a classical r-matrix? Funct. Anal. Appl., 4, 1983.
* [50] M. A. Semenov-Tian-Shansky. Dressing transformations and Poisson Lie group actions. Publ. RIMS, Kyoto University, 21:1237–1260, 1985.
* [51] J. Sniatycki and A. Weinstein. Reduction and quantization for singular momentum map. Letters in mathematical physics, 1983.
* [52] J-M. Souriau. Geométrie de l’espace de phases, calcul des variations et mécanique quantique. Faculté des Sciences de Marseille, 1965.
* [53] T. Tjin. An introduction to quantized Lie groups and algebras. Int. J. Mod. Phys. A, 7:6175–6213, 1992.
* [54] I. Vaisman. Lectures on the Geometry of Poisson Manifolds. Birkhäuser, 1994.
* [55] A. Weinstein. The local structure of Poisson Manifolds. Journal of Differential Geometry, 1983.
* [56] A. Weinstein. Some remarks on dressing transformations. J. Fac. Sci. Univ. Tokyo, 1988.
* [57] H. Weyl. Quantenmechanik und Gruppentheorie. Z. Physics, 46:1–46, 1927.
* [58] E. P. Wigner. Quantum corrections for thermodynamic equilibrium. Phys. Rev., 40:749–759, 1932.
* [59] N. M. J. Woodhouse. Geometric quantization. Oxford Mathematical Monographs. Clarendon Press, 2nd edition, 1992.
* [60] P. Xu. Quantum groupoids. Comm. Math. Phys., 216:539–581, 2001. arXiv:math/9905192v2 [math.QA].
|
arxiv-papers
| 2012-03-19T14:16:15 |
2024-09-04T02:49:28.759577
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chiara Esposito",
"submitter": "Chiara Esposito",
"url": "https://arxiv.org/abs/1203.4112"
}
|
1203.4295
|
# The Inhomogeneous Hall’s Ray
D.J. Crisp, W. Moran and A.D. Pollington
(Date: March 2012)
## 1\. Introduction
The expression
${\mathcal{M}}_{+}(\alpha,\beta)=\liminf_{q\to\infty}q||q\alpha-\beta||$
measures how well multiples of a fixed irrational $\alpha>0$ approximate a
real number $\beta$. A similar concept is defined by Rockett and Szüsz ([26]
Ch. 4, §9), where they consider the slight variant,
${\mathcal{M}}(\alpha,\beta)$, (the _two-sided_ case) with the initial $q$
replaced by $|q|$. It is evident that (see, for example, [23, 20])
${\mathcal{M}}(\alpha,\beta)=\min\bigl{(}{\mathcal{M}}_{+}(\alpha,\beta),{\mathcal{M}}_{+}(\alpha,-\beta)\bigr{)}.$
We define
(1) $\displaystyle{\mathcal{S}}_{+}(\alpha)$
$\displaystyle=\\{{\mathcal{M}}_{+}(\alpha,\beta):\beta\in{\mathbf{R}}^{+}\\}$
$\displaystyle{\mathcal{S}}(\alpha)$
$\displaystyle=\\{{\mathcal{M}}(\alpha,\beta):\beta\in{\mathbf{R}}^{+}\\}.$
We refer to the first set as the _(one-sided) inhomogeneous approximation
spectrum_ of $\alpha$.
${\mathcal{M}}_{+}(\alpha,\beta)$ and the corresponding spectrum have been
considered in precisely this form by various authors, [18, 19, 11, 7], and the
ideas relate to inhomogeneous minima of binary quadratic forms [2, 7, 4, 5, 6,
1, 3]. In the celebrated paper ([15]), Hall showed that the _Lagrange
spectrum_ ,
${\mathcal{L}}=\\{{\mathcal{M}}_{+}(\alpha,0):\alpha\in{\mathbf{R}}\\}$,
contains an interval $[0,\mu_{H}]$ ($\mu_{H}>0$) subsequently called _Hall’s
Ray_. The precise value of $\mu_{H}$ has been determined by Freiman ([13]) in
a heroic calculation; we refer the reader to [10], where this result is
discussed in detail. Our aim here is to prove the existence of an interval
$[0,\mu_{\alpha}]$ in the inhomogeneous spectrum for all irrationals $\alpha$,
though without a precise value for the maximum endpoint of the interval. It is
clear that the result fails for rational $\alpha$.
Since ${\mathcal{M}}_{+}(\alpha,\beta)={\mathcal{M}}_{+}(\alpha,\beta+1)$, the
values of $\beta$ may be restricted to the unit interval $[0,1)$. Similarly,
we may assume without loss of generality that $0\leq\alpha<1$. The key theorem
of this paper is the following:
###### Theorem 1.
For $\alpha$ irrational, the set ${\mathcal{S}}_{+}(\alpha)$ contains an
interval of the form $[0,\mu_{\alpha}]$ for some $\mu_{\alpha}>0$.
Once this is established, it is straightforward to extend to the two-sided
case, and to binary quadratic forms.
### 1.1. History
As far as we are aware, the first work on inhomogeneous minima dates back to
Minkowski [21] who expressed his results in terms of binary quadratic forms.
He showed that if $a,b,c,d$ are real numbers with $\Delta=ad-bc\neq 0$ then,
for any real numbers $\lambda$ and $\mu$, there are integers $m,n$ such that
$|(am-bn-\lambda)(cm-dn-\mu)|\leq\frac{1}{4}\Delta.$
This implies that $\inf_{q}|q|||q\alpha-\beta||\leq\frac{1}{4}$ for all
$\alpha,\beta$. The same conclusion is true for $\mathcal{M}(\alpha,\beta)$
but this requires more work. In fact Khintchine [17] proved that
$\mathcal{M}_{+}(\alpha,\beta)\leq\frac{1}{3}$, and the result with
$\frac{1}{4}$ replacing $\frac{1}{3}$ is claimed by Cassels as derivable from
his methods in [8].
Khintchine [16] showed that there exists $\delta>0$ such that, for any
$\alpha$, there exists $\beta$ for which
$\mathcal{M}(\alpha,\beta)\geq\delta.$
In fact, like Minkowski, he deals with the infimum rather than $\liminf$.
Fukusawa gave an explicit value for $\delta$ of $1/457$ and this was
subsequently improved by Davenport ($\delta=1/73.9$) [12] and by Prasad
($\delta=3/32$) [25]. These papers are of special significance because they
develop a methodology for handling calculations of values of
$\mathcal{M}_{+}(\alpha,\beta)$ that has been the cornerstone of much
subsequent work, and underlies the techniques used in this paper.
Far too many authors have contributed to the understanding of
${\mathcal{M}}(\alpha,\beta)$ and $\mathcal{M}_{+}(\alpha,\beta)$ for us to
reference all of the papers here. As far as we are aware, the first ray
results occur in [14], Satz XIII, where it is shown that if, in a semi-regular
continued fraction expansion of $\alpha$, the partial quotients tend to
$\infty$ then ${\mathcal{S}}(\alpha)$ contains the interval $[0,\frac{1}{4}]$.
Barnes obtains essentially the same result in [2] though he states a weaker
one: that, for each $t\in[0,\frac{1}{4}]$, there are uncountably many
$\alpha$’s and $\beta$’s with ${\mathcal{M}}(\alpha,\beta)=t$.
The predominant methodology for handling problems of this kind, originating
with Davenport [12], invokes some form of continued fraction expansion of
$\alpha$ and a corresponding digit expansion of $\beta$. We will use this
methodology but choose to use the negative continued fraction because of the
simple and “decimal”-like geometrical interpretation of the expansion of
$\beta$ associated with it (which we call the _Davenport Expansion_). Use of
the regular continued fraction is possible, and was first done by Prasad [25],
but makes the construction less intuitive and more complicated from our
perspective, because divisions of subintervals alternate in direction. The
general machinery for the regular continued fraction is well-exposed in
Rockett and Szüsz [26]. Cassels also uses the Davenport expansion ideas in his
paper [8], without attribution, where he shows that, except for special cases,
$\mathcal{M}_{+}(\alpha,\beta)\leq\frac{4}{11}$. Several authors have
contributed to refinement of the technique, including Sós [27], and Cusick,
Rockett and Szüsz [9]. These authors ascribe the origin of the technique to
Cassels in [8].
Almost all of the work for this paper, including a more complicated proof of
the main theorem, was done in the early 1990’s, and versions of it have been
circulating privately since then. Its ideas and results have been used and
cited in various places, in particular, in [23, 24].
## 2\. The negative continued fraction expansion for $\alpha$
Here we briefly describe the features needed from the theory of the negative
continued fraction. For a more complete discussion of the corresponding
concepts for the regular continued fraction, see [26] or for the more general
semi-regular continued fraction see Perron [22]. For $0<\alpha<1$, let
$\alpha_{1}=\alpha$, $a_{1}=\lceil\frac{1}{\alpha_{1}}\rceil$, and define,
recursively,
$a_{i}=\left\lceil\frac{1}{\alpha_{i}}\right\rceil\qquad\text{and}\qquad\alpha_{i+1}=a_{i}-\frac{1}{\alpha_{i}}.$
so that $a_{i}\geq 2$ and $0<\alpha_{i+1}<1$, for all $i$. Evidently, $\alpha$
has the continued fraction expansion
$\alpha=\cfrac{1}{a_{1}-\cfrac{1}{a_{2}-\cfrac{1}{a_{3}-\cfrac{1}{\ddots}}}},$
abbreviated as $\alpha=\langle a_{1},a_{2},a_{3},\ldots\rangle$. The numbers
$\alpha_{i}$ are called the $i$th _complete quotients_ of $\alpha$ and satisfy
$\alpha_{i}=\langle a_{i},a_{i+1},a_{i+2},\ldots\rangle.$
Since $\alpha$ is irrational, the _partial quotients_ $a_{i}$ are greater than
$2$ for infinitely many indices $i$, and so there is a unique sequence
$a^{\prime}_{1},a^{\prime}_{2},a^{\prime}_{3},\ldots$ of positive integers
such that
(2)
$a_{1},a_{2},a_{3},\ldots=a^{\prime}_{1}+1,\underbrace{2,\ldots,2}_{a^{\prime}_{2}-1},a^{\prime}_{3}+2,\underbrace{2,\ldots,2}_{a^{\prime}_{4}-1},a^{\prime}_{5}+2,\underbrace{2,\ldots,2}_{a^{\prime}_{6}-1},a^{\prime}_{7}+2,\ldots.$
It will be necessary occasionally to discuss the usual continued fraction
expansion of $\alpha$, now expressible as
(3)
$\alpha=\cfrac{1}{a^{\prime}_{1}+\cfrac{1}{a^{\prime}_{2}+\cfrac{1}{a^{\prime}_{3}+\cfrac{1}{\ddots}}}}.$
Eventually, we will split the proof of Theorem 1 into two cases, corresponding
to whether or not the sequence $(a_{n}^{\prime})$ is bounded.
We make use of the (negative continued fraction) _convergents_ $p_{i}/q_{i}$
to $\alpha$:
(4) $\frac{p_{i}}{q_{i}}=\langle a_{1},a_{2},\ldots,a_{i}\rangle,$
satisfying the recurrence relations
(5) $p_{i+1}=a_{i+1}p_{i}-p_{i-1}\qquad\text{and}\qquad
q_{i+1}=a_{i+1}q_{i}-q_{i-1}$
where $i\geq 1$ and $p_{0},\ q_{0}=0,\ 1$. Easily established are the
following simple properties:
(6) $\displaystyle 1$ $\displaystyle=p_{i}q_{i-1}-q_{i}p_{i-1}$ (7)
$\displaystyle\alpha$
$\displaystyle=\frac{(a_{i}-\alpha_{i+1})p_{i-1}-p_{i-2}}{(a_{i}-\alpha_{i+1})q_{i-1}-q_{n-2}}=\frac{p_{i}-\alpha_{i+1}p_{i-1}}{q_{i}-\alpha_{i+1}q_{i-1}}.$
Moreover, $q_{i-1}/q_{i}=\overline{\alpha}_{i}$ where
(8) $\overline{\alpha}_{i}=\langle a_{i},a_{i-1},\ldots,a_{1}\rangle.$
Since $q_{0}=1$, the identity
(9)
$q_{i}=\frac{1}{\overline{\alpha}_{1}\overline{\alpha}_{2}\ldots\overline{\alpha}_{i}}$
follows.
This section concludes with a brief description of the _Ostrowski expansion_
(see [26]) for positive integers. Any given integer $q\geq 1$ can be written
as a sum of the form
(10) $q=\sum^{n}_{k=1}c_{k}q_{k-1}$
where
(11) $c_{n}\geq 1\qquad\text{and}\qquad 0\leq c_{k}\leq
a_{k}-1\qquad\text{for}\qquad 1\leq k\leq n.$
A greedy algorithm is used to determine the coefficients $c_{n}$.
It is not hard to verify that
(12)
$q_{k}-1=(a_{1}-2)q_{0}+(a_{2}-2)q_{1}+\cdots+(a_{k-1}-2)q_{k-2}+(a_{k}-1)q_{k-1}.$
This last identity yields that, for no pair of indices $i$ and $j$, is there a
consecutive subsequence of coefficients of the form
(13)
$(c_{i},c_{i+1},\dots,c_{j})=(a_{i}-1,a_{i+1}-2,a_{i+2}-2,\ldots,a_{j-1}-2,a_{j}-1).$
The basic facts about the Ostrowski expansion are described in the following
lemma.
###### Lemma 2.1.
Each integer $q\geq 1$ has a unique expansion of the form (10) such that the
constraint (11) holds and no consecutive sub-sequence of coefficients is of
the form (13).
## 3\. The Davenport expansion of $\beta$
We now describe the _Davenport Expansion_ for the elements $\beta$ of the
interval $[0,1)$. While the expansion is analogous to that used in [11], we
remind the reader that it is based on a different continued fraction
algorithm. This approach results in a “decimal”-like geometry of the Davenport
expansion in the negative continued fraction case which makes more intuitive
the invocation of Hall’s theorem on sums of Cantor sets [15] later. This is a
key component of the proof in the bounded case.
For $0\leq\beta<1$, let $\beta_{1}=\beta$ and define, inductively,
$b_{i}=\left\lfloor\frac{\beta_{i}}{\alpha_{i}}\right\rfloor\qquad\text{and}\qquad\beta_{i+1}=\frac{\beta_{i}}{\alpha_{i}}-b_{i}.$
so that $0\leq b_{i}\leq a_{i}-1$ and $0\leq\beta_{i+1}<1$. The convergent sum
$\beta=\sum_{k=1}^{\infty}b_{k}D_{k}$ is called the _Davenport expansion_ of
$\beta$ or the _Davenport sum_ of the sequence $(b_{k})$ relative to $\alpha$.
The integers $b_{i}$ are the _Davenport coefficients_. In the same way as in
the decimal expansion $0.999\ldots$ is identified with $1.000\ldots$, we
identify
(14) $b_{1},b_{2},\ldots,b_{i},a_{i}-1,a_{i+1}-2,a_{i+2}-2,\ldots,\text{ with
}b_{1},b_{2},\ldots,b_{i}+1,0,0,\ldots$
for $b_{i}<a_{i}-1$, since their Davenport sums are the same.
Figure 1 gives an illustration of the geometry of the situation for the case
when $\alpha=\langle 5,3,5,3,\ldots\rangle$. The interval $[0,1)$ is
subdivided by the numbers $n\alpha\pmod{1}$, $(n=1,2,3,4)$ into $5$ intervals,
the first four of which are “long” and the last “short” since $5\alpha>1$.
When we allow $n$ to range up to $13$, each long interval is then subdivided
into $3$ intervals with the same pattern in each: $2$ “long” intervals and $1$
“short” interval, whereas the “short” interval is divided into just $1$ “long”
interval and $1$ “short” interval. This pattern of “long” and “short”
intervals is repeated at finer and finer resolutions as $n$ increases,
reflecting, in this example, the periodic structure of the continued fraction.
This structure corresponds to a “decimal” expansion with restrictions on
digits, involving dependencies on the preceding digits. The general case is
described below.
Figure 1. The “Long-Short” Picture for $\alpha=\langle 5,3,5,3,\dots\rangle$
From the inductive step in the Davenport expansion,
$\beta_{i}=b_{i}\alpha_{i}+\beta_{i+1}\alpha_{i}$
and, as a result,
(15)
$\beta_{i}=b_{i}\alpha_{i}+b_{i+1}\alpha_{i}\alpha_{i+1}+\ldots+b_{j}(\alpha_{i}\alpha_{i+1}\ldots\alpha_{j})+\beta_{j+1}(\alpha_{i}\alpha_{i+1}\ldots\alpha_{j})$
for all $j\geq i$. Note that $\beta_{i}$ is the location of $\beta$ in the
rescaled copy of the (long) interval in which it is contained. We define
$D_{1}=1,\qquad D_{i}=\alpha_{1}\alpha_{2}\ldots\alpha_{i}$
and write
$\beta_{i}D_{i-1}=b_{i}D_{i}+b_{i+1}D_{i+1}+\cdots+b_{j}D_{j}+\beta_{j+1}D_{j}.$
$D_{i}$ is the length of the long intervals at the $i$th level, and
$D_{i}-D_{i+1}$ is the length of the short intervals at that level.
The following result is straightforward.
###### Theorem 2.
Let $\beta=\sum^{\infty}_{k=1}b_{k}D_{k}$ where $(b_{i})$ is a sequence of
positive integers. Then $0\leq\beta<1$ and $(b_{i})$ are the Davenport
coefficients of $\beta$ if and only if $b_{i}<a_{i}$ for all $i\geq 1$ and no
block of the form
(16) $a_{i}-1,a_{i+1}-2,a_{i+2}-2,\dots,a_{j-1}-2,a_{j}-1$
or of the form
(17) $a_{i}-1,a_{i+1}-2,a_{i+2}-2,a_{i+3}-2,\ldots$
occurs in $(b_{i})$.
The exceptional cases in this result; when $b_{i},b_{i+1},\ldots,b_{j}$ is of
the form $a_{i}-1,a_{i+1}-2,a_{i+2}-2,\dots,a_{j-1}-2,a_{j}-1$, correspond to
the missing long intervals in the short intervals one level higher. As in the
example in Figure 1, each short interval has one fewer long interval at the
next level. In the general geometric picture, $a_{1}-1$ multiples of $\alpha$
subdivide the unit interval into $a_{1}$ intervals, the first $a_{1}-1$ of
which have length $\alpha$ and the last of length $1-(a_{1}-1)\alpha$. The
next multiple (modulo 1) is $\alpha_{1}\alpha_{2}=a_{1}\alpha-1$. This
subdivides each of the long intervals at the previous level into $a_{2}-2$
intervals of the same length followed by a short interval. The final short
interval of the initial subdivision is subdivided into $a_{2}-2$ long
intervals followed by a short interval. This pattern is repeated at all finer
resolutions with the appropriate partial quotients.
By means of the Davenport expansion, we can describe the integer pairs $(p,q)$
for which $0<q\alpha-p<1$. It is straightforward to see that if
$q=\sum^{n}_{k=1}c_{k}q_{k-1}$ is the Ostrowski expansion of $q$ then
(18) $p=\sum^{n}_{k=1}c_{k}p_{k-1},\quad i\geq 1.$
###### Lemma 3.1.
1. (1)
Let $q\geq 1$ be an integer with Ostrowski expansion as in (10) and let $p$ be
defined by (18). Then $0<q\alpha-p<1$ and
$q\alpha-p=\sum^{\infty}_{k=1}b_{k}D_{k}$
is the Davenport expansion of $q\alpha-p$, where $(b_{i})$ is the sequence
$c_{1},c_{2},\ldots,c_{n},0,0,0,\ldots$.
2. (2)
Let $0<\beta<1$ and let $(b_{i})$ be the Davenport coefficients of $\beta$.
Then there are integers $q\geq 1$ and $p$ such that $\beta=q\alpha-p$ if and
only if there is $n\geq 1$ such that $b_{i}=0$ for all $i>n$. Further, if that
is so then $q=\sum^{n}_{k=1}b_{k}q_{k-1}$ and $p=\sum^{n}_{k=1}b_{k}p_{k-1}$.
## 4\. Calculation of ${\mathcal{M}}_{+}(\alpha,\beta)$ via the Davenport
Expansion
The Davenport expansion will be used to calculate
${\mathcal{M}}_{+}(\alpha,\beta)$. Again we stress that the underlying ideas
are not really new, being essentially contained in the work of Davenport,
Cassels, Sós, and others. Accordingly, we omit much of the justification and
instead aim to provide geometrical insights.
To begin, let $0\leq\beta<1$ and let $(b_{i})$ be the Davenport coefficients
of $\beta$. We define
$\displaystyle Q_{n}$ $\displaystyle=\sum^{n}_{k=1}b_{k}q_{k-1}$
$\displaystyle Q^{\prime}_{n}$
$\displaystyle=\begin{cases}Q_{n}+q_{n-1}&\text{ if $Q_{n}<q_{n}-q_{n-1}$}\\\
Q_{n}+q_{n-1}-q_{n}&\text{ if $Q_{n}\geq q_{n}-q_{n-1}$}\end{cases}$
for all $n\geq 1$. The two cases here correspond to when $\beta$ lies in a
long or a short interval, respectively, at the appropriate level of the
decomposition of the interval. If $\beta$ is in a short interval, then the
right endpoint of that interval occurred earlier in the decomposition; hence
the $q_{n}-q_{n-1}$ term.
The next two lemmas are relatively straightforward consequences of these
definitions and ideas.
###### Lemma 4.1.
1. (1)
$0\leq Q_{n}<q_{n}$ for all $n\geq 1$ and $Q_{n}\geq q_{n-1}$ if and only if
$b_{n}\neq 0$.
2. (2)
$Q_{n}\geq Q_{n-1}$ for all $n\geq 2$ and $Q_{n-1}=Q_{n}$ if and only if
$b_{n}=0$.
3. (3)
$0\leq Q^{\prime}_{n}<q_{n}$ for all $n\geq 1$ and $Q^{\prime}_{n}\geq
q_{n-1}$ if and only if $Q_{n}<q_{n}-q_{n-1}$.
4. (4)
$Q^{\prime}_{n}\geq Q^{\prime}_{n-1}$ for all $n\geq 2$ and
$Q^{\prime}_{n-1}=Q^{\prime}_{n}$ if and only if $Q_{n}\geq q_{n}-q_{n-1}$.
5. (5)
The inequality $Q_{n}\geq q_{n}-q_{n-1}$ holds if and only if there is some
index $m$ with $1\leq m\leq n$ such that the sequence
$b_{m},b_{m+1},\ldots,b_{n}$ is equal to
$a_{m}-1,a_{m+1}-2,a_{m+2}-2,\ldots,a_{n}-2.$
The last condition, $Q_{n}\geq q_{n}-q_{n-1}$, occurs if the point $\beta$ is
inside a short interval.
The integers $Q_{n}$ and $Q^{\prime}_{n}$ are used to define quantities
$\lambda_{n}(\beta)$ and $\rho_{n}(\beta)$, the significance of which will be
evident from the following lemma.
###### Definition 4.1.
Let $0\leq\beta<1$ and let $\beta_{1},\beta_{2},\beta_{3},\ldots$ be the
sequence of numbers generated by applying the Davenport expansion algorithm to
$\beta$. We define
(19) $\lambda_{n}(\beta)=Q_{n}D_{n}\beta_{n+1}$
and
(20) $\rho_{n}(\beta)=\begin{cases}Q^{\prime}_{n}D_{n}(1-\beta_{n+1})&\text{
if $Q_{n}<q_{n}-q_{n-1}$}\\\
Q^{\prime}_{n}D_{n}(1-\alpha_{n+1}-\beta_{n+1})&\text{ if $Q_{n}\geq
q_{n}-q_{n-1}$}\end{cases}$
for all $n\geq 1$.
Recall that $Q_{n}$ is the “count” of $q\alpha$ that corresponds the left
endpoint of the interval at level $n$ that contains $\beta$, and that $D_{n}$
is the length of a long interval at that level. It follows that
$D_{n}\beta_{n+1}$ is the distance to $\beta$ from the left endpoint of the
interval at level $n$ containing $\beta$. In similar vein, $\rho_{n}(\beta)$
is the count for the right endpoint of that interval multiplied by the
distance from $\beta$ to that endpoint. The next lemma is straightforward from
the geometrical picture of the interval decompositions.
###### Lemma 4.2.
Let $n<m$, $0<\beta<1$, and $(b_{i})$ be the Davenport coefficients of
$\beta$, with $b_{i}\neq 0$ for infinitely many $i$. Then
1. (1)
$\lambda_{n}(\beta)=Q_{n}||Q_{n}\alpha-\beta||\qquad\text{and}\qquad\rho_{n}(\beta)=Q^{\prime}_{n}||Q^{\prime}_{n}\alpha-\beta||.$
2. (2)
If $b_{n}=0$ then $\lambda_{n}(\beta)=\lambda_{n-1}(\beta)$. In other words if
$\beta$ is in the first interval of the decomposition at level $n$, then
$Q_{n}\alpha$ is $Q_{n+1}\alpha$ modulo 1.
3. (3)
If $b_{n}\neq 0$ and $b_{m}\neq 0$ and $b_{i}=0$ for all $i$ which satisfy
$n<i<m$ then $q_{n-1}D_{m}\leq\lambda_{n}(\beta)<q_{n}D_{m-1}$.
4. (4)
If $Q_{n}\geq q_{n}-q_{n-1}$ then $\rho_{n}(\beta)=\rho_{n-1}(\beta)$. In
other words, if $\beta$ is a short interval (namely a rightmost) at level $n$
then $Q_{n}^{\prime}\alpha$ is equal to $Q_{n}\alpha$ modulo 1.
5. (5)
If $Q_{n}<q_{n}-q_{n-1}$ and $Q_{m}<q_{m}-q_{m-1}$ and $Q_{i}\geq
q_{i}-q_{i-1}$ for all $i$ which satisfy $n<i<m$ then
$q_{n-1}D_{m}(1-\alpha_{m+1})\leq\rho_{n}(\beta)<q_{n}D_{m-1}(1-\alpha_{m})$
unless $m=n+1$ in which case
$q_{n-1}D_{n+1}(1-\alpha_{n+2})\leq\rho_{n}(\beta)<q_{n}D_{n}$.
The next lemma is a key step in calculating ${\mathcal{M}}_{+}(\alpha,\beta)$
in terms of $\lambda_{n}(\beta)$ and $\rho_{n}(\beta)$.
###### Lemma 4.3.
For $n\geq 1$,
(21)
$\min\\{\lambda_{n}(\beta),\rho_{n}(\beta),\lambda_{n+1}(\beta),\rho_{n+1}(\beta)\\}$
is a lower bound for the infimum of the set $\\{q||q\alpha-\beta||:\;q_{n}\leq
q<q_{n+1}\\}$.
###### Proof.
We sketch the proof of the result. The diagram showing the key ideas is given
in Figure 2.
Figure 2. The Approximations of $\beta$
Write $I_{n}$ and $I_{n+1}$ for the intervals prescribed by the Davenport
expansion at level $n$ and $n+1$ that contain $\beta$:
$I_{n}=[Q_{n}\alpha,Q_{n}^{\prime}\alpha]$,
$I_{n+1}=[Q_{n+1}\alpha,Q_{n+1}^{\prime}\alpha]$. The obvious candidates for
the smallest values of $q||q\alpha-\beta||$ for $q_{n}\leq q\leq q_{n+1}$ are
the cases $q=Q_{n+1}$ or $q=Q_{n+1}^{\prime}$ — the left and right endpoints
of the interval $I_{n+1}$ at level $n+1$ containing $\beta$. It is clear from
fairly straightforward size considerations that they do better than any
$q\alpha\in I_{n}\ (q_{n}\leq q\leq q_{n+1})$. It is also clear that the
candidates $q=Q_{n}$ and $q=Q^{\prime}_{n}$ are better than any
$q\alpha\not\in I_{n}(q_{n}\leq q\leq q_{n+1})$ since $Q_{n}<q_{n}$. ∎
The key equation for calculation of ${\mathcal{M}}_{+}(\alpha,\beta)$ is in
the following theorem, which captures the important ingredient of the
preceding lemma.
###### Theorem 3.
If $0<\beta<1$ and no integers $q\geq 1$ and $p$ satisfy $\beta=q\alpha-p$
then
(22)
${\mathcal{M}}_{+}(\alpha,\beta)=\min\left\\{\liminf_{n\to\infty}\lambda_{n}(\beta),\;\liminf_{n\to\infty}\rho_{n}(\beta)\right\\}.$
For completeness, we note that, in Theorem 3, we have not dealt with the
possibility that $\beta$ is of the form $q\alpha-p$ where $q$ and $p$ are
positive integers. In this case, we have
(23)
${\mathcal{M}}_{+}(\alpha,\beta)=\liminf_{q^{\prime}\to\infty}q^{\prime}||q^{\prime}\alpha-q\alpha-p||=\liminf_{q^{\prime}\to\infty}q^{\prime}||(q^{\prime}-q)\alpha||$
and consequently
(24)
${\mathcal{M}}_{+}(\alpha,\beta)=\liminf_{q^{\prime}\to\infty}(q^{\prime}-q)||(q^{\prime}-q)\alpha||={\mathcal{M}}_{+}(\alpha,0).$
The quantity ${\mathcal{M}}_{+}(\alpha,0)$ it is, of course, the homogeneous
approximation constant of $\alpha$.
## 5\. The Unbounded Case
In this section we dispense quickly and relatively straightforwardly with the
case where $\alpha$ has unbounded partial quotients ($a^{\sharp}_{n}$) in its
ordinary continued fraction, before turning to the much more difficult case of
bounded partial quotients. We write
$\mathcal{M}_{+}(\alpha)=\sup_{\beta}{\mathcal{M}}_{+}(\alpha,\beta).$
The following theorem is the key result of this section.
###### Theorem 4.
If $\alpha$ has unbounded partial quotients in its ordinary continued fraction
then
$\\{\mathcal{M}_{+}(\alpha,\beta):\beta\in{\mathbf{R}}\\}=[0,\mathcal{M}_{+}(\alpha)].$
In this case, we avoid the problems of long sequences of $2$s in the negative
continued fraction by making use of the Davenport expansion of $\beta$ with
respect to $\alpha$ using the ordinary continued fraction. This theory is
described in Rockett and Szüsz [26] with a different notation. The notation we
use is largely that of Cassels [8] with $\sharp$ appended to indicate use of
the ordinary continued fraction but with $D^{\sharp}_{n}$ denoting the
quantity he refers to as $\epsilon_{n}$.
Note that, when $\beta=n\alpha+m$ for $n$ and $m$ integers,
$\mathcal{M}(\alpha,\beta)=0$. Accordingly, we restrict attention to $\beta$
not of this form.
Set $\alpha=[0;a^{\sharp}_{1},a^{\sharp}_{2},...]$ and let $(n_{k})$ be a
sequence of indices on which the partial quotients are strictly monotonically
increasing. Now let $0<c<\mathcal{M}_{+}(\alpha)$ and choose $\beta$ for which
$c<\mathcal{M}_{+}(\alpha,\beta)\leq\mathcal{M}_{+}(\alpha)$. Let its
Davenport coefficients be $(b^{\sharp}_{j})$ in the ordinary continued
fraction. We will construct a sequence $(c^{\sharp}_{j})$ so that
$c^{\sharp}_{j}=b^{\sharp}_{j}$ except on a subsequence of the $n_{k}$ which
will be chosen sufficiently sparse for our purposes.
Since $\beta=\sum_{k=1}^{\infty}b^{\sharp}_{k}D^{\sharp}_{k}$, where
$D^{\sharp}_{k}=q^{\sharp}_{k-1}\alpha-p^{\sharp}_{k-1}$, we put
$\lambda^{\sharp}_{n}(\beta)=Q^{\sharp}_{n}\|Q^{\sharp}_{n}\alpha-\beta\|$
The ordinary case of (22) (see[8] or [26]) gives
(25)
$\lambda^{\sharp}_{n}(\beta)=(\sum_{k=1}^{n}b^{\sharp}_{k}q^{\sharp}_{k-1})|\sum_{k={n+1}}^{\infty}b^{\sharp}_{k}D^{\sharp}_{k}|\\\
=q^{\sharp}_{n}|D^{\sharp}_{n}|(b^{\sharp}_{n}\frac{q^{\sharp}_{n-1}}{q^{\sharp}_{n}}+b^{\sharp}_{n-1}\frac{q^{\sharp}_{n-2}}{q^{\sharp}_{n-1}}\frac{q^{\sharp}_{n-1}}{q^{\sharp}_{n}}+...)|b^{\sharp}_{n+1}\frac{D^{\sharp}_{n+1}}{D^{\sharp}_{n}}+b^{\sharp}_{n+2}\frac{D^{\sharp}_{n+2}}{D^{\sharp}_{n}}+...|$
Note that
$q^{\sharp}_{n}|D^{\sharp}_{n}|=[a^{\sharp}_{n},a^{\sharp}_{n+1},...]/[a^{\sharp}_{n},a^{\sharp}_{n-1},a^{\sharp}_{n-2},...,a^{\sharp}_{2},a^{\sharp}_{1}]$,
and so is absolutely bounded above and away from zero. For this choice of
$\beta$, this product is always at least $1/30$ and so the second two terms in
the product are each at least $1/60$. Changing the value of $b^{\sharp}_{n}$
by 1 will change the value of $\lambda^{\sharp}_{n}$ by at most
$1/(a^{\sharp}_{n}-1)$, so by choosing $n=n_{k}$ and adjusting the value of
$b^{\sharp}_{n_{k}}$ to $c_{n_{k}}$, we replace $\beta$ by
$\widetilde{\beta}$, so that
$c<\min(\lambda_{n}(\widetilde{\beta}),\lambda_{n-1}(\widetilde{\beta}))<c+2/a^{\sharp}_{n}.$
By making this change at the indices $n_{k}$ (so that
$a^{\sharp}_{n_{k}}\to\infty$), and putting $c^{\sharp}_{n}=b^{\sharp}_{n}$
elsewhere, we obtain a number $\gamma=\sum_{k}c^{\sharp}_{k}D^{\sharp}_{k}$
for which
$\mathcal{M}_{+}(\alpha,\gamma)=c,$
since the effect of these changes for other $\lambda_{n}$ is smaller than that
at $n=n_{k}$ or $n=n_{k-1}$. In fact we have:
###### Lemma 5.1.
Let $\beta$ have Davenport coefficients $(b^{\sharp}_{i})$ in the ordinary
continued fraction. Given any $\epsilon>0$ and $k$ sufficiently large, there
is an $M=M(k)<k/2$ and $N=N(k)$ such that if $m\not\in(k-M,k+N)$ then any
change in $b^{\sharp}_{k}$ will not change $\lambda^{\sharp}_{m}(\beta)$ or
$\rho^{\sharp}_{m}(\beta)$ by more than $\epsilon$.
###### Proof.
This follows quickly by (25), since $\alpha$ must have infinitely many partial
quotients in its continued fraction expansion which are larger than $2$. If
$k$ is sufficiently large then there are at least $-\log{\epsilon}/\log 2$
such terms $a^{\sharp}_{n}$ in $n\in[k/2,k)$ and at least
$-\log{\epsilon}/\log 2$ such terms $a^{\sharp}_{n}$ in $(k,k+N]$.
Consequently any change in $b_{k}$ will make a variation in the value of
$\lambda^{\sharp}_{m}(\beta)$ and $\rho^{\sharp}_{m}(\beta)$ less than
$\epsilon$.
We now choose a sequence of the $n_{k}$ which are sufficiently sparse that
these intervals do not overlap. Choose $c_{n_{k}}$ so that
$c<\min_{j\in[n_{k}-M(n_{k}),n_{k}+N(n_{k}]}(min(\lambda^{\sharp}_{j}(\gamma),{\rho^{\sharp}_{j}}(\gamma))<c+2/a^{\sharp}_{n_{k}}.$
This is clearly possible using the fact that changing $b_{k}$ by 1 increases
or decreases the expression in (25) by no more than $1/(a^{\sharp}_{n}-1)$.
This completes the proof of the fact that for such well approximable $\alpha$
the spectrum consists of a single ray. ∎
## 6\. The Bounded Case
In the light of results of the previous section, we restrict attention from
this point to the case where the ordinary continued has bounded partial
quotients ($a^{\sharp}_{n}$). This translates in the case of the negative
continued fraction to the sequence $a_{1},a_{2},a_{3},\ldots$ being bounded,
with least upper bound $M$, and the lengths of the blocks of consecutive $2$’s
also being bounded with least upper bound $N-1\geq 0$. Then it follows from
equations (4) and (5) that
(26) $\frac{1}{M}<\overline{\alpha}_{i}<\frac{N}{N+1}$
hold for all $i\geq 1$. We choose $L$ to be the smallest integer such that
(27)
$\Bigl{(}\frac{N}{N+1}\Bigr{)}^{L}\leq\frac{(1-\frac{N}{N+1})(1-\frac{N^{2}}{(N+1)^{2}})}{M^{N}(M^{2}-1)}.$
The numbers $N$ and $L$ will figure significantly in the proof in the bounded
case.
### 6.1. Computation of $\mathcal{M}_{+}(\alpha,\beta)$
We will define a collection of $\beta$’s, in terms of their Davenport
coefficients, which $\beta$ have the property that for some subsequence
$(k(i))$ of positive integers
(28)
${\mathcal{M}}_{+}(\alpha,\beta)=\liminf_{i\to\infty}\lambda_{k(i)}(\beta).$
This enables us to work with just the $\lambda_{k(i)}$ rather than the
$\rho_{n}$ and simplifies the rest of the proof of our main theorem. We assume
throughout the remainder of the proof of the bounded partial quotient case
that $\beta\neq n\alpha+m$ for some integers $n$ and $m$.
We record some simple results in the following lemma.
###### Lemma 6.1.
1. (1)
For $i<j$,
(29)
$q_{i}D_{j}=\frac{\alpha_{i+1}\alpha_{i+2}\ldots\alpha_{j}}{1-\overline{\alpha}_{i}\alpha_{i+1}}.$
2. (2)
Let $r$ and $s$ be positive integers satisfying $r\geq sL$. Then
$q_{u}D_{v-1}<q_{n-1}D_{m}(1-\alpha_{m+1})<q_{n-1}D_{m}$
whenever $u$, $v$, $n$ and $m$ are positive integers with $u+r<v$ and $n<m\leq
n+s+N$.
###### Proof.
The first part is a simple calculation. For the second part, note that the
right inequality is obviously true since $0<\alpha_{m+1}<1$. To prove the left
inequality we observe that (29) implies
$q_{u}D_{v-1}=\frac{\alpha_{u+1}\alpha_{u+2}\dots\alpha_{v-1}}{1-\overline{\alpha}_{u}\alpha_{u+1}}.$
Using (26), (8) and $u+r<v$, we have
$q_{u}D_{v-1}<\frac{R^{v-u-1}}{1-R^{2}}\leq\frac{R^{r}}{1-R^{2}},$
where $R=N/(N+1)$. Similarly,
$\displaystyle q_{n-1}D_{m}(1-\alpha_{m+1})$
$\displaystyle=\frac{\alpha_{n}\alpha_{n+1}\dots\alpha_{m}(1-\alpha_{m+1})}{1-\overline{\alpha}_{n-1}\alpha_{n}}$
$\displaystyle>\frac{M^{-(s+N+1)}(1-R)}{1-M^{-2}}.$
The lemma is, therefore, true if
$\frac{R^{r}}{1-R^{2}}\leq\frac{M^{-(s+N+1)}(1-R)}{1-M^{-2}}$
Since $r\geq sL$ and $R<1$ and $s\geq 1$ and $R^{L}M<1$ we have
$R^{r}M^{s-1}<R^{sL}M^{s-1}<R^{L}$ and the result follows immediately from the
definition of $L$. ∎
###### Theorem 5.
Choose positive integers $r$ and $s$ with $r\geq sL$, and an increasing
sequence of indices $(k(i))$ with $k(i+1)>k(i)+r$. Let $0<\beta<1$ with
Davenport coefficients $(b_{i})$ satisfy:
1. (1)
for each $i\geq 1$ the sequence $b_{k(i)+1},b_{k(i)+2},\ldots,b_{k(i)+r}$ is a
block of $r$ zeros;
2. (2)
there is no block of $N+s$ consecutive zeros in $(b_{n})$ between $k(i)+r$ and
$k(i+1)$,
3. (3)
$\beta$ is not in short intervals at level $n$ for $N+s$ consecutive values of
$n$, in other words the Davenport coefficients of $\beta$ contain no sequence
of the form $a_{j}-1,a_{j+1}-2,\ldots,a_{j+N+s-1}-2$.
then
${\mathcal{M}}_{+}(\alpha,\beta)=\liminf_{i\to\infty}\lambda_{k(i)}(\beta).$
###### Proof.
By Theorem 3, it is enough to show that
$\lambda_{n}(\beta)\geq\lambda_{k(i)}(\beta)\qquad\text{or}\qquad\lambda_{n}(\beta)\geq\lambda_{k(i+1)}(\beta)$
and
(30) $\rho_{n}(\beta)\geq\lambda_{k(i)}(\beta)$
for all integers $n$ with $k(i)\leq n<k(i+1)$, for $i$ sufficiently large. We
choose $i>i_{0}$ to ensure that some $b_{j}\neq 0$ for some $j<i_{0}$ and that
$\beta$ has appeared in a long interval before that stage. If this were not
possible $\beta$ would be a multiple of $\alpha$ modulo 1. Now fix $n$ between
$k(i)$ and $k(i+1)$. We will liberally use the fact stated in Lemma 4.2 that
we can move back and forth between $\lambda_{n}(\beta)$ and
$\lambda_{m}(\beta)$ provided the intervening $b_{k}$ are all zero. Similarly,
at the other extreme, we could move back and forward between $\rho_{n}(\beta)$
and $\rho_{m}(\beta)$ provided that at the intervening levels $\beta$ is in
short intervals.
Choose $u\leq k(i)<v$ to be such that $b_{j}=0$ if $u<j<v$ and to be the
extreme integers with that property. We observe that $v-u>r$. It follows from
Lemma 4.2 that
$\lambda_{k(i)}<q_{u}D_{v-1}.$
If $n<k(i)+r$ then $\lambda_{k(i)}=\lambda_{n}$. If not, then $b_{n}$ is
followed by a block of at most $N+s$ zeros unless $b_{m}=0$ for all $m$ with
$n<m<k(i+1)$, in which case $\lambda_{k(i+1)}=\lambda_{n}$. If
$\lambda_{k(i)}\neq\lambda_{n}\neq\lambda_{k(i+1)}$ then
$q_{n-1}D_{m}\leq\lambda_{n}(\beta),$
for some $m\leq n+N+s$. That $\lambda_{k(i)}(\beta)\leq\lambda_{n}(\beta)$
follows from
$q_{u}D_{v-1}\leq q_{n-1}D_{m}.$
which follows immediately from $m\leq n+s+N$ and $u+r<v$.
The argument to show that (30) holds when $k(i)\leq n<k(i+1)$ is similar but
uses the fact that $\beta$ is not in a long sequence of consecutive short
intervals. ∎
### 6.2. Elements of ${\mathcal{S}}_{+}(\alpha)$
Now we give a construction for certain elements of ${\mathcal{S}}_{+}(\alpha)$
using Theorem 5. First we impose additional constraints on the sequence
$(k(i))$ so that the limits of the sequences $\overline{\alpha}_{k(i)}$ (8)
and $\alpha_{k(i)+1}$ both exist. Moreover, the limits lie strictly between
$0$ and $1$, since (26) and (8) hold for all $i\geq 1$ and $0<1/M<N/(N+1)<1$.
The collection of $\beta$ to be described in terms of their Davenport
expansions will be the ones for which $\mathcal{M}(\alpha,\beta)$ are in the
Hall’s Ray.
###### Definition 6.1.
We choose $(K(i))$ be an increasing sequence of indices with gaps
$K(i+1)-K(i)$ tending to infinity such that the limits
(31) $\displaystyle a_{1}^{-},a_{2}^{-},a_{3}^{-},\ldots$
$\displaystyle=\lim_{i\to\infty}a_{K(i)},a_{K(i)-1},\ldots,a_{2},a_{1}\ldots$
$\displaystyle a_{1}^{+},a_{2}^{+},a_{3}^{+},\ldots$
$\displaystyle=\lim_{i\to\infty}a_{K(i)+1},a_{K(i)+2},a_{K(i)+3},\ldots,$
exist; that is, that in each case the sequence of integers eventually becomes
constant. The existence of such a sequence follows quickly by a diagonal
argument from the finiteness of the alphabet from which the $a_{i}$’s are
chosen.
We write
(32) $\alpha^{-}=\langle
a_{1}^{-},a_{2}^{-},a_{3}^{-},\ldots\rangle\qquad\text{and}\qquad\alpha^{+}=\langle
a_{1}^{+},a_{2}^{+},a_{3}^{+},\ldots\rangle.$
The following lemma is a straightforward consequence of the properties of the
sequence $a_{1},a_{2},a_{3},\ldots$
###### Lemma 6.2.
Each of the sequences $\alpha^{-}$ and $\alpha^{+}$ have all of their terms
less than or equal to $M$ and contain no block of $N$ consecutive $2$’s.
Evidently, the numbers $\alpha^{-}$ and $\alpha^{+}$ are irrational with $0<\
\alpha^{\pm}<1$, and the partial quotients of their regular continued fraction
expansions satisfy (26).
All of the theory in the preceding sections is applicable to $\alpha^{-}$ or
$\alpha^{+}$ in place of $\alpha$. We introduce the following notation. For
$i\geq 1$, define
(33) $\alpha^{-}_{i}=\langle
a^{-}_{i},a^{-}_{i+1},a^{-}_{i+2},\ldots\rangle\qquad\text{and}\qquad\alpha^{+}_{i}=\langle
a^{+}_{i},a^{+}_{i+1},a^{+}_{i+2},\ldots\rangle$
and set
(34)
$D^{-}_{i}=\alpha^{-}_{1}\alpha^{-}_{2}\ldots\alpha^{-}_{i}\qquad\text{and}\qquad
D^{+}_{i}=\alpha^{+}_{1}\alpha^{+}_{2}\ldots\alpha^{+}_{i}.$
It follows from (32), (33), and the discussion above that
$\alpha^{-}_{k}=\lim_{i\to\infty}\overline{\alpha}_{K(i)-k+1}\qquad\text{and}\qquad\alpha^{+}_{k}=\lim_{i\to\infty}\alpha_{K(i)+k}.$
Hence
$\displaystyle D^{-}_{k}$
$\displaystyle=\lim_{i\to\infty}\overline{\alpha}_{K(i)}\overline{\alpha}_{K(i)-1}\ldots\overline{\alpha}_{K(i)-k+1}=\lim_{i\to\infty}\frac{q_{K(i)-k}}{q_{K(i)}}$
$\displaystyle D^{+}_{k}$
$\displaystyle=\lim_{i\to\infty}\alpha_{K(i)+1}\alpha_{K(i)+2}\ldots\alpha_{K(i)+k}=\lim_{i\to\infty}\frac{D_{K(i)+k}}{D_{K(i)}}.$
The next lemma, a crucial one in the proof, makes use of these identities.
###### Lemma 6.3.
Let $(b_{i})$ be the Davenport coefficients of a number $\beta\in[0,1]$ for
which both of the limits
$\displaystyle b^{-}_{1},b^{-}_{2},b^{-}_{3},\ldots$
$\displaystyle=\lim_{i\to\infty}b_{K(i)},b_{K(i)-1},\ldots,b_{1},0,0,0,\ldots$
$\displaystyle b^{+}_{1},b^{+}_{2},b^{+}_{3},\ldots$
$\displaystyle=\lim_{i\to\infty}b_{K(i)+1},b_{K(i)+2},b_{K(i)+3},\ldots$
exist and let
$\beta^{-}=\sum^{\infty}_{k=1}b^{-}_{k}D^{-}_{k}\qquad\text{and}\qquad\beta^{+}=\sum^{\infty}_{k=1}b^{+}_{k}D^{+}_{k}.$
Then
$\lim_{i\to\infty}\lambda_{K(i)}(\beta)=\frac{\beta^{-}\beta^{+}}{1-\alpha^{-}\alpha^{+}}.$
###### Proof.
By definition
$\lambda_{K(i)}(\beta)=Q_{K(i)}D_{K(i)}\beta_{K(i)+1}=\frac{Q_{K(i)}}{q_{K(i)}}q_{K(i)}D_{K(i)}\beta_{K(i)+1}$
and, by (33),
$\lim_{i\to\infty}q_{K(i)}D_{K(i)}=\frac{1}{1-\alpha^{-}\alpha^{+}}.$
In consequence, it is sufficient to observe that
$\lim_{i\to\infty}\frac{Q_{K(i)}}{q_{K(i)}}=\beta^{-}\qquad\text{and}\qquad\lim_{i\to\infty}\beta_{K(i)+1}=\beta^{+}.$
This is a straightforward consequence of the fact that
$D_{K(i)}\beta_{K(i)+1}=\sum^{\infty}_{k=1}b_{K(i)+k}D_{K(i)+k}$
and a corresponding expression for the first limit. ∎
Now we define two Cantor-like subsets of $[0,1)$ in terms of their Davenport
expansions.
###### Definition 6.2.
1. (1)
$\beta\in E(\alpha,s)$ if and only if in its Davenport coefficents $(b_{i})$
no block $b_{i},b_{i+1},\dots,b_{i+s}$ consists solely of zeros or is of the
form
(35) $a_{i}-2,a_{i+1}-2,\dots,a_{i+s-1}-2,a_{i+s}-1.$
Note that this does not preclude tail sequences of the form
$a_{i}-1,a_{i+1}-2,a_{i+2}-2,\ldots$.
2. (2)
$\beta\in F(\alpha,s)$ if and only if in the sequence
$b_{1},b_{2},b_{3},\dots$ no block $b_{i},b_{i+1},\dots,b_{i+s}$ consists
solely of zeros or is of the form
$a_{i}-1,a_{i+1}-2,a_{i+2}-2,\dots,a_{i+s}-2.$
We note that both of $F(\alpha,s)$ and $E(\alpha,s)$ are closed subsets of
$[0,1]$.
We now state and prove the main result of this section.
###### Theorem 6.
Let $r$ and $s$ be positive integers which satisfy $s\geq N$ and $r\geq sL$
and let $\alpha^{-}$ and $\alpha^{+}$ be defined by (33) and $\alpha^{+}_{r}$
by (33) and $D^{+}_{r}$ by (34). For all $e\in E(\alpha^{-},s)$ and $f\in
F(\alpha^{+}_{r+1},s)$ there is some $\beta$ with $0<\beta<1$ such that
(36)
${\mathcal{M}}_{+}(\alpha,\beta)=\frac{efD^{+}_{r}}{1-\alpha^{-}\alpha^{+}}.$
###### Proof.
We will exhibit appropriate Davenport expansions of $\beta$ to achieve this
result for $f\in F(\alpha^{+}_{r+1},s)$ and $e\in E(\alpha^{-},s)$.
Let $e\in E(\alpha^{-},s)$ and $f\in F(\alpha^{+}_{r+1},s)$. We shall prove
there is a $\beta$ with $0<\beta<1$ which satisfies (36) by constructing its
Davenport coefficients $(b_{i})$. Specifically, we shall construct
$b_{1},b_{2},b_{3},\ldots$ so that the limits
$\displaystyle b^{-}_{1},b^{-}_{2},b^{-}_{3},\ldots$
$\displaystyle=\lim_{i\to\infty}b_{K(i)},b_{K(i)-1},\ldots,b_{1},0,0,0,\ldots$
$\displaystyle b^{+}_{1},b^{+}_{2},b^{+}_{3},\ldots$
$\displaystyle=\lim_{i\to\infty}b_{K(i)+1},b_{K(i)+2},b_{K(i)+3},\ldots$
exist and
(37) $e=\sum^{\infty}_{k=1}b^{-}_{k}D^{-}_{k}\qquad\text{and}\qquad
fD^{+}_{r}=\sum^{\infty}_{k=1}b^{+}_{k}D^{+}_{k}.$
Lemma 6.3 then yields:
(38)
$\lim_{i\to\infty}\lambda_{K(i)}(\beta)=\frac{efD^{+}_{r}}{1-\alpha^{-}\alpha^{+}}.$
We describe sequences $b^{+}_{1},b^{+}_{2},b^{+}_{3},\ldots$ and
$b^{-}_{1},b^{-}_{2},b^{-}_{3},\ldots$ for which (38) holds. Let
$f_{1},f_{2},f_{3},\ldots$ be the Davenport coefficients of $f$ with respect
to $\alpha^{+}_{r+1}$ and observe that
$f=\sum^{\infty}_{k=1}f_{k}\alpha^{+}_{r+1}\alpha^{+}_{r+2}\ldots\alpha^{+}_{r+k}.$
Multiplication by $D^{+}_{r}$ gives
$fD^{+}_{r}=\sum^{\infty}_{k=1}f_{k}D^{+}_{r+k}.$
and therefore the right hand formula in (37) holds if we define
(39)
$b^{+}_{1},b^{+}_{2},b^{+}_{3},\ldots=\underbrace{0,\ldots,0}_{r},f_{1},f_{2},f_{3},\ldots.$
It is easily seen that these satisfy the appropriate conditions for a
Davenport expansion.
For a number $e\in E(\alpha^{-},s)$, we let $e_{1},e_{2},e_{3},\ldots$ be the
$\alpha^{-}$-expansion of $e$ and as above we observe that
$e=\sum^{\infty}_{k=1}e_{k}D^{-}_{k}.$
The left hand formula in (37) then holds if we set
$b^{-}_{1},b^{-}_{2},b^{-}_{3},\ldots=e_{1},e_{2},e_{3},\ldots.$
Next, we specify enough of the sequence $b_{1},b_{2},b_{3},\ldots$ to ensure
that (39) and (37) hold. At this point Figure 3 illustrates definition of the
various pieces of the sequence.
Figure 3. The definition of the sequence $b_{i}$.
For this purpose, we choose a positive integer $i_{0}$ and sequences of
integers $(u(i))^{\infty}_{i=i_{0}}$ and $(v(i))^{\infty}_{i=i_{0}}$ such that
$K(i)\leq u(i)<u(i)+N<v(i)\leq K(i+1)$ for all $i\geq i_{0}$ and
$\lim_{i\to\infty}u(i)-K(i)=\infty\qquad\text{and}\qquad\lim_{i\to\infty}K(i+1)-v(i)=\infty.$
Such sequences exist since the differences $K(i+1)-K(i)$ tend to infinity as
$i$ increases. Furthermore we can also assume that, for all $i\geq i_{0}$,
$\displaystyle a_{K(i)+1},a_{K(i)+2},\ldots,a_{u(i)}$
$\displaystyle=a^{+}_{1},a^{+}_{2},\ldots,a^{+}_{u(i)-K(i)}$ $\displaystyle
a_{K(i+1)},a_{K(i+1)-1},\ldots,a_{v(i)}$
$\displaystyle=a^{-}_{1},a^{-}_{2},\ldots,a^{-}_{K(i+1)-v(i)+1}.$
We ensure (39) and (37) hold by defining
$\displaystyle b_{K(i)+1},b_{K(i)+2},\ldots,b_{u(i)}$
$\displaystyle=b^{+}_{1},b^{+}_{2},\ldots,b^{+}_{u(i)-K(i)}$ $\displaystyle
b_{K(i+1)},b_{K(i+1)-1},\ldots,b_{v(i)}$
$\displaystyle=b^{-}_{1},b^{-}_{2},\ldots,b^{-}_{K(i+1)-v(i)+1}$
for all $i\geq i_{0}$.
Before completing the specification of $(b_{j})$ we further restrict $i_{0}$
and the sequences $(u(i))^{\infty}_{i=i_{0}}$ and $(v(i))^{\infty}_{i=i_{0}}$.
$b_{u(i)}\neq 0\qquad\text{and}\qquad b_{v(i)}\neq 0$
for all $i\geq i_{0}$. This is relatively easy to arrange from the properties
of the $K(i)$ in relation to the Davenport expansion, and of the sequences
$(u(i))$ and $(v(i))$. To complete the specification of $(b_{j})$ we introduce
one more sequence. We choose the sequence $(w(i))^{\infty}_{i=i_{0}}$ so that
$u(i)<w(i)\leq u(i)+N\text{ and }a_{w(i)}\geq 3\quad(i\geq i_{0}).$
Such a choice is clearly possible.
We can now unambiguously define
$b_{j}=\begin{cases}0&\text{ if $1\leq j\leq K(i_{0})$}\\\ a_{j}-2&\text{ if
$u(i)<j<w(i)$ for some $i\geq i_{0}$}\\\ a_{j}-3&\text{ if $j=w(i)$ for some
$i\geq i_{0}$}\\\ a_{j}-2&\text{if $w(i)<j<v(i)$ for some $i\geq
i_{0}$.}\end{cases}$
It is not hard to verify that $0\leq b_{i}<a_{i}$ for all $i\geq 1$ and since
$b_{w(i)}=a_{w(i)}-3$ for all $i\geq i_{0}$ it is also clear that no
subsequence $b_{i},b_{i+1},b_{i+2},\ldots$ is of the form (17). It is easy to
check that $(b_{i})$ are Davenport coefficients by showing that no block
$b_{i},b_{i+1},\ldots,b_{j}$ is of the form (16).
Now we observe that the hypotheses of Theorem 5 holds with $k(i)=K(i)$ for all
$i$ to complete the proof. ∎
### 6.3. Cantor dissections for $E(\alpha,s)$ and $F(\alpha,s)$
Our eventual aim is to show that if the integer $s$ is large enough then the
product of the two sets $E(\alpha^{-},s)$ and $F(\alpha^{+}_{r+1},s)$, where
$r\geq 1$, contains an interval. Towards that aim we describe each of these
two sets in terms of Cantor dissections. We do this for a generic $\alpha$
rather than $\alpha^{-}$ and $\alpha^{+}$ at this stage. We collect together a
few definitions.
###### Definition 6.3.
1. (1)
$H(\alpha,s)$ and $G(\alpha,s)$ are the smallest closed intervals containing
$F(\alpha,s)$ and $E(\alpha,s)$ respectively.
2. (2)
For each sequence $\mathbf{c}_{n}=c_{1},c_{2},\ldots,c_{n}$ of positive
integers,define:
$\displaystyle S(\mathbf{c}_{n})$ $\displaystyle=\sum^{n}_{k=1}c_{k}D_{k},$
$\displaystyle F(\mathbf{c}_{n})$
$\displaystyle=\\{\gamma=\sum_{k=1}^{\infty}b_{k}D_{k}\in
F(\alpha,s):b_{k}=c_{k},\ (k=1,2,\ldots,n)\\}$
where $\sum_{k=1}^{\infty}b_{k}D_{k}$ is the Davenport expansion of $\gamma$.
Denote by $C(\mathbf{c}_{n})$ the smallest closed interval which contains
$F(\mathbf{c}_{n})$. Observe that $C(\mathbf{c}_{n})$ may be the empty set.
3. (3)
when $(\mathbf{c}_{n})\neq\emptyset$,
$C(\mathbf{c}_{n})=[\underline{C}(\mathbf{c}_{n}),\overline{C}(\mathbf{c}_{n})]$
where
$\underline{C}(\mathbf{c}_{n})=\inf C(\mathbf{c}_{n}),\ \overline{C}=\sup
C(\mathbf{c}_{n})\text{ and
}|C(\mathbf{c}_{n}|=\overline{C}(\mathbf{c}_{n})-\underline{C}(\mathbf{c}_{n}).$
We allow the possibility that $n=0$ in which case $C(\;)=H(\alpha,s)$.
The dissection of $H(\alpha,s)$ to obtain $F(\alpha,s)$ begins by replacing
$C(\;)=H(\alpha,s)$ with the collection of intervals
$\\{C(0),C(1),\ldots,C(a_{1}-1)\\}.$
The $n$-th stage of the dissection replaces each non-empty interval
$C(\mathbf{c}_{n})$ by the collection of intervals
(40) $\\{C(\mathbf{c}_{n+1}):\;0\leq c_{n+1}<a_{n+1}\\}.$
From the definition of $C(\mathbf{c}_{n})$ it is clear that it is the smallest
closed interval containing the collection of intervals (40). Moreover the
restrictions on the digits results in gaps between all of these. As an
illustration, note that if $C(\mathbf{c}_{n})$ has the last $n-1$ $c_{k}$ all
equal to $0$, then at the next level $C(\mathbf{c}_{n},0)=\emptyset$. The same
kind of phenomenon occurs at the opposite end because of the restriction on
the number of terms of the form $a_{i}-2$. It is clear that this is a Cantor
dissection that produces $F(\alpha,s)$, and we have
$\begin{gathered}S(\mathbf{c}_{n})+D_{n+s+1}\leq\underline{C}(\mathbf{c}_{n})\leq\overline{C}(\mathbf{c}_{n})\leq
S(\mathbf{c}_{n})+D_{n}\\\
C(\mathbf{c}_{n+1})\subset[S(\mathbf{c}_{n})+c_{n+1}D_{n+1}+D_{n+s+2},\;S(\mathbf{c}_{n})+(c_{n+1}+1)D_{n+1}].\end{gathered}$
Clearly $|C(\mathbf{c}_{n})|\leq D_{n}$, and it is evident that
$F(\alpha,s)=\bigcap^{\infty}_{n=1}\bigcup\\{C(\mathbf{c}_{n})\neq\emptyset:\;0\leq
c_{i}<a_{i}\\}.$
Now we obtain more precise estimates of the values of the endpoints
$\underline{C}(\mathbf{c}_{n})$ and $\overline{C}(\mathbf{c}_{n})$.
###### Lemma 6.4.
Let $s\geq N$, $C(\mathbf{c}_{n})\neq\emptyset$, $t$ the largest integer with
$0\leq t\leq n$ such that all of $c_{n-t+1},c_{n-t+2},\ldots,c_{n}$ are zero
and $u$ the unique integer with $0\leq u\leq n$ such that
$c_{n-u+1},c_{n-u+2},\ldots,c_{n}$ is equal to
$a_{n-u+1}-1,a_{n-u+2}-2,a_{n-u+3}-2,\ldots,a_{n}-2.$
Then
$\underline{C}(\mathbf{c}_{n})<\left\\{\begin{aligned}
&S(\mathbf{c}_{n})+D_{n+s}\quad&&\text{if $t=0$}\\\
&S(\mathbf{c}_{n})+D_{n+1}+D_{n+s}\quad&&\text{if $t>0$}\end{aligned}\right.$
and
$\overline{C}(\mathbf{c}_{n})>\left\\{\begin{aligned}
&S(\mathbf{c}_{n})+D_{n}-D_{n+s-N}\quad&&\text{if $u=0$}\\\
&S(\mathbf{c}_{n})+D_{n+N+1}\quad&&\text{if $u>0$}\end{aligned}\right.$
###### Proof.
Write $C=C(\mathbf{c}_{n})$ and note that
$\underline{C}=\underline{C}(\mathbf{c}_{n})$ is the number $\beta$ whose
Davenport coefficients $(b_{i})$ are of the form
(41)
$c_{1},c_{2},\ldots,c_{n},\underbrace{0,\ldots,0}_{s-t},1,\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\ldots.$
Note that $t\leq s$ else $c_{1},c_{2},\ldots,c_{n}$ ends with more than $s$
consecutive zeros and $C=\emptyset$. In other words,
$\underline{C}=\sum^{n}_{k=1}c_{k}D_{k}+D_{n+s-t+1}+D_{n+2s-t+2}+D_{n+3s-t+3}+\ldots,$
and
$\underline{C}\leq\left\\{\begin{aligned}
&S(\mathbf{c}_{n})+D_{n+s+1}+D_{n+2s+2}+D_{n+3s+3}+\ldots\quad&&\text{if
$t=0$}\\\
&S(\mathbf{c}_{n})+D_{n+1}+D_{n+s+2}+D_{n+2s+3}+\ldots\quad&&\text{if
$t>0$}\end{aligned}\right.$
Further, since $s\geq N$ we know
$D_{n+s}>D_{n+s+1}+D_{n+2s+2}+D_{n+3s+3}+\ldots,$
and
$D_{n+s}>D_{n+s+2}+D_{n+2s+3}+D_{n+3s+4}+\ldots,$
and the truth of the first statement of the lemma is evident.
We describe the Davenport expansion of
$\overline{C}=\overline{C}(\mathbf{c}_{n})$ next. Let $k(0)=n-u$ and
inductively define the sequence $k(1),k(2),k(3),\ldots$ by choosing $k(i)$ to
be the largest integer such that
$k(i-1)+2\leq k(i)\leq k(i-1)+s+1\qquad\text{and}\qquad a_{k(i)}\geq 3.$
This is possible by the properties of $a_{n}$ as enunciated in Lemma 6.2 for
$\alpha=\alpha^{-}$ and $\alpha=\alpha^{+}$. Further,
$k(i)\geq k(i-1)+s-N+2$
because if not $a_{k}=2$ for all $k$ between and including $k(i-1)+s-N+2$ and
$k(i-1)+s+1$ contrary to the definition of $N$. Now $\overline{C}$ is the
number $\beta$ whose Davenport coefficients $(b_{i})$ are defined by
$b_{i}=\begin{cases}c_{i}&\text{ if $i\leq k(0)$}\\\ a_{i}-1&\text{ if
$i=k(j)+1$ for some $j\geq 0$}\\\ a_{i}-2&\text{ if $k(j)+1<i<k(j+1)$ for some
$j\geq 0$}\\\ a_{i}-3&\text{ if $i=k(j)$ for some $j\geq 1$.}\end{cases}$
These are clearly Davenport coefficients, and the sequence contains no block
$b_{i},b_{i+1},\ldots,b_{i+s}$ of the form (35) nor does it contain a block of
$b_{i},b_{i+1},\ldots,b_{i+s}$ consisting solely of zeros. We conclude that
$\beta\in F(\alpha,s)$. It is also fairly clear that the sequence
$b_{1},b_{2},b_{3},\ldots$ begins with $c_{1},c_{2},\ldots,c_{n}$.
It remains to show that no other element of $C(\mathbf{c}_{n})$ is larger than
$\beta$. If that were the case, and there were some $\beta^{\prime}\in
C(\mathbf{c}_{n})$ with Davenport coefficients $(b^{\prime}_{i})$ such that
$\beta^{\prime}>\beta$. However, the form of the definition of $\beta$
prohibits any possible increase in the values of the Davenport coefficients
while staying a member of $F(\alpha,s)$ and starting with
$c_{1},c_{2},\ldots,c_{n}$. Evidently
$\underline{C}=\sum^{\infty}_{k=1}b_{k}D_{k}$. By truncating this series at
the term with index $k(1)+1$ and making some minor rearrangements we find that
$\overline{C}>\sum^{k(0)}_{l=1}c_{l}D_{l}+\sum^{k(1)+1}_{l=k(0)+1}(a_{l}-2)D_{l}+D_{k(0)+1}-D_{k(1)}+D_{k(1)+1}.$
We consider two cases. First we suppose $u=0$ and hence $k(0)=n$. In this
case, since
$D_{n}<\sum^{k(1)+1}_{l=n+1}(a_{l}-2)D_{l}+D_{n+1}+D_{k(1)+1}$
we obtain $\overline{C}>S(\mathbf{c}_{n})+D_{n}-D_{k(1)}$. It is easy to
deduce from (41) with $i=1$ that $D_{k(1)}>D_{n+s-N}$ and the second statement
of the lemma is proved. Now we suppose $u>0$ and hence $k(0)<n$. Since
$k(1)\geq n+1$,
$\overline{C}>\sum^{n}_{l=1}c_{l}D_{l}+\sum^{k(1)+1}_{l=n+1}(a_{l}-2)D_{l}-D_{k(1)}+D_{k(1)+1}.$
As $a_{k(1)}\geq 3$ and $a_{k(1)+1}\geq 2$ we have
$\overline{C}>\sum^{n}_{l=1}c_{l}D_{l}+\sum^{k(1)-1}_{l=n+1}(a_{l}-2)D_{l}+D_{k(1)+1}.$
The definition of $N$ implies there is some $i$ with $n+1\leq i\leq n+N+1$
such that $a_{i}\geq 3$ and so
$\sum^{n+N+1}_{l=n+1}(a_{l}-2)D_{l}\geq D_{n+N+1}.$
It follows that if $k(1)-1\geq n+N+1$ then the second statement of the lemma
is true. If on the other hand $k(1)<n+N+1$ then $D_{k(1)+1}\geq D_{n+N+1}$ and
again the truth of the second statement is clear. This completes the proof the
lemma. ∎
A key remark about the $F(\alpha,s)$ construction, that will not be true for
the case if $E(\alpha,s)$, is that, at least generically, the deleted
intervals resulting from the “zeros” condition and the “$a_{n}-2$” condition
in this Cantor construction have the property that their left and right
endpoints respectively are $S(c_{1},...,c_{n}+1)$.
Next we deal with the set $E(\alpha,s)$. This is a little more complicated;
the two restrictions on the Davenport coefficients of the elements of
$E(\alpha,s)$ no longer correspond to a single gap in the dissection of
$G(\alpha,s)$. We make the following definition.
###### Definition 6.4.
$A(\;)=[0,1-D_{1}]$ and $B(\;)=[1-D_{1},1]$. For each sequence
$\mathbf{c}_{n}=c_{1},c_{2},\ldots,c_{n}$ of positive integers, we define
$A(\mathbf{c}_{n})$ to be the smallest closed interval containing
$E(\alpha,s)\cap[S(\mathbf{c}_{n}),S(\mathbf{c}_{n})+D_{n}-D_{n+1}]$, and
$B(\mathbf{c}_{n})$ to be the smallest closed interval containing
$E(\alpha,s)\cap[S(\mathbf{c}_{n})+D_{n}-D_{n+1},S(\mathbf{c}_{n})+D_{n}]$.
The dissection of $G(\alpha,s)$ begins by replacing $G(\alpha,s)$ with the
pair of intervals $A(\;)$ and $B(\;)$. The next step is the substitution
$\displaystyle A(\;)$
$\displaystyle\to\\{A(0),B(0),A(1),B(1),\ldots,A(a_{1}-3),B(a_{1}-3),A(a_{1}-2)\\}$
$\displaystyle B(\;)$ $\displaystyle\to\\{B(a_{1}-2),\;A(a_{1}-1)\\}.$
The $n$-th step of the dissection is
(42) $\displaystyle\emptyset\neq A(\mathbf{c}_{n})$
$\displaystyle\to\\{A(\mathbf{c}_{n+1}):\;0\leq c_{n+1}\leq a_{n+1}-2\\}$
$\displaystyle\qquad\cup\\{B(\mathbf{c}_{n+1}):\;0\leq c_{n+1}\leq
a_{n+1}-3\\}$ $\displaystyle\emptyset\neq A(\mathbf{c}_{n})$
$\displaystyle\to\\{B(c_{1},c_{2},\ldots,c_{n},a_{n+1}-2),\;A(c_{1},c_{2},\ldots,c_{n},a_{n+1}-1)\\}.$
where again we use the notation $\mathbf{c}_{n}=c_{1},c_{2},\ldots,c_{n})$.
We note that $A(\mathbf{c}_{n})$ and $B(\mathbf{c}_{n})$ are the smallest
closed intervals containing the collections (42) at the previous level. This
follows because
$S(\mathbf{c}_{n})+D_{n}-D_{n+1}=S(c_{1},c_{2},\ldots,c_{n},a_{n+1}-2)+D_{n+1}-D_{n+2}.$
For the moment, we write
$A=A(\mathbf{c}_{n})\qquad B=B(\mathbf{c}_{n})\qquad S=S(\mathbf{c}_{n}).$
Now let $\beta\in E(\alpha,s)$ and suppose its sequence of Davenport
coefficients $(b_{i})$ begins with $c_{1},c_{2},\ldots,c_{n}$. Then the block
$b_{n+1},b_{n+2},\ldots,b_{n+s+1}$ does not consist entirely of zeros, and so
$b_{i}\geq 1$ and $\beta\geq S+D_{i}$ for some $i$ with $n+1\leq i\leq n+s+1$.
Hence $\beta\geq S+D_{n+s+1}$. Since $A$ is the smallest closed interval
containing all such numbers $\beta$ which also satisfy $\beta\leq
S+D_{n}-D_{n+1}$ it follows that
(43) $S+D_{n+s+1}\leq\underline{A}\leq\overline{A}\leq S+D_{n}-D_{n+1}.$
In particular $|A|\leq D_{n}$. If, on the other hand, $\beta>S+D_{n}-D_{n+1}$,
there is $i\geq n+1$ such that the block $b_{n+1},b_{n+2},\ldots,b_{i}$ is of
the form
$a_{n+1}-2,a_{n+2}-2,\ldots,a_{i-1}-2,a_{i}-1.$
We conclude that
(44) $S+D_{n}-D_{n+1}+D_{n+s+1}\leq\underline{B}\leq\overline{B}\leq S+D_{n}.$
In fact,
$\displaystyle A(\mathbf{c}_{n+1})$
$\displaystyle\subset[S+c_{n+1}D_{n+1}+D_{n+s+2},\;S+(c_{n+1}+1)D_{n+1}-D_{n+2}]$
$\displaystyle B(\mathbf{c}_{n+1})$
$\displaystyle\subset[S+(c_{n+1}+1)D_{n+1}-D_{n+2}+D_{n+s+2},\;S+(c_{n+1}+1)D_{n+1}].$
We note that all such intervals where $0\leq c_{n+1}<a_{n+1}$ are disjoint,
and since
(45)
$E(\alpha,s)=\bigcap^{\infty}_{n=1}\bigcup\\{A(\mathbf{c}_{n})\neq\emptyset,\;B(\mathbf{c}_{n})\neq\emptyset:\;0\leq
c_{i}<a_{i}\\},$
it is totally disconnected. Again the gaps arise because of the constraints on
digits in the definition of $E(\alpha,s)$.
Now we need to find estimates for the endpoints of the intervals
$A(\mathbf{c}_{n})$ and $B(\mathbf{c}_{n})$, just as we have for
$C(\mathbf{c}_{n})$ in Lemma 6.4.
###### Lemma 6.5.
Let $s\geq N$ and $A(\mathbf{c}_{n})\neq\emptyset$. Then
$\underline{A}(\mathbf{c}_{n})<S(\mathbf{c}_{n})+D_{n}-D_{n+1}-D_{n+3N}$
and
$\overline{A}(\mathbf{c}_{n})=S(\mathbf{c}_{n})+D_{n}-D_{n+1}.$
Further,
$\underline{A}(\mathbf{c}_{n})<S(\mathbf{c}_{n})+D_{n+s}$
whenever $n=0$ or $B(c_{1},c_{2},\ldots,c_{n-1},c_{n}-1)\neq\emptyset$.
###### Proof.
Write $A=A(\mathbf{c}_{n})$. We note first that $\underline{A}$ is the number
$\beta$ whose Davenport coefficients $(b_{i})$ are of the form
$c_{1},c_{2},\ldots,c_{n},\underbrace{0,\ldots,0}_{s-t},1,\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\ldots.$
where $t$ is the largest integer with $0\leq t\leq n$ for which
$c_{n-t+1},c_{n-t+2},\ldots,c_{n}$ are all zero, and observe that there is
some $j$ satisfying
$n+1\leq j\leq n+s-t+N+1$
such that $b_{j}\leq a_{j}-3$ and $b_{i}=a_{i}-2$ for all $i$ with $n+1\leq
i\leq j-1$. It follows that
(46)
$\underline{A}\leq\sum^{n}_{k=1}c_{k}D_{k}+\sum^{j-1}_{k=n+1}(a_{k}-2)D_{k}+(a_{j}-3)D_{j}+D_{j}.$
Since $j\leq n+2N$,
$\underline{A}\leq\sum^{n}_{k=1}c_{k}D_{k}+\sum^{n+2N}_{k=n+1}(a_{k}-2)D_{k}.$
We know
$\sum^{n+2N}_{k=n+1}(a_{k}-2)D_{k}=D_{n}-D_{n+1}-\sum^{\infty}_{k=n+2N+1}(a_{k}-2)D_{k}$
and since the definition of $N$ implies there is some $k$ with $n+2N<k\leq
n+3N$ such that $a_{k}\geq 3$ we conclude that (46) does not exceed
$D_{n}-D_{n+1}-D_{n+3N}$. The truth of the first statement of the lemma is now
evident.
Now we redefine $\beta$ as
$\beta=S(\mathbf{c}_{n})+D_{n}-D_{n+1}$
and observe that it has Davenport coefficients
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-2,a_{n+2}-2,a_{n+3}-2,\ldots.$
This contains no block $b_{i},b_{i+1},\ldots,b_{j}$ of the form (16) nor a
block $b_{i},b_{i+1},\ldots,b_{i+s}$ of the form (35) and so $\beta\in
E(\alpha,s)$.
Now suppose $n=0$ or $B(c_{1},c_{2},\ldots,c_{n}-1)$ is non-empty. If
$B(c_{1},c_{2},\ldots,c_{n}-1)\neq\emptyset$ then $c_{n}\geq 1$ and so $t=0$.
Obviously $t$ is also zero if $n=0$. As a result,
$\underline{A}=\sum^{n}_{k=1}c_{k}D_{k}+D_{n+s+1}+D_{n+2s+2}+D_{n+3s+3}+\ldots.$
Because $s\geq N$ we know that
$D_{n+s}\geq D_{n+s+1}+D_{n+2s+2}+D_{n+3s+3}+\ldots,$
and this is enough to complete the proof. ∎
###### Lemma 6.6.
Let $s\geq N$ and $B(\mathbf{c}_{n})\neq\emptyset$. Then
$\displaystyle\underline{B}(\mathbf{c}_{n})$
$\displaystyle<S(\mathbf{c}_{n})+D_{n}-D_{n+1}+D_{n+2}+D_{n+s},$
$\displaystyle\overline{B}(\mathbf{c}_{n})$
$\displaystyle=S(\mathbf{c}_{n})+D_{n}.$
Further,
$\underline{B}(\mathbf{c}_{n})<S(\mathbf{c}_{n})+D_{n}-D_{n+1}+D_{n+s}$
whenever $n=0$ or $c_{n}\neq a_{n}-2$ and $A(\mathbf{c}_{n})$ is non-empty.
###### Proof.
As usual, we write $B=B(\mathbf{c}_{n})$ and observe that $B$ contains the
number $\beta$ whose Davenport coefficients $b_{1},b_{2},b_{3},\ldots$ are
equal to
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-1,\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\ldots.$
Therefore
$\underline{B}\leq\sum^{n}_{k=1}c_{k}D_{k}+(a_{n+1}-1)D_{n+1}+D_{n+s+2}+D_{n+2s+3}+D_{n+3s+4}+\ldots.$
We note that
$D_{n+s}\geq D_{n+s+2}+D_{n+2s+3}+D_{n+3s+4}+\ldots.$
The first inequality of the lemma then follows since
$a_{n+1}D_{n+1}=D_{n}+D_{n+2}$.
For the second statement of the lemma we consider $\overline{B}=\beta$ where
$\beta$ is
(47) $\beta=S(\mathbf{c}_{n})+D_{n},$
so that $\beta=\sum^{\infty}_{k=1}c_{k}D_{k}$ where $b_{1},b_{2},b_{3},\ldots$
is the sequence
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-1,a_{n+2}-2,a_{n+3}-2,a_{n+4}-2,\ldots,$
and the rest is clear.
Now suppose either $n=0$ or $A(\mathbf{c}_{n})\neq\emptyset$ and $c_{n}\neq
a_{n}-2$. In this case $\underline{B}$ is the number $\beta$ whose Davenport
coefficient sequence $(b_{i})$ begins with
(48)
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-2,a_{n+2}-2,\ldots,a_{n+s-1}-2,a_{n+s}-1$
and continues with
(49)
$\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\underbrace{0,\ldots,0}_{s},1,\ldots.$
Clearly $(b_{i})$ is a sequence of Davenport coefficients and $\beta\in
E(\alpha,s)$. Further, since $(b_{i})$ begins with (48),
$\beta\geq\sum^{n}_{k=1}c_{k}D_{k}+\sum^{n+s-1}_{k=n+1}(a_{k}-2)D_{k}+(a_{n+s}-1)D_{n+s}.$
As the sequence of Davenport coefficients of $\underline{B}$ begins with (48)
and continues with (49),
$\underline{B}=\sum^{n}_{k=1}c_{k}D_{k}+\sum^{n+s}_{k=n+1}(a_{k}-2)D_{k}+D_{n+s}+D_{n+2s+1}+D_{n+3s+2}+\ldots.$
By using the appropriate identities of Section 3 we obtain
$\underline{B}=\sum^{n}_{k=1}c_{k}D_{k}+D_{n}-D_{n+1}+D_{n+s+1}+D_{n+2s+1}+D_{n+3s+2}+D_{n+4s+3}+\ldots.$
The usual arguments yield
$D_{n+s}\geq D_{n+s+1}+D_{n+2s+1}+D_{n+3s+2}+D_{n+4s+3}+\ldots$
and the truth of the final statement of the lemma is clear. ∎
### 6.4. Application of Hall’s Theorem
As mentioned in the introduction to the last section, we shall now use a
theorem of Hall, namely Theorem 2.2 in [15], to show that if $s$ is large
enough then the product of the sets $E(\alpha,s)$ and $F(\alpha,s)$ contains
an interval. This idea was used in the context of inhomogeneous diophantine
approximation by Cusick, Moran and Pollington, see [11]. The actual statement
of Hall’s theorem [15] concerns the sum of Cantor sets but, as Hall points
out, his result can be applied to products by taking logarithms. Specifically,
we have
$\log(E(\alpha,s).F(\alpha,s))=\log E(\alpha,s)\;+\;\log F(\alpha,s)$
and since the logarithm function is continuous and strictly increasing, it
maps the Cantor dissections of $G(\alpha,s)$ and $H(\alpha,s)$ to Cantor
dissections of $\log G(\alpha,s)$ and $\log H(\alpha,s)$, respectively.
Before applying Hall’s theorem we need to check that his Condition 1 holds.
This condition states that if, in going from level $n$ to $n+1$, an interval
$C$ is replaced by two disjoint intervals $C_{1}$ and $C_{2}$ with an open
interval $C_{12}$ between them, so that $C_{1}\cup C_{12}\cup C_{2}=C$, then
the length of $C_{12}$ should not exceed the minimum of $|C_{1}|$ and
$|C_{2}|$. We note, as Hall does in his discussion of bounded continued
fractions, that the transition from the $n$th to the $(n+1)$th stage of the
Cantor dissections leading to the sets $F(\alpha,s)$ and $E(\alpha,s)$ can be
done by iteratively removing just one “middle” interval at a time. To verify
Condition 1 of Hall, it is enough to show that for any pair of adjacent
intervals formed at the $n$th stage of the Cantor dissection to produce either
$F(\alpha,s)$ or $E(\alpha,s)$, the minimum of their lengths exceeds the
length of the removed interval.
We can now verify this for the Cantor dissection for $\log F(\alpha,s)$.
###### Lemma 6.7.
There is an integer $s_{0}\geq N$ such that if $s\geq s_{0}$ and if $C_{1}$
and $C_{2}$ are non-empty neighbouring intervals arising at the same stage of
the Cantor dissection for $F(\alpha,s)$ then
(50) $|\log C_{12}|\leq\min\\{|\log C_{1}|,|\log C_{2}|\\}$
where $C_{12}$ is the open interval lying between $C_{1}$ and $C_{2}$.
###### Proof.
Let $s\geq N$ and let $C_{1}$ and $C_{2}$ and $C_{12}$ be as described. We
assume without loss of generality that $C_{1}$ lies to the left of $C_{2}$.
Our aim is to show that if $s$ is large enough then the number
$|\log C_{12}|=\log\underline{C_{2}}-\log\overline{C_{1}}$
is less than or equal to both
$|\log C_{1}|=\log\overline{C_{1}}-\log\underline{C_{1}}\text{ and }|\log
C_{2}|=\log\overline{C_{2}}-\log\underline{C_{2}}.$
By rearranging and using the properties of logarithms we reduce this statement
to
(51)
$\underline{C_{1}}\;\underline{C_{2}}\leq\overline{C_{1}}\;\overline{C_{1}}\qquad\text{and}\qquad\underline{C_{2}}\;\underline{C_{2}}\leq\overline{C_{1}}\;\overline{C_{2}}.$
Note that, since
$4\underline{C_{1}}\underline{C_{2}}=(\underline{C_{1}}+\underline{C_{2}})^{2}-(\underline{C_{1}}-\underline{C_{2}})^{2},$
to prove the first of the inequalities in (51) it is enough to show
$\underline{C_{1}}+\underline{C_{2}}<2\;\overline{C_{1}},$
and we concentrate on this.
Since $C_{1}$ and $C_{2}$ arise at the same stage of the dissection and
$C_{1}$ lies to the left of $C_{2}$ we write
$C_{1}=C(\mathbf{c}_{n})\qquad\text{and}\qquad
C_{2}=C(c_{1},c_{2},\ldots,c_{n-1},c^{\prime}_{n})$
where $c^{\prime}_{n}>c_{n}$. The key fact here is that
$C(c_{1},c_{2},\ldots,c_{n},c)$ is empty only for the extreme values of $c$,
because of the conditions that describe $F(\alpha,s)$. Hence
$c^{\prime}_{n}=c_{n}+1$.
We write
$S_{1}=S(\mathbf{c}_{n})\qquad\text{and}\qquad
S_{2}=S(c_{1},c_{2},\ldots,c_{n-1},c^{\prime}_{n}).$
Note that $S_{2}=S_{1}+D_{n}$. Assume $t$ is the largest integer with $0\leq
t\leq n$ such that all of $c_{n-t+1},c_{n-t+2},\ldots,c_{n}$ are zero and $u$
the unique integer with $0\leq u\leq n$ such that
$c_{n-u+1},c_{n-u+2},\ldots,c_{n}$ is equal to (45). We denote the
corresponding integers for $C_{2}$ by $t^{\prime}$ and $u^{\prime}$,
respectively. We know $u^{\prime}=0$ else
$c_{1},c_{2},\ldots,c_{n-1},c^{\prime}_{n}$ ends with
$a_{n-u+1}-1,a_{n-u+2}-2,a_{n-u+3}-2,\ldots,a_{n-1}-2,a_{n}-1$
implying that $C_{2}=\emptyset$. Similarly, $t^{\prime}=0$ since
$c^{\prime}_{n}\geq 1$. Hence
$\overline{C_{1}}>S_{1}+D_{n}-D_{n+s-N}\qquad\text{and}\qquad\underline{C_{2}}<S_{2}+D_{n+s}.$
and
$\underline{C_{1}}<S_{1}+D_{n+1}+D_{n+s}\qquad\text{and}\qquad\overline{C_{2}}>S_{2}+D_{n+N+1}-D_{n+s-N}.$
We are now ready to consider the inequalities in (51). The inequalities above
imply that
$2\;\overline{C_{1}}-(\underline{C_{1}}+\underline{C_{2}})>S_{1}+2D_{n}-2D_{n+s-N}-S_{2}-D_{n+1}-2D_{n+s}.$
Further, $S_{2}=S_{1}+D_{n}$ and $D_{n+s-N}\geq D_{n+s}$ and thus
$2\;\overline{C_{1}}-(\underline{C_{1}}+\underline{C_{2}})>D_{n}-D_{n+1}-4D_{n+s-N}.$
Since (26) holds for all $i\geq 1$ we know there is some $s_{0}\geq N$ such
that
$1-\alpha_{n+1}-4\;\alpha_{n+1}\alpha_{n+2}\ldots\alpha_{n+s-N}>0$
and hence
$D_{n}-D_{n+1}-4D_{n+s-N}>0$
if $s\geq s_{0}$. Note that the size of $s_{0}$ is independent of $n$. It
follows that if $s\geq s_{0}$ then
$\underline{C_{1}}+\underline{C_{2}}<2\;\overline{C_{1}}$ and we have the
desired result.
For the second inequality in (51) we observe that
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>(S_{1}+D_{n}-D_{n+s-N})(S_{2}+D_{n+N+1}-D_{n+s-N})-(S_{2}+D_{n+s})^{2}.$
Since $S_{1}\geq 0$ and $S_{2}\geq D_{n}$ and $D_{n+s-N}\geq D_{n+s}$ we have
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>(D_{n}-D_{n+s-N})(D_{n}+D_{n+N+1}-D_{n+s-N})-(D_{n}+D_{n+s-N})^{2}$
and hence
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>D_{n}(D_{n+N+1}-4D_{n+s-N})-D_{n+N+1}D_{n+s-N}.$
Clearly $D_{n}>D_{n+N+1}$ and therefore it suffices to show that if $s$ is
large enough then
$D_{n+N+1}-4D_{n+s-N}>D_{n+s-N}$
or equivalently
$1>5\;\alpha_{n+N+2}\alpha_{n+N+3}\ldots\alpha_{n+s-N}.$
As above, this is an easy consequence of (26). ∎
We can now verify that Hall’s Condition 1 holds for the dissection for $\log
E(\alpha,s)$.
###### Lemma 6.8.
There is an integer $s_{0}\geq N$ such that if $s\geq s_{0}$ and if $C_{1}$
and $C_{2}$ are non-empty neighbouring intervals arising at the same stage of
the Cantor dissection which produces $E(\alpha,s)$ then
$|\log C_{12}|\leq\min\\{|\log C_{1}|,|\log C_{2}|\\}$
where $C_{12}$ is the open interval lying between $C_{1}$ and $C_{2}$.
###### Proof.
Let $s\geq N$ and let $C_{1}$ and $C_{2}$ and $C_{12}$ be as described. We may
assume without loss of generality that $C_{1}$ lies to the left of $C_{2}$. We
know from proof of Lemma 6.7 that it is sufficient to prove the inequalities
(52)
$\underline{C_{1}}\;\underline{C_{2}}\leq\overline{C_{1}}\;\overline{C_{1}}\text{
and
}\underline{C_{2}}\;\underline{C_{2}}\leq\overline{C_{1}}\;\overline{C_{2}}$
hold when $s$ is large enough. We can also make use of statement (51).
We consider two possibilities for $C_{1}$. We suppose first that
(53) $C_{1}=A(\mathbf{c}_{n}).$
In this case $B(c_{1},c_{2},\ldots,c_{n})\neq\emptyset$ and therefore
$C_{2}=B(\mathbf{c}_{n}).$
To see this, we produce a number $\beta$ that belongs to $B(\mathbf{c}_{n})$.
To this end we note that in the Cantor dissection of $G(\alpha,s)$ the
intervals
$A(c_{1},c_{2},\ldots,c_{n-1},a_{n}-2)\text{ and
}A(c_{1},c_{2},\ldots,c_{n-1},a_{n}-1)$
have no right neighbours since they result from the dissection of
$A(c_{1},c_{2},\ldots,c_{n-1})$ and $B(c_{1},c_{2},\ldots,c_{n-1})$,
respectively. Therefore, either $n=0$ or $c_{n}\leq a_{n}-3$. It follows from
the proof of Lemma 6.5 that $\overline{C}_{1}$ lies in $E(\alpha,s)$ and has
Davenport coefficients
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-2,a_{n+2}-2,a_{n+3}-2,\ldots.$
Now let $\beta=\sum^{\infty}_{k=1}b_{k}D_{k}$ where $b_{1},b_{2},b_{3},\ldots$
is the sequence
$c_{1},c_{2},\ldots,c_{n},a_{n+1}-1,a_{n+2}-2,a_{n+3}-2,a_{n+4}-2,\ldots.$
It is straightforward again to check that $\beta\in E(\alpha,s)$. It now
follows from (44) that $\beta$ belongs to an interval of the form
$A(c^{\prime}_{1},c^{\prime}_{2},\ldots,c^{\prime}_{n})$ or
$B(c^{\prime}_{1},c^{\prime}_{2},\ldots,c^{\prime}_{n})$. By observing that
$\beta=S(\mathbf{c}_{n})+D_{n}$
and applying the inequalities in (43), it can be seen that the only
possibility is $\beta\in B(\mathbf{c}_{n})\neq\emptyset$.
We can now apply Lemmas 6.5 and 6.6 to $C_{1}$ and $C_{2}$. As usual, it is
convenient to write $S=S(\mathbf{c}_{n})$. Lemma 6.5 implies
$\underline{C_{1}}<S+D_{n}-D_{n+1}-D_{n+3N}\qquad\text{and}\qquad\overline{C_{1}}=S+D_{n}-D_{n+1}$
and Lemma 6.6 implies
$\underline{C_{2}}<S+D_{n}-D_{n+1}+D_{n+s}\qquad\text{and}\qquad\overline{C_{2}}=S+D_{n}.$
It follows that
$2\;\overline{C_{1}}-(\underline{C_{1}}+\underline{C_{2}})>D_{n+3N}-D_{n+s}.$
Since (26) holds for all $i\geq 1$ we know there is some $s_{0}\geq 3N+1$ such
that if $s\geq s_{0}$ then
$1>\alpha_{n+3N+1}\alpha_{n+3N+2}\ldots\alpha_{n+s}.$
We emphasis that the size of $s_{0}$ does not depend on $n$. For such a choice
of $s_{0}$ we have $D_{n+3N}>D_{n+s}$ and hence
$\underline{C_{1}}+\underline{C_{2}}<2\;\overline{C_{1}}$ for all $s\geq
s_{0}$. An application of (51) gives first inequality in (52) for $s\geq
s_{0}$.
For the second inequality in (52) we observe that
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>(S+D_{n}-D_{n+1})(S+D_{n})-(S+D_{n}-D_{n+1}+D_{n+s})^{2}.$
Since $S\geq 0$ it follows that
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>(D_{n}-D_{n+1})(D_{n+1}-2D_{n+s})-D_{n+s}^{2}.$
Therefore it suffices to show there is some $s_{0}$ (which does not depend on
$n$) such that
$D_{n}-D_{n+1}>D_{n+s}\qquad\text{and}\qquad D_{n+1}-2D_{n+s}>D_{n+s}$
or equivalently
$1>\alpha_{n+1}+\alpha_{n+1}\alpha_{n+2}\ldots\alpha_{n+s}\qquad\text{and}\qquad
1>3\;\alpha_{n+2}\alpha_{n+3}\ldots\alpha_{n+s}$
for all $s\geq s_{0}$. This is easily done with the help of (26).
The other possibility for $C_{1}$ is that
$C_{1}=B(\mathbf{c}_{n}).$
It is easy to see that $\beta=\sum^{\infty}_{k=1}b_{k}D_{k}\in
A(c_{1},c_{2},\ldots,c_{n-1},c_{n}+1)$ where $b_{1},b_{2},b_{3},\ldots$ is the
sequence
$c_{1},c_{2},\ldots,c_{n-1},c_{n}+1,a_{n+1}-2,a_{n+2}-2,a_{n+3}-2,\ldots,$
and so $A(c_{1},c_{2},\ldots,c_{n-1},c_{n}+1)\neq\emptyset$. Therefore
$C_{2}=A(c_{1},c_{2},\ldots,c_{n-1},c_{n}+1).$
Again we apply Lemmas 6.5 and 6.6 to $C_{1}$ and $C_{2}$. This time we write
$S_{1}=S(\mathbf{c}_{n})\qquad\text{and}\qquad
S_{2}=S(c_{1},c_{2},\ldots,c_{n-1},c_{n}+1).$
Note that $S_{2}=S_{1}+D_{n}$. Lemma 6.6 implies
$\underline{C_{1}}<S_{1}+D_{n}-D_{n+1}+D_{n+2}+D_{n+s}\qquad\text{and}\qquad\overline{C_{1}}=S_{1}+D_{n}$
and Lemma 6.5 implies
$\underline{C_{2}}<S_{2}+D_{n+s}\qquad\text{and}\qquad\overline{C_{2}}=S_{2}+D_{n}-D_{n+1}.$
These combine to yield
$2\;\overline{C_{1}}-(\underline{C_{1}}+\underline{C_{2}})>D_{n+1}-D_{n+2}-2D_{n+s}.$
Using (26) we know there is some $s_{0}\geq 1$ (which does not depend on $n$)
such that
$1>\alpha_{n+2}+2\;\alpha_{n+2}\alpha_{n+3}\ldots\alpha_{n+s}$
and hence $D_{n+1}>D_{n+2}+2D_{n+s}$ for all $s\geq s_{0}$. As a result
$\underline{C_{1}}+\underline{C_{2}}<2\;\overline{C_{1}}$ if $s\geq s_{0}$ and
using (51) we conclude that the first inequality in (52) holds if $s$ is large
enough.
To see that the second inequality in (52) is true we note that
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>(S_{1}+D_{n})(S_{2}+D_{n}-D_{n+1})-(S_{2}+D_{n+s})^{2}.$
Since $S_{2}=S_{1}+D_{n}$ and $S\geq 0$ it follows that
$\overline{C_{1}}\;\overline{C_{2}}-\underline{C_{2}}\;\underline{C_{2}}>D_{n}(D_{n}-D_{n+1}-2D_{n+s})-D_{n+s}^{2}.$
Therefore it suffices to show there is some $s_{0}$ (which does not depend on
$n$) such that
$D_{n}-D_{n+1}-2D_{n+s}>D_{n+s}$
or equivalently
$1>\alpha_{n+1}+3\;\alpha_{n+1}\alpha_{n+2}\ldots\alpha_{n+s}$
for all $s\geq s_{0}$. Again this is easily done with the help of (26). ∎
These sequence of lemmas leads to the following key precursor to the main
result.
###### Theorem 7.
There is an integer $s_{0}\geq N$ such that if $s\geq s_{0}$ and $R=N/(N+1)$
and
$P_{1}=\frac{R^{2s}}{(1-R^{(s-N)})^{2}}\qquad\text{and}\qquad
P_{2}=1-R^{(s-N)}$
then $P_{2}\geq P_{1}$ and the interval $[P_{1},P_{2}]$ lies in the product of
the sets $E(\alpha,s)$ and $F(\alpha,s)$.
###### Proof.
The proof applies Theorem 2.2 in Hall’s paper [15] to the sum
(54) $\log E(\alpha,s)\;+\;\log F(\alpha,s).$
It is appropriate to outline why this is possible. In the last section we
showed that the sets $E(\alpha,s)$ and $F(\alpha,s)$ are the result of Cantor
dissections of the intervals $G(\alpha,s)$ and $H(\alpha,s)$. By applying the
logarithm function it follows that the sets $\log E(\alpha,s)$ and $\log
F(\alpha,s)$ are the result of Cantor dissections of the intervals $\log
G(\alpha,s)$ and $\log H(\alpha,s)$. We know from Lemmas 6.7 and 6.8 that
these dissections satisfy Condition 1 in Hall’s paper, if $s$ is large enough.
In other words, there is some $s_{0}\geq N$ such that for all $s\geq s_{0}$
Hall’s theorem applies to the sum (54). Note that since $R<1$ we can choose
$s_{0}$ so that we also have $P_{2}>P_{1}$.
Hall’s theorem implies that the sum (54) contains the interval
$[\log x_{2}+\log y_{2}-2\min\\{\log x_{2}-\log x_{1},\;\log y_{2}-\log
y_{1}\\},\;\log x_{2}+\log y_{2}]$
where
$x_{1}=\underline{G}(\alpha,s)\qquad x_{2}=\overline{G}(\alpha,s)\qquad
y_{1}=\underline{H}(\alpha,s)\qquad y_{2}=\overline{H}(\alpha,s).$
It follows immediately that the product of $E(\alpha,s)$ and $F(\alpha,s)$
contains the interval
$[x_{2}y_{2}(\max\\{x_{1}/x_{2},y_{1}/y_{2}\\})^{2},\;x_{2}y_{2}].$
To prove the lemma it suffices to show
(55) $x_{2}y_{2}(\max\\{x_{1}/x_{2},y_{1}/y_{2}\\})^{2}\leq
P_{1}\qquad\text{and}\qquad x_{2}y_{2}\geq P_{2}.$
To this end we observe that $\overline{G}(\alpha,s)=\overline{B}(\;)$ and
$\overline{H}(\alpha,s)=\overline{C}(\;)$ and hence Lemma 6.4 and 6.6 imply
$x_{2}=1$ and $y_{2}>1-D_{s-N}$. Therefore $x_{2}y_{2}>1-D_{s-N}$. We know
$D_{s-N}=\alpha_{1}\alpha_{2}\ldots\alpha_{s-N}$ and since (26) holds for all
$i\geq 1$ it is easy to see that the second inequality in (55) is true.
For the first inequality in (55) we observe that
$\underline{G}(\alpha,s)=\underline{A}(\;)$ and
$\underline{H}(\alpha,s)=\underline{C}(\;)$ and hence Lemmas 6.4 and 6.5 imply
$x_{1}<D_{s}$ and $y_{1}<D_{s}$. Thus
$x_{1}/x_{2}<D_{s}\qquad\text{and}\qquad y_{1}/y_{2}<D_{s}/(1-D_{s-N}).$
Clearly $x_{2}y_{2}<1$ and it follows that
$x_{2}y_{2}(\max\\{x_{1}/x_{2},y_{1}/y_{2}\\})^{2}<\frac{D_{s}^{2}}{(1-D_{s-N})^{2}}.$
The truth of the first inequality in (6.4) can now be seen by expressing
$D_{s-N}$ and $D_{s}$ in terms of the numbers $\alpha_{i}$ and applying (26).
∎
Finally, we return to the sets $E(\alpha^{-},s)$ and $F(\alpha^{+}_{r+1},s)$,
where $r\geq 1$. Recall that $\alpha^{-}$ and $\alpha^{+}$ are defined by (33)
and $\alpha^{+}_{r+1}$ by (33). We know from Lemma 6.2 that $\alpha^{-}$ and
$\alpha^{+}$ satisfy all the constraints we have placed on $\alpha$. Clearly
the same is true of $\alpha^{+}_{r+1}$. We can, therefore, replace
$E(\alpha,s)$ and $F(\alpha,s)$ in Theorem 6.7 with $E(\alpha^{-},s)$ and
$F(\alpha^{+}_{r+1},s)$, respectively. In this manner we obtain the following
corollary.
###### Corollary 1.
There is an integer $s_{0}\geq N$ such that if $s\geq s_{0}$ and $R=N/(N+1)$
and
$P_{1}=\frac{R^{2s}}{(1-R^{(s-N)})^{2}}\qquad\text{and}\qquad
P_{2}=1-R^{(s-N)}$
then $P_{2}\geq P_{1}$ and the product of the sets $E(\alpha^{-},s)$ and
$F(\alpha^{+}_{r+1},s)$, where $r\geq 1$, contains the interval
$[P_{1},P_{2}]$.
### 6.5. The existence of Hall’s ray
In this section, we prove the existence of a Hall’s ray in the set
${\mathcal{S}}_{+}(\alpha)$ in (1); that is we prove Theorem 1.
###### Proof of Theorem 1.
The proof of this theorem consists of showing that the set
${\mathcal{S}}_{+}(\alpha)$ contains a chain of intersecting intervals whose
endpoints converge to zero. We shall construct the chain with the help of
Theorem 6 and the Corollary to Theorem 7.
Let $s_{0}\geq N$ be the integer mentioned in the Corollary to Theorem 7 and
define $r_{0}$ to be the smallest integer which is greater than or equal to
$s_{0}L$. Note that since $L\geq 1$ we have $s_{0}=\lfloor r_{0}/L\rfloor$
where as usual $\lfloor x\rfloor$ denotes the largest integer which is less
than or equal to $x$. Now let $r$ be an integer with $r\geq r_{0}$ and put
$s=\lfloor r/L\rfloor$. Since $r/s\geq L$ we can apply Theorem 6. Thus for
every number $x$ in the product of the sets $E(\alpha^{-},s)$ and
$F(\alpha^{+}_{r+1},s)$ there is some $\beta$ with $0<\beta<1$ such that
${\mathcal{M}}^{+}(\alpha,\beta)=\frac{xD^{+}_{r}}{1-\alpha^{-}\alpha^{+}}.$
Because $r\geq r_{0}$ we know that $s\geq s_{0}$. Therefore Theorem 7 implies
$P_{1}\leq P_{2}$ and the product of the sets $E(\alpha^{-},s)$ and
$F(\alpha^{+}_{r+1},s)$ contains the interval $[P_{1},P_{2}]$ where
$P_{1}=\frac{R^{2s}}{(1-R^{(s-N)})^{2}}\qquad\text{and}\qquad
P_{2}=1-R^{(s-N)}$
and $R=N/(N+1)$. It follows that for every number $\mu$ in the interval
(56)
$\left[\frac{P_{1}D^{+}_{r}}{1-\alpha^{-}\alpha^{+}}\;,\;\frac{P_{2}D^{+}_{r}}{1-\alpha^{-}\alpha^{+}}\right]$
there is some $\beta$ with $0<\beta<1$ such that $M(\alpha,\beta)=\mu$. In
other words the interval (56) lies in the set ${\mathcal{S}}^{+}(\alpha)$.
Since $r$ was any integer with $r\geq r_{0}$ we conclude that
${\mathcal{S}}^{+}(\alpha)$ contains a chain of intervals.
By choosing $s_{0}$ large enough we can ensure that the intervals just
mentioned intersect. To this end let $s^{\prime}=\lfloor(r+1)/L\rfloor$ and
set
$P^{\prime}_{1}=\frac{R^{2s^{\prime}}}{(1-R^{(s^{\prime}-N)})^{2}}\qquad\text{and}\qquad
P^{\prime}_{2}=1-R^{(s^{\prime}-N)}.$
Note that $s^{\prime}\geq s$. According to the argument above, the interval
for the integer $r+1$ is
$\left[\frac{P^{\prime}_{1}D^{+}_{r+1}}{1-\alpha^{-}\alpha^{+}}\;,\;\frac{P^{\prime}_{2}D^{+}_{r+1}}{1-\alpha^{-}\alpha^{+}}\right].$
It will overlap the interval (56) if both the inequalities
(57)
$\frac{P^{\prime}_{1}D^{+}_{r+1}}{1-\alpha^{-}\alpha^{+}}\;\leq\;\frac{P_{2}D^{+}_{r}}{1-\alpha^{-}\alpha^{+}}\qquad\text{and}\qquad\frac{P_{1}D^{+}_{r}}{1-\alpha^{-}\alpha^{+}}\;\leq\;\frac{P^{\prime}_{2}D^{+}_{r+1}}{1-\alpha^{-}\alpha^{+}}$
hold. These inequalities become $P^{\prime}_{1}\alpha^{+}_{r+1}\leq P_{2}$ and
$P_{1}\leq P^{\prime}_{2}\alpha^{+}_{r+1}$ and, substituting for $P_{1}$,
$P_{2}$, $P^{\prime}_{1}$ and $P^{\prime}_{2}$ and rearranging, we have
(58)
$R^{2s^{\prime}}\alpha^{+}_{r+1}\leq(1-R^{(s^{\prime}-N)})^{2}(1-R^{(s-N)})$
and
(59) $R^{2s}\leq(1-R^{(s-N)})^{2}(1-R^{(s^{\prime}-N)})\alpha^{+}_{r+1}.$
Now we observe that $R<1$ and hence the quantities $R^{2s}$ and $R^{(s-N)}$
and $R^{2s^{\prime}}$ and $R^{(s^{\prime}-N)}$ all converge to zero as $s$ and
$s^{\prime}$ increase to infinity. Since $s^{\prime}\geq s\geq s_{0}$ and the
term $\alpha^{+}_{r+1}$ satisfies $1/M<\alpha^{+}_{r+1}<N/(N+1)$, it is clear
that by choosing $s_{0}$ sufficiently large we can ensure that (57) always
holds. We conclude as indicated that $s_{0}$ can be chosen so that successive
members in the chain of intervals in ${\mathcal{S}}^{+}(\alpha)$ intersect one
another. Evidently the endpoints of the interval (56). converge to zero as $r$
increases to infinity. ∎
## References
* [1] E. S. Barnes. The inhomogeneous minima of binary quadratic forms. IV. Acta Math., 92:235–264, 1954.
* [2] E. S. Barnes. On linear inhomogeneous diophantine approximation. J. London Math. Soc., 31:73–79, 1956.
* [3] E. S. Barnes. The inhomogeneous minima of indefinite quadratic forms. J. Aust. Math. Soc., 2:9–10, 1961.
* [4] E. S. Barnes and H. P. F. Swinnerton-Dyer. The inhomogeneous minima of binary quadratic forms. I. Acta Math., 87:259–323, 1952.
* [5] E. S. Barnes and H. P. F. Swinnerton-Dyer. The inhomogeneous minima of binary quadratic forms. II. Acta Math., 88:279–316, 1952.
* [6] E. S. Barnes and H. P. F. Swinnerton-Dyer. The inhomogeneous minima of binary quadratic forms. III. Acta Math., 92:199–234, 1954.
* [7] P. E. Blanksby. Various Problems in Inhomogeneous Diophantine Approximation. PhD thesis, The University of Adelaide, 1967.
* [8] J.W.S. Cassels. Über $\liminf_{x\to+\infty}x|\theta x+\alpha-y|$. Mathematische Annalen, 127(1):288–304, December 1954.
* [9] Thomas W. Cusick, Andrew M. Rockett, and Peter Szűsz. On inhomogeneous diophantine approximation. Journal of Number Theory, 48:259–283, 1994.
* [10] T.W. Cusick and M.E. Flahive. The Markoff and Lagrange Spectra. Number 30 in Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1989.
* [11] T.W. Cusick, W. Moran, and A.D Pollington. Hall’s ray in inhomogeneous diophantine approximation. J. Aust. Math. Soc. (Series A), 60:42–50, 1996.
* [12] H. Davenport. On a theorem of Khintchine. Proceedings of the London Mathematical Society, 52:65–80, 1950\.
* [13] G.A. Freiman. The initial point of Hall’s ray. Kalinin. Gos. Univ. Moscow, (Russian), pages 87–120, 1973.
* [14] V. S. Fukasawa. Über die Grössenordnung des absoluten Betrages von einer linearen inhomogenen Form, I. Japanese Journal of Mathematics, 3:2–26, 1926.
* [15] M. Hall, Jr. On the sum and product of continued fractions. Ann. Math., 48:966–993, 1947.
* [16] A. Khintchine. Über eine Klasse linearer diophantischer Approximatinen. Rendiconti del Circolo Matematico di Palermo, 50:170–195, 1926\.
* [17] A. Khintchine. Neuer Beweis und Verallgemeinerung eines Hurwitzschen Satzes. Math. Annalen, 111:631–637, 1935.
* [18] T. Komatsu. On inhomogeneous diophantine approximation with some quasi-periodic expressions. Acta Mathematica Hungarica, 85(4):311–330, December 1999.
* [19] T. Komatsu. On inhomogeneous diophantine approximation with some quasi-periodic expressions, II. Journal de théorie des nombres de Bordeaux, 11(2):331–344, 1999\.
* [20] Takao Komatsu. On inhomogeneous continued fraction expansions and inhomogeneous diophantine approximation. Journal of Number Theory, 62(1):192 – 212, 1997.
* [21] H. Minkowski. Ueber die Annäherung an eine reelle Grösse durch rationale Zahlen. Mathematische Annalen, 54:91–124, 1901.
* [22] Oscar Perron. Die Lehre von den Kettenbrüchen : Elementare Kettenbrüche, volume 1. B.G. Teubner Verlagsellschaft, Stuttgart, 3rd edition, 1954.
* [23] C.G. Pinner. More on inhomogeneous diophantine approximation. Journal de théorie des nombres de Bordeaux, 13:539–557, 2001\.
* [24] Chistopher G. Pinner and Dan Wolczuk. On the inhomogeneous Hall’s ray of period 1 quadratics. Experimental Math., 10(4):487–495, 2001.
* [25] V. S. Prasad. On a theorem of Khintchine. Proceedings of the London Mathematical Society, 53:310–330, 1951\.
* [26] Andrew M. Rockett and Peter Szüsz. Continued Fractions. World Scientic, Singapore and Teaneck, NJ, 1992. ISBN 981-02-1047-1.
* [27] V. T. Sós. On the theory of diophantine approximations, II. Acta Math. Acad, Sci Hung., 9:229–241, 1958.
|
arxiv-papers
| 2012-03-20T00:18:50 |
2024-09-04T02:49:28.786903
|
{
"license": "Public Domain",
"authors": "D. J. Crisp, W. Moran and A. D. Pollington",
"submitter": "William Moran Bill Moran",
"url": "https://arxiv.org/abs/1203.4295"
}
|
1203.4337
|
# Necessary and sufficient conditions for boundedness of commutators of the
general fractional integral operators on weighted Morrey spaces
Zengyan Si Zengyan Si
School of Mathematics and Information Science
Henan Polytechnic University
Jiaozuo 454000
P. R. China sizengyan@yahoo.cn and Fayou Zhao∗ Fayou Zhao (Corresponding
author)
Department of Mathematics
Shanghai University
Shanghai 200444
P. R. China zhaofayou2008@yahoo.com.cn
###### Abstract.
We prove that $b$ is in $Lip_{\beta}(\beta)$ if and only if the commutator
$[b,L^{-\alpha/2}]$ of the multiplication operator by $b$ and the general
fractional integral operator $L^{-\alpha/2}$ is bounded from the weighed
Morrey space $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where $0<\beta<1$,
$0<\alpha+\beta<n,1<p<{n}/({\alpha+\beta})$,
${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$
$\omega^{{q}/{p}}\in A_{1}$ and $r_{\omega}>\frac{1-k}{p/q-k},$ and here
$r_{\omega}$ denotes the critical index of $\omega$ for the reverse Hölder
condition.
###### Key words and phrases:
commutator; weighted Lipschitz function; weighted Morrey space; fractional
integrals.
###### 2000 Mathematics Subject Classification:
42B20; 42B35
The second author is the corresponding author. The research was supported by
Shanghai Leading Academic Discipline Project (Grant No. J50101).
## 1\. Introduction and main results
Suppose that $L$ is a linear operator on $L^{2}({{{\mathbb{R}}}^{n}})$ which
generates an analytic semigroup $e^{-tL}$ with a kernel $p_{t}(x,y)$
satisfying a Gaussian upper bound, that is,
(1) $|p_{t}(x,y)|\leq\frac{C}{t^{{n}/{2}}}e^{-c\frac{|x-y|^{2}}{t}}$
for $x,y\in{{{\mathbb{R}}}^{n}}$ and all $t>0$. Since we assume only upper
bound on heat kernel $p_{t}(x,y)$ and no regularity on its space variables,
this property (1) is satisfied by a class of differential operator, see [1]
for details.
For $0<\alpha<n,$ the general fractional integral $L^{-\alpha/2}$ of the
operator $L$ is defined by
$L^{-\frac{\alpha}{2}}f(x)=\frac{1}{\Gamma(\frac{\alpha}{2})}\int_{0}^{\infty}e^{-tL}f\frac{dt}{t^{-\alpha/2+1}}(x).$
Note that if $L=-\Delta$ is the Laplacian on ${{{\mathbb{R}}}^{n}}$, then
$L^{-\alpha/2}$ is the classical fractional integral $I_{\alpha}$ which plays
important roles in many fields. Let $b$ be a locally integrable function on
${{{\mathbb{R}}}^{n}}$, the commutator of $b$ and $L^{-\alpha/2}$ is defined
by
$[b,L^{-\alpha/2}]f(x)=b(x)L^{-\alpha/2}f(x)-L^{-\alpha/2}(bf)(x).$
For the special case of $L=-\Delta$, many results have been produced.
Paluszyński [7] obtained that $b\in Lip_{\beta}({{{\mathbb{R}}}^{n}})$ if the
commutator $[b,I_{\alpha}]$ is bounded from $L^{p}({{{\mathbb{R}}}^{n}})$ to
$L^{r}({{{\mathbb{R}}}^{n}})$, where $1<p<r<\infty,0<\beta<1$ and
$1/p-1/r=(\alpha+\beta)/n$ with $p<n/(\alpha+\beta)$. Shirai [9] proved that
$b\in Lip_{\beta}({{{\mathbb{R}}}^{n}})$ if and only if the commutator
$[b,I_{\alpha}]$ is bounded from the classical Morrey spaces
$L^{p,\lambda}({{{\mathbb{R}}}^{n}})$ to $L^{q,\lambda}({{{\mathbb{R}}}^{n}})$
for $1<p<q<\infty,\ 0<\alpha,\ 0<\beta<1$ and
$0<\alpha+\beta=(1/p-1/q)(n-\lambda)<n$ or
$L^{p,\lambda}({{{\mathbb{R}}}^{n}})$ to $L^{q,\mu}({{{\mathbb{R}}}^{n}})$ for
$1<p<q<\infty,\ 0<\alpha,\ 0<\beta<1,\ 0<\alpha+\beta=(1/p-1/q)<n,\
0<\lambda<n-(\alpha+\beta)p$ and $\mu/q=\lambda/p$. Wang [12] established some
weighted boundedness of properties of commutator $[b,I_{\alpha}]$ on the
weighted Morrey spaces $L^{p,k}$ under appropriated conditions on the weight
$\omega$, where the symbol $b$ belongs to (weighted) Lipschitz spaces. The
weighted Morrey space was first introduced by Komori and Shirai [5]. For the
general case, Wang [13] proved that if $b\in
Lip_{\beta}({{{\mathbb{R}}}^{n}})$, then the commutator $[b,I_{\alpha}]$ is
bounded from $L^{p}(\omega^{p})$ to $L^{q}(\omega^{q})$, where $0<\beta<1,\
0<\alpha+\beta<n,\ 1<p<n/(\alpha+\beta),1/p-1/q=(\alpha+\beta)/n$ and
$\omega^{q}\in A_{1}$.
The purpose of this paper is to give necessary and sufficient conditions for
boundedness of commutators of the general fractional integrals with $b\in
Lip_{\beta}(\omega)$ (the weighted Lipschitz space). Our theorems are the
following:
###### Theorem 1.1.
Let $0<\beta<1$, $0<\alpha+\beta<n,1<p<\frac{n}{\alpha+\beta}$,
${1}/{q}={1}/{p}-({\alpha+\beta})/{n},$ $0\leq
k<\min\\{{p}/{q},{p\beta}/{n}\\}$ and $\omega^{q}\in A_{1}$. Then we have
(a) If $b\in Lip_{\beta}({{{\mathbb{R}}}^{n}}),$ then $[b,L^{-\alpha/2}]$ is
bounded from $L^{p,k}(\omega^{p},\omega^{q})$ to $L^{q,kq/p}(\omega^{q})$;
(b) If $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega^{p},\omega^{q})$ to
$L^{q,kq/p}(\omega^{q})$, then $b\in Lip_{\beta}({{{\mathbb{R}}}^{n}})$.
###### Theorem 1.2.
Let $0<\beta<1$, $0<\alpha+\beta<n,1<p<\frac{n}{\alpha+\beta}$,
${1}/{q}={1}/{p}-({\alpha+\beta})/{n},$ $0\leq k<{p}/{q},$
$\omega^{{q}/{p}}\in A_{1}$ and $r_{\omega}>\frac{1-k}{p/q-k},$ where
$r_{\omega}$ denotes the critical index of $\omega$ for the reverse Hölder
condition. Then we have
(a) If $b\in Lip_{\beta}(\omega),$ then $[b,L^{-\alpha/2}]$ is bounded from
$L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$;
(b) If $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, then $b\in
Lip_{\beta}(\omega)$.
Our results not only extend the results of [12] from $(-\triangle)$ to a
general operator $L$, but also characterize the (weighted) Lipschitz spaces by
means of the boundedness of $[b,L^{-\alpha/2}]$ on the weighted Morrey spaces,
which extend the results of [12] and [13]. The basic tool is based on a
modification of sharp maximal function $M^{\sharp}_{L}$ introduced by [6].
Throughout this paper all notation is standard or will be defined as needed.
Denote the Lebesgue measure of $B$ by $|B|$ and the weighted measure of $B$ by
$\omega(B)$, where $\omega(B)=\int_{B}\omega(x)dx$. For a measurable set $E$,
denote by $\chi_{E}$ the characteristic function of $E$. For a real number
$p$, $1<p<\infty$, let $p^{\prime}$ be the dual of $p$ such that
$1/p+1/{p^{\prime}}=1$. The letter $C$ will be used for various constants, and
may change from one occurrence to another.
## 2\. Some preliminaries
A non-negative function $\omega$ defined on ${{{\mathbb{R}}}^{n}}$ is called
weight if it is locally integral. A weight $\omega$ is said to belong to the
Muckenhoupt class $A_{p}({{{\mathbb{R}}}^{n}})$ for $1<p<\infty$, if there
exists a constant $C$ such that
$\left(\frac{1}{|B|}\int_{B}\omega(x)dx\right)\left(\frac{1}{|B|}\int_{B}\omega(x)^{-\frac{1}{p-1}}dx\right)^{p-1}\leq
C.$
for every ball $B\subset{{{\mathbb{R}}}^{n}}$. The class
$A_{1}({{{\mathbb{R}}}^{n}})$ is defined replacing the above inequality by
$\left(\frac{1}{|B|}\int_{B}\omega(x)dx\right)\leq
C\mathop{\textup{essinf}}_{x\in B}\,\omega(x).$
When $p=\infty,\omega\in A_{\infty},$ if there exist positive constants
$\delta$ and $C$ such that given a ball $B$ and $E$ is a measurable subset of
$B$, then
$\frac{\omega(E)}{\omega(B)}\leq C\left(\frac{|E|}{|B|}\right)^{\delta}.$
A weight function $\omega$ belongs to $A_{p,q}$ for $1<p<q<\infty$ if for
every ball $B$ in ${{{\mathbb{R}}}^{n}}$, there exists a positive constant $C$
which is independent of $B$ such that
$\left(\frac{1}{|B|}\int_{B}\omega(x)^{q}dx\right)^{\frac{1}{q}}\left(\frac{1}{|B|}\int_{B}\omega(x)^{-p^{\prime}}dx\right)^{\frac{1}{p^{\prime}}}\leq
C.$
From the definition of $A_{p,q}$, we can get that
(2) $\omega\in A_{p,q}\ if\ and\ only\ if\ \omega^{q}\in
A_{1+q/{p^{\prime}}}.$
Since $\omega^{q/p}\in A_{1}$, then by (2), we have $\omega^{1/p}\in A_{p,q}$.
A weight function $\omega$ belongs to the reverse Hölder class $RH_{r}$ if
there exist two constants $r>1$ and $C>0$ such that the following reverse
Hölder inequality
$\left(\frac{1}{|B|}\int_{B}\omega(x)^{r}dx\right)^{\frac{1}{r}}\leq
C\frac{1}{|B|}\int_{B}\omega(x)dx$
holds for every ball $B$ in ${{{\mathbb{R}}}^{n}}$.
It is well known that if $\omega\in A_{p}$ with $1\leq p<\infty$, then there
exists $r>1$ such that $\omega\in RH_{r}.$ It follows from Hölder s inequality
that $\omega\in RH_{r}$ implies $\omega\in RH_{s}$ for all $1<s<r.$ Moreover,
if $\omega\in RH_{r},r>1,$ then we have $\omega\in RH_{r+\epsilon}$ for some
$\epsilon>0.$ We thus write $r_{w}=\sup\\{r>1:\omega\in RH_{r}\\}$ to denote
the critical index of $\omega$ for the reverse Hölder condition. For more
details on Muchenhoupt class $A_{p,q}$, we refer the reader to [3], [10] and
[11].
###### Definition 2.1.
([5]) Let $1\leq p<\infty$ and $0\leq k<1$. Then for two weights $\mu$ and
$\nu$, the weighted Morrey space is defined by
$L^{p,k}(\mu,\nu)=\\{f\in
L_{loc}^{p}(\mu):\|f\|_{L^{p,k}(\mu,\nu)}<\infty\\},$
where
$\|f\|_{L^{p,k}(\mu,\nu)}=\sup_{B}\left(\frac{1}{\nu(B)^{k}}\int_{B}|f(x)|^{p}\mu(x)dx\right)^{\frac{1}{p}}.$
and the supremum is taken over all balls $B$ in ${{{\mathbb{R}}}^{n}}$.
If $\mu=\nu,$ then we have the classical Morrey space $L^{p,k}(\mu)$ with
measure $\mu$. When $k=0,$ then $L^{p,k}(\mu,\nu)=L^{p}(\mu)$ is the Lebesgue
space with measure $\mu$.
###### Definition 2.2.
([2]) Let $1\leq p<\infty$, $0<\beta<1$, and $\omega\in A_{\infty}$. A locally
integral function $b$ is said to be in $Lip^{p}_{\beta}(\omega)$ if
$\|b\|_{Lip^{p}_{\beta}(\omega)}=\sup_{B}\frac{1}{\omega(B)^{\beta/n}}\left(\frac{1}{\omega(B)}\int_{B}|b(x)-b_{B}|^{p}\omega(x)^{1-p}dx\right)^{\frac{1}{p}}\leq
C<\infty,$
where $b_{B}={|B|^{-1}}\int_{B}b(y)dy$ and the supremum is taken over all ball
$B\subset R^{n}.$ When $p=1,$ we denote $Lip^{p}_{\beta}(\omega)$ by
$Lip_{\beta}(\omega).$
Obviously, for the case $\omega=1$, then the $Lip^{p}_{\beta}(\omega)$ space
is the classical $Lip^{p}_{\beta}$ space.
###### Remark 2.1.
Let $\omega\in A_{1}$, García-Cuerva [2] proved that the spaces
$\|f\|_{Lip^{p}_{\beta}(\omega)}$ coincide, and the norm of
$||\cdot||_{Lip^{p}_{\beta}(\omega)}$ are equivalent with respect to different
values of provided that $1\leq p<\infty.$
Given a locally integrable function $f$ and $\beta$, $0\leq\beta<n$, define
the fractional maximal function by
$M_{\beta,r}f(x)=\sup_{x\in B}\left(\frac{1}{|B|^{1-{\beta
r}/{n}}}\int_{B}|f(y)|^{r}dy\right)^{\frac{1}{r}},\quad r\geq 1,$
when $0<\beta<n$. If $\beta=0$ and $r=1$, then $M_{0,\ 1}f=Mf$ denotes the
usual Hardy-Littlewood maximal function.
Let $\omega$ be a weight. The weighted maximal operator $M_{\omega}$ is
defined by
$M_{\omega}f(x)=\sup_{x\in B}\frac{1}{\omega(B)}\int_{B}|f(y)|dy.$
The fractional weighted maximal operator $M_{\beta,r,\omega}$ is defined by
$M_{\beta,r,\omega}f(x)=\sup_{x\in B}\left(\frac{1}{\omega(B)^{1-{\beta
r}/{n}}}\int_{B}|f(y)|^{r}\omega(y)dy\right)^{\frac{1}{r}},$
where $0\leq\beta<n$ and $r\geq 1$. For any $f\in
L^{p}({{{\mathbb{R}}}^{n}}),\ p\geq 1,$ the sharp maximal function
$M^{\sharp}_{L}f$ associated the generalized approximations to the identity
$\\{e^{-tL},\ t>0\\}$ is given by Martell [6] as follows:
$M^{\sharp}_{L}f(x)=\sup_{x\in
B}\frac{1}{|B|}\int_{B}|f(y)-e^{-t_{B}L}f(y)|dy,$
where $t_{B}=r^{2}_{B}$ and $r_{B}$ is the radius of the ball $B$. For
$0<\delta<1$, we introduce the $\delta-$sharp maximal operator
$M_{L,\delta}^{\sharp}$ as
$M_{L,\delta}^{\sharp}f=M_{L}^{\sharp}(|f|^{\delta})^{1/\delta},$
which is a modification of the sharp maximal operator $M^{\sharp}$ of
Fefferman and Stein ([10]). Set $M_{\delta}f=M(|f|^{\delta})^{1/\delta}$.
Using the same methods as those of [10] and [8], we can get
###### Lemma 2.1.
Assume that the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$ which satisfies
the upper bound (1). Let $\lambda>0$ and $f\in L^{p}({{{\mathbb{R}}}^{n}})$
for some $1<p<\infty$. Suppose that $\omega\in A_{\infty}$, then for every
$0<\eta<1$, there exists a real number $\gamma>0$ independent of $\gamma,\ f$
such that we have the following weighted version of the local good $\lambda$
inequality, for $\eta>0$, $A>1$,
$\omega\\{x\in{{{\mathbb{R}}}^{n}}:M_{\delta}f>A\lambda,M_{L,\delta}^{\sharp}f(x)\leq\gamma\lambda\\}\leq\eta\omega\\{x\in{{{\mathbb{R}}}^{n}}:M_{\delta}f(x)>\lambda\\}.$
where $A>1$ is a fixed constant which depends only on $n$.
If $\mu,\nu\in A_{\infty},1<p<\infty,0\leq k<1$, then
(3) $\|f\|_{L^{p,k}(\mu,\nu)}\leq\|M_{\delta}f\|_{L^{p,k}(\mu,\nu)}\leq
C\|M^{\sharp}_{{}_{L},\delta}f\|_{L^{p,k}(\mu,\nu)}.$
In particular, when $\mu=\nu=\omega$ and $\omega\in A_{\infty},$ we have
(4) $\|f\|_{L^{p,k}(\omega)}\leq\|M_{\delta}f\|_{L^{p,k}(\omega)}\leq
C\|M^{\sharp}_{{}_{L},\delta}f\|_{L^{p,k}(\omega)}.$
## 3\. proof of theorem 1.1
To prove Theorem 1.1, we need the following lemmas.
###### Lemma 3.1.
([1]) Assume that the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$ which
satisfies the upper bound (1). Then for $0<\alpha<1,$ the difference operator
$L^{-\frac{\alpha}{2}}-e^{-tL}L^{-\frac{\alpha}{2}}$ has an associated kernel
$K_{\alpha,t}(x,y)$ which satisfies
$K_{\alpha,t}(x,y)\leq\frac{C}{|x-y|^{n-\alpha}}\frac{t}{|x-y|^{2}}.$
###### Lemma 3.2.
([12]) Let $0<\alpha+\beta<n$, $1<p<n/{(\alpha+\beta)},\
1/q=1/p-(\alpha+\beta)/n$ and $\omega\in A_{1}$. Then for every $0<k<p/q$ and
$1<r<p$, we have
$\|M_{\alpha+\beta,r}f\|_{L^{q,kq/p}(\omega^{q})}\leq
C\|f\|_{L^{p,q}(\omega^{p},\omega^{q})}.$
###### Lemma 3.3.
([5]) Let $0<\beta<n$, $1<p<n/{\beta},\ 1/s=1/p-\beta/n$ and $\omega\in
A_{p,s}$. Then for every $0<k<p/s$, we have
$\|M_{\beta,1}f\|_{L^{s,ks/p}(\omega^{s})}\leq
C\|f\|_{L^{p,k}(\omega^{p},\omega^{s})}.$
###### Lemma 3.4.
([12]) Let $0<\alpha+\beta<n$, $1<p<n/{(\alpha+\beta)},\ 1/q=1/p-\alpha/n$,
$1/s=1/q-\beta/n$ and $\omega^{q}\in A_{1}$. Then for every $0<k<p/s$, we have
$\|M_{\beta,1}f\|_{L^{s,ks/p}(\omega^{s})}\leq
C\|f\|_{L^{q,kq/p}(\omega^{q},\omega^{s})}.$
###### Lemma 3.5.
Let $0<\alpha+\beta<n$, $1<p<n/{(\alpha+\beta)},\ 1/q=1/p-\alpha/n$,
$1/s=1/q-\beta/n$ and $\omega^{q}\in A_{1}$. Then for every $0<k<p\beta/n$, we
have
$\|L^{-\alpha/2}f\|_{L^{q,kq/p}(\omega^{q},\omega^{s})}\leq
C\|f\|_{L^{p,k}(\omega^{p},\omega^{s})}.$
###### Proof.
As before, we know that $L^{-\alpha/2}f(x)\leq CI_{\alpha}(|f|)(x)$ for all
$x\in{{{\mathbb{R}}}^{n}}.$ Together with the result (cf. [12]), that is,
$\|I_{\alpha}f\|_{L^{q,kq/p}(\omega^{q},\omega^{s})}\leq
C\|f\|_{L^{p,k}(\omega^{p},\omega^{s})},$
we can get the desired result. ∎
###### Remark 3.1.
Using the boundedness property of $I_{\alpha}$, we also know $L^{-\alpha/2}$
is bounded from $L^{1}$ to weak $L^{n/(n-\alpha)}$. It is easy to check that
Lemma 3.2-3.5 also hold when $k=0$.
The following lemma plays an important role in the proof of Theorem 1.1.
###### Lemma 3.6.
Let $0<\delta<1,\ 0<\alpha<n,\ 0<\beta<1$ and $b\in
Lip_{\beta}({{{\mathbb{R}}}^{n}}).$ Then for all $r>1$ and for all
$x\in{{{\mathbb{R}}}^{n}},$ we have
$\displaystyle M^{\sharp}_{L,\delta}([b,L^{-\alpha/2}]f)(x)$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}({{{\mathbb{R}}}^{n}})}\left(M_{\beta,1}(L^{-\alpha/2}f)(x)+M_{\alpha+\beta,r}f(x)+M_{\alpha+\beta,1}f(x)\right).$
The same method of proof as that of Lemma 4.6 (see below), we omit the
details.
Proof of Theorem 1.1. We first prove $(a)$. We only prove Theorem 1.1 in the
case $0<\alpha<1$. For the general case $0<\alpha<n$, the method is the same
as that of [1]. We omit the details.
For $0<\alpha+\beta<n$ and $1<p<n/(\alpha+\beta)$, we can find a number $r$
such that $1<r<p$. By Eq.(4) and Lemma 3.6, we obtain
$\displaystyle\|[b,\ L^{-\alpha/2}]f\|_{L^{q,kq/p}(\omega^{q})}$
$\displaystyle\leq$ $\displaystyle C\|M^{\sharp}_{L,\delta}([b,\
L^{-\alpha/2}]f)\|_{L^{q,kq/p}(\omega^{q})}$ $\displaystyle\leq$
$\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\left(\|M_{\beta,1}(L^{-\alpha/2}f)\|_{L^{q,kq/p}(\omega^{q})}\right.$
$\displaystyle\quad+$
$\displaystyle\|M_{\alpha+\beta,r}f\|_{L^{q,kq/p}(\omega^{q})}+\left.\|M_{\alpha+\beta,1}f\|_{L^{q,kq/p}(\omega^{q})}\right).$
Let $1/{q_{1}}=1/p-\alpha/n$ and $1/q=1/{q_{1}}-\beta/n$. Since $\omega^{q}\in
A_{1}$, then by Eq.(2), we have $\omega\in A_{p,q}$. Since $0<k<\min\\{p/q,\
p\beta/n\\}$, by Lemmas 3.2–3.5, we yield that
$\displaystyle\|[b,\ L^{-\alpha/2}]f\|_{L^{q,kq/p}(\omega^{q})}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}({{{\mathbb{R}}}^{n}})}\left(\|L^{-\alpha/2}f\|_{L^{{q_{1}},k{q_{1}}/p}(\omega^{{q_{1}}},\
\omega^{q})}+\|f\|_{L^{p,k}(\omega^{p},\ \omega^{q})}\right)$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}({{{\mathbb{R}}}^{n}})}\|f\|_{L^{p,k}(\omega^{p},\
\omega^{q})}.$
Now we prove $(b)$. Let $L=-\Delta$ be the Laplacian on
${{{\mathbb{R}}}^{n}}$, then $L^{-{\alpha/2}}$ is the classical fractional
integral $I_{\alpha}$. Let $k=0$ and weight $\omega\equiv 1,$ then
$L^{p,k}(\omega^{p},\omega^{q})=L^{p}$ and
$L^{q,kq/p}(\omega^{q},\omega)=L^{q}.$ From [7], the $(L^{p},L^{q})$
bounedness of $[b,I_{\alpha}]$ implies that $b\in
Lip_{\beta}({{{\mathbb{R}}}^{n}})$.
Thus Theorem 1.1 is proved. ∎
## 4\. proof of theorem 1.2
We also need some Lemmas to prove Theorem 1.2.
###### Lemma 4.1.
([12]) Let
$0<\alpha+\beta<n,1<p<\frac{n}{\alpha+\beta},{1}/{q}={1}/{p}-{\alpha}/{n},{1}/{s}={1}/{q}-{\beta}/{n}$
and $\omega^{s/p}\in A_{1}.$ Then if $0<k<p/s$ and
$r_{\omega}>\frac{1}{p/q-k}$, we have
$\|M_{\beta,1}f\|_{L^{s,ks/p}(\omega^{s/p},\omega)}\leq
C\|f\|_{L^{q,kq/p}(\omega^{q/p},\omega)}.$
###### Lemma 4.2.
([12]) Let $0<\alpha<n,1<p<{n}/{\alpha},{1}/{q}={1}/{p}-{\alpha}/{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q$ and $r_{\omega}>\frac{1-k}{p/q-k}$,
we have
$\|M_{\alpha,1}f\|_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C\|f\|_{L^{p,k}(\omega)}.$
###### Lemma 4.3.
([12]) Let $0<\alpha<n,1<p<{n}/{\alpha},{1}/{q}={1}/{p}-{\alpha}/{n}$,
$0<k<p/q$, $\omega\in A_{\infty}.$ For any $1<r<p,$ we have
$\|M_{\alpha,r,\omega}f\|_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C\|f\|_{L^{p,k}(\omega)}.$
###### Lemma 4.4.
Let $0<\alpha<n,1<p<{n}/{\alpha},{1}/{q}={1}/{p}-{\alpha}/{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q$ and $r_{\omega}>\frac{1-k}{p/q-k}$,
we have
$\|L^{-\alpha/2}f\|_{L^{q,kq/p}(\omega^{q/p},\ \omega)}\leq
C\|f\|_{L^{p,k}(\omega)}.$
###### Proof.
Since the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$ which satisfies the
upper bound (1), it is easy to check that $L^{-\alpha/2}f(x)\leq
CI_{\alpha}(|f|)(x)$ for all $x\in{{{\mathbb{R}}}^{n}}.$ Using the boundedness
property of $I_{\alpha}$ on weighted Morrey space (cf. [12]), we have
$\|L^{-\alpha/2}f\|_{L^{q,kq/p}(\omega^{q/p},\
\omega)}\leq\|I_{\alpha}f\|_{L^{q,kq/p}(\omega^{q/p},\ \omega)}\leq
C\|f\|_{L^{p,k}(\omega)},$
where $1<p<{n}/{\alpha}$ and ${1}/{q}={1}/{p}-{\alpha}/{n}.$ ∎
###### Remark 4.1.
It is easy to check that the above lemmas also hold for $k=0$.
###### Lemma 4.5.
Assume that the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$ which satisfies
the upper bound (1), and let $b\in Lip_{\beta}(\omega),\ \omega\in A_{1}.$
Then, for every function $f\in L^{p}({{{\mathbb{R}}}^{n}}),\ p>1,\
x\in{{{\mathbb{R}}}^{n}},$ and $1<r<\infty,$ we have
$\sup_{x\in B}\frac{1}{|B|}\int_{B}|e^{-t_{B}L}(b(y)-b_{B})f(y)|dy\leq
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)M_{\beta,r,\omega}f(x).$
###### Proof.
Fix $f\in L^{p}({{{\mathbb{R}}}^{n}}),1<p<\infty$ and $x\in B.$ Then
$\displaystyle\frac{1}{|B|}\int_{B}|e^{-t_{B}L}((b(\cdot)-b_{B})f)(y)|dy$
$\displaystyle\leq$
$\displaystyle\frac{1}{|B|}\int_{B}\int_{{{{\mathbb{R}}}^{n}}}|p_{t_{B}}(y,z)\|(b(z)-b_{B})f(z)|dzdy$
$\displaystyle\leq$
$\displaystyle\frac{1}{|B|}\int_{B}\int_{2B}|p_{t_{B}}(y,z)\|(b(z)-b_{B})f(z)|dzdy$
$\displaystyle\quad+$
$\displaystyle\frac{1}{|B|}\int_{B}\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus
2^{k}B}|p_{t_{B}}(y,z)\|(b(z)-b_{B})f(z)|dzdy$ $\displaystyle\doteq$
$\displaystyle\mathcal{M}+\mathcal{N}.$
It follows from $y\in B$ and $z\in 2B$ that
$|p_{t_{B}}(y,z)|\leq Ct^{-{n}/{2}}_{B}\leq C\frac{1}{|2B|}.$
Thus, Hölder’s inequality and Definition 2.2 lead to
$\displaystyle\mathcal{M}$ $\displaystyle\leq$ $\displaystyle
C\frac{1}{|2B|}\int_{2B}|(b(z)-b_{B})f(z)|dz$ $\displaystyle\leq$
$\displaystyle
C\frac{1}{|2B|}\left(\int_{2B}\|b(z)-b_{B}|^{r^{\prime}}\omega(z)^{1-r^{\prime}}dz\right)^{\frac{1}{r^{\prime}}}\left(\int_{2B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\frac{1}{|2B|}\omega(2B)^{\frac{\beta}{n}+\frac{1}{r^{\prime}}}\omega(2B)^{\frac{1}{r}}\left(\frac{1}{\omega(2B)}\int_{2B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\frac{1}{|2B|}\omega(2B)^{\frac{\beta}{n}+1}\left(\frac{1}{\omega(2B)}\int_{2B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)\left(\frac{1}{\omega(2B)^{1-\frac{\beta
r}{n}}}\int_{2B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)M_{\beta,r,\omega}f(x).$
Moreover, for any $y\in B$ and $z\in 2^{k+1}B\setminus 2^{k}B$, we have
$|y-z|\geq 2^{k-1}r_{B}$ and $|p_{t_{B}}|\leq
C\frac{e^{-c2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}$.
$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\frac{1}{|B|}\int_{B}\sum_{k=1}^{\infty}\int_{2^{k+1}B\setminus
2^{k}B}|p_{t_{B}}(y,z)\|(b(z)-b_{B})f(z)|dzdy$ $\displaystyle\leq$
$\displaystyle
C\sum_{k=1}^{\infty}\frac{e^{-c2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|(b(z)-b_{B})f(z)|dz$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}\frac{e^{-c2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|(b(z)-b_{2^{k+1}B})f(z)|dz$
$\displaystyle\quad+$ $\displaystyle
C\sum_{k=1}^{\infty}\frac{e^{-c2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|(b_{2^{k+1}B}-b_{2B})f(z)|dz$
$\displaystyle\doteq$ $\displaystyle\mathcal{N}_{1}+\mathcal{N}_{2}.$
We will estimate the values of terms $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$,
respectively.
Using Hölder’s inequality and Remark 2.1, we have
$\displaystyle\mathcal{N}_{1}$ $\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}\frac{e^{-c2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}$
$\displaystyle\times\left(\int_{2^{k+1}B}|b(z)-b_{B}|^{r^{\prime}}\omega(z)^{1-r^{\prime}}dz\right)^{\frac{1}{r^{\prime}}}\left(\int_{2^{k+1}B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}2^{(k+1)n}e^{-c2^{2(k-1)}}$
$\displaystyle\times\|b\|_{Lip_{\beta}(\omega)}\frac{\omega(2^{k+1}B)}{|2^{k+1}B|}\left(\frac{1}{\omega(2^{k+1}B)^{1-\beta
r/n}}\int_{2^{k+1}B}|f(z)|^{r}\omega(z)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)M_{\beta,r,\omega}f(x).$
Since $\omega\in A_{1},$ by the Hölder inequality, we get
$\displaystyle\mathcal{N}_{2}$ $\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}2^{(k+1)n}e^{-c2^{2(k-1)}}\frac{k}{|2^{k+1}B|^{1-\beta
r/n}}\omega(x)\|b\|_{Lip_{\beta}(\omega)}\int_{2^{k+1}B}|f(z)|dz$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}k2^{(k+1)n}e^{-c2^{2(k-1)}}\omega(x)\|b\|_{Lip_{\beta}(\omega)}\left(\frac{1}{|2^{k+1}B|^{1-\beta
r/n}}\int_{2^{k+1}B}|f(z)|^{r}dz\right)^{\frac{1}{r}}$ $\displaystyle=$
$\displaystyle C\sum_{k=1}^{\infty}k2^{(k+1)n}e^{-c2^{2(k-1)}}$
$\displaystyle\times\omega(x)\|b\|_{Lip_{\beta}(\omega)}\left(\frac{\omega(2^{k+1}B)^{1-\beta
r/n}}{|2^{k+1}B|^{1-\beta r/n}}\frac{1}{\omega(2^{k+1}B)^{1-\beta
r/n}}\int_{2^{k+1}B}|f(z)|^{r}dz\right)^{\frac{1}{r}}$ $\displaystyle\leq$
$\displaystyle
C\sum_{k=1}^{\infty}k2^{(k+1)n}e^{-c2^{2(k-1)}}\omega(x)\|b\|_{Lip_{\beta}(\omega)}\left(\frac{1}{\omega(2^{k+1}B)^{1-\beta
r/n}}\int_{2^{k+1}B}|f(z)|^{r}\omega(x)dz\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)M_{\beta,r,\omega}f(x).$
Thus Lemma 4.5 is proved. ∎
###### Lemma 4.6.
Let $0<\alpha<1,$ $\omega\in A_{1}$ and $b\in Lip_{\beta}(\omega).$ Then for
all $r>1$ and for all $x\in{{{\mathbb{R}}}^{n}},$ we have
$\displaystyle M^{\sharp}_{L,\delta}([b,L^{-\alpha/2}]f)(x)\leq
C\|b\|_{Lip_{\beta}(\omega)}$
$\displaystyle\times\left(\omega(x)^{1+\frac{\beta}{n}}M_{\beta,1}(L^{-\alpha/2}f)(x)+\omega(x)^{1-\frac{\alpha}{n}}M_{\alpha+\beta,r,\omega}f(x)+\omega(x)^{1+\frac{\beta}{n}}M_{\alpha+\beta,1}f(x)\right).$
###### Proof.
For any given $x\in{{{\mathbb{R}}}^{n}},$ fix a ball $B=B(x_{0},r_{B})$ which
contains $x.$ We decompose $f=f_{1}+f_{2},$ where $f_{1}=f\chi_{2B}.$ Observe
that
$[b,L^{-\alpha/2}]f(x)=(b-b_{B})L^{-\alpha/2}f-L^{-\alpha/2}(b-b_{B})f_{1}-L^{-\alpha/2}(b-b_{B})f_{2}$
and
$e^{-t_{B}L}([b,L^{-\alpha/2}]f)=e^{-t_{B}L}[(b-b_{B})L^{-\alpha/2}f-L^{-\alpha/2}(b-b_{B})f_{1}-L^{-\alpha/2}(b-b_{B})f_{2}].$
Then
$\displaystyle\left(\frac{1}{|B|}\int_{B}\left|[b,L^{-\alpha/2}]f(y)-e^{-t_{B}L}[b,L^{-\alpha/2}]f(y)\right|^{\delta}dy\right)^{1/\delta}$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{1}{|B|}\int_{B}\left|(b(y)-b_{B})L^{-\alpha/2}f(y)dy\right|^{\delta}\right)^{1/\delta}$
$\displaystyle+C\left(\frac{1}{|B|}\int_{B}\left|L^{-\alpha/2}(b(y)-b_{B})f_{1})(y)|dy\right|^{\delta}\right)^{1/\delta}$
$\displaystyle+C\left(\frac{1}{|B|}\int_{B}\left|e^{-t_{B}L}((b(y)-b_{B})L^{-\alpha/2}f)(y)|dy\right|^{\delta}\right)^{1/\delta}$
$\displaystyle+C\left(\frac{1}{|B|}\int_{B}\left|e^{-t_{B}L}L^{-\alpha/2}((b(y)-b_{B})f_{1}(y))|dy\right|^{\delta}\right)^{1/\delta}$
$\displaystyle+C\left(\frac{1}{|B|}\int_{B}\left|(L^{-\alpha/2}-e^{-t_{B}L}L^{-\alpha/2})((b(y)-b_{B})f_{2})(y)|dy\right|^{\delta}\right)^{1/\delta}$
$\displaystyle\doteq$ $\displaystyle I+II+III+IV+V.$
We are going to estimate each term, respectively. Fix $0<\delta<1$ and choose
a real number $\tau$ such that $1<\tau<2$ and $\tau^{\prime}\delta<1$. Since
$\omega\in A_{1},$ then it follows from Hölder’s inequality that
$\displaystyle I$ $\displaystyle\leq$ $\displaystyle
C\left(\frac{1}{|B|}\int_{B}\left|(b(y)-b_{B})\right|^{\tau\delta}dy\right)^{\frac{1}{\tau\delta}}\left(\int_{B}\left|L^{-\alpha/2}f(y)\right|^{\tau^{\prime}\delta}dy\right)^{\frac{1}{\tau^{\prime}\delta}}$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{1}{|B|}\int_{B}\left|(b(y)-b_{B})\right|dy\right)\left(\int_{B}\left|L^{-\alpha/2}f(y)\right|dy\right)$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\frac{1}{|B|}\omega(B)^{1+{\beta}/{n}}\left(\int_{B}\left|L^{-\alpha/2}f(y)\right|dy\right)$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)^{1+\beta/n}M_{\beta,1}(L^{-\alpha/2}f)(x).$
For II, using Hölder’s inequality and Kolmogorov’s inequality(see[3], p.485),
then we deduce that
$\displaystyle II$ $\displaystyle\leq$ $\displaystyle
C\frac{1}{|B|}\int_{B}|L^{-\alpha/2}(b(y)-b_{B})f_{1})(y)|dy$
$\displaystyle\leq$ $\displaystyle
C\frac{1}{|B|}|B|^{\frac{\alpha}{n}}\|L^{-\alpha/2}(b(y)-b_{2B})f_{1}\|_{L^{\frac{n}{n-\alpha},\infty}}$
$\displaystyle\leq$ $\displaystyle
C\frac{1}{|B|^{1-\frac{\alpha}{n}}}\int_{B}(b(y)-b_{2B})f_{1}(y)dy$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)^{1-\frac{\alpha}{n}}M_{\alpha+\beta,r,\omega}f(x).$
Using Hölder’s inequality and Lemma 4.5, we obtain that
$III\leq
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)M_{\beta,r,\omega}(L^{-\alpha/2}f)(x).$
For IV, using the estimate in II, we get
$\displaystyle IV$ $\displaystyle\leq$
$\displaystyle\frac{C}{|B|}\int_{B}\int_{2B}|p_{t_{B}}(y,z)\|b(z)-b_{B}\|f(z)|dzdy$
$\displaystyle\leq$
$\displaystyle\frac{C}{|2B|}\int_{2B}L^{-\alpha/2}((b(z)-b_{B}))f(z)|dz$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)^{1-\frac{\alpha}{n}}M_{\alpha+\beta,r,\omega}f(x).$
By virtue of Lemma 3.1, we have
$\displaystyle V$ $\displaystyle\leq$
$\displaystyle\frac{C}{|B|}\int_{B}\int_{(2B)^{c}}|K_{\alpha,t_{B}}(y,z)\|(b(z)-b_{B})f(z)|dzdy$
$\displaystyle\leq$
$\displaystyle\frac{C}{|B|}\sum_{k=1}^{\infty}\int_{2^{k}r_{B}\leq|x_{0}-z|<2^{k+1}r_{B}}\frac{1}{|x_{0}-z|^{n-\alpha}}\frac{r^{2}_{B}}{|x_{0}-z|^{2}}|(b(z)-b_{B})f(z)|dz$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}2^{-2k}\frac{1}{|2^{k+1}B|^{1-\frac{\alpha}{n}}}\int_{2^{k+1}B}|(b(z)-b_{B})f(z)|dz$
$\displaystyle\leq$ $\displaystyle
C\sum_{k=1}^{\infty}2^{-2k}\frac{1}{|2^{k+1}B|^{1-\frac{\alpha}{n}}}\int_{2^{k+1}B}|(b(z)-b_{2^{k+1}B})f(z)|dz$
$\displaystyle+C\sum_{k=1}^{\infty}2^{-2k}(b_{2^{k+1}B}-b_{B})\frac{1}{|2^{k+1}B|^{1-\frac{\alpha}{n}}}\int_{2^{k+1}B}|f(z)|dz$
$\displaystyle\doteq$ $\displaystyle VI+VII.$
Making use of the same argument as that of II, we have
$VI\leq
C\|b\|_{Lip_{\beta}(\omega)}\omega(x)^{1-{\alpha}/{n}}M_{\alpha+\beta,r,\omega}f(x).$
Note that $\omega\in A_{1},$
$|b_{2^{k+1}B}-b_{2B}|\leq
Ck\,\omega(x)\|b\|_{Lip_{\beta}(\omega)}\omega(2^{k+1}B)^{{\beta}/{n}}.$
So, the value of $VII$ can be controlled by
$C\|b\|_{Lip_{\beta}(\omega)}\omega(x)^{1+{\beta}/{n}}M_{\alpha+\beta,1}f(x).$
Combining the above estimates for I–V, we finish the proof of Lemma 4.6. ∎
Proof of Theorem 1.2. We first prove $(a)$. As before, we only prove Theorem
1.2 in the case $0<\alpha<1$. For $0<\alpha+\beta<n$ and
$1<p<n/(\alpha+\beta)$, we can find a number $r$ such that $1<r<p$. By Lemma
4.6, we obtain
$\displaystyle\|[b,\ L^{-\alpha/2}]f\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\
\omega)}$ $\displaystyle\leq$ $\displaystyle C\|M^{\sharp}_{L,\delta}([b,\
L^{-\alpha/2}]f)\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\ \omega)}$
$\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\left(\|\omega(\cdot)^{1+\frac{\beta}{n}}M_{\beta,1}(L^{-\alpha/2}f)\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\
\omega)}\right.$ $\displaystyle\quad+$
$\displaystyle\|\omega(\cdot)^{1-\frac{\alpha}{n}}M_{\alpha+\beta,r,\omega}f\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\
\omega)}$ $\displaystyle\quad+$
$\displaystyle\left.\|\omega(\cdot)^{1+\frac{\beta}{n}}M_{\alpha+\beta,1}f\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\
\omega)}\right)$ $\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\left(\|M_{\beta,1}(L^{-\alpha/2}f)\|_{L^{q,kq/p}(\omega^{q/p},\
\omega)}\right.$ $\displaystyle\quad+$
$\displaystyle\|M_{\alpha+\beta,r,\omega}f\|_{L^{q,kq/p}(\omega)}+\left.\|M_{\alpha+\beta,1}f\|_{L^{q,kq/p}(\omega^{q/p},\
\omega)}\right).$
Let $1/{q_{1}}=1/p-\alpha/n$ and $1/q=1/{q_{1}}-\beta/n$. Lemmas 4.1–4.4 yield
that
$\displaystyle\|[b,\ L^{-\alpha/2}]f\|_{L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\
\omega)}$ $\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\left(\|L^{-\alpha/2}f\|_{L^{{q_{1}},k{q_{1}}/p}(\omega^{{q_{1}}/p},\
\omega)}+\|f\|_{L^{p,k}(\omega)}\right)$ $\displaystyle\leq$ $\displaystyle
C\|b\|_{Lip_{\beta}(\omega)}\|f\|_{L^{p,k}(\omega)}.$
Now we prove (b). Let $L=-\Delta$ be the Laplacian on ${{{\mathbb{R}}}^{n}}$,
then $L^{-{\alpha/2}}$ is the classical fractional integral $I_{\alpha}$. We
use the same argument as Janson [4]. Choose $Z_{0}\in{{{\mathbb{R}}}^{n}}$ so
that $|Z_{0}|=3.$ For $x\in B(Z_{0},2),$ $|x|^{-\alpha+n}$ can be written as
the absolutely convergent Fourier series, $|x|^{-\alpha+n}=\sum_{m\in
Z_{n}}a_{m}e^{i<\nu_{m},x>}$ with $\sum_{m}|a_{m}|<\infty$ since
$|x|^{-\alpha+n}\in C^{\infty}(B(Z_{0},2))$. For any
$x_{0}\in{{{\mathbb{R}}}^{n}}$ and $\rho>0,$ let $B=B(x_{0},\rho)$ and
$B_{Z_{0}}=B(x_{0}+Z_{0}\rho,\rho),$
$\displaystyle\int_{B}|b(x)-b_{B_{Z_{0}}}|dx=\frac{1}{|B_{Z_{0}}|}\int_{B}\left|\int_{B_{Z_{0}}}(b(x)-b(y))dy\right|dx$
$\displaystyle=$
$\displaystyle\frac{1}{\rho^{n}}\int_{B}s(x)\left(\int_{B_{Z_{0}}}(b(x)-b(y))|x-y|^{-\alpha+n}|x-y|^{n-\alpha}dy\right)dx,$
where $s(x)=\overline{\textup{sgn}\,(\int_{B_{Z_{0}}}(b(x)-b(y))dy)}.$ Fix
$x\in B$ and $y\in B_{Z_{0}}$, then ${(y-x)}/{\rho}\in B_{Z_{0},2}$, hence,
$\displaystyle\frac{\rho^{-\alpha+n}}{\rho^{n}}\int_{B}s(x)\left(\int_{B_{Z_{0}}}\left(b(x)-b(y)\right)|x-y|^{-\alpha+n}\left(\frac{|x-y|}{\rho}\right)^{n-\alpha}dy\right)dx$
$\displaystyle=$ $\displaystyle\rho^{-\alpha}\sum_{m\in
Z^{n}}a_{m}\int_{B}s(x)\left(\int_{B_{Z_{0}}}\left(b(x)-b(y)\right)|x-y|^{n-\alpha}e^{i<\nu_{m},y/\rho>}dy\right)e^{-i<\nu_{m},x/\rho>}dx$
$\displaystyle\leq$ $\displaystyle\rho^{-\alpha}\left|\sum_{m\in
Z^{n}}|a_{m}|\int_{B}s(x)[b,L^{-{\alpha/2}}]\left(\chi_{B_{Z_{0}}}e^{i<\nu_{m},\cdot/\rho>}\right)\chi_{B}(x)e^{-i<\nu_{m},x/\rho>}dx\right|$
$\displaystyle\leq$ $\displaystyle\rho^{-\alpha}\sum_{m\in
Z^{n}}|a_{m}|\|[b,L^{-{\frac{\alpha}{2}}}](\chi_{B_{Z_{0}}}e^{i<\nu_{m},\cdot/\rho>})\|_{L^{q,0}(\omega^{1-(1-\alpha/n)q},\omega)}\left(\int_{B}\omega(x)^{q^{\prime}(\frac{1}{{q^{\prime}}}-\frac{\alpha}{n})}dx\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\leq$ $\displaystyle C\rho^{-\alpha}\sum_{m\in
Z^{n}}|a_{m}|\|\chi_{B_{Z_{0}}}\|_{L^{p,0}(\omega)}\left(\int_{B}\omega(x)^{{q^{\prime}}(1/{q^{\prime}}-\alpha/n)}dx\right)^{\frac{1}{q^{\prime}}}$
$\displaystyle\leq$ $\displaystyle
C\omega(B)^{1/p+1/{q^{\prime}}-\alpha/n}=C\omega(B)^{1+\beta/n}.$
This implies that $b\in Lip_{\beta}(\omega)$. Thus, $(b)$ is proved. ∎
## References
* [1] X. T. Duong and L. Yan, On commutators of fractional integral, Proc. Amer. Math. Soc., 132 (2004), 3549–3557.
* [2] J. Garcia-Guerva, Weighted $H^{p}$ spaces, Dissertations Math., 162(1979), 1–63.
* [3] J. Garc a-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
* [4] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Math., 16 (1978), 263–270.
* [5] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282(2009), 219-231.
* [6] J. M. Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math., 161 (2004), 113–145.
* [7] M. Paluszyński, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44 (1995), 1–17.
* [8] C. Pérze, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 128(1995), 163–185. MR 1317714
* [9] S. Shirai, Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on classical Morrey spaces, Hokkaido Math. J., 35 (2006), 683–696.
* [10] E. M. Stein, Harmonic Analysis: Real-Variable methods, Orthogonality, and Oscillatory Integrals, Princeton New Jersey. Princeton Univ Press, 1993.
* [11] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
* [12] H. Wang, On some commutator theorems for fractional integral operators on the weighted morrey spaces, arXiv:1010.2638v1 [math. CA]
* [13] H. Wang, Some estimates for the commutators of fractional integrals associated to operators with Gaussian kenerl bounds, arXiv:1102.4380v1 [math. CA]
|
arxiv-papers
| 2012-03-20T08:24:56 |
2024-09-04T02:49:28.802609
|
{
"license": "Public Domain",
"authors": "Zengyan Si and Fayou Zhao",
"submitter": "Zengyan Si",
"url": "https://arxiv.org/abs/1203.4337"
}
|
1203.4361
|
# Building micro-soccer-balls with evaporating colloidal fakir drops
Álvaro G. Marín a.marin@unibw.de Physics of Fluids Group, Faculty of Science
and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands. Arturo Susarrey-Arce Catalytic Processes and Materials, Faculty
of Science and Technology, University of Twente. Mesoscale Chemical Systems,
Faculty of Science and Technology
University of Twente Hanneke Gelderblom Physics of Fluids Group, Faculty of
Science and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands. Arie van Houselt Leon Lefferts Catalytic Processes and
Materials, Faculty of Science and Technology, University of Twente. Han
Gardeniers Mesoscale Chemical Systems, Faculty of Science and Technology
University of Twente Detlef Lohse Physics of Fluids Group, Faculty of
Science and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands. Jacco H. Snoeijer Physics of Fluids Group, Faculty of Science
and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands.
###### Abstract
Evaporation-driven particle self-assembly can be used to generate three-
dimensional microstructures. We present a new method to create these colloidal
microstructures, in which we can control the amount of particles and their
packing fraction. To this end, we evaporate colloidal dispersion droplets on a
special type of superhydrophobic micro-structured surface, on which the
droplet remains in Cassie-Baxter state during the entire evaporative process.
The remainders of the droplet consist of a massive spherical cluster of the
microspheres, with diameters ranging from a few tens up to several hundreds of
microns. We present scaling arguments to show how the final particle packing
fraction of these balls depends on the dynamics of the droplet evaporation.
###### pacs:
Evaporation-driven particle self-assembly is an ideal mechanism for
constructing micro- and nanostructures at scales where direct manipulation is
impossible. For example, in colloidal dispersion droplets with pinned contact
lines, evaporation gives rise to the so-called coffee-stain effect Deegan _et
al._ (1997): a capillary flow drags the particles towards the contact line to
form a ring-shaped stain. Such a flow not only aggregates the particles, but
is also able to organize them in crystalline phases Fan and Stebe (2004);
Bigioni _et al._ (2006); Harris _et al._ (2007); Marín _et al._ (2011).
Similar mechanisms such as the “convective assembly” Velikov _et al._ (2002);
Meng _et al._ (2006) are currently successfully used to produce two-
dimensional colloidal crystal films. To obtain three-dimensional clusters of
micro-particles, colloidal dispersion droplets which are suspended in
emulsions Velev _et al._ (2000); Manoharan _et al._ (2003); Dinsmore _et
al._ (2002) or kept in Leidenfrost levitation Tsapis _et al._ (2005) are
used. With these techniques, new colloidal structures arise from the
geometrical constraints during the drying Lauga and Brenner (2004). This
problem of organization of particles into a spherical topography dates back to
the days of the first models of the atom and has been extensively studied by
Bausch et al. Bausch _et al._ (2003); Lipowsky _et al._ (2005). The main
drawback of these three-dimensional assembly techniques, however, is the lack
of control on both the amount of particles and the particle arrangement in the
remaining structures.
Figure 1: (a) A droplet of colloidal solution is left to evaporate on a
superhydrophobic surface. As the solvent evaporates, the particle packing
fraction increases. Once all the solvent has completely evaporated the
colloidal particles have aggregated to form a spherical particle conglomerate:
a colloidal supraball. (b) Top view of the resultant compact colloidal
supraball left on the superhydrophobic surface after evaporation. The
micropillars forming the structure are seen as circular objects around the
supraball.
In this work, we devise a new, controlled way of generating on-demand self-
assembled spherical micro-structures via droplet evaporation on a
superhydrophobic surface (see figure 1). We present scaling arguments to
predict the particle arrangement in the microstructures formed, based on the
dynamics of the evaporation process. To generate the microstructures, we
evaporate colloidal dispersion droplets on a special type of superhydrophobic
substrates. In most of the cases, a liquid Cassie-Baxter state drop
evaporating on a superhydrophobic surface will eventually suffer a wetting
transition into a Wenzel state, i.e. it will get impaled into the structure
and loose its spherical shape Reyssat _et al._ (2008); Tsai _et al._ (2010).
Here, however, we use a surface that combines overhanging pillared structures
Tuteja _et al._ (2008) with a hierarchical nano-structure (figure 2c). These
surface properties impose a huge energy barrier for the wetting transition to
occur, and therefore the droplet will remain almost floating over the
structure in a Cassie-Baxter state during its entire life Susarrey-Arce _et
al._ (2012). A typical result can be observed in figure 1: a water droplet
containing $1\mu m$ polystyrene particles (concentration 0.08% weight and
initial volume $5\mu l$) evaporates on the superhydrophobic surface at room
temperature and 30% humidity. After a typical evaporation time of 45 minutes,
the solvent is completely evaporated and only the colloids are left upon the
substrate. Remarkably, the particles ($\sim 10^{7}$ in this particular case)
are not just lying scattered over the substrate but they have aggregated and
form a spherical macro-cluster resting on top of the micro-pillars, which we
call _colloidal supraball_. We do not observe shell formation and buckling
during the evaporation of the droplets Tsapis _et al._ (2005): the supraballs
we obtain are solid, and present a high mechanical resistance and stability.
When looking closer at the surface of these particular colloidal supraballs as
shown in figure 2d, one can identify crystalline flat patches which resemble
the pentagonal patches present in a soccer ball.
To understand the final structure of these supraballs, it will turn out
crucial to understand the dynamics of the droplet evaporation. The fact that
we do not observe shell formation, suggests that the particles do not
influence the droplet evaporation. To test this hypothesis, we compare the
evaporation dynamics to that of a liquid drop that does not contain any
particles. The evaporative mass loss from such a drop is typically governed by
the diffusion of vapor molecules in the surrounding air Deegan _et al._
(1997); Popov (2005); Gelderblom _et al._ (2011). For diffusion-limited
evaporation, the rate of volume change of the drop is given by
$\frac{dV}{dt}\sim D^{\prime}R,$ (1)
where $R$ is the drop radius, and $D^{\prime}=D_{va}\Delta c/\rho$, with
$D_{va}$ the diffusion constant for vapor in air, $\Delta c$ the vapor
concentration difference between drop surface and the surroundings and $\rho$
the liquid density Marín _et al._ (2011). One might have expected the
evaporation rate from the drop surface to be proportional to the droplet
surface area $\sim R^{2}$. However, the vapor concentration gradient is
proportional to $1/R$, and therefore the total evaporation rate is
proportional to $R$ Eggers and Pismen (2010). If the droplet evaporates with a
constant contact angle, we find that, since $V\sim R^{3}$,
$R(t)\sim[D^{\prime}(t_{f}-t)]^{1/2}.$ (2)
Here $t_{f}$ is the total droplet lifetime in case no particles are present,
for which the drop radius reaches zero. In the present case the drop radius
saturates at a finite radius, $R_{ball}$, at a time
$\hat{t}=t_{f}-R_{ball}^{2}/D^{\prime}$, corresponding to the moment where the
particles become densely packed. In figure 3 we plot the colloidal droplet
radius versus $t_{f}-t$. Our experimental data for different number of
particles are in very good agreement with the $1/2$-power law. This confirms
that the particles do not influence the evaporation process, until the final
radius $R_{ball}$ is reached. The scaling (1) implies that the speed with
which the interface is moving inwards, is given by $dR/dt\sim D^{\prime}/R$.
Hence, the interface speed increases dramatically as the droplet shrinks and
the maximum speed reached in the experiment will be determined by the final
radius $R_{ball}$. As we will show further on, this increase in interface
speed determines the particle packing inside the supraballs.
Figure 2: (a) Tilted view of the supraball in contact with the microstructure.
(b) Detail of the contact area. (c) Magnified view of the micropillars forming
the microstructure. (d) Close-up of the supraball surface. The distribution of
crystalline patches resemble the pentagons in a soccer-ball.
The final size of the ball depends on the number of particles inside the drop.
This can be tuned by manipulating either the initial particle concentration or
the droplet size. In our experiment, the ball size was in the range
$100<R_{ball}/R_{p}<1000$, with $R_{p}$ the particle radius. Clearly, the
exact final size of the ball will not only depend on the amount of particles
in the system but also on their packing fraction. We define the packing
fraction as
$\Phi\equiv N\left(\frac{R_{p}}{R_{ball}}\right)^{3},$ (3)
where $N$ is the total number of particles in the droplet. The final supraball
radius $R_{ball}$ is accurately determined from Scanning Electron Microscope
(SEM) images. If the packing fractions were identical for all supraballs, one
would expect that $R_{ball}/R_{p}\sim N^{1/3}$, as depicted in figure 4a
111All quantitative results shown in this paper have been performed with
colloids of $1\mu m$ diameter (non-surface-modified fluorescent microspheres
supplied by Thermo Scientific), but the same qualitative behavior has been
observed for $0.2$ and $2\mu m$. The colloidal solutions were always prepared
with deionized water.. However, in figure 4b we see that the packing fraction
strongly depends on the number of particles in the system.
Figure 3: The radius of the droplet plotted against $t_{f}-t$ with $t_{f}$ the
lifetime of the droplet and $t$ the actual time. The triangle indicates the
$1/2$-power law, the dots represent the data sets for 7 different experiments,
where the number of particles was varied. For a certain $\hat{t}<t_{f}$ the
final ball size $R_{ball}$ is reached. The final time was extrapolated as
$t_{f}=\hat{t}+R_{ball}^{2}/D^{\prime}$.
As the number of particles increases, the packing fraction approaches that of
a perfect Hexagonal Close Packing configuration, in which case one would find
$\Phi=0.74$ Kepler (1966), hence, we have an _ordered_ particle packing inside
the balls. On the other hand, the supraballs with a smaller amount of
particles show remarkably low packing fractions, even below the Random Close
Packing (RCP) limit ($\Phi=0.64$) Weitz (2004), corresponding to a
_disordered_ particle arrangement. The balls which show packing fractions
below the RCP limit contain several empty cavities. Remarkably, the final
configuration reached, seems to depend on the number of particles in the
system. In figure 4b we indicated the critical number of particles
$N_{c}\approx 3\cdot 10^{6}$ when the packing fraction reaches that of a RCP.
For $N<N_{c}$ we get a loose, _disordered_ particle packing in the supraball,
whereas for $N>N_{c}$ we get a densely packed, _ordered_ supraball.
What causes the transition from ordered to disordered packings, and what
determines the critical number of particles? To explain this, we follow a
similar approach as in Marín et al. Marín _et al._ (2011): we compare the
time scale on which particles can arrange by diffusion to the hydrodynamic
time scale for the particle transport by convection, given by the inward
motion of the liquid-air interface. If the diffusion time is small compared to
the hydrodynamic time, particles can arrange into an ordered packing. The
diffusive time scale is $t_{d}=R_{p}^{2}/D$, with $R_{p}$ the particle radius
and $D$ the diffusivity of the particles in the liquid Marín _et al._ (2011).
The hydrodynamic time-scale is $t_{h}=L/\left|\frac{dR}{dt}\right|$. Here
$R(t)$ is the droplet radius, and $L$ is the typical inter-particle distance,
which depends on the particle concentration as $L=N^{-1/3}R$, as long as the
solution is dilute ($L\gg R_{p}$). We define the ratio of both time scales as:
$\mathcal{A}(t)\equiv\frac{t_{d}}{t_{h}}=\left|\frac{dR(t)}{dt}\right|\frac{t_{d}}{L}=\frac{D^{\prime}}{D}N^{1/3}\left(\frac{R_{p}}{R(t)}\right)^{2},$
(4)
where in the last step we used (1) to replace $dR/dt\sim D^{\prime}/R$.
Figure 4: (a) Supraball to microparticle diameter $R_{ball}/R_{p}$ plotted
against the total amount of particles $N$ in the system. (b) Packing fraction
$\Phi$ strongly depends on $N$. Blue dots represent experimental measurements
and the red solid line corresponds the most efficient particle packing
$\Phi=0.74$ (hexagonal close packed), the dashed line marks $\Phi=0.64$,
random close packed. $N_{c}$ is the critical number of particles, above which
we find an ordered ball structure
From (4) we observe that $\mathcal{A}(t)$ increases as the droplet radius
becomes smaller during the evaporation (see figure 3), until the limit
$R=R_{ball}$ is reached. A cross-over between the time-scales is reached when
the hydrodynamic time becomes equal to the diffusion time, hence when
$\mathcal{A}=1$. If the cross-over is reached when $R\gg R_{ball}$, the amount
of crystalline clusters is still very small. From this point in time onwards
the interface speed is too high for the particles to further arrange in a
crystalline way Marín _et al._ (2011). Instead, they are pushed together in a
random arrangement, with a low packing fraction. If the cross-over is reached
when $R\leq R_{ball}$, the particle packing is already dense and ordered, and
we find a high packing fraction. For all droplets the evaporative mass loss,
and hence the decrease in radius, is the same (see figure 3), hence, the
moment when the particle packing becomes sufficiently dense for particles to
arrange depends solely on the number of particles in the droplet. If $N$ is
high ($N>N_{c}$), this moment is reached relatively early, i.e. well before
$\mathcal{A}=1$, and we get an ordered particle packing inside the supraballs.
Using that $R_{ball}/R_{p}\sim N^{1/3}$ and considering $\mathcal{A}=1$, we
find from (4) the critical number of particles above which we obtain ordered
supraballs
$N_{c}\sim\left(\frac{D^{\prime}}{D}\right)^{3}.$ (5)
This result emphasizes that the transition is governed by two diffusion
processes: the diffusion of vapor, determining the speed of evaporation,
versus the diffusion of particles inside the drop. The ratio of diffusion
constants selects the critical number of particles. In our experiment
$D^{\prime}=3\times 10^{-10}$ m2/s and $D=2\times 10^{-13}$ m2/s, from which
we find that $N_{c}\sim 10^{9}$. This is 2 to 3 orders of magnitudes larger
than the experimentally observed $N_{c}$. However, in the preceding analysis
we have neglected all prefactors, and we emphasize that the result is strongly
(to the third power) dependent on the experimental parameters included in
$D^{\prime}$, i.e. humidity, liquid density, diffusivity of vapor, and
saturated vapor concentration.
To verify whether the final packing fraction indeed depends on the spacing
between the particles the moment the cross-over time is reached, we go back to
our experimental data. We define the time-dependent packing fraction as
$N(R_{p}/R(t))^{3}$. As the droplet evaporates, this packing fraction will
increase until it reaches its final value $\Phi$. At the cross-over, defined
by ${\cal A}=1$, the droplets will have a packing fraction $\Phi^{*}$. If this
$\Phi^{*}$ is low, the amount of crystalline clusters is still very small. On
the other hand, if $\Phi^{*}$ is high, we expect crystalline clusters to have
formed already. After the cross-over time, the interface moves too fast to
allow for further ordering, and it just presses the ordered particle clusters
closer together. In figure 5 we show that droplets with a high $\Phi^{*}$ have
a high $\Phi$: when $\Phi^{*}\gtrapprox 0.1$ we obtain a final packing
fraction above the RCP limit.
Figure 5: Final packing fraction $\Phi_{f}$ versus the packing fraction at the
cross-over time $\Phi^{*}$. Droplets below a certain $\Phi^{*}$ have a too low
packing fraction at the cross-over time to achieve final packing fractions
above the RCP limit. The particle packing can not only be obtained from the
value of the packing fraction, but it can also directly be observed from the
SEM images of the surface of the supraballs.
We cannot predict the critical $\Phi^{*}$ theoretically. However, we can,
retrospectively, use the experimental critical $\Phi^{*}$ to compute $N_{c}$.
Using that $R_{ball}/R_{p}\sim 0.1N^{1/3}$ at the cross-over, we obtain
$N_{c}\sim 10^{7}$, which is in the same order of magnitude as our
experimental results; see figure 4b.
The particle packing in the supraballs can not only be assessed by measuring
the packing fraction, but it can also directly be seen in SEM images from the
surface of the colloidal supraballs, as shown in figure 5. The size of the
soccer-ball-like crystalline patches on the surface of the ordered supraballs
depends on the ball size: bigger balls will show larger patches due to the
reduced curvature at their surfaces. To explain the size of the crystalline
domains, we hypothesize that a crystalline patch will bend radially no more
than a particle size. Then, it follows by simple trigonometry that the size of
a patch $S$ will be related to the ball radius $R_{ball}$ and the particle
size $R_{p}$ via: $S/R_{ball}=\arccos(1-R_{p}/R_{ball})$. This expression
predicts a typical patch size of $\sim 15\mu m$ for a ball with
$R_{ball}=100\mu m$ and $R_{p}=1\mu m$, which is in the right order of
magnitude as one can observe in figure 2d.
In conclusion, in this Letter we devise a simple technique to create spherical
colloidal supraballs relying only on droplet evaporation over a robust
superhydrophobic surface. The supraballs show a highly ordered structure if
the number of particles inside the drop is large enough to trigger early
particle clustering. The critical number of particles required to obtain an
ordered particle packing inside the balls depends on the parameters driving
the droplet evaporation (through $D^{\prime}$) and the diffusivity of the
particles. Hence, by controlling the humidity and ambient temperature the
supraball packing fraction and hence size can be controlled. Massive
fabrication of micro-compact-supraballs could easily be achieved by simply
spraying a colloidal solution over the micro-structure in a controlled
atmosphere. By tuning the wetting properties of the particles one could also
be able to generate the well-known _colloidosomes_ Dinsmore _et al._ (2002)
using the same proposed technique.
###### Acknowledgements.
We gratefully acknowledge Vicenzo Vitelli, Martin van Hecke, Devaraj van der
Meer, Chao Sun and many others for their useful comments and their support.
The authors also acknowledge financial support by the NWO grant No. 700.10.408
and No. 700.58.041.
## References
* Deegan _et al._ (1997) R. D. Deegan, O. Bakajin, T. Dupont, G. Huber, S. Nagel, and T. Witten, Nature 389, 827 (1997).
* Fan and Stebe (2004) F. Fan and K. J. Stebe, Langmuir 20, 3062 (2004).
* Bigioni _et al._ (2006) T. P. Bigioni, X. M. Lin, T. T. Nguyen, E. I. Corwin, T. A. Witten, and H. M. Jaeger, Nature Materials 5, 265 (2006).
* Harris _et al._ (2007) D. J. Harris, H. Hu, J. Conrad, and J. Lewis, Phys. Rev. Lett. 98, 148301 (2007).
* Marín _et al._ (2011) A. G. Marín, H. Gelderblom, D. Lohse, and J. H. Snoeijer, Phys. Rev. Lett. 107 (2011).
* Velikov _et al._ (2002) K. P. Velikov, C. G. Christova, R. P. A. Dullens, and A. van Blaaderen, Science 296, 106 (2002).
* Meng _et al._ (2006) L. Meng, H. Wei, A. Nagel, J. Benjamin, L. Scriven, and D. Norris, Nano Letters 6, 2249 (2006).
* Velev _et al._ (2000) O. D. Velev, A. M. Lenhoff, and E. W. Kaler, Science 287 (2000).
* Manoharan _et al._ (2003) V. Manoharan, M. Elsesser, and D. Pine, Science 301, 483 (2003).
* Dinsmore _et al._ (2002) A. Dinsmore, M. Hsu, M. Nikolaides, M. Marquez, A. Bausch, and D. Weitz, Science 298, 1006 (2002).
* Tsapis _et al._ (2005) N. Tsapis, E. Dufresne, S. Sinha, C. Riera, J. Hutchinson, L. Mahadevan, and D. Weitz, Phys. Rev. Lett. 94 (2005).
* Lauga and Brenner (2004) E. Lauga and M. Brenner, Phys. Rev. Lett. 93, 238301 (2004).
* Bausch _et al._ (2003) A. Bausch, M. Bowick, A. Cacciuto, A. Dinsmore, M. Hsu, D. Nelson, M. Nikolaides, A. Travesset, and D. Weitz, Science 299, 1716 (2003).
* Lipowsky _et al._ (2005) P. Lipowsky, M. Bowick, J. Meinke, D. Nelson, and A. Bausch, Nature Materials 4, 407 (2005).
* Reyssat _et al._ (2008) M. Reyssat, J. Yeomans, and D. Quéré, Europhys. Lett. 81, 26006 (2008).
* Tsai _et al._ (2010) P. Tsai, R. Lammertink, M. Wessling, and D. Lohse, Physical Review Letters 104, 116102 (2010).
* Tuteja _et al._ (2008) A. Tuteja, W. Choi, J. Mabry, G. McKinley, and R. Cohen, Proceedings of the National Academy of Sciences 105, 18200 (2008).
* Susarrey-Arce _et al._ (2012) A. Susarrey-Arce, A. G. Marín, H. Nair, L. Lefferts, J. Gardeniers, D. Lohse, and A. van Houselt, Submitted to Soft Matter (2012).
* Popov (2005) Y. O. Popov, Phys. Rev. E 71, 036313 (2005).
* Gelderblom _et al._ (2011) H. Gelderblom, A. G. Marín, H. Nair, A. van Houselt, L. Lefferts, J. Snoeijer, and D. Lohse, Phys. Rev. E 83, 026306 (2011).
* Eggers and Pismen (2010) J. Eggers and L. M. Pismen, Phys. Fluids 22, 112101 (2010).
* Note (1) All quantitative results shown in this paper have been performed with colloids of $1\mu m$ diameter (non-surface-modified fluorescent microspheres supplied by Thermo Scientific), but the same qualitative behavior has been observed for $0.2$ and $2\mu m$. The colloidal solutions were always prepared with deionized water.
* Kepler (1966) J. Kepler, _Strena seu de nive sexangula, 1611_ (English translation, Clarendon Press, Oxford, 1966).
* Weitz (2004) D. A. Weitz, Science 303, 968 (2004).
|
arxiv-papers
| 2012-03-20T09:58:11 |
2024-09-04T02:49:28.810050
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alvaro G. Marin, Arturo Susarrey-Arce, Hanneke Gelderblom, Arie van\n Houselt, Leon Lefferts, Han Gardeniers, Detlef Lohse and Jacco Snoeijer",
"submitter": "Alvaro Marin",
"url": "https://arxiv.org/abs/1203.4361"
}
|
1203.4384
|
# Disembodiment of Physical Properties by Pre- and Post-Selections
Chang-Ling Zou Xu-Bo Zou xbz@ustc.edu.cn F.-W. Sun fwsun@ustc.edu.cn Guang-
Can Guo Key Lab of Quantum Information, University of Science and Technology
of China, Hefei 230026
###### Abstract
The detailed study of disembodiment of physical properties by pre- and post-
selection is present. A criterion is given to disembody physical properties
for single particle with multiple degrees of freedom. It is shown that the
non-commute operators can also be well separated in different paths. We
generalize the disembodiment to entangled particles, and found that the
disembodiment can happen under special conditions due to the entanglement.
###### pacs:
42.55.Sa, 05.45.Mt, 42.25.-p,42.60.Da
## I Introduction
Since the foundation of quantum physics, the controversies about the self-
consistency, completeness, causality and locality in quantum physics have not
stopped. It has shown that the quantum physics is wacky and hard to
interpreted, but have been confirmed by experiments for nearly one century.
Physicists are keeping trying to understand this wacky quantum worlds with a
lot of paradoxes, such as the EPR paradox, Zeno paradox _etc_. These paradoxes
reveal the contradiction of intuition of people and the truth of nature, and
usually inspires the new ideas and new interpretations, further improve the
development of quantum physics.
One of the paradoxes, the pre- and post-selection (PPS) which was firstly
proposed by Aharonov, Bergmann, and Lebowitz (ABL) abl ; PT , questioned about
the quantum arrow of time. Further studies on PPS leads to numbers of
paradoxes, such as the famous Aharonov, Alber and Vaidman (AAV) paradox which
revealed that the outcome of measurement can be widely out of the range of the
eigenvalues of a system through PPS aav . The paradox about PPS is not only of
theoretical interests on foundations of quantum mechanics, but also offers a
powerful tool in experiments. The quantum state estimation and precision
measurement based on the PPS has been proposed, and successfully demonstrated
in experiments Hosten ; Lundeen . Very recently, Aharonov, Popescu and
Skrzypczyk (APS) proposed that the Cheshire cats, i.e. the “body” and “grin”
of cat can be surprisedly separated through appropriate selections of initial
and final states Cheshire . It is very potential for further application in
theoretical and experimental studies twin , but the details of the embodiment
of physics properties is still waiting to be explored.
In this paper, we present a general treatment to disembodiment of physical
properties by PPS and give a criterion to disembody physical properties for
single particle with multiple degrees of freedom. It is shown that the non-
commute operators can also be separated successfully. We generalize the
disembodiment to entangled particles, and found that the disembodiment can
happen for special conditions due to the entanglement.
## II The Cheshire cats and Weak Measurement
The quantum mechanics is time symmetric, that the initial state is as
important as the final state. Thus, we can prepare particles in selected
initial state $\left|\Psi\right\rangle$ which called the pre-selection, and
then postselect the ensemble of the particles corresponding to a final state
$\left|\Phi\right\rangle$ through the measurement in detectors. It has been
shown in Ref.Cheshire , in the specific ensembles with PPS, the photon number
and the polarization can be separated in different paths. However, the
polarization and photon number cannot be directly readout through traditional
collapse measurement, where the quantum state is changed significantly. In
this case, we cannot distinguish whether the Cheshire cats is found. If we
resorted to the weak measurement that the ancilla measuring device weakly
coupling to the system, therefore the disturbance to the state of system
induced by the measurements can be neglected. The Hamiltonian of the weak
measurement reads
$H_{I}=\hbar gAO,$ (1)
where $g$ is the interaction strength, $A$ is the ancilla, and $O$ is the
observer operator of the system. In the case of the PPS, the average outcome
(also called Weak Value) of the observer should be
$\left\langle
O\right\rangle_{w}=\frac{\left\langle\Phi|O|\Psi\right\rangle}{\left\langle\Phi|\Psi\right\rangle},$
(2)
with $\left\langle\Phi|\Psi\right\rangle\neq 0$ that the initial and final
state are not orthogonal to each other. The $\left\langle O\right\rangle_{w}$
is amplified if the $\left\langle\Phi|\Psi\right\rangle$ is a very small,
which can greatly enhance the measurement precision but with the scarifies of
counts at detector.
Here we want to separate the properties described by operator $O^{j}$ with
superscript $j=1,2,3\ldots$, for different quantum properties. Disembodiment
of a physics properties requires that the expect value of operator $O^{j}$ is
nonzero only in one output, for example, in one output path for a photon.
Additionally, in this path, the expect value of other operator should be zero.
Without loss of generality, to separate operators $O^{j}$ in path $j$, the PPS
must satisfy
$\left\langle
O_{i}^{j}\right\rangle_{w}=\frac{\left\langle\Phi|O_{i}^{j}|\Psi\right\rangle}{\left\langle\Phi|\Psi\right\rangle}=a_{i}\delta_{ij},$
(3)
where the subscript $i$ denote the path, $a_{i}$ are non-zero numbers and
$\delta_{ij}$ is Kronecker delta.
## III Single Particle
For any single particle system, the quantum states can be present by a vector
in the Hilbert space with the basis $\\{\left|b_{i}^{jk}\right\rangle\\}$,
where $i=1,\ldots,p$ and $j=1,\ldots,q$ stand for different paths and physical
properties, and $k=1,2,\ldots,d$ show the degrees of freedom of the $j$th
physical properties. For path $i$, the PPS can be present as
$\left\langle\Phi_{i}\right|=\\{x_{i}^{1},x_{i}^{2},...,x_{i}^{n}\\}$ and
$\left|\Psi_{i}\right\rangle=\\{y_{i}^{1},y_{i}^{2},...,y_{i}^{n}\\}^{T}$,
where the dimension $n=q\times d$. The condition for disembodiment in path $i$
becomes
$\left\langle\Phi_{i}\right|O_{i}^{j}\left|\Psi_{i}\right\rangle=a_{i}\delta_{ij},$
(4)
with the weak value unnormalized. Or we can write the equation in the matrix
form as
$\sum_{kl}(O_{i}^{j})_{kl}x_{i}^{k}y_{i}^{l}=a_{i}\delta_{ij},$ (5)
where $(O_{i}^{j})_{kl}=\left\langle
b_{i}^{jk}\right|O_{i}^{j}\left|b_{i}^{jl}\right\rangle$. Combining all
operators $j=1,\ldots,q$, we can write the tensor in the matrix form as
$\displaystyle
M_{i}\\{x_{i}^{1}y_{i}^{1},x_{i}^{1}y_{i}^{2},\ldots,x_{i}^{n}y_{i}^{n}\\}$
(6) $\displaystyle=$ $\displaystyle\\{0,\ldots,0,a_{i},0,\ldots,0\\}^{T},$ (7)
where the dimension of $M_{i}$ is $q\times n^{2}$. In the present case, the
physical properties and operators are the same for all paths, thus we have
$M_{i}=M$ independent on the path. Suppose there are $m$ operators to separate
($m\leq p$), then we should solve the linearized equations
$\displaystyle M\overrightarrow{v_{1}}$ $\displaystyle=$
$\displaystyle\\{a_{1},0,\ldots,0\\}^{T},$ $\displaystyle
M\overrightarrow{v_{2}}$ $\displaystyle=$
$\displaystyle\\{0,a_{2},0,\ldots,0\\}^{T}\text{.}$ $\displaystyle\ldots$
The criterion of the existance of solutions to above equations is
$\mathrm{rank}(M)=m.$
Applying this criteria, we study two examples of photon with polarization
degree of freedom.
Example I: Single photon with the polarization $\left|+\right\rangle$ or
$\left|-\right\rangle$ in two paths $\left|1\right\rangle$ or
$\left|2\right\rangle$. The operator of the which photon number operator is
$O_{i}^{1}=I_{i}=diag\\{1,1\\},$ (9)
and the polarization operator is
$O_{i}^{2}=\sigma_{i}^{z}=diag\\{1,-1\\}.$ (10)
Then, we have the matrix $M=\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)$, with $\mathrm{rank}(M)=2$. Thus, we can separate the
photon number and polarization in two different paths. We need to solve the
equations
$\displaystyle M\left(\begin{array}[]{c}x_{1}^{1}y_{1}^{1}\\\
x_{1}^{2}y_{1}^{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}1\\\ 0\end{array}\right)\text{,}$ (15)
$\displaystyle M\left(\begin{array}[]{c}x_{2}^{1}y_{2}^{1}\\\
x_{2}^{2}y_{2}^{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}0\\\ 1\end{array}\right)\text{,}$ (20)
and we have $x_{1}^{1}y_{1}^{1}=x_{1}^{2}y_{1}^{2}=\frac{1}{2}$ and
$x_{2}^{1}y_{2}^{1}=-x_{2}^{2}y_{2}^{2}=\frac{1}{2}$. There are infinite
choices of pre-selections and post-selections. Let the pre-selection state be
$\left\langle\Psi\right|=\\{x_{1}^{1},x_{1}^{2},x_{2}^{1},x_{2}^{2}\\}=\\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\\}$,
then
$\left|\Phi\right\rangle=\\{y_{1}^{1},y_{1}^{2},y_{2}^{1},y_{2}^{2}\\}^{T}=\\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\\}^{T}$.
This is exactly the case in the original Cheshire Cats paper by Aharonov et
al. Cheshire .
Example II: Single photon with polarization $\left|+\right\rangle$ or
$\left|-\right\rangle$ in four path
$\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle$ or
$\left|4\right\rangle$. We want to separate the photon number operator $I$ and
the polarization operators $\sigma^{x},\sigma^{y}$ and $\sigma^{z}$ in
different paths. Thus, we have
$\displaystyle I_{i}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),$ (23)
$\displaystyle\sigma^{x}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),$ (26)
$\displaystyle\sigma^{y}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right),$ (29)
$\displaystyle\sigma^{z}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),$ (32)
The matrix form is
$M=\left(\begin{array}[]{cccc}1&0&0&1\\\ 0&1&1&0\\\ 0&-i&i&0\\\
1&0&0&-1\end{array}\right)\text{.}$ (33)
In path $j$, the equation should be satisfied
$M\left(\begin{array}[]{c}x_{j}^{1}y_{j}^{1}\\\ x_{j}^{1}y_{j}^{2}\\\
x_{j}^{2}y_{j}^{1}\\\
x_{j}^{2}y_{j}^{2}\end{array}\right)=\left(\begin{array}[]{c}\delta_{1j}\\\
\delta_{2j}\\\ \delta_{3j}\\\ \delta_{4j}\end{array}\right).$ (34)
Since $\mathrm{\det}(M)=-4i$, thus $\mathrm{rank}(M)=4$, we can separate all
four operators in four paths. One set of the solutions is
$\displaystyle x_{1}^{1}y_{1}^{1}$ $\displaystyle=$ $\displaystyle
x_{1}^{2}y_{1}^{2}=\frac{1}{2},$ $\displaystyle x_{2}^{1}y_{2}^{2}$
$\displaystyle=$ $\displaystyle x_{2}^{2}y_{2}^{1}=\frac{1}{2},$
$\displaystyle x_{3}^{1}y_{3}^{2}$ $\displaystyle=$ $\displaystyle-
x_{3}^{2}y_{3}^{1}=\frac{i}{2},$ $\displaystyle x_{4}^{1}y_{4}^{1}$
$\displaystyle=$ $\displaystyle-x_{4}^{2}y_{4}^{2}=\frac{1}{2}.$ (35)
Thus, we can have $\left\langle\Psi\right|=\\{1,1,1,1,1,1,1,1\\}$ and
$\left|\Phi\right\rangle=\\{1,1,1,1,i,-i,1,-1\\}^{T}.$
## IV Entangled Particles
Now, we turn to consider the two-particle system. If there is no entanglement,
the two-particle system is only the simple extension of single particle
systems with more degrees of freedom. Take the two photons in four path
(photon A in path $\left|1\right\rangle$ or $\left|2\right\rangle$; photon B
in path $\left|3\right\rangle$ or $\left|4\right\rangle$ ) with the the
polarization is entangled
($\left|H\right\rangle_{A}\left|V\right\rangle_{B}+\left|H\right\rangle_{B}\left|V\right\rangle_{A}$).
Thus, the Hilbert space of state can be represent in the basis
$\displaystyle\\{\left|1H\right\rangle\left|3V\right\rangle,\left|1H\right\rangle\left|4V\right\rangle,\left|1V\right\rangle\left|3H\right\rangle,\left|1V\right\rangle\left|4H\right\rangle,$
$\displaystyle\left|2H\right\rangle\left|3V\right\rangle,\left|2H\right\rangle\left|4V\right\rangle,\left|2V\right\rangle\left|3H\right\rangle,\left|2V\right\rangle\left|4H\right\rangle.\\}$
(36)
Since the polarization is entangled, the dimensional of the state space is 8\.
The operators of the photon number are
$\displaystyle I_{1}^{A}$ $\displaystyle=$ $\displaystyle
diag\\{1,1,1,1,0,0,0,0\\},$ $\displaystyle I_{2}^{A}$ $\displaystyle=$
$\displaystyle diag\\{0,0,0,0,1,1,1,1\\},$ $\displaystyle I_{3}^{B}$
$\displaystyle=$ $\displaystyle diag\\{1,0,1,0,1,0,1,0\\},$ $\displaystyle
I_{4}^{B}$ $\displaystyle=$ $\displaystyle diag\\{0,1,0,1,0,1,0,1\\}.$ (37)
and operators of photon polarization are
$\displaystyle\sigma_{1}^{A}$ $\displaystyle=$ $\displaystyle
diag\\{1,1,-1,-1,0,0,0,0\\},$ $\displaystyle\sigma_{2}^{A}$ $\displaystyle=$
$\displaystyle diag\\{0,0,0,0,1,1,-1,-1\\},$ $\displaystyle\sigma_{3}^{B}$
$\displaystyle=$ $\displaystyle diag\\{-1,0,1,0,-1,0,1,0\\},$
$\displaystyle\sigma_{4}^{B}$ $\displaystyle=$ $\displaystyle
diag\\{0,-1,0,1,0,-1,0,1\\}.$ (38)
The matrix form is
$M=\left(\begin{array}[]{cccccccc}1&1&1&1&0&0&0&0\\\ 0&0&0&0&1&1&1&1\\\
1&0&1&0&1&0&1&0\\\ 0&1&0&1&0&1&0&1\\\ 1&1&-1&-1&0&0&0&0\\\
0&0&0&0&1&1&-1&-1\\\ -1&0&1&0&-1&0&1&0\\\
0&-1&0&1&0&-1&0&1\end{array}\right).$ (39)
We find that, for the expecting value for operator as
$\overrightarrow{e}=\\{1,0,0,1,1,0,0,1\\}^{T},$ (40)
the solution to
$M\overrightarrow{v}=\overrightarrow{e}$ (41)
does not exist. However, for
$\overrightarrow{e}=\\{1,0,0,1,1,0,0,-1\\}^{T},$ (42)
the disembodiment can happen. This is due to the polarization entanglement
between the two photons, where the expecting values should be $1$ and $-1$
respectively. So, we can separate the physics properties even for entangled
particles.
## V Discussion
(1) From the analysis for single particle above, the disembodiment of physical
properties is not restricted to the system with two degrees of freedom. The
physics properties can always be separated through particular PPS ensemble
according to the criterion. In addition, the selected initial and final states
for disembodiment are not sole.
(2) The disembodiment is not restricted to separation physical properties in
different paths. It can be extended to any other degree of freedom that we can
address in experiment, such as internal degree of atoms.
(3) Potential application of disembodiment would be selectively measure the
parameters with different operators. For instance, for a system interact with
ancilla $H_{I}=\hbar A(g_{1}O^{1}+g_{2}O^{2})$, where $O^{1}$ and $O^{2}$ are
independent to each other, the observer $O^{1}$ and $O^{2}$ can be selectively
measured through disembodiment by only one ancilla.
(4) In all above analysis, the PPS is perfect regardless the actual
preparation and detection process. The real experimental situation, the
imperfect devices give errors in state preparation and detection. This type of
error is stationary and can be modified through transformations when the
imperfection of devices are calibrated shen . Noting that, the effect of noise
is random that can not be estimated through transformation.
## VI Conclusions
In summary, we have studied the disembodiment of physical properties by pre-
and post-selection in detail. We give a criterion to disembody physical
properties for single particle with multiple degrees of freedom. It is shown
that the non-commute operator can also be separated in different paths. We
generalize the disembodiment to entangled particles and find that the
disembodiment can happen for special conditions due to the entanglement.
Acknowledgements This work was supported by the 973 Program under Grant
2011CB921200 and Grant 2011CBA00200, the Natural Science Foundation of China
under Grant 11004184, and in part by the Knowledge Innovation Project of the
Chinese Academy of Science.
## References
* (1) Y. Aharonov, P. G. Bergmann, and J. Lebowitz, Phys. Rev. 134, B1410 1964 .
* (2) Y. Aharonov, S. Popescu, and J. Tollaksen, Phys. Today 63(11), 27 (2010).
* (3) Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
* (4) O. Hosten and P. Kwiat, Science 319, 787 (2008).
* (5) J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011)
* (6) Y. Aharonov, S. Popescu, and P. Skrzypczyk, arXiv: 1202.0631 (2012).
* (7) I. Ibnouhsein, and A. Grinbaum, arXiv: 1202.4894 (2012).
* (8) C. Shen and L.-M. Duan, arXiv:1201.4379 (2012).
|
arxiv-papers
| 2012-03-20T10:55:06 |
2024-09-04T02:49:28.818800
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chang-Ling Zou, Xu-Bo Zou, F.-W. Sun, and Guang-Can Guo",
"submitter": "Fangwen Sun",
"url": "https://arxiv.org/abs/1203.4384"
}
|
1203.4407
|
# Commutator Theorems for Fractional Integral Operators on Weighted Morrey
Spaces
Zengyan Si Zengyan Si
School of Mathematics and Information Science
Henan Polytechnic University
Jiaozuo 454000
P. R. China sizengyan@yahoo.cn
###### Abstract.
Let $L$ be the infinitesimal generator of an analytic semigroup on
$L^{2}(R^{n})$ with Gaussican kernel bounds, and let $L^{-\alpha/2}$ be the
fractional integrals of $L$ for $0<\alpha<n.$ For any locally integrable
function $b$, The commutators associated with $L^{-\alpha/2}$ are defined by
$[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x)$. When
$b\in BMO(\omega)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the
necessary and sufficient conditions for the boundedness of $[b,L^{-\alpha/2}]$
on weighted Morrey spaces respectively.
###### Key words and phrases:
weighted $BMO$ spaces, weighted Morrey spaces, fractional integrals.
###### 2000 Mathematics Subject Classification:
Primary 42B20, 42B25. Secondary 46B70, 47B38.
## 1\. INTRODUCTION AND MAIN RESULTS
Morrey [morrey] introduced the classical Morrey spaces to investigate the
local behavior of solutions to second order elliptic partial differential
equations. Chiarenza and Frasca [chiarenzafrasca] established the boundedness
of the Hardy-Littlewood maximal operator, the fractional operator and a
singular integral operator on the Morrey spaces. On the other hand, Coifman
and Fefferman [coifman], Muckenhoupt [muckenhoupt] studied the boundedness of
these operator on weighted $L^{p}$ spaces. Motivated by these work, Komori and
Shirai [ks] introduced the following weighted Morrey space and investigated
the boundedness of classical operators in harmonic analysis, that is, the
Hardy-Littlewood maximal operator, a Calderón-Zygmund operator, the fractional
integral operator, etc.
Let $1\leq p<\infty$ and $0\leq k<1$. Then for two weights $\mu$ and $\nu$,
the weighted Morrey space is defined by
$L^{p,k}(\mu,\nu)=\\{f\in
L_{loc}^{p}(\mu):||f||_{L^{p,k}(\mu,\nu)}<\infty\\},$
where
$||f||_{L^{p,k}(\mu,\nu)}=\sup_{Q}\left(\frac{1}{\nu(Q)^{k}}\int_{Q}|f(x)|^{p}\mu(x)dx\right)^{\frac{1}{p}}.$
and the supremum is taken over all cubes $Q$ in $R^{n}$.
If $\mu=\nu,$ then we have the classical Morrey space $L^{p,k}(\mu)$ with
measure $\mu$. When $k=0,$ then $L^{p,k}(\mu,\nu)=L^{p}(\mu)$ is the Lebesgue
space with measure $\mu$.
Suppose that $L$ is a linear operator on $L^{2}(R^{n})$ which generates an
analytic semigroup $e^{-tL}$ with a kernel $p_{t}(x,y)$ satisfying a Gaussian
upper bound, that is,
(1.1) $|p_{t}(x,y)|\leq\frac{C}{t^{\frac{n}{2}}}e^{-c\frac{|x-y|^{2}}{t}}$
for $x,y\in R^{n}$ and all $t>0.$
For $0<\alpha<n,$ the fractional integral $L^{-\alpha/2}$ of the operator $L$
is defined by
$L^{-\alpha/2}f(x)=\frac{1}{\Gamma(\frac{\alpha}{2})}\int_{0}^{\infty}e^{-tL}(f)\frac{dt}{t^{-\alpha/2+1}}(x).$
Note that if $L=-\Delta$ is the Laplacian on $R^{n}$, then $L^{-\alpha/2}$ is
the classical fractional integral $I_{\alpha}$ which plays important roles in
many fields. It is well known that $I_{\alpha}$ is bounded from $L^{p}(R^{n})$
to $L^{q}(R^{n})$ for all $p>1,1/q=1/p-\alpha/n>0$ and is also of weak type
$(1,n/(n-\alpha)).$
Let $1\leq p<\infty$ and $\omega$ be a weight function. A locally integral
function $b$ is said to be in $BMO_{p}(\omega)$ if
$||b||_{BMO_{p}(\omega)}=\sup_{Q}\left(\frac{1}{\omega(Q)}\int_{Q}|b(x)-b_{Q}|^{p}\omega(x)^{1-p}dx\right)^{\frac{1}{p}}\leq
C<\infty,$
where $b_{Q}=\frac{1}{|Q|}\int_{Q}b(y)dy$ and the supremum is taken over all
cube $Q\in R^{n}.$
Let $\omega\in A_{1}$, García-Cuerva [garcia] proved that the spaces
$BMO_{p}(\omega)$ coincide, and the norm of $||\cdot||_{BMO_{p}(\omega)}$ are
equivalent with respect to different values of provided that $1\leq p<\infty.$
Let $b$ be a locally integrable function on $R^{n}$, we consider the
commutator $[b,L^{-\alpha/2}]$ defined by
$[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x).$
Chanillo [chanillo] proved that the commutator $[b,I_{\alpha}]$ of the
multiplication operator by $b\in BMO$ is bounded on $L^{p}$ for $1<p<\infty.$
Duong and Yan [duongyan] proved $[b,L^{-\alpha/2}]$ is bounded from $L^{p}$ to
$L^{q},$ where $b\in BMO,1<p<n/\alpha,1/q=1/p-\alpha/n,0<\alpha<n.$
Mo and Lu [molu] proved the multilinear commutator generated by $\vec{b}$ and
$L^{-\alpha/2}$ is bounded from $L^{p}$ to $L^{q},$ where
$1<p<n/\alpha,1/q=1/p-\alpha/n,0<\alpha<1$,
$\vec{b}=(b_{1},\cdots,b_{m}),b_{i}\in BMO,$ for $i=1,\cdots,m.$
Lu, Ding and Yan [ludingyan] proved $[b,I_{\alpha}]$ is bounded from $L^{p}$
to $L^{q}$ if and only if $b\in BMO.$
Wang [wang] proved that $[b,I_{\alpha}]$ is bounded from $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega),$ where $b\in BMO(\omega)$,
$0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,0<k<p/q$ and $\omega^{q/p}\in
A_{1}.$
Inspired by the above results, we study the boundedness properties of the
commutator $[b,L^{-\alpha/2}]$ on weighted Morrey spaces in this work. The
main theorems are stated as follows.
###### Theorem 1.1.
Let $0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,$ $0\leq k<p/q,$
$\omega^{\frac{q}{p}}\in A_{1}$ and $r_{\omega}>\frac{1-k}{p/q-k},$ where
$r_{\omega}$ denotes the critical index of $\omega$ for the reverse Hölder
condition. Then the following conditions are equivalent:
$(a)$ $b\in BMO(\omega).$
$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega)$ to
$L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$.
Specially, when $k=0$ in Theorem 1.1, we get
###### Corollary 1.1.
Let $0<\alpha<n,1<p<n/\alpha,1/q=1/p-\alpha/n,$ $\omega^{\frac{q}{p}}\in
A_{1}$ and $r_{\omega}>\frac{q}{p},$ where $r_{\omega}$ denotes the critical
index of $\omega$ for the reverse Hölder condition. Then the following
conditions are equivalent:
$(a)$ $b\in BMO(\omega).$
$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p}(\omega)$ to
$L^{q}(\omega^{1-(1-\alpha/n)q})$.
Further more, if $L=-\Delta$ is the Laplacian, Then the following conditions
are equivalent:
$(a^{{}^{\prime}})$ $b\in BMO(\omega).$
$(b^{{}^{\prime}})$ $[b,I_{\alpha}]$ is bounded from $L^{p}(\omega)$ to
$L^{q}(\omega^{1-(1-\alpha/n)q})$.
###### Theorem 1.2.
Let $0<\alpha<n,0\leq k<p/q,1/q=1/p-\alpha/n,$ and $1<r,s<\infty$ such that
$1<rs<p<n/\alpha$, $\omega^{rs}\in A_{p/rs,q/rs}.$ Then the following
conditions are equivalent:
$(a)$ $b\in BMO.$
$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p,k}(\omega^{p},\omega^{q})$ to
$L^{q,kq/p}(\omega^{q})$.
Specially, when $k=0$ in Theorem 1.2, we obtain
###### Corollary 1.2.
Let $0<\alpha<n,1/q=1/p-\alpha/n,$ and $1<r,s<\infty$ such that
$1<rs<p<n/\alpha$, $\omega^{rs}\in A_{p/rs,q/rs}.$ Then the following
conditions are equivalent:
$(a)$ $b\in BMO.$
$(b)$ $[b,L^{-\alpha/2}]$ is bounded from $L^{p}(\omega^{p})$ to
$L^{q}(\omega^{q})$.
Further more, if $L=-\Delta$ is the Laplacian, Then the following conditions
are equivalent:
$(a^{{}^{\prime}})$ $b\in BMO.$
$(b^{{}^{\prime}})$ $[b,I_{\alpha}]$ is bounded from $L^{p}(\omega^{p})$ to
$L^{q}(\omega^{q})$.
###### Remark 1.1.
It is easy to see that our results extend the results in
[chanillo],[duongyan],[ludingyan],[wang] significantly.
## 2\. PREREQUISITE MATERIAL
Let us first recall some definitions.
###### Definition 2.1.
The Hardy-Littlewood maximal operator $M$ is defined by
$M(f)(x)=\sup_{x\in Q}\frac{1}{|Q|}\int_{Q}|f(y)|dy.$
Let $\omega$ be a weight. The weighted maximal operator $M_{\omega}$ is
defined by
$M_{\omega}(f)(x)=\sup_{x\in Q}\frac{1}{\omega(Q)}\int_{Q}|f(y)|\omega(y)dy.$
For $0<\alpha<n,r\geq 1,$ the fractional maximal operator $M_{\alpha,r}$ is
defined by
$M_{\alpha,r}(f)(x)=\sup_{x\in Q}\left(\frac{1}{|Q|^{1-\frac{\alpha
r}{n}}}\int_{Q}|f(y)|^{r}dy\right)^{\frac{1}{r}};$
and the fractional weighted maximal operator $M_{\alpha,r,\omega}$ is defined
by
$M_{\alpha,r,\omega}(f)(x)=\sup_{x\in
Q}\left(\frac{1}{\omega(Q)^{1-\frac{\alpha
r}{n}}}\int_{Q}|f(y)|^{r}\omega(y)dy\right)^{\frac{1}{r}}.$
For any $f\in L^{p}(R^{n}),p\geq 1,$ the sharp maximal function
$M^{\sharp}_{L}f$ associated the generalized approximations to the identity
$\\{e^{-tL},t>0\\}$ is given by
$M^{\sharp}_{L}f(x)=\sup_{x\in
Q}\frac{1}{|Q|}\int_{Q}|f(y)-e^{-t_{Q}L}f(y)|dy$
where $t_{Q}=r^{2}_{Q}$ and $r_{Q}$ is the radius of the ball $Q$.
In the above definitions, the supremum is taken over all cubes $Q$ containing
$x.$
###### Definition 2.2.
A weight function $\omega$ is in the Muckenhoupt class $A_{p}$ with
$1<p<\infty$ if for every cube $Q$ in $R^{n}$, there exists a positive
constant $C$ which is independent of $Q$ such that
$\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{-\frac{1}{p-1}}dx\right)^{p-1}\leq
C.$
When $p=1,\omega\in A_{1},$ if
$\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)\leq C{\texttt{ess}\,\inf}_{x\in
Q}\,\omega(x).$
When $p=\infty,\omega\in A_{\infty},$ if there exist positive constants
$\delta$ and $C$ such that given a cube $Q$ and $E$ is a measurable subset of
$Q$, then
$\frac{\omega(E)}{\omega(Q)}\leq C\left(\frac{|E|}{|Q|}\right)^{\delta}.$
###### Definition 2.3.
A weight function $\omega$ belongs to $A_{p,q}$ for $1<p<q<\infty$ if for
every cube $Q$ in $R^{n}$, there exists a positive constant $C$ which is
independent of $Q$ such that
$\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{q}dx\right)^{\frac{1}{q}}\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{-p^{{}^{\prime}}}dx\right)^{\frac{1}{p^{{}^{\prime}}}}\leq
C.$
where $p^{{}^{\prime}}$ denotes the conjugate exponent of $p>1,$ that is
$1/p+1/p^{{}^{\prime}}=1.$
###### Definition 2.4.
A weight function $\omega$ belongs to the reverse Hölder class $RH_{r}$ if
there exist two constants $r>1$ and $C>0$ such that the following reverse
Hölder inequality
$\left(\frac{1}{|Q|}\int_{Q}\omega(x)^{r}dx\right)^{\frac{1}{r}}\leq
C\left(\frac{1}{|Q|}\int_{Q}\omega(x)dx\right)$
holds for every cube $Q$ in $R^{n}$.
It is well known that if $\omega\in A_{p}$ with $1\leq p<\infty$, then there
exists $r>1$ such that $\omega\in RH_{r}.$ It follows from Hölder s inequality
that $\omega\in RH_{r}$ implies $\omega\in RH_{s}$ for all $1<s<r.$ Moreover,
if $\omega\in RH_{r},r>1,$ then we have $\omega\in RH_{r+\epsilon}$ for some
$\epsilon>0.$ We thus write $r_{w}=\sup\\{r>1:\omega\in RH_{r}\\}$ to denote
the critical index of $\omega$ for the reverse Hölder condition.
We will make use of the following lemmas. We first provide a weighted version
of the local good $\lambda$ inequality for $M^{\sharp}_{L}$ which allow us to
obtain an analog of the classical Fefferman-Stein(see [fefferman, coifman])
estimate on weighted Morrey spaces.
###### Lemma 2.1.
([martell]) Assume that the semigroup $e^{-tL}$ has a kernel $p_{t}(x,y)$
which satisfies the upper bound (1.1). Take $\lambda>0,f\in L^{1}_{0}(R^{n})$
and a ball $Q_{0}$ such that there exists $x_{0}\in Q_{0}$ with
$Mf(x_{0})\leq\lambda.$ Then, for every $\omega\in A_{\infty},0<\eta<1,$ we
can find $\gamma>0$(independent of $\lambda,Q_{0},f,x_{0}$) and constant
$C_{\omega,}r>0$(which only depend on $\omega$) .
$\omega\\{x\in Q_{0}:Mf>A\lambda,M^{\sharp}_{L}f(x)\leq\gamma\lambda\\}\leq
C_{\omega}\eta^{r}\omega(Q_{0}).$
where $A>1$ is a fixed constant which depends only on $n.$
As a consequence, by using the standard arguments, we have the following
estimates:
For every $f\in L^{p,k}(\mu,\nu)$, with $1<p<\infty.$ If $\mu,\nu\in
A_{\infty},1<p<\infty,0\leq k<1.$
$||f||_{L^{p,k}(\mu,\nu)}\leq||Mf||_{L^{p,k}(\mu,\nu)}\leq
C||M^{\sharp}_{L}f||_{L^{p,k}(\mu,\nu)}$
In particular, when $\mu=\nu=\omega$ and $\omega\in A_{\infty},$ we have
$||f||_{L^{p,k}(\omega)}\leq||Mf||_{L^{p,k}(\omega)}\leq
C||M^{\sharp}_{L}f||_{L^{p,k}(\omega)}$
###### Lemma 2.2.
([wang]) Let
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q$ and $r_{\omega}>\frac{1-k}{p/q-k}$,
we have
$||M_{\alpha,1}f||_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C||f||_{L^{p,k}(\omega)}.$
The same conclusion still hold for $I_{\alpha}$.
###### Lemma 2.3.
([wang]) Let
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and
$\omega^{q/p}\in A_{1}.$ Then if $0<k<p/q,1<r<p$ and
$r_{\omega}>\frac{1-k}{p/q-k}$, we have
$||M_{r,\omega}f||_{L^{q,kq/p}(\omega^{q/p},\omega)}\leq
C||f||_{L^{q,kq/p}(\omega^{q/p},\omega)}.$
###### Lemma 2.4.
([wang])
$0<\alpha<n,1<p<\frac{n}{\alpha},\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$,
$0<k<p/q$, $\omega\in A_{\infty}.$ For any $1<r<p,$ we have
$||M_{\alpha,r,\omega}f||_{L^{q,kq/p}(\omega)}\leq C||f||_{L^{p,k}(\omega)}.$
|
arxiv-papers
| 2012-03-20T12:24:18 |
2024-09-04T02:49:28.824469
|
{
"license": "Public Domain",
"authors": "Zengyan Si",
"submitter": "Zengyan Si",
"url": "https://arxiv.org/abs/1203.4407"
}
|
1203.4412
|
# Amplitude analysis of $\gamma n\\!\to\\!\pi^{-}p$ data above 1 GeV
W. Chen1, H. Gao1, W. J. Briscoe2, D. Dutta3, A. E. Kudryavtsev4,2, M.
Mirazita5, M. W. Paris6, P. Rossi5, S. Stepanyan7, I. I. Strakovsky2, V. E.
Tarasov4, R. L. Workman2
1Duke University, Durham, NC 27708, USA 2The George Washington University,
Washington, DC 20052, USA 3Mississippi State University, Mississippi State,
MS 39762, USA 4Institute of Theoretical and Experimental Physics, Moscow,
117259 Russia 5INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy
6Theory Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
7Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
###### Abstract
We report a new extraction of nucleon resonance couplings using $\pi^{-}$
photoprodution cross sections on the neutron. The world database for the
process $\gamma n\to\pi^{-}p$ above 1 GeV has quadrupled with the addition of
new differential cross sections from the CEBAF Large Acceptance Spectrometer
(CLAS) at Jefferson Lab in Hall B. Differential cross sections from CLAS have
been improved with a new final-state interaction determination using a
diagrammatic technique taking into account the $NN$ and $\pi$N final-state
interaction amplitudes. Resonance couplings have been extracted and compared
to previous determinations. With the addition of these new cross sections
significant changes are seen in the high-energy behavior of the SAID cross
sections and amplitudes.
###### pacs:
13.60.Le, 24.85.+p, 25.10.+s, 25.20.-x
## I Introduction
High-precision data and new analysis techniques for $\gamma N\to\pi N$ are
beginning to have a transformative impact on our understanding of the $N$ and
$\Delta$ resonance spectrum. With the arrival of new and improved measurements
of single- and double-polarization quantities, fits have become highly
constrained. As a result, some multipole amplitudes and their underlying
resonant components have changed significantly. This is particularly true for
the neutron-target sector, where, until recently, there were few data on which
to base fits and from which to extract $n\gamma$ photo-decay amplitudes.
The radiative decay width of the neutral states may be extracted from
$\pi^{-}$ and $\pi^{0}$ photoproduction off a neutron, which involves a bound
neutron target (typically the deuteron) and requires the use of a model-
dependent nuclear correction. As a result, our knowledge of neutral resonance
decays is less precise compared to the charged ones.
The existing database contains mainly $\gamma n\\!\to\\!\pi^{-}p$ differential
cross sections. Many of these are old bremsstrahlung measurements with limited
angular coverage and broad energy binning. In several cases, the systematic
uncertainties have not been given. At lower energies, there are data sets for
the inverse $\pi^{-}$ photoproduction reaction: $\pi^{-}p\to\gamma n$. This
process is free from complications associated with a deuteron target. However,
the disadvantage of using this reaction is the high background from the 5 to
500 times larger cross section for $\pi^{-}p\to\pi^{0}n\to\gamma\gamma n$.
Here we explore the effect of adding CLAS differential cross sections for
$\gamma n\\!\to\\!\pi^{-}p$, extracted from $\gamma d\\!\to\\!\pi^{-}pp$ CLAS
, to the full SAID database. Measurements extend from 1.05 to 3.5 GeV in the
photon energy. The present cross section set has quadrupled the world database
for $\gamma n\\!\to\\!\pi^{-}p$ above 1 GeV, which allows for fits covering
the region up to 2.7 GeV. We will show that these new data require large
adjustments of our fits.
In the next section, Sec. II, we will give a brief overview of the available
experimental data. A discussion of the final-state interaction (FSI)
calculations is given in Sec. III. The new CLAS data are compared with fits
and older measurements in Sec. IV. In Sec. V, we discuss the fits and the
extraction of resonance parameters. Finally, in Sec. VI, we summarize our
findings and discuss the potential impact of future measurements and partial-
wave analysis (PWA).
## II Data Set
Due to lack of neutron targets, the database for the reactions $\gamma
n\\!\to\\!\pi^{0}n$ and $\gamma n\\!\to\\!\pi^{-}p$ is small compared to
single-pion photoproduction reactions using proton targets, $\gamma
p\to\pi^{+}n$ and $\gamma p\to\pi^{0}p$. Previous $\gamma N\to\pi N$
measurements are available in the SAID database world_data .
Only $364$ data-points are available for $\gamma n\\!\to\\!\pi^{0}n$ below 2.7
GeV. For $\gamma n\\!\to\\!\pi^{-}p$, the situation is especially dire in the
photon energy range above 1 GeV. There are only 360 data points, half of which
come from polarized measurements. Below 1 GeV, there are significant numbers
of $\gamma n\\!\to\\!\pi^{-}p$ data, coming mainly from Meson Factories
(LAMPF, TRIUMF, and PSI) via inverse pion photoproduction $\pi^{-}p\to\gamma
n$. Overall, there are $2093$ data points, 17% of which are from polarized
measurements. Some differential cross sections for the $\pi^{-}p\to\gamma n$
have been measured at BNL AGS, using the Crystal Ball multiphoton
spectrometer. Measurements were made at 18 pion momenta from 238 to 748
MeV/$c$, corresponding to Eγ from 285 to 769 MeV aziz . These data have been
used to evaluate neutron multipoles in the vicinity of the $N(1440)1/2^{+}$
resonance.
We have recently considered the effect of the beam-asymmetry data ($\Sigma$)
of $\vec{\gamma}n\to\pi^{-}p$ graal1 and $\vec{\gamma}n\to\pi^{0}n$ graal2
from GRAAL on our fits to neutron-target data sn11 . These include $216$
$\Sigma$ measurements of $\pi^{0}n$ covering Eγ=703–1475 MeV and
$\theta$=53–164∘ plus $99$ $\Sigma$ measurements of $\pi^{-}p$ for Eγ=753–1439
MeV and $\theta$=33–163∘. Predictions for $\gamma n\\!\to\\!\pi^{0}n$ were
qualitatively different from the measurements over a wide angular range above
a center-of-mass (CM) energy of 1650 MeV.
In 2009, the CLAS Collaboration at Jefferson Lab reported a detailed study of
the reaction $\gamma n\\!\to\\!\pi^{-}p$ using a high statistics
photoproduction experiment on deuterium CLAS . This data set added $855$
differential cross sections between 1.05 and 3.5 GeV, and pion production
center-of-mass angles between 32∘ and 157∘, to the existing data base. The
overall systematic uncertainty varies between 5.8%, at the lowest photon
energy, and up to 9.4% at the highest photon energy. Details of the data
processing and analysis can be found in Ref. Wei . An improvement in the FSI
has been made since the original publication CLAS .
Chen, et al. CLAS estimated FSI corrections according to the Glauber
formulation trans and this correction was found to be about 20%. The
uncertainty of the Glauber calculation for the FSI correction was estimated to
be 5% in Ref. zhu ; zhu1 . To study the model uncertainty in calculating the
FSI correction, another calculation using the approach of Ref. laget was
adopted. Both methods agreed within 10%. A 10% systematic uncertainty to the
differential cross section was assigned for the FSI correction CLAS .
In a further study of the FSI corrections for the $\gamma n\to\pi^{-}p$ cross
section determination from the deuteron data, we used a diagrammatic technique
PRC2011 , including the fact that CLAS does not detect protons with momenta
less than 200 MeV/$c$. A short description of the FSI formalism is given in
Sec. III.
## III FSI calculations
### III.1 Amplitudes
Calculations of the $\gamma d\\!\to\\!\pi^{-}pp\,$ differential cross sections
with the FSI taken into account, were done in a model represented by the
diagrams in Fig. 1. These diagrams correspond to the IA [Fig. 1(a)], $pp$-FSI
[Fig. 1(b)], and $\pi$N-FSI [Fig. 1(c)] amplitudes, denoted by $M_{a}$,
$M_{b}$, and $M_{c}$, respectively. The resulting amplitude $M_{\gamma d}$
reads
$M_{\gamma d}=M_{a}+M_{b}+M_{c},$ $M_{a,c}\\!=M^{(1)}_{a,c}+M^{(2)}_{a,c}.$
(1)
Figure 1: Feynman diagrams for the leading components of the $\gamma
d\\!\to\\!\pi^{-}pp$ amplitude. (a) IA, (b) $pp$-FSI, and (c) $\pi$N-FSI.
Filled black circles show FSI vertices. Wavy, dashed, solid, and double lines
correspond to the photons, pions, nucleons, and deuterons, respectively.
IA and $\pi$N-FSI diagrams [Figs. 1(a),(c)] include also the cross-terms
between the final protons. The terms in Eq. (1) depend on the elementary
$\gamma N\\!\to\\!\pi N$ amplitudes and deuteron wave function (DWF). The
terms $M_{b}$ and $M_{c}$ depend also on the $N\\!N\\!\to\\!N\\!N$ and $\pi
N\\!\to\\!\pi N$ amplitudes, respectively.
The $\gamma N\\!\to\\!\pi N$ amplitudes were expressed through four
independent Chew-Goldberger-Low-Nambu (CGLN) amplitudes CGLN $F_{1-4}$, which
were generated by the SAID code, using the George Washington University (GW)
Data Analysis Center (DAC) pion photoproduction multipoles SAID02 ; pr_PWA .
The $N\\!N$\- and $\pi N$-scattering amplitudes were calculated, using the
results of GW $N\\!N$ NN_PWA and $\pi N$ piN_PWA PWAs. The DWF was taken
from the Bonn potential (full model) BonnCD . The elementary amplitudes are
dependent on the momenta of the external and intermediate particles in Fig. 1.
Thus, Fermi motion is taken into account in the $\gamma d\\!\to\\!\pi^{-}pp$
amplitude $M_{\gamma d}$. Details of calculations of the amplitudes
$M_{a,b,c}$ in Eq. (1) are given in Ref. PRC2011 .
### III.2 FSI Correction
We extract the $\gamma n\\!\to\\!\pi^{-}p\,$ cross section from the deuteron
data in the quasi-free (QF) kinematical region of the $\gamma
d\\!\to\\!\pi^{-}pp\,$ reaction with fast and slow protons $p_{1}$ and
$p_{2}$, respectively, where the $\gamma d\\!\to\\!\pi^{-}pp$ cross section is
dominated by the IA amplitude $M^{(1)}_{a}$, i.e., $M_{\gamma d}\approx
M^{(1)}_{a}$, while the cross term $M^{(2)}_{a}$ and the FSI amplitudes
$M_{b,c}$ are relatively small. This consideration is addressed in the
analysis of the CLAS Wei ; CLAS data for the reaction $\gamma
d\\!\to\\!\pi^{-}pp$ with kinematical cuts
$|\mbox{\boldmath$p$}_{2}|<200~{}MeV/c<|\mbox{\boldmath$p$}_{1}|$,
corresponding to the CLAS experimental conditions.
In the QF approximation, the $\gamma d\\!\to\\!\pi^{-}pp$ and $\gamma
n\\!\to\\!\pi^{-}p\,$ differential cross sections for unpolarized particles
are related to each other in a known way BL .
$\frac{d\sigma_{\gamma
d}^{QF}}{d\mbox{\boldmath$p$}_{2}\,d\Omega}=\frac{E_{\gamma}^{\,\prime}}{E_{\gamma}}\,\rho(p_{2})\,\frac{d\sigma_{\gamma
n}}{d\Omega}.$ (2)
Here: $\Omega$ is the solid angle of relative motion in the $\pi^{-}p_{1}$
system; $E_{\gamma}$ and
$E_{\gamma}^{\,\prime}=(1+\beta\cos\theta_{2})E_{\gamma}$ are the photon
laboratory energies for the reactions $\gamma d\\!\to\\!\pi^{-}pp$ and $\gamma
n\\!\to\\!\pi^{-}p\,$, respectively; $\beta=p_{2}/E_{2}$ ($\theta_{2}$) is the
laboratory velocity (polar angle) of spectator proton $p_{2}\,$; $\rho(p)$ is
the square of DWF and $\int\\!\rho(p)\,d\mbox{\boldmath$p$}=1$. Let
$d\sigma_{\gamma d}^{QF}\\!/d\Omega$ and $d\sigma_{\gamma d}\\!/d\Omega$ be
the deuteron cross section, integrated over $\mbox{\boldmath$p$}_{2}$ in a
small region $|\mbox{\boldmath$p$}_{2}|<p_{\,max}$ and obtained with the
amplitudes $M_{\gamma d}\\!=M^{(1)}_{a}$ and $M_{a}\\!+M_{b}\\!+M_{c}$,
respectively. Then, from Eq. (2) (see details in Ref. PRC2011 ) we obtain
$\frac{d\bar{\sigma}_{\gamma
n}^{\,exp}}{d\Omega}(\bar{E}_{\gamma},\theta)=c^{-1}R^{-1}\\!(E_{\gamma},\theta)\frac{d\sigma_{\gamma
d}^{exp}}{d\Omega}(E_{\gamma},\theta),$
$c=\mkern-22.0mu\int\limits_{p\,<p_{max}}\mkern-22.0mu\rho(\mbox{\boldmath$p$})\,d\mbox{\boldmath$p$},~{}~{}~{}~{}~{}R(E_{\gamma},\theta)=\frac{d\sigma_{\gamma
d}\\!/d\Omega}{d\sigma_{\gamma d}^{QF}\\!/d\Omega},$ (3)
were $d\bar{\sigma}_{\gamma n}^{\,exp}/d\Omega$ is the neutron cross section,
extracted from the deuteron data. Here: $\theta$ is the polar angle of the
outgoing pion in the $\pi^{-}p_{1}$ frame; $c=c(p_{max})\leq 1$ is the
“effective number” of neutrons with momenta $p<p_{max}$ in the deuteron; $R$
is the correction factor for FSI effects as well as for the “suppressed”
amplitude $M^{(2)}_{a}$. The factor $R$ depends on $E_{\gamma}$ and $\theta$
as well as on the kinematical cuts applied.
The neutron cross section $d\bar{\sigma}_{\gamma n}/d\Omega$ (3) is averaged
over the photon energy $E_{\gamma}^{\prime}\sim E_{\gamma}$, and
$\bar{E}_{\gamma}$ is some “effective” $E_{\gamma}^{\prime}$ value in the
range $E_{\gamma}(1\pm\beta)$. For small values of $p_{max}$ we have $\beta\ll
1$ and $\bar{E}_{\gamma}\approx E_{\gamma}$.
We applied FSI corrections PRC2011 dependent on the Eγ and $\theta$. As an
illustration, Fig. 2 shows the FSI correction factor $R$ for the present
$\gamma n\\!\to\\!\pi^{-}p$ differential cross sections as a function of the
pion production angle in the CM frame for different energies over the range of
the CLAS experiment. Overall, the FSI correction factor $R<1$, while the
effect, i.e., the $(1-R)$ value, is less than 10% and the behavior is very
smooth vs. pion production angle.
The contribution of FSI calculations PRC2011 to the overall systematics is
estimated to be 2% (3%) below (above) 1800 MeV. Above 2700 MeV, our estimation
of systematic uncertainty due to the FSI calculations is 5%. Then we added FSI
systematics to the overall experimental systematics in quadrature.
Figure 2: FSI correction factor $R$ for $\gamma n\\!\to\\!\pi^{-}p$ as a
function of $\theta$, where $\theta$ is the production angle of $\pi^{-}$ in
the CM frame. The present calculations (solid circles) are shown for five
energies: (a) $E_{\gamma}$ = 1100 MeV, (b) 1500 MeV, (c) 1900 MeV, (d) 2300
MeV, and (e) 2700 MeV. There are no uncertaintes given.
## IV Results
Since the CLAS results for the $\gamma n\\!\to\\!\pi^{-}p$ differential cross
sections consist of $855$ experimental points, they are not tabulated in this
or the previous CLAS publication, but are available in the SAID database
world_data along with their uncertainties and the energy binning.
Specific examples of agreement with previous measurements are displayed in
Fig. 3, where we compare differential cross sections obtained here with those
from SLAC sf74 , DESY be73 , and Yerevan ab80 , at energies common to those
experiments. Previous measurements used a modified Glauber approach and the
procedure of unfolding the Fermi motion of the neutron target. The CLAS data
and the results from SLAC, DESY, and Yerevan appear to agree well at these
energies. Unfortunately, there are no measurements for $\pi^{-}p\to\gamma n$
to compare at these energies.
Figure 3: (Color online) Differential cross sections for $\gamma
n\\!\to\\!\pi^{-}p$ as a function of $\theta$, where $\theta$ is the
production angle of $\pi^{-}$ in the CM frame. The present data (solid
circles) are shown for six energy bins: (a) $E_{\gamma}$ = 1150 MeV, (b) 1200
MeV, (c) 1250 MeV, (d) 1350 MeV, (e) 1550 MeV, and (f) 1900 MeV. Previous data
are shown for experiments at SLAC sf74 (open circles); DESY be73 (open
squares), and Yerevan ab80 (crosses). Plotted uncertainties are statistical
only. Solid (dash-dotted) lines correspond to the GB12 (SN11 sn11 ) solution.
Thick solid (dashed) lines give GZ12 solution (MAID07 maid , which terminates
at $W$ = 2 GeV).
While agreement with previous measurements is generally good, the new data
extend to higher energies with more complete angular coverage and are more
constraining in the PWA, as is evident in Fig. 3.
A more complete comparison of the CLAS data with fits and predictions is given
in Fig. 4. It is interesting to note that the data appear to have fewer
angular structures than the earlier fits.
## V Amplitude Analysis of Data
We have included the new cross sections from the CLAS experiment in a number
of multipole analyses covering incident photon energies up to 2.7 GeV, using
the full SAID database, in order to gauge the influence of these measurements,
as well as their compatibility with previous measurements.
Table 1: $\chi^{2}$ comparison of fits to pion photoproduction data up to 2.7 GeV. Results are shown for six different SAID solutions (current GB12 and GZ12 with previous SN11 sn11 , SP09 du1 , FA06 pr_PWA , SM02 SAID02 , and SM95 sm95 ). Solution | Energy limit | $\chi^{2}$/NData | NData
---|---|---|---
| (MeV) | |
GZ12 | 2700 | 1.95 | 26179
GB12 | 2700 | 2.09 | 26179
SN11 | 2700 | 2.08 | 25553
SP09 | 2700 | 2.05 | 24912
SM02 | 2000 | 2.01 | 17571
SM95 | 2000 | 2.37 | 13415
In Table 1, we compare the new GB12 and GZ12 solutions with four previous SAID
fits (SN11 sn11 , SP09 du1 , SM02 SAID02 , and SM95 sm95 ). The overall
$\chi^{2}$ has remained stable against the growing database, which has
increased by a factor of 2 since 1995 (most of this increase coming from data
from photon-tagging facilities).
In fitting the data, the stated experimental systematic uncertainties have
been used as an overall normalization adjustment factor for the angular
distributions SAID02 ; sn11 . Presently, the pion photoproduction database
below Eγ = 2.7 GeV consists of 26179 data points that have been fit in the
GB12 (GZ12) solution with $\chi^{2}$ = 54832 (50998). The contribution to the
total $\chi^{2}$ in the GB12 (GZ12) analyses of the $626$ new CLAS $\gamma
n\\!\to\\!\pi^{-}p$ data points (e.g., those data points up to Eγ = 2.7 GeV)
is 1580 (1190). This should be compared to a starting $\chi^{2}$ = 45636 for
the new CLAS data using predictions from our previous SN11 solution.
The solution, GB12, uses the same fitting form as our recent SN11 solution
sn11 , which incorporated the neutron-target $\Sigma$ data from GRAAL graal1 ;
graal2 . This fit form was motivated by a multi-channel K-matrix approach,
with an added phenomenological term proportional to the $\pi N$ reaction cross
section. A second fit, GZ12, used instead the recently proposed form mc12
based on a unified Chew-Mandelstam parameterization of the GW DAC fits to both
$\pi N$ elastic scattering and photoproduction. This form explicitly includes
contributions from channels $\pi N$, $\pi\Delta$, $\rho N$, and $\eta N$, as
determined in the SAID elastic $\pi N$ scattering analysis.
Resonance couplings, extracted as in Ref. sn11 , are listed in Table 2 and
compared to the previous SN11 determinations and the Particle Data Group (PDG)
averages PDG . Couplings for the $N(1440)1/2^{+}$, $N(1520)3/2^{-}$, and
$N(1675)5/2^{-}$ are reasonably close to the SN11 estimates. The value of
$A_{1/2}$ found for the $N(1535)1/2^{-}$, using the GB12 fit, is very close to
the SN11 determination. Using the GZ12 fit, however, the result is somewhat
larger in magnitude ($-85\pm 15$). A similar feature was found for the proton
couplings, using this form, in Ref. mc12 . Using this alternate form, a
determination of the $N(1650)1/2^{-}$ $A_{1/2}$ was difficult and resulted in
a value, lower in magnitude by about 50%, compared to the value from GB12
listed in Table 2. For this reason, we consider the uncertainty associated
with this state to be a lower limit only. No value was quoted for the
$N(1720)3/2^{+}$ state. As can be seen in Figs. 5 – 7, the two different fit
forms GB12 and GZ12, though similar in shape, have opposite signs for the
imaginary parts of corresponding multipoles (${}_{n}E^{1/2}_{1+}$ and
${}_{n}M^{1/2}_{1+}$) in the neighborhood of the resonance position, and even
the sign can not be determined. This is in line with the PDG estimates, which
also fail to give signs for the couplings to this state.
Figure 4: (Color online) The differential cross section for $\gamma
n\\!\to\\!\pi^{-}p$ below Eγ = 2.7 GeV versus pion CM angle. Solid (dash-
dotted) lines correspond to the GB12 (SN11 sn11 ) solution. Dashed lines give
the MAID07 maid predictions. Experimental data are from the current (filled
circles). Plotted uncertainties are statistical.
Figure 5: (Color online) Neutron multipole I=1/2 amplitudes from threshold to
$W$ = 2.43 GeV ($E_{\gamma}$ = 2.7 GeV). Solid (dash-dotted) lines correspond
to the GB12 (SN11 sn11 ) solution. Thick solid (dashed) lines give GZ12
solution (MAID07 maid , which terminates at $W$ = 2 GeV). Vertical arrows
indicate resonance energies, $W_{R}$, and horizontal bars show full ($\Gamma$)
and partial ($\Gamma_{\pi N}$) widths associated with the SAID $\pi N$
solution SP06 piN_PWA .
Figure 6: (Color online) Notation of the multipoles is the same as in Fig. 5.
Figure 7: (Color online) Notation of the multipoles is the same as in Fig. 5. Table 2: Resonance parameters for N∗ states from the SAID fit to the $\pi N$ data piN_PWA (second column) and neutron helicity amplitudes $A_{1/2}$ and $A_{3/2}$ (in [(GeV)${}^{-1/2}\times 10^{-3}$] units) from the GB12 solution (first row), previous SN11 sn11 solution (second row), and average values from the PDG10 PDG (third row). †See text. Resonance | $\pi N$ SAID | $A_{1/2}$ | $A_{3/2}$
---|---|---|---
$N(1535)1/2^{-}$ | $W_{R}$=1547 MeV | $-$58$\pm$6† |
| $\Gamma$=188 MeV | $-$60$\pm$3 |
| $\Gamma_{\pi}/\Gamma$=0.36 | $-$46$\pm$27 |
$N(1650)1/2^{-}$ | $W_{R}$=1635 MeV | $-$40$\pm$10† |
| $\Gamma$=115 MeV | $-$26$\pm$8 |
| $\Gamma_{\pi}/\Gamma$=1.00 | $-$15$\pm$21 |
$N(1440)1/2^{+}$ | $W_{R}$=1485 MeV | 48$\pm$4 |
| $\Gamma$=284 MeV | 45$\pm$15 |
| $\Gamma_{\pi}/\Gamma$=0.79 | 40$\pm$10 |
$N(1520)3/2^{-}$ | $W_{R}$=1515 MeV | $-$46$\pm$6 | $-$115$\pm$5
| $\Gamma$=104 MeV | $-$47$\pm$2 | $-$125$\pm$2
| $\Gamma_{\pi}/\Gamma$=0.63 | $-$59$\pm$9 | $-$139$\pm$11
$N(1675)5/2^{-}$ | $W_{R}$=1674 MeV | $-$58$\pm$2 | $-$80$\pm$5
| $\Gamma$=147 MeV | $-$42$\pm$2 | $-$60$\pm$2
| $\Gamma_{\pi}/\Gamma$=0.39 | $-$43$\pm$12 | $-$58$\pm$13
$N(1680)5/2^{+}$ | $W_{R}$=1680 MeV | 26$\pm$4 | $-$29$\pm$2
| $\Gamma$=128 MeV | 50$\pm$4 | $-$47$\pm$2
| $\Gamma_{\pi}/\Gamma$=0.70 | 29$\pm$10 | $-$33$\pm$9
## VI Summary and Conclusion
A comprehensive set of differential cross sections at 26 energies for
negative-pion photoproduction on the neutron, via the reaction $\gamma
d\\!\to\\!\pi^{-}pp$, have been determined with a JLab tagged-photon beam for
incident photon energies from 1.05 to 3.5 GeV. To accomplish a state-of-the-
art analysis, we included new FSI corrections using a diagrammatic technique,
taking into account a kinematical cut with momenta less (more) than 200
MeV/$c$ for slow (fast) outgoing protons.
The updated PWAs examined mainly the effect of new CLAS neutron-target data on
the SAID multipoles and resonance parameters. These new data have been
included in a SAID multipole analysis, resulting in new SAID solutions, GB12
and GZ12. A major accomplishment of this CLAS experiment is a substantial
improvement in the $\pi^{-}$-photoproduction database, adding $855$ new
differential cross sections quadrupling the world database for $\gamma
n\\!\to\\!\pi^{-}p$ above 1 GeV. Comparison to earlier SAID fits, and a lower-
energy fit from the Mainz group, shows that the new solutions are much more
satisfactory at higher energies.
On the experimental side, further improvements in the PWAs await more data,
specifically in the region above 1 GeV, where the number of measurements for
this reaction is small. Of particular importance in all energy regions is the
need for data obtained involving polarized photons and polarized targets. Due
to the closing of hadron facilities, new $\pi^{-}p\to\gamma n$ experiments are
not planned and only $\gamma n\\!\to\\!\pi^{-}p$ measurements are possible at
electromagnetic facilities using deuterium targets. Our agreement with
existing $\pi^{-}$ photoproduction measurements leads us to believe that these
photoproduction measurements are reliable despite the necessity of using a
deuterium target. We hope that new CLAS $\Sigma$-beam asymmetry measurements
for $\vec{\gamma}n\to\pi^{-}p$, at Eγ = 910 up to 2400 MeV and pion production
angles from 20∘ to 140∘ (1200 data) in the CM frame, will soon dar provide
further constraints for the neutron multipoles.
###### Acknowledgements.
The authors acknowledge helpful comments and preliminary fits from R. A. Arndt
in the early stages of this work. Then the authors are thankful to E. Pasyuk
for useful discussions. We acknowledge the outstanding efforts of the CLAS
Collaboration who made the experiment possible. This work was supported in
part by the U.S. Department of Energy Grants DE–FG02–99ER41110 and
DE–FG02—03ER41231, by the Russian RFBR Grant No. 02–0216465, by the Russian
Atomic Energy Corporation “Rosatom” Grant NSb–4172.2010.2, the National
Science Foundation, and the Italian Istituto Nazionale di Fisica Nucleare.
## References
* (1) W. Chen, el al. (CLAS Collaboration), Phys. Rev. Lett. 103, 012301 (2009).
* (2) W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Institute of Nuclear Studies of The George Washington University Database;
http://gwdac.phys.gwu.edu/analysis/pr_analysis.html .
* (3) A. Shafi, S. Prakhov, I. I. Strakovsky, W. J. Briscoe, B. M. K. Nefkens, et al. (Crystal Ball Collaboration), Phys. Rev. C 70, 035204 (2004).
* (4) G. Mandaglio, et al. (GRAAL Collaboration), Phys. Rev. C 82, 045209 (2010).
* (5) R. Di Salvo, et al. (GRAAL Collaboration), Eur. Phys. J. A 42, 151 (2009).
* (6) R. L. Workman, W. J. Briscoe, M. W. Paris, and I. I. Strakovsky, Phys. Rev. C 85, 025201 (2012).
* (7) W. Chen, Ph.D. Thesis, Duke University, 2010.
* (8) H. Gao, R. J. Holt, and V.R. Pandharipande, Phys. Rev. C 54, 2779 (1996).
* (9) L. Y. Zhu, et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. Lett. 91, 022003 (2003).
* (10) L. Y. Zhu, et al. (Jefferson Lab Hall A Collaboration), Phys. Rev. C 71, 044603 (2005).
* (11) J. M. Laget, Phys. Rev. C 73, 044003 (2006).
* (12) V. E. Tarasov, W. J. Briscoe, H. Gao, A. E. Kudryavtsev, and I. I. Strakovsky, Phys. Rev. C 84, 035203 (2011).
* (13) G. F. Chew, M. L. Goldberger, F. E. Low, and Y. Nambu, Phys. Rev. 106, 1345 (1957).
* (14) R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 66, 055213 (2002).
* (15) M. Dugger, J. P. Ball, P. Collins, E. Pasyuk, B. G. Ritchie, R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, et al. (CLAS Collaboration), Phys. Rev. C 76, 025211 (2007).
* (16) R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 76, 025209 (2007).
* (17) R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 74, 045205 (2006).
* (18) R. Machleidt, K. Holinde, and C. Elster, Phys. Rept. 149, 1 (1987).
* (19) I. Blomqvist and J. M. Laget, Nucl. Phys. A280, 405 (1977).
* (20) P. E. Scheffler, et al., Nucl. Phys. B75, 125 (1974).
* (21) P. Benz, et al., Nucl. Phys. B65, 158 (1973).
* (22) L. O. Abrahamian, et al., Sov. J. Nucl. Phys. 32, 69 (1980).
* (23) D. Drechsel, S. S. Kamalov, and L. Tiator, Eur. Phys. J. A 34, 69 (2007); http://www.kph.uni-mainz.de/MAID/.
* (24) M. Dugger, J. P. Ball, P. Collins, E. Pasyuk, B. G. Ritchie, R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, et al. (CLAS Collaboration), Phys. Rev. C 79, 065206 (2009).
* (25) R. A. Arndt, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 53, 430 (1996).
* (26) R. L. Workman, M. W. Paris, W. J. Briscoe, and I. I. Strakovsky, submitted to Phys. Rev. C (2012); arXiv:1202.0845 [hep–ph].
* (27) K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010).
* (28) D. Sokhan, Ph.D. Thesis, Edinburgh University, 2009.
|
arxiv-papers
| 2012-03-20T12:44:05 |
2024-09-04T02:49:28.830287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "W. Chen (Duke U.), H. Gao (Duke U.), W. J. Briscoe (GW), D. Dutta\n (MSU), A. E. Kudryavtsev (ITEP/GW), M. Mirazita (INFN), M. Paris (LANL), P.\n Rossi (INFN), S. Stepanyan (JLab), I. I. Strakovsky (GW), V. E. Tarasov\n (ITEP), and R. L. Workman (GW)",
"submitter": "Igor Strakovsky",
"url": "https://arxiv.org/abs/1203.4412"
}
|
1203.4579
|
# Metrics and norms used for obtaining sparse solutions to underdetermined
Systems of Linear Equations
Leoni Dalla and George K. Papageorgiou
###### Abstract
This paper focuses on defining a measure, appropriate for obtaining optimally
sparse solutions to underdetermined systems of linear equations.111The
following work done, was within the completion of my master thesis titled
“Algorithms for the computation of sparse solutions of undefined systems of
equations” at the department of Mathematics, University of Athens which was
assigned to me in association with the department of Informatics and
Telecommunications, National and Kapodistrian University of Athens. The
general idea is the extension of metrics in n-dimensional spaces via the
Cartesian product of metric spaces.
## 1 Introduction
In general topology, mathematicians have long ago defined measures that had
seen minimum usage (if not at all) in applications. Later on, the development
of Measure Theory was mandatory for the progression of applied mathematics and
other sciences too. Along with the progress in computer sciences came the
demand defining measures of unusual nature and uncovering the properties they
obey.
In signal (image or sound) processing a usual problem that arises, is how to
transfer a signal using a sparse (economical, but sufficient) representation
[2, 3]. Given a specific matrix $\mathbf{A}$ of dimension $m\times n,\
m,n\in\mathbb{N}$ with $m<n$ (underdetermined) and a vector $\mathbf{b},$ find
among all, the sparsest or (a less sparse) solution
$\mathbf{x}\in\mathbb{R}^{n}$ of the linear system $\mathbf{Ax=b}$. This is
the simplest form of the problem, which means that the noise of the signal is
not included (noiseless problem).
Since an undefined system of linear equations has infinite number of
solutions, they need to be filtered, using additional functions, in order to
obtain solutions of a certain type according to specific criteria. Functions
that measure “energy”, like the $l_{2}$ norm, are used in many occasions, yet
measuring sparsity requires a measure of “sparsity”, i.e. a different function
[2, 3].
The optimization task is minimizing the number of nonzero coordinates of a
vector in $n$-dimensions, i.e. finding a sparse representative of the signal.
The number of nonzero coordinates of a vector $\mathbf{x}$ is known to be the
number of elements included in the set of nonzero values of a vector, which is
called support of the vector, i.e. $supp\\{\mathbf{x}\\}$. Also, in recent
work of Donoho and Elad the measure was “used” under the symbol of norm
$\|\mathbf{x}\|_{0}=\\#\\{i:x_{i}\neq 0\\},$ but it is clear that it does not
satisfy the norm properties [2, 3].
In the following paper, we begin posing some examples in everyday life, where
different measures are needed in order to figure out distances. After a short
review on metric spaces follows the definition of $p$-metrics in a Cartesian
product space. The next section is of main interest, since we equip
$\mathbb{R}^{n}$ with metrics and prove that the discrete metric could be
obtained as a limit of a $p$-metric [2]. In addition, follows a review on
norms and the correlation between norms and metrics. Finally, we conclude with
a comparison of functions, on the quest for a convex one, suitable for
optimization tasks.
## 2 Measures in everyday life
In everyday life people subconsciously use measures in order to figure out
distances, albeit those measures are not always well defined. Given an
arbitrary set of points, a matter of most concern is to measure the distances
between those points. However, the measure that we use in every problem is
different and depends on the scale we would like to use, as well as the
structure of the setting.
The distance between Athens and New York is measured using the geodesic line
between those points, i.e. the shortest route between two points on Earth’s
surface. (Fig. 1).
Figure 1: Geodesic distance between Athens - New York: $7920\ km$ Figure 2:
Travel distance between Athens - Thessaloniki: $502\ km$ Figure 3: Distance in
the area of Manhattan, borough of New York City.
In case of a road trip, the travel distance depends on the road’s structure
and does not coincide with shortest distance between those two points (towns)
(Fig. 2).
Furthermore, the existence of distances that differ from our perception of the
shortest path cannot pass unnoticed. The distance that a person has to travel
in the area of Manhattan (borough of New York City) in order to move from
Times Square to the junction of 57th Street with 9th Avenue depends on the
structure of the setting (Fig. 3).
Another measure, used in order to define distances between compact sets, e.g.
the distance between two islands, is the Hausdorff distance (named after Felix
Hausdorff) between the whole sets $K$ and $A$ defined
$h(K,A)=max\\{\max_{\kappa\in K}{\min_{\alpha\in
A}|\kappa-\alpha|},\max_{\alpha\in A}\min_{\kappa\in K}|\kappa-\alpha|\\}.$
The latter represents, e.g. the minimum distance one has to travel in order to
move from any village of the island of Andros (or Kefallonia) to any village
of the island of Kefallonia (or Andros) (Fig. 4).
Figure 4: Distance between Kefallonia - Andros
So far, we have seen cases where the concept of distance needs to be
mathematically defined in order to understand, develop and solve problems
arising from very different settings. Hence, we should define the means needed
in order to measure in a wide variety of cases.
## 3 Metric spaces
###### Definition 1
Let $X$ be an arbitrary nonempty set. Metric222Symbolize $d,\rho$ or $\sigma$.
(or distance) in $X,$ is a map $\rho:X\times X\longrightarrow\mathbb{R}$
obeying the following properties:
1. 1.
$\rho(x,y)\geq 0,\ \forall x,y\in X$ and $\rho(x,y)=0\Leftrightarrow x=y$
2. 2.
$\rho(x,y)=\rho(y,x),\ \forall x,y\in X,$ (Symmetric property)
3. 3.
$\rho(x,y)\leq\rho(x,z)+\rho(z,y),\ \forall x,y,z\in X,$ (Triangular
inequality)
The elements of the set are called points, the real nonnegative number
$\rho(x,y)$ is called the distance between $x,y\in X$ and the pair $(X,\rho)$
metric space.
Consequently, a set equipped with a metric, automatically obtains the
structure of a topological space 333A topological space doesn’t have to be a
metric space.. We now define the open and the closed ball of center $x_{0}\in
X$ and radius $r>0,$ notions necessary for the topological description of a
metric space.
###### Definition 2
Let $(X,d)$ be a metric space, $x_{0}\in X$ and $r>0$. The set
$S_{d}(x_{0},r)=\\{x\in X:d(x,x_{0})<r\\}$ is called an open ball of center
$x_{0}$ and radius $r$.
###### Definition 3
Let $(X,d)$ be a metric space, $x_{0}\in X$ and $r\geq 0$. The set
$\widetilde{S}_{d}(x_{0},r)=\\{x\in X:d(x,x_{0})\leq r\\}$ is called a closed
ball of center $x_{0}$ and radius $r$.
###### Definition 4
The set $A\subseteq X$ is called an open set, if for every $\alpha\in A$ there
exists $r>0$ such that $S_{d}(\alpha,r)\subseteq A$.
###### Definition 5
The set $B\subseteq X$ is called a closed set, if its complement $X\setminus
B$ is an open set.
###### Definition 6
The set $\Gamma$ $\subseteq X$ is bounded if there exists $x_{0}\in X$ and
$r>0,$ such that $\Gamma\subseteq S_{d}(x_{0},r).$
#### Examples of Metric spaces:
* •
The most common metrics to use in $\mathbb{R}^{n}$ are $d_{1},\ d_{2}$ between
its points $\mathbf{x}=(x_{1},x_{2},...,x_{n}),$
$\mathbf{y}=(y_{1},y_{2},...,y_{n}).$
The metric $\mathbf{d_{1}}$ (Manhattan metric) in $\mathbb{R}^{n}$ is defined
as
$d_{1}(\mathbf{x},\mathbf{y})=\sum_{i=1}^{n}|x_{i}-y_{i}|.$
Thus, in $\mathbb{R}^{2}$:
$d_{1}(\mathbf{x},\mathbf{y})=|x_{1}-y_{1}|+|x_{2}-y_{2}|$ and the closed ball
is respectively
$\widetilde{S}_{d_{1}}(\mathbf{0},r)=\\{\mathbf{x}\in\mathbb{R}^{2}:d_{1}(\mathbf{x,0})\leq
r\\}$, (Fig. 5 \- i), with $r=1$ ).
The metric $\mathbf{d_{2}}$ (Euclidean metric) in $\mathbb{R}^{n}$ is defined
as
$d_{2}(\mathbf{x},\mathbf{y})=\Big{(}\sum_{i=1}^{n}(x_{i}-y_{i})^{2}\Big{)}^{\frac{1}{2}}.$
Thus, in $\mathbb{R}^{2}$:
$d_{2}(\mathbf{x,y})=\big{(}(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\big{)}^{\frac{1}{2}}$
and the closed ball is respectively
$\widetilde{S}_{d_{2}}(\mathbf{0},r)=\\{\mathbf{x}\in\mathbb{R}^{2}:d_{2}(\mathbf{x,0})\leq
r\\},$ (Fig. 5 \- ii), with $r=1$ ).
* •
In every nonempty set $X$ the metric $\mathbf{\sigma_{0}}$ (discrete metric)
between points $x,y\in X$ is defined as
$\sigma_{0}(x,y)=\begin{cases}1,&\text{ for $x\neq y,$}\\\ 0,&\text{ for
$x=y$.}\\\ \end{cases}$ (1)
Thus, the closed ball is
$\widetilde{S}_{\sigma_{0}}(x_{0},r)=\begin{cases}\\{x_{0}\\},&0<r<1\\\
X,&r\geq 1\end{cases},$ ( Fig. 5 \- iii), with
$X=(\alpha,\beta)\subseteq\mathbb{R}$ ).
Figure 5: i)
$\widetilde{S}_{d_{1}}(\mathbf{0},1)=\\{\mathbf{x}\in\mathbb{R}^{2}:d_{1}(\mathbf{x},\mathbf{0})\leq
1\\}.$ ii) The closed ball
$\widetilde{S}_{d_{2}}(\mathbf{0},1)=\\{\mathbf{x}\in\mathbb{R}^{2}:d_{2}(\mathbf{x},\mathbf{0})\leq
1\\}$ of the Euclidean metric in $\mathbb{R}^{2}$ is the unit circle. iii)
($a$) For $x_{0}\in X=(a,b)\subseteq\mathbb{R}$ and $0<r<1,$
$\widetilde{S}_{\sigma_{0}}(x_{0},r)=\\{x_{0}\\}.$ iii) ($b$) For $x_{0}\in
X=(a,b)\subseteq\mathbb{R}$ and $r\geq 1$ the closed ball is the entire set
$X,$ i.e. $\widetilde{S}_{\sigma_{0}}(x_{0},r)=(a,b).$
### 3.1 Cartesian product space
If the set $X$ is arbitrary and not of a specific structure (e.g. vector
space), the discrete metric (1) seems to be the only available choice.
Given the metric spaces $(X_{i},\rho_{i}),\ i=1,2...,n$ we define the
$p$-metrics ($p\geq 1$)444For $0<p<1$ the triangular inequality does not hold,
hence we do not define a metric. in the Cartesian product $X=X_{1}\times
X_{2}\times...\times X_{n}$ for $\mathbf{x}=(x_{1},...,x_{n}),\
\mathbf{y}=(y_{1},...,y_{n})\in X$:
$d_{(\rho_{1},\rho_{2},...,\rho_{n};\
p)}\big{(}\mathbf{x},\mathbf{y}\big{)}=\begin{cases}\Big{(}\sum_{i=1}^{n}\big{(}\rho_{i}(x_{i},y_{i})\big{)}^{p}\Big{)}^{\frac{1}{p}},\
&\text{for $1\leq p<+\infty$}\\\ \max\\{\rho_{i}(x_{i},y_{i}),\
i=1,2,...,n\\},&\text{for $p=+\infty$}\\\ \end{cases}$ (2)
In case $X_{1}=X_{2}=...=X_{n}=Y$, i.e. $X=Y^{n}$ and
$\rho_{1}=\rho_{2}=...=\rho_{n}=\rho$, we denote $d_{(\rho;\ p)}(\cdot\ ,\
\cdot)$ instead of $d_{(\rho_{1},\rho_{2},...,\rho_{n};\ p)}(\cdot\ ,\
\cdot).$
Metrics in (2) are compatible with the ones that already exist in
$X_{1},X_{2},...,X_{n}$ according to the following sense. Let
$(y_{1},...,y_{n})\in X$ be an arbitrary fixed point. Coinciding $x_{i}\in
X_{i}$ with $(y_{1},...,x_{i},...,y_{n})\in X$ we have
$d_{(\rho_{1},\rho_{2},...,\rho_{n};\ p)}\big{(}(y_{1},...,x_{i},...,y_{n}),\
\big{(}y_{1},...,x^{\prime}_{i},...,y_{n})\big{)}=\rho_{i}(x_{i},x^{\prime}_{i}),\
\text{for}\ p\geq 1,$
where $i$ is the index corresponding to the metric space $(X_{i},\rho_{i}).$
### 3.2 $\mathbb{R}^{n}\ (n\geq 1)$ equipped with metrics
Due to the discrete nature of computers, our main interest is the set $X$ of
the metric space to be a vectored space or subspace. In most of the
applications the space that appears is $\mathbb{R}^{n}$ or subsets of this
space. The axiomatic foundation of the set $\mathbb{R}$ of real numbers, gives
us the latitude to define the metric $\sigma_{|\cdot|}(x,y)=|x-y|,\ x,\
y\in\mathbb{R},$ where $|\cdot|$ stands for the absolute value of a real
number. More generally we may take the metrics $\sigma_{s}(x,y)=|x-y|^{s},$
for $0<s\leq 1\ (\sigma_{1}=\sigma_{|\cdot|}).$ At this point it is important
to consider that
$\lim_{s\rightarrow 0^{+}}\sigma_{s}(x,y)=\sigma_{0}(x,y).$ (3)
In case of the set
$X\subseteq\mathbb{R}\times\mathbb{R}\times...\times\mathbb{R}=\mathbb{R}^{n}$,
emerge the $p$-metrics resulting from $(\mathbb{R},\sigma_{|\cdot|})$,
$(\mathbb{R},\sigma_{s})$ for $0<s<1$ and $(\mathbb{R},\sigma_{0})$ for points
$\mathbf{x}=(x_{1},...,x_{n}),\
\mathbf{y}=(y_{1},...,y_{n})\in\mathbb{R}^{n},$ respectively. Alternatively, a
combination of $\sigma$-metrics is also possible in order to measure in a
different way among subsets of $X,$ however the latter choice lacks in
practice.
Analytically we use the following metrics:
#### Usual metrics in $\mathbb{R}^{n}$:
* •
Let $\mathbb{R}$ equipped with the metric $\sigma_{|\cdot|}=|x-y|.$ It follows
that $\mathbb{R}^{n}$ is equipped with the metric $d_{(\sigma_{|\cdot|};\ p)}$
which according to (2) leads to:
$d_{(\sigma_{|\cdot|};\
p)}\big{(}\mathbf{x},\mathbf{y}\big{)}=\begin{cases}\Big{(}\sum_{i=1}^{n}|x_{i}-y_{i}|^{p}\Big{)}^{\frac{1}{p}},\
&\text{for $1\leq p<+\infty$}\\\ \max\\{|x_{i}-y_{i}|,\
i=1,...,n\\},&\text{for $p=+\infty$}\\\ \end{cases}$ (4)
Thus, for $p=2$ we have the Euclidean metric in $\mathbb{R}^{n}$ (Fig. 6).
Figure 6: $\widetilde{S}_{d_{(\sigma_{|\cdot|};\
p)}}(\mathbf{0},1)=\\{\mathbf{x}\in\mathbb{R}^{2}:\ d_{(\sigma_{|\cdot|};\
p)}(\mathbf{x},\mathbf{0})\leq 1\\}$ for different values of $p$. Notice that
while $p$ increases, the ball of our space tends to be
$\widetilde{S}_{d_{(\sigma_{|\cdot|};\ +\infty)}}(\mathbf{0},1),$ whereas $p$
decreases to $1,$ tends to be $\widetilde{S}_{d_{(\sigma_{|\cdot|};\
1)}}(\mathbf{0},1).$
* •
Let $\mathbb{R}$ equipped with the metric $\sigma_{s}=|x-y|^{s}$ for $0<s\leq
1$. It follows that $\mathbb{R}^{n}$ is equipped with the metric
$d_{(\sigma_{s};\ p)}$ that according to (2) leads to:
$d_{(\sigma_{s};\
p)}\big{(}\mathbf{x},\mathbf{y}\big{)}=\begin{cases}\Big{(}\sum_{i=1}^{n}\big{(}\sigma_{s}(x_{i},y_{i})\big{)}^{p}\Big{)}^{\frac{1}{p}},\
&\text{for $1\leq p<+\infty$}\\\ \max\\{\sigma_{s}(x_{i},y_{i}),\
i=1,...,n\\},&\text{for $p=+\infty$}\\\ \end{cases}$ (5)
Specifically, for $p=1$ we have:
$d_{(\sigma_{s};\ 1)}\big{(}\mathbf{x},\
\mathbf{y}\big{)}=\sum_{i=1}^{n}|x_{i}-y_{i}|^{s},\ 0<s\leq 1.$ (6)
#### Discrete metric in $\mathbb{R}^{n}$:
Let $\mathbb{R}$ equipped with the metric
$\sigma_{0}(x,y)=\begin{cases}1,&\text{ for $x\neq y$}\\\ 0,&\text{ for
$x=y$}\\\ \end{cases}$, thus $\mathbb{R}^{n}$ is equipped with the metric
$d_{(\sigma_{0};\ p)}$ that according to (2) results to:
$d_{(\sigma_{0};\
p)}\big{(}\mathbf{x},\mathbf{y}\big{)}=\begin{cases}\Big{(}\sum_{i=1}^{n}\big{(}\sigma_{0}(x_{i},y_{i})\big{)}^{p}\Big{)}^{\frac{1}{p}},\
&\text{for $1\leq p<+\infty$}\\\ \max\\{\sigma_{0}(x_{i},y_{i}),\
i=1,...,n\\},&\text{for $p=+\infty$}\\\ \end{cases}$ (7)
Hence considering the case $p=1$ we have
$d_{(\sigma_{0};\ 1)}(\mathbf{x},\ \mathbf{y})=\\#\\{i:x_{i}\neq y_{i}\\}.$
(8)
As $d_{(\sigma_{s};\ 1)}$ is of most importance in sparse theory, we denote it
as $d_{s}$, if not to be confused with any other metric and use the symbolism
$\widetilde{S}_{d_{s}}$ for the closed ball respectively. Finally, combining
equation (8), with both (3) and (6) we obtain:
$\lim_{s\rightarrow 0^{+}}d_{s}(\mathbf{x},\mathbf{y})=\lim_{s\rightarrow
0^{+}}\Big{(}\sum_{i=1}^{n}|x_{i}-y_{i}|^{s}\Big{)}=\\#\\{i:x_{i}\neq
y_{i}\\}=d_{0}(\mathbf{x},\mathbf{y}).$ (9)
The final equation indicates the behaviour of closed balls. In (Fig. 7) it can
be easily seen that in $\mathbb{R}^{2}$ and for $r\in(0,1)$ ($r=0.5$) the
balls $\widetilde{S}_{s}(\mathbf{0},r)$ decrease, i.e. for $0<s^{\prime}<s\leq
1$ we have
$\widetilde{S}_{s^{\prime}}(\mathbf{0},r)\subset\widetilde{S}_{s}(\mathbf{0},r)$
and finally tend to be the ball
$\widetilde{S}_{0}(\mathbf{0},r)=\bigcap\limits_{0<s\leq
1}\widetilde{S}_{s}(\mathbf{0},r).$ For $r\in[1,2)$ ($r=1.5$) a relation of
subset does not exist between the balls
$\widetilde{S}_{s^{\prime}}(\mathbf{0},r)$ and
$\widetilde{S}_{s}(\mathbf{0},r)$ for $s^{\prime}<s,$ however
$\widetilde{S}_{s}(\mathbf{0},r)$ decrease and tends to coincide with the axis
while $s\rightarrow 0^{+},$ i.e. the ball $\widetilde{S}_{0}(\mathbf{0},r).$
For $r\in[2,+\infty)$ ($r=3$) the balls increase and for $0<s^{\prime}<s\leq
1$ we have
$\widetilde{S}_{s^{\prime}}(\mathbf{0},r)\supset\widetilde{S}_{s}(\mathbf{0},r)$
until they finally fill the whole space, while
$\widetilde{S}_{0}(\mathbf{0},r)=\bigcup\limits_{0<s\leq
1}\widetilde{S}_{s}(\mathbf{0},r).$
Equation (9) indicates a desirable measure of sparsity, defined as
$d_{0}(\mathbf{x},\mathbf{0})=\\#\\{i:x_{i}\neq 0\\},$ (10)
measures the number of nonzero coordinates555Also called support of a vector
and denoted as $supp\\{\mathbf{x}\\}.$ of a vector and belongs to the family
of metrics.
Figure 7:
$\widetilde{S}_{s}(\mathbf{0},r)=\\{\mathbf{x}=(x,y)\in\mathbb{R}^{2}:\sigma_{s}(\mathbf{x},\mathbf{0})\leq
r\\},\ 0\leq s\leq 1:$ i) For $0<r<1,\
\widetilde{S}_{0}(\mathbf{0},r)=\\{(0,0)\\}.$ ii) For $1\leq r<2,\
\widetilde{S}_{0}(\mathbf{0},r)=\\{\mathbf{x}=(x,y)\in\mathbb{R}^{2}:x=0$ or
$y=0\\}.$ iii) For $r\geq 2,\ \widetilde{S}_{0}(\mathbf{0},r)=\mathbb{R}^{2}.$
#### Alternative metrics in $\mathbb{R}^{n}$:
Another measure constructed by a combination of different metrics (2) enables
us to measure each subset differently. At its simplest form we state an
example in $\mathbb{R}^{2}.$ Let the metric space
$(\mathbb{R},\sigma_{0})\times(\mathbb{R},\sigma_{|\cdot|})$ and set
$\mathbf{x}=(x_{1},x_{2}),\ \mathbf{y}=(y_{1},y_{2})\in\mathbb{R}^{2}.$
$d_{(\sigma_{0},\sigma_{|\cdot|};\
p)}(\mathbf{x},\mathbf{y})=\Big{(}(\sigma_{0}(x_{1},y_{1}))^{p}+|x_{2}-y_{2}|^{p}\Big{)}^{1/p},\
1\leq p<+\infty$
Thus, for $p=1$:
$d_{(\sigma_{0},\sigma_{|\cdot|};\
1)}(\mathbf{x},\mathbf{y})=\sigma_{0}(x_{1},y_{1})+|x_{2}-y_{2}|$ (11)
Consequently, the closed ball of center $\mathbf{0}$ and radius $r$ are
(Fig.8):
$\widetilde{S}(\mathbf{0},r)=\\{\mathbf{x}\in\mathbb{R}^{2}:\sigma_{0}(x_{1},0)+|x_{2}|\leq
r\\}=\begin{cases}-r\leq x_{2}\leq r,\ &x_{1}=0,\\\ 1-r\leq x_{2}\leq
r-1,&x_{1}\neq 0.\end{cases}$ (12)
Figure 8:
$\widetilde{S}(\mathbf{0},r)=\\{\mathbf{x}=(x,y)\in\mathbb{R}^{2}:\sigma_{0}(x,0)+|y|\leq
r\\}$ i) For $0<r=0.5<1,\ \widetilde{S}(\mathbf{0},0.5)=\\{-1/2\leq y\leq
1/2,\ x=0\\}.$ ii) For $r=1,\ \widetilde{S}(\mathbf{0},1)=\\{-1\leq y\leq 1,\
x=0\ or\ y=0,\ x\neq 0\\}.$ iii) For $r=1.5>1,\
\widetilde{S}(\mathbf{0},1.5)=\\{-3/2\leq y\leq 3/2,\ x=0\ or\ -1/2\leq y\leq
1/2,\ x\neq 0\\}.$
## 4 Normed spaces
###### Definition 7
Vector (linear) space is called the trio ($X$, +, $\cdot$), where $X$ is a
nonempty set, $+\ :\ X\times X\longrightarrow X$ an inner operation (addition)
and $\cdot\ :\ F\times X\longrightarrow X$666The field $F=\mathbb{C}$ or
$\mathbb{R}.$ an outer operation (scalar product) that obey the following
properties:
1. 1.
$x+y=y+x,\ \forall x,y\in X$,
2. 2.
$(x+y)+z=x+(y+z),\ \forall x,y,z\in X$,
3. 3.
There exists $0\in X$ such that $x+0=0+x=x,\ \forall x\in X,$
4. 4.
For all $x\in X$ there exists $-x\in X$ such that $x+(-x)=(-x)+x=0,$
5. 5.
$\lambda(x+y)=\lambda x+\lambda y,\ \forall x,y\in X$ and $\lambda\in F,$
6. 6.
$(\lambda+\mu)x=\lambda x+\mu x,\ \forall x\in X$ and $\lambda,\ \mu\in F,$
7. 7.
$\lambda(\mu x)=(\lambda\mu)x,\ \forall x\in X$ and $\lambda,\ \mu\in F,$
8. 8.
$1x=x,\ \forall x\in X.$
The elements of a vectored space are called vectors.
###### Definition 8
Let $X$ be a vector space over a field of numbers $F$. The set $A\subseteq X$
is called convex, if for every pair $x,y\in X$ and every $t\in[0,1]$, the
element $tx+(1-t)y$ belongs to the set $A$ as well.
###### Definition 9
A real function $f:A\rightarrow\mathbb{R}$ defined over a convex subset of a
linear space $X$ is called convex, if for every $x,y\in A$ and $t\in[0,1]$,
$f\big{(}tx+(1-t)y\big{)}\leq tf(x)+(1-t)f(y).$
###### Definition 10
A real function $f:A\rightarrow\mathbb{R}$ defined over a convex subset of a
linear space $X$ is called concave, if for every $x,y\in A$ and $t\in[0,1]$,
$f\big{(}tx+(1-t)y\big{)}\geq tf(x)+(1-t)f(y).$
Let $\mathbb{R}$ be the vector space. Thus, the absolute value obeys the
following properties:
1. 1.
$|x|\geq 0,$ $x\in\mathbb{R}$ and $|x|=0\Leftrightarrow x=0$
2. 2.
$|\lambda x|=|\lambda||x|,\ x\in\mathbb{R},\ \lambda\in\mathbb{R},$ (positive
homogeneous)
3. 3.
$|x+y|\leq|x|+|y|,\ x,\ y\in\mathbb{R},$ (triangular inequality)
Therefore the function $f(x)=|x|$ is positive homogeneous, convex and $f(x)>0$
for $x\neq 0.$ A norm is the generalization of the absolute value in higher-
dimensional vector spaces.
###### Definition 11
Let $(X,+,\cdot)$ be a real vector space. The map
$\|\cdot\|:X\longrightarrow\mathbb{R}$ is called norm if it obeys the
following properties:
1. 1.
$\|x\|\geq 0,\ \ \forall x\in X$ and $\|x\|=0\Leftrightarrow\ x=0,$
2. 2.
$\|\lambda x\|=|\lambda|\|x\|,\ \ \forall x\in X$ and $\lambda\in\mathbb{R},$
(positive homogeneous)
3. 3.
$\|x+y\|\leq\|x\|+\|y\|,\ \ \forall x,y\in X,$ (triangular inequality)
The pair $(X,\|\cdot\|$) is called a normed space.
It follows that $f(x)=\|x\|$ is also a positive homogeneous, convex function
with $f(x)>0$ for $x\neq 0.$ It is not difficult to see that if
$\|x-y\|=f(x-y)=d(x,y)$ for $x,y\in X,$ then $d$ is a metric in $X$ with
$d(x,0)=\|x\|.$ Moreover, if $\rho$ is a metric in a vector space $X$
satisfying the additional properties $\rho(x+z,y+z)=\rho(x,y),\ x,y\in X$ and
$\rho(\lambda x,0)=|\lambda|\rho(x,0)$ with $x\in X,\ \lambda\in\mathbb{R}$
(positive homogeneous), then $\rho(x,0)=\|x\|$ is a norm. However, we will see
that some of the metrics defined do not derive from norms.
Likewise metric spaces $X=X_{1}\times X_{2}\times...\times X_{n},$ the
$p-$norms in $(X_{i},\|\cdot\|_{i})$ are defined.
#### Examples of normed spaces for $X=\mathbb{R}^{n}$:
* •
The $\mathbf{p-}$norms for $1<p<+\infty$:
$\|\mathbf{x}\|_{p}=\Big{(}\sum_{i=1}^{n}|x_{i}|^{p}\Big{)}^{\frac{1}{p}}.$
* •
The Euclidean norm ($p=2$):
$\|\mathbf{x}\|_{2}=\Big{(}\sum_{i=1}^{n}x_{i}^{2}\Big{)}^{\frac{1}{2}}.$
* •
The addition norm ($\mathbf{1-}$norm) $\|\ \|_{\mathbf{1}}$ and the norm $\|\
\|_{\mathbf{\infty}}$ respectively:
$\|\mathbf{x}\|_{1}=\sum_{i=1}^{n}|x_{i}|,$
$\|\mathbf{x}\|_{\infty}=max\\{|x_{i}|:i=1,2,...,n\\}.$
At this point we should emphasise that $1\leq p\leq+\infty,$ so that
$\|\cdot\|_{p}$ is a norm in $\mathbb{R}^{n}$, which could be easily proved
using the Minkowski inequality.
Because of the demand for an optimization function, we set
$f_{p}(\mathbf{x})=\Big{(}\sum_{i=1}^{n}|x_{i}|^{p}\Big{)}^{\frac{1}{p}},\
p>0$ (13)
which is positive homogeneous for every $p>0,$ whilst convex only for $p\geq
1.$ For $0<p<1$ the function is partially concave, hence the triangular
inequality is not satisfied (Fig. 9).
Metric $\sigma_{s}(x,y)=|x-y|^{s},\ x,\ y\in\mathbb{R}$ is transposition
invariant, though not positive homogeneous for $s\neq 1.$ Hence,
$d_{(\sigma_{s};\ p)}(\mathbf{x},\mathbf{0})$ for $0<s<1$ in $\mathbb{R}^{n}$
are not norms (Fig. 10 for $p=1$).
From another point of view, the geometric interpretation gives us a clear
image of all above. For $p\geq 1$ the closed balls are convex sets, unlike for
$0<p<1$ . Suppose $(X,\|\cdot\|)$ is a normed vector space, the balls
$\widetilde{S}_{\|\cdot\|}(x,r)$ are always convex sets. Indeed, if
$y,z\in\widetilde{S}_{\|\cdot\|}(x,r)$ then $\|y-x\|,\ \|z-x\|\leq r,$ thus
for $\lambda\in(0,1),\ \|\lambda y+(1-\lambda
z)-x\|=\|\lambda(y-x)+(1-\lambda)(z-x)\|\leq\lambda r+(1-\lambda)r\leq r,$
i.e. $\lambda y+(1-\lambda)z\in\widetilde{S}_{\|\cdot\|}(x,r),$ hence the set
$\widetilde{S}_{\|\cdot\|}(x,r)$ is convex.
Remark: The property of convexity is of great importance. Suppose that
$(X,\|\cdot\|)$ is a normed vector space and let $K\subseteq X$ be a convex
and symmetric ($K=-K$) open set, such that $R,r>0$ exist and
$S_{\|\cdot\|}(0,r)\subseteq K\subseteq S_{\|\cdot\|}(0,R).$ Thus $K$
constitutes the unit ball of another norm, i.e.
$\|x\|_{K}=inf\\{\lambda>0:x\in\lambda K\\}$ (Minkowski functional) which is
topological equivalent to the initial.
Figure 9: Function $f(x,y)=\big{(}|x|^{p}+|y|^{p}\big{)}^{\frac{1}{p}}$ for
$p>0:$ i) - ii) For $p\geq 1$ those functions are convex. iii) - iv) For
$0<p<1$ functions are not convex. i) - iv) All functions are positive
homogeneous for all $p>0$.
Figure 10: Function $g(x,y)=|x|^{s}+|y|^{s}$ for $s>0:$ i) - ii) For $s\geq 1$
the functions are convex. iv) - v) For $0<s<1$ functions are not convex. ii) -
v) For $s\neq 1$ the functions are not positive homogeneous.
## References
* [1] Herbert Amann, Joachim Escher, “Analysis I”, Birkhäuser Verlag, 1998.
* [2] Alfred M. Bruckstein, David L. Donoho, Micheal Elad, “From Sparse Solutions of Equations to Sparse Modeling of signals and Images”, SIAM Review Vol.51, No. 1, 2009.
* [3] David L. Donoho, Michael Elad, “Optimally Sparse Representation in General (non-Orthogonal) Dictionaries via $l^{1}$ minimization”, Proceedings, National Academy of Sciences, Vol. 100, pp. 2197-2202, 2003.
* [4] Stephen Boyd, Lieven Vandenberghe, “Convex Optimization”, Cambridge University Press, 2004.
The University of Athens
Department of Mathematics
Panepistemiopolis 15784
Athens
Greece
Email:
ldalla@math.uoa.gr
ge99210@hotmail.com
|
arxiv-papers
| 2012-03-20T20:23:44 |
2024-09-04T02:49:28.849679
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leoni Dalla, George K. Papageorgiou",
"submitter": "Leoni Dalla",
"url": "https://arxiv.org/abs/1203.4579"
}
|
1203.4802
|
A Reference-Free Algorithm for Computational Normalization of Shotgun
Sequencing Data
C. Titus Brown1,2,∗, Adina Howe2, Qingpeng Zhang1, Alexis B. Pyrkosz3, Timothy
H. Brom1
1 Computer Science and Engineering, Michigan State University, East Lansing,
MI, USA
2 Microbiology and Molecular Genetics, Michigan State University, East
Lansing, MI, USA
3 USDA Avian Disease and Oncology Laboratory, East Lansing, MI, USA
$\ast$ E-mail: ctb@msu.edu
## Abstract
Deep shotgun sequencing and analysis of genomes, transcriptomes, amplified
single-cell genomes, and metagenomes has enabled investigation of a wide range
of organisms and ecosystems. However, sampling variation in short-read data
sets and high sequencing error rates of modern sequencers present many new
computational challenges in data interpretation. These challenges have led to
the development of new classes of mapping tools and de novo assemblers. These
algorithms are challenged by the continued improvement in sequencing
throughput. We here describe digital normalization, a single-pass
computational algorithm that systematizes coverage in shotgun sequencing data
sets, thereby decreasing sampling variation, discarding redundant data, and
removing the majority of errors. Digital normalization substantially reduces
the size of shotgun data sets and decreases the memory and time requirements
for de novo sequence assembly, all without significantly impacting content of
the generated contigs. We apply digital normalization to the assembly of
microbial genomic data, amplified single-cell genomic data, and transcriptomic
data. Our implementation is freely available for use and modification.
## Author Summary
## Introduction
The ongoing improvements in DNA sequencing technologies have led to a new
problem: how do we analyze the resulting large sequence data sets quickly and
efficiently? These data sets contain millions to billions of short reads with
high error rates and substantial sampling biases [1]. The vast quantities of
deep sequencing data produced by these new sequencing technologies are driving
computational biology to extend and adapt previous approaches to sequence
analysis. In particular, the widespread use of deep shotgun sequencing on
previously unsequenced genomes, transcriptomes, and metagenomes, has resulted
in the development of several new approaches to de novo sequence assembly [2].
There are two basic challenges in analyzing short-read sequences from shotgun
sequencing. First, deep sequencing is needed for complete sampling. This is
because shotgun sequencing samples randomly from a population of molecules;
this sampling is biased by sample content and sample preparation, requiring
even deeper sequencing. A human genome may require 100x coverage or more for
near-complete sampling, leading to shotgun data sets 300 GB or larger in
size[3]. Since the lowest abundance molecule determines the depth of coverage
required for complete sampling, transcriptomes and metagenomes containing rare
population elements can also require similarly deep sequencing.
The second challenge to analyzing short-read shotgun sequencing is the high
error rate. For example, the Illumina GAII sequencer has a 1-2% error rate,
yielding an average of one base error in every 100 bp of data [1]. The total
number of errors grows linearly with the amount of data generated, so these
errors usually dominate novelty in large data sets [4]. Tracking this novelty
and resolving errors is computationally expensive.
These large data sets and high error rates combine to provide a third
challenge: it is now straightforward to generate data sets that cannot easily
be analyzed [5]. While hardware approaches to scaling existing algorithms are
emerging, sequencing capacity continues to grow faster than computational
capacity [6]. Therefore, new algorithmic approaches to analysis are needed.
Many new algorithms and tools have been developed to tackle large and error-
prone short-read shotgun data sets. A new class of alignment tools, most
relying on the Burrows-Wheeler transform, has been created specifically to do
ultra-fast short-read alignment to reference sequence [7]. In cases where a
reference sequence does not exist and must be assembled de novo from the
sequence data, a number of new assemblers have been written, including ABySS,
Velvet, SOAPdenovo, ALLPATHS, SGA, and Cortex [8, 9, 10, 3, 11, 12]. These
assemblers rely on theoretical advances to store and assemble large amounts of
data [13, 14]. As short-read sequencing has been applied to single cell
genomes, transcriptomes, and metagenomes, yet another generation of assemblers
has emerged to handle reads from abundance-skewed populations of molecules;
these tools, including Trinity, Oases, MetaVelvet, Meta-IDBA, and Velvet-SC,
adopt local models of sequence coverage to help build assemblies [15, 16, 17,
18, 19]. In addition, several ad hoc strategies have also been applied to
reduce variation in sequence content from whole-genome amplification [20, 21].
Because these tools all rely on k-mer approaches and require exact matches to
construct overlaps between sequences, their performance is very sensitive to
the number of errors present in the underlying data. This sensitivity to
errors has led to the development of a number of error removal and correction
approaches that preprocess data prior to assembly or mapping [22, 23, 24].
Below, we introduce “digital normalization”, a single-pass algorithm for
elimination of redundant reads in data sets. Critically, no reference sequence
is needed to apply digital normalization. Digital normalization is inspired by
experimental normalization techniques developed for cDNA library preparation,
in which hybridization kinetics are exploited to reduce the copy number of
abundant transcripts prior to sequencing[25, 26]. Digital normalization works
after sequencing data has been generated, progressively removing high-coverage
reads from shotgun data sets. This normalizes average coverage to a specified
value, reducing sampling variation while removing reads, and also removing the
many errors contained within those reads. This data and error reduction
results in dramatically decreased computational requirements for de novo
assembly. Moreover, unlike experimental normalization where abundance
information is removed prior to sequencing, in digital normalization this
information can be recovered from the unnormalized reads.
We present here a fixed-memory implementation of digital normalization that
operates in time linear with the size of the input data. We then demonstrate
its effectiveness for reducing compute requirements for de novo assembly on
several real data sets. These data sets include E. coli genomic data, data
from two single-cell MD-amplified microbial genomes, and yeast and mouse
mRNAseq.
## Results
### Estimating sequencing depth without a reference assembly
Short-read assembly requires deep sequencing to systematically sample the
source genome, because shotgun sequencing is subject to both random sampling
variation and systematic sequencing biases. For example, 100x sampling of a
human genome is required for recovery of 90% or more of the genome in contigs
$>$ 1kb [3]. In principle much of this high-coverage data is redundant and
could be eliminated without consequence to the final assembly, but determining
which reads to eliminate requires a per-read estimate of coverage. Traditional
approaches estimate coverage by mapping reads to an assembly. This presents a
chicken-and-egg problem: to determine which regions are oversampled, we must
already have an assembly!
We may calculate a reference-free estimate of genome coverage by looking at
the k-mer abundance distribution within individual reads. First, observe that
k-mers, DNA words of a fixed length $k$, tend to have similar abundances
within a read: this is a well-known property of k-mers that stems from each
read originating from a single source molecule of DNA. The more times a region
is sequenced, the higher the abundance of k-mers from that region would be. In
the absence of errors, average k-mer abundance could be used as an estimate of
the depth of coverage for a particular read (Figure 1, “no errors” line).
However, when reads contain random substitution or indel errors from
sequencing, the k-mers overlapping these errors will be of lower abundance;
this feature is often used in k-mer based error correction approaches [24].
For example, a single substitution will introduce $k$ low-abundance k-mers
within a read. (Figure 1, “single substitution error” line). However, for
small $k$ and reads of length $L$ where $L>3k-1$, a single substitution error
will not skew the median k-mer abundance. Only when multiple substitution
errors are found in a single read will the median k-mer abundance be affected
(Figure 1, “multiple substitution errors”).
Using a fixed-memory CountMin Sketch data structure to count k-mers (see
Methods and [27]), we find that median k-mer abundance correlates well with
mapping-based coverage for artificial and real genomic data sets. There is a
strong correlation between median k-mer abundance and mapping-based coverage
both for simulated 100-base reads generated with 1% error from a 400kb
artificial genome sequence ($r^{2}=0.79$; also see Figure 2a), as well as for
real short-read data from E. coli ($r^{2}=0.80$, also see Figure 2b). This
correlation also holds for simulated and real mRNAseq data: for simulated
transcriptome data, $r^{2}=0.93$ (Figure 3a), while for real mouse
transcriptome data, $r^{2}=0.90$ (Figure 3b). Thus the median k-mer abundance
of a read correlates well with mapping-based estimates of read coverage.
### Eliminating redundant reads reduces variation in sequencing depth
Deeply sequenced genomes contain many highly covered loci. For example, in a
human genome sequenced to 100x average coverage, we would expect 50% or more
of the reads to have a coverage greater than 100. In practice, we need many
fewer of these reads to assemble the source locus.
Using the median k-mer abundance estimator discussed above, we can examine
each read in the data set progressively to determine if it is high coverage.
At the beginning of a shotgun data set, we would expect many reads to be
entirely novel and have a low estimated coverage. As we proceed through the
data set, however, average coverage will increase and many reads will be from
loci that we have already sampled sufficiently.
Suppose we choose a coverage threshold $C$ past which we no longer wish to
collect reads. If we only keep reads whose estimated coverage is less than
$C$, and discard the rest, we will reduce the average coverage of the data set
to $C$. This procedure is algorithmically straightforward to execute: we
examine each read’s estimated coverage, and retain only those whose coverage
is less than $C$. The following pseudocode provides one approach:
for read in dataset:
if estimated_coverage(read) < C:
accept(read)
else:
discard(read)
where accepted reads contribute to the $\tt estimated\\_coverage$ function.
Note that for any data set with an average coverage $>2C$, this has the effect
of discarding the majority of reads. Critically, low-coverage reads,
especially reads from undersampled regions, will always be retained.
The net effect of this procedure, which we call digital normalization, is to
normalize the coverage distribution of data sets. In Figure 4a, we display the
estimated coverage of an E. coli genomic data set, a S. aureus single-cell MD-
amplified data set, and an MD-amplified data set from an uncultured
Deltaproteobacteria, calculated by mapping reads to the known or assembled
reference genomes (see [19] for the data source). The wide variation in
coverage for the two MDA data sets is due to the amplification procedure [28].
After normalizing to a k-mer coverage of 20, the high coverage loci are
systematically shifted to an average mapping coverage of 26, while lower-
coverage loci remain at their previous coverage. This smooths out coverage of
the overall data set.
At what rate are sequences retained? For the E. coli data set, Figure 5 shows
the fraction of sequences retained by digital normalization as a function of
the total number of reads examined when normalizing to C=20 at k=20. There is
a clear saturation effect showing that as more reads are examined, a smaller
fraction of reads is retained; by 5m reads, approximately 50-100x coverage of
E. coli, under 30% of new reads are kept. This demonstrates that as expected,
only a small amount of novelty (in the form of either new information, or the
systematic accumulation of errors) is being observed with increasing
sequencing depth.
### Digital normalization retains information while discarding both data and
errors
The 1-2% per-base error rate of next-generation sequencers dramatically affect
the total number of k-mers. For example, in the simulated genomic data of
200x, a 1% error rate leads to approximately 20 new k-mers for each error,
yielding 20-fold more k-mers in the reads than are truly present in the genome
(Table 1, row 1). This in turn dramatically increases the memory requirements
for tracking and correcting k-mers [4]. This is a well-known problem with de
Bruijn graph approaches, in which erroneous nodes or edges quickly come to
dominate deep sequencing data sets.
When we perform digital normalization on such a data set, we eliminate the
vast majority of these k-mers (Table 1, row 1). This is because we are
accepting or rejecting entire reads; in going from 200x random coverage to 20x
systematic coverage, we discard 80% of the reads containing 62% of the errors
(Table 1, row 1). For reads taken from a skewed abundance distribution, such
as with MDA or mRNAseq, we similarly discard many reads, and hence many errors
(Table 1, row 2). In fact, in most cases the process of sequencing fails to
recover more true k-mers (Table 1, middle column, parentheses) than digital
normalization discards (Table 1, fourth column, parentheses).
The net effect of digital normalization is to retain nearly all real k-mers,
while discarding the majority of erroneous k-mers – in other words, digital
normalization is discarding data but not information. This rather dramatic
elimination of erroneous k-mers is a consequence of the high error rate
present in reads: with a 1% per-base substitution error rate, each 100-bp read
will have an average of one substitution error. Each of these substitution
errors will introduce up to $k$ erroneous k-mers. Thus, for each read we
discard as redundant, we also eliminate an average of $k$ erroneous k-mers.
We may further eliminate erroneous k-mers by removing k-mers that are rare
across the data set; these rare k-mers tend to result from substitution or
indel errors [24]. We do this by first counting all the k-mers in the accepted
reads during digital normalization. We then execute a second pass across the
accepted reads in which we eliminate the 3’ ends of reads at low-abundance
k-mers. Following this error reduction pass, we execute a second round of
digital normalization (a third pass across the data set) that further
eliminates redundant data. This three-pass protocol eliminates additional
errors and results in a further decrease in data set size, at the cost of very
few real k-mers in genomic data sets (Table 2).
Why use this three-pass protocol rather than simply normalizing to the lowest
desired coverage in the first pass? We find that removing low-abundance k-mers
after a single normalization pass to $C\approx 5$ removes many more real
k-mers, because there will be many regions in the genome that by chance have
yielded 5 reads with errors in them. If these erroneous k-mers are removed in
the abundance-trimming step, coverage of the corresponding regions is
eliminated. By normalizing to a higher coverage of 20, removing errors, and
only then reducing coverage to 5, digital normalization can retain accurate
reads for most regions. Note that this three-pass protocol is not considerably
more computationally expensive than the single-pass protocol: the first pass
discards the majority of data and errors, so later passes are less time and
memory intensive than the first pass.
Interestingly, this three-pass protocol removed many more real k-mers from the
simulated mRNAseq data than from the simulated genome – 351 of 48,100 (0.7%)
real k-mers are lost from the mRNAseq, vs 4 of 399,981 lost (.000001%) from
the genome (Table 2). While still only a tiny fraction of the total number of
real k-mers, the difference is striking – the simulated mRNAseq sample loses
k-mers at almost 1000-fold the rate of the simulated genomic sample. Upon
further investigation, all but one of the lost k-mers were located within 20
bases of the ends of the source sequences; see Figure 6. This is because
digital normalization cannot distinguish between erroneous k-mers and k-mers
that are undersampled due to edge effects. In the case of the simulated
genome, which was generated as one large chromosome, the effect is negligible,
but the simulated transcriptome was generated as 100 transcripts of length
500\. This added 99 end sequences over the genomic simulation, which in turn
led to many more lost k-mers.
While the three-pass protocol is effective at removing erroneous k-mers, for
some samples it may be too stringent. For example, the mouse mRNAseq data set
contains only 100m reads, which may not be enough to thoroughly sample the
rarest molecules; in this case the abundance trimming would remove real k-mers
as well as erroneous k-mers. Therefore we used the single-pass digital
normalization for the yeast and mouse transcriptomes. For these two samples we
can also see that the first-pass digital normalization is extremely effective,
eliminating essentially all of the erroneous k-mers (Table 1, rows 4 and 5.)
### Digital normalization scales assembly of microbial genomes
We applied the three-pass digital normalization and error trimming protocol to
three real data sets from Chitsaz et al (2011) [19]. The first pass of digital
normalization was performed in 1gb of memory and took about 1 min per million
reads. For all three samples, the number of reads remaining after digital
normalization was reduced by at least 30-fold, while the memory and time
requirements were reduced 10-100x.
Despite this dramatic reduction in data set size and computational
requirements for assembly, both the E. coli and S. aureus assemblies
overlapped with the known reference sequence by more than 98%. This confirms
that little or no information was lost during the process of digital
normalization; moreover, it appears that digital normalization does not
significantly affect the assembly results. (Note that we did not perform
scaffolding, since the digital normalization algorithm does not take into
account paired-end sequences, and could mislead scaffolding approaches.
Therefore, these results cannot directly be compared to those in Chitsaz et
al. (2011) [19].)
The Deltaproteobacteria sequence also assembled well, with 98.8% sequence
overlap with the results from Chitsaz et al. Interestingly, only 30kb of the
sequence assembled with Velvet-SC in Chitsaz et al. (2011) was missing, while
an additional 360kb of sequence was assembled only in the normalized samples.
Of the 30kb of missing sequence, only 10% matched via TBLASTX to a nearby
Deltaproteobacteria assembly, while more than 40% of the additional 360kb
matched to the same Deltaproteobacteria sample. Therefore these additional
contigs likely represents real sequence, suggesting that digital normalization
is competitive with Velvet-SC in terms of sensitivity.
### Digital normalization scales assembly of transcriptomes
We next applied single-pass digital normalization to published yeast and mouse
mRNAseq data sets, reducing them to 20x coverage at k=20 [15]. Digital
normalization on these samples used 8gb of memory and took about 1 min per
million reads. We then assembled both the original and normalized sequence
reads with Oases and Trinity, two de novo transcriptome assemblers (Table 4)
[16, 15].
For both assemblers the computational resources necessary to complete an
assembly were reduced (Table 4), but normalization had different effects on
performance for the different samples. On the yeast data set, time and memory
requirements were reduced significantly, as for Oases running on mouse.
However, while Trinity’s runtime decreased by a factor of three on the
normalized mouse data set, the memory requirements did not decrease
significantly. This may be because the mouse transcriptome is 5-6 times larger
than the yeast transcriptome, and so the mouse mRNAseq is lower coverage
overall; in this case we would expect fewer errors to be removed by digital
normalization.
The resulting assemblies differed in summary statistics (Table 5). For both
yeast and mouse, Oases lost 5-10% of total transcripts and total bases when
assembling the normalized data. However, Trinity gained transcripts when
assembling the normalized yeast and mouse data, gaining about 1% of total
bases on yeast and losing about 1% of total bases in mouse. Using a local-
alignment-based overlap analysis (see Methods) we found little difference in
sequence content between the pre- and post- normalization assemblies: for
example, the normalized Oases assembly had a 98.5% overlap with the
unnormalized Oases assembly, while the normalized Trinity assembly had a 97%
overlap with the unnormalized Trinity assembly.
To further investigate the differences between transcriptome assemblies caused
by digital normalization, we looked at the sensitivity with which long
transcripts were recovered post-normalization. When comparing the normalized
assembly to the unnormalized assembly in yeast, Trinity lost only 3% of the
sequence content in transcripts greater than 300 bases, but 10% of the
sequence content in transcripts greater than 1000 bases. However, Oases lost
less than 0.7% of sequence content at 300 and 1000 bases. In mouse, we see the
same pattern. This suggests that the change in summary statistics for Trinity
is caused by fragmentation of long transcripts into shorter transcripts, while
the difference for Oases is caused by loss of splice variants. Indeed, this
loss of splice variants should be expected, as there are many low-prevalence
splice variants present in deep sequencing data [29]. Interestingly, in yeast
we recover more transcripts after digital normalization; these transcripts
appear to be additional splice variants.
The difference between Oases and Trinity results show that Trinity is more
sensitive to digital normalization than Oases: digital normalization seems to
cause Trinity to fragment long transcripts. Why? One potential issue is that
Trinity only permits k=26 for assembly, while normalization was performed at
k=20; digital normalization may be removing 26-mers that are important for
Trinity’s path finding algorithm. Alternatively, Trinity may be more sensitive
than Oases to the change in coverage caused by digital normalization.
Regardless, the strong performance of Oases on digitally normalized samples,
as well as the high retention of k-mers (Table 1) suggests that the primary
sequence content for the transcriptome remains present in the normalized
reads, although it is recovered with different effectiveness by the two
assemblers.
## Discussion
### Digital normalization dramatically scales de novo assembly
The results from applying digital normalization to read data sets prior to de
novo assembly are extremely good: digital normalization reduces the
computational requirements (time and memory) for assembly considerably,
without substantially affecting the assembly results. It does this in two
ways: first, by removing the majority of reads without significantly affecting
the true k-mer content of the data set. Second, by eliminating these reads,
digital normalization also eliminates sequencing errors contained within those
reads, which otherwise would add significantly to memory usage in assembly
[4].
Digital normalization also lowers computational requirements by eliminating
most repetitive sequence in the data set. Compression-based approaches to
graph storage have demonstrated that compressing repetitive sequence also
effectively reduces memory and compute requirements [30, 11]. Note however
that eliminating many repeats may also have its negatives (discussed below).
Digital normalization should be an effective preprocessing approach for most
assemblers. In particular, the de Bruijn graph approach used in many modern
assemblers relies on k-mer content, which is almost entirely preserved by
digital normalization (see Tables 1 and 2) [2].
### A general strategy for normalizing coverage
Digital normalization is a general strategy for systematizing coverage in
shotgun sequencing data sets by using per-locus downsampling, albeit without
any prior knowledge of reference loci. This yields considerable theoretical
and practical benefits in the area of de novo sequencing and assembly.
In theoretical terms, digital normalization offers a general strategy for
changing the scaling behavior of sequence assembly. Assemblers tend to scale
poorly with the number of reads: in particular, de Bruijn graph memory
requirements scale linearly with the size of the data set due to the
accumulation of errors, although others have similarly poor scaling behavior
(e.g. quadratic time in the number of reads) [2]. By calculating per-locus
coverage in a way that is insensitive to errors, digital normalization
converts genome assembly into a problem that scales with the complexity of the
underlying sample - i.e. the size of the genome, transcriptome, or metagenome.
Digital normalization also provides a general strategy for applying online or
streaming approaches to analysis of de novo sequencing data. The basic
algorithm presented here is explicitly a single-pass or streaming algorithm,
in which the entire data set is never considered as a whole; rather, a partial
“sketch” of the data set is retained and used for progressive filtering.
Online algorithms and sketch data structures offer significant opportunities
in situations where data sets are too large to be conveniently stored,
transmitted, or analyzed [31]. This can enable increasingly efficient
downstream analyses. Digital normalization can be applied in any situation
where the abundance of particular sequence elements is either unimportant or
can be recovered more efficiently after other processing, as in assembly.
The construction of a simple, reference-free measure of coverage on a per-read
basis offers opportunities to analyze coverage and diversity with an assembly-
free approach. Genome and transcriptome sequencing is increasingly being
applied to non-model organisms and ecological communities for which there are
no reference sequences, and hence no good way to estimate underlying sequence
complexity. The reference-free counting technique presented here provides a
method for determining community and transcriptome complexity; it can also be
used to progressively estimate sequencing depth.
More pragmatically, digital normalization also scales existing assembly
techniques dramatically. The reduction in data set size afforded by digital
normalization may also enable the application of more computationally
expensive algorithms such as overlap-layout-consensus assembly approaches to
short-read data. Overall, the reduction in data set size, memory requirements,
and time complexity for contig assembly afforded by digital normalization
could lead to the application of more complex heuristics to the assembly
problem.
### Digital normalization drops terminal k-mers and removes isoforms
Our implementation of digital normalization does discard some real
information, including terminal k-mers and low-abundance isoforms. Moreover,
we predict a number of other failure modes: for example, because k-mer
approaches demand strict sequence identity, data sets from highly polymorphic
organisms or populations will perform more poorly than data sets from low-
variability samples. Digital normalization also discriminates against highly
repetitive sequences. We note that these problems traditionally have been
challenges for assembly strategies: recovering low-abundance isoforms from
mRNAseq, assembling genomes from highly polymorphic organisms, and assembling
across repeats are all difficult tasks, and improvements in these areas
continue to be active areas of research [32, 33, 34]. Using an alignment-based
approach to estimating coverage, rather than a k-mer based approach, could
provide an alternative implementation that would improve performance on
errors, splice variants, and terminal k-mers. Our current approach also
ignores quality scores; a “q-mer” counting approach as in Quake, in which
k-mer counts are weighted by quality scores, could easily be adapted [24].
Another concern for normalizing deep sequencing data sets is that, with
sufficiently deep sequencing, sequences with many errors will start to accrue.
This underlies the continued accumulation of sequence data for E. coli
observed in Figure 5. Assemblers may be unable to distinguish between this
false sequence and the error-free sequences, for sufficiently deep data sets.
This accumulation of erroneous sequences is again caused by the use of k-mers
to detect similarity, and is one reason why exploring local alignment
approaches (discussed below) may be a good future direction.
### Applying assembly algorithms to digitally normalized data
The assembly problem is challenging for several reasons: many formulations are
computationally complex (NP-hard), and practical issues of both genome content
and sequencing, such as repetitive sequence, polymorphisms, short reads and
high error rates, challenge assembly approaches [35]. This has driven the
development of heuristic approaches to resolving complex regions in
assemblies. Several of these heuristic approaches use the abundance
information present in the reads to detect and resolve repeat regions; others
use pairing information from paired-end and mate-pair sequences to resolve
complex paths. Digital normalization aggressively removes abundance
information, and we have not yet adapted it to paired-end sequencing data
sets; this could and should affect the quality of assembly results! Moreover,
it is not clear what effect different coverage (C) and k-mer (k) values have
on assemblers. In practice, for at least one set of k-mer size $k$ and
normalized coverage $C$ parameters, digital normalization seems to have little
negative effect on the final assembled contigs. Further investigation of the
effects of varying $k$ and $C$ relative to specific assemblers and assembler
parameters will likely result in further improvements in assembly quality.
A more intriguing notion than merely using digital normalization as a pre-
filter is to specifically adapt assembly algorithms and protocols to digitally
normalized data. For example, the reduction in data set size afforded by
digital normalization may make overlap-layout-consensus approaches
computationally feasible for short-read data [2]. Alternatively, the quick and
inexpensive generation of contigs from digitally normalized data could be used
prior to a separate scaffolding step, such as those supported by SGA and
Bambus2 [14, 36]. Digital normalization offers many future directions for
improving assembly.
### Conclusions
Digital normalization is an effective demonstration that much of short-read
shotgun sequencing is redundant. Here we have shown this by normalizing
samples to 5-20x coverage while recovering complete or nearly complete contig
assemblies. Normalization is substantially different from uniform
downsampling: by doing downsampling in a locus-specific manner, we retain low
coverage data. Previously described approaches to reducing sampling variation
rely on ad hoc parameter measures and/or an initial round of assembly and have
not been shown to be widely applicable [20, 21].
We have implemented digital normalization as a prefilter for assembly, so that
any assembler may be used on the normalized data. Here we have only
benchmarked a limited set of assemblers – Velvet, Oases, and Trinity – but in
theory digital normalization should apply to any assembler. De Bruijn and
string graph assemblers such as Velvet, SGA, SOAPdenovo, Oases, and Trinity
are especially likely to work well with digital normalized data, due to the
underlying reliance on k-mer overlaps in these assemblers.
### Digital normalization is widely applicable and computationally convenient
Digital normalization can be applied de novo to any shotgun data set. The
approach is extremely computationally convenient: the runtime complexity is
linear with respect to the data size, and perhaps more importantly it is
single-pass: the basic algorithm does not need to look at any read more than
once. Moreover, because reads accumulate sub-linearly, errors do not
accumulate quickly and overall memory requirements for digital normalization
should grow slowly with data set size. Note also that while the algorithm
presented here is not perfectly parallelizable, efficient distributed k-mer
counting is straightforward and it should be possible to scale digital
normalization across multiple machines [37].
The first pass of digital normalization is implemented as an online streaming
algorithm in which reads are examined once. Streaming algorithms are useful
for solving data analysis problems in which the data are too large to easily
be transmitted, processed, or stored. Here, we implement the streaming
algorithm using a fixed memory data “sketch” data structure, CountMin Sketch.
By combining a single-pass algorithm with a fixed-memory data structure, we
provide a data reduction approach for sequence data analysis with both
(linear) time and (constant) memory guarantees. Moreover, because the false
positive rate of the CountMin Sketch data structure is well understood and
easy to predict, we can provide data quality guarantees as well. These kinds
of guarantees are immensely valuable from an algorithmic perspective, because
they provide a robust foundation for further work [31]. The general concept of
removing redundant data while retaining information underlies “lossy
compression”, an approach used widely in image processing and video
compression. The concept of lossy compression has broad applicability in
sequence analysis. For example, digital normalization could be applied prior
to homology search on unassembled reads, potentially reducing the
computational requirements for e.g. BLAST and HMMER without loss of
sensitivity. Digital normalization could also help merge multiple different
read data sets from different read technologies, by discarding entirely
redundant sequences and retaining only sequences containing “new” information.
These approaches remain to be explored in the future.
## Methods
All links below are available electronically through
ged.msu.edu/papers/2012-diginorm/, in addition to the archival locations
provided.
### Data sets
The E. coli, S. aureus, and Deltaproteobacteria data sets were taken from
Chitsaz et al. [19], and downloaded from bix.ucsd.edu/projects/singlecell/.
The mouse data set was published by Grabherr et al. [15] and downloaded from
trinityrnaseq.sf.net/. All data sets were used without modification. The
complete assemblies, both pre- and post-normalization, for the E. coli, S.
aureus, the uncultured Deltaproteobacteria, mouse, and yeast data sets are
available from ged.msu.edu/papers/2012-diginorm/.
The simulated genome and transcriptome were generated from a uniform AT/CG
distribution. The genome consisted of a single chromosome 400,000 bases in
length, while the transcriptome consisted of 100 transcripts of length 500.
100-base reads were generated uniformly from the genome to an estimated
coverage of 200x, with a random 1% per-base error. For the transcriptome, 1
million reads of length 100 were generated from the transcriptome at relative
expression levels of 10, 100, and 1000, with transcripts assigned randomly
with equal probability to each expression group; these reads also had a 1%
per-base error.
### Scripts and software
All simulated data sets and all analysis summaries were generated by Python
scripts, which are available at github.com/ged-lab/2012-paper-diginorm/.
Digital normalization and k-mer analyses were performed with the khmer
software package, written in C++ and Python, available at github.com/ged-
lab/khmer/, tag ’2012-paper-diginorm’. khmer also relies on the screed package
for loading sequences, available at github.com/ged-lab/screed/, tag
’2012-paper-diginorm’. khmer and screed are Copyright (c) 2010 Michigan State
University, and are free software available for distribution, modification,
and redistribution under the BSD license.
Mapping was performed with bowtie v0.12.7 [38]. Genome assembly was done with
velvet 1.2.01 [9]. Transcriptome assembly was done with velvet 1.1.05/oases
0.1.22 and Trinity, head of branch on 2011.10.29. Graphs and correlation
coefficients were generated using matplotlib v1.1.0, numpy v1.7, and ipython
notebook v0.12 [39]. The ipython notebook file and data analysis scripts
necessary to generate the figures are available at github.com/ged-
lab/2012-paper-diginorm/.
### Analysis parameters
The khmer software uses a CountMin Sketch data structure to count k-mers,
which requires a fixed memory allocation [27]. In all cases the memory usage
was fixed such that the calculated false positive rate was below 0.01. By
default k was set to 20.
Genome and transcriptome coverage was calculated by mapping all reads to the
reference with bowtie (-a --best --strata) and then computing the per-base
coverage in the reference. Read coverage was computed by then averaging the
per-base reference coverage for each position in the mapped read; where reads
were mapped to multiple locations, a reference location was chosen randomly
for computing coverage. Median k-mer counts were computed with khmer as
described in the text. Artificially high counts resulting from long stretches
of Ns were removed after the analysis. Correlations between median k-mer
counts and mapping coverage were computed using numpy.corrcoef; see calc-r2.py
script.
### Normalization and assembly parameters
For Table 3, the assembly k parameter for Velvet was k=45 for E. coli; k=41
for S. aureus single cell; and k=39 for Deltaproteobacteria single cell.
Digital normalization on the three bacterial samples was done with -N 4 -x
2.5e8 -k 20, consuming 1gb of memory. Post-normalization k parameters for
Velvet assemblies were k=37 for E. coli, k=27 for S. aureus, and k=27 for
Deltaproteobacteria. For Table 4, the assembly k parameter for Oases was k=21
for yeast and k=23 for mouse. Digital normalization on both mRNAseq samples
was done with -N 4 -x 2e9 -k 20, consuming 8gb of memory. Assembly of the
single-pass normalized mRNAseq was done with Oases at k=21 (yeast) and k=19
(mouse).
### Assembly overlap and analysis
Assembly overlap was computed by first using NCBI BLASTN to build local
alignments for two assemblies, then filtering for matches with bit scores $>$
200, and finally computing the fraction of bases in each assembly with at
least one alignment. Total fractions were normalized to self-by-self BLASTN
overlap identity to account for BLAST-specific sequence filtering. TBLASTX
comparison of the Deltaproteobacteria SAR324 sequence was done against another
assembled SAR324 sequence, acc AFIA01000002.1.
### Compute requirement estimation
Execution time was measured using real time from Linux bash ’time’. Peak
memory usage was estimated either by the ’memusg’ script from
stackoverflow.com, peak-memory-usage-of-a-linux-unix-process, included in the
khmer repository; or by the Torque queuing system monitor, for jobs run on
MSU’s HPC system. While several different machines were used for analyses,
comparisons between unnormalized and normalized data sets were always done on
the same machine.
## Acknowledgments
We thank Chris Hart, James M. Tiedje, Brian Haas, Jared Simpson, Scott Emrich,
and Russell Neches for their insight and helpful technical commentary.
## References
* 1. Metzker M (2010) Sequencing technologies - the next generation. Nat Rev Genet 11: 31-46.
* 2. Miller J, Koren S, Sutton G (2010) Assembly algorithms for next-generation sequencing data. Genomics 95: 315-27.
* 3. Gnerre S, Maccallum I, Przybylski D, Ribeiro F, Burton J, et al. (2011) High-quality draft assemblies of mammalian genomes from massively parallel sequence data. Proc Natl Acad Sci U S A 108: 1513-8.
* 4. Conway T, Bromage A (2011) Succinct data structures for assembling large genomes. Bioinformatics 27: 479-86.
* 5. Sboner A, Mu X, Greenbaum D, Auerbach R, Gerstein M (2011) The real cost of sequencing: higher than you think! Genome Biol 12: 125.
* 6. Stein L (2010) The case for cloud computing in genome informatics. Genome Biol 11: 207.
* 7. Trapnell C, Salzberg S (2009) How to map billions of short reads onto genomes. Nat Biotechnol 27: 455-7.
* 8. Simpson J, Wong K, Jackman S, Schein J, Jones S, et al. (2009) Abyss: a parallel assembler for short read sequence data. Genome Res 19: 1117-23.
* 9. Zerbino D, Birney E (2008) Velvet: algorithms for de novo short read assembly using de bruijn graphs. Genome Res 18: 821-9.
* 10. Li Y, Hu Y, Bolund L, Wang J (2010) State of the art de novo assembly of human genomes from massively parallel sequencing data. Hum Genomics 4: 271-7.
* 11. Simpson J, Durbin R (2012) Efficient de novo assembly of large genomes using compressed data structures. Genome Res 22: 549-56.
* 12. Iqbal Z, Caccamo M, Turner I, Flicek P, McVean G (2012) De novo assembly and genotyping of variants using colored de bruijn graphs. Nat Genet 44: 226-32.
* 13. Compeau P, Pevzner P, Tesler G (2011) How to apply de bruijn graphs to genome assembly. Nat Biotechnol 29: 987-91.
* 14. Simpson J, Durbin R (2010) Efficient construction of an assembly string graph using the fm-index. Bioinformatics 26: i367-73.
* 15. Grabherr M, Haas B, Yassour M, Levin J, Thompson D, et al. (2011) Full-length transcriptome assembly from rna-seq data without a reference genome. Nat Biotechnol 29: 644-52.
* 16. Schulz M, Zerbino D, Vingron M, Birney E (2012) Oases: robust de novo rna-seq assembly across the dynamic range of expression levels. Bioinformatics 28: 1086-92.
* 17. Namiki T, Hachiya T, Tanaka H, Sakakibara Y (2011) MetaVelvet: An extension of Velvet assembler to de novo metagenome assembly from short sequence reads. ACM Conference on Bioinformatics, Computational Biology and Biomedicine .
* 18. Peng Y, Leung H, Yiu S, Chin F (2011) Meta-idba: a de novo assembler for metagenomic data. Bioinformatics 27: i94-101.
* 19. Chitsaz H, Yee-Greenbaum J, Tesler G, Lombardo M, Dupont C, et al. (2011) Efficient de novo assembly of single-cell bacterial genomes from short-read data sets. Nat Biotechnol 29: 915-21.
* 20. Rodrigue S, Malmstrom R, Berlin A, Birren B, Henn M, et al. (2009) Whole genome amplification and de novo assembly of single bacterial cells. PLoS One 4: e6864.
* 21. Woyke T, Sczyrba A, Lee J, Rinke C, Tighe D, et al. (2011) Decontamination of mda reagents for single cell whole genome amplification. PLoS One 6: e26161.
* 22. Medvedev P, Scott E, Kakaradov B, Pevzner P (2011) Error correction of high-throughput sequencing datasets with non-uniform coverage. Bioinformatics 27: i137-41.
* 23. Chaisson M, Pevzner P, Tang H (2004) Fragment assembly with short reads. Bioinformatics 20: 2067-74.
* 24. Kelley D, Schatz M, Salzberg S (2010) Quake: quality-aware detection and correction of sequencing errors. Genome Biol 11: R116.
* 25. Bonaldo M, Lennon G, Soares M (1996) Normalization and subtraction: two approaches to facilitate gene discovery. Genome Res 6: 791-806.
* 26. Soares M, Bonaldo M, Jelene P, Su L, Lawton L, et al. (1994) Construction and characterization of a normalized cdna library. Proc Natl Acad Sci U S A 91: 9228-32.
* 27. Cormode G, Muthukrishnan S (2004) An improved data stream summary: the count-min sketch, and its applications. Journal of Algorithms .
* 28. Spits C, Caignec CL, Rycke MD, Haute LV, Steirteghem AV, et al. (2006) Whole-genome multiple displacement amplification from single cells. Nat Protoc 1: 1965-70.
* 29. Pickrell J, Pai A, Gilad Y, Pritchard J (2010) Noisy splicing drives mrna isoform diversity in human cells. PLoS Genet 6: e1001236.
* 30. Pinho A, Pratas D, Garcia S (2012) Green: a tool for efficient compression of genome resequencing data. Nucleic Acids Res 40: e27.
* 31. Muthukrishnan S (2005) Data streams: algorithms and applications. Foundations and trends in theoretical computer science. Now Publishers. URL http://books.google.com/books?id=415loiMd_c0C.
* 32. Do H, Choi K, Preparata F, Sung W, Zhang L (2008) Spectrum-based de novo repeat detection in genomic sequences. J Comput Biol 15: 469-87.
* 33. Novak P, Neumann P, Macas J (2010) Graph-based clustering and characterization of repetitive sequences in next-generation sequencing data. BMC Bioinformatics 11: 378.
* 34. Gu W, Castoe T, Hedges D, Batzer M, Pollock D (2008) Identification of repeat structure in large genomes using repeat probability clouds. Anal Biochem 380: 77-83.
* 35. Nagarajan N, Pop M (2009) Parametric complexity of sequence assembly: theory and applications to next generation sequencing. J Comput Biol 16: 897-908.
* 36. Koren S, Treangen T, Pop M (2011) Bambus 2: scaffolding metagenomes. Bioinformatics 27: 2964-71.
* 37. Schatz M (2009) Cloudburst: highly sensitive read mapping with mapreduce. Bioinformatics 25: 1363-9.
* 38. Langmead B, Trapnell C, Pop M, Salzberg S (2009) Ultrafast and memory-efficient alignment of short dna sequences to the human genome. Genome Biol 10: R25.
* 39. Pérez F, Granger BE (2007) IPython: a System for Interactive Scientific Computing. Comput Sci Eng 9: 21-29.
## Figure Legends
Figure 1: Representative rank-abundance distributions for 20-mers from
100-base reads with no errors, a read with a single substitution error, and a
read with multiple substitution errors.
Figure 2: Mapping and k-mer coverage measures correlate for simulated genome
data and a real E. coli data set (5m reads). Simulated data $r^{2}=0.79$; E.
coli $r^{2}=0.80$.
Figure 3: Mapping and k-mer coverage measures correlate for simulated
transcriptome data as well as real mouse transcriptome data. Simulated data
$r^{2}=0.93$; mouse transcriptome $r^{2}=0.90$.
Figure 4: Coverage distribution of three microbial genome samples, calculated
from mapped reads (a) before and (b) after digital normalization (k=20, C=20).
Figure 5: Fraction of reads kept when normalizing the E. coli dataset to C=20
at k=20.
Figure 6: K-mers at the ends of sequences are lost during digital
normalization.
## Tables
Table 1: Digital normalization to C=20 removes many erroneous k-mers from sequencing data sets. Numbers in parentheses indicate number of true k-mers lost at each step, based on reference. Data set | True 20-mers | 20-mers in reads | 20-mers at C=20 | % reads kept
---|---|---|---|---
Simulated genome | 399,981 | 8,162,813 | 3,052,007 (-2) | 19%
Simulated mRNAseq | 48,100 | 2,466,638 (-88) | 1,087,916 (-9) | 4.1%
E. coli genome | 4,542,150 | 175,627,381 (-152) | 90,844,428 (-5) | 11%
Yeast mRNAseq | 10,631,882 | 224,847,659 (-683) | 10,625,416 (-6,469) | 9.3%
Mouse mRNAseq | 43,830,642 | 709,662,624 (-23,196) | 43,820,319 (-13,400) | 26.4%
Table 2: Three-pass digital normalization removes most erroneous k-mers. Numbers in parentheses indicate number of true k-mers lost at each step, based on known reference. Data set | True 20-mers | 20-mers in reads | 20-mers remaining | % reads kept
---|---|---|---|---
Simulated genome | 399,981 | 8,162,813 | 453,588 (-4) | 5%
Simulated mRNAseq | 48,100 | 2,466,638 (-88) | 182,855 (-351) | 1.2%
E. coli genome | 4,542,150 | 175,627,381 (-152) | 7,638,175 (-23) | 2.1%
Yeast mRNAseq | 10,631,882 | 224,847,659 (-683) | 10,532,451 (-99,436) | 2.1%
Mouse mRNAseq | 43,830,642 | 709,662,624 (-23,196) | 42,350,127 (-1,488,380) | 7.1%
Table 3: Three-pass digital normalization reduces computational requirements for contig assembly of genomic data. Data set | N reads pre/post | Assembly time pre/post | Assembly memory pre/post
---|---|---|---
E. coli | 31m / 0.6m | 1040s / 63s (16.5x) | 11.2gb / 0.5 gb (22.4x)
S. aureus single-cell | 58m / 0.3m | 5352s / 35s (153x) | 54.4gb / 0.4gb (136x)
Deltaproteobacteria single-cell | 67m / 0.4m | 4749s / 26s (182.7x) | 52.7gb / 0.4gb (131.8x)
Table 4: Single-pass digital normalization to C=20 reduces computational requirements for transcriptome assembly. Data set | N reads pre/post | Assembly time pre/post | Assembly memory pre/post
---|---|---|---
Yeast (Oases) | 100m / 9.3m | 181 min / 12 min (15.1x) | 45.2gb / 8.9gb (5.1x)
Yeast (Trinity) | 100m / 9.3m | 887 min / 145 min (6.1x) | 31.8gb / 10.4gb (3.1x)
Mouse (Oases) | 100m / 26.4m | 761 min/ 73 min (10.4x) | 116.0gb / 34.6gb (3.4x)
Mouse (Trinity) | 100m / 26.4m | 2297 min / 634 min (3.6x) | 42.1gb / 36.4gb (1.2x)
Table 5: Digital normalization has assembler-specific effects on transcriptome assembly. Data set | Contigs $>$ 300 | Total bp $>$ 300 | Contigs $>$ 1000 | Total bp $>$ 1000
---|---|---|---|---
Yeast (Oases) | 12,654 / 9,547 | 33.2mb / 27.7mb | 9,156 / 7,345 | 31.2mb / 26.4mb
Yeast (Trinity) | 10,344 / 12,092 | 16.2mb / 16.5mb | 5,765 / 6,053 | 13.6 mb / 13.1mb
Mouse (Oases) | 57,066 / 49,356 | 98.1mb / 84.9mb | 31,858 / 27,318 | 83.7mb / 72.4mb
Mouse (Trinity) | 50,801 / 61,242 | 79.6 mb / 78.8mb | 23,760 / 24,994 | 65.7mb / 59.4mb
|
arxiv-papers
| 2012-03-21T18:58:45 |
2024-09-04T02:49:28.865900
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Titus Brown, Adina Howe, Qingpeng Zhang, Alexis B. Pyrkosz, Timothy\n H. Brom",
"submitter": "C. Titus Brown",
"url": "https://arxiv.org/abs/1203.4802"
}
|
1203.4938
|
# Advanced Programming Platform for efficient use of Data Parallel Hardware
Luis Cabellos Institute of Physics of Cantabria (IFCA), CSIC-UC
Santander, 39005, Spain
Email: cabellos@ifca.unican.es
###### Abstract
Graphics processing units (GPU) had evolved from a specialized hardware
capable to render high quality graphics in games to a commodity hardware for
effective processing blocks of data in a parallel schema. This evolution is
particularly interesting for scientific groups, which traditionally use mainly
CPU as a work horse, and now can profit of the arrival of GPU hardware to HPC
clusters. This new GPU hardware promises a boost in peak performance, but it
is not trivial to use. In this article a programming platform designed to
promote a direct use of this specialized hardware is presented. This platform
includes a visual editor of parallel data flows and it is oriented to the
execution in distributed clusters with GPUs. Examples of application in two
characteristic problems, Fast Fourier Transform and Image Compression, are
also shown.
## I Introduction
The game industry saw in the 2000s the revolution of the programmable shaders.
Programmable shaders insert specific code in the 3D graphics pipeline in order
to have customized effects 111Effects not available in the graphics hardware..
Initially programmable shaders allowed to change the pixels color, but soon
they evolved to be able to modify also the geometry, and currently they have
the flexibility to interact with almost all graphical elements, achieving the
so called General-Purpose computing on graphics processing units (GPGPU).
The use of GPGPU allow developers to program general processing applications
using graphics hardware. The game companies soon noticed the problem with the
programmable shaders: the graphics are the objective of visual artists, but
programming is a hard and time consuming process that is not taught in art
academies. And programmers do not have the experience, nor the knowledge to
get the best visual result programming customized effects with graphics
hardware. The solution reached in the game industry was to develop powerful
tools, easy to use for the artist but flexible enough to get all the results
that a programmer can implement directly. These tools are based on a data-flow
programming design that allows the visual artist to create special effects and
display them on-screen during editing, exactly as they will appear along game
execution. The tools include the possibility of directed edition showing
immediately the effects of a proposed change, and also the intermediate state
of data (as graphics primitives) in the workflow and so to understand the
result of the change as a whole product or as a sum of transformations.
Scientific communities have increasing needs for processing large amounts of
data as fast as possible[1]. Under this demand, they see GPGPU hardware with
its high peak processing power as a valuable resource to handle for their
workflows [2], but they are now in a similar situation to the the game
industry with the programmable shaders: it is not trivial to program GPU
resources for use in real applications. Efficient use of GPUs is difficult
because although it is possible to use them in almost every kind of
algorithms, only a few of them are executed more efficiently that in a CPU
with a more general architecture. The input data needs to be sent from the CPU
to the GPU and results returned back, but there is a limited memory bandwidth
between both processors. Also the internal bus in a GPU is more powerful, but
it has a different organization of memory caches and locations compared to a
CPU, and programs need to take this organization into account to benefit. Also
the GPU has the advantage of large parallelism thanks to hardware replication,
but it has limited and strict pipelines limiting context switching between
tasks without suffering performance degradation.
Data-flow programming is a paradigm that constructs applications as directed
graphs[3][4]. The vertex of the graphs are processes and the edges between
vertexes define the input/output of such processes and the path the data
should travel. The applications in this paradigm are defined changing the set
of vertexes and creating the network of edges between them. Starting from a
clever definition of vertexes and changing only the edges, it is possible to
create many different applications under this paradigm.
The data-flow programming paradigm is close to the preferred work methodology
used by scientific communities. They usually share the data from experimental
sources and define scientific workflows to analyze that data. Scientific
groups with better computing skills also share computational services to
analyze data. However researchers do not usually apply the data-flow paradigm
directly to program their applications, they rather compose them through
successive filter and processing steps starting from the raw data obtained
from the experimental setup.
However, there are several solutions to compose workflows in a scientific
programming context, starting with the popular and powerful LabView software
[5], the Kepler System [6], or others adapted to a Grid computational
infrastructure[7].
The proposal presented here aims to use the data-flow paradigm starting at the
basic level, constructing modules from a well defined set of processes,
conceived as orthogonal components, including data parallelism, and that
communicate between them to build applications.
This Distributed Programming Platform for Data Parallel Algorithms will also
benefit of powerful visual tools like those already developed under the data-
flow paradigm [8]. These tools will include a user friendly editor to program
data-flow applications when the algorithm fits into a data parallelism model,
and also a service able to execute the resultant program flows in the most
efficient way using computers with one or more GPUs.
The architecture explained in this paper is designed to allow the user to
build the flow once and be able to execute it with different data sets and on
different distributed computing hardware offering GPU resources.
## II Implementation
### II-A GPU Framework
The first step in the development of a Data-Parallel Platform is the selection
of a GPGPU platform. Currently there are are two major platforms available:
CUDA and OpenCL[9]. CUDA stands for Compute Unified Device Architecture, and
it is a computing engine developed by Nvidia Corporation enabling access to
Nvidia GPUs as a GPGPU platform[10]. OpenCL is an specification made by the
Khronos Group, of an open standard for general purpose parallel
programming[11]. Although both platforms are presented to use GPU hardware as
GPGPU platforms, they also support manycore and multicore hardware, so they
are able to execute code in both CPUs and GPUs, provided the corresponding
driver.
CUDA offers higher quality libraries and also includes better development
tools, like a debugger and an emulator. Applications in CUDA require less
setup code, and the performance compares favorably with OpenCL[12].
Both platforms use the C language as their reference model, and have similar
memory and concurrency characteristics, so converting programs between both
platforms is not difficult.
On the other hand, OpenCL has a clear advantage over CUDA: while CUDA is
designed to work with NVIDIA hardware, OpenCL, as an open standard, has
already drivers implemented for NVIDIA, ATI and Intel hardware, both for GPUs
and CPUs.
From the point of view of a potential user of the Data-Parallel Platform, this
is the most relevant argument. In fact, potential users of the Data-Parallel-
Platform are not interested in the tools that the developers will use, nor in
the setup details, but are critically concerned about availability of the
platform for their existing hardware. Based on these arguments, OpenCL was
selected as the platform to build the Data-Parallel Platform.
The Data-Parallel Platform uses OpenCL in two different ways. Firstly, the
OpenCL software development kit is used to execute the program flows in the
GPU hardware, including manycore and multicore hardware when available
Secondly, the OpenCL C programming language is employed to codify the behavior
inside the vertexes of the Data-Parallel Program graphs. As it will be shown
below, there is a direct translation between the Data-Parallel Platform
vertexes and OpenCL C source code. A simple example can be seen comparing
middle and bottom sections in Table I.
for( int i = 0 ; i < MAX ; i++ ){
z[i]=x[i]+y[i];
}
---
__kernel adder( global float * x, global float * y,
global float * z ){
int i = get_global_id(0);
z[i]=x[i]+y[i];
}
"adder":{
"body":
"int i = get_global_id(0);\nz[i]=x[i]+y[i];\n",
"io":{
"x":{ "data":"float", "type":"InputPoint"},
"y":{ "data":"float", "type":"InputPoint"},
"z":{ "data":"float", "type":"OutputPoint"}}
}
TABLE I: Comparative of the implementation of a loop in the different
platforms. Top: the loop in C-like pseudocode. Middle: the loop using an
OpenCL kernel. Bottom: the loop in Data-Parallel JSON format.
This strategy can be seen as a complexity reduction of the access to GPU
programming[13]: hand coding complexity for the final user is much reduced as
OpenCL functions input and output parameters are limited. However it implies
also a limitation on the complexity of the problem being coded, and addressed.
OpenCL support two execution models, data parallel and task parallel
programming models. The Data-Parallel Platform will use only the data parallel
programming model of OpenCL to offer data-flow programming to users. In this
data parallel model, a sequence of instructions is applied to multiple
elements in memory. Each one of these elements is called a work-item and the
parallelism is achieved executing the sequence of instructions at the same
time over all the work-items. In the Data-Parallel Platform a one-to-one bind
between the work-item in memory and the kernel currently executed is
established. In this way the input data-flow in a Data-Parallel program is
split into chunks of work-items, then executed in parallel using OpenCL and
finally the result is re-joined to compose the output data-flow.
### II-B The Data Parallel Model
The use of the data parallel model in OpenCL and the division in blocks of
work-items requires that the Data-Parallel Programs are strictly Directed
Acyclic Graphs (DAGs). This requirement avoids return edges, that would
complicate the parallelism of blocks of elements if a vertex would have to
wait for the output of a posterior vertex.
Figure 1: Data-Parallel Platform workflow scheme. The graph shows the two
different ways to use the Data-Parallel Platform: 1) As a library from a user
application; 2) from the Data-Parallel Platform Editor, either running
directly in the Data-Parallel Platform Server, or, in the future, as a job
system.
The Data-Parallel Platform is structured as several layers or components in
order to allow the execution of Data-Parallel programs in different ways:
direct execution, scheduled execution on a queue, or integrated in existing
applications using a library. These different possibilities are represented in
figure 1.
The most basic example would start with the creation of a Data-Parallel
Program using the Data-Parallel Editor (see below), then selecting the input
files to be processed, and finally executing the program in a Data-Parallel
Server (also presented later).
The same Data-Parallel Program created using the editor can be executed by a
program using the functions from the Data-Parallel Platform library. Execution
using a queue system is being implemented under a wider scope, in a
Distributed Data-Parallel Platform including a Data-Parallel Scheduler acting
as a batch system for Data-Parallel Programs222The whole set of tools
described here is being further developed under the name of the Skema
Platform..
### II-C A Visual Editor of Data-Parallel Programs
A Data-Parallel Program Editor has been implemented as a visual tool following
the Blender333Blender is the free open source 3D content creation suite.
Compositor style. The edition of a complete Data-Parallel Program has two
parts. The first one is the definition of the nodes: individual nodes are
created and can be modified individually, including its input/output set and
its body main program in OpenCL C Programming Language. The second part is the
definition of the data flow between instances of the nodes. Nodes must be
instantiated and arranged into a processing network.
Figure 2 shows how a Data-Parallel Program appears in the visual editor.
Figure 2: The Visual Editor showing a basic Data-Parallel Program. The data
flows from left to right with a floating number as input and a floating number
as result.
A Data-Parallel Program is a data-flow application created to be executed in
the Data-Parallel Platform. Its main components are the following ones:
The available data types in the Data-Parallel Platform: OpenCL 1.0 [11] data
types are used, including scalar and vector data types.
Points attached to vertexes in the Data-Parallel Programs. The set of points
of a Node define the possible communication channels between instances of that
node.
A node defines the behavior of the graph vertexes in a Data-Parallel Program.
It is composed of a set of Input/Output points (at least one of each type) and
a main program body coded using the OpenCL C Programming Language
specification.
An instance of a node is a vertex in the Data-Parallel Program. In the example
shown in the previous figure 2 there are three instances of three different
nodes.
An arrow is an edge between two instances or vertexes. Specifically, an arrow
connects an output point of an instance with a compatible input point from a
different instance. The points are compatible if they have the same base
scalar type444The base type of an scalar is that scalar data type. The base
type of a vector data type is the scalar element of the vector. A point from
an instance without a connected arrow is named an unassigned point or a free
point.
A Program is the directed acyclic graph of instances and arrows than can be
executed in the Data-Parallel Platform. The diagram shown in figure 2 shows a
basic Data-Parallel Program.
A stream is a continuous flow of data with a defined type and related to a
free point of a Data-Parallel Program. The execution of a program requires one
or more input streams and one or more output streams. The figure 3 describes
the execution of a Data-Parallel program over an input stream to generate an
output stream.
The Data-flow is the set including the input stream and the output stream in a
Data-Parallel Program execution.
Figure 3: Representation of the execution of a Data-Parallel Program. The
Data-Parallel Program get chunks of data from an input stream, executes the
programming code included in the nodes in parallel for each of the elements of
that chunk, and generates an output stream composed of the results re-joined
in adequate order.
Continuing with the editing process, once the nodes, instances and arrows
between instances of a Data-Parallel Program are defined, all the
corresponding information is exported to a JSON[14] file. This file stores
this information in a format used in the execution of Data-Parallel programs,
either for sending it to a Data-Parallel Server or for connecting to the Data-
Parallel Platform Library. The Table II shows a basic example of exported
information corresponding to the Data-Parallel Program described in Figure 2.
The corresponding JSON file is used by the Data-Parallel Platform to execute
the program flow.
"kernels":{
"adder":{
"body":"int i=get_global_id(0);
z[id]=x[i]+y[i];",
"io":{
"x":{"data":"float","type":"InputPoint"},
"y":{"data":"float","type":"InputPoint"},
"z":{"data":"float","type":"OutputPoint"}}},
"fan":{
"body":"int i=get_global_id(0);
x[i]=z[i].x;
y[i]=z[i].y;",
"io":{
"x":{"data":"float","type":"OutputPoint"},
"y":{"data":"float","type":"OutputPoint"},
"z":{"data":"float2","type":"InputPoint"}}},
"rot":{
"body":"int i=get_global_id(0);\ny[i]=x[i]<<16;",
"io":{
"x":{"data":"float","type":"InputPoint"},
"y":{"data":"float","type":"OutputPoint"}}}},
"nodes":[[0,{"kernel":"fan"}],
[1,{"kernel":"rot"}],
[2,{ "kernel":"adder"}]],
"arrows":[{"output":[0,"x"],"input":[2,"x"]},
{"output":[1,"y"],"input":[2,"y"]},
{"output":[0,"y"],"input":[1,"x"]}]
---
TABLE II: JSON format corresponding to a Data-Parallel program. This basic
example corresponds to the program previously shown in Figure 2, composed of
three nodes and three instances, and the corresponding data-flow between
instances composed of three edges.
### II-D The Data-Parallel Server
The Data-Parallel Server is the module in the platform that executes the Data-
Parallel programs on an input data-flow to obtain an output data-flow. For
that reason it is the only module that actually requires the OpenCL driver and
also direct access to the associated hardware. The server is in charge of
communicating the state of the OpenCL platform, the state of the GPGPU
hardware and its characteristics, and also the running progress of Data-
Parallel programs. It uses a simplified approach for external communication
based on REST (Representational State Transfer)[15] with HTTP networking
protocol and JSON documents as information exchange format. The Data-Parallel
Server currently is not RESTful as not all the REST architectural elements are
implemented, in particular the layered and cacheable properties, although it
is planned to include these features when possible in future developments to
improve its scalability.
Figure 4: Distributed Run Protocol. Client starts a running instance in the
server, to execute the desired program, using a web API in http with JSON as
data language; once the instance is created, the client sends and receives
data from the server via the TCP protocol.
The Data-Parallel Server executes the Data-Parallel Programs using a simple
Run Protocol to connect with the clients. As shown in Figure 4, this protocol
defines the order of the steps to execute a program over a data-flow: send the
Data-Parallel Program to the server, initialize the execution of the program
and finally send the input data-flow and receive back the output data-flow. It
is important to notice that the first step could be skipped if the program is
transferred previously. For this purpose, an unique ID can be associated with
the JSON representation of the program, a program ID. This option may save a
significant time if the same Data-Parallel Program is to be executed with
different input streams.
## III Examples
In order to show the capabilities of our Data-Parallel Platform two working
examples are presented below. The first example computes the discrete Fourier
transform using the Cooley-Tukey algorithm. The second example is a simple
lossy image compression application using a visual vector quantization of
blocks.
Both examples were tested in a server node running the Data-Parallel Server
and a desktop computer running the client programs. The server node was a
Megware Computer solution equipped with an Intel Xeon X5550 2.66GHz Quad Core
processor and 4GB memory. It had four NVIDIA Tesla C1060 GPUs installed.
The desktop computer was an HP Proliant ML330 G6. Both computers are
interconnected using a Gigabit Ethernet LAN.
### III-A Discrete Fourier Transform Example
The discrete Fourier transform (DFT) is a mathematical transform of a signal
between discrete domains used for Fourier analysis. The DFT is widely used in
signal processing, to analyze the frequencies of a signal, data compression
eliminating frequencies with less information in a signal, polynomial
multiplication and convolutions. All these applications depend upon an
efficient calculation of this transformation, so it is a good example to test
the speedup of this DFT using the Data-Parallel Platform on GPU hardware.
The Cooley-Tukey[16] algorithm was used to implement the Fast Fourier
Transform. This algorithm, one of the most used in DFT, recursively calculates
a DFT of N elements using two DFT of sizes $N1$ and $N2$ having $N=N1\cdot
N2$. When $N$ is highly composite 555Highly composite numbers are numbers
which factors completely into small prime numbers. the DFT computation time
can be reduced from $O(N^{2})$ to $O(N\cdot logN)$. In particular, the radix-2
Cooley-Tukey algorithm was used, where the decimation of the DFT is done with
two interleaved DFT of size $N/2$ in each recursive step.
Using this radix-2 decimation, the computation of the last $k$ steps can be
sent to the Data-Parallel Platform Server, parallelizing the calculus of a
large number of DFT of size $2^{k}$. The $2^{k}$ DFT is computed in a simple
Program Node and the DFT flow is the input flow of the Data-Parallel Program.
Figure 5: Comparing DFT on CPU versus GPU execution. The size of the data used
span from 20 Kbytes to 10 MBytes. Three FFT sizes 2, 4 and 8 values in both
types of tests (CPU and GPU) are used. All GPU tests remain well below 2
seconds in execution time while the CPU tests increase up to $8\sim\ 10$
seconds.
Figure 5 presents the result of executing the Data-Parallel Program with a
flow of DFT with sizes 2, 4 and 8 and the execution of the same data with a
CPU implementation of the Cooley-Tukey algorithm for the same sizes. The time
increase is linear in both implementations, as it should be, but it can be
seen that the Data-Parallel Program is approximately five times faster,
although it has to send the data over the local network to the Data-Parallel
Server.
The Data-Parallel Program has also a remarkable advantage over the full CPU
implementation. While the Data Parallel platform is executing the DFT
calculations, the CPU usage in the local computer is only around $10\%$ and
corresponding to I/O time; in contrast the CPU implementation requires $\sim
90\%$ CPU usage.
### III-B A second example: Image Block Compression
(a) Original
(b) Luminance compressed
(c) Result
Figure 6: Image compression from (a) original image to (c) compressed result.
The block compression is done in the (b) luminance of the image, the colour
layer is scaled down 1/4th from the full picture. Uncompressed size is $\sim
770$ Kbytes, and compressed size is $\sim 80$ Kbytes
The purpose of image compression is to represent images using less data in
order to save storage costs or transmission time. A raw image, without
compression, can be quite large, usually up to several megabytes, and
compression can reduce the file size very significantly. The image data can be
compressed in such way that the exact original data to can be recovered from
the compressed data, or losing information in the image data but reaching
better compression rates.
Lossy compression is usually based on techniques that remove details that
humans do not notice. In this example a lossy image compression has been
implemented using the Data-Parallel platform and employing well known methods.
The first method used is the conversion of red-green-blue image data to a
chrome-luminance representation, followed by color sub-sampling to scale 1/4
the from the full size using the fact that the human eye is more sensitive
towards light intensity variation than color variation 666Nobody will notice
the colour downscale. The second method applied, explained in references [17]
[18], divides the image luminance in blocks of $4x4$ pixels, and determines a
code book of N representative mean blocks. With this code book image blocks
are encoded, using intensity deviation, instead of using the full information
of the 16 pixels.
The implemented algorithm uses the following five steps:
1. 1.
Convert to Chrome + Luminance representation
2. 2.
Downscale Chrome layer
3. 3.
Calculate Directional Derivative of Luminance
4. 4.
Apply k-means for calculate codebook
5. 5.
Compress Luminance in blocks
Steps 1, 2 and 3 are calculated in the Data-Parallel Platform, while the
creation of the code book (step 4) is made in the CPU with the returned data.
Once the code book is created, the information is sent back to the GPU to
calculate the compressed data.
The compression is noticeable, as can be seen in Figure 6, both in quality and
in the reduction factor, but the objective with this example is to show how
the Data-Parallel Platform can be used to build a working application. The
sections of the program executed in the GPU were created using the visual
editor, and during the execution of the image compression this work was
distributed on a running Data-Parallel Server using the Data-Parallel Library.
## IV Conclusions and Outlook
A Data-Parallel Platform has been designed supporting the use of GPGPU on
clusters allowing to access to the power of GPUs as a service, with the
advantages this means for the implementation of work-flows and schedulers.
Difficulties of GPGPU programming are reduced thanks to a clear programming
model using the OpenCL platform and modeling the problems using DAGs, and
offering a visual editor tool for final users powerful enough to exploit the
GPGPU syntax.
There is an increasing trend to use GPUs specialized processors as common
building blocks of supercomputers. China’s Tianhe-1A supercomputer achieved in
October 2010 the number one in the TOP500 ranking using graphics chips, and in
the march 2011 3 of the top 5 supercomputers[19] were using mixed
architectures with both CPUs and GPUs. Although the increase in performance
thanks to the use of GPUs seems very high, with up to a 20x factor, GPUs
require specialized programing, and the lack of advanced programming tools and
languages with limited features is a problem[20].
The Data-Parallel Platform presented does not aim to be the best tool for
performance, and it’s not yet fully completed to offer all the characteristics
planned, but it is a solution prepared to allow distributed computing with
GPGPU hardware.
Many improvements will be required to make it a production tool. For example,
regarding performance of program executions, the gap when using a cascade of
instances due to inefficient movement of data between them, has to be solved
Graph theory must be revisited in order to further optimize the Data-Parallel
Programs. In particular to understand how to split a Data-Parallel Program
into several concurrent flows. There is also the possibility of include
characteristics of other distributed solutions in the Data-Parallel Platform,
like high availability, large scalability or Map/Reduce technologies.
Also the design of a job system for Data-Parallel Programs running on Data-
Parallel servers, as part of a Distributed Data-Parallel Platform, will allow
a better scale of applications and a better use of GPGPU resources, especially
in computer clusters with GPU hardware.
## Acknowledgment
This works was supported by the Ministry of Science and Innovation of Spain
and their National Scientific Research, Development and Technological
Innovation Plan (National R&D&i Plan) at the University of Cantabria.
## References
* [1] T. Hey and A. Trefethen, _The Data Deluge: An e-Science Perspective_. John Wiley & Sons, Ltd, 2003, pp. 809–824. [Online]. Available: http://dx.doi.org/10.1002/0470867167.ch36
* [2] Z. Fan, F. Qiu, A. Kaufman, and S. Yoakum-Stover, “Gpu cluster for high performance computing,” in _Proceedings of the 2004 ACM/IEEE conference on Supercomputing_ , ser. SC ’04. Washington, DC, USA: IEEE Computer Society, 2004, pp. 47–. [Online]. Available: http://dx.doi.org/10.1109/SC.2004.26
* [3] Arvind and D. E. Culler, _Dataflow architectures_. Palo Alto, CA, USA: Annual Reviews Inc., 1986, pp. 225–253. [Online]. Available: http://dl.acm.org/citation.cfm?id=17814.17824
* [4] A. L. Davis and R. M. Keller, “Data flow program graphs,” _IEEE Computer_ , vol. 15, no. 2, pp. 26–41, 1982.
* [5] G. W. Johnson, _LabVIEW Graphical Programming: Practical Applications in Instrumentation and Control_ , 2nd ed. McGraw-Hill School Education Group, 1997.
* [6] B. Ludäscher, I. Altintas, C. Berkley, D. Higgins, E. Jaeger, M. Jones, E. A. Lee, J. Tao, and Y. Zhao, “Scientific workflow management and the kepler system,” in _Concurr. Comput. : Pract. Exper_ , 2005, p. 2006.
* [7] I. Foster, “What is the Grid? - a three point checklist,” _GRIDtoday_ , vol. 1, no. 6, Jul. 2002. [Online]. Available: http://www-fp.mcs.anl.gov/~foster/Articles/WhatIsTheGrid.pdf
* [8] W. M. Johnston, J. R. P. Hanna, and R. J. Millar, “Advances in dataflow programming languages,” _ACM Comput. Surv._ , vol. 36, pp. 1–34, March 2004\. [Online]. Available: http://doi.acm.org/10.1145/1013208.1013209
* [9] M. Harris and D. Göddeke. General-purpose computation on graphics hardware. [Online]. Available: http://gpgpu.org
* [10] _NVIDIA CUDA Compute Unified Device Architecture: Programming Guide_ , 2nd ed., Nvidia Corporation, July 2008.
* [11] _The OpenCL Specification, Version 1.0_ , Rev. 33 ed., Khronos Group Std., April 2009. [Online]. Available: http://www.khronos.org/registry/cl/specs/opencl-1.0.33.pdf
* [12] K. Karimi, N. G. Dickson, and F. Hamze, “A performance comparison of cuda and opencl,” _CoRR_ , vol. abs/1005.2581, 2010.
* [13] S.-Z. Ueng, M. Lathara, S. S. Baghsorkhi, and W.-M. W. Hwu, “Cuda-lite: Reducing gpu programming complexity,” in _Languages and Compilers for Parallel Computing_ , J. N. Amaral, Ed. Berlin, Heidelberg: Springer-Verlag, 2008, pp. 1–15.
* [14] D. Crockford, “The application/json Media Type for JavaScript Object Notation (JSON),” RFC 4627 (Informational), Internet Engineering Task Force, Jul. 2006\. [Online]. Available: http://www.ietf.org/rfc/rfc4627.txt
* [15] R. T. Fielding and R. N. Taylor, “Principled design of the modern web architecture,” _ACM Trans. Internet Technol._ , vol. 2, pp. 115–150, May 2002. [Online]. Available: http://doi.acm.org/10.1145/514183.514185
* [16] J. Cooley and J. Tukey, “An algorithm for the machine calculation of complex fourier series,” _Mathematics of Computation_ , vol. 19, no. 90, pp. 297–301, 1965.
* [17] N. M. Nasrabadi and R. A. King, “Image coding using vector quantization: a review,” _Communications, IEEE Transactions on_ , vol. 36, no. 8, pp. 957–971, 1988. [Online]. Available: http://dx.doi.org/10.1109/26.3776
* [18] G. Qiu, “A fast algorithm for constructing image identification.”
* [19] H. Meuer, E. Strohmaier, J. Dongarra, and H. Simon, “Top500 supercomputer sites list,” TOP500.Org, Tech. Rep., June 2001. [Online]. Available: http://top500.org/lists/2011/06
* [20] P. Varhol, “Gpu vs. cpu computing: When your time is on the line, you need both types of processors.” _Desktop Engineering_ , September 2010. [Online]. Available: http://www.deskeng.com/articles/aaayet.htm
|
arxiv-papers
| 2012-03-22T09:54:58 |
2024-09-04T02:49:28.880374
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Luis Cabellos",
"submitter": "Luis Cabellos",
"url": "https://arxiv.org/abs/1203.4938"
}
|
1203.5018
|
# Electromagnetic Energy, Momentum, and Angular Momentum in an Inhomogeneous
Linear Dielectric
Michael E. Crenshaw and Thomas B. Bahder US Army Aviation and Missile
Research, Development, and Engineering Center, Redstone Arsenal, AL 35898, USA
###### Abstract
In a previous work, Optics Communications 284 (2011) 2460–2465, we considered
a dielectric medium with an anti-reflection coating and a spatially uniform
index of refraction illuminated at normal incidence by a quasimonochromatic
field. Using the continuity equations for the electromagnetic energy density
and the Gordon momentum density, we constructed a traceless, symmetric
energy–momentum tensor for the closed system. In this work, we relax the
condition of a uniform index of refraction and consider a dielectric medium
with a spatially varying index of refraction that is independent of time,
which essentially represents a mechanically rigid dielectric medium due to
external constraints. Using continuity equations for energy density and for
Gordon momentum density, we construct a symmetric energy–momentum matrix,
whose four-divergence is equal to a generalized Helmholtz force density four-
vector. Assuming that the energy-momentum matrix has tensor transformation
properties under a symmetry group of space-time coordinate transformations, we
derive the global conservation laws for the total energy, momentum, and
angular momentum.
## I Introduction
Starting with the hydrodynamic continuity equation, Umov BIUmov obtained an
expression for energy continuity in a continuous spatial flow of an
electromagnetic field in 1874 BIHeaviside ; BIPoynting ; BIMicrophysics . A
decade later, Poynting BIPoynting derived a similar energy continuity
equation as a general theorem of the macroscopic Maxwell equations. Poynting’s
theorem is generally preferred because it can be derived directly from the
macroscopic Maxwell equations
$\nabla\times{\bf B}-\frac{n^{2}}{c}\frac{\partial{\bf E}}{\partial t}=0$ (1)
$\nabla\times{\bf E}+\frac{1}{c}\frac{\partial{\bf B}}{\partial t}=0$ (2)
$\nabla\cdot({n^{2}\bf{E}})=0$ (3) $\nabla\cdot{\bf{B}}=0$ (4)
of classical continuum electrodynamics. Here, the Maxwell equations are
written in Heaviside–Lorentz units for a nonmagnetic linear medium in the
absence of free charges and currents. We have assumed that the index of
refraction, $n=n({\bf r})$, depends on position, occupies a finite region of
3-dimensional space, and is independent of time. Poynting’s theorem can be
derived by subtracting the scalar product of the Maxwell–Ampère law, Eq. (1),
with ${\bf E}$ from the scalar product of Faraday’s law, Eq. (2), with ${\bf
B}$ to obtain
$\frac{n^{2}}{c}\frac{\partial{\bf E}}{\partial t}\cdot{\bf
E}+\frac{1}{c}\frac{\partial{\bf B}}{\partial t}\cdot{\bf B}$
$+(\nabla\times{\bf E})\cdot{\bf B}-(\nabla\times{\bf B})\cdot{\bf E}=0.$ (5)
Upon application of a vector identity and the definition of the energy
density,
$\rho_{e}=\frac{1}{2}(n^{2}{\bf E}^{2}+{\bf B}^{2}),$ (6)
the preceding equation becomes Poynting’s theorem
$\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot c({\bf E}\times{\bf B})=0$
(7)
and defines Poynting’s energy flux vector ${\bf S}_{P}=c({\bf E}\times{\bf
B})$.
In a previous work BICB , we considered a dielectric medium with an index of
refraction that was time-independent, spatially uniform, and covered with a
thin gradient-index antireflection coating. For this case, we showed that the
energy
$U=\int_{\sigma}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf B}^{2}\right)dv$ (8)
and the Gordon BIGord momentum
${\bf G}_{G}=\int_{\sigma}\frac{1}{c}(n{\bf E}\times{\bf B})dv$ (9)
are the conserved electromagnetic quantities with integration performed over
all space $\sigma$. We constructed the corresponding traceless, symmetric
energy–momentum tensor, whose four–divergence provided the continuity
equations for the energy and momentum densities of the closed system. This
result incorporated a condition of $\nabla n/n\ll 1/\lambda$ corresponding to
unimpeded flow in the absence of external forces.
In the current work, we consider the case of an inhomogeneous dielectric
medium in which the condition $\nabla n/n\ll 1/\lambda$ no longer holds. The
assumption that the index of refraction is independent of time necessarily
implies that the dielectric medium is mechanically rigid RigidFootnote and
that no momentum is transferred to the dielectric medium. In other words, at
each spatial point, there is effectively an external field that acts as a
constraint that holds the dielectric in place. Therefore, the system that we
are considering is not a closed system and the continuity equations and global
conservation equations will reflect this feature. The external field that acts
as a constraint is not a field that we impose, instead it arises as a result
of the way in which we arrange terms in the continuity equations. We construct
the energy and momentum continuity equations and find that the spatial
gradient of the refractive index appears in the continuity equations in a
generalized Helmholtz force density. This generalized Helmholtz force density
is a spatial and time-varying field that provides the constraint for the
dielectric medium to be mechanically rigid. We then write the continuity
equation as a four divergence of a symmetric energy–momentum matrix. Assuming
that the energy-momentum matrix has tensor transformation properties under a
symmetry group of space-time coordinate transformations, we apply the four-
dimensional divergence theorem to derive global conservation laws for the
total energy, momentum, and angular momentum.
## II Energy and Momentum Continuity Equations
The energy–momentum tensor is a concise way to represent the local continuity
of energy and momentum of a field. While simple in concept, the form of the
energy–momentum tensor for the electromagnetic field in a dielectric has been
at the center of the century-long Abraham-Minkowski controversy BIPfeifer .
The tensor that was proposed by Minkowski BIMin in 1908 is not diagonally
symmetric, a fact that is adverse to conservation of angular momentum BILL .
Abraham BIAbr subsequently proposed a symmetric tensor at the expense of a
phenomenological force. The disagreeable properties of the Minkowski and
Abraham energy-momentum tensors are manifestations of underlying conservation
issues: neither the Minkowski momentum nor the Abraham momentum is conserved.
In this section, we derive the energy and momentum continuity equations for an
inhomogeneous dielectric and construct the corresponding tensor continuity
equation and energy–momentum tensor. In the following section, we will obtain
the globally conserved quantities—the total energy, the total momentum, and
the total angular momentum system, from the energy momentum tensor.
Starting from the Maxwell Eqs.(1)–(4), we use the continuity of the
electromagnetic energy density and the Gordon momentum to obtain a symmetric
stress-energy tensor. We begin with the temporal derivatives of the energy
density and the Gordon momentum density,
$\frac{n}{c}\frac{\partial}{\partial t}\frac{1}{2}\left(n^{2}{\bf E}^{2}+{\bf
B}^{2}\right)=\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}\cdot n{\bf
E}+\frac{n}{c}\frac{\partial{\bf B}}{\partial t}\cdot{\bf B}$ (10)
$\frac{n}{c}\frac{\partial}{\partial t}(n{\bf E}\times{\bf
B})=\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}\times{\bf B}+n{\bf
E}\times\frac{n}{c}\frac{\partial{\bf B}}{\partial t}.$ (11)
The Gordon momentum in Eq. (11) has been scaled by $c$ so that the equations
are in the same units. Next, we apply the vector identity
$\nabla\times(\psi{\bf a})=\nabla\psi\times{\bf a}+\psi\nabla\times{\bf a}$
(12)
to Faraday’s law and write the macroscopic Maxwell equations, Eqs. (1) and
(2), as
$\frac{n}{c}\frac{\partial(n{\bf E})}{\partial t}=\nabla\times{\bf B}$ (13)
$\frac{n}{c}\frac{\partial{\bf B}}{\partial t}=-\nabla\times(n{\bf
E})+\frac{\nabla n}{n}\times n{\bf E}.$ (14)
The variant form of Maxwell’s equations, Eqs. (13) and (14), are
mathematically equivalent to the original versions, Eqs. (1) and (2),
respectively. Substituting the Maxwell equations, Eqs. (13) and (14), into
Eqs. (10) and (11), we produce the energy continuity equation
$\frac{n}{c}\frac{\partial\rho_{e}}{\partial t}+\nabla\cdot(n{\bf E}\times{\bf
B})=\frac{\nabla n}{n}\cdot(n{\bf E}\times{\bf B})$ (15)
and the (Gordon) momentum continuity equation
$\frac{n}{c}\frac{\partial}{\partial t}(n{\bf E}\times{\bf B})+\nabla\cdot{\bf
W}+n{\bf E}\left(\nabla\cdot n{\bf E}\right)=$ $\left(n{\bf
E}\times\frac{\nabla n}{n}\right)\times n{\bf E}.$ (16)
Here, we have used Eq. (3) and the definition of the Maxwell stress tensor
BIJackson ; BILL
$W_{ij}=\left(-nE_{i}nE_{j}-B_{i}B_{j}+\frac{1}{2}\left(n{\bf E}\cdot n{\bf
E}+{\bf B}\cdot{\bf B}\right)\delta_{ij}\right).$ (17)
We can write the energy continuity equation, Eq. (15), and momentum continuity
equation, Eq. (16), as the matrix differential equation
$\bar{\partial}_{\beta}T^{\alpha\beta}=f^{\alpha}$ (18)
with summation over repeated indices. The quantities that appear in Eq. (18)
are a four-divergence operator
$\bar{\partial}_{\alpha}=\left(\frac{n}{c}\frac{\partial}{\partial
t},\partial_{x},\partial_{y},\partial_{z}\right),$ (19)
the array
$T^{\alpha\beta}=\left[\begin{matrix}\rho_{e}&c{g}_{{\rm G}_{1}}&c{g}_{{\rm
G}_{2}}&c{g}_{{\rm G}_{3}}\cr c{g}_{{\rm G}_{1}}&W_{11}&W_{12}&W_{13}\cr
c{g}_{{\rm G}_{2}}&W_{21}&W_{22}&W_{23}\cr c{g}_{{\rm
G}_{3}}&W_{31}&W_{32}&W_{33}\cr\end{matrix}\right],$ (20)
and a generalized force density four-vector
$f^{\alpha}=\left(\nabla n\cdot({\bf E}\times{\bf B}),-{\bf f}_{H}\right),$
(21)
where
${\bf f}_{H}=-nE^{2}\nabla n+2n{\bf E}\left({\bf E}\times\nabla
n\right)+n^{2}{\bf E}(\nabla\cdot{\bf E}).$ (22)
Using Eq. (3), the last two terms on the right side of Eq. (22) cancel, so
that
${\bf f}_{H}=-n{E^{2}}\,\nabla{n}$ (23)
in the absence of free charges.
The array in Eq. (20) appears to have the properties of an energy–momentum
tensor. By construction, the operator defined in Eq. (19) applied to the rows
of the array in Eq. (20) generates continuity equations for the
electromagnetic energy and the momentum. The same operation applied to the
columns
$\bar{\partial}_{\alpha}T^{\alpha\beta}=f^{\beta},$ (24)
generates the same continuity equations by symmetry
$T^{\alpha\beta}=T^{\beta\alpha}.$ (25)
The array has a vanishing trace
$T^{\alpha}_{\alpha}=0$ (26)
corresponding to massless particles BIJackson ; BILL .
The new feature of this result is the appearance of the four-vector
$f^{\alpha}$ in the continuity equation (18). The appearance of the four-
vector $f^{\alpha}$ in the divergence of the stress energy tensor in Eq. (18)
is a result of the fact that the system is not closed. When the gradient of
the index of refraction is non-zero, there is a back-action on the field
altering its spatial properties, so the field does not experience “unimpeded
flow”. It is possible to define an effective stress-energy tensor that takes
into account the back-action on the field by introducing
$\partial_{\alpha}t^{\alpha\beta}=f^{\beta}.$ (27)
The continuity of energy and momentum can then be expressed by
$\partial_{\alpha}(T^{\alpha\beta}+t^{\alpha\beta})=0$ (28)
where the tensor $t^{\alpha\beta}$ contains the influence of the inhomogeneity
of the material and is zero for homogeneous dielectrics.
## III Global Conservation Equations
If we assume that $T^{\alpha\beta}$ transforms as a tensor under some symmetry
group of space-time coordinate transformations, then we can apply the four-
dimensional divergence theorem to obtain global conservation equations.
Integrating Eq. (18) over a four-volume, $d\Omega=cd{\bar{t}}\;d^{3}x\equiv
cd{\bar{t}}\;dv$ (where ${\bar{t}}=t/n$), between hypersurfaces BILL of
constant time at $\bar{t}_{1}=t_{1}/n$ and $\bar{t}_{2}=t_{2}/n$, we have
$\int_{\bar{t}_{1}}^{\bar{t}_{2}}\bar{\partial}_{\beta}T^{\alpha
0}\;cd\bar{t}\;dv=\int_{\bar{t}_{1}}^{\bar{t}_{2}}f^{\alpha}\;cd\bar{t}\;dv,$
(29)
where the integrals over $dv=d^{3}x$ are 3-dimensional volume integrals over
the volume containing the field. Applying the four-divergence theorem results
in
$\int\left(T^{\alpha 0}(t_{2})-T^{\alpha
0}(t_{1})\right)\,dv=c\int\;\frac{1}{n}f^{\alpha}dt\;dv.$ (30)
When $\alpha=1,2,3$, Eq. (30) reduces to
${\bf G}(t_{2})-{\bf G}(t_{1})=\int\,E^{2}\,\nabla n\,dt\;dv$ (31)
where ${\bf G}(t)$ is the Gordon momentum at time $t$, and the time
integration is between $t=t_{1}$ and $t=t_{2}$. Equation (31) shows that the
Gordon momentum of the field is not constant, i.e., there is a source or sink
of momentum provided by the external constraint field ${\bf f}_{H}$. For the
case $\alpha=0$, Eq. (30) gives
$\int\left(\rho_{e}(t_{2})-\rho_{e}(t_{1})\right)dv=\int\,\frac{\nabla
n}{n}\cdot({\bf E}\times{\bf B})\,dt\;dv$ (32)
where $\rho_{e}$ is given by Eq. (6). The left side of Eq. (32) is the
difference in total energy at two different times, $U(t_{2})-U(t_{1})$. We see
that a non-zero spatial gradient in the index means that the field can gain or
loose energy. The results given in Eqs. (31) and (32) assume that the matrix
$T^{\alpha\beta}$ in Eq. (18) transforms as a tensor under some symmetry group
of space-time coordinate transformations. Note that when $\nabla n=0$, Eqs.
(31) and (32) show that the Gordon momentum ${\bf G}(t)$ and the energy $U(t)$
are constants.
In recent years, there is intense interest in angular momentum carried by the
electromagnetic field, see the recent review and references cited therein
TwistedPhotons ; BIJHL . However, the angular momentum carried by the
electromagnetic field in a dielectric environment is no less unsettled than
the linear momentum case BIang . We can define the four-tensor of angular
momentum density in terms of our energy–momentum tensor as BILL
$m^{\alpha\beta\gamma}=\frac{1}{c}\left(x^{\alpha}\,T^{\gamma\beta}-x^{\beta}\,T^{\gamma\alpha}\right)=-m^{\beta\alpha\gamma}.$
(33)
Continuity of angular momentum is given by
$\bar{\partial}_{\gamma}m^{\alpha\beta\gamma}=\frac{1}{c}\left(x^{\alpha}\,f^{\beta}-x^{\beta}\,f^{\alpha}\right).$
(34)
The divergence of $m^{\alpha\beta\gamma}$ is not zero, thereby indicating that
there is a source or sink of angular momentum density, due to the gradient of
the index of refraction.
Once again, we assume that there exists a symmetry group of space-time
coordinate transformations, so that $m^{\alpha\beta\gamma}$ is a tensor. As
above, we can then use the four-divergence theorem to obtain
$\int\left(m^{\alpha\beta 0}(t_{2})-m^{\alpha\beta
0}(t_{1})\right)\,dv=\int\frac{1}{n}\left(x^{\alpha}f^{\beta}-x^{\beta}f^{\alpha}\right)dt\,dv$
(35)
where the time integral is between the two times $t_{1}$ and $t_{2}$ and where
the volume integral $dv$ is over the portion of three-dimensional space
containing the field and includes the region where $n(\bf r)>1$. When $\alpha$
and $\beta$ take values $i,j=1,2,3$, we have
$m^{ij0}(t)=x^{i}f^{j}-x^{j}f^{i}$ (36)
where $f^{i}$ are the components defined in Eq. (23). Equation (35) gives the
change in total angular momentum of the field, $\Delta{\bf M}$, between time
$t_{1}$ and $t_{2}$, and can be written as
$\Delta{\bf M}=\int\left({\bf m}(t_{2})-{\bf m}(t_{1})\right)\,dv=\int
E^{2}\left({\bf r}\times\nabla n\right)dt\,dv$ (37)
where the ${\bf m}(t)$ is the angular momentum density at position ${\bf r}$
at time $t$, defined by
${\bf m}(t)={\bf r}\times{\bf g_{G}}(t)$ (38)
and is defined in terms of the Gordon momentum density
${\bf g_{G}}(t)=n{\bf E}\times{\bf B}/c.$ (39)
Equation (37) shows that when ${\bf r}\times\nabla n$ is non-zero, then the
total angular momentum of the field (as expressed through the angular momentum
density) can change. Once again, we remind the reader that we have assumed the
index of refraction as isotropic, constant in time, but varying in position.
Equation (37) essentially shows that a particular spatial distribution of
${\bf r}\times\nabla n$ can lead to a back-action on the field that can alter
the field angular momentum. Indeed this has been exploited in a number of
recent experiments TwistedPhotons .
## IV Summary
In a previous work, we considered a dielectric medium with an index of
refraction that was time-independent and spatially uniform, and the dielectric
was covered with a thin gradient-index antireflection coating BICB . We found
that the total energy and the Gordon momentum were conserved quantities
(constant in time). In this work we have relaxed these conditions to include a
dielectric medium that has a spatially varying index of refraction that is
constant in time. The fact that the index of refraction is constant in time
essentially means that the dielectric medium is mechanically rigid and subject
to an external constraint. This constraint means that we are not dealing with
a closed system. Using the continuity equations for energy density and
momentum density, we derived a symmetric energy-momentum tensor for the
electromagnetic field. Due to the fact that the system is not closed, the
divergence of the stress-energy tensor is not zero, but equal to a generalized
Helmholtz force vector that represents the constraint. Similarly, the
divergence of the angular momentum density is not zero, due to the external
constraint of a time independent index of refraction. We found that the total
energy, Gordon momentum, and total angular momentum are not conserved because
of the external constraint on time independence of the index of refraction.
However, the time dependence of the total energy, Gordon momentum, and total
angular momentum is related to the constraint, which is proportional to the
gradient of the index of refraction.
Finally, we note that time $t$ has been renormalized to ${\bar{t}}=t/n({\bf
r})$ in the intermediate steps used to obtain the symmetric energy-momentum
tensor and its continuity relation given by its four-divergence in Eq. (18).
However, the final conservation laws of total energy, total momentum and total
angular momentum, given by Eqs. (32), (31), and (37), respectively, are
expressed in terms of the unrenormalized time $t$.
## References
* (1) N. Umow, “Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen” Zeitschrift für Mathematik und Physik XIX, 97–114 (1874).
* (2) J. H. Poynting, “On the transfer of energy in the electromagnetic field,” Phil. Trans. 175, 343–361 (1884).
* (3) O. Heaviside, Electromagnetic Theory, 3 vols. 1893-1912, multiple reprints including (Forgotten Books, 2010).
* (4) J. Z. Buchwald, From Maxwell to Microphysics, (Univ of Chicago Press, 1985).
* (5) M. E. Crenshaw and T. B. Bahder, “Energy–momentum tensor for the electromagnetic field in a dielectric,” Opt. Comm. 284, 2460–2465 (2011).
* (6) J. P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A 8, 14-21 (1973).
* (7) The assumption of a rigid dielectric medium does not violate relativistic principles, however, it leads to a non-zero four divergence of the stress–energy tensor for the electromagnetic field, since the electromagnetic field is not a closed system (because it is interacting with the dielectric medium).
* (8) For a recent review, see: R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007).
* (9) H. Minkowski, Natches. Ges. Wiss. Göttingen, 53, (1908).
* (10) L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th. ed., (Elsevier, 2006).
* (11) M. Abraham, Rend. Circ. Mat. Palermo 28, 1 (1909); 30, 33 (1910).
* (12) J. D. Jackson, Classical Electrodynamics, 2nd ed., (Wiley, 1975).
* (13) For a recent review, see: J. P. Torres and L. Torner, Editors, Twisted Photons: Applications of Light with Orbital Angular Momentum, WILEY-VCH Verlag & Co., Germany (2011).
* (14) Y. S. Jiang, Y. T. He, and F. Li, “Electromagnetic Orbiytal Angular Momentum in Remote Sensing,” PIERS Proceedings, Moscow, Russia, 1330–1337 (2009).
* (15) M. Padgett, S. M. Barnett, and R. Loudon, “The angular momentum of light inside a dielectric,” J. Mod. Optics 50, 1555-1562 (2003).
|
arxiv-papers
| 2012-03-22T15:39:47 |
2024-09-04T02:49:28.889726
|
{
"license": "Public Domain",
"authors": "Michael E. Crenshaw and Thomas B. Bahder",
"submitter": "Michael Crenshaw",
"url": "https://arxiv.org/abs/1203.5018"
}
|
1203.5051
|
# Analysing Temporally Annotated Corpora with CAVaT
###### Abstract
We present CAVaT, a tool that performs Corpus Analysis and Validation for
TimeML. CAVaT is an open source, modular checking utility for statistical
analysis of features specific to temporally-annotated natural language
corpora. It provides reporting, highlights salient links between a variety of
general and time-specific linguistic features, and also validates a temporal
annotation to ensure that it is logically consistent and sufficiently
annotated. Uniquely, CAVaT provides analysis specific to TimeML-annotated
temporal information. TimeML111See www.TimeML.org. TimeML has recently become
an ISO standard [Group, 2008]. is a standard for annotating temporal
information in natural language text. In this paper, we present the reporting
part of CAVaT, and then its error-checking ability, including the workings of
several novel TimeML document verification methods. This is followed by the
execution of some example tasks using the tool to show relations between
times, events, signals and links. We also demonstrate inconsistencies in a
TimeML corpus (TimeBank) that have been detected with CAVaT.
Analysing Temporally Annotated Corpora with CAVaT
Leon Derczynski, Robert Gaizauskas
---
University of Sheffield
211 Portobello, S1 4DP, UK
L.Derczynski@dcs.shef.ac.uk, R.Gaizauskas@dcs.shef.ac.uk
Abstract content
## 1\. Introduction
In essence, TimeML mandates the mark up of expressions referring to _times_ ,
expressions denoting _events_ and expressions _signalling_ temporal relations
between times and events or events and events; it also allows _links_ to be
added between entities, which are labelled with the temporal relation holding
between them.
Existing TimeML tools can be divided into two categories: those which produce
or alter mark-up, for example to assist annotation, and those that perform
analysis. Only a few tools have as yet been developed for TimeML, mostly
focusing on the annotation task, such as TTK [Verhagen and Pustejovsky, 2008],
which does not support analysis. From the second category, in the absence of
other software, the TimeML-using community is restricted to generic XML
analysis tools, such as Xaira [Burnard and Dodd, 2003] or LT-XML111From
http://www.ltg.ed.ac.uk/software/ltxml, as well as similar format-specific
tools (TEI). These generic corpus tools are powerful applications, but require
substantial effort to apply to TimeML data.
We have constructed CAVaT (Corpus Analysis and Validation for TimeML) to
process collections of temporally annotated documents. CAVaT’s functionality
is divided into two main parts; an integrated browsing and report generation
system, and a modular extensible error checking and corpus validation
framework.
In this paper, we first describe the technical aspects of the tool. We then
present the reporting part of CAVaT, and then its error-checking ability,
followed by the execution of some example tasks using the tool. We present an
overview of the tool’s operation and capabilities in Section 2. This includes
details of the corpus loading and folding process (Section 2.1), report
generation, and also a detailed explanation of the advanced validation modules
that are included with CAVaT (Section 2.3). A brief syntax summary is
presented in Section 3; the full guide is on the CAVaT website222Available at
http://code.google.com/p/cavat/. Next, in Section 4, we present sample queries
and output. In Section 5, we show inconsistencies and observations in a TimeML
corpus (TimeBank) that have been detected with CAVaT. Finally, Section 6
summarises the tool and discusses future work.
## 2\. Overview of the tool
CAVaT is an open source tool, constructed from a set of Python modules and a
database. It uses NLTK333See http://www.nltk.org/ and MySQL444See
http://www.mysql.com/. The interface is a text-based interactive prompt, and
all operations are performed with text commands. Command syntax strives to be
simple, flexible and close to natural language. After loading and pre-
processing a TimeML corpus, one can analyse it using built-in reporting
functions, and perform data validation with one of many checking components.
### 2.1. Preprocessing
CAVaT can work on any TimeML-annotated corpus that is stored as a collection
of uncompressed files in a single directory, by importing it to a set of
database tables. The corpus is initially processed by an XML parser (using
Python’s minidom and expat implementations), which retrieves document level
data as well as all temporally annotated information, and places it into a
MySQL database. Temporally annotated data includes all TimeML tags and their
attributes, as well any enclosed tokens for EVENT, SIGNAL and TIMEX3 tags.
In TimeML, events are represented with the EVENT tag, and temporal expressions
with the TIMEX3 tag. These intervals are the elements which CAVaT and the rest
of this paper assume as temporal primitives, unless otherwise stated. Temporal
relations between intervals are described with the TLINK tag, and temporal
signals with SIGNAL. See Figure 1 for an example.
Figure 1: Example text and TimeML annotation.
Automatically classifying the type of temporal relation between intervals is
currently a difficult problem in temporal processing of text [Mani et al.,
2006, Lapata and Lascarides, 2006, Hepple et al., 2007]. The task is often
made simpler by reducing the number of temporal link classes. TimeML includes
before and after relations, though one may simply reverse the arguments of a
before relation to turn it into an after one — so, _June 2008 was before
August 2009_ is equivalent to _August 2009 was after June 2008_. It is thus
possible to convert all links of one of these types to the other. We call this
technique folding. Given a set of mappings, the 13 TimeML relations can be
reduced.
Table 1: Mappings between TimeML relations that can be applied in order to reduce the size of the relation set; when applying the transformation in the table, TLINK argument order is swapped. Original relation type | Folds to relation
---|---
after | before
is_included | includes
iafter | ibefore
begun_by | begins
ended_by | ends
during_inv | simultaneous
during | simultaneous
simultaneous | simultaneous
CAVaT offers three folds:
* •
CAVaT fold – Collapses all inverse relations, such as mapping included_by to
includes (see Table 1).
* •
SputLink fold – The mapping introduced by Marc Verhagen, included in TTK
[Verhagen, 2005].
* •
Compact fold – Reduces TimeML’s link relation set to 3 classes, using mappings
defined in ?).
The first two are lossless, in that no temporal information is removed by the
folding process. The third is lossy. It is possible to perform a lossy fold
by, for example, reducing the TimeML begun_by relationship to one of includes.
### 2.2. Querying
The reporting part of CAVaT makes analysis of TimeML corpora simpler and
easier than working directly with a set of XML documents, allowing flexible
queries, and catering for inquiries specific to temporally-annotated data.
The development of CAVaT has been driven by investigations of TimeML corpora.
Many of the operations performed against a corpora had common elements, often
centred around the retrieval of class distributions or token frequencies. A
tool for TimeML corpus research could encompass all the required operations,
while providing access to a larger range of reports.
CAVaT uses a report generation system where one can view any number of pre-
defined features that match conditions of the user’s choosing. Queries can
produce reports at varying levels of granularity – one may choose to examine
data at sentence, document or corpus level. Reports can output counts,
distributions, lists or text extracts. Example queries are listed in Section
4. Data such as part-of-speech information, tense, aspect, and event
recurrence are captured by attributes described by TimeML, and any data like
this (annotated by tags and their attributes) can be queried. In addition,
properties specific to temporal data but not directly present in mark-up are
implemented, including:
* •
Event / event instance abstraction In some cases, one piece of text may refer
to two separate events (an example is given later in Section 5.1.). To permit
annotation of this, TimeML’s EVENT tag is placed around the text, and then
event instances are specified using one or more MAKEINSTANCE tags. Data
relating to a piece of event text, such as part of speech, polarity and
modality, are described in the MAKEINSTANCE tag. However, we would often like
to see the part of speech data for an event; indeed, when discussing temporal
entities, the term “event” is often used in place of “event instance”. Thus,
CAVaT implicitly translates between these two related tags when requested; for
example, when one asks to see event modality or cardinality.
* •
Signalled links TLINKs may indicate a textual signal that suggests the type of
relationship between their arguments. For example, in _Lydia ate dinner before
leaving the house_ , the word _before_ acts as a signal, ordering two events.
As signals are explicit indicators of temporal association, and correctly
typing a temporal link is difficult, it is useful to be able to quickly
identify which links employ a signal.
* •
Signal text and TLINKs As SIGNAL text referenced from a TLINK may be thought
of as that TLINK’s signal text, CAVaT permits queries that specify signal text
as an attribute of a TLINK.
* •
Text position and lemma Although not part of the TimeML annotation schema,
CAVaT logs text position (by sentence number and word number), and maintains
lemmas of text found within tags.
One may view a particular TLINK’s location in the original document, showing
the link’s arguments and their relation type. This helps understand the
context of a single TLINK. For example, one may often see many links to a
single document date, or discover that most links have arguments within the
same paragraph – something not immediately obvious to humans while browsing
the TLINK markup, and unclear with generic corpus tools.
### 2.3. Checking
Temporal annotation is a complex task, and as a result, a relatively small
amount of text has been annotated to date. The largest TimeML corpus is
TimeBank [Pustejovsky et al., 2003], with less than 200 documents, and around
65000 tokens. Because of the complexity of temporal annotation, errors can
arise beyond those that may be detected using an XML DTD. CAVaT is both a
reporting and validation tool, and seeks to automatically detect high-level
and complex errors that are rarely immediately obvious. Part of the motivation
behind this part of the tool is similar to that of writing unit tests that
highlight bugs in an application: to improve quality by automatically
detecting previously seen errors. In this section we detail some checks that
CAVaT can perform on a TimeML corpus.
Error checks are defined as Python modules, so that one may describe a
detection method for an error case and share it with other researchers without
modifying CAVaT’s core code. The modules inherit from the cavatModule class;
documentation is in the source code, and one may view a list of available
modules with the command check list.
#### 2.3.1. Inconsistent closure
It is possible to create an inconsistent configuration of temporal links. For
example, we may have $A$ before $B$ and $B$ includes $A$; this is clearly not
possible, as includes stipulates that the start point of $A$ occur after the
start point of $B$ (see Figure 2). While this example is fairly clear, it may
not be at all clear to human annotators that a partial temporal link
annotation could imply an inconsistent configuration.
Figure 2: With time flowing from left to right, this represents $A$ before $B$
and $B$ includes $C$. It is not possible for $C$ and $A$ to be the same
interval.
Automatically checking the consistency of a temporal network is hard. TimeML’s
relations are based on those of ?), and it is difficult to guarantee the
consistency of networks formed using the latter set of relations [Vilain et
al., 1989, Tsang, 1987]. We re-state the problem in a more simple fashion, as
follows. Intervals are represented by pairs of endpoints, and we define
intervals and the TimeML relations between them in terms of relations between
these points. Our model uses only simultaneous ($=$) and before ($<$)
relations.
The consistency checker works in a similar way to the closure algorithm in ?).
It maintains an agenda and database. Assertions are taken from the agenda and
used to infer further assertions when combined with assertions in the
database. We initially process intervals in the document (taken from TLINK
arguments) – for each interval $I$ we add $I_{start}<I_{end}$ to the database.
We then generate initial data for the agenda based on TLINKs in the document
and a mapping for each TLINK to one or more assertions, listed in Table 2.
The only inference rules needed with our minimal set of relations are:
If $x=y$ then $y=x$
If $x=y$ and $y=z$ then $x=z$
If $x<y$ and $y<z$ then $x<z$
We can take items from the agenda. For each such item, we compare it against
the database, and deduce new relations using the above rules. If a newly
generated relation conflicts with anything in the agenda or database, then the
document is inconsistent. Otherwise, we will move the item from the agenda to
the database, and add newly generated relations to the agenda. If we can clear
the agenda, then the document is consistent; otherwise, it is not. Whether we
add new relations to the top or bottom of the agenda (achieving depth- or
breadth-first search, respectively) is irrelevant to the success of the
algorithm, though computational performance differences have not been
measured.
Our baselines are the results of the tlink_loop test (Section 2.3.3.) and also
the results of closure success according to SputLink [Verhagen, 2005]. This
algorithm detected all known inconsistencies in TimeBank, and found one more;
full details are later in Section 5.2. A test TimeML corpus is included with
CAVaT for verifying that the consistency checker works, which alternative
implementations may use for validation.
Table 2: Mapping from TimeML relation types to a simple point-based temporal algebra. The TimeML relation is of the form $a$ relation $b$. Where multiple relations are given, all hold. Similar to the table listed in [Verhagen, 2005]. TimeML relation type | Relation added to agenda
---|---
before | $a_{end}<b_{start}$
after | $b_{end}<a_{start}$
iafter | $b_{end}=a_{start}$
ibefore | $a_{end}=b_{start}$
includes | $a_{start}<b_{start}$, $b_{end}<a_{end}$
is_included | $b_{start}<a_{start}$, $a_{end}<b_{end}$
begins | $a_{start}=b_{start}$, $a_{end}<b_{end}$
begun_by | $a_{start}=b_{start}$, $b_{end}<a_{end}$
ends | $a_{end}=b_{end}$, $b_{start}<a_{start}$
ended_by | $b_{end}=a_{end}$, $a_{start}<b_{start}$
simultaneous | $a_{start}=b_{start}$, $a_{end}=b_{end}$
identity | $a_{start}=b_{start}$, $b_{end}=a_{end}$
during | $a_{start}=b_{start}$, $a_{end}=b_{end}$
during_inv | $a_{start}=b_{start}$, $a_{end}=b_{end}$
Below is sample output from a consistency check:
cavat> check consistent in 3
# Temporal graph consistency checker v1 loaded
# Checking wsj_0927.tml (id 3)
! Inconsistent closure - could not assert
(ei2415_2 < ei2414_1)
#### 2.3.2. Disconnected sub-graph detection
After inferring a temporal closure [Verhagen, 2005] of a document, one is
usually left with a single interconnected temporal graph, where nodes are
TIMEX3s or EVENTs and edges represent TLINKs. However, disconnected groups of
links may exist post-closure. This should be brought to the attention of the
user; it often suggests that annotating a small number of additional links can
greatly increase the amount of data inferable though closure, and that an
annotation is incomplete.
CAVaT’s sub-graph identification module, split_graph, works by processing
TLINKs from a document sequentially. We maintain a list of sets that will hold
interconnected intervals, beginning with an empty list. For each TLINK, we
check to see if either of its arguments (which are both intervals) can be
found in any set in our list. If one argument can but the other cannot, the
new interval is added to the same set as the found interval. If they are both
found in the same set, no action is taken. If they are found in different
sets, those two sets are merged. If neither TLINK argument can be found
anywhere, a new set holding both intervals is created. This process is
repeated until all TLINKs have been processed, at which point each set in the
list represents an independent sub-graph of connected intervals.
The module will then report statistics about the graph(s) found in the
specified document. These include:
Count of sub-graphs, intervals and TLINKs;
The number of “isolated” sub-graphs – that is, those described by only one
temporal link – and the proportion of intervals/links used to describe all
these isolated sub-graphs;
Mean and maximum sub-graph size, and the proportion of the document’s
intervals that are in the largest sub-graph;
The entropy of sub-graph sizes, which acts as a “fracturedness” measure,
showing how far the document is from having one single totally connected
temporal graph including all TLINKs;
The distribution of sub-graph sizes.
Even though sub-graphs are populated by processing the two intervals of a
TLINK at the same time, it is possible to have a sub-graph containing just one
node, in the case of a TLINK loop (Section 2.3.3.). Note that a document
containing intervals but no temporal links between them is marked as “un-
fractured”, as this check ignores any items not referenced at least once by a
temporal link.
Here is sample output from an attempt to identify disconnected sub-graphs:
cavat> check split_graph in 3
# Split graph detection v1 loaded
# Checking wsj_0927.tml (id 3)
Subgraphs found: 13 - composed of 69 nodes and linked
by 65 TLINKS.
Isolated subgraphs, that contain just one TLINK: 5
(making up 38.5% of all subgraphs / consuming 14.5%
of all nodes / described by 7.7% of all TLINKs);
Mean graph size 5.3 nodes; largest subgraph (size 35)
has 50.7% of all nodes.
Entropy of subgraph sizes: 0.448277644573
2 nodes: ( 5) .....
3 nodes: ( 4) ....
4 nodes: ( 3) ...
35 nodes: ( 1) .
#### 2.3.3. Superfluous TLINKs
Some TLINKs in TimeML corpora have been specified that associate an event with
itself. For example:
<TLINK lid="l67" relType="IDENTITY" eventInstanceID="ei1241"
relatedToEventInstance="ei1241" />
In this case, the only information conveyed is that ei1241 is identical to
itself, making this a redundant TLINK. CAVaT includes a check that will
identify TLINKs where both arguments reference the same event instance or
event. Although such TLINK loops might be detected by consistency checking,
those which specify a SIMULTANEOUS or IDENTITY relation will not.
Below is sample output, showing some superfluous TLINKs:
cavat> check tlink_loop in 165 159 143
# TLINK loop checker v1 loaded
# Checking ABC19980304.1830.1636.tml (id 165)
TLINK ID l23 may be a loop (eventID match), type
INCLUDES, event ei286 / ei288 - check document
manually
# Checking wsj_1013.tml (id 159)
TLINK ID l107 loops directly (instanceID match), type
IDENTITY, event ei2495 / ei2495
# Checking wsj_0586.tml (id 143)
TLINK ID l192 loops directly (instanceID match), type
BEFORE, event ei1404 / ei1404
#### 2.3.4. Orphaned object details
There is not yet a definition for TimeML annotation completeness, that states
a minimal satisfactory level of annotation for a document. In the absence of
such a definition, it is not a mistake to annotate entities without attaching
them to anything else in the document. However, we believe that wherever
possible, every interval should be connected to at least one other interval,
and that the annotation of entities that do not contribute or relate to any
other annotated information is superfluous. For example, if one chooses to
mark text as a temporal signal, a related link or event instance should
reference the signal. In this example, if the signal conveys no temporal
information, it should not be annotated.
To this end, CAVaT includes a module that is aware of five cases which
describe objects attached to nothing else, and reports such orphan objects.
Firstly, any TIMEX3 that is not related by any link is deemed to be
independent. Also, any event instance (from MAKEINSTANCE) that is not
referenced by a link is also orphaned. Next, an EVENT that is never
instantiated is unattached, as instantiation is required by current TimeML
syntax before EVENTs can be linked to anything else. Instances that come from
non-existent or mislabelled EVENTs are also orphans. Finally, SIGNALs that are
not referenced by any link or event instance (as in our example above) are
included in the list of orphaned objects.
Here is the sample output from a check for orphans:
cavat> check orphans in wsj_0927.tml
# Orphaned tag detection v1 loaded
TIMEX3 t104 not in any link
TIMEX3 t131 not in any link
### 2.4. Limitations
CAVaT is currently limited in the number of objects (based on TimeML tags)
that it can store for a single corpus. Objects are stored in MySQL tables, and
these are limited by the operating system’s maximum file size limit. The
maximum number of corpora that CAVaT can stored is restricted to the operating
system limit of files in a single directory.
## 3\. Syntax
Here we briefly introduce CAVaT’s basic top-level commands, and some of their
more useful features. A full specification of CAVaT’s syntax is available at
http://code.google.com/p/cavat.
### 3.1. Corpus manipulation
Commands for manipulating TimeML corpora within CAVaT begin with corpus. One
may view a list of available corpora with corpus list, and use a name from the
resulting list to select a corpus for querying or checking with the corpus use
command. It is also possible to view any notes attached to the currently
selected corpus by using corpus info. Before one can use a corpus, though, a
directory of TimeML files must be imported into CAVaT, using corpus import.
One may also opt to fold the corpus on import (see Section 2.1.); a note will
be attached to the database if this has been done.
### 3.2. Querying
The show command generates reports from the current corpus. Reports focus on
one tag type, and give information about its attributes. One can view all
values for a tag with “list” reports, or the distribution of values with
“distribution” reports, or simply see how many instances of that tag use a
particular field with “state” reports.
The general format for report generation is:
show <report type> of <tag> <field> [as <format>]
From the above example, <tag> corresponds to a TimeML tag, and is one of
event, instance, timex3, signal, tlink, slink or alink. As well as the
attributes available for each tag, the following extra values for <field> are
available:
For TLINKs, signaltext refers to the text enclosed by the start and end tags
of an associated signal;
For EVENTs, one may reference all the attributes of a MAKEINSTANCE tag too;
In TLINKs, SLINKs and ALINKs the arguments are referred to as arg1 and arg2,
so that the CAVaT user does not have to worry about the implicit indication of
interval type present in attribute names.
Reports are available in multiple formats. These can be specified by adding as
<format> to the end of a show query.
screen \- The default choice, screen gives text formatted for display in a
fixed-width font.
csv \- Output as comma separated values.
tex \- TeX table format, including caption.
The TeX output of an example report, showing the distribution of TLINK
relTypes in TimeBank v1.2, can be generated with show distribution of tlink
reltype as tex and is shown in Table 3.
Table 3: Distribution of Tlink reltype Tlink reltype | Frequency | Proportion
---|---|---
BEFORE | 1408 | 21.9%
IS_INCLUDED | 1357 | 21.1%
AFTER | 897 | 14.0%
IDENTITY | 743 | 11.6%
SIMULTANEOUS | 671 | 10.5%
INCLUDES | 582 | 9.07%
DURING | 302 | 4.71%
ENDED_BY | 177 | 2.76%
ENDS | 76 | 1.18%
BEGUN_BY | 70 | 1.09%
BEGINS | 61 | 0.95%
IAFTER | 39 | 0.608%
IBEFORE | 34 | 0.53%
DURING_INV | 1 | 0.0156%
Total | 6418 |
One may also specify a subset of a corpus to be used for reporting, using a
simple where clause. For example, one may ask:
cavat> show state of tlink signalid where reltype is after
to see how many TLINKs of type after use a signal; or, one may ask:
cavat> show distribution of tlink reltype where signalid is not filled
to find out which relTypes are most likely in TLINKs that do not specify a
signal. As part of CAVaT’s goal to be easy to use and close to natural
language, there are multiple valid syntaxes for filled/unfilled attributes.
### 3.3. Browsing
The ability to examine annotated entities in a TimeML corpus is required as
part of investigative research. To enable this, CAVaT includes the browse
command.
Browsing allows the user to select a document (with browse doc, followed by a
document ID or filename), and then view any tag within that document.
Associated data is also shown; for example, if one browses an EVENT tag, any
related MAKEINSTANCE tags will also be listed. One may view the tag in three
formats – screen, the default; csv, as two rows of comma-separated values (the
first with attribute names as column headings); or timeml, giving valid TimeML
for the requested object. The syntax for these is the same as that for show
commands; simply append as <output type> to the browse command.
The document selected for browsing is also used as the default document for
checks, which are detailed in Section 2.3.
## 4\. Example tasks
Below are some examples of using CAVaT to address real research problems. All
are based on TimeBank v1.2.
### 4.1. Show all temporal links that employ a signal
As part of research toward better automatic TLINK annotation, we wanted to
know what proportion of TLINKs in a corpus had been annotated as employing a
signal.
cavat> show state of tlink signalid
Count State of Tlink signalid
===========================================
718 signalid filled (11.2%)
5700 signalid unfilled (88.8%)
The state keyword here treats signalID as having two states – filled or
unfilled. The TLINK’s signalid field will either be empty/absent or contain a
reference to a signal annotated in text; for this task, we do not care which
specific signal is being referenced.
### 4.2. Dealing with ambiguous “part of speech” values
Many instances of events in TimeBank assert pos="other". This is a problem
when, e.g., using WordNet to lemmatise event strings. The distribution in
Table 4 can be created with the command:
cavat> show distribution of event pos
Table 4: Distribution of Event part-of-speech Event pos | Frequency | Proportion
---|---|---
VERB | 5122 | 64.5%
NOUN | 2225 | 28.0%
OTHER | 299 | 3.77%
ADJECTIVE | 266 | 3.35%
PREPOSITION | 28 | 0.353%
Total | 7940 |
After this, we would like to view event text that is classified as other,
using the following query:
cavat> show list of event text where pos is other
#10.86
#39.8 million
#54.8 million
$1
$1.05
(truncated)
The result suggests that there are at least some numeric values for these
event tokens, as well as the more typical verbs. This led to the substitution
of all currency and numeric event strings with representative tokens, as a
feature for a CRF classifier, yielding a performance increase in TLINK
classification (in unpublished results).
### 4.3. Which signals does the before relation use?
Sometimes, particular relation types are strongly suggested by related
signals. To determine the signal texts used with before TLINKs, one may query:
cavat> show distribution of tlink signaltext where reltype is before
Table 5: Distribution of Tlink signal text when Reltype is “before” Signal text | Frequency | Proportion
---|---|---
before | 24 | 31.6%
Previously | 10 | 13.2%
by | 7 | 9.21%
already | 6 | 7.89%
Earlier | 6 | 7.89%
until | 5 | 6.58%
then | 4 | 5.26%
followed by | 2 | 2.63%
prior to | 2 | 2.63%
_Other signals, frequency 1_ | 10 | 13.2%
Total | 76 |
From the results in Table 5, we can see that the token “before” suggests a
before relation, but that the majority of annotated before relations do not
employ this signal (from Table 3, there are a total of 1408 such relations,
only 24 of which use the signal). This indicates that building a relation
classifier that relies solely on such signals will not be useful.
### 4.4. Superfluous TLINK checking
One may want to find instances where a link has been made between an entity
and itself. We have an error checking module for this, tlink_loop:
cavat> check tlink_loop in WSJ910225-0066.tml
TLINK ID l383 matches, type IS_INCLUDED, event ei1482
TLINK ID l376 matches, type AFTER, event ei1454
TLINK ID l345 matches, type AFTER, event ei1356
One can explicitly query in all to search the entire corpus for similar mis-
annotations.
## 5\. Validation of a sample corpus
As we can now load and process any TimeML corpus, and have a set of advanced
validation tests, it is logical to test existing TimeML annotated corpora and
examine them. In this section, we present the results of running CAVaT’s check
modules on TimeBank v1.2. This corpus is not new and has been amended and
improved by the community [Boguraev et al., 2007], so may contain many fewer
errors than freshly annotated documents.
### 5.1. Checking for loops
We used the tlink_loop module (Section 2.3.3.) on the corpus. This identifies
TLINKs where both arguments are the same event or event instance.
Of TimeBank’s 183 documents, 19 have at least one TLINK containing such a
loop, and there are 26 loops in total. Of these loops, 10 are on TLINKs of
type simultaneous or identity. Such TLINKs will not make a graph inconsistent,
but are certainly redundant. The remaining 16 loops of other types will cause
an inconsistency. All but one of the loops found are temporal links where both
arguments reference the same event instance; only one references two different
instances of the same event (TLINK L23, in document
ABC19980304.1830.1636.tml). The TimeML in question is as follows:
But they still have <EVENT eid="e28" class="I_ACTION">catching</EVENT> up to
do two hundred and thirty four Americans have <EVENT eid="e30"
class="OCCURRENCE">flown</EVENT> in space, only twenty six of them women.
<MAKEINSTANCE eventID="e30" eiid="ei286" tense="PRESENT" aspect="PERFECTIVE"
polarity="POS" cardinality="234" pos="VERB"/>
<MAKEINSTANCE eventID="e30" eiid="ei288" tense="PRESENT" aspect="PERFECTIVE"
polarity="POS" cardinality="26" pos="VERB"/>
<TLINK lid="l23" relType="INCLUDES" eventInstanceID="ei286"
relatedToEventInstance="ei288"/>
In this case, the annotation suggests that during the flying in space of 234
Americans, 26 women flew, which is a correct interpretation of the text. CAVaT
recommends the manual examination of eventID loops upon their detection. All
the other tags reported by this check indicate redundant or incorrect
annotations.
### 5.2. Checking for consistent graphs
Since the consistency checker uses a novel method (see Section 2.3.1), we
verified its output by comparing it with that of SputLink and CAVaT’s loop
detection, and finding explanations for every inconsistency. A small test
corpus of TimeML documents is also included with CAVaT for assuring the
accuracy of this tool.
SputLink would not report an inconsistency with a TLINK loop that was not of
type simultaneous or identity; many of the inconsistent documents were found
faulty by both SputLink and CAVaT. Some documents had an erroneous initial
TLINK configuration; most faults were subtler than this, and their discovery
required a closure attempt.
### 5.3. Checking for split graphs
The split_graph module checks for documents whose temporal graphs contain sets
disconnected TLINKs. No single document in TimeBank has a fully-connected
temporal graph, with a path traceable between every interval. The “best-
connected” document (least fractured) is wsj_0144.tml, which has 34 intervals
split into only two subgraphs; one containing 32 intervals, the other two.
The most fractured document is wsj_1033.tml, which is split into 12 sub-graphs
having a mean graph size of only 2.7 intervals (a single TLINK connected to no
other creates a graph of size 2). Despite having 32 intervals in total to
connect, the largest sub-graphs in this document include only 4 intervals.
### 5.4. Replication
The results above can be simply replicated by downloading CAVaT v1, gathering
a copy of TimeBank v1.2555LDC catalogue number LDC2006T08, importing the
data/timeml/ subdirectory of TimeBank, and running check _test_ in all in
CAVaT, where _test_ is the name of the desired test module.
## 6\. Conclusion and future work
We have described CAVaT, a language-independent tool which adds a layer of
abstraction between TimeML markup and human researchers, making data easier to
analyse, and patterns easier to spot. It also helps identify trouble spots in
annotations.
TimeML corpora are only available at this time in Romanian [Forăscu et al.,
2007] and English; this makes multilingual testing of the tool difficult.
However, the markup is not language-specific, and results are likely to be
equally useful across many languages; this may be shown using test corpora
released for TempEval 2666http://www.timeml.org/tempeval2/, which will include
English, Italian, Spanish, Chinese, Korean and French.
##### Future work
CAVaT may be able to provide repair suggestions. These may include fixes for
inconsistent graphs, as well as suggestions for missing fields based on
lexical resources, third-party tools or heuristics. The modular error checks
allow creation of an open database of TimeML validations, to help improve the
integrity of all TimeML corpora. Check modules that match the output of rule-
based high confidence tools such as S2T [Verhagen and Pustejovsky, 2008] can
be added.
The consistency checker is “a TimeML closure engine that uses the precise
relations behind the scenes” [Verhagen, 2005]. Therefore, it may be used to
empirically discover how often incorrect links are introduced in closure, when
compared with the existing leading closure tool, SputLink.
##### Acknowledgements
The first author would like to acknowledge the UK Engineering and Physical
Science Research Council for support in the form of a doctoral studentship,
and Marc Verhagen of Brandeis University for useful comments on the temporal
closure process.
## References
* Allen, 1983 J.F. Allen. 1983\. Maintaining Knowledge about Temporal Intervals. Communications of the ACM, 26(11).
* Boguraev et al., 2007 B. Boguraev, J. Pustejovsky, R. Ando, and M. Verhagen. 2007\. Timebank evolution as a community resource for timeml parsing. Language Resources and Evaluation, 41(1):91–115.
* Burnard and Dodd, 2003 L. Burnard and T. Dodd. 2003\. Xara: an XML aware tool for corpus searching. In Proceedings of Corpus Linguistics, pages 142–4.
* Forăscu et al., 2007 C. Forăscu, R. Ion, and D. Tufiş. 2007\. Semi-automatic Annotation of the Romanian TimeBank 1.2. In Proceedings of the RANLP Workshop on Computer-aided language processing, pages 978–954.
* Group, 2008 ISO－TimeML Working Group. 2008\. 24617-1: 2008. Language resources management–Semantic annotation framework (SemAF)–Part1: Time and events. ISO/TC 37/SC 4/WG 2.
* Hepple et al., 2007 M. Hepple, A. Setzer, and R. Gaizauskas. 2007\. USFD: preliminary exploration of features and classifiers for the TempEval-2007 tasks. In Proceedings of the 4th International Workshop on Semantic Evaluations, pages 438–441. Association for Computational Linguistics.
* Lapata and Lascarides, 2006 M. Lapata and A. Lascarides. 2006\. Learning sentence-internal temporal relations. Journal of Artificial Intelligence Research, 27(1):85–117.
* Mani et al., 2006 I. Mani, M. Verhagen, B. Wellner, C.M. Lee, and J. Pustejovsky. 2006\. Machine learning of temporal relations. In Proceedings of the 44th annual meeting of the Association for Computational Linguistics, page 760.
* Pustejovsky et al., 2003 J. Pustejovsky, P. Hanks, R. Sauri, A. See, R. Gaizauskas, A. Setzer, D. Radev, B. Sundheim, D. Day, L. Ferro, et al. 2003\. The TimeBank corpus. In Corpus Linguistics, pages 647–656.
* Setzer et al., 2005 A. Setzer, R. Gaizauskas, and M. Hepple. 2005\. The role of inference in the temporal annotation and analysis of text. Language Resources and Evaluation, 39(2):243–265.
* Tsang, 1987 E.P.K. Tsang. 1987\. The consistent labeling problem in temporal reasoning. In Proc. AAAI Conference, Seattle, pages 251–255.
* Verhagen and Pustejovsky, 2008 M. Verhagen and J. Pustejovsky. 2008\. Temporal processing with the TARSQI toolkit. Proceedings of CoLing: Posters and Demonstrations, pages 189–192.
* Verhagen, 2005 M. Verhagen. 2005\. Temporal closure in an annotation environment. Language Resources and Evaluation, 39(2):211–241.
* Vilain et al., 1989 M. Vilain, H. Kautz, and P. Van Beek. 1989\. Constraint propagation algorithms for temporal reasoning: A revised report. Readings in qualitative reasoning about physical systems, 373:381.
|
arxiv-papers
| 2012-03-22T17:45:39 |
2024-09-04T02:49:28.896977
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5051"
}
|
1203.5055
|
11institutetext: University of Sheffield, Regent Court, Sheffield S1 4DP, UK
# Using Signals to Improve Automatic Classification of Temporal Relations
Leon Derczynski Robert Gaizauskas
###### Abstract
Temporal information conveyed by language describes how the world around us
changes through time. Events, durations and times are all temporal elements
that can be viewed as intervals. These intervals are sometimes temporally
related in text. Automatically determining the nature of such relations is a
complex and unsolved problem. Some words can act as “signals” which suggest a
temporal ordering between intervals. In this paper, we use these signal words
to improve the accuracy of a recent approach to classification of temporal
links.
## 1 Introduction
The ability to order events, and the ability to determine which information is
valid at a given time, are important in practical NLP. Effective automation of
tasks such as summarisation and question answering require information
extraction methods that can interpret information about time stored in
documents.
One difficult problem in temporal information extraction is the ordering of
events. Although accurate event ordering has been the topic of much research
[1, 10, 5, 8], work using the temporal signals present in text – for example,
phrases such as _after_ , _for the duration of_ and _while_ – has been
limited, and often only yields a minimal benefit [11]. Clearly these words
contain temporal ordering information that human readers can access. This
paper investigates the augmentation of a recent, high-performance temporal
link classifier with information about temporal signals.
Our hypothesis is that signals provide information useful to TLINK
classification. We also present data on signal usage within a temporally
annotated corpus, in an attempt to gauge the likelihood of their being helpful
and establish an upper bound on performance. After replicating existing work
as a basis for comparison, we add signal-specific features and show how they
lead to an improvement in classifier performance.
In this paper, we begin by describing the temporal annotation schema we have
chosen to use (TimeML [12]) and provide a definition of temporal signals in
the context of this paper (Sect. 2. In Sect. 3, we describe firstly how
results from a previous experiment by Mani et al. [10] are replicated, and
then detail the introduction of signal information into our system. Following
this in Sect. 4 we detail our results, provide analysis in Sect. 5, and
conclude in Sect. 6.
## 2 Background
Here we will introduce the annotation used in this work, introduce problems
with temporal signals, and cover some of the relevant literature.
### 2.1 Temporal Annotation
In order to capture temporal information well, a sophisticated annotation
schema is required. We use the TimeML schema [12], which includes tags for
event and time expression annotation (<EVENT> and <TIMEX3> respectively), as
well as temporal relations between intervals (<TLINK>) and signal phrases
(<SIGNAL>). The two largest resources of TimeML annotated text are TimeBank
[13] and the AQUAINT TimeML corpus111Available for download from
http://timeml.org/site/timebank/timebank.html., which we merge to form a
corpus for this work.
### 2.2 Temporal Links
Temporal links (or TLINKs) describe a temporal relation between two intervals,
each of which is either an event or a time expression. Allen [2] describes a
set of relation types in terms of the interval endpoints. As our work is based
on TimeML-annotated data, we use the set of TimeML relations, which are
similar to Allen’s. Each temporal link can optionally reference a signal.
### 2.3 Signals
Signals in TimeML are used to indicate multiple occurrences of events
(temporal quantification) and also to mark words that indicate the type of
relation between two intervals. For event ordering we are only interested in
this latter use of signals. “A University Grammar of English” [14] lists a
subset of these words in Sect. 10.5, “Time Relaters”.
For example, in the sentence _John smiled after he ate_, the word _after_
specifies an event ordering. This example could be represented in TimeML as
follows:
John <EVENT id="e1"> smiled </EVENT> <SIGNAL id="s1"> after </SIGNAL>
he <EVENT id="e2"> ate </EVENT> .
<TLINK id="l1" eventID="e1" relatedToEvent="e2"
relType="AFTER" signalID="s1" />
TimeML allows us to associate text that suggests an event ordering (a signal)
with a TLINK. To avoid confusion, it is worthwhile clarifying our use of the
term “signal”. We use SIGNAL in capitals for tags of this name in TimeML, and
signal/signal word/signal phrase for a word or words in discourse that
describe the temporal ordering of an event pair. Examples of the signals found
in TimeBank are provided in Table 1. It is important to note that not every
occurrence of text such as _after_ is a temporal signal. What is not shown due
to space constraints is that a temporal signal such as _after_ may be used by
(for example) 39 TLINKs labelled after, 17 labelled before, and four labelled
includes; the signal text alone does not infer a single interpretation.
Phrase | Corpus freq. | Occurrences as signal | Likelihood of being signal
---|---|---|---
subsequently | 3 | 3 | 100%
after | 72 | 67 | 93%
’s | 10 | 8 | 80%
follows | 4 | 3 | 75%
before | 33 | 23 | 70%
until | 36 | 25 | 69%
during | 19 | 13 | 68%
as soon as | 3 | 2 | 67%
Table 1: A sample of phrases most likely to be annotated as a signal when they
occur in TimeBank, which occur more than once in the corpus. All corpus data
in this paper was provided by the CAVaT command-line tool [6].
### 2.4 Previous work
When temporally ordering events, it is intuitively likely that signal
information may be useful. The trend in previous automated TLINK
classification work has not been to directly target signals as a primary
source of ordering information, although other attributes of annotated TLINKs
and EVENTs have been exploited as training features. For example, the best
known automatic TimeML annotation tool (TARSQI [15]) performs no SIGNAL
annotation. Lapata and Lascarides [7] worked with signals, using a restricted
reference list of signal tokens instead of drawing signal text from human-
annotated data. This work was only on same-sentence temporal links. Their
accuracy at temporal relation classification was 70.7%. Bethard and Martin [3]
included some features that described signals, where the compl-word feature
(the signal text) was the 8th strongest in their set of features for temporal
relation classification. However, this work has a number of limitations.
First, it only uses the signal word and a simple relation type suggestion as
features. It is also restricted to verb-clause construction TLINKs. Finally,
The classifier only has to choose from a set of three TLINK classes (before,
overlap, after).
## 3 Method
To explore the question of whether signal information can be successfully
exploited for TLINK classification, we proceed as follows. First we re-
implement a well-known TLINK relation classifier with state-of-the-art
accuracy. Then we add various signal-related features to the classifier to
investigate their impact on classification performance. The approach we have
replicated as closely as possible is from Mani et al. [9]. In brief, the
method was as follows.
Firstly, the set of possible relation types was reduced by applying a mapping.
For example, as a before b and b after a describe the same ordering between
events a and b, we can flip the argument order in any after relation to
convert it to a before relation. This simplifies training data and provides
more examples per temporal relation class. Secondly, the following information
from each TLINK is used as features: event class, aspect modality, tense,
negation, event string for each event, as well as two boolean features
indicating whether both events have the same tense or same aspect. Thirdly, we
trained and evaluated the predictive accuracy of the maximum entropy
classifier from Carafe222Available at http://sourceforge.net/projects/carafe/.
using 10-fold cross-validation.
Corpus | Total TLINKs | With SIGNAL | Without SIGNAL
---|---|---|---
TimeBank v1.2 | 6418 | 718 | (11.2%) | 5700
AQUAINT TimeML v1.0 | 5365 | 178 | (3.3%) | 5187
ATC (combined) | 11783 | 896 | (7.6%) | 10887
ATC event-event | 6234 | 319 | (5.1%) | 5915
Table 2: TLINKs and signals in our data.
TLINK data came from the union of TimeBank v1.2a and the AQUAINT TimeML
corpora. As the corpus used in the previous work by Mani et al. (TimeBank
v1.2a) is not publicly available, we used TimeBank v1.2. This use of a
publicly-available version of TimeBank instead of a private custom version was
the only change from the previous method. In this work we only examine event-
event links, which make up 52.9% of all TLINKs in our corpus (See Table 2).
We will later (Sect. 3.2) add features that require data to be separated into
test and training sets, with more sophistication required than that available
in Carafe’s maximum entropy classifier; thus, as well as performing 10-fold
cross-validation (XV), we also split all event-event TLINKs into a training
set of 4156 instances and an evaluation set of 2078 instances.
| Predictive accuracy | Baseline
---|---|---
Mani et al. results | 61.79% | 51.6%
Replicated results with our tools (10-fold XV) | 60.32% | 53.34%
Replicated results with our tools (train/test) | 60.04% | 53.34%
Table 3: Results from replicating one of MITRE’s TLINK classification
experiments.
### 3.1 Replicating Previous Work
Table 3 shows results from replicating the previous experiment on event-event
TLINKs. The baseline listed is the most-common-class in the training data. We
achieved a similar score of 60.32% accuracy compared to 61.79% in the previous
work. The differences may be attributed to the non-standard corpus that they
use. The TLINK distribution over a merger of TimeBank v1.2 and the AQUAINT
corpus differs from that listed in the paper.
### 3.2 Introducing Signals to the Feature Set
To add information about signals to our training instances, we use the extra
features described below; the two arguments of a TLINK are represented by e1
and e2.
Signal phrase. This shows the actual text that was marked up as a SIGNAL. From
this, we can start to guess temporal orderings based on signal phrases.
However, just using the phrase is insufficient. For example, the two sentences
_Run before sleeping_ and _Before sleeping, run_ are temporally equivalent, in
that they both specify two events in the order run-sleep, signalled by the
same word _before_.
Textual order of e1/e2. The textual ordering of linked events can be reversed
without affecting temporal order. Thus, it is important to know the textual
order of events and their signals even when we know a temporal ordering. This
feature assumes that the order event-signal-event is most prevalent in text;
values are either e1-e2 or e2-e1.
Textual order of signal and e1, signal and e2. These features describe the
textual ordering of both TLINK arguments and a related signal. It will also
help us see how the arguments of TLINKs that employ a particular signal tend
to be textually distributed.
Textual distance between e1/e2. Sentence and token count between e1 and e2.
Textual distance from e1/e2 to SIGNAL. If we allow a signal to influence the
classification of a TLINK, we need to be certain of its association with the
link’s events. Distances are measured in tokens.
TLINK class given SIGNAL phrase. Most likely TLINK classification in the
training data given this signal phrase (or empty if the phrase has not been
seen). Referred to as signal hint.Referred to as signal hint.
## 4 Results
Moving to a feature set which adds SIGNAL information, including signal-event
word order/distance data, 61.46% predictive accuracy is reached. The increase
is small when compared to 60.32% accuracy without this information, but TLINKs
that employ a SIGNAL in are a minority in our corpus (possibly due to under-
annotation). It would be interesting to see the performance difference when
classifying only TLINKs that use a SIGNAL.
There are in total 11783 TLINKs in the combined corpus, of which 7.6% are
annotated including a SIGNAL; for just TimeBank v1.2, the figure is higher at
11.2% (see Table 2). The proportion of signalled TLINKs in our data is lowest
at 5.1%.
Predictive accuracy | XV | Split
---|---|---
Baseline (most common class) | 53.34% | 53.34%
Without signal features | 60.32% | 60.04%
With basic signal features | 61.46% | 60.81%
With signal features including hint | n/a | 61.98%
Table 4: TLINK classification with and without signal features, using both
10-fold cross validation and a one-third/two-thirds split between evaluation
and training data.
The results of extending the feature set over a split of signalled and un-
signalled links is shown in Table 5, from a one-third/two-thirds
evaluation/training split.
Predictive accuracy | Unsignalled links | Signalled links
---|---|---
Baseline | 52.68% | 64.21%
Plain features | 62.05% | 55.65%
Plain + signal features | 62.05% | 69.57%
Plain + signal features + hint | 62.05% | 41.72%
Table 5: Predictive accuracy from Carafe’s maximum entropy classifier, using
features that do or do not include signal information, over signalled and non-
signalled TLINKs in ATC. The baseline is accuracy when the most-common-class
is always assigned.
## 5 Analysis
From Table 1 we can estimate the probability that a word or word sequence can
be annotated as a SIGNAL associated with a TLINK. This may be of use when
annotating signals, especially in the AQUAINT TimeML corpus. In any case,
given that our feature set might only be helpful to 5.1% of event-event links
in the ATC corpus (Table 2), the maximum performance increase at predicting
signalled links can be estimated.
Let us suppose that we have perfect signal discrimination and association.
Suppose our extra features do not help TLINKs without SIGNALs, and that the
increase in performance is due solely to better accuracy classification of
TLINKs that use signals. Let accuracy at classifying this signalled minority
be $a$. Given a proportion of signalled TLINKs $s$, and predictive accuracy of
our classifier when using features that do not depend on signals $P_{n}$ (from
Table 4):
$\displaystyle P_{n}(1-s)+as=0.6146$ (1) $\displaystyle\mathbf{a=0.8381}$ (2)
Thus, we may be classifying signalled TLINKs at over 80% accuracy when using
the augmented features. This indicates a significant increase in predictive
accuracy for signalled event-event TLINKs from the previous accuracy of
60.32%. This is a target for classification of signal-employing TLINKs.
It is hard to determine an external upper bound for the classification of
signal-employing TLINKs because inter-annotator agreement (IAA) figures are
only available for TimeBank, and not at this level of detail. However, we can
see from [4] that TLINK IAA reached 0.55. One would have to refer to the
original annotator data and identify those TLINKs which were marked as
employing a signal to determine an IAA value just for TLINKs with an
associated SIGNAL. IAA for signals was 0.77.
We have hypothesised that adding features to represent signals in TLINK
classification will lead to an increase in predictive accuracy. To test this,
we repeat the above experiment, which compared features includign and
excluding signal information. Data was divided into TLINKs that employ a
signal, and those that do not. We expected to see similar prediction accuracy
from both feature sets when classifying TLINKs that do not use signals. The
baseline was the most common class in the dataset.
If there is no performance difference between feature sets when classifying
TLINKs that _do_ use signals our hypothesis is incorrect, or the features we
used are bad representation. If signals are helpful, and our features capture
information useful for temporal ordering, we expect a performance difference
when evaluating signalled TLINKs. Results in Table 5 support our hypothesis
that signals are useful, but we are performing nowhere near the maximum level
suggested above. Data sparsity is a problem here, as the combined corpus only
contains 319 suitable TLINKs, and both source corpora evidence of signal
under-annotation. The results also suggest that the signal hint feature was
not helpful; this is the same result found in [3].
Exploring the strongest feature set (basic+signals; no hint), attempting to
combat the data sparsity problem, we used 10-fold XV instead of a split;
results are in Table 6. This shows a distinct improvement in the predictive
accuracy of signalled TLINKs using this feature set over the features in
previous work.
Predictive accuracy | Baseline | Plain features | Plain and signal features
---|---|---|---
Unsignalled links | 52.68% | 61.81% | 61.81%
Only signalled links | 62.41% | 60.32% | 82.19%
Table 6: TLINK predictive accuracy using 10-fold cross validation over
signalled and non-signalled TLINKs
## 6 Conclusion
When learning to classify signalled TLINKs, there is a significant increase in
predictive accuracy when features describing signals are used. This suggests
that signals are useful when it comes to providing information for classifying
temporal links, and also that the features we have used to describe them are
effective.
Future work is focused on improving signal and TLINK annotation. We need to
explore how to discriminate whether or not a string is used as a temporal
signal in text. Next, after finding a temporal signal, we need to determine
which intervals it temporally connects. Finally, we can attempt to annotate a
temporal link based on the signal. Once finished, we can integrate all this
into existing temporal annotation tools.
## References
* [1] D. Ahn, S.F. Adafre, and MD Rijke. Towards task-based temporal extraction and recognition. In Dagstuhl Seminar Proceedings, volume 5151, 2005.
* [2] J.F. Allen. Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11):832–843, 1983.
* [3] S. Bethard, J.H. Martin, and S. Klingenstein. Timelines from text: Identification of syntactic temporal relations. In International Conference on Semantic Computing, pages 11–18, 2007.
* [4] B. Boguraev, J. Pustejovsky, R. Ando, and M. Verhagen. TimeBank Evolution as a Community Resource for TimeML Parsing. Language Resources and Evaluation, 41(1):91–115, 2007.
* [5] N. Chambers and D. Jurafsky. Jointly combining implicit constraints improves temporal ordering. In Proceedings of the EMNLP, pages 698–706. ACL, 2008.
* [6] L. Derczynski and R. Gaizauskas. Analysing Temporally Annotated Corpora with CAVaT. In LREC, 2010.
* [7] M. Lapata and A. Lascarides. Inferring sentence-internal temporal relations. In Proceedings of the NAACL, 2004.
* [8] H. Llorens, E. Saquete, and B. Navarro. TIPSem (English and Spanish): Evaluating CRFs and Semantic Roles in TempEval-2. In Proceedings of SemEval-2010. ACL.
* [9] I. Mani, M. Verhagen, B. Wellner, C.M. Lee, and J. Pustejovsky. Machine learning of temporal relations. In Proceedings of the 44th annual meeting of the ACL, page 760. ACL, 2006.
* [10] I. Mani, B. Wellner, M. Verhagen, and J. Pustejovsky. Three approaches to learning TLINKS in TimeML. Technical report, CS-07-268, Brandeis University, 2007.
* [11] C. Min, M. Srikanth, and A. Fowler. LCC-TE: A hybrid approach to temporal relation identification in news text. In Proceedings of SemEval-2007, pages 219–222. ACL, 2007.
* [12] J. Pustejovsky, B. Ingria, R. Sauri, J. Castano, J. Littman, R. Gaizauskas, A. Setzer, G. Katz, and I. Mani. The Specification Language TimeML. The Language of Time: A Reader. Oxford University Press, 2004.
* [13] J. Pustejovsky, R. Sauri, R. Gaizauskas, A. Setzer, L. Ferro, et al. The TimeBank Corpus. In Corpus Linguistics, pages 647–656, 2003.
* [14] R. Quirk and S. Greenbaum. A University Grammar of English. London, 1973.
* [15] M. Verhagen and J. Pustejovsky. Temporal Processing with the TARSQI Toolkit. In Coling 2008: Posters and Demonstrations, pages 189–192, 2008\.
|
arxiv-papers
| 2012-03-22T17:50:08 |
2024-09-04T02:49:28.906321
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5055"
}
|
1203.5060
|
# USFD2: Annotating Temporal Expresions and TLINKs for TempEval-2
Leon Derczynski
Dept of Computer Science
University of Sheffield
Regent Court
211 Portobello
Sheffield S1 4DP, UK
leon@dcs.shef.ac.uk Robert Gaizauskas
Dept of Computer Science
University of Sheffield
Regent Court
211 Portobello
Sheffield S1 4DP, UK
robertg@dcs.shef.ac.uk
###### Abstract
We describe the University of Sheffield system used in the TempEval-2
challenge, USFD2. The challenge requires the automatic identification of
temporal entities and relations in text.
USFD2 identifies and anchors temporal expressions, and also attempts two of
the four temporal relation assignment tasks. A rule-based system picks out and
anchors temporal expressions, and a maximum entropy classifier assigns
temporal link labels, based on features that include descriptions of
associated temporal signal words. USFD2 identified temporal expressions
successfully, and correctly classified their type in 90% of cases. Determining
the relation between an event and time expression in the same sentence was
performed at 63% accuracy, the second highest score in this part of the
challenge.
## 1 Introduction
The TempEval-2 [Pustejovsky and Verhagen (2009] challenge proposes six tasks.
Our system tackles three of these: task A – identifying time expressions,
assigning TIMEX3 attribute values, and anchoring them; task C – determining
the temporal relation between an event and time in the same sentence; and task
E – determining the temporal relation between two main events in consecutive
sentences. For our participation in the task, we decided to employ both rule-
and ML-classifier-based approaches. Temporal expressions are dealt with by
sets of rules and regular expressions, and relation labelling performed by
NLTK’s111See http://www.nltk.org/ . maximum entropy classifier with rule-based
processing applied during feature generation. The features (described in full
in Section 2) included attributes from the TempEval-2 training data
annotation, augmented by features that can be directly derived from the
annotated texts. There are two main aims of this work: (1) to create a rule-
based temporal expression annotator that includes knowledge from work
published since GUTime [Mani and Wilson (2000] and measure its performance,
and (2) to measure the performance of a classifier that includes features
based on temporal signals.
Our entry to the challenge, USFD2, is a successor to USFD [Hepple et al.
(2007]. In the rest of this paper, we will describe how USFD2 is constructed
(Section 2), and then go on to discuss its overall performance and the impact
of some internal parameters on specific TempEval tasks. Regarding classifiers,
we found that despite using identical feature sets across relation
classification tasks, performance varied significantly. We also found that
USFD2 performance trends with TempEval-2 did not match those seen when
classifiers were trained on other data while performing similar tasks. The
paper closes with comments about future work.
## 2 System Description
The TempEval-2 training and test sets are partitioned into data for entity
recognition and description, and data for temporal relation classification. We
will first discuss our approach for temporal expression recognition,
description and anchoring, and then discuss our approach to two of the
relation labelling tasks.
### 2.1 Identifying, describing and anchoring temporal expressions
Task A of TempEval-2 requires the identification of temporal expressions (or
timexes) by defining a start and end boundary for each expression, and
assigning an ID to it. After this, systems should attempt to describe the
temporal expression, determining its type and value (described below).
Our timex recogniser works by building a set of n-grams from the data to be
annotated ($1\leq n\leq 5$), and comparing each n-gram against a hand-crafted
set of regular expressions. This approach has been shown to achieve high
precision, with recall increasing in proportion to ruleset size [Han et al.
(2006, Mani and Wilson (2000, Ahn et al. (2005]. The recogniser chooses the
largest possible sequence of words that could be a single temporal expression,
discarding any sub-parts that independently match any of our set of regular
expressions. The result is a set of boundary-pairs that describe temporal
expression locations within documents. This part of the system achieved 0.84
precision and 0.79 recall, for a balanced f1-measure of 0.82.
The next part of the task is to assign a type to each temporal expression.
These can be one of TIME, DATE, DURATION, or SET. USFD2 only distinguishes
between DATE and DURATION timexes. If the words _for_ or _during_ occur in the
three words before the timex, the timex ends with an _s_ (such as in _seven
years_), or the timex is a bi-gram whose first token is _a_ (e.g. in _a
month_), then the timex is deemed to be of type DURATION; otherwise it is a
DATE. These three rules for determining type were created based on observation
of output over the test data, and are correct 90% of the time with the
evaluation data.
The final part of task A is to provide a value for the timex. As we only
annotate DATEs and DURATIONs, these will be either a fixed calendrical
reference in the format YYYY-MM-DD, or a duration in according to the TIMEX2
standard [Ferro et al. (2005]. Timex strings of _today_ or _now_ were assigned
the special value PRESENT_REF, which assumes that _today_ is being used in a
literal and not figurative manner, an assumption which holds around 90% of the
time in newswire text [Ahn et al. (2005] such as that provided for TempEval-2.
In an effort to calculate a temporal distance from the document creation time
(DCT), USFD2 then checks to see if numeric words (e.g. _one_ , _seven
hundred_) are in the timex, as well as words like _last_ or _next_ which
determine temporal offset direction. This distance figure supplies either the
second parameter to a DURATION value, or helps calculate DCT offset. Strings
that describe an imprecise amount, such as _few_ , are represented in duration
values with an X, as per the TIMEX2 standard. We next search the timex for
temporal unit strings (e.g. _quarter_ , _day_). This helps build either a
duration length or an offset. If we are anchoring a date, the offset is
applied to DCT, and date granularity adjusted according to the coarsest
temporal primitive present – for example, if DCT is 1997-06-12 and our timex
is _six months ago_ , a value of 1997-01 is assigned, as it is unlikely that
the temporal expression refers to the day precisely six months ago, unless
followed by the word _today_.
Where weekday names are found, we used Baldwin’s 7-day window [Baldwin (2002]
to anchor these to a calendrical timeline. This technique has been found to be
accurate over 94% of the time with newswire text [Mazur and Dale (2008]. Where
dates are found that do not specify a year or a clear temporal direction
marker (e.g., _April 17_ vs. _last July_), our algorithm counts the number of
days between DCT and the next occurrence of that date. If this is over a limit
$f$, then the date is assumed to be last year. This is a very general rule and
does not take into account the tendency of very-precisely-described dates to
be closer to DCT, and far off dates to be loosely specified. An $f$ of 14 days
gives the highest performance based on the TempEval-2 training data.
Anchoring dates / specifying duration lengths was the most complex part of
task A and our naïve rule set was correct only 17% of the time.
### 2.2 Labelling temporal relations
Table 1: Features used by USFD2 to train a temporal relation classifier. Feature | Type
---|---
_For events_ |
Tense | String
Aspect | String
Polarity | pos or neg
Modality | String
_For timexes_ |
Type | Timex type
Value | String
_Describing signals_ |
Signal text | String
Signal hint | Relation type
Arg 1 before signal? | Boolean
Signal before Arg 2? | Boolean
_For every relation_ |
Arguments are same tense | Boolean
Arguments are same aspect | Boolean
Arg 1 before Arg 2? | Boolean
_For every interval_ |
Token number in sentence / 5 | Integer
Text annotated | String
Interval type | event or timex
Our approach for labelling temporal relations (or TLINKs) is based on NLTK’s
maximum entropy classifier, using the feature sets initially proposed in ?).
Features that describe temporal signals have been shown to give a 30%
performance boost in TLINKs that employ a signal [Derczynski and Gaizauskas
(2010]. Thus, the features in ?) are augmented with those used to describe
signals detailed in ?), with some slight changes. Firstly, as there are no
specific TLINK/signal associations in the TempEval-2 data (unlike TimeBank
[Pustejovsky et al. (2003]), USFD2 needs to perform signal identification and
then associate signals with a temporal relation between two events or timexes.
Secondly, a look-up list is used to provide TLINK label hints based on a
signal word. A list of features employed by USFD2 is in Table 1.
We used a simplified version of the approach in ?) to identify signal words.
This involved the creation of a list of signal phrases that occur in TimeBank
with a frequency of 2 or more, and associating a signal from this list with a
temporal entity if it is in the same sentence and clause. The textually
nearest signal is chosen in the case of conflict.
Table 2: A sample of signals and the TempEval-2 temporal relation they suggest. Signal phrase | Suggested relation
---|---
previous | after
ahead of | before
so far | overlap
thereafter | before
in anticipation of | before
follows | after
since then | before
soon after | after
as of | overlap-or-after
throughout | overlap
As this list of signal phrases only contained 42 entries, we also decided to
define a “most-likely” temporal relation for each signal. This was done by
imagining a short sentence of the form _event1 – signal – event2_ , and
describing the type of relation between event 1 and event 2. An excerpt from
these entries is shown in Table 2. The hint from this table was included as a
feature. Determining whether or not to invert the suggested relation type
based on word order was left to the classifier, which is already provided with
word order features. It would be possible to build these suggestions from data
such as TimeBank, but a number of problems stand in the way; the TimeML and
TempEval-2 relation types are not identical, word order often affects the
actual relationship type suggested by a signal (e.g. compare _He ran home
before he showered_ and _Before he ran home, he showered_), and noise in mined
data is a problem with the low corpus occurrence frequency of most signals.
This approach was used for both the intra-sentence timex/event TLINK labelling
task and also the task of labelling relations between main events in adjacent
sentences.
## 3 Discussion
USFD2’s rule-based element for timex identification and description performs
well, even achieving above-average recall despite a much smaller rule set than
comparable and more complex systems. However, the temporal anchoring component
performs less strongly. The “all-or-nothing” metric employed for evaluating
the annotation of timex values gives non-strict matches a zero score (e.g. if
the expected answer is 1990-05-14, no reward is given for 1990-05) even if
values are close, which many were.
In previous approaches that used a maximum entropy classifier and comparable
feature set [Mani et al. (2006, Derczynski and Gaizauskas (2010], the accuracy
of event-event relation classification was higher than that of event-timex
classification. Contrary to this, USFD2’s event-event classification of
relations between main events of successive sentences (Task E) was less
accurate than the classification of event-timex relations between events and
timexes in the same sentence (Task C). Accuracy in Task C was good (63%),
despite the lack of explicit signal/TLINK associations and the absence of a
sophisticated signal recognition and association mechanism. This is higher
than USFD2’s accuracy in Task E (45%) though the latter is a harder task, as
most TempEval-2 systems performed significantly worse at this task than
event/timex relation classification.
Signal information was not relied on by many TempEval 2007 systems (?)
discusses signals to some extent but the system described only includes a
single feature – the signal text), and certainly no processing of this data
was performed for that challenge. USFD2 begins to leverage this information,
and gives very competitive performance at event/timex classification. In this
case, the signals provided an increase from 61.5% to 63.1% predictive accuracy
in task C. The small size of the improvement might be due to the crude and
unevaluated signal identification and association system that we implemented.
The performance of classifier based approaches to temporal link labelling
seems to be levelling off – the 60%-70% relation labelling accuracy of work
such as ?) has not been greatly exceeded. This performance level is still the
peak of the current generation of systems. Recent improvements, while
employing novel approaches to the task that rely on constraints between
temporal link types or on complex linguistic information beyond that
describable by TimeML attributes, still yield marginal improvements (e.g. ?)).
It seems that to break through this performance “wall”, we need to continue to
innovate with and discuss temporal relation labelling, using information and
knowledge from many sources to build practical high-performance systems.
## 4 Conclusion
In this paper, we have presented USFD2, a novel system that annotates temporal
expressions and temporal links in text. The system relies on new hand-crafted
rules, existing rule sets, machine learning and temporal signal information to
make its decisions. Although some of the TempEval-2 tasks are difficult, USFD2
manages to create good and useful annotations of temporal information. USFD2
is available via Google Code222See http://code.google.com/p/usfd2/ ..
## Acknowledgments
Both authors are grateful for the efforts of the TempEval-2 team and
appreciate their hard work. The first author would like to acknowledge the UK
Engineering and Physical Science Research Council for support in the form of a
doctoral studentship.
## References
* [Ahn et al. (2005] D. Ahn, S.F. Adafre, and MD Rijke. 2005\. Towards task-based temporal extraction and recognition. In Dagstuhl Seminar Proceedings, volume 5151.
* [Baldwin (2002] J.A. Baldwin. 2002\. Learning temporal annotation of French news. Ph.D. thesis, Georgetown University.
* [Cheng et al. (2007] Y. Cheng, M. Asahara, and Y. Matsumoto. 2007\. Temporal relation identification using dependency parsed tree. In Proceedings of the 4th International Workshop on Semantic Evaluations, pages 245–248.
* [Derczynski and Gaizauskas (2010] L. Derczynski and R. Gaizauskas. 2010\. Using signals to improve automatic classification of temporal relations. In Proceedings of the ESSLLI StuS. Submitted.
* [Ferro et al. (2005] L. Ferro, L. Gerber, I. Mani, B. Sundheim, and G. Wilson. 2005\. TIDES 2005 standard for the annotation of temporal expressions. Technical report, MITRE.
* [Han et al. (2006] B. Han, D. Gates, and L. Levin. 2006\. From language to time: A temporal expression anchorer. In Temporal Representation and Reasoning (TIME), pages 196–203.
* [Hepple et al. (2007] M. Hepple, A. Setzer, and R. Gaizauskas. 2007\. USFD: preliminary exploration of features and classifiers for the TempEval-2007 tasks. In Proceedings of SemEval-2007, pages 438–441.
* [Mani and Wilson (2000] I. Mani and G. Wilson. 2000\. Robust temporal processing of news. In Proceedings of the 38th Annual Meeting on ACL, pages 69–76. ACL.
* [Mani et al. (2006] I. Mani, M. Verhagen, B. Wellner, C.M. Lee, and J. Pustejovsky. 2006\. Machine learning of temporal relations. In Proceedings of the 21st International Conference on Computational Linguistics, page 760. ACL.
* [Mazur and Dale (2008] P. Mazur and R. Dale. 2008\. What’s the date? High accuracy interpretation of weekday. In 22nd International Conference on Computational Linguistics (Coling 2008), Manchester, UK, pages 553–560.
* [Min et al. (2007] C. Min, M. Srikanth, and A. Fowler. 2007\. LCC-TE: a hybrid approach to temporal relation identification in news text. In Proceedings of the 4th International Workshop on Semantic Evaluations, pages 219–222.
* [Pustejovsky and Verhagen (2009] J. Pustejovsky and M. Verhagen. 2009\. SemEval-2010 task 13: evaluating events, time expressions, and temporal relations (TempEval-2). In Proceedings of the Workshop on Semantic Evaluations, pages 112–116. ACL.
* [Pustejovsky et al. (2003] J. Pustejovsky, P. Hanks, R. Sauri, A. See, R. Gaizauskas, A. Setzer, D. Radev, D. Day, L. Ferro, et al. 2003\. The Timebank Corpus. In Corpus Linguistics, volume 2003, page 40.
* [Yoshikawa et al. (2009] K. Yoshikawa, S. Riedel, M. Asahara, and Y. Matsumoto. 2009\. Jointly identifying temporal relations with markov logic. In IJCNLP: Proceedings of 47th Annual Meeting of the ACL, pages 405–413.
|
arxiv-papers
| 2012-03-22T17:59:22 |
2024-09-04T02:49:28.912555
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5060"
}
|
1203.5062
|
# An Annotation Scheme for Reichenbach’s Verbal Tense Structure
Leon Derczynski Robert Gaizauskas
Department of Computer Science
University of Sheffield, UK
{leon,robertg}@dcs.shef.ac.uk
###### Abstract
In this paper we present RTMML, a markup language for the tenses of verbs and
temporal relations between verbs. There is a richness to tense in language
that is not fully captured by existing temporal annotation schemata. Following
Reichenbach we present an analysis of tense in terms of abstract time points,
with the aim of supporting automated processing of tense and temporal
relations in language. This allows for precise reasoning about tense in
documents, and the deduction of temporal relations between the times and
verbal events in a discourse. We define the syntax of RTMML, and demonstrate
the markup in a range of situations.
## 1 Introduction
In his 1947 account, Reichenbach offered an analysis of the tenses of verbs,
in terms of abstract time points. Reichenbach details nine tenses (see Table
1). The tenses detailed by Reichenbach are past, present or future, and may
take a simple, anterior or posterior form. In English, these apply to single
verbs and to verbal groups (e.g. _will have run_ , where the main verb is
_run_).
To describe a tense, Reichenbach introduces three abstract time points.
Firstly, there is the speech time, $S$. This represents the point at which the
verb is uttered or written. Secondly, event time $E$ is the time that the
event introduced by the verb occurs. Thirdly, there is reference time $R$;
this is an abstract point, from which events are viewed. In Example 1, speech
time $S$ is when the author created the discourse (or perhaps when the reader
interpreted it). Reference time $R$ is _then_ – an abstract point, before
speech time, but after the event time $E$, which is the leaving of the
building. In this sentence, one views events from a point in time later than
they occurred.
_By then, she had left the building._
While we have rich annotation languages for time in discourse, such as
TimeML111http://www.timeml.org; ?). and TCNL222See ?)., none can mark the time
points in this model, or the relations between them. Though some may provide a
means for identifying speech and event times in specific situations, there is
nothing similar for reference times. All three points from Reichenbach’s model
are sometimes necessary to calculate the information used in these rich
annotation languages; for example, they can help determine the nature of a
temporal relation, or a calendrical reference for a time. We will illustrate
this with two brief examples.
_By April 26 ${}^{\textrm{th}}$, it was all over._
In Example 1, there is an anaphoric temporal expression describing a date. The
expression is ambiguous because we cannot position it absolutely without an
agreed calendar and a particular year. This type of temporal expression is
interpreted not with respect to speech time, but with respect to reference
time [Ahn et al., 2005]. Without a time frame for the sentence (presumably
provided earlier in the discourse), we cannot determine which year the date is
in. If we are able to set bounds for $R$ in this case, the time in Example 1
will be the April $26^{th}$ adjacent to or contained in $R$; as the word _by_
is used, we know that the time is the April $26^{th}$ following $R$, and can
normalise the temporal expression, associating it with a time on an absolute
scale.
Temporal link labelling is the classification of relations between events or
times. We might say an event of _the airport closed_ occurred after another
event of _the aeroplane landed_ ; in this case, we have specified the type of
temporal relation between two events. This task is difficult to automate
[Verhagen et al., 2010]. There are clues in discourse that human readers use
to temporally relate events or times. One of these clues is tense. For
example:
_John told me the news, but I had already sent the letter._
Example 1 shows a sentence with two verb events – _told_ and _had sent_. Using
Reichenbach’s model, these share their speech time $S$ (the time of the
sentence’s creation) and reference time $R$, but have different event times.
In the first verb, reference and event time have the same position. In the
second, viewed from when John told the news, the letter sending had already
happened – that is, event time is before reference time. As reference time $R$
is the same throughout the sentence, we know that the letter was sent before
John mentioned the news. Describing $S$, $E$ and $R$ for verbs in a discourse
and linking these points with each other (and with times) is the only way to
ensure correct normalisation of all anaphoric and deictic temporal
expressions, as well as enabling high-accuracy labelling of some temporal
links.
Some existing temporal expression normalisation systems heuristically
approximate reference time. GUTime [Mani and Wilson, 2000] interprets the
reference point as “the time currently being talked about”, defaulting to
document creation date. Over 10% of errors in this system were directly
attributed to having an incorrect reference time, and correctly tracking
reference time is the only way to resolve them. TEA [Han et al., 2006]
approximates reference time with the most recent time temporally before the
expression being evaluated, excluding noun-modifying temporal expressions;
this heuristic yields improved performance in TEA when enabled, showing that
modelling reference time helps normalisation. HeidelTime [Strötgen and Gertz,
2010] uses a similar approach to TEA but does not exclude noun-modifying
expressions.
The recently created WikiWars corpus of TIMEX2 annotated text prompted the
comment that there is a “need to develop sophisticated methods for temporal
focus tracking if we are to extend current time-stamping technologies” [Mazur
and Dale, 2010]. Resources that explicitly annotate reference time will be
direct contributions to the completion of this task.
?) describe how to relate events based on a “perspective” which is calculated
from the reference and event times of an event pair. They construct a natural
language generation system that requires accurate reference times in order to
correctly write stories. ?) also found reference point management critical to
medical summary generation.
These observations suggest that the ability to automatically determine
reference time for verbal expressions is useful for a number of computational
language processing tasks. Our work in this area – in which we propose an
annotation scheme including reference time – is a first step in this
direction.
In Section 2 we describe some crucial points of Reichenbach’s model and the
requirements of an annotation schema for tense in natural language. We also
show how to reason about speech, event and reference times. Then, in Section
3, we present an overview of our markup. In Section 4 we give examples of
annotated text (fictional prose and newswire text that we already have another
temporal annotation for), event ordering and temporal expression
normalisation. Finally we conclude in Section 5 and discuss future work.
## 2 Exploring Reichenbach’s model
Each tensed verb can be described with three points; speech time, event time
and reference time. We refer to these as $S$, $E$ and $R$ respectively. Speech
time is when the verb is uttered. Event time is when the action described by
the verb occurs. Reference time is a viewpoint from where the event is
perceived. A summary of the relative positions of these points is given in
Table 1.
While each tensed verb involves a speech, event and reference time, multiple
verbs may share one or more of these points. For example, all narrative in a
news article usually has the same speech time (that of document creation).
Further, two events linked by a temporal conjunction (e.g. _after_) are very
likely to share the same reference time.
_Relation_ | _Reichenbach’s Tense Name_ | _English Tense Name_ | _Example_
---|---|---|---
E$<$R$<$S | Anterior past | Past perfect | _I had slept_
E=R$<$S | Simple past | Simple past | _I slept_
R$<$E$<$S | Posterior past | | _I expected that .._
R$<$S=E | | | _I would sleep_
R$<$S$<$E | | |
E$<$S=R | Anterior present | Present perfect | _I have slept_
S=R=E | Simple present | Simple present | _I sleep_
S=R$<$E | Posterior present | Simple future | _I will sleep (Je vais dormir)_
S$<$E$<$R | Anterior future | Future perfect | _I will have slept_
S=E$<$R | | |
E$<$S$<$R | | |
S$<$R=E | Simple future | Simple future | _I will sleep (Je dormirai)_
S$<$R$<$E | Posterior future | | _I shall be going to sleep_
Table 1: Reichenbach’s tenses; from ?)
From Table 1, we can see that conventionally English only distinguishes six
tenses. Therefore, some English tenses will suggest more than one arrangement
of $S$, $E$ and $R$. Reichenbach’s tense names suffer from this ambiguity too,
but to a much lesser degree. When following Reichenbach’s tense names, it is
the case that for past tenses, $R$ always occurs before $S$; in the future,
$R$ is always after $S$; and in the present, $S$ and $R$ are simultaneous.
Further, “anterior” suggests $E$ before $R$, “simple” that $R$ and $E$ are
simultaneous, and “posterior” that $E$ is after $R$. The flexibility of this
model permits the full set of available tenses [Song and Cohen, 1988], and
this is sufficient to account for the observed tenses in many languages.
Our goal is to define an annotation that can describe $S$, $E$ and $R$
(speech, event and reference time) throughout a discourse. The lexical
entities that these times are attached to are verbal events and temporal
expressions. Therefore, our annotation needs to locate these entities in
discourse, and make the associated time points available.
### 2.1 Special properties of the reference point
The reference point $R$ has two special uses. When sentences or clauses are
combined, grammatical rules require tenses to be adjusted. These rules operate
in such a way that the reference point is the same in all cases in the
sequence. Reichenbach names this principle permanence of the reference point.
Secondly, when temporal expressions (such as a TimeML TIMEX3 of type DATE, but
not DURATION) occur in the same clause as a verbal event, the temporal
expression does not (as one might expect) specify event time $E$, but instead
is used to position reference time $R$. This principle is named positional use
of the reference point.
### 2.2 Context and the time points
In the linear order that events and times occur in discourse, speech and
reference points persist until changed by a new event or time. That is, the
reference time from one sentence will roll over to the next sentence, until it
is repositioned explicitly by a tensed verb or time. To cater for subordinate
clauses in cases such as reported speech, we add a caveat – $S$ and $R$
persist as a discourse is read in textual order, for each context. We can
define a context as an environment in which events occur, such as the main
body of the document, reported speech, or the conditional world of an _if_
clause [Hornstein, 1990]. For example:
_Emmanuel had said “This will explode!”, but changed his mind._
Here, _said_ and _changed_ share speech and reference points. Emmanuel’s
statement occurs in a separate context, which the opening quote instantiates,
ended by the closing quote (unless we continue his reported speech later), and
begins with an $S$ that occurs at the same time as _said_ ’s $E$. This
persistence must be explicitly stated in RTMML.
### 2.3 Capturing the time points with TimeML
TimeML is a rich, developed standard for temporal annotation. There exist
valuable resources annotated with TimeML that have withstood significant
scrutiny. However TimeML does not address the issue of annotating
Reichenbach’s tense model with the goal of understanding reference time or
creating resources that enable detailed examination of the links between
verbal events in discourse.
Although TimeML permits the annotation of tense for <EVENT>s, it is not
possible to unambiguously map its tenses to Reichenbach’s model. This
restricts how well we can reason about verbal events using TimeML-annotated
documents. Of the usable information for mapping TimeML annotations to
Reichenbach’s time points, TimeML’s tense attribute describes the relation
between $S$ and $E$, and its aspect attribute can distinguish between
PERFECTIVE and NONE – that is, between $E<R$ and a conflated class of
$(E=R)\vee(R<E)$. Cases where $R<E$ are often awkward in English (as in Table
1), and may even lack a distinct syntax; the French _Je vais dormir_ and _Je
dormirai_ both have the same TimeML representation and both translate to _I
will sleep_ in English, despite having different time point arrangements.
It is not possible to describe or build relations to reference points at all
in TimeML. It may be possible to derive the information about $S$, $E$ and $R$
directly represented in our scheme from a TimeML annotation, though there are
cases – especially outside of English – where it is not possible to capture
the full nuance of Reichenbach’s model using TimeML. An RTMML annotation
permits simple reasoning about reference time, and assist the labelling of
temporal links between verb events in cases where TimeML’s tense and aspect
annotation is insufficient. This is why we propose an annotation, and not a
technique for deriving $S$, $E$, and $R$ from TimeML.
## 3 Overview of RTMML
The annotation schema RTMML is intended to describe the verbal event structure
detailed in ?), in order to permit the relative temporal positioning of
reference, event, and speech times. A simple approach is to define a markup
that only describes the information that we are interested in, and can be
integrated with TimeML. For expositional clarity we use our own tags but it is
possible (with minor modifications) to integrate them with TimeML as an
extension to the standard.
Our procedure is as follows. Mark all times and verbal events (e.g. TimeML
TIMEX3s and those EVENTs whose lexical realisation is a verb) in a discourse,
as $T_{1}..T_{n}$ and $V_{1}..V_{n}$ respectively. We mark times in order to
resolve positional uses of the reference point. For each verbal event $V_{i}$,
we may describe or assign three time points $S_{i}$, $E_{i}$, and $R_{i}$.
Further, we will relate $T$, $S$, $E$ and $R$ points using disjunctions of the
operators $<$, $=$ and $>$. It is not necessary to define a unique set of
these points for each verb – in fact, linking them across a discourse helps us
temporally order events and track reference time. We can also define a
“discourse creation time,” and call this $S_{D}$.
_John said, “Yes, we have left”._ If we let _said_ be $V_{1}$ and _left_ be
$V_{2}$:
$S_{1}=S_{D}$
From the tense of $V_{1}$ (simple past), we can say:
$R_{1}=S_{1}$
$E_{1}<R_{1}$
As $V_{2}$ is reported speech, it is true that:
$S_{2}=E_{1}$
Further, as $V_{2}$ is anterior present:
$R_{2}=S_{2}$
$E_{2}<R_{2}$
As the $=$ and $<$ relations are transitive, we can deduce an event ordering
$E_{2}<E_{1}$.
### 3.1 Annotation schema
The annotation language we propose is called RTMML, for Reichenbach Tense
Model Markup Language. We use standoff annotation. This keeps the text
uncluttered, in the spirit of _ISO LAF_ and _ISO SemAF-Time_. Annotations
reference tokens by index in the text, as can be seen in the examples below.
Token indices begin from zero. We explicitly state the segmentation plan with
the <seg> element, as described in ?) and _ISO DIS 24614-1 WordSeg-1_.
The general speech time of a document is defined with the <doc> element, which
has one or two attributes: an ID, and (optionally) @time. The latter may have
a normalised value, formatted according to TIMEX3 [Boguraev et al., 2005] or
TIDES [Ferro et al., 2005], or simply be the string now.
Each <verb> element describes a tensed verbal group in a discourse. The
@target attribute references token offsets; it has the form target="#token0"
or target="#range(#token7,#token10)" for a 4-token sequence. Comma-separated
lists of offsets are valid, for situations where verb groups are non-
contiguous. Every verb has a unique value in its @id attribute. The tense of a
verb group is described using the attributes @view (with values _simple_ ,
_anterior_ or _posterior_) and @tense (_past_ , _present_ or _future_).
The <verb> element has optional @s, @e and @r attributes; these are used for
directly linking a verb’s speech, event or reference time to a time point
specified elsewhere in the annotation. One can reference document creation
time with a value of doc or a temporal expression with its id (for example,
t1). To reference the speech, event or reference time of other verbs, we use
hash references to the event followed by a dot and then the character s, e or
r; e.g., v1’s reference time is referred to as #v1.r.
As every tensed verb always has exactly one $S$, $E$ and $R$, and these points
do not hold specific values or have a position on an absolute scale, we do not
attempt to directly annotate them or place them on an absolute scale. One
might think that the relations should be expressed in XML links; however this
requires reifying time points when the information is stored in the relations
between time points, so we focus on the relations between these points for
each <verb>. To capture these internal relations (as opposed to relations
between the $S$, $E$ and $R$ of different verbs), we use the attributes se, er
and sr. These attributes take a value that is a disjunction of $<$, $=$ and
$>$.
Time-referring expressions are annotated using the <timerefx> element. This
has an @id attribute with a unique value, and a @target, as well as an
optional @value which works in the same was as the <doc> element’s @time
attribute.
Relation name | Interpretation
---|---
positions | $T_{a}=R_{b}$
same_timeframe | $R_{a}=R_{b}[,R_{c},..R_{x}]$
reports | $E_{a}=S_{b}$
Table 2: The meaning of a certain link type between verbs or times a and b.
<rtmml>
Yesterday, John ate well.
<seg type="token" />
<doc time="now" />
<timerefx xml:id="t1" target="
#token0" />
<verb xml:id="v1" target="#token3"
view="simple" tense="past"
sr=">" er="=" se=">"
r="t1" s="doc" />
</rtmml>
In this example, we have defined a time _Yesterday_ as t1 and a verbal event
_ate_ as v1. We have categorised the tense of v1 within Reichenbach’s
nomenclature, using the verb element’s @view and @tense attributes.
Next, we directly describe the reference point of v1, as being the same as the
time t1. Finally, we say that this verb is uttered at the same time as the
whole discourse – that is, $S_{v1}=S_{D}$. In RTMML, if the speech time of a
verb is not otherwise defined (directly or indirectly) then it is $S_{D}$. In
cases of multiple voices with distinct speech times, if a speech time is not
defined elsewhere, a new one may be instantiated with a string label; we
recommend the formatting _s_ , _e_ or _r_ followed by the verb’s ID.
This sentence includes a positional use of the reference point, annotated in
v1 when we say r="t1". To simplify the annotation task, and to verbosely
capture a use of the reference point, RTMML permits an alternative annotation
with the <rtmlink> element. This element takes as arguments a relation and a
set of times and/or verbs. Possible relation types are positions,
same_timeframe (annotating permanence of the reference point) and reports for
reported speech; the meanings of these are given in Table 2. In the above
markup, we could replace the <verb> element with the following:
<verb xml:id="v1" target="#token3"
view="simple" tense="past"
sr=">" er="=" se=">" s="doc" />
<rtmlink xml:id="l1" type="POSITIONS">
<link source="#t1" />
<link target="#v1" />
</rtmlink>
When more than two entities are listed as targets, the relation is taken as
being between an optional source entity and each of the target entities.
Moving inter-verbal links to the <rtmlink> element helps fulfil _TEI p5_ and
the _LAF_ requirements that referencing and content structures are separated.
Use of the <rtmlink> element is not compulsory, as not all instances of
positional use or permanence of the reference point can be annotated using it;
Reichenbach’s original account gives an example in German.
### 3.2 Reasoning and inference rules
Our three relations $<$, $=$ and $>$ are all transitive. A minimal annotation
is acceptable. The $S$, $E$ and $R$ points of all verbs, $S_{D}$ and all $T$s
can represent nodes on a graph, connected by edges labelled with the relation
between nodes.
To position all times in a document with maximal accuracy, that is, to label
as many edges in such a graph as possible, one can generate a closure by means
of deducing relations. An agenda-based algorithm is suitable for this, such as
the one given in ?).
### 3.3 Integration with TimeML
To use RTMML as an ISO-TimeML extension, we recommend that instead of
annotating and referring to <timerefx>s, one refers to <TIMEX3> elements using
their tid attribute; references to <doc> will instead refer to a <TIMEX3> that
describes document creation time. The attributes of <verb> elements (except
xml:id and target) may be be added to <MAKEINSTANCE> or <EVENT> elements, and
<rtmlink>s will refer to event or event instance IDs.
## 4 Examples
In this section we will give developed examples of the RTMML notation, and
show how it can be used to order events and position events on an external
temporal scale.
### 4.1 Annotation example
Here we demonstrate RTMML annotation of two short pieces of text.
#### 4.1.1 Fiction
From _David Copperfield_ by Charles Dickens: _When he had put up his things
for the night he took out his flute, and blew at it, until I almost thought he
would gradually blow his whole being into the large hole at the top, and ooze
away at the keys._
<doc time="1850" mod="BEFORE" />
<!-- had put -->
<verb xml:id="v1"
target="#range(#token2,#token3)"
view="anterior" tense="past" />
<!-- took -->
<verb xml:id="v2" target="#token11"
view="simple" tense="past" />
<!-- blew -->
<verb xml:id="v3" target="#token17"
view="simple" tense="past" />
<!-- thought -->
<verb xml:id="v4" target="#token24"
view="simple" tense="past" />
<!-- would gradually blow -->
<verb xml:id="v5"
target="#range(#token26,#token28)"
view="posterior" tense="past"
se="=" er=">" sr=">"
r="#v4.e" />
<!-- ooze -->
<verb xml:id="v6"
target="#range(#token26,#token28)"
view="posterior" tense="past"
se="=" er=">" sr=">" />
<rtmlink xml:id="l1"
type="SAME_TIMEFRAME">
<link target="#v1" />
<link target="#v2" />
<link target="#v3" />
<link target="#v4" />
</rtmlink>
<rtmlink xml:id="l2"
type="SAME_TIMEFRAME">
<link target="#v5" />
<link target="#v6" />
</rtmlink>
Figure 1: RTMML for a passage from David Copperfield.
We give RTMML for the first five verbal events from Example 4.1.1 RTMML in
Figure 1. The fifth, v5, exists in a context that is instantiated by v4; its
reference time is defined as such. We can use one link element to show that
v2, v3 and v4 all use the same reference time as v1. The temporal relation
between event times of v1 and v2 can be inferred from their shared reference
time and their tenses; that is, given that v1 is anterior past and v2 simple
past, we know $E_{v1}<R_{v1}$ and $E_{v2}=R_{v2}$. As our <rtmlink> states
$R_{v1}=R_{v2}$, then $E_{v1}<E_{v2}$. Finally, v5 and v6 happen in the same
context, described with a second same_timeframe link.
#### 4.1.2 Editorial news
From an editorial piece in TimeBank [Pustejovsky et al., 2003]
(AP900815-0044.tml): _Saddam appeared to accept a border demarcation treaty he
had rejected in peace talks following the August 1988 cease-fire of the eight-
year war with Iran._
<doc time="1990-08-15T00:44" />
<!-- appeared -->
<verb xml:id="v1" target="#token1"
view="simple" tense="past" />
<!-- had rejected -->
<verb xml:id="v2"
target="#range(#token9,#token10)"
view="anterior" tense="past" />
<rtmlink xml:id="l1"
type="SAME_TIMEFRAME">
<link target="#v1" />
<link target="#v2" />
</rtmlink>
Here, we relate the simple past verb _appeared_ with the anterior past (past
perfect) verb _had rejected_ , permitting the inference that the first verb
occurs temporally after the second. The corresponding TimeML (edited for
conciseness) is:
Saddam <EVENT eid="e74" class="I_STATE">
appeared</EVENT> to accept a border
demarcation treaty he had <EVENT eid="e77"
class="OCCURRENCE">rejected</EVENT>
<MAKEINSTANCE eventID="e74" eiid="ei1568"
tense="PAST" aspect="NONE" polarity="POS"
pos="VERB"/>
<MAKEINSTANCE eventID="e77" eiid="ei1571"
tense="PAST" aspect="PERFECTIVE"
polarity="POS" pos="VERB"/>
In this example, we can see that the TimeML annotation includes the same
information, but a significant amount of other annotation detail is present,
cluttering the information we are trying to see. Further, these two <EVENT>
elements are not directly linked, requiring transitive closure of the network
described in a later set of <TLINK> elements, which are omitted here for
brevity.
### 4.2 Linking events to calendrical references
RTMML makes it possible to precisely describe the nature of links between
verbal events and times, via positional use of the reference point. We will
link an event to a temporal expression, and suggest a calendrical reference
for that expression, allowing the events to be placed on a calendar. Consider
the below text, from wsj_0533.tml in TimeBank.
_At the close of business Thursday, 5,745,188 shares of Connaught and C$44.3
million face amount of debentures, convertible into 1,826,596 common shares,
had been tendered to its offer._
<doc time="1989-10-30" />
<!-- close of business Thursday -->
<timerefx xml:id="t1"
target="#range(#token2,#token5)" />
<!-- had been tendered -->
<verb xml:id="v1"
target="#range(#token25,#token27)"
view="anterior" tense="past" />
<rtmlink xml:id="l1" target="#t1 #v1">
<link target="#t1" />
<link target="#v1" />
</rtmlink>
This shows that the reference time of v1 is t1. As v1 is anterior, we know
that the event mentioned occurred before _close of business Thursday_.
Normalisation is not a task that RTMML addresses, but there are existing
methods for deciding which Thursday is being referenced given the document
creation date [Mazur and Dale, 2008]; a time of day for _close of business_
may be found in a gazetteer.
### 4.3 Comments on annotation
As can be seen in Table 1, there is not a one-to-one mapping from English
tenses to the nine specified by Reichenbach. In some annotation cases, it is
possible to see how to resolve such ambiguities. Even if view and tense are
not clearly determinable, it is possible to define relations between $S$, $E$
and $R$; for example, for arrangements corresponding to the simple future,
$S<E$. In cases where ambiguities cannot be resolved, one may annotate a
disjunction of relation types; in this example, we might say “$S<R$ or $S=R$”
with sr="<=".
Contexts seem to have a shared speech time, and the $S-R$ relationship seems
to be the same throughout a context. Sentences which contravene this (e.g.
_“By the time I ran, John will have arrived”_) are rather awkward.
RTMML annotation is not bound to a particular language. As long as a
segmentation scheme (e.g. WordSeg-1) is agreed and there is a compatible
system of tense and aspect, the model can be applied and an annotation
created.
## 5 Conclusion and Future Development
Being able to recognise and represent reference time in discourse can help in
disambiguating temporal reference, determining temporal relations between
events and in generating appropriately tensed utterances. A first step in
creating computational tools to do this is to develop an annotation schema for
recording the relevant temporal information in discourse. To this end we have
presented RTMML, our annotation for Reichenbach’s model of tense in natural
language.
We do not intend to compete with existing languages that are well-equipped to
annotate temporal information in documents; RTMML may be integrated with
TimeML. What is novel in RTMML is the ability to capture the abstract parts of
tense in language. We can now annotate Reichenbach’s time points in a document
and then process them, for example, to observe interactions between temporal
expressions and events, or to track reference time through discourse. This is
not directly possible with existing annotation languages.
There are some extensions to Reichenbach’s model of the tenses of verbs, which
RTMML does not yet cater for. These include the introduction of a reference
interval, as opposed to a reference point, from ?), and Comrie’s suggestion of
a second reference point in some circumstances [Comrie, 1985]. RTMML should
cater for these extensions.
Further, we have preliminary annotation tools and have begun to create a
corpus of annotated texts that are also in TimeML corpora. This will allow a
direct evaluation of how well TimeML can represent Reichenbach’s time points
and their relations. To make use of Reichenbach’s model in automatic
annotation, given a corpus, we would like to apply machine learning techniques
to the RTMML annotation task. Work in this direction should enable us to label
temporal links and to anchor time expressions with complete accuracy where
other systems have not succeeded.
## 6 Acknowledgements
The authors would like to thank David Elson for his valuable comments. The
first author would also like to acknowledge the UK Engineering and Physical
Science Research Council’s support in the form of a doctoral studentship.
## References
* [Ahn et al., 2005] D. Ahn, S.F. Adafre, and MD Rijke. 2005\. Towards task-based temporal extraction and recognition. In Dagstuhl Seminar Proceedings, volume 5151.
* [Boguraev et al., 2005] B. Boguraev, J. Castano, R. Gaizauskas, B. Ingria, G. Katz, B. Knippen, J. Littman, I. Mani, J. Pustejovsky, A. Sanfilippo, et al. 2005\. TimeML 1.2. 1: A Formal Specification Language for Events and Temporal Expressions.
* [Comrie, 1985] B. Comrie. 1985\. Tense. Cambridge University Press.
* [Dowty, 1979] D.R. Dowty. 1979\. Word meaning and Montague grammar. Kluwer.
* [Elson and McKeown, 2010] D. Elson and K. McKeown. 2010\. Tense and Aspect Assignment in Narrative Discourse. In Proceedings of the Sixth International Conference in Natural Language Generation.
* [Ferro et al., 2005] L. Ferro, L. Gerber, I. Mani, B. Sundheim, and G. Wilson. 2005\. TIDES 2005 standard for the annotation of temporal expressions. Technical report, MITRE.
* [for Standardization, 2009a] International Organization for Standardization. 2009a. ISO DIS 24612 LRM - Language Annotation Framework (LAF). ISO/TC 37/SC 4/WG 2.
* [for Standardization, 2009b] International Organization for Standardization. 2009b. ISO DIS 24614-1 LRM - Word Segmentation of Text - Part 1: Basic Concepts and General Principles (WordSeg-1). ISO/TC 37/SC 4/WG 2.
* [for Standardization, 2009c] International Organization for Standardization. 2009c. ISO DIS 24617-1 LRM - Semantic Annotation Framework - Part 1: Time and Events (SemAF-Time). ISO/TC 37/SC 4/WG 2.
* [Han et al., 2006] B. Han, D. Gates, and L. Levin. 2006\. From language to time: A temporal expression anchorer. In Temporal Representation and Reasoning (TIME), pages 196–203.
* [Hornstein, 1990] N. Hornstein. 1990\. As time goes by: Tense and universal grammar. MIT Press.
* [Lee and Romary, 2010] K. Lee and L. Romary. 2010\. Towards Interoperability of ISO Standards for Language Resource Management. In International Conference on Global Interoperability for Language Resources.
* [Mani and Wilson, 2000] I. Mani and G. Wilson. 2000\. Robust temporal processing of news. In Proceedings of the 38th Annual Meeting on ACL, pages 69–76. ACL.
* [Mani et al., 2005] I. Mani, J. Pustejovsky, and R. Gaizauskas. 2005\. The language of time: a reader. Oxford University Press, USA.
* [Mazur and Dale, 2008] P. Mazur and R. Dale. 2008\. What’s the date?: high accuracy interpretation of weekday names. In Proceedings of the 22nd International Conference on Computational Linguistics-Volume 1, pages 553–560. ACL.
* [Mazur and Dale, 2010] P. Mazur and R. Dale. 2010\. WikiWars: A New Corpus for Research on Temporal Expressions. In Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing, pages 913–922. ACL.
* [Portet et al., 2009] F. Portet, E. Reiter, A. Gatt, J. Hunter, S. Sripada, Y. Freer, and C. Sykes. 2009\. Automatic generation of textual summaries from neonatal intensive care data. Artificial Intelligence, 173(7-8):789–816.
* [Pustejovsky et al., 2003] J. Pustejovsky, P. Hanks, et al. 2003\. The TimeBank corpus. In Corpus Linguistics, volume 2003, page 40.
* [Reichenbach, 1947] H. Reichenbach. 1947\. The Tenses of Verbs. Elements of Symbolic Logic, pages 287–98.
* [Setzer et al., 2005] A. Setzer, R. Gaizauskas, and M. Hepple. 2005\. The role of inference in the temporal annotation and analysis of text. Language Resources and Evaluation, 39(2):243–265.
* [Song and Cohen, 1988] F. Song and R. Cohen. 1988\. The interpretation of temporal relations in narrative. In Proceedings of the 7th National Conference of AAAI.
* [Strötgen and Gertz, 2010] J. Strötgen and M. Gertz. 2010\. HeidelTime: High quality rule-based extraction and normalization of temporal expressions. In Proceedings of the 5th Workshop on Semantic Evaluation, pages 321–324. ACL.
* [Verhagen et al., 2010] M. Verhagen, R. Saurí, T. Caselli, and J. Pustejovsky. 2010\. SemEval-2010 task 13: TempEval-2. In Proceedings of the 5th Workshop on Semantic Evaluation, pages 57–62. ACL.
|
arxiv-papers
| 2012-03-22T18:05:26 |
2024-09-04T02:49:28.918440
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5062"
}
|
1203.5066
|
# A Corpus-based Study of Temporal Signals
###### Abstract
Automatic temporal ordering of events described in discourse has been of great
interest in recent years. Event orderings are conveyed in text via various
linguistic mechanisms including the use of expressions such as “before”,
“after” or “during” that explicitly assert a temporal relation – temporal
signals. In this paper, we investigate the role of temporal signals in
temporal relation extraction and provide a quantitative analysis of these
expressions in the TimeBank annotated corpus.
A Corpus-based Study of Temporal Signals
Leon Derczynski and Robert Gaizauskas
---
University of Sheffield
211 Portobello, S1 4DP, UK
L.Derczynski@dcs.shef.ac.uk, R.Gaizauskas@dcs.shef.ac.uk
Abstract content
## 1\. Introduction
The task of automatically determining the temporal relations that hold between
events described in a text is a research challenge that has increasingly
occupied researchers in computational language processing [Setzer and
Gaizauskas, 2000, Pustejovsky et al., 2004, Verhagen et al., 2009, Verhagen et
al., 2010]. The mechanisms used to convey temporal relational information in
text are complex and include tense, textual ordering, as well as specific
lexical cues; and of course readers and writers bring to bear lexical and
world knowledge, informing them of likely event sequences and inter-
relationships.
Of the mechanisms that play a part in conveying temporal relational
information, one that has been under-investigated is the use of expressions,
typically adverbials or conjunctions, which overtly signal temporal relations
– words or phrases such as after, during and as soon as. Very few of the teams
participating in the recent TempEval challenges [Verhagen et al., 2009,
Verhagen et al., 2010] exploited these words as features in their automated
temporal relation classification systems. Certainly no detailed study of these
words and their potential contribution to the task of temporal relation
detection has been carried out to date, despite their demonstrable utility
[Derczynski and Gaizauskas, 2010b]. This paper begins to address this
deficiency. Using the TimeBank corpus, a corpus of news wire texts annotated
with TimeML [Pustejovsky et al., 2003], in which a class of expressions
referred to as temporal signals is explicitly annotated, we set out to answer
the following questions:
1. 1.
What proportion of temporal relations annotated in TimeBank have an associated
temporal signal? That is, are explicitly signalled using a signal word or
phrase?
2. 2.
Of the expressions which can function as temporal signals, what proportion of
their usage in the TimeBank corpus is as a temporal signal? E.g. how ambiguous
are these expressions in terms of their role as temporal signals?
3. 3.
Of the occurrences of these expressions as temporal signals, how ambiguous are
they with respect to the temporal relation they convey?
The following paper provides provisional answers to these questions –
provisional as one of the difficulties we encountered was significant under-
annotation of temporal signals in TimeBank. We have addressed this to some
extent, but more work remains to be done. Nonetheless we believe the current
study provides important insights into the behaviour of temporal signals and
how they may be exploited by computational systems carrying out the temporal
relation detection task.
The remainder of the paper is divided into three parts. In section two we give
a more detailed characterisation of temporal signals, further describe
TimeBank and TimeML and discuss prior related work. In section three we
describe the additional annotation work we have done on TimeBank and present
the quantitative analysis that provides answers to the questions framed above.
The fourth section considers, on a case by case basis, specific examples of
expressions which are highly ambiguous as regards their role as temporal
signals and discusses their behaviour in detail.
## 2\. Temporal Signals
### 2.1. Linguistic Characterisation
Signal expressions explicitly indicate the existence and nature of a temporal
relation between two events or states or between an event or state and a time
point or interval. Hence a temporal signal has two arguments, which are the
temporal ”entities” that are related. One of these arguments may be deictic
instead of directly attached to an event or time; anaphoric temporal
references are also permitted. For example, the temporal function and
arguments of _after_ in _He slept after a long day at work_ are clear and
available in the immediately surrounding text. With After that, he swiftly
finished his meal and left we must look back to the antecedent of that to
locate the second argument.
Sometimes a signal will appear to be missing an argument; for example,
sentence-initial signals with only one event in the sentence (_“Later, they
subsided.”_). These relate an event in their sentence with the discourse’s
current temporal focus – for example, document creation time, the previous
sentence’s main event, or reference time [Reichenbach, 1947, Dowty, 1979]. In
a more complex case, such as Example 2.1., we suggest that two temporal links
are present. First, _Later_ is attached to the current focus, as is
_surveyed_. Secondly, _after_ describes the relation between _the storm_ and
_surveyed_.
_It rained heavily. Later, after the storm, we surveyed the damage._
Sometimes a signal may appear to only take one argument, when the other is
(implicitly) reference time. For example, _afterwards_ and _after that_ are
temporally equivalent, though _afterwards_ only takes one extra argument.
Signal surface forms have a compound structure consisting of a head and an
optional qualifier. The head describes the temporal operation of the signal
phrase and the qualifier modifies or clarifies this operation. An example of
an unqualified signal expression is _after_ , which provides information about
the nature of a temporal link, but does not say anything about the absolute or
relative magnitude of the temporal separation of its arguments. We can
elaborate on this with phrases which give qualitative information about the
relative size of temporal separation between events (such as _very shortly
after_), or which give a specific separation between events using a duration
as a modifying phrase (e.g. _two weeks after_).
### 2.2. TimeML and TimeBank
TimeML [Pustejovsky et al., 2004] is a temporal annotation language. It may be
used to annotate events, time expressions or timex’s (times, dates,
durations), temporal relations between events and times (such as before or
during), and signal expressions – words or phrases (such as conjunctions,
adverbials) that provide information about temporal relations. TimeBank
[Pustejovsky et al., 2003] is currently the largest TimeML-annotated gold
standard corpus available, including over 6 000 temporal relation annotations,
as well as events, times and signals. It consists of around 65 000 tokens of
English newswire text.
TimeML offers the following definition of temporal signal. From the annotation
guidelines111See http://timeml.org/site/publications/timeMLdocs
/annguide_1.2.1.pdf .:
> A signal is a textual element that makes explicit the relation holding
> between two entities (timex and event, timex and timex, or event and event).
> Signals are generally:
>
> * •
>
> Temporal prepositions: on, in, at, from, to, before, after, during, etc.
>
> * •
>
> Temporal conjunctions: before, after, while, when, etc.
>
> * •
>
> Prepositions signaling modality: to.
>
> * •
>
> Special characters: “-” and “/”, in temporal expressions denoting ranges.
>
>
In cases where a specific duration occurs as part of a complex qualifier-head
temporal signal, e.g. two weeks after, TimeBank has followed the convention
that the signal head alone is annotated as a signal and the qualifier is
annotated as a TIMEX of type duration.
### 2.3. Previous Work
Signals help create well-structured discourse. Temporal signals can provide
context shifts and orderings [Hitzeman, 1997]. These signal expressions
therefore work as discourse segmentation markers [Ho-Dac and Péry-Woodley,
2008]. It has been shown that correctly including such explicit markers make
texts easier for human readers to process [Bestgen and Vonk, 1999].
Some prior work has approached linguistic characterisation of signals. Brée et
al. [Brée and Smit, 1986] performed a study of temporal conjunctions and
prepositions and suggested rules for discriminating temporal from non-temporal
uses of signal expressions that fall into these classes. However, this work is
purely theoretical and not a corpus-based study. Schlüter [Schlüter, 2001]
identifies signal expressions used with the present perfect and compares their
frequency in British and US English. Vlach [Vlach, 1993] presents a semantic
framework that deals with duratives when used as signal modifiers (see Section
2.1.). Brée et al. [Brée et al., 1993] later describe the ambiguity of nine
temporal prepositions in terms of their roles as temporal signals. Our work
differs from the literature in that is it the first to be based on gold
standard annotations of temporal semantics and that it encompasses all
temporal signal expressions, not just those of a particular grammatical class.
Intuitively, signal expressions contain temporal ordering information that
human readers can access easily. Once temporal conjunctions are identified,
existing semantic formalisms may be applied to discourse semantics [Dowty,
1979]. It is however ambiguous which temporal expression they attempt to
convey [Hitzeman, 2005]. Our work quantifies this ambiguity for a subset of
expressions.
Previous work applying temporal signals has been related to the labeling of
temporal links [Min et al., 2007] and question answering [Pustejovsky et al.,
2005, Saquete et al., 2009]. In particular, Lapata and Lascarides [Lapata and
Lascarides, 2006] remove the temporal signal from sentences containing two
temporally connected clauses and attempt to learn sentence-level temporal
relations using the orderings suggested by the removed signal as training
data. Directly applying signals to the temporal relation identification task,
Derczynski and Gaizauskas [Derczynski and Gaizauskas, 2010b] halved the error
rate of TLINK classification for TLINKs that have a signal by adding features
describing signals. This raised classification accuracy from 62% to 82%.
## 3\. Signals in TimeBank
In this section, we give a detailed profiling of temporal signals in the
TimeBank corpus. Statistics are generated using the CAVaT [Derczynski and
Gaizauskas, 2010a] tool for TimeML-annotated corpus analysis.
First, we note that in TimeML signals may be divided into three classes based
on the type of relation they signal: temporal (tlink), sub-ordinating (slink)
or aspectual (alink). The distribution of signals by class in Timebank is
shown in Table 1. For the rest of the paper we discuss temporal signals only.
Annotated SIGNAL elements | 758
---|---
Signals used by a TLINK | 721
Signals used by an ALINK | 1
Signals used by a SLINK | 39
TLINKs that use a SIGNAL | 787
Signals used by more than one TLINK | 54
Table 1: How <SIGNAL> elements are used in TimeBank. TLINKs per signal | Number of signals
---|---
1 | 597
2 | 41
3 | 12
5 | 1
Table 2: The number of TLINKs associated with each temporal signal word/phrase, in TimeBank. Signals not used on TLINKs (e.g. those used on aspectual or subordinate links, or for event cardinality) are excluded. The distribution is Zipfian. Part of speech | Frequency | Proportion
---|---|---
IN | 521 | 77.3%
RB | 73 | 10.8%
WRB | 53 | 7.9%
JJ | 14 | 2.1%
RBR | 5 | 0.7%
VBG | 4 | 0.6%
CC | 2 | 0.3%
RP | 1 | 0.1%
JJR | 1 | 0.1%
Table 3: Distribution of part-of-speech in signals and the first word of multiword signals, using the Penn Treebank tag set. Expression | Count in corpus | As signal | Proportion as signals | After curation | Proportion
---|---|---|---|---|---
in | 1214 | 161 | 13.3% | |
after | 72 | 56 | 77.8% | 66 | 91.7%
for | 621 | 52 | 8.4% | |
if | 65 | 37 | 56.9% | |
when | 62 | 35 | 56.5% | 56 | 90.3%
on | 344 | 33 | 9.6% | |
until | 36 | 25 | 69.4% | 36 | 100.0%
before | 33 | 23 | 69.7% | 30 | 90.9%
by | 356 | 20 | 5.6% | |
from | 366 | 19 | 5.2% | |
since | 31 | 17 | 54.8% | 18 | 58.1%
through | 69 | 15 | 21.7% | |
as | 271 | 14 | 5.2% | |
over | 59 | 14 | 23.7% | |
already | 32 | 13 | 40.6% | 13 | 40.6%
ended | 21 | 13 | 61.9% | |
during | 19 | 13 | 68.4% | |
at | 311 | 11 | 3.5% | |
previously | 19 | 11 | 57.9% | 16 | 84.2%
within | 23 | 8 | 34.8% | |
s | 10 | 8 | 80.0% | |
later | 15 | 7 | 46.7% | |
earlier | 50 | 6 | 12.0% | |
while | 39 | 6 | 15.4% | 9 | 23.1%
then | 23 | 5 | 21.7% | |
once | 15 | 5 | 33.3% | |
still | 35 | 4 | 11.4% | |
following | 15 | 4 | 26.7% | |
meanwhile | 14 | 4 | 28.6% | 9 | 64.3%
at the same time | 6 | 4 | 66.7% | |
to | 1600 | 3 | 0.2% | |
into | 63 | 3 | 4.8% | |
follows | 4 | 3 | 75.0% | |
subsequently | 3 | 3 | 100.0% | |
followed | 10 | 2 | 20.0% | 4 | 40.0%
former | 16 | 0 | 0.0% | 12 | 75.0%
Table 4: Frequency of candidate signal expressions in TimeBank. We include counts of how often these occur as signal expressions both before and after manual curation. Signal Expression | TLINK count | after | before | begins | begun_by | during | ended_by | ends | iafter | ibefore | includes | is_included | simultaneous
---|---|---|---|---|---|---|---|---|---|---|---|---|---
after | 76 | 62 | 3 | 4 | | | | 5 | 2 | | | |
when | 57 | 16 | 3 | 1 | 2 | 1 | | | 1 | 1 | 9 | 9 | 14
until | 37 | 4 | 7 | 1 | | | 21 | 1 | 1 | 2 | | |
before | 36 | 1 | 28 | 2 | | | 1 | 2 | 1 | 1 | | |
since | 19 | 9 | 1 | 2 | 7 | | | | | | | |
already | 13 | | 6 | | | | | | | | 4 | 3 |
previously | 18 | 6 | 12 | | | | | | | | | |
while | 9 | | | | | | | | | | | | 9
meanwhile | 9 | | 1 | | | 2 | | | | | | 1 | 5
followed | 4 | 2 | 2 | | | | | | | | | |
former | 12 | | 12 | | | | | | | | | |
Table 5: Signal expressions and the TimeML relations that they can denote,
ordered as per Table 4 for comparison. Counts do not match because a single
signal expression can support more than one temporal link.
### 3.1. Additional Annotation
Upon examination of the non-annotated instances of words that often occur as a
temporal signal (such as _after_) it became evident that TimeBank’s signals
are under-annotated. As we are certain of some annotation errors in the source
data, we revisited the original annotations. A subset of signal words was
selected for re-annotation. This set consisted of signals that were ambiguous
(occurred temporally close to 50% of the time) or that we expected contained,
based on informal observations, would yield a number of missed temporal
annotations. All temporal instances of these words were re-annotated with
TimeML, adding EVENTs, TIMEX3s and TLINKs where necessary to create a
signalled TLINK.
A single annotator checked the source documents and annotated 70 extra
signals, as well as adding 34 events, 1 temporal expression and 49 extra
temporal links.
### 3.2. Proportion of Temporal Relations with Signals
TimeBank contains 6 418 TLINKs (6 467 after re-annotation) of which 718 (787)
are explicitly indicated by a temporal signal – 11.2% (12.2%). This provides
an answer to the first question we posed in Section 1. Thus while ability to
successfully detect temporal signals will not solve the problem of assigning
temporal relations, it is likely to make a noticeable difference (see
Derczynski and Gaizauskas [Derczynski and Gaizauskas, 2010b]). Perhaps of more
interest is that so few temporal relations are explicitly signalled – we must
look elsewhere for an explanations of how temporal relations are conveyed in
natural language.
While many TLINKs do not have any associated temporal signal it is also the
case that some temporal signals are associated with more than one TLINK. Table
2 shows details of just how signals are being used by TLINKs.
### 3.3. Temporal vs Non-temporal Uses
The semantic function that a temporal signal expression performs is that of
relating two temporal entities. However, the words that can function as
temporal signals also play other roles.
Table 4 details the distribution of expressions that are found as temporal
signals more than twice (after re-annotation) in TimeBank. The most frequent
signal word was “in”, accounting for 24.8% of all signal-using TLINKs.
However, only 13.3% of occurrences of the word “in” have a temporal sense. The
word “after” is far more likely (91.7% of all occurrences) to have a temporal
sense. In total TimeBank contains 62 unique signal words and phrases (ignoring
case) and of these, over half (36) are also found in Table 4 .
As an aside, note that any thought that temporal signals might be easily
picked out based on word class may be dispelled by examining the distribution
of parts-of-speech possessed by temporal signals – see Table 3.
### 3.4. Relation Ambiguity
The nature of the temporal relation described by a signal is not constant,
though each signal tends to describe a particular relation type most often.
Table 5 gives an excerpt of data showing which temporal relations are made
explicit by each signal expression. The variation in relation type associated
with a signal is not as great as it might appear as the assignment of temporal
relation type has an element of arbitrariness – one may choose to annotate a
before or after relation for the same event pair by simply reversing the
temporal link’s argument order, for example. Nevertheless, it is possible to
draw useful information from the table; for example, one can see that
_meanwhile_ is much more likely to suggest some sort of temporal overlap
between events than an ordering where arguments occur discretely.
## 4\. Per-expression details
We chose to curate signal annotations in TimeBank for a subset of candidate
signal expressions (as described in Section 3.1.). During this curation, we
attempted to determine distinguishing features that could aid automatic
discrimination of temporal from non-temporal sense of the expressions. Details
of our findings are given below.
#### Previously
TimeBank contains eight instances of the word that were not annotated as a
signal. Of these, all were being used as temporal signals. The word only takes
one event or time as its direct argument, which is placed temporally before an
event or time that is in focus. For example:
_“X reported a third-quarter loss, citing a previously announced capital
restructuring program”_
In this sentence, the second argument of _previously_ is _“announced”_ , which
is temporally situated before its first argument (_“reported”_). When
_previously_ occurs at the top of a section, the temporal element that has
focus is either document creation time or, if one has been specified in
previous discourse, the time currently in focus.
#### After
Of the nineteen instances of this word not annotated as temporal, only three
were actually non-temporal. The cases that were non-temporal were a different
sense of the word. The temporal signals are adverbial, with a temporal
function. Two non-temporal cases used a positional sense. The last case was in
a phrasal verb _to go after_ ; _“whether we would go after attorney’s fees”_.
#### When
There are 35 annotated and 27 non-annotated occurrences of this phrase. It
indicates either an overlap between intervals, or a point relation that
matches an interval’s start. Twenty-three of the twenty-seven non-annotated
occurrences are used as temporal signals. Two of the remaining four are in
negated phrases and not used to link an interval pair. for example, _“did not
say when the reported attempt occurred”_. The other two are used in context
setting phrases, e.g. _“we think he is someone who is capable of rational
judgements when it comes to power”_, which are not temporal in nature.
#### While
The cases of _while_ that have not been annotated as a signal – the majority
class, 33 to 6 – are often used in a contrastive sense. This does suggest that
the connected events have some overlap, often between statives. For example,
_“But while the two Slavic neighbours see themselves as natural partners,
their relations since the breakup of the Soviet Union have been bedeviled”_.
As two states described in the same sentences are likely to temporally overlap
and any events or times outside or bounding these states will be related to
the state, it is unlikely that any contribution to TLINK annotation would be
made by linking the two states with a “roughly simultaneous” relation; the
closest suitable label is TempEval’s overlap relation [Verhagen et al., 2010].
_“nor can the government easily back down on promised protection for a
privatized company while it proceeds with …”_
The cases of _while_ that were not of this sense were easier to annotate.
Sometimes it was used as a temporal expression; _“for a while”_. Other times,
it was not used in a contrastive sense, but instead as irrealis – see Example
4.. The four cases of non-contrastive usage were annotated as temporal
signals.
[.PP [.IN before ] [.NP [.Det the ] [.NNS wars ] ] ]
Figure 1: An example of the common syntactic surroundings of a _before_
signal.
#### Before
Three of the ten negative examples are correctly annotated. They are _before_
in the spatial sense of “in front of” (as in _“The procedures are to go before
the Security Council next week”_) and also a logical before that does not link
instantiated or specific events (_“ before taxes”_). The remaining seven
unannotated examples of the word are all temporal signals. These directly
precede either an NP describing a nominalised event, or directly precede a
subordinate clause (e.g. [IN before, S] – see Figure 1).
#### Until
All eleven non-annotated instances of _until_ should have been annotated as
temporal signals. This word suggests a TimeML ibefore relation, unless
qualified otherwise by something like “not until” or “at least until”.
#### Already
There were thirteen positive examples of _already_. All of the non-annotated
examples had a non-temporal sense as per our description of temporal signals.
The word tends to be used for emphasis, but can also suggest a broad “before
DCT” position, which goes without saying for any past and present tensed
events. As _already_ can be removed without changing the temporal links
present in a sentence, we have not annotated any more examples of this beyond
the thirteen present in TimeBank.
#### Meanwhile
This word tends to refer to a reference or event time introduced earlier in
discourse, often from the same sentence. As well as a temporal sense, it can
have a contrastive “despite”-like meaning. _Meanwhile_ tends to refer more to
previous actions, instead of states specified in immediately prior sentences.
Sometimes _meanwhile_ is used with no previous temporal reference. In these
cases, the implicit argument is DCT. Five of the ten non-annotated _meanwhile_
s were temporal signals.
#### Again
This word shows recurrence and is always used for this purpose where it occurs
in TimeBank not annotated as a temporal signal. No instances of _“again”_ were
annotated.
[.NX [.NX [.NN founder ] ] [.CC and ] [.NX [.JJ former ] [.NN chairman ] ] ]
Figure 2: Example of a non-annotated signal (_former_) from TimeBank’s
wsj_0778.tml.
#### Former
This word indicates a state that persisted before DCT or current speech time
and has now finished. Generally the construction that is found is an NP, which
contains an optional determiner, followed by _former_ and then a substituent
NP which may be annotated as an EVENT of class state. This configuration
suggests a TLINK that places the event before the state’s utterance.
_“The San Francisco sewage plant was named in honour of former President
Bush.”_
In Example 4., there is a state-class event – _President_ – that at one time
has applied to the named entity _Bush_. The signal expression _former_
indicates that this state terminated before the time of the sentence’s
utterance.
Three-quarters of the non-annotated instances of _former_ in TimeBank are
temporal signals.
#### Recently
Although _recently_ is a temporal adverb, it can only be to applied simple or
anterior tensed verbs (using Reichenbach’s tense nomenclature). In the corpus,
these are only seen in reported speech or of verbal events that happened
before DCT. _Recently_ adds a qualitative distance between event and utterance
time, but is of reduced use when we can already use tense information. The
phrase “Until recently” appears awkward when cast as a temporal signal but can
be interpreted as “before DCT”, with the interval’s endpoint being close to
DCT. In this case, recently functions as a temporal expression, not a signal.
Only one of the non-annotated _recently_ s in TimeBank is a temporal signals.
The exception, _“More recently”_ , includes a comparative and is annotated as
a TIMEX3; both this phrase and, e.g., _“less recently”_ suggest a relation to
a previously-mentioned (and in-focus) past event. As a result, we posit that
_recently_ behaves as an abstract temporal point (as seen in the behaviour of
_“until recently”_). Structures such as _[comparative] recently_ may be
interpreted as a qualified temporal signal, as they convey information about
the relative ordering of the event that they dominate vent compared with a
previously mentioned interval.
## 5\. Conclusion
We have provided a characterisation of temporal signal expressions. In an
analysis of the TimeBank corpus we have shown what proportion of temporal
relations are explicitly signalled by these expressions and have given
quantitative descriptions of how ambiguous these phrases are, both regarding
their temporal/non-temporal senses and the type of temporal relation that they
convey.
### 5.1. Acknowledgments
The first author would like to acknowledge the UK Engineering and Physical
Science Research Council’s support in the form of a doctoral studentship.
## References
* Bestgen and Vonk, 1999 Y. Bestgen and W. Vonk. 1999\. Temporal Adverbials as Segmentation Markers in Discourse Comprehension. Journal of Memory and Language, 42(1):74–87.
* Brée and Smit, 1986 D.S. Brée and R.A. Smit. 1986\. Temporal relations. Journal of Semantics, 5(4):345.
* Brée et al., 1993 D.S. Brée, A. Feddag, and I. Pratt. 1993\. Towards a formalization of the semantics of some temporal prepositions. Time & Society, 2(2):219.
* Derczynski and Gaizauskas, 2010a L. Derczynski and R. Gaizauskas. 2010a. Analysing Temporally Annotated Corpora with CAVaT. In Proceedings of the Language Resources and Evaluation Conferece.
* Derczynski and Gaizauskas, 2010b L. Derczynski and R. Gaizauskas. 2010b. Using signals to improve automatic classification of temporal relations. Proceedings of the ESSLLI StuS.
* Dowty, 1979 D.R. Dowty. 1979\. Word meaning and Montague grammar: The semantics of verbs and times in generative semantics and in Montague’s PTQ, volume 7. Springer.
* Hitzeman, 1997 J. Hitzeman. 1997\. Semantic partition and the ambiguity of sentences containing temporal adverbials. Natural Language Semantics, 5(2):87–100.
* Hitzeman, 2005 J. Hitzeman. 2005\. Text type and the position of a temporal adverbial within the sentence. In Proceedings of the 2005 international conference on Annotating, extracting and reasoning about time and events, pages 29–40. Springer-Verlag.
* Ho-Dac and Péry-Woodley, 2008 L.M. Ho-Dac and M.P. Péry-Woodley. 2008\. Temporal adverbials and discourse segmentation revisited. In Multidisciplinary Approaches to Discourse.
* Lapata and Lascarides, 2006 M. Lapata and A. Lascarides. 2006\. Learning sentence-internal temporal relations. Journal of Artificial Intelligence Research, 27(1):85–117.
* Min et al., 2007 C. Min, M. Srikanth, and A. Fowler. 2007\. LCC-TE: A hybrid approach to temporal relation identification in news text. In Proceedings of SemEval-2007, pages 219–222. ACL.
* Pustejovsky et al., 2003 J. Pustejovsky, P. Hanks, R. Sauri, A. See, R. Gaizauskas, A. Setzer, D. Radev, B. Sundheim, D. Day, L. Ferro, et al. 2003\. The TimeBank corpus. In Corpus Linguistics, volume 2003, page 40.
* Pustejovsky et al., 2004 J. Pustejovsky, B. Ingria, R. Sauri, J. Castano, J. Littman, R. Gaizauskas, A. Setzer, G. Katz, and I. Mani. 2004\. The specification language TimeML. The Language of Time: A Reader. Oxford University Press, Oxford.
* Pustejovsky et al., 2005 J. Pustejovsky, R. Knippen, J. Littman, and R. Saurí. 2005\. Temporal and Event Information in Natural Language Text. Language Resources and Evaluation, 39(2):123–164.
* Reichenbach, 1947 H. Reichenbach. 1947\. The tenses of verbs. Elements of Symbolic Logic.
* Saquete et al., 2009 E. Saquete, J.L. Vicedo, P. Martınez-Barco, R. Munoz, and H. Llorens. 2009\. Enhancing QA systems with complex temporal question processing capabilities. Journal of Artificial Intelligence Research, 35(1):755–811.
* Schlüter, 2001 N. Schlüter. 2001\. Temporal specification of the present perfect: a corpus-based study. Language and Computers, 36(1):307–315.
* Setzer and Gaizauskas, 2000 A. Setzer and R. Gaizauskas. 2000\. Annotating events and temporal information in newswire texts. In Proceedings of the Second International Conference on Language Resources And Evaluation, pages 1287–1294, Athens.
* Verhagen et al., 2009 M. Verhagen, R. Gaizauskas, F. Schilder, M. Hepple, J. Moszkowicz, and J. Pustejovsky. 2009\. The TempEval Challenge: Identifying Temporal Relations in Text. Journal of Language Resources and Evaluation, 43(2):161–179.
* Verhagen et al., 2010 M. Verhagen, R. Saurí, T. Caselli, and J. Pustejovsky. 2010\. SemEval-2010 task 13: TempEval-2. In Proceedings of the 5th International Workshop on Semantic Evaluation, pages 57–62. ACL.
* Vlach, 1993 F. Vlach. 1993\. Temporal adverbials, tenses and the perfect. Linguistics and Philosophy, 16(3):231–283.
|
arxiv-papers
| 2012-03-22T18:08:47 |
2024-09-04T02:49:28.926606
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5066"
}
|
1203.5073
|
# USFD at KBP 2011: Entity Linking, Slot Filling and Temporal Bounding
Amev Burman, Arun Jayapal, Sathish Kannan, Madhu Kavilikatta
Ayman Alhelbawy, Leon Derczynski, Robert Gaizauskas
Natural Language Processing Group
Department of Computer Science
University of Sheffield
Sheffield, S1 4DP, UK
## 1 Introduction
This paper describes the University of Sheffield’s entry in the 2011 TAC KBP
entity linking and slot filling tasks [Ji et al., 2011]. We chose to
participate in the monolingual entity linking task, the monolingual slot
filling task and the temporal slot filling tasks. Our team consisted of five
MSc students, two PhD students and one more senior academic. For the MSc
students, their participation in the track formed the core of their MSc
dissertation project, which they began in February 2011 and finished at the
end of August 2011. None of them had any prior experience in human language
technologies or machine learning before their programme started in October
2010. For the two PhD students participation was relevant to their ongoing PhD
research. This team organization allowed us to muster considerable manpower
without dedicated external funding and within a limited period time; but of
course there were inevitable issues with co-ordination of effort and of
getting up to speed. The students found participation to be an excellent and
very enjoyable learning experience.
Insofar as any common theme emerges from our approaches to the three tasks it
is an effort to learn from and exploit data wherever possible: in entity
linking we learn thresholds for nil prediction and acquire lists of name
variants from data; in slot filling we learn entity recognizers and relation
extractors; in temporal slot filling we use time and event annotators that are
learned from data.
The rest of this paper describes our approach and related investigations in
more detail. Sections 2 and 3 describe in detail our approaches to the EL and
SF tasks respectively, and Section 4 summarises our temporal slot filling
approach.
## 2 Entity Linking Task
Figure 1: Flow Chart of the Initial Approach for Entity Linking
The entity linking task is to associate a queried named entity mention, as
contextualized within a given document, with a knowledge base (KB) node in a
provided knowledge base which describes the same real world entity. If there
is no such node the entity should be linked to Nil. There are three main
challenges in this task. The first challenge is the ambiguity and multiplicity
of names: the same named entity string can occur in different contexts with
different meaning (e.g. Norfolk can refer to a city in the United States or
the United Kingdom); furthermore, the same named entity may be denoted using
various strings, including, e.g. acronyms (USA) and nick names (Uncle Sam).
The second challenge is that the queried named entity may not be found in the
knowledge base at all. The final challenge is to cluster all Nil linked
mentions.
### 2.1 System Processing
Our system consists of four stage model, as shown in Figure 1:
Candidate Generation: In this stage, all KB nodes which might possibly be
linked to the query entity are retrieved.
Nil Predictor: In this stage, a binary classifier is applied to decide whether
the query mention should be linked to a KB node or not.
Candidate Selection: In this stage, for each query mention that is to be
linked to the KB, one candidate from the candidate set is selected as the link
for the query mention.
Nil Mention Clustering: In this stage, all Nil linked query mentions are
clustered so that each cluster contains all mentions that should be linked to
a single KB node, i.e. pertain to the same entity.
#### 2.1.1 Candidate Generation
The main objective of the candidate generation process is to reduce the search
space of potential link targets from the full KB to a small subset of
plausible candidate nodes within it. The query mention is used, both as a
single phrase and as the set of its constituent tokens, to search for the
query string in the titles and body text of the KB node.
##### Variant name extraction
We extracted different name forms for the same named entity mention from a
Wikipedia dump. Hyper-links, redirect pages and disambiguation pages are used
to associate different named entity mentions with the same entity [Reddy et
al., 2010, Varma et al., 2009]. This repository of suggested name variants is
then used in query expansion to extend the queries regarding a given entity to
all of its possible names. Since the mention of the entity is not yet
disambiguated, it is not necessary for all suggested name variants to be
accurate.
##### Query Generation
We generated sets of queries according to two different strategies. The first
strategy is based on name variants, using the previously built repository of
Wikipedia name variants. The second strategy uses additional named entity (NE)
mentions for query expansion: the Stanford NE recognizer [Finkel et al., 2005]
is used to find NE mentions in the query document, and generates a query
containing the query entity mention plus all the NE mentions found in the
query document,
Data Set | $\alpha$ | Precision | Recall | F1 | Accuracy
---|---|---|---|---|---
TAC 2009 | 5.9 | 59.42 | 80.18 | 68.26 | 68.01
TAC 2010 | 5.9 | 75.36 | 77.65 | 76.48 | 78.36
TAC 2011 | 5.9 | 74.83 | 66.90 | 70.64 | 72.22
Table 1: Performance of Nil Predictor using highest score candidate Data Set | $\beta$ | Precision | Recall | F1 | Accuracy
---|---|---|---|---|---
TAC 2009 | 0.16 | 54.74 | 64.84 | 59.36 | 61.91
TAC 2010 | 0.16 | 57.92 | 62.35 | 60.06 | 62.40
TAC 2011 | 0.16 | 54.69 | 31.14 | 39.68 | 52.71
Table 2: Performance of Nil Predictor using difference between two highest
scored candidates
##### Retrieval
After query generation, we performed document retrieval using Lucene. All
knowledge base nodes, titles, and wiki-text were included in the Lucene index.
Documents are represented as in the Vector Space Model (VSM). For ranking
results, we use the default Lucene similarity function which is closely
related to cosine similarity .
#### 2.1.2 Nil Prediction
In many cases, a named entity mention is not expected to appear in the
knowledge base. We need to detect these cases and mark them with a NIL link.
The NIL link is assigned after generating a candidate list (see Varma et al.
[Varma et al., 2009], Radford et al. [Radford et al., 2010]).
If the generated candidate list is empty, then the query mention is linked to
NIL. If the candidate list is not empty, we use two techniques to find a
candidate. The first just chooses the highest ranked candidate, i.e. the
highest scoring candidate using the Lucene similarity score. If the the
highest scoring candidate score is above some threshold $\alpha$ then the
candidate is selected and if it is under the threshold, the predictor links
the mention to NIL. The second technique calculates the difference between the
scores of two highest scoring candidates, then compares this difference with
some threshold $\beta$; if the difference exceeds the threshold the highest
scoring candidate is selected, otherwise the query mention is linked to NIL..
Results of these two techniques are shown in Tables 1 and 2.
Levenshtein distance | RI | Precision | Recall | F1
---|---|---|---|---
0 | 99.98 | 95.06 | 99.15 | 97.06
1 | 99.98 | 92.22 | 99.21 | 95.59
Table 3: Performance of Nil Clustering
##### Parameter Setting for Nil Matching
To find the best thresholds, a Naïve Bayes classifier is trained using the TAC
2010 training data. We created training data as follows. For each query in the
training set, we generate a candidate list and the highest scoring document is
used as a feature vector. If it is the correct candidate then the output is
set to true else the output set to false. From this set of instances a
classifier is learned to get the best threshold.
#### 2.1.3 Candidate Selection
The candidate selection stage will run only on a non-empty candidate list,
since an empty candidate list means linking the query mention to NIL. For each
query, the highest-scoring candidate is selected as the correct candidate.
#### 2.1.4 Nil Clustering
A simple clustering technique is applied. The Levenshtein distance is measured
between the different mentions and if the distance is under a threshold
$\alpha$, the mentions are grouped into the same cluster. Two experiments are
carried out and results are presented in Table 3. As shown clustering
according to the string equality achieves better results than allowing a
distance of one.
##### Data Set:
The TAC2011 data set contains 2250 instances of which 1126 must be linked to
“Nil”. In the gold standard, the 1126 Nil instances are clustered into 814
clusters. Only those 1126 instances are sent to the clustering module to check
its performance separately, regardless of Nil predictor performance.
##### Evaluation Metric:
“All Pair Counting Measures” are used to evaluate the similarity between two
clustering algorithm’s results. This metric examines how likely the algorithms
are to group or separate a pair of data points together in different clusters.
These measures are able to compare clusterings with different numbers of
clusters.
The Rand index [Rand, 1971] computes similarity between the system output
clusters (output of the clustering algorithm) and the clusters found in a gold
standard. So, the Rand index measures the percentage of correct decisions –
pairs of data points that are clustered together in both system output and
gold standard, or, clustered in different clusters in both system output and
gold standard – made by the algorithm. It can be computed using the following
formula:
$RI=\frac{Tp+Tn}{Tp+Tn+Fp+Fn}$
### 2.2 Evaluation
In this section we provide a short description of different runs and their
results. All experiments are evaluated using the B-Cubed+ and micro average
scoring metrics. In our experimental setup, a threshold $\alpha=5.9$ is used
in Nil-Predictor and Levenshtein distance = 0 is used for Nil clustering. The
standard scorer released by the TAC organizers is used to evaluate each run,
with results in Table 4. Different query schemes are used in different runs as
follows.
Wiki-text is not used, with search limited to nodes titles only. The search
scheme used in this run uses query mention only.
Wiki-text is used. The search scheme used in this run uses the query mention
and the different name variants for the query mention.
Wiki-text is used, The search scheme used in this run uses the query mention
and the different name variants for the query mention. Also, it uses the query
document named entities recognized by the NER system to search within the
wiki-text of the node.
Run | Micro Average | B^3 Precision | B^3 Recall | B^3 F1
---|---|---|---|---
1 | 49.2 | 46.2 | 48.8 | 47.5
2 | 43.6 | 40.8 | 43.2 | 42.0
3 | 46.8 | 44.0 | 45.6 | 44.8
Table 4: Results of Three runs for entity linking task
## 3 Slot Filling Task
There are a number of different features that can describe an entity. For an
organisation, one might talk about its leaders, its size, and place of origin.
For a person, one might talk about their gender, their age, or their religious
alignment. These feature types can be seen as ‘slots’, the values of which can
be used to describe an entity.
The slot-filling task is to find values for a set of slots for each of a given
list of entities, based on a knowledge base of structured data and a source
collection of millions of documents of unstructured text. In this section, we
discuss our approach to slot filling.
### 3.1 System Processing
Our system is structured as a pipeline. For each entity/slot pair, we begin by
selecting documents that are likely to bear slot values, using query
formulation (Section 3.1.2) and then information retrieval (Section 3.1.1)
steps. After this, we examine the top ranking returned texts and, using
learned classifiers, attempt to extract all standard named entity mentions
plus mentions of other entity types that can occur as slot values (Section
3.1.3). Then we run a learned slot-specific relation extractor over the
sentences containing an occurrence of the target entity and an entity of the
type required as a value for the queried slot, yielding a list of candidate
slot values (Section 3.1.4). We then rank these candidate slot values and
return a slot value, or list of slot values in the case of list-valued slots,
from the best candidates (Section 3.1.5).
#### 3.1.1 Preprocessing and Indexing
Information Retrieval (IR) was used to address the tasks of Slot Filling (SF)
and Entity Linking (EL) primarily because it helps in choosing the right set
of documents and hence reduces the number of documents that need to be
processed further down the pipeline. Two variations of IR were used in the SF
task: document retrieval (DR) and passage retrieval (PR).
The documents were parsed to extract text and their parent elements using JDOM
and then indexed using Lucene. We used Lucene’s standard analyzer for indexing
and stopword removal. The parent element of the text is used as field name.
This gives the flexibility of searching the document using fields and document
structure as well as just body [Baeza-Yates et al., 1999]. Instead of
returning the text of the document, the pointers or paths of the document were
returned when a search is performed. For searching and ranking, Lucene’s
default settings were used.
For passage retrieval, various design choices were considered [Roberts and
Gaizauskas, 2004] and a two stage process was selected. In the two stage
process, the original index built for DR is used to retrieve the top $n$
documents and the plain text (any text between two SGML elements) is extracted
as a separate passage. A temporary mini-index is then built on the fly from
these passages. From the temporary index, the top $n$ _passages_ are retrieved
for a given query. Instead of returning the text of the passages, the location
of the passage (element retrieval) in the document is returned as a passage
offset within a document referenced by a file system pointer. Two versions of
passage systems were created, one that removes stop-words while indexing and
searching and other that keeps the stop words. For ranking, Lucene’s default
settings were used.
Finally the IR system and the query formulation strategies were evaluated on
the DR task to determine the optimal number of top ranked documents to
retrieve for further processing down the pipeline and for PR. This evaluation
is further discussed in Section 3.2 below.
#### 3.1.2 Query Formulation
This step generates a query for the IR system that attempts to retrieve the
best documents for a given entity and slot type.
##### Variant name extraction
Variant names are the alternate names of an entity (persons or organizations
only for the slot filling task in 2011) which are different from their formal
name. These include various name forms such as stage names, nick names and
abbreviations. Many people have an alias; some people even have more than one
alias. In several cases people are better known to the world by their alias
names rather than their original name. For example, Tom Cruise is well known
to the world as an actor, but his original name is Thomas Cruise Mapother IV.
Alias names are very helpful to disambiguate the named entity, but in some
cases the alias names are also shared among multiple people. For example, MJ
is the alias name for both Michael Jackson (Pop Singer) and Michael Jordan
(Basketball player).
Variant name forms are used for query formulation. The methods used in the
slot filling task for extracting variant name forms from a Wikipedia page are:
Extract all the name attributes from the infobox, such as nickname, birth
name, stage name and alias name.
Extract the title and all bold text from the first paragraph of the article
page.
Extract the abbreviations of the entity name by finding patterns like “(ABC)”
consisting of all capital letters appearing after the given entity name. For
example, TCS is an abbreviation of the entity Tata Consultancy Service in case
of the following pattern _Tata Consultancy Service, (TCS)_.
Extract all redirect names that refer to the given entity. For example, the
name ‘King of Pop’ automatically redirects to the entity named ‘Michael
Jackson’.
In the case of ambiguous names extract all the possible entity names that
share the same given name from the disambiguation page.
A variant name dictionary was created by applying all the above methods to
every entity in the Wikipedia dump. Each line of the dictionary contains the
entity article title name as in Wikipedia followed by one of the variant name
forms. This dictionary is then used at query time to find the variant name
forms of the given entity.
##### Slot keyword collection
The query formulation stage deals with developing a query to retrieve the
relevant documents or passages for each slot of each entity. Our approach is
as follows:
Collect manually (by referring to public sources such as Wikipedia) a list of
keywords for each slot query. Some example keywords for the
per:countries_of_residence slot query are ‘house in’, ‘occupies’, ‘lodges in’,
‘resides in’, ‘home in’, ‘grew up in’ and ‘brought up in’.
Extract all the alternate names of the given entity name the variant name
dictionary (Section 3.1.2).
Formulate a query for each slot of an entity by including terms for entity
mention, variant names and keywords collected for the slot query in the first
step. These terms are interconnected by using Boolean operators.
The formulated query is then fed into the IR component and the top $n$
documents retrieved.
#### 3.1.3 Entity Identification
Given the top $n$ documents returned by the previous phase of the system, the
next task is to identify potential slot values. To do this we used entity
recognizers trained over existing annotated datasets plus some additional
datasets we developed. For a few awkward slot value types we developed regular
expression based matchers to identify candidate slot fills. We have also
developed a restricted coreference algorithm for identifying coreferring
entity mentions, particularly mentions coreferring with the query (target)
entity,
##### Named Entity Recognition
The Stanford Named Entity Recognition (NER) tool [Finkel et al., 2005] was
used to find named entities. It is a supervised learning conditional random
field based approach which comes with a pre-trained model for three entity
classes. Because we needed a broader range of entity classes we re-trained the
classifier using the MUC6 and MUC7 datasets 111LDC refs. LDC2001T02,
LDC2003T13 and NLTK [Bird et al., 2009] gazetteers. Training the classifier
was not straightforward as the source data had to be reformatted into the
format recognized by Stanford NER. The MUC datasets provided training data for
the entities Location, Person, Organization, Time, Person, Money, Percent,
Date, Number and Ordinal. More classes were added to the MUC training dataset
since the slot-filling task required nationality, religion, country, state,
city and cause-of-death slot fill types to be tagged as well. For country,
state and city, which can be viewed as sub-types of type location we semi-
automatically adapted the MUC training data by finding all location entities
in the data, looking them up in a gazetteer and then manually adding their
sub-type. For nationalities, causes of death and religion, we extracted lists
of nationalities, causes of death and religions from Wikipedia. In the case of
nationality and causes of death we searched for instances of these in the MUC
data and then labelled them to provide training data. For religion, however,
because there were so few instances in the MUC corpus and because of issues in
training directly on Wikipedia text, we used a post-classifier list matching
technique to identify religions.
The trained classifier was used to identify and tag all mentions of the entity
types it knew about in the documents and/or passages returned by the search
engine. These tagged documents were then passed on to the co-reference
resolution system. After some analysis we discovered that in some cases the
target entity supplied in the quey was not being correctly tagged by the
entity tagger. Therefore we added a final phase to our entity identifier in
which all occurrences of the target entity were identified and tagged with the
correct type, regardless of whether they had or had not been tagged correctly
by the CRF entity tagger. s
##### Restricted Co-reference Resolution
To identify the correct slot fill for an entity requires not just identifying
mentions which are of the correct slot fill type but of ensuring that the
mention stands in the appropriate relation to the target entity – so, to find
Whistler’s mother requires not only finding entities of type PERSON, but also
determining that the person found stands the relation “mother-of” to Whistler.
Our approach to relation identification, described in the next section, relies
on the relation being expressed in a sentence in which both the candidate slot
fill and the target entity occur. However, since references to the target
entity or to the slot fill may be anaphoric, ability to perform coreference
resolution is required.
Off-the-shelf co-reference resolvers, such as the Stanford CRF-based
coreference tool, proved too slow to complete slot-filling runs in a
reasonable timeframe. Therefore, we designed a custom algorithm to do limited
heuristic coreference to suit the slot-filling task. Our algorithm is limited
in two ways. First, it only considers coreferring references to the target
entity and ignores any coreference to candidate slot fills or between any
other entities in the text. Second, only a limited set of anaphors is
considered. In the case of target entities of type PERSON the only anaphors
considered are personal and possessive pronouns such as _he_ , _she_ , _his_
and _her_. In these cases it also helps to identify whether the target entity
is male or female. We trained the maximum entropy classifier provided with
NLTK with a list of male names and female names also from NLTK. The last and
second to last characters for each name were taken as features for training
the classifier. Based on the output produced by the classifier, the system
decides whether certain pronouns are candidate anaphors for resolving with the
target entity. For example, when the output produced by the classifier for the
PERSON entity Michael Jackson is male, only mentions of he and his will be
considered as candidate anaphors.
When the target entity is of type ORGANIZATION, only the pronoun it or common
nouns referring to types of organization, such as company, club, society,
guild, association, etc. are considered as potential anaphors. A list of such
organization nouns is extracted from GATE.
For both PERSONs and ORGANIZATIONs, when candidate anaphors are identified the
algorithm resolves them to the target entity if a tagged mention of the target
entity is the textually closest preceding tagged mention of an entity of the
target entity type. For example, he will be coreferred with Michael Jackson if
a tagged instance of Michael Jackson, or something determined to corefer to
it, is the closest preceding mention of a male entity of type PERSON. If an
intervening male person is found, then no coreference link is made. When
coreference is established, the anaphor – either pronoun or common noun – is
labelled as ‘target entity”.
This approach to coreference massively reduces the complexity of the
generalized coreference task, making it computationally tractable within the
inner loop of processing multiple documents per slot per target entity.
Informal evaluation across a small number of manually examined documents
showed the algorithm performed quite well.
#### 3.1.4 Candidate Slot Value Extraction
The next sub-task is to extract candidate slot fills by determining if the
appropriate relation holds between a mention of the target entity and a
mention of an entity of the appropriate type for the slot. For example if the
slot is date_of_birth and the target entity is Michael Jackson then does the
date_of_birth relation hold between some textual mention of the target entity
Michael Jackson (potentially an anaphor labelled as target entity) and some
textual mention of an entity tagged as type DATE.
The general approach we took was to select all sentences that contained both a
target entity mention as well as a mention of the slot value type and run a
binary relation detection classifier to detect relations between every
potentially related target entity mention-slot value type mention in the
sentence. If the given relation is detected in the sentence, the slot value
for the relation (e.g. the entity string) is identified as a candidate value
for the slot of the target entity.
Run | Recall | Precision | F1 | Retrieval | Co-ref? | Slot extractor
---|---|---|---|---|---|---
1 | 1.38% | 2.43% | 0.0176 | document | no | BoW
2 | 5.08% | 4.84% | 0.0496 | document | yes | BoW + ngram
3 | 1.16% | 2.97% | 0.0167 | passage | yes | BoW + ngram
Table 5: Slot filling results for USFD2011.
##### Training the Classifiers
A binary relation detection classifier needed to be trained for each type of
slot. Since there is no data explicitly labelled with these relations we used
a distant supervision approach (see, e.g., ?)). This relied on an external
knowledge base – the infoboxes from Wikipedia – to help train the classifiers.
In this approach, the fact names from the Wikipedia infoboxes were mapped to
the KBP. These known slot value pairs from the external knowledge base were
used to extract sentences that contain the target entity and the known slot
value. These formed positive instances. Negative instances were formed from
sentences containing the target entity and an entity mention of the
appropriate type for the slot fill, but whose value did not match the value
taken from the infobox (e.g. a DATE, but not the date of birth as specified in
the infobox for the target entity). The classifiers learned from this data
were then used on unknown data to extract slot value pairs.
##### Feature Set
Once the positive and negative training sentences were extracted, the next
step was to extract feature sets from these sentences which would then be used
by machine learning algorithms to train the classifiers. Simple lexical
features and surface features were included in the feature set. Some of the
features used include:
Bag of Words: all words in the training data not tagged as entities were used
as binary features whose value is 1 or 0 for the instance depending on whether
they occur in sentence from which the training instance is drawn.
Words in Window: like Bag of Words but only words between the target entity
and candidate slot value mentions plus two words before and after are taken as
features.
N-grams: like bag of words, but using bi-grams instead of unigrams
Token distance: one of three values – short ($<=3$), medium ($>3$ and $<=6$)
or long ($>6$) – depending on the distance in tokens between the the target
entity and candidate slot value mentions.
Entity in between: binary feature indicating whether there is another entity
of the same type between the candidate slot value mention and the target
entity.
Target first: binary feature indicating whether the target entity comes before
the candidate slot value in the sentence?
We experimented with both the Naive Bayes and Maximum Entropy classifiers in
the NLTK. For technical reasons could not get the maximum entropy classifier
working in time for the official test runs, so our submitted runs used the
Naive Bayes classifiers, which is almost certainly non-optimal given the non-
independence of the features.
#### 3.1.5 Slot Value Selection
The final stage in our system is to select which candidate slot value (or slot
values in the case of list-valued slots) to return as the correct answer from
the candidate slot values extracted by the relation extractor in the previous
stage. To do this we rank the candidates identified in the candidate slot
value extraction stage. Two factors are considered in ranking the candidates:
(1) the number of times a value has been extracted, and (2) the confidence
score provided for each candidate by the relation extractor classifier. If any
value in the list of possible slot values occurs more than three times, then
the system uses the number of occurrences as a ranking factor. Otherwise, the
system uses the confidence score as a ranking factor. In the first case
candidate slot values are sorted on the basis of number of occurrences. In the
second case values are sorted on the basis of confidence score. In both cases
the top n value(s) are taken as the correct slot value(s) for the given slot
query. We use $n=1$ for single-valued slots $n=3$ for list-valued slots.
Once the system selects the final slot value(s), the final results are written
to a file in the format required by the TAC guidelines.
### 3.2 Evaluation
We evaluated both overall slot-filling performance, and also the performance
of our query formulation / IR components in providing suitable data for slot-
filling.
#### 3.2.1 Overall
We submitted three runs: one with document-level retrieval, no coreference
resolution, and bag-of-words extractor features; a second with document-level
retrieval, coreference resolution, and n-gram features; a third with passage-
level retrieval, coreference resolution, and n-gram features. Our results are
in Table 5.
#### 3.2.2 Query Formulation/Document Retrieval Evaluation
Slots | TD | NQ | LC | SC | LR | SR
---|---|---|---|---|---|---
All | 5 | 742 | 0.468 | 0.252 | 0.954 | 0.252
All | 10 | 742 | 0.534 | 0.307 | 1.558 | 0.307
All | 20 | 742 | 0.589 | 0.358 | 2.434 | 0.358
All | 50 | 742 | 0.616 | 0.391 | 3.915 | 0.391
Table 6: Coverage and Redundancy Analysis for All Entities and All Slots. *TD
= Top Docs, NQ = No of Queries, LC = Lenient Coverage, SC= Strict Coverage, LR
= Lenient Redundancy and SR = Strict Redundancy.
We evaluated query formulation and document retrieval using the coverage and
redundancy measures introduced by ?), originally developed for question
answering. Coverage is the proportion of questions for which answers can be
found from the documents or passages retrieved, while redundancy is the
average number of documents or passages retrieved which contain answers for a
given question or query. These measures may be directly carried over to the
slot filling task, where we treat each slot as a question.
The evaluation used the 2010 TAC-KBP data for all entities and slots; results
are shown in Table 6. Strict and lenient versions of each measure were used,
where for the strict measure both document ID and response string must match
those in the gold standard, while for the lenient only the response string
must match, i.e. the slot fill must be correct but the document in which it is
found need not be one which has been judged to contain a correct slot fill.
This follows the original strict and lenient measures implemented in the tool
we used to assist evaluation, IR4QA [Sanka, 2005].
The results table shows a clear increase in all measures as the number of top
ranked documents is increased. With the exception of lenient redundancy, the
improvement in the scores from the top 20 to the 50 documents is not very big.
Furthermore if 50 documents are processing through the entire system as
opposed to 20, the additional 30 documents will both more than double
processing times per slot and introduce many more potential distractors for
the correct slot fill (See Section 3.1.5). For these reasons we chose to limit
the number of documents passed on from this stage in the processing to 20 per
slot. Note that this bounds our slot fill performance to just under 60%.
#### 3.2.3 Entity Extraction and Coreference Evaluation
We evaluated our entity extractor as follows. We selected one entity and one
slot for entities of type ORGANIZATION and one for entities of type PERSON and
gathered the top 20 documents returned by our query formulation and document
retrieval system for each of these entity-slot pairs. We manually annotated
all candidate slot value across these two twenty document sets to provide a
small gold standard test set. For candidate slot fills in documents matching
the ORGANIZATION query, overall F-measure for the entity identifier was 78.3%
while for candidate slot fills in documents matching the PERSON query, overall
F-measure for the entity identifier was 91.07%. We also manually evaluated our
coreference approach over the same two document sets and arrived at an
F-measure of 73.07% for coreference relating to the ORGANIZATION target entity
and 90.71% for coreference relating to the PERSON target entity. We are still
analyzing the wide difference in performance of both entity tagger and
coreference resolver when processing documents returned in response to an
ORGANIZATION query as compared to documents returned in response to a PERSON
query.
#### 3.2.4 Candidate Slot Value Extraction Evaluation
| DATE | PER | LOC | ORG
---|---|---|---|---
+ve | 97.5 | 95 | 92.5 | 83.33
-ve | 87.5 | 95 | 97.5 | 85
Table 7: Estimated % Training Instances Correct Dataset | DATE | PER | LOC | ORG
---|---|---|---|---
Training | 82.34 | 78.44 | 66.29 | 80
Handpicked +ve | 62.5 | 40 | 36.37 | 45.45
Handpicked -ve | 100 | 75 | 71.42 | 88.89
Table 8: % Slot Values Correctly Extracted
To evaluate our candidate slot value extraction process we did two separate
things. First we assessed the quality of training data provided by our distant
supervision approach. Since it was impossible to check all the training data
produced manually we randomly sampled 40 positive examples for each of four
slot types (slots expecting DATEs, PERSONs, LOCATIONs and ORGANIZATIONs) and
40 negative examples for each of four slot types. Results of this evaluation
are in Table 7.
In addition to evaluating the quality of the training data we generated, we
did some evaluation to determine the optimal feature set combination. Ten fold
cross validation figures for the optimal feature set over the training data
are shown in the first row in Table 8, again for a selection of one slots from
each of four slot types . Finally we evaluated the slot value extraction
capabilities on a small test set of example sentences selected from the source
collection to ensure they contained the target entity and the correct answer,
as well as some negative instances, and manually processed to correctly
annotate the entities within them (simulating perfect upstream performance).
Results are shown in rows2 and 3 of Table 8. The large difference in
performance between the ten fold cross validation figures over the training
and the evaluation against the small handpicked and annotated gold standard
from the source collection may be due to the fact that the training data was
Wikipedia texts while the test set is news texts and potentially other text
types such as blogs; however, the handpicked test set is very small (70
sentences total) so generalizations may not be warranted.
## 4 Temporal Filling Task
The task is to detect upper and lower bounds on the start and end times of a
state denoted by an entity-relation-filler triple. This results in four dates
for each unique filler value. There are two temporal tasks, a full temporal
bounding task and a diagnostic temporal bounding task. We provide the filler
values for the full task, and TAC provides the filler values and source
document for the diagnostic task. Our temporal component did not differentiate
between the two tasks; for the full task, we used output values generated by
our slot-filling component.
We approached this task by annotating source documents in TimeML [Pustejovsky
et al., 2003], a modern standard for temporal semantic annotation. This
involved a mixture of off-the-shelf components and custom code. After
annotating the document, we attempted to identify the TimeML event that best
corresponded to the entity-relation-filler triple, and then proceeded to
detect absolute temporal bounds for this event using TimeML temporal relations
and temporal expressions. We reasoned about the responses gathered by this
exercise to generate a date quadruple as required by the task.
In this section, we describe our approach to temporal filling and evaluate its
performance, with subsequent failure analysis.
### 4.1 System Processing
We divide our processing into three parts: initial annotation, selection of an
event corresponding to the persistence of the filler’s value, and temporal
reasoning to detect start and finish bounds of that state.
#### 4.1.1 TimeML Annotation
Our system must output absolute times, and so we are interested in temporal
expressions in text, or TIMEX3 as they are in TimeML. We are also interested
in events, as these may signify the start, end or whole persistence of a
triple. Finally we need to be able to determine the nature of the relation
between these times and events; TimeML uses TLINKs to annotate these
relationships.
We used a recent version of HeidelTime [Strötgen and Gertz, 2010] to create
TimeML-compliant temporal expression (or timex) annotations on the selected
document. This required a document creation time (DCT) reference to function
best. For this, we built a regular-expression based DCT
extractor222https://bitbucket.org/leondz/add-dct and used it to create a DCT
database of every document in the source collection (this failed for one of
the 1 777 888 documents; upon manual examination, the culprit contained no
hints of its creation time).
The only off-the-shelf TimeML event annotation tool found was Evita [Saurí et
al., 2005], which requires some preprocessing. Specifically, explicit sentence
tokenisation, verb group and noun group annotations need to be added. For our
system we used the version of Evita bundled with
TARSQI333http://timeml.org/site/tarsqi/toolkit/. Documents were preprocessed
with the ANNIE VP Chunker in GATE444http://gate.ac.uk/. We annotated the
resulting documents with Evita, and then stripped the data out, leaving only
TimeML events and the timexes from the previous step.
At this point, we loaded our documents into a temporal annotation analysis
tool, CAVaT [Derczynski and Gaizauskas, 2010], to simplify access to
annotations. Our remaining task is temporal relation annotation. We divided
the classes of entity that may be linked into two sets, as per TempEval
[Verhagen et al., 2010]: intra-sentence event-time links, and inter-sentence
event-event links with a 3-sentence window. Then, two classifiers were learned
for these types of relation using the TimeBank corpus555LDC catalogue entry
LDC2006T08. as training data and the linguistic tools and classifiers in
NLTK666http://www.nltk.org/. Our feature set was the same used as Mani et al.
[Mani et al., 2007] which relied on surface data available from any TimeML
annotation.
#### 4.1.2 Event Selection
To find the timexes that temporally bound a triple, we should first find
events that occur during that triple’s persistence. We call this task “event
selection”. Our approach was simple. In the first instance we looked for a
TimeML event whose text matched the filler. Failing that, we looked for
sentences containing the filler, and chose an event in the same sentence. If
none were found, we took the entire document text and tried to match a
simplified version of the filler text anywhere in the document; we then
returned the closest event to any mention. Finally, we tried to find the
closest timex to the filler text. If there was still nothing, we gave up on
the slot.
Slot name | Count
---|---
per:title | 73
per:member_of | 36
per:employee_of | 33
org:subsidiaries | 29
per:schools_attended | 17
per:cities_of_residence | 14
per:stateorprovinces_of_residence | 13
per:spouse | 11
per:countries_of_residence | 11
org:top_members/employees | 10
Total | 247
Table 9: Distribution of slot types in the available training and sample data.
#### 4.1.3 Temporal Reasoning
Given a TimeML annotation and an event, our task is now to find which timexs
exist immediately before and after the event. We can detect these by
exploiting the commutative and transitive nature of some types of temporal
relation. To ensure that as many relations as possible are created between
events and times, we perform pointwise temporal closure over the initial
automatic annotation with CAVaT’s consistency tool, ignoring inconsistent
configurations. Generating temporal closures is computationally intensive. We
reduced the size of the dataset to be processed by generating isolated groups
of related events and times with CAVaT’s subgraph modules, and then computed
the closure over just these “nodes”.
We now have an event representing the slot filler value, and a directed graph
of temporal relations connecting it to times and events, which have been
decomposed into start and end points. We populate the times as follows:
$T_{1}$: Latest timex before event start
$T_{2}$: Earliest timex after event start
$T_{3}$: Latest timex before event termination
$T_{4}$: Earliest timex after event termination
Slot Type | Score
---|---
per:stateorprovinces_of_residence | 0.583
per:employee_of | 0.456
per:countries_of_residence | 0.787
per:member_of | 0.534
per:title | 0.529
org:top_members/employees | 0.571
per:spouse | 0.535
per:cities_of_residence | 0.744
Weighted overall score | 0.552
Table 10: Scores of our temporal system over the union of temporal sample and
training data.
Timex bounds are simply the start and end points of an annotated TIMEX3
interval. We resolve these to calendar references that specify dates in cases
where their granularity is greater than one day; for example, using 2006-06-01
and 2006-06-30 for the start and end of a 2006-06 timex. Arbitrary points are
used for season bounds, which assume four seasons of three months each, all in
the northern hemisphere. If no bound is found in the direction that we are
looking, we leave that value blank.
### 4.2 Evaluation
Testing and sample data were available for the temporal tasks777LDC catalogue
entries LDC2011E47 and LDC2011E49.. These include query sets, temporal slot
annotations, and a linking file describing which timexes were deemed related
to fillers. The distribution of slots in this data is given in Table 9. To
test system efficacy we evaluated output performance with the provided entity
query sets against these temporal slot annotations. Results are in Table 10,
including per-slot performance.
Retrieval level | Precision | Recall | F1
---|---|---|---
Document | 1.52% | 0.70% | 0.96%
Paragraph | 0.14% | 0.79% | 0.24%
Table 11: Full temporal slot-filling results
Results for the full slot-filling task are given in Table 11. This relies on
accurate slot values as well as temporal bounding. An analysis of our approach
to the diagnostic temporal task, perhaps using a corpus such as TimeBank,
remains for future work.
## 5 Conclusion
We set out to build a framework for experimentation with knowledge base
population. This framework was created, and applied to multiple KBP tasks. We
demonstrated that our proposed framework is effective and suitable for
collaborative development efforts, as well as useful in a teaching
environment. Finally we present results that, while very modest, provide
improvements an order of magnitude greater than our 2010 attempt [Yu et al.,
2010].
## References
* [Baeza-Yates et al., 1999] R. Baeza-Yates, B. Ribeiro-Neto, et al. 1999\. Modern Information Retrieval. ACM press New York.
* [Bird et al., 2009] S. Bird, E. Klein, and E. Loper. 2009\. Natural Language Processing with Python - Analyzing Text with the Natural Language Toolkit. O’Reilly Media.
* [Derczynski and Gaizauskas, 2010] L. Derczynski and R. Gaizauskas. 2010\. Analysing Temporally Annotated Corpora with CAVaT. In Proceedings of the 7th LREC, pages 398–404.
* [Finkel et al., 2005] J.R. Finkel, T. Grenager, and C. Manning. 2005\. Incorporating non-local information into information extraction systems by Gibbs sampling. In Proceedings of ACL, pages 363–370.
* [Ji et al., 2011] H. Ji, R. Grishman, H.T. Dang, X.S. Li, K. Griffit, and J. Ellis. 2011\. Overview of the TAC2011 Knowledge Base Population Track. In Proc. Text Analytics Conference.
* [Mani et al., 2007] I. Mani, B. Wellner, M. Verhagen, and J. Pustejovsky. 2007\. Three approaches to learning TLINKS in TimeML. Technical report, Technical Report CS-07-268, Brandeis University.
* [Mintz et al., 2009] M. Mintz, S. Bills, R. Snow, and D. Jurafsky. 2009\. Distant supervision for relation extraction without labeled data. In Proceedings of the Joint ACL-IJCNLP Conference, pages 1003–1011.
* [Pustejovsky et al., 2003] J. Pustejovsky, J. Castano, R. Ingria, R. Saurí, R. Gaizauskas, A. Setzer, G. Katz, and D. Radev. 2003\. TimeML: Robust specification of event and temporal expressions in text. In IWCS-5 Fifth International Workshop on Computational Semantics.
* [Radford et al., 2010] W. Radford, B. Hachey, J. Nothman, M. Honnibal, and J.R. Curran. 2010\. Document-level Entity Linking: CMCRC at TAC 2010. In Proc. Text Analytics Conference.
* [Rand, 1971] W.M. Rand. 1971\. Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, pages 846–850.
* [Reddy et al., 2010] K. Reddy, K. Kumar, S. Krishna, P. Pingali, and V. Varma. 2010\. Linking Named Entities to a Structured Knowledge Base. In Proceedings of 11th International Conference on Intelligent Text Processing and Computational Linguistics.
* [Roberts and Gaizauskas, 2004] I. Roberts and R. Gaizauskas. 2004\. Evaluating Passage Retrieval Approaches for Question Answering. In S. McDonald and J. Tait, editors, Advances in Information Retrieval: Proceedings of the 26th European Conference on Information Retrieval (ECIR’04), Lecture Notes in Computer Science, Vol. 2997, pages 72–84, Sunderland, April. Springer.
* [Sanka, 2005] A. Sanka. 2005\. Passage retrieval for question answering. Master’s thesis, University of Sheffield.
* [Saurí et al., 2005] R. Saurí, R. Knippen, M. Verhagen, and J. Pustejovsky. 2005\. Evita: a robust event recognizer for QA systems. In Proceedings of EMNLP, pages 700–707.
* [Strötgen and Gertz, 2010] J. Strötgen and M. Gertz. 2010\. HeidelTime: High quality rule-based extraction and normalization of temporal expressions. In Proceedings of SemEval-2010, pages 321–324.
* [Varma et al., 2009] V. Varma, V. Bharat, S. Kovelamudi, P. Bysani, GSK Santosh, K. Kumar, and N. Maganti. 2009\. IIIT Hyderabad at TAC 2009. In Proc. Text Analytics Conference.
* [Verhagen et al., 2010] M. Verhagen, R. Sauri, T. Caselli, and J. Pustejovsky. 2010\. SemEval-2010 task 13: TempEval-2. In Proceedings of SemEval-2010, pages 57–62.
* [Yu et al., 2010] J. Yu, O. Mujgond, and R. Gaizauskas. 2010\. The University of Sheffield System at TAC KBP 2010. In Proc. Text Analytics Conference.
|
arxiv-papers
| 2012-03-22T18:34:19 |
2024-09-04T02:49:28.934412
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Amev Burman, Arun Jayapal, Sathish Kannan, Madhu Kavilikatta, Ayman\n Alhelbawy, Leon Derczynski, Robert Gaizauskas",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5073"
}
|
1203.5076
|
# Massively Increasing TIMEX3 Resources: A Transduction Approach
###### Abstract
Automatic annotation of temporal expressions is a research challenge of great
interest in the field of information extraction. Gold standard temporally-
annotated resources are limited in size, which makes research using them
difficult. Standards have also evolved over the past decade, so not all
temporally annotated data is in the same format. We vastly increase available
human-annotated temporal expression resources by converting older format
resources to TimeML/TIMEX3. This task is difficult due to differing annotation
methods. We present a robust conversion tool and a new, large temporal
expression resource. Using this, we evaluate our conversion process by using
it as training data for an existing TimeML annotation tool, achieving a 0.87
F1 measure – better than any system in the TempEval-2 timex recognition
exercise.
Keywords: Temporal Information Processing, TimeML, temporal expression, corpus
creation
Massively Increasing TIMEX3 Resources: A Transduction Approach
Leon Derczynski∗, Héctor Llorens†, Estela Saquete†
---
∗University of Sheffield
S1 4DP, UK
leon@dcs.shef.ac.uk
†Universidad de Alicante
03690, Spain
hector,stela@dlsi.ua.es
Abstract content
## 1\. Introduction
In this paper, we introduce a tool for unifying temporal annotations produced
under different standards and show how it can be improved to cope with wide
variations in language. We then apply our enhanced tool to existing annotated
corpora to generate a TIMEX3 corpus larger than the sum of all existing TIMEX3
corpora by an order of magnitude and show that this resource is useful for
automatic temporal annotation.
Temporal expressions (timexes) are a basic part of time in language. They
refer to a period or specific time, or temporally reify recurrences. Durations
such as _“two weeks”_ typically have a quantifier and a unit. Dates or times
such as _“next Thursday”_ and _“July 21 2008”_ can be anchored to a calendar
and have set beginning and end bounds; sets like _“every summer”_ indicate a
recurrence pattern.
Dates can be further broken down into deictic and absolute expressions.
Absolute temporal expressions can be directly placed on a calendar without
further information, whereas deictic temporal expressions need some external
(perhaps anaphoric) information to be resolved. For example, _“April 19”_ is
deictic, because its year depends on the context in which it appears.
After a decade of development, there are two main standards with which to
annotate timexes. TIMEX2 [Ferro et al. (2004] is dedicated to timex
annotation. TimeML [Pustejovsky et al. (2005] is a later standard for all
aspects of temporal annotation. It defines TIMEX3 for timex annotation, and
introduces other entities such as events and temporal relations.
Manual creation of fully temporally annotated resources is a complex and
intensive task [Setzer and Gaizauskas (2001]. This has lead to only small
corpora being available. The largest two corpora in TimeML, the current
standard for temporal annotation, total about 260 newswire documents including
just over 2 000 gold standard TIMEX3 annotations. Automatic annotation is also
a difficult task, compounded by the scarcity of annotated training data. To
this end, some recent work has explored the complex issue of converting TIMEX2
corpora to TIMEX3 [Saquete and Pustejovsky (2011].
The current state of affairs is that we have small TIMEX3 resources, much
larger TIMEX2 resources, and a proof-of-concept tool for mapping from TIMEX2
to TIMEX3. Because data sparsity has limited automatic TimeML and TIMEX3
annotation systems, we assume that increasing the volume of TIMEX3 data will
help the performance of such systems. We will do this via conversion of
multiple TIMEX2 resources. Our research questions are as follows:
1. 1.
What practical issues are there in converting large-scale TIMEX2 resources to
TIMEX3?
2. 2.
How can we evaluate the success of such a conversion?
3. 3.
Does extra training data help automatic timex annotation?
We answer these questions in this paper. In Section 2. we introduce the
corpora and an existing format conversion tool and in Section 3. describe how
we enhance it to perform its task more accurately. We use the tool to create
the largest current TIMEX3 resource, described in Section 4.. We then show how
this new training data can be used with a state-of-the-art TIMEX3 annotation
system to improve automatic annotation performance in Section 5. and finally
conclude in Section 6.
## 2\. Background
Manual temporal annotation is a complex, tiring and error-prone process
[Setzer and Gaizauskas (2001]. The abstract notion of temporality and the
requirement to make formal annotations using time have lead to in-depth
annotation schemas accompanied by detailed annotation guidelines. This makes
the generation of temporally annotated resources expensive.
Temporal expressions generally fall in to one of four categories. These are:
* •
Absolute — Where the text explicitly states an unambiguous time. Depending on
the granularity of the interval, the text includes enough information to
narrow a point or interval directly down to one single occurrence. This is in
contrast to a time which, while precise and maybe easy for humans to pin onto
a calendar, relies on an external reference. For example, _Thursday October
1st, 2009_ would be considered absolute, but _The week after next_ would not -
the information is not all explicit or held in the same place; this latter
expression implies reliance on some external time point.
* •
Deictic — Cases where, given a known time of utterance, one can determine the
period being referred to. These time expressions, specify a temporal distance
and direction from the utterance time. One might see a magazine bulletin begin
with _Two weeks ago, we were still in Saigon._ ; this expression relies on an
unclear speech time, which one could safely assume was the date the article
was written. More common examples include _tomorrow_ and _yesterday_ , which
are both offset from the time of their utterance.
* •
Anaphoric — Anaphoric temporal expressions have three parts – temporal
distance (e.g. 4 days), temporal direction (past or future), and an anchor
that the distance and direction are applied from. The anchor, for anaphoric
temporal expressions (sometimes also known as relative temporal expressions),
is an abstract discourse-level point. Example phrases include _the next week_
, _that evening_ or _a few hours later_ , none of which can be anchored even
when their time of utterance is known.
* •
Duration — A duration describes an interval bounded by start and end times.
These might be implicit (_during next week_), where the reader must use world
knowledge to deduce start and end points and their separation distance, or
explicit (_From 8pm to 11.20pm this evening_). Durations generally include a
time unit as their head token. This type of temporal expression is easily
confused with relative expressions; for example, in _“The plane was flying for
seven days”_ , the timex _“seven days”_ acts as a duration, whereas in _“I
will have finished this in seven days”_ , the same timex refers to a point
seven days after the utterance.
The TIDES TIMEX2 standard [Ferro et al. (2004], preceded by the STAG timex
descriptions [Setzer (2001], formally defines how to determine what
constitutes a temporal expression in discourse and further defines an encoding
for temporal expressions. A simple TIMEX2 annotation is shown in Example 2..
The Yankees had just finished <TIMEX2 val="1998-10-02TEV">a draining
evening</TIMEX2> with a 4-0 decision over the Rangers
The TIMEX2 standard is designed to be the sole temporal annotation applied to
a document, and it introduces just one annotation element: <TIMEX2>. As a
result, complex time-referring expressions made of contiguous words are
labelled as a single TIMEX2, perhaps with specific sub-parts annotated as
nested (or “embedded”) TIMEX2s. This is shown in Example 2..
before <TIMEX2 VAL="1999-W23">the week of <TIMEX2 VAL="1999-06-07">the
seventh</TIMEX2> until <TIMEX2 VAL="1999-06-11">the eleventh</TIMEX2>
</TIMEX2>
Later, TIMEX3 was introduced as the next iteration of this timex annotation
scheme. As part of TimeML [Pustejovsky et al. (2005], which is a rich
annotation schema designed to capture a complete range of temporal
information, TIMEX3 focuses on minimally-expressed timexes. This means that
entities that would have been a nested or event-based temporal expressions are
represented as atomic temporal expressions and separate events, the relations
between which are described with TimeML TLINKs. In Example 2., what would have
been a single event-based temporal expression under TIMEX3 is broken down into
an event and a timex which are co-ordinated by a temporal signal.
until <TIMEX3 tid="t31" type="DURATION" value="P90D" temporalFunction="false"
functionInDocument="NONE">90 days</TIMEX3> <SIGNAL sid="s16">after</SIGNAL>
their <EVENT eid="e32" class="OCCURRENCE" stem="issue">issue</EVENT> date.
TimeML removed nested and conjoined timexes, preferring a finer annotation
granularity where timexes are events are separate entities with explicit
relations defined between them. The work in this paper centres on applying a
transducer to TIMEX2 resources to bring them into a LAF-compliant format [Ide
and Romary (2002] (our TIMEX3 annotations are valid ISO-TimeML [Pustejovsky et
al. (2010]). The resulting corpora will further the state-of-the-art in
temporal information extraction.
### 2.1. Comparable Work
The two most similar previous papers cover generation of TIMEX3 from TIMEX2
resources, and creation of TIMEX3 resources. In this section, we describe them
and how our work differs.
Saquete and Pustejovsky [Saquete and Pustejovsky (2011] describe a technique
for converting TIMEX2 to TIMEX3 annotations and present the T2T3 tool as an
implementation of it. As some things annotated as TIMEX2s were no longer
considered parts of temporal expressions in TimeML and instead assigned to
other functions, T2T3 generates not only TIMEX3s but also any extra TimeML
elements. T2T3 is evaluated using TimeBank [Pustejovsky et al. (2003] and 50
ACE TERN documents. This work was novel, but its practical evaluation limited
to the TimeBank corpus and a small selection from the ACE TERN data. In terms
of temporal expressions, there is not much more diversity to be found in
TimeBank, which is often used as the sole training and testing resource for
temporal information processing systems. Although it is new data, only a small
sample of the ACE data was used for the original evaluation of T2T3. In our
work, we greatly increase the volume and variety of text converted from
TIMEX2, creating a more robust, enhanced tool that works beyond a
demonstration dataset.
TimeBank [Pustejovsky et al. (2003] and the AQUAINT TimeML corpus111See
http://www.timeml.org/timebank/timebank.html. comprise around 250 TimeML
annotated documents. These newswire corpora have annotations for temporal
entities other than timexes in addition to a total of 2 023 TIMEX3
annotations. While mature and gold-standard annotated, existing TimeML corpora
(TimeBank and the AQUAINT TimeML corpus222See http://www.timeml.org/.) are
limited in size and scope, and larger resources are required to advance the
state of the art. Our contribution is that we introduce new high-quality
automatically-generated resources, derived from gold-standard annotations.
These comprise large numbers of new timex, event and relation annotations,
covering a wider range of forms of expression.
Resource name | Type | Words | Annotations
---|---|---|---
TimeBank v1.2 | TIMEX3 | 68.5K | 1 414
AQUAINT | TIMEX3 | 34.1K | 609
TempEval-2 test | TIMEX3 | 5.5K | 81
TimenEval | TIMEX3 | 7.9K | 214
_Total_ | | 116K | _3 289_
WikiWars | TIMEX2 | 120K | 2 681
ACE 2004 TERN | TIMEX2 | 54.6K | 8 047
ACE 2005 | TIMEX2 | 260K | 5 483
TIDES dialogue | TIMEX2 | 31.6K | 3 541
_Total_ | | 466K | _19 752_
Table 1: A summary of publicly-available TIMEX2- and TIMEX3-annotated corpora
for English.
### 2.2. TIMEX2 Datasets
There are a few TIMEX2-standard datasets available, both new and old. In this
section, we describe the publicly-available TIMEX2-annotated corpora.
Corpora are still produced in TIMEX2 format [Mazur and Dale (2010, Strötgen
and Gertz (2011]. It is less complex than TimeML and if one is only concerned
with temporal expressions, one may annotate these adequately without requiring
annotation of temporal signals, events and relations. This gives the situation
where similar information is annotated in incompatible formats, impeding the
work of those interested in TimeML annotation.
TIMEX2 resources were produced in volume for the ACE TERN tasks [Ferro (2004]
and temporal information extraction research conducted shortly after. These
contained no other temporal annotations (e.g. for events). Considerable
investment was made in developing annotation guidelines and resources and as a
result some very large and well-annotated corpora are available in TIMEX2
format. For example, the ACE 2004 Development Corpus333See LDC catalogue refs.
LDC2005T07 & LDC2010T18. contains almost 900 documents including approximately
8 000 TIMEX2 annotations. For a discussion of the nuances of these resources
and this standard, see Mazur and Dale [Mazur and Dale (2010].
The ACE2005 corpus444See LDC catalogue ref. LDC2006T06. [Strassel et al.
(2008] includes text of multiple genres annotated with a variety of entity
types, including timexes. The corpus contains text from broadcast news,
newswire, web logs, broadcast conversation, usenet discussions, and
conversational telephone speech – a much wider range of genres than existing
English TIMEX3 resources (which are almost exclusively newswire).
As part of an effort to diversify the genres of timex-annotated corpora,
WikiWars [Mazur and Dale (2010] is a 20-document corpus of Wikipedia articles
about significant wars, annotated to TIMEX2. Document length provides
interesting challenges regarding tracking frame of temporal reference and co-
reference, and the historical genre provides a wide range of temporal
granularities (from seconds to millenia) as well as a wealth of non-
contemporary timexes.
Finally, the TIDES Parallel Temporal corpus contains transcriptions of
conversations about arranging dates. The conversations were originally in
Spanish and comprised that language’s part of the Enthusiast corpus [Suhm et
al. (1994], which were later translated into English (by humans). These
dialogues thus comprise a parallel corpus rich in temporal language, where
both languages are fully annotated according to TIMEX2. Utterances in this
case tend to have a high ratio of timexes per sentence, and the language used
to describe times is heavily context-dependent compared to newswire. For
example, dates and times are often referred to by only numbers (_“How about
the ninth? Or the tenth?”_ without an accompanying explicit temporal unit.
A summary of timex-annotated English corpora is given in Table 1. Aside from
TimeBank and AQUAINT, other relevant TIMEX3 corpora are the TempEval-2
international evaluation exercise dataset [Verhagen et al. (2010] and the
TimenEval TIMEX3 dataset [Llorens et al. (2012].
<timex2 ID="TTRACY_20050223.1049-T1" VAL="FUTURE_REF" ANCHOR_VAL="2005-02-23T10:49:00" ANCHOR_DIR="AFTER">
<timex2_mention ID="TTRACY_20050223.1049-T1-1">
<extent>
<charseq START="1768" END="1787">the next month or so</charseq>
</extent>
</timex2_mention>
</timex2>
Figure 1: Example ACE2005 corpus standoff annotation.
### 2.3. Applications
Here we discuss three applications of the resulting TIMEX3 resource: improved
timex recognition, improved timex interpretation and temporal annotation of
the semantic web.
Annotating non-newswire texts is problematic with only newswire training data,
and solving this problem has practical benefits. TIMEX3 annotated resources
are almost exclusively newswire, and the breadth of genres covered by TIMEX2
resources should help with this problem. These previous datasets cover a wide
variety of genres, as opposed to existing TIMEX3 resources, which are (with
the partial exception of three TimenEval documents) all newswire. The limited
variation in forms of expression given a single genre reduces performance of
timex recognition systems trained on such data when applied to other genres.
Thus, our addition of TIMEX3 annotations in new genres should permit
improvements in timex annotation performance in more general contexts.
The ability to automatically build a formal representation of a temporal
expression from a phrase in text is improved with more source data. After a
timex’s extents have been determined, the next annotation step is to interpret
it in context and build a standardised representation of timex’s semantics,
such as an ISO 8601 compliant specification of a calendrical time or date.
This is called timex normalisation. In the small existing datasets, newswire,
dates, times and durations are expressed in a limited manner. The diversity of
temporal expression phrases grows with the volume of annotated timex
resources. Building a complete and high-performance temporal expression
normalisation system therefore requires a large and diverse resource.
The Semantic web poses a tough temporal annotation problem [Wilks (2008]. To
temporally annotate the semantic web, one requires both a standard and also
tools capable of performing reliable annotation on data with extremely
variable quality. Annotation standards have been proposed – TIMEX3 is suitable
for temporal expressions, and OWL-TIME [Hobbs and Pan (2004] is a temporal
ontology suitable for the semantic web. When it comes to dealing with text
quality on the web, even semi-structured resources such as Wikipedia pose
challenges [Völkel et al. (2006, Maynard et al. (2009, Wang et al. (2010]. For
example, dates are often expressed inconsistently on Wikipedia as well as
other phrases used to express durations, times and sets, both in article text
and infoboxes. While a capable timex normalisation system should be able to
handle variances in this kind of expression, the lack of formal timex
annotation can make for slow work. Thanks to WikiWars, our final TIMEX3
resource includes a significant amount of Wikipedia data, annotated and
normalised in TIMEX3 format. This paves the way for the creation of data-
driven systems that are capable of formally annotating Wikipedia (and other
resources) for the semantic web.
## 3\. Method
The original T2T3 tool worked well with a subset of the ACE TERN corpus and
TimeBank. However, upgrades were needed to cope with linguistic variations in
new text. In this section, we detail our handling of the source datasets and
our solutions to linguistic and technical shortcomings of the original T2T3
when applied to these datasets.
Our general approach has three stages. Firstly, we pre-process the source
documents into a uniform format. Then, we run T2T3 over each document
individually. Finally, we wrap the resulting annotations in TimeML header and
footer and validate them. This process produces a corpus based on gold-
standard annotations, though cannot be said to be gold-standard as the
machine-generated annotation transductions have not all been manually checked
and corrected. To compensate for this, we release the corpus as version 1.0,
and will provide future releases repairing mis-annotations as they are found.
Our development cycle consisted of processing source documents with T2T3 and
then validating the output using a TimeML corpus analysis tool [Derczynski and
Gaizauskas (2010]. We would then compare the structure of the source documents
with the consequent TimeML. Any errors or mis-conversions prompted
modifications to T2T3. Converting WikiWars proved an especially useful
challenge due to the variety of non-English text and encodings found within.
In this section we describe our TIMEX2 corpus preprocessing, the enhancements
made to T2T3, and the validation process.
#### 3.0.1. Preprocessing
The target format for T2T3 to work with is plain Unicode text, containing
TIMEX2 annotations delimited by <TIMEX2> tags. The following work needed to be
done to bring source corpora into this format. All meta-information and other
XML tags are stripped. In the case of the ACE2005 data, standoff annotations
such as in Example 1. Along with the source documents, these annotations were
merged in to form inline TIMEX2 elements. Finally, all documents were (where
possible) converted to UTF8 or UTF16, with unrecognised entities removed.
Wikiwars documents were the hardest to map, having more than one encoding, but
these contain words from almost twenty languages in total with more than seven
different writing systems.
[.walked [.Neil Armstrong ] [.on [.the moon ] ] ]
Figure 2: A chunk of a sentence, dependency parsed in order to find which word
to annotate as an event.
### 3.1. Running T2T3
#### 3.1.1. Signalled event-based times
Some longer TIMEX2s position a timex relative to an event by means of a co-
ordinating phrase with temporal meaning. This co-ordinating phrase is known as
a temporal signal. To convert this into TimeML, the event and signal need to
be identified, allowing shortening of annotation to just the timex according
to the standard. For this, we use an approach that first identifies the signal
(according to the definition and investigation of temporal signals provided in
?)) and then determines which parts of the remaining parts of the phrase
(“chunks”) are a TimeML TIMEX3 and EVENT.
This procedure constitutes handling a special case (also the majority case) of
event-based times, where an event provides a deictic reference required to
normalise the time expression. Example 3.1.1. is a single TIMEX2, whereas the
only TIMEX3 in the phrase is _Tuesday_.
_“The Tuesday after the party”_
The example might look like this as a TIMEX2:
<TIMEX2 VAL="2012-03-20">The Tuesday after the party</TIMEX2>
and as follows in (slightly simplified) TimeML:
The
<TIMEX3 tid="t1" value="2012-03-20">Tuesday</TIMEX3>
<SIGNAL sid="s1">after</SIGNAL>
the
<EVENT eid="e1" type="OCCURRENCE">party</EVENT>
<TLINK timeID="t1" relType="AFTER"
relatedEventID="e1" signalID="s1" />
Example 3.1.1. shows the expansion of a signalled event-based TIMEX2 into
TimeML EVENT, SIGNAL, TLINK and TIMEX3 annotations. One may unpack Example
3.1.1. as follows: _the party_ is an event, _Tuesday_ a TIMEX3 and _after_ a
temporal signal that explicitly connects the TIMEX3 and event, using a TimeML
TLINK.
To achieve this kind of unpacking, it is critical to first select the signal
correctly and then subdivide the remainder of the TIMEX2 annotation in order
to determine the event and timex elements. We approach this as follows.
1. 1.
From a closed class of temporal signal phrases, find a phrase that co-
ordinates the TIMEX2. Our strategy in the case that there is more than one
candidate is this. Based on a corpus-based survey of temporal signal phrase
meanings [Derczynski and Gaizauskas (2011], we prefer monosemous words (giving
preference to the most frequently-occurring ones) followed by polysemous words
in descending order of likelihood of being a temporal signal. This gives us at
most one signal annotation.
2. 2.
Split the original timex into (up to) three contiguous chunks: pre-signal
words, signal phrase, and post-signal words.
3. 3.
Make the timex chunk the shortest one that has a timex measure word (such as
_“day”_), removing tailing or prefixing prepositions and articles. If there is
no such matching chunk, make the first chunk the timex chunk.
4. 4.
The event chunk contains an event word. Annotate the word that dominates this
chunk, based on a dependency parse [De Marneffe et al. (2006] – see Figure 2.
5. 5.
Add an untyped TLINK between the event and timex, supported by the signal.
For example, in _“the 30 years since Neil Armstrong walked on the moon”_ , we
split on the monosemous signal word _since_ (and not _on_). The time chunk is
initially _the 30 years_ , from which we remove _the_ to end up with _30
years_ – the destination TIMEX3, given the same value as in TIMEX2 (a
duration, P30Y). The remainder is dependency parsed (Figure 2) and the
dominant word, _walked_ , annotated as an event.
[.SBAR [.PP twenty days later.NP than [.S the termination notice.NP ] [.VP ’s
[.NP delivery ] ] ] ]
Figure 3: Constituent parse of a long TIMEX2. Timex length | Frequency
---|---
1 | 654
2 | 426
3 | 226
4 | 81
5 | 22
6 | 1
$\geq$7 | 0
Table 2: Distribution of token-length of TIMEX3s in TimeBank.
#### 3.1.2. Nested expressions
As discussed in Section 2., TIMEX2 produces larger annotations than 3, which
may be nested (as in Example 2.). T2T3 does not handle these. They need to be
mapped to multiple TIMEX3 annotations, perhaps with an associated anchorTimeID
attribute or temporal relation.
Following from that above example, given the text of _“the week of the
seventh”_ , the destination TimeML annotation is to describe a week-long
duration, two single specific days, and the temporal relations between all
three. This would look as follows:
<TIMEX3 tid="t1" type="DATE" value="1999-W23">the
week</TIMEX3>
<SIGNAL sid="s1">of</SIGNAL>
<TIMEX3 tid="t2" type="DATE" value="1999-06-07">
the seventh</TIMEX3>
<TLINK timeID="t1" relType="INCLUDED\_BY"
relatedToTime="t2" signalID="s1" />
We reach this automatically by:
1. 1.
Finding all the TIMEX2s in the scope of the outer one which do not have any
children, and mapping them to TIMEX3;
2. 2.
Searching for co-ordinating phrases indicating temporal relations, and
annotating those as signals;
3. 3.
Break the string into chunks, with boundaries based on tokens and sub-element
– new TIMEX3 and SIGNAL – bounds;
4. 4.
Select the chunk most likely to be a timex corresponding to the TIMEX2 VAL
attribute, preferring chunks containing temporal measure words (such as
_week_) and chunks near the front, and convert it to TIMEX3;
5. 5.
Insert TLINK annotations to co-ordinate the new elements, based on value clues
and signal-suggested orderings (relation type is left blank when ambiguous).
#### 3.1.3. Brevity
We automatically trim long timexes. TIMEX3 annotations are minimal – that is,
including the minimal set of words that can describe a temporal expression –
where TIMEX2 can include whole phrases. Even after reducing the long
annotations that contain temporal substructres, a significant amount of text
can remain in some cases. To handle this, we implement reduction of long
TIMEX2s into just the TIMEX3-functional part. This is done by measuring the
distribution of TIMEX3 token lengths in gold standard corpora, and determining
a cut-off point. This distribution is shown in Table 2. Any TIMEX2s of six
tokens or more that have no yet been handled by the algorithms mentioned above
are syntactically parsed. They are then reduced to the largest same-
constituent chunk that is shorter than six tokens and contains a temporal
measure word, with preference given to the leftmost arguments.
Example 3.1.3. shows a long TIMEX2.
_twenty days later than the termination notice’s delivery_
This produces the constituent tree shown in Figure 3. In this case, the four
chunks below the root node are considered first; the NP contains a temporal
measure word (_days_) and so the TIMEX2 annotation over the whole string is
reduced to a TIMEX3 over just _“twenty days later”_.
#### 3.1.4. Technical changes
To speed up processing, we moved to NLTK555See http://www.nltk.org/. for PoS
tagging, which is a maximum entropy based tagger trained on the Penn Treebank.
We also stopped doing lemmatisation, as in practise it is never used. For a
further speedup, we implemented a timex phrase PoS tagging cache; this reduced
execution times by two orders of magnitude.
The tool has generally become more robust and now handles a greater range of
texts, providing more precise TimeML annotation. Our work has resulted in a
publicly available tool, downloadable from a public Mercurial repository666See
http://bitbucket.org/leondz/t2t3..
### 3.2. Document post-processing
After conversion of the body text to TIMEX3, each document is designated at
TimeML by giving it an XML header and wrapping the text in <TimeML> elements.
Each document is then processed by a DOM parser to check for basic validity,
then strictly evaluated compared to the TimeML XSD to check for representation
errors, and finally verified at a high level with CAVaT [Derczynski and
Gaizauskas (2010]. This results in consistent and TimeML-valid documents.
Corpus name | TIMEX2s | TIMEX3s
---|---|---
WikiWars | 2 681 | 2 676
ACE 2004 TERN | 8 047 | 7 638
ACE 2005 | 5 483 | 5 199
TIDES parallel dialogue | 3 481 | 3 290
Table 3: Annotation counts after conversion to TIMEX3. Corpus | DATE | DUR. | TIME | SET
---|---|---|---|---
TimeBank | 1 164 | 175 | 63 | 12
AQUAINT | 495 | 69 | 27 | 14
TempEval-2 | 984 | 170 | 42 | 12
TimenEval | 121 | 34 | 43 | 16
WikiWars | 2 323 | 230 | 93 | 30
ACE 2004 | 4 697 | 947 | 1 759 | 235
ACE 2005 | 3 024 | 628 | 1 406 | 141
TIDES dialogue | 1 684 | 141 | 1 402 | 63
_Total_ | _14 492_ | _2 394_ | _4 835_ | _523_
Table 4: Distribution of timex types in all TIMEX3 corpora. Corpus | TempEval-2 | Entity-based
---|---|---
Train | Test | P | R | F1 | P | R | F1
TBAQ | T2T3 | 0.84 | 0.35 | 0.49 | 64.4% | 32.9% | 43.6%
TBAQ + T2T3 | 80/20 split | 0.84 | 0.74 | 0.79 | 72.1% | 71.0% | 71.5%
TBAQ train (80%) | TBAQ test (20%) | 0.93 | 0.80 | 0.86 | 85.2% | 69.8% | 76.7%
T2T3 + TBAQ train | TBAQ test (20%) | 0.87 | 0.87 | 0.87 | 80.1% | 80.1% | 80.1%
Table 5: Timex recognition results. TBAQ corresponds to the merger of the
TimeBank 1.2 and AQUAINT TimeML corpora.
## 4\. Resource creation
In this section we describe the results of converting all the aforementioned
corpora. Table 3 shows the volume of TIMEX3 and other TimeML element
annotations created by conversion from TIMEX2. The TIMEX2 counts are generally
slightly lower; this may be caused by the removal of nested temporal
expressions, where the outer timex annotation is removed during conversion. To
show the timex composition of the resultant corpora, Table 4 shows the
distribution of timex type in native and in converted corpora; they introduce
18 803 new TIMEX3 annotations.
Of these timexes, 4 343 are in “web-grade” data – that is, data taken from
blogs, forums newsgroups and Wikipedia. These include 2 676 from Wikiwars
(Wikipedia) and the remainder from ACE2005 – 675 from newsgroups, 20 from
community forums and 972 from crawled web text. This is a significant resource
for developing automatic methods to accurately and consistently annotate
temporal information for the semantic web.
## 5\. Evaluation
We evaluate the impact of our new resources by measuring the performance of a
state-of-the-art timex recogniser, TIPSem-B [Llorens et al. (2010]. It
achieves competitive performance when trained over the TimeBank and AQUAINT
corpora. We extend its training set to include our newly generated data. Our
evaluation includes timex annotation (both recognition and interpretation)
performance on:
1. 1.
New (T2T3) data when trained on prior data (TimeBank + AQUAINT) to show the
“difficulty” of the new data given current TIMEX3 training resources;
2. 2.
A mixture of prior and T2T3 data, with an 80/20 training/test split, to show
how the recognition method handles the new data;
3. 3.
Prior data with an 80/20 training/test split, as a baseline measure;
4. 4.
As above, but with all of the T2T3 data added to the training set, to see its
impact on the TIMEX3 task as previously posed;
Performance is reported using both entity recognition precision and recall
(strict), as well as the TempEval-2 scorer, which uses a token-based metric
instead of entity-based matching (see ?) for details). Results are given in
Table 5.
### 5.1. General recognition
The timex recognition performance on our T2T3 data of systems trained using
prior newswire-only corpora was low, with an F1 measure below 50%. This
suggests that existing resources are not sufficient to develop generic timex
recognition models that are effective outside the newswire genre. However,
existing recognition methods are capable of adapting to the new corpora given
some of it as training data; an 80/20 training/test split of combined
newswire/T2T3 timexes gave F1 measures in the seventies.
### 5.2. Improving performance on newswire
It is useful to measure performance on a TempEval-2-like task – recognising
timexes in the TimeBank/AQUAINT TimeML corpora. To this end, we set an 80/20
training/test split of TBAQ (TimeBank + AQUAINT) and measured system
performance on a model learned from the training data. The large
T2T3-generated resource is then added to the training data, the recognition
model re-learned, and performance evaluated. As shown by the results, the
larger set of more-diverse training data provides an improvement over the
TimeBank set. Recall rises considerably at the cost of some precision, under
both evaluation metrics. This matches with what one might expect given a much
wider range of expression forms in the training data. The final result for
TempEval-2 F1 measure is greater than the best score achieved during the
TempEval-2 evaluation task.
## 6\. Conclusion
Identifying and overcoming issues with TIMEX2/TIMEX3 conversion, we have
created a robust tool for converting TIMEX2 resources to TimeML/TIMEX3. Using
this, we have generated a TIMEX3 resource with an order of magnitude more
annotations than all previous resources put together. The resource contains
new information about temporal expressions, and is helpful for training
automatic timex annotation systems.
We have made both the transduction tool and the TIMEX3 annotated results
available, as part of a public repository. Version 1.0 of the data is packaged
as a single release available on the project web page (distribution licenses
apply).
### 6.1. Future Work
As TIMEX2 resources exist in languages other than English (particularly
Spanish and German), T2T3 can be enhanced to cater for these as well as
English.
Further, the extensive diversity of temporal expression phrasings found in the
corpora introduced in this paper coupled with their TIMEX3 annotations is a
significant boon to those working on the problem of timex normalisation.
### 6.2. Acknowledgments
The authors would like to thank Lisa Ferro of the MITRE Corporation for kind
help with corralling TIMEX2 resources. The first author would also like to
acknowledge the support of the UK Engineering and Physical Science Research
Council in the form of a doctoral training grant. This paper has been also
supported by the Spanish Government, in projects TIN-2009-13391-C04-01,
MESOLAP TIN2010-14860, PROMETEO/2009/119 and ACOMP/2011/001.
## References
* De Marneffe et al. (2006 M.C. De Marneffe, B. MacCartney, and C.D. Manning. 2006\. Generating typed dependency parses from phrase structure parses. In Proceedings of the 4th International Conference on Language Resources and Evaluation, volume 6, pages 449–454.
* Derczynski and Gaizauskas (2010 L. Derczynski and R. Gaizauskas. 2010\. Analysing Temporally Annotated Corpora with CAVaT. In Proceedings of the 7th International Conference on Language Resources and Evaluation, pages 398–404.
* Derczynski and Gaizauskas (2011 L. Derczynski and R. Gaizauskas. 2011\. A Corpus-based Study of Temporal Signals. In Proceedings of the 6th Corpus Linguistics Conference.
* Ferro et al. (2004 L. Ferro, L. Gerber, I. Mani, B. Sundheim, and G. Wilson. 2004\. TIDES 2005 standard for the annotation of temporal expressions. MITRE, April.
* Ferro (2004 L. Ferro. 2004\. TERN evaluation task overview and corpus. In Time Expression Recognition and Normalization Workshop.
* Hobbs and Pan (2004 J.R. Hobbs and F. Pan. 2004\. An ontology of time for the semantic web. ACM Transactions on Asian Language Information Processing (TALIP), 3(1):66–85.
* Ide and Romary (2002 N. Ide and L. Romary. 2002\. Standards for Language Resources. In Proceedings of the 3rd Language Resources and Evaluation Conference, pages 839–844.
* Llorens et al. (2010 H. Llorens, E. Saquete, and B. Navarro-Colorado. 2010\. TIPSem (English and Spanish): Evaluating CRFs and Semantic Roles in TempEval-2. In Proceedings of the 5th International Workshop on Semantic Evaluation, pages 284–291. ACL.
* Llorens et al. (2012 H. Llorens, L. Derczynski, R. Gaizauskas, and E. Saquete. 2012\. TIMEN: An Open Temporal Expression Normalisation Resource. In Proceedings of the 7th International Conference on Language Resources and Evaluation.
* Maynard et al. (2009 D. Maynard, A. Funk, and W. Peters. 2009\. Sprat: a tool for automatic semantic pattern-based ontology population. In Proceedings of the International Conference for Digital Libraries and the Semantic Web.
* Mazur and Dale (2010 P. Mazur and R. Dale. 2010\. WikiWars: a new corpus for research on temporal expressions. In Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing, pages 913–922. Association for Computational Linguistics.
* Pustejovsky et al. (2003 J. Pustejovsky, P. Hanks, R. Sauri, A. See, R. Gaizauskas, A. Setzer, D. Radev, B. Sundheim, D. Day, L. Ferro, et al. 2003\. The TimeBank corpus. In Corpus Linguistics, volume 2003, page 40.
* Pustejovsky et al. (2005 J. Pustejovsky, B. Ingria, R. Sauri, J. Castano, J. Littman, R. Gaizauskas, A. Setzer, G. Katz, and I. Mani. 2005\. The specification language TimeML. The language of time: A reader, pages 545–557.
* Pustejovsky et al. (2010 J. Pustejovsky, K. Lee, H. Bunt, and L. Romary. 2010\. ISO-TimeML: An International Standard for Semantic Annotation. In Proceedings of the 7th conference on International Language Resources and Evaluation.
* Saquete and Pustejovsky (2011 E. Saquete and J. Pustejovsky. 2011\. Automatic transformation from TIDES to TimeML annotation. Language Resources and Evaluation.
* Setzer and Gaizauskas (2001 A. Setzer and R. Gaizauskas. 2001\. A pilot study on annotating temporal relations in text. In Proceedings of the workshop on Temporal and spatial information processing, pages 1–8. Association for Computational Linguistics.
* Setzer (2001 A. Setzer. 2001\. Temporal information in newswire articles: an annotation scheme and corpus study. Ph.D. thesis, University of Sheffield, Department of Computer Science.
* Strassel et al. (2008 S. Strassel, M. Przybocki, K. Peterson, Z. Song, and K. Maeda. 2008\. Linguistic resources and evaluation techniques for evaluation of cross-document automatic content extraction. In Proceedings of the 6th International Conference on Language Resources and Evaluation.
* Strötgen and Gertz (2011 J. Strötgen and M. Gertz. 2011\. WikiWarsDE: A German Corpus of Narratives Annotated with Temporal Expressions. In Proceedings of GSCL 2011: Conference of the German Society for Computational Linguistics and Language Technology.
* Suhm et al. (1994 B. Suhm, L. Levin, N. Coccaro, J. Carbonell, R. Isotani, A. Lavie, L. M Ayfield, and C. P. Rose. 1994\. Speech-language Integration in a Multi-lingual Speech Translation System. In Proceedings of the Workshop on Integration of Natural Language and Speech Processing, AAAI.
* Verhagen et al. (2010 M. Verhagen, R. Sauri, T. Caselli, and J. Pustejovsky. 2010\. SemEval-2010 task 13: TempEval-2. In Proceedings of the 5th International Workshop on Semantic Evaluation, pages 57–62. Association for Computational Linguistics.
* Völkel et al. (2006 M. Völkel, M. Krötzsch, D. Vrandecic, H. Haller, and R. Studer. 2006\. Semantic wikipedia. In Proceedings of the 15th international conference on World Wide Web, pages 585–594. ACM.
* Wang et al. (2010 Y. Wang, M. Zhu, L. Qu, M. Spaniol, and G. Weikum. 2010\. Timely YAGO: Harvesting, querying, and visualizing temporal knowledge from Wikipedia. In Proceedings of the 13th International Conference on Extending Database Technology, pages 697–700. ACM.
* Wilks (2008 Y. Wilks. 2008\. The semantic web: Apotheosis of annotation, but what are its semantics? Intelligent Systems, IEEE, 23(3):41–49.
|
arxiv-papers
| 2012-03-22T18:45:07 |
2024-09-04T02:49:28.944569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski and H\\'ector Llorens and Estela Saquete",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5076"
}
|
1203.5084
|
# A Data Driven Approach to Query Expansion in Question Answering
Leon Derczynski, Jun Wang, Robert Gaizauskas and Mark A. Greenwood
Department of Computer Science
University of Sheffield
Regent Court, 211 Portobello
Sheffield S1 4DP UK
{aca00lad, acp07jw, r.gaizauskas, m.greenwood}@shef.ac.uk
###### Abstract
Automated answering of natural language questions is an interesting and useful
problem to solve. Question answering (QA) systems often perform information
retrieval at an initial stage. Information retrieval (IR) performance,
provided by engines such as Lucene, places a bound on overall system
performance. For example, no answer bearing documents are retrieved at low
ranks for almost 40% of questions.
In this paper, answer texts from previous QA evaluations held as part of the
Text REtrieval Conferences (TREC) are paired with queries and analysed in an
attempt to identify performance-enhancing words. These words are then used to
evaluate the performance of a query expansion method.
Data driven extension words were found to help in over 70% of difficult
questions. These words can be used to improve and evaluate query expansion
methods. Simple blind relevance feedback (RF) was correctly predicted as
unlikely to help overall performance, and an possible explanation is provided
for its low value in IR for QA.
## 1 Introduction
The task of supplying an answer to a question, given some background
knowledge, is often considered fairly trivial from a human point of view, as
long as the question is clear and the answer is known. The aim of an automated
question answering system is to provide a single, unambiguous response to a
natural language question, given a text collection as a knowledge source,
within a certain amount of time. Since 1999, the Text Retrieval Conferences
have included a task to evaluate such systems, based on a large pre-defined
corpus (such as AQUAINT, containing around a million news articles in English)
and a set of unseen questions.
Many information retrieval systems perform document retrieval, giving a list
of potentially relevant documents when queried – Google’s and Yahoo!’s search
products are examples of this type of application. Users formulate a query
using a few keywords that represent the task they are trying to perform; for
example, one might search for “eiffel tower height” to determine how tall the
Eiffel tower is. IR engines then return a set of references to potentially
relevant documents.
In contrast, QA systems must return an exact answer to the question. They
should be confident that the answer has been correctly selected; it is no
longer down to the user to research a set of document references in order to
discover the information themselves. Further, the system takes a natural
language question as input, instead of a few user-selected key terms.
Once a QA system has been provided with a question, its processing steps can
be described in three parts - Question Pre-Processing, Text Retrieval and
Answer Extraction:
#### 1\. Question Pre-Processing
TREC questions are grouped into series which relate to a given target. For
example, the target may be “Hindenburg disaster” with questions such as “What
type of craft was the Hindenburg?” or “How fast could it travel?”. Questions
may include pronouns referencing the target or even previous answers, and as
such require processing before they are suitable for use.
#### 2\. Text Retrieval
An IR component will return a ranked set of texts, based on query terms.
Attempting to understand and extract data from an entire corpus is too
resource intensive, and so an IR engine defines a limited subset of the corpus
that is likely to contain answers. The question should have been pre-processed
correctly for a useful set of texts to be retrieved – including anaphora
resolution.
#### 3\. Answer Extraction (AE)
Given knowledge about the question and a set of texts, the AE system attempts
to identify answers. It should be clear that only answers within texts
returned by the IR component have any chance of being found.
Reduced performance at any stage will have a knock-on effect, capping the
performance of later stages. If questions are left unprocessed and full of
pronouns (e.g.,“When did it sink?”) the IR component has very little chance of
working correctly – in this case, the desired action is to retrieve documents
related to the Kursk submarine, which would be impossible.
IR performance with a search engine such as Lucene returns no useful documents
for at least 35% of all questions – when looking at the top 20 returned texts.
This caps the AE component at 65% question “coverage”. We will measure the
performance of different IR component configurations, to rule out problems
with a default Lucene setup.
For each question, answers are provided in the form of regular expressions
that match answer text, and a list of documents containing these answers in a
correct context. As references to correct documents are available, it is
possible to explore a data-driven approach to query analysis. We determine
which questions are hardest then concentrate on identifying helpful terms
found in correct documents, with a view to building a system than can
automatically extract these helpful terms from unseen questions and supporting
corpus. The availability and usefulness of these terms will provide an
estimate of performance for query expansion techniques.
There are at least two approaches which could make use of these term sets to
perform query expansion. They may occur in terms selected for blind RF (non-
blind RF is not applicable to the TREC QA task). It is also possible to build
a catalogue of terms known to be useful according to certain question types,
thus leading to a dictionary of (known useful) expansions that can be applied
to previously unseen questions. We will evaluate and also test blind relevance
feedback in IR for QA.
## 2 Background and Related Work
The performance of an IR system can be quantified in many ways. We choose and
define measures pertinent to IR for QA. Work has been done on relevance
feedback specific to IR for QA, where it is has usually be found to be
unhelpful. We outline the methods used in the past, extend them, and provide
and test means of validating QA relevance feedback.
### 2.1 Measuring QA Performance
This paper uses two principle measures to describe the performance of the IR
component. _Coverage_ is defined as the proportion of questions where at least
one answer bearing text appears in the retrieved set. _Redundancy_ is the
average number of answer bearing texts retrieved for each question [2004].
Both these measures have a fixed limit $n$ on the number of texts retrieved by
a search engine for a query. As redundancy counts the number of texts
containing correct answers, and not instances of the answer itself, it can
never be greater than the number of texts retrieved.
The TREC reference answers provide two ways of finding a correct text, with
both a regular expression and a document ID. Lenient hits (retrievals of
answer bearing documents) are those where the retrieved text matches the
regular expression; strict hits occur when the document ID of the retrieved
text matches that declared by TREC as correct _and_ the text matches the
regular expression. Some documents will match the regular expression but not
be deemed as containing a correct answer (this is common with numbers and
dates [1999]), in which case a lenient match is found, but not a strict one.
The answer lists as defined by TREC do not include every answer-bearing
document – only those returned by previous systems and marked as correct.
Thus, false negatives are a risk, and strict measures place an approximate
lower bound on the system’s actual performance. Similarly, lenient matches can
occur out of context, without a supporting document; performance based on
lenient matches can be viewed as an approximate upper bound [2005].
### 2.2 Relevance Feedback
Relevance feedback is a widely explored technique for query expansion. It is
often done using a specific measure to select terms using a limited set of
ranked documents of size $r$; using a larger set will bring term distribution
closer to values over the whole corpus, and away from ones in documents
relevant to query terms. Techniques are used to identify phrases relevant to a
query topic, in order to reduce noise (such as terms with a low corpus
frequency that relate to only a single article) and query drift [2005, 1996].
In the context of QA, Pizzato [2006] employs blind RF using the AQUAINT corpus
in an attempt to improve performance when answering factoid questions on
personal names. This is a similar approach to some content in this paper,
though limited to the study of named entities, and does not attempt to examine
extensions from the existing answer data.
Monz [2003] finds a negative result when applying blind feedback for QA in
TREC 9, 10 and 11, and a neutral result for TREC 7 and 8’s ad hoc retrieval
tasks. Monz’s experiment, using $r=10$ and standard Rocchio term weighting,
also found a further reduction in performance when $r$ was reduced (from 10 to
5). This is an isolated experiment using just one measure on a limited set of
questions, with no use of the available answer texts.
Robertson [1992] notes that there are issues when using a whole document for
feedback, as opposed to just a single relevant passage; as mentioned in
Section 3.1, passage- and document-level retrieval sets must also be compared
for their performance at providing feedback. Critically, we will survey the
intersection between words known to be helpful and blind RF terms based on
initial retrieval, thus showing exactly how likely an RF method is to succeed.
## 3 Methodology
We first investigated the possibility of an IR-component specific failure
leading to impaired coverage by testing a variety of IR engines and
configurations. Then, difficult questions were identified, using various
performance thresholds. Next, answer bearing texts for these harder questions
were checked for words that yielded a performance increase when used for query
expansion. After this, we evaluated how likely a RF-based approach was to
succeed. Finally, blind RF was applied to the whole question set. IR
performance was measured, and terms used for RF compared to those which had
proven to be helpful as extension words.
### 3.1 IR Engines
A QA framework [2004a] was originally used to construct a QA system based on
running a default Lucene installation. As this only covers one IR engine in
one configuration, it is prudent to examine alternatives. Other IR engines
should be tested, using different configurations. The chosen additional
engines were: Indri, based on the mature INQUERY engine and the Lemur toolkit
[2003]; and Terrier, a newer engine designed to deal with corpora in the
terabyte range and to back applications entered into TREC conferences [2005].
We also looked at both passage-level and document-level retrieval. Passages
can be defined in a number of ways, such as a sentence, a sliding window of
$k$ terms centred on the target term(s), parts of a document of fixed (and
equal) lengths, or a paragraph. In this case, the documents in the AQUAINT
corpus contain paragraph markers which were used as passage-level boundaries,
thus making “passage-level” and “paragraph-level” equivalent in this paper.
Passage-level retrieval may be preferable for AE, as the number of potential
distracters is somewhat reduced when compared to document-level retrieval
[2004].
The initial IR component configuration was with Lucene indexing the AQUAINT
corpus at passage-level, with a Porter stemmer [1980] and an augmented version
of the CACM [1976] stopword list.
Indri natively supports document-level indexing of TREC format corpora.
Passage-level retrieval was done using the paragraph tags defined in the
corpus as delimiters; this allows both passage- and document-level retrieval
from the same index, according to the query.
All the IR engines were unified to use the Porter stemmer and the same CACM-
derived stopword list.
The top $n$ documents for each question in the TREC2004, TREC2005 and TREC2006
sets were retrieved using every combination of engine, and
configuration111Save Terrier / TREC2004 / passage-level retrieval; passage-
level retrieval with Terrier was very slow using our configuration, and could
not be reliably performed using the same Terrier instance as document-level
retrieval.. The questions and targets were processed to produce IR queries as
per the default configuration for the QA framework. Examining the top 200
documents gave a good compromise between the time taken to run experiments
(between 30 and 240 minutes each) and the amount one can mine into the data.
Tabulated results are shown in Table 1 and Table 2. Queries have had anaphora
resolution performed in the context of their series by the QA framework. AE
components begin to fail due to excess noise when presented with over 20
texts, so this value is enough to encompass typical operating parameters and
leave space for discovery [2006].
A failure analysis (FA) tool, an early version of which is described by
[2005], provided reporting and analysis of IR component performance. In this
experiment, it provided high level comparison of all engines, measuring
coverage and redundancy as the number of documents retrieved, $n$, varies.
This is measured because a perfect engine will return the most useful
documents first, followed by others; thus, coverage will be higher for that
engine with low values of $n$.
| Coverage | Redundancy
---|---|---
| Year | Len. | Strict | Len. | Strict
Lucene | 2004 | 0.686 | 0.636 | 2.884 | 1.624
2005 | 0.703 | 0.566 | 2.780 | 1.155
2006 | 0.665 | 0.568 | 2.417 | 1.181
Indri | 2004 | 0.690 | 0.554 | 3.849 | 1.527
2005 | 0.694 | 0.512 | 3.908 | 1.056
2006 | 0.691 | 0.552 | 3.373 | 1.152
Terrier | 2004 | - | - | - | -
2005 | - | - | - | -
2006 | 0.638 | 0.493 | 2.520 | 1.000
Table 1: Performance of Lucene, Indri and Terrier at paragraph level, over top 20 documents. This clearly shows the limitations of the engines. | Coverage | Redundancy
---|---|---
| Year | Len. | Strict | Len. | Strict
Indri | 2004 | 0.926 | 0.837 | 7.841 | 2.663
2005 | 0.935 | 0.735 | 7.573 | 1.969
2006 | 0.882 | 0.741 | 6.872 | 1.958
Terrier | 2004 | 0.919 | 0.806 | 7.186 | 2.380
2005 | 0.928 | 0.766 | 7.620 | 2.130
2006 | 0.983 | 0.783 | 6.339 | 2.067
Table 2: Performance of Indri and Terrier at document level IR over the
AQUAINT corpus, with $n=20$
### 3.2 Identification of Difficult Questions
Once the performance of an IR configuration over a question set is known, it’s
possible to produce a simple report listing redundancy for each question. A
performance reporting script accesses the FA tool’s database and lists all the
questions in a particular set with the strict and lenient redundancy for
selected engines and configurations. Engines may use passage- or document-
level configurations.
Data on the performance of the three engines is described in Table 2. As can
be seen, the coverage with passage-level retrieval (which was often favoured,
as the AE component performs best with reduced amounts of text) languishes
between 51% and 71%, depending on the measurement method. Failed anaphora
resolution may contribute to this figure, though no deficiencies were found
upon visual inspection.
Not all documents containing answers are noted, only those checked by the NIST
judges [2004]. Match judgements are incomplete, leading to the potential
generation of false negatives, where a correct answer is found with complete
supporting information, but as the information has not been manually flagged,
the system will mark this as a failure. Assessment methods are fully detailed
in Dang et al. [2006]. Factoid performance is still relatively poor, although
as only 1.95 documents match per question, this may be an effect of such false
negatives [2003]. Work has been done into creating synthetic corpora that
include exhaustive answer sets [2004, 2003, 2005], but for the sake of
consistency, and easy comparison with both parallel work and prior local
results, the TREC judgements will be used to evaluate systems in this paper.
Mean redundancy is also calculated for a number of IR engines. Difficult
questions were those for which no answer bearing texts were found by either
strict or lenient matches in any of the top $n$ documents, using a variety of
engines. As soon as one answer bearing document was found by an engine using
any measure, that question was deemed _non-difficult_. Questions with mean
redundancy of zero are marked _difficult_ , and subjected to further analysis.
Reducing the question set to just difficult questions produces a TREC-format
file for re-testing the IR component.
### 3.3 Extension of Difficult Questions
The documents deemed relevant by TREC must contain some useful text that can
help IR engine performance. Such words should be revealed by a gain in
redundancy when used to extend an initially difficult query, usually signified
by a change from zero to a non-zero value (signifying that relevant documents
have been found where none were before). In an attempt to identify where the
useful text is, the relevant documents for each difficult question were
retrieved, and passages matching the answer regular expression identified. A
script is then used to build a list of terms from each passage, removing words
in the question or its target, words that occur in the answer, and stopwords
(based on both the indexing stopword list, and a set of stems common within
the corpus). In later runs, numbers are also stripped out of the term list, as
their value is just as often confusing as useful [1999]. Of course, answer
terms provide an obvious advantage that would not be reproducible for
questions where the answer is unknown, and one of our goals is to help query
expansion for unseen questions. This approach may provide insights that will
enable appropriate query expansion where answers are not known.
Performance has been measured with both the question followed by an extension
(Q+E), as well as the question followed by the target and then extension
candidates (Q+T+E). Runs were also executed with just Q and Q+T, to provide
non-extended reference performance data points. Addition of the target often
leads to gains in performance [2005], and may also aid in cases where anaphora
resolution has failed.
Some words are retained, such as titles, as including these can be inferred
from question or target terms and they will not unfairly boost redundancy
scores; for example, when searching for a “Who” question containing the word
“military”, one may want to preserve appellations such as “Lt.” or “Col.”,
even if this term appears in the answer.
This filtered list of extensions is then used to create a revised query file,
containing the base question (with and without the target suffixed) as well as
new questions created by appending a candidate extension word.
Results of retrievals with these new question are loaded into the FA database
and a report describing any performance changes is generated. The extension
generation process also creates custom answer specifications, which replicate
the information found in the answers defined by TREC.
This whole process can be repeated with varying question difficulty
thresholds, as well as alternative $n$ values (typically from 5 to 100),
different engines, and various question sets.
### 3.4 Relevance Feedback Performance
Now that we can find the helpful extension words (HEWs) described earlier,
we’re equipped to evaluate query expansion methods. One simplistic approach
could use blind RF to determine candidate extensions, and be considered
potentially successful should these words be found in the set of HEWs for a
query. For this, term frequencies can be measured given the top $r$ documents
retrieved using anaphora-resolved query $Q$. After stopword and question word
removal, frequent terms are appended to $Q$, which is then re-evaluated. This
has been previously attempted for factoid questions [2005] and with a limited
range of $r$ values [2003] but not validated using a set of data-driven terms.
We investigated how likely term frequency (TF) based RF is to discover HEWs.
To do this, the proportion of HEWs that occurred in initially retrieved texts
was measured, as well as the proportion of these texts containing at least one
HEW. Also, to see how effective an expansion method is, suggested expansion
terms can be checked against the HEW list.
We used both the top 5 and the top 50 documents in formulation of extension
terms, with TF as a ranking measure; 50 is significantly larger than the
optimal number of documents for AE (20), without overly diluting term
frequencies.
Problems have been found with using entire documents for RF, as the topic may
not be the same throughout the entire discourse [1992]. Limiting the texts
used for RF to paragraphs may reduce noise; both document- and paragraph-level
terms should be checked.
## 4 Results
Once we have HEWs, we can determine if these are going to be of significant
help when chosen as query extensions. We can also determine if a query
expansion method is likely to be fruitful. Blind RF was applied, and assessed
using the helpful words list, as well as RF’s effect on coverage.
| Engine
---|---
Year | Lucene Para | Indri Para | Indri Doc | Terrier Doc
2004 | 76 | 72 | 37 | 42
2005 | 87 | 98 | 37 | 35
2006 | 108 | 118 | 59 | 53
Table 3: Number of difficult questions, as defined by those which have zero
redundancy over both strict and lenient measures, at $n=20$. Questions seem to
get harder each year. Document retrieval yields fewer difficult questions, as
more text is returned for potential matching. Engine
---
| Lucene | Indri | Terrier
Paragraph | 226 | 221 | -
Document | - | 121 | 109
Table 4: Number of difficult questions in the 2006 task, as defined above,
this time with $n=5$. Questions become harder as fewer chances are given to
provide relevant documents.
### 4.1 Difficult Question Analysis
| Match type
---|---
Strict | Lenient
Year | 2004 | 39 | 49
2005 | 56 | 66
2006 | 53 | 49
Table 5: Common difficult questions (over all three engines mentioned above) by year and match type; $n=20$. Difficult questions used | 118
---|---
Variations tested | 6683
Questions that benefited | 87 (74.4%)
Helpful extension words (strict) | 4973
Mean helpful words per question | 42.144
Mean redundancy increase | 3.958
Table 6: Using Terrier Passage / strict matching, retrieving 20 docs, with
TREC2006 questions / AQUAINT. Difficult questions are those where no strict
matches are found in the top 20 IRT from just one engine.
The number of difficult questions found at $n=20$ is shown in Table 3.
Document-level retrieval gave many fewer difficult questions, as the amount of
text retrieved gave a higher chance of finding lenient matches. A comparison
of strict and lenient matching is in Table 5.
Extensions were then applied to difficult questions, with or without the
target. The performance of these extensions is shown in Table 6. Results show
a significant proportion (74.4%) of difficult questions can benefit from being
extended with non-answer words found in answer bearing texts.
| 2004 | 2005 | 2006
---|---|---|---
HEW found in IRT | 4.17% | 18.58% | 8.94%
IRT containing HEW | 10.00% | 33.33% | 34.29%
RF words in HEW | 1.25% | 1.67% | 5.71%
Table 7: “Helpful extension words”: the set of extensions that, when added to the query, move redundancy above zero. $r=5$, $n=20$, using Indri at passage level. | $r$ |
---|---|---
| 5 | 50 | Baseline
Rank | Doc | Para | Doc | Para |
5 | 0.253 | 0.251 | 0.240 | 0.179 | 0.312
10 | 0.331 | 0.347 | 0.331 | 0.284 | 0.434
20 | 0.438 | 0.444 | 0.438 | 0.398 | 0.553
50 | 0.583 | 0.577 | 0.577 | 0.552 | 0.634
Table 8: Coverage (strict) using blind RF. Both document- and paragraph-level
retrieval used to determine RF terms.
### 4.2 Applying Relevance Feedback
Identifying HEWs provides a set of words that are useful for evaluating
potential expansion terms. Using simple TF based feedback (see Section 3.4), 5
terms were chosen per query. These words had some intersection (see Table 7)
with the extension words set, indicating that this RF may lead to performance
increases for previously unseen questions. Only a small number of the HEWs
occur in the initially retrieved texts (IRTs), although a noticeable
proportion of IRTs (up to 34.29%) contain at least one HEW. However, these
terms are probably not very frequent in the documents and unlikely to be
selected with TF-based blind RF. The mean proportion of RF selected terms that
were HEWs was only 2.88%. Blind RF for question answering fails here due to
this low proportion. Strict measures are used for evaluation as we are
interested in finding documents which were not previously being retrieved
rather than changes in the distribution of keywords in IRT.
Document and passage based RF term selection is used, to explore the effect of
noise on terms, and document based term selection proved marginally superior.
Choosing RF terms from a small set of documents ($r=5$) was found to be
marginally better than choosing from a larger set ($r=50$). In support of the
suggestion that RF would be unlikely to locate HEWs, applying blind RF
consistently hampered overall coverage (Table 8).
## 5 Discussion
_Question:_
---
Who was the nominal leader after the overthrow?
_Target:_ Pakistani government overthrown in 1999
Extension word | Redundancy
Kashmir | 4
Pakistan | 4
Islamabad | 2.5
_Question:_ Where did he play in college?
_Target:_ Warren Moon
Extension word | Redundancy
NFL | 2.5
football | 1
_Question:_ Who have commanded the division?
_Target:_ 82nd Airborne division
Extension word | Redundancy
Gen | 3
Col | 2
decimated | 2
officer | 1
Table 9: Queries with extensions, and their mean redundancy using Indri at
document level with $n=20$. Without extensions, redundancy is zero.
HEWs are often found in answer bearing texts, though these are hard to
identify through simple TF-based RF. A majority of difficult questions can be
made accessible through addition of HEWs present in answer bearing texts, and
work to determine a relationship between words found in initial retrieval and
these HEWs can lead to coverage increases. HEWs also provide an effective
means of evaluating other RF methods, which can be developed into a generic
rapid testing tool for query expansion techniques. TF-based RF, while finding
some HEWs, is not effective at discovering extensions, and reduces overall IR
performance.
There was not a large performance change between engines and configurations.
Strict paragraph-level coverage never topped 65%, leaving a significant number
of questions where no useful information could be provided for AE.
The original sets of difficult questions for individual engines were small –
often less than the 35% suggested when looking at the coverage figures.
Possible causes could include:
Difficult questions being defined as those for which average redundancy is
zero: This limit may be too low. To remedy this, we could increase the
redundancy limit to specify an arbitrary number of difficult questions out of
the whole set.
The use of both strict and lenient measures: It is possible to get a lenient
match (thus marking a question as non-difficult) when the answer text occurs
out of context.
Reducing $n$ from 20 to 5 (Table 4) increased the number of difficult
questions produced. From this we can hypothesise that although many search
engines are succeeding in returning useful documents (where available), the
distribution of these documents over the available ranks is not one that
bunches high ranking documents up as those immediately retrieved (unlike a
perfect engine; see Section 3.1), but rather suggests a more even distribution
of such documents over the returned set.
The number of candidate extension words for queries (even after filtering) is
often in the range of hundreds to thousands. Each of these words creates a
separate query, and there are two variations, depending on whether the target
is included in the search terms or not. Thus, a large number of extended
queries need to be executed for each question run. Passage-level retrieval
returns less text, which has two advantages: firstly, it reduces the scope for
false positives in lenient matching; secondly, it is easier to scan result by
eye and determine why the engine selected a result.
Proper nouns are often helpful as extensions. We noticed that these cropped up
fairly regularly for some kinds of question (e.g. “Who”). Especially useful
were proper nouns associated with locations - for example, adding “Pakistani”
to a query containing the word Pakistan lifted redundancy above zero for a
question on President Musharraf, as in Table 9. This reconfirms work done by
Greenwood [2004b].
## 6 Conclusion and Future Work
IR engines find some questions very difficult and consistently fail to
retrieve useful texts even with high values of $n$. This behaviour is common
over many engines. Paragraph level retrieval seems to give a better idea of
which questions are hardest, although the possibility of false negatives is
present from answer lists and anaphora resolution.
Relationships exist between query words and helpful words from answer
documents (e.g. with a military leadership themes in a query, adding the term
“general” or “gen” helps). Identification of HEWs has potential use in query
expansion. They could be used to evaluate RF approaches, or associated with
question words and used as extensions.
Previous work has ruled out relevance feedback in particular circumstances
using a single ranking measure, though this has not been based on analysis of
answer bearing texts. The presence of HEWs in IRT for difficult questions
shows that guided RF may work, but this will be difficult to pursue. Blind RF
based on term frequencies does not increase IR performance. However, there is
an intersection between words in initially retrieved texts and words data
driven analysis defines as helpful, showing promise for alternative RF methods
(e.g. based on TFIDF). These extension words form a basis for indicating the
usefulness of RF and query expansion techniques.
In this paper, we have chosen to explore only one branch of query expansion.
An alternative data driven approach would be to build associations between
recurrently useful terms given question content. Question texts could be
stripped of stopwords and proper nouns, and a list of HEWs associated with
each remaining term. To reduce noise, the number of times a particular
extension has helped a word would be counted. Given sufficient sample data,
this would provide a reference body of HEWs to be used as an aid to query
expansion.
## References
* [2003] Allan, J., J. Callan, K. Collins-Thompson, B. Croft, F. Feng, D. Fisher, J. Lafferty, L. Larkey, TN Truong, P. Ogilvie, et al. 2003\. The Lemur toolkit for language modeling and information retrieval.
* [1996] Allan, J. 1996\. Incremental relevance feedback for information filtering. In Research and Development in IR, pages 270–278.
* [1999] Baeza-Yates, R. and B. Ribeiro-Neto. 1999\. Modern Information Retrieval. Addison Wesley.
* [2004] Bilotti, M.W., B. Katz, and J. Lin. 2004\. What works better for question answering: Stemming or morphological query expansion. Proc. IR for QA Workshop at SIGIR 2004.
* [2004] Bilotti, M.W. 2004\. Query Expansion Techniques for Question Answering. Master’s thesis, Massachusetts Institute of Technology.
* [2006] Dang, H.T., J. Lin, and D. Kelly. 2006\. Overview of the TREC 2006 QA track. Proc. 15th Text REtrieval Conf..
* [2006] Greenwood, M.A., M. Stevenson, and R. Gaizauskas. 2006\. The university of sheffield’s trec 2006 q&a experiments. In Proc. 15th Text REtrieval Conf.
* [2004a] Greenwood, M.A. 2004a. Answerfinder: Question answering from your desktop. In Proc. 7th Annual Colloquium for the UK SIG for Computational Linguistics (CLUK ’04).
* [2004b] Greenwood, M.A. 2004b. Using pertainyms to improve passage retrieval for questions requesting information about a location. In Proc. Workshop on IR for QA (SIGIR 2004).
* [1976] Jones, K.S. and C.J. van Rijsbergen. 1976\. IR test collections. J. of Documentation, 32(1):59–75.
* [2005] Lin, J. and B. Katz. 2005\. Building a reusable test collection for question answering. J. American Society for Information Science and Technology.
* [2003] Monz, C. 2003\. From Document Retrieval to Question Answering. ILLC Dissertation Series 2003, 4.
* [2005] Ounis, I., G. Amati, V. Plachouras, B. He, C. Macdonald, and D. Johnson. 2005\. Terrier IR Platform. Proc. 27th European Conf. on IR (ECIR 05), Santiago de Compostela, Spain, pages 517–519.
* [2006] Pizzato, L.A., D. Molla, and C. Paris. 2006\. Pseudo-relevance feedback using named entities for question answering. Australasian Language Technology Workshop (ALTW2006), pages 83–90.
* [1980] Porter, M. 1980\. An Algorithm for Suffix Stripping Program. Program, 14(3):130–137.
* [2004] Roberts, I and R Gaizauskas. 2004\. Evaluating passage retrieval approaches for question answering. In Proc. 26th European Conf. on IR.
* [1992] Robertson, S.E., S. Walker, M. Hancock-Beaulieu, A. Gull, and M. Lau. 1992\. Okapi at TREC. In Text REtrieval Conf., pages 21–30.
* [2005] Roussinov, D. and W. Fan. 2005\. Discretization based learning approach to information retrieval. In Proc. 2005 Conf. on Human Language Technologies.
* [2005] Roussinov, D., M. Chau, E. Filatova, and J.A. Robles-Flores. 2005\. Building on redundancy: Factoid question answering, robust retrieval and the other . In Proc. 14th Text REtrieval Conf.
* [2005] Sanka, Atheesh. 2005\. Passage retrieval for question answering. Master’s thesis, University of Sheffield.
* [2003] Tellex, S., B. Katz, J. Lin, A. Fernandes, and G. Marton. 2003\. Quantitative evaluation of passage retrieval algorithms for question answering. Proc. 26th Annual Int’l ACM SIGIR Conf. on R&D in IR, pages 41–47.
* [2003] Voorhees, E. and L. P. Buckland, editors. 2003\. Proc. 12th Text REtrieval Conf..
|
arxiv-papers
| 2012-03-22T19:19:02 |
2024-09-04T02:49:28.953439
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leon Derczynski, Jun Wang, Robert Gaizauskas and Mark A. Greenwood",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1203.5084"
}
|
1203.5120
|
# Pair density waves and vortices in an elongated two-component Fermi gas
Ran Wei Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China Laboratory of Atomic and Solid State Physics,
Cornell University, Ithaca, NY, 14850 Erich J. Mueller Laboratory of Atomic
and Solid State Physics, Cornell University, Ithaca, NY, 14850
###### Abstract
We study the vortex structures of a two-component Fermi gas experiencing a
uniform effective magnetic field in an anisotropic trap that interpolates
between quasi-one dimensional (1D) and quasi-two dimensional (2D). At a fixed
chemical potential, reducing the anisotropy (or equivalently increasing the
attractive interactions or increasing the magnetic field) leads to
instabilities towards pair density waves, and vortex lattices. Reducing the
chemical potential stabilizes the system. We calculate the phase diagram, and
explore the density and pair density. The structures are similar to those
predicted for superfluid Bose gases. We further calculate the paired fraction,
showing how it depends on chemical potential and anisotropy.
###### pacs:
67.85.Lm, 03.75.Ss, 05.30.Fk, 74.25.Uv
Introduction — Quantized vortices play an essential role in understanding the
behavior of type-II superconductors and superfluids such as 3He. In cold
gases, these vortices were the smoking gun for superfluidity Zwierlein2005 .
Here we study how confinement influences the vortex structures in a trapped
gas of ultracold fermions. We use the microscopic Bogoliubov-de-Gennes (BdG)
equations, and consider anisotropic traps that interpolate between quasi-one
dimensional (1D) and quasi-two dimensional (2D).
The behavior of topological defects in confined geometries can be quite rich.
A good example is rotating bosons in anisotropic traps Sinha2005 , where one
sees multiple transitions in the structure of vortex lattices as the
parameters are changed. Most intriguing, in the quasi-1D limit one sees a
“roton” spectrum which softens as the rotation rate increases, signaling an
instability to form a snake-like density wave. With recent experimental
developments Lin2009nature , we expect these structures can soon be explored
in Bose gases, and related studies will be undertaken in Fermi gases. In the
Fermi gas, we find parallels to all of the predicted boson physics. The single
particle instability which drives density waves in the Bose case becomes a
collective instability for the fermions, and instead drives pair density waves
agterberg2008 . For a range of parameters we even find that the order
parameter has the form predicted by Larkin and Ovchinnikov LO1965 for a
polarized gas.
In very different contexts, studies of vortices in confined geometries lead to
a number of interesting and important results such as “non-Hermitian” quantum
mechanical analogies confinedvortices , and the destruction of
superconductivity via phase slips superconductingwire . Generically, reducing
the dimensionality enhances fluctuations, leading to novel effects.
Driven partially by increased computer power and partially by interest in the
BCS-BEC crossover, a number of research groups have recently produced
Bogoliubov-de-Gennes (BdG) or density functional calculations of single
vortices singlevortex , and vortex lattices vortexlattice . These have largely
been 2D or three dimensional (3D) calculations, with translational symmetry
along the magnetic field. The numerical challenges of these calculations come
from the large basis set needed to describe the single particle states. By
truncating to the lowest Landau level, one can greatly simplify the problem
LLLBdG . As we explain below this limit is experimentally relevant
experimentLLL ; Aidelsburger2011 .
Model — We start from the Hamiltonian of a spin balanced two-component Fermi
gas, with a total number of particles
$N=\int(\Psi_{\uparrow}^{\dagger}\Psi_{\uparrow}+\Psi_{\downarrow}^{\dagger}\Psi_{\downarrow})d\bm{r}$
and chemical potential $\tilde{\mu}$,
$\displaystyle\mathcal{K}=\int\biggl{(}\sum_{\sigma=\uparrow,\downarrow}\Psi_{\sigma}^{\dagger}H_{0}\Psi_{\sigma}+H_{\rm
int}\biggr{)}d\bm{r}-\tilde{\mu}N,$ (1)
where the single particle Hamiltonian
$H_{0}=(p_{x}-By)^{2}/2m+p_{y}^{2}/2m+p_{z}^{2}/2m+V(\bm{r})$, describes a
neutral atom of mass $m$ and momentum $\bm{p}$ experiencing a uniform
effective magnetic field $B$ in the $z$ direction (Landau gauge), where the
harmonic trap is
$V(\bm{r})=\frac{1}{2}m(\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})$, and the
inter-component interaction $H_{\rm
int}=-g\Psi_{\uparrow}^{\dagger}\Psi_{\downarrow}^{\dagger}\Psi_{\downarrow}\Psi_{\uparrow}$,
is attractive, with the coupling constant related to $s$-wave scattering
lengths $a_{s}$ via $g=-4\pi\hbar^{2}a_{s}/m$ ($g>0$) regulation . We do not
treat the case where $g<0$, in which the physics is more involved Zwerger2004
. This single particle Hamiltonian is readily engineered in cold atoms either
by using two counter-propagating Raman beams with spatially dependent detuning
Lin2009nature or rotating the gas in anisotropic traps where the rotation
rate approaches the weakest trapping frequency dalibard2002 . When
$\omega_{z}$ is large, this model can be tuned from quasi-1D to quasi-2D by
changing $\omega_{y}$.
(A) lowest Landau level — Following Sinha et.al Sinha2005 , the single
particle Hamiltonian is readily diagonalized, with eigenstates labeled by
three quantum numbers $K,n,n^{\prime}$, and energies given by
$\displaystyle E_{nn^{\prime}}(K)={\cal
E}K^{2}+n\hbar\omega_{z}+n^{\prime}\hbar\tilde{\omega}_{c},$ (2)
where the effective cyclotron frequency is
$\tilde{\omega}_{c}=\sqrt{\omega_{y}^{2}+\omega_{c}^{2}}$, the cyclotron
frequency is $\omega_{c}=B/m$, the characteristic energy of motion in the $x$
direction is ${\cal E}=\hbar\omega_{y}^{2}/4\tilde{\omega}_{c}$, and we have
neglected the zero-point energy. The dimensionless wave-number
$K=\sqrt{2}\tilde{\ell}k$ labels the momentum $k$ along the $x$ direction,
where the effective magnetic length is
$\tilde{\ell}=\sqrt{\hbar/m\tilde{\omega}_{c}}$. The discrete quantum numbers
$n$ and $n^{\prime}$ corresponds to the number of nodes in the $z$ and $y$
directions. In the absence of confinement in the $y$ direction, ${\cal
E}\rightarrow 0$, and we recover degenerate Landau levels. Hence, we refer to
$n$ as the Landau level index. If the interaction energy per particle $\langle
H_{\rm int}/N\rangle$ and the characteristic “kinetic energy” $\langle{\cal
E}K^{2}\rangle$ are small compared to $\hbar\tilde{\omega}_{c}$ and
$\hbar\omega_{z}$, one can truncate to the lowest eigenstates with
$n=n^{\prime}=0$, which are of the form
$\phi_{K}(\bm{r})=\frac{1}{\sqrt{\pi\tilde{\ell}d_{z}L}}e^{i\frac{Kx}{\sqrt{2}\tilde{\ell}}}e^{-\frac{(y-y_{K})^{2}}{2\tilde{\ell}^{2}}}e^{-\frac{z^{2}}{2d_{z}^{2}}},$
(3)
where $y_{K}=\sqrt{2}\omega_{c}K\tilde{\ell}/2\tilde{\omega}_{c}$,
$d_{z}=\sqrt{\hbar/m\omega_{z}}$ and $L$ is the length in the $x$ direction.
The conditions allowing us to truncate to the lowest Landau level constrain
the 3D density $n_{3D}$ and magnetic field strength $B$. For example, the
condition $\langle H_{\rm
int}/N\rangle\ll\hbar\tilde{\omega}_{c}\sim\hbar\omega_{z}$ requires
$n_{3D}\ll\hbar\tilde{\omega}_{c}/g\sim\hbar\omega_{z}/g$. The other
condition, ${\cal E}K^{2}\ll\hbar\tilde{\omega}_{c}$, requires $B\gg
m\omega_{y}$. While such fields are challenging to produce in cold atoms, they
are not completely unreasonable. In a very recent experiment performed by
I.Bloch’s group Aidelsburger2011 , the density is $n_{3D}\sim 10^{13}$cm-3,
and the cyclotron frequency is $\omega_{c}\sim 100$kHz. Since this experiment
involves coupled “wires”, it is natural to use them for quasi-1D studies.
Note, the magnetic field is “staggered” in that experiment, while we consider
the uniform case.
Letting $a_{K}$ annihilate the state in Eq. (3), one has an effective 1D
model,
$\displaystyle H/{\cal
E}=\sum_{K,\sigma}(K^{2}-\mu)a_{K\sigma}^{\dagger}a_{K\sigma}+\beta\sum_{q}f^{\dagger}(q)f(q),$
(4)
where $f(q)\equiv\sum_{K}e^{-1/8(2K-q)^{2}}a_{q-K\downarrow}a_{K\uparrow}$,
the dimensionless chemical potential is $\mu=\tilde{\mu}/{\cal E}$ and the
effective interaction parameter is
$\beta=-\frac{2mg}{\pi\hbar^{2}L}(\frac{\omega_{z}}{\tilde{\omega}_{c}})^{1/2}(\frac{\tilde{\omega}_{c}}{\omega_{y}})^{2}$.
From the definition of $\beta$, one sees that increasing the interaction
strength $g$ has the same effect as increasing the magnetic field $B$,
increasing the $z$-confinement $\omega_{z}$, or reducing the $y$-confinement
$\omega_{y}$. In the following, we will investigate the properties of the
confined Fermi gas by studying Eq.(4). One can show that the interaction in
Eq.(4) is equivalent to $\beta\sum_{q}f^{\dagger}(q)f(q)=\beta\int
d\bm{r}F^{\dagger}(\bm{r})F(\bm{r})$, where
$F(\bm{r})=\sum_{q}f(q)\phi_{q}(\bm{r})$.
(B) Bogoliubov de Gennes approach — We introduce the pair field
$\Delta_{q}=\beta\langle f(q)\rangle$, and its transform
$\Delta(\bm{r})=\beta\langle F(\bm{r})\rangle$. We neglect the fluctuation
$(f^{\dagger}(q)-\Delta_{q}^{*}/\beta)(f(q)-\Delta_{q}/\beta)$ to reduce
Eq.(4) to a bilinear form,
$\displaystyle H/{\cal E}$ $\displaystyle=$
$\displaystyle\sum_{K,\sigma}(K^{2}-\mu)a_{K\sigma}^{\dagger}a_{K\sigma}$ (5)
$\displaystyle+$
$\displaystyle\sum_{q}\left(\Delta_{q}^{*}f(q)+\Delta_{q}f^{\dagger}(q)-|\Delta_{q}|^{2}/\beta\right).$
Given $\Delta_{q}$, one can diagonalize $H$, and then impose self-consistency.
For arbitrary $\Delta_{q}$, this process is unwieldy vortexlattice . We here
introduce two approximations which make the numerical calculations more
efficient. First, we assume $\Delta_{q}$ is non-vanishing only when the
central momentum of the paired fermions is $q=nK_{0}$, where $n=0,\pm 1,\pm
2,...$. The characteristic wave-number $K_{0}$ is taken to be a variational
parameter. This is equivalent to assuming $\Delta(\bm{r})$ is periodic in the
$x$ direction and treating the wavelength variationally. Second, we restrict
ourselves to consider the symmetric pair field: $\Delta_{q}=\Delta_{-q}$. This
implies a spatially symmetric field $\Delta(\bm{r})=\Delta(-\bm{r})$. Under
these assumptions, the Hamiltonian is reduced to
$\displaystyle H/{\cal E}$ $\displaystyle=$
$\displaystyle\sum_{K,\sigma}(K^{2}-\mu)a_{K\sigma}^{\dagger}a_{K\sigma}-\sum_{n}|\Delta_{|n|K_{0}}|^{2}/\beta$
(6) $\displaystyle+$
$\displaystyle\sum_{n}\left(\Delta_{|n|K_{0}}^{*}f(nK_{0})+\Delta_{|n|K_{0}}f^{\dagger}(nK_{0})\right).$
Since Eq.(6) will be calculated by taking the continuum limit
$\sum_{K}\rightarrow(\sqrt{2}L/4\pi\tilde{\ell})\int dK$ (see Supplemental
materials), it is useful to introduce a positive parameter
$\alpha=-\sqrt{2}L\beta/4\pi\tilde{\ell}$ to characterize the effective
attractive interaction. For small $\alpha$, we find $\Delta_{|n|K_{0}}\neq 0$
for only a few values of $n$. We define $\xi$ to be the number of nonzero
$\Delta_{|n|K_{0}}$. The various phases can be distinguished by looking at the
pair density $|\langle\Psi_{\uparrow}\Psi_{\downarrow}\rangle|^{2}$ and/or the
particle density $\langle\Psi_{\uparrow}^{\dagger}\Psi_{\uparrow}\rangle$ (see
Fig.1(b)). The features are clearest in the pair density. If more than one
$\Delta_{nK_{0}}$ is nonzero, we have either a pair density wave or vortices.
For example, the case $\xi=3$ ($\Delta_{0}\neq 0,\Delta_{\pm K_{0}}\neq 0$),
as illustrated in Fig.1(b), corresponds to a pair density wave where
$|\langle\Psi_{\uparrow}\Psi_{\downarrow}\rangle|^{2}$ has corrugations. The
case $\xi=2$ ($\Delta_{0}=0,\Delta_{\pm K_{0}}\neq 0$), consists of a single
row of vortices. Larger $\xi$, for example in Fig.3, corresponds to a vortex
lattice. The case $\xi=2$ gives an order parameter which can formally be
identified with the Larkin-Ovchinnikov (LO) state LO1965 (see also FF1964 ).
Here, $\Delta_{K}$ is nonzero except when $K=\pm K_{0}$. Defining an effective
1D order parameter $\Delta^{1D}(x)=\sum_{K}e^{iKx}\Delta_{K}$, we have
$\Delta^{1D}(x)=2\Delta_{K_{0}}\cos K_{0}x$. Note that unlike the LO state,
the physical order parameter
$\Delta(\bm{r})=\sum_{K}\Delta_{K}\phi_{K}(\bm{r})$, is not a simple cosine.
Also note that unlike LO’s model, here we assume both spin states have equal
chemical potentials. Instead of being driven by the polarization, our
instability towards a paired density wave is driven by the form of the
effective 1D interaction.
When $\xi=1$ ($\Delta_{0}\neq 0$), Eq.(6) can be analyzed analytically (see
Supplemental materials – A). One readily obtains the gap equation,
$\displaystyle\frac{1}{\alpha}$ $\displaystyle=$
$\displaystyle\int\frac{e^{-K^{2}}}{2\epsilon_{K}}dK,$ (7)
and the number equation,
$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}L}{4\pi\tilde{\ell}}\int(1-\frac{\epsilon_{0}}{\epsilon_{K}})dK.$
(8)
where $\epsilon_{K}=\sqrt{\epsilon_{0}^{2}+|\Delta_{0}|^{2}e^{-K^{2}}}$ and
$\epsilon_{0}=K^{2}-\mu$.
Unlike the traditional case, the integrand in the RHS of Eq.(7) has a factor
$e^{-K^{2}}$ in the numerator, which dominates the behavior of the integrand
for $K\gg 1$. If $\mu\gg 1$ (meaning in physics units $\tilde{\mu}\gg{\cal
E}$), and $\Delta_{0}$ is sufficiently small, the integrand in Eq.(7) is
bimodal. There is a gentle peak of height $\frac{1}{2\mu}$ and width $1$
centered at $K=0$, and a sharp peak of height
$\frac{e^{-\mu/2}}{2|\Delta_{0}|}$ and width
$\frac{|\Delta_{0}|e^{-\mu/2}}{\sqrt{\mu}}$ centered at $K=\sqrt{\mu}$. The
power-law tails of this sharp peak give a contribution to the integral which
scales as $A\frac{e^{-\mu}}{\sqrt{\mu}}\log|\Delta_{0}|$ as
$\Delta_{0}\rightarrow 0$, where $A$ is a constant. Solving Eq.(7) in this
regime yields an extremely small order parameter. In this weak pairing limit,
our numerics are unstable and the vortex lattices are better treated by
expanding the energies in power of $\Delta_{0}$ abrikosov1957 .
Another instructive limit is $\mu<0$ and $N/L\rightarrow 0$, where the
behavior is dominated by two-body physics. Eq.(7) then becomes the Schrödinger
equation of a two-body problem in momentum space Salpeter1951 , i.e.,
$\alpha=2/\int e^{-K^{2}}/({K^{2}-\mu})dK$, where the two-body binding energy
$\eta$ is identified with twice the chemical potential, $\eta=2\mu(\alpha)$.
Figure 1: (color online) (a): The structure of phase diagram as a function of
$\alpha$ and $\mu$. The value of $\xi$ (the number of nonzero
$\Delta_{|n|K_{0}}$) is denoted in each region. The two black solid curves are
the boundaries of two continuous transitions: $\xi=0\leftrightarrow\xi=1$ and
$\xi=1\leftrightarrow\xi=3$. They show a fairly good agreement with numerics.
(b): The structures of pair density
$|\langle\Psi_{\uparrow}\Psi_{\downarrow}\rangle|^{2}$ and density
$\langle\Psi_{\uparrow}^{\dagger}\Psi_{\uparrow}\rangle$ in the corresponding
regions. The color key is shown in Fig.3.
Phase diagram — We numerically minimize the energy by studying Eq.(6) (see
Supplemental materials – B). We find discrete jumps in $\xi$ as a function of
the dimensionless attractive interaction $\alpha$ and the dimensionless
chemical potential $\mu$. The resulting phase diagram is shown in Fig.1(a).
The darkest red region ($\xi=0$) is the vacuum with no particles. Increasing
$\alpha$ and/or $\mu$ brings one to a quasi-1D superfluid state. This state,
characterized by $\xi=1$, has no vortices and is translational invariant in
the $x$ direction. The $\xi=0$ to $\xi=1$ transition is continuous with
$\Delta_{0}\rightarrow 0$ and $N/L\rightarrow 0$ at the boundary. Further
increasing $\alpha$ and/or $\mu$ leads to an instability towards a $\xi=3$
state (the narrow yellow region). This state breaks translational symmetry.
The transition is continuous, and the boundary can be found via a linear
stability analysis of the $\xi=1$ state (see Supplemental materials – C). At
larger $\alpha$ and/or $\mu$, there is a discontinuous transition to a state
with $\xi=2$. This sequence of instabilities closely mirrors what is found in
calculations for Bose gases Sinha2005 .
Pair fraction — It is useful to put these results in the context of the BCS-
BEC crossover. In 3D Fermi gases one thinks of the superfluid with $\mu<0$ as
being formed from tightly bound bosonic pairs, analogous to 4He. The
superfluid with $\mu>0$ is instead thought of within a BCS picture where
diffuse pairs are formed by atoms at the Fermi surface. One can continuously
tune between these two idealized limits by taking $\mu$ through zero: the size
of the pairs varies continuously. Our approach to gaining insight into
analogies with the 3D BCS-BEC crossover is to study the pair fraction
$P=2N_{\rm pair}/N$ pair , as in Fig.2. While some of the qualitative features
of the 3D crossover persist in our effective 1D model, many of the details
differ.
Figure 2: (color online) The pair fraction $P=2N_{\rm pair}/N$ versus $\alpha$
with $\mu=-1,0,1$. The exponential small $P$ for $\mu=1$ at $\alpha\rightarrow
0$ is reminiscent of the BCS limit, and the large value of $P$ for $\mu=-1$ at
$\alpha\approx 1.5$ is analogous to the BEC limit. The kink on each curve
corresponds to the $\xi=3\leftrightarrow\xi=2$ phase transition.
To understand this figure, one must note that in a quasi-1D system the ratio
of the interaction to the kinetic energy is inverse proportional to the
density, thus the strongly interacting regime can be reached by making the
density small, or by making $\alpha$ large. The density increases
monotonically with $\mu$, but varies in a more complicated fashion with
$\alpha$. For small $\alpha$ and $\mu>0$ we find $\partial
N/\partial\alpha<0$, while for large $\alpha$ and/or $\mu<0$ we find $\partial
N/\partial\alpha>0$. At fixed $\alpha$, the pair fraction decreases with $\mu$
(consistent with $\partial N/\partial\mu>0$).
The top curve in Fig.2, representing $\mu=-1$, starts at $P=1$, roughly when
$\alpha=1.5$. Such a large value of $P$ is reminiscent of the BEC limit. The
density vanishes here, then grows as $\alpha$ increases. For $\mu=-1$, the
pair fraction decreases with $\alpha$, except for a small kink, corresponding
to the first order $\xi=3\leftrightarrow\xi=2$ phase transition.
On the contrary, for $\mu=1$, $P$ grows with $\alpha$. As $\alpha\rightarrow
0$, $P$ becomes exponentially small, as is predicted by the BCS theory. After
a sharp rise, driven both by increasing $\alpha$ and decreasing $N$, the pair
fraction levels out.
Each curve displays a kink, corresponding to the $\xi=3\leftrightarrow\xi=2$
phase transition. As $\alpha$ increases to the region $\xi=2$, one row of
vortices enters the elongated superfluid. This transition is accompanied by
density modulations.
To summarize we find that for $\mu>0$ and small $\alpha$ the system behaves
analogously to the BCS limit, while for $\mu<0$ and $\alpha\sim|\mu|$ the
system behaves more like the BEC limit. The density vanishes if $\mu<0$ and
$\alpha\lesssim|\mu|$. For most of our parameter range, we observe physics
analogous to the crossover regime.
Figure 3: (color online) The profile of density (left panel) and pair density
(right panel) at $\alpha=65,\mu=2$, where the dimensionless coordinates are
${\rm X}=x/\sqrt{2}\tilde{\ell},{\rm Y}=y/\sqrt{2}\tilde{\ell}$. The color key
is shown on the top.
Vortex lattice — With increasing $\alpha$, the number of Fourier components
$\xi$ increases, and the width in the $y$ direction grows. We illustrate the
large $\alpha$ limit in Fig.3 by calculating the density and the pair density
of the state with $\mu=2,\alpha=65$ and $\xi=7$. Only “faint” vortices are
seen in the density (left panel). Unpaired fermions fill the vortex cores
leading to very poor contrast. On the contrary, one sees a clear stretched
triangular lattice in the pair density (right panel). The lattice spacing is
$\sim 2\pi\sqrt{2}\tilde{\ell}/K_{0}$ and the size of the vortex core is
$\sim\tilde{\ell}$. Note the dimensionless wave-number $K_{0}$ varies slightly
with $\alpha$ but is of order $2$. The vortex lattice is slightly deformed
from a regular triangular lattice, but we expect this deformation to disappear
in the quasi-2D limit ($\alpha\rightarrow\infty$).
Observation — Since the density depletion in the vortex core is highly
suppressed, directly imaging the vortices through phase contrast or absorption
imaging would be challenging. Coherent Bragg scattering of light may be a
promising route for increasing the sensitivity of such optical probes
Sciamarella2001 . One can also study the structures of pair density through
photoassociation photoassociation , where the paired state is transformed to a
bound molecular state after illuminated with light.
Summary — We have studied the two-component Fermi gases in elongated
geometries. Truncating the BdG equations to the lowest Landau level, we
investigate the vortex structures that emerge as the trap evolves from
quasi-1D and quasi-2D. We calculate the phase diagram and find instabilities
towards pair density waves and vortex lattices. We explore the structures of
density and pair density, and calculate the pair fraction. We hope our results
can soon be explored in experiment.
Acknowledgements — We thank S. S. Natu and S. Baur for carefully reading the
manuscript. R. W. is supported by CSC, the CAS, and the National Fundamental
Research Program (under Grant No. 2011CB921304). This material is based upon
work supported by the National Science Foundation under Grant No. PHY-1068165.
## References
* (1) M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Nature (London) 435, 1047 (2005).
* (2) S. Sinha and G. V. Shlyapnikov, Phys. Rev. Lett. 94, 150401 (2005).
* (3) Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, and I. B. Spielman, Nature (London) 462, 628 (2009).
* (4) D. F. Agterberg, and, H. Tsunetsugu, Nat. Phys. 4, 639 (2008).
* (5) A. I. Larkin, and Y. N. Ovchinnikov, Soviet Phys. JETP 20, 762 (1965).
* (6) K. Kim, and D. R. Nelson, Phys. Rev. B, 64, 054508 (2001); W. Hofstetter, I. Affleck, D. Nelson, and U. Schollwöck, Europhys. Lett. 66, 178 (2004); I. Affleck, W. Hofstetter, D. R. Nelson, and U. Schollwöck, J. Stat. Mech.: Theor. Exp. 2004, P10003 (2004).
* (7) J. S. Langer, and Q. Amberaogar, Phys. Rev. 64, 498 (1967).
* (8) P. Fulde, and R. A. Ferrell, Phys. Rev. 135, A550 (1964).
* (9) A. A. Abrikosov, Soviet Phys. JETP 5, 1174 (1957).
* (10) Strictly speaking, this coupling constant $g$ should be regulized as $-1/g=m/4\pi\hbar^{2}a_{s}-1/V\sum_{\bf{k}}1/2\epsilon_{\bf{k}}$, where $V$ is the system volume and $\epsilon_{\bf{k}}$ is the excitation energy. However, as long as $a_{s}$ is small compared to the transverse confinement, this regularization does not change the effective 1D model Olshanii1998 .
* (11) J. N. Fuchs, A. Recati, and W. Zwerger, Phys. Rev. Lett. 93, 090408 (2004).
* (12) P. Rosenbusch, D. S. Petrov, S. Sinha, F. Chevy, V. Bretin, Y. Castin, G. Shlyapnikov, and J. Dalibard, Phys. Rev. Lett. 88, 250403 (2002).
* (13) A. Bulgac, and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003); N. Nygaard, G. M. Bruun, C. W. Clark, and D. L. Feder, ibid. 90, 210402 (2003); M. Machida, and T. Koyama, ibid. 94, 140401 (2005); R. Sensarma, M. Randeria, and T.-L. Ho, ibid. 96, 090403 (2006); M. Takahashi, T. Mizushima, M. Ichioka, and K. Machida, ibid. 97, 180407 (2006); Hui Hu, Xia-Ji Liu, and Peter D. Drummond, ibid. 98, 060406 (2007); N. Nygaard, G. M. Bruun, B. I. Schneider, C. W. Clark, and D. L. Feder, Phys. Rev. A 69, 053622 (2004); M. Machida, Y. Ohashi, and T. Koyama ibid. 74, 023621 (2006); H. J. Warringa, and A. Sedrakian, ibid. 84, 023609 (2011).
* (14) D. L. Feder, Phys. Rev. Lett. 93, 200406 (2004); G. Tonini, F. Werner, and Y. Castin, Eur. Phys. J. D 39, 283 (2006); A. Bulgac, Y.-L. Luo, P. Magierski, K. J. Roche, and Y. Yu, Science 332, 1288 (2011).
* (15) H. Akera, A. H. MacDonald, S. M. Girvin, and M. R. Norman, Phys. Rev. Lett. 67, 2375 (1991); G. Möller and N. R. Cooper, ibid. 99, 190409 (2007); H. Zhai, R. O. Umucalılar, and M. Ö. Oktel, ibid. 104, 145301 (2010).
* (16) V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell, Phys. Rev. Lett. 92, 040404 (2004); V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, ibid. 92, 050403 (2004).
* (17) M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
* (18) E. E. Salpeter, Phys. Rev. 84, 1226 (1951).
* (19) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
* (20) D. Sciamarella, and Y. Pomeau, Journal of Low Temp. Phys. 123, 35, (2001)
* (21) K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, Rev. Mod. Phys. 78, 483 (2006); G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet, Phys. Rev. Lett. 95, 020404 (2005).
* (22) We define the number of fermions and paired fermions as $N=\frac{\sqrt{2}L}{2\pi\tilde{\ell}}\int_{-K_{0}/2}^{K_{0}/2}\sum_{n}\langle a_{K-nK_{0}\uparrow}^{\dagger}a_{K-nK_{0}\uparrow}\rangle dK$ and $N_{\rm pair}=\frac{\sqrt{2}L}{4\pi\tilde{\ell}}\int_{-K_{0}/2}^{K_{0}/2}\sum_{n,n^{\prime}}|\langle a_{-K+nK_{0}\downarrow}a_{K-n^{\prime}K_{0}\uparrow}\rangle|^{2}dK$.
## I Supplemental materials
### I.1 A – Derivation of gap equation and number equation
Here we analyze the special case where $\xi=1$, corresponding to a 1D model
with translational invariance: $\Delta_{q}=0$ unless $q=0$. Under these
circumstances, Eq.(6) simplifies to
$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{K,\sigma}(K^{2}-\mu)a_{K\sigma}^{\dagger}a_{K\sigma}$ (9)
$\displaystyle+$
$\displaystyle\sum_{K}\left(\Delta_{0}^{*}f(0)+\Delta_{0}f^{\dagger}(0)\right)-|\Delta_{0}|^{2}/\beta.$
where we have introduced the dimensionless Hamiltonian
$\mathcal{H}_{0}=H/{\cal E}$, and
$f(0)=\sum_{K}e^{-\frac{1}{2}K^{2}}a_{-K\downarrow}a_{K\uparrow}$.
$\mathcal{H}_{0}$ can be diagonalized in terms of non-interacting Bogoliubov
quasi-particle operators $\xi_{K},\chi_{K}$ by the transformation
$\displaystyle\left(\begin{array}[]{c}a_{K\uparrow}\\\
a^{\dagger}_{-K\downarrow}\\\
\end{array}\right)=\left(\begin{array}[]{cc}u_{K}&-v_{K}^{*}\\\
v_{K}&u_{K}^{*}\\\ \end{array}\right)\left(\begin{array}[]{c}\xi_{K}\\\
\chi_{K}^{\dagger}\\\ \end{array}\right)$ (16)
yielding the diagonalized Hamiltonian,
$\displaystyle\mathcal{H}_{0}=\sum_{K}\left(\epsilon_{K}(\xi_{K}^{\dagger}\xi_{K}+\chi_{K}^{\dagger}\chi_{K})+\epsilon_{0}-\epsilon_{K}\right)-\frac{|\Delta_{0}|^{2}}{\beta}$
(17)
where
$\displaystyle
u_{K}=\sqrt{\frac{\epsilon_{K}+\epsilon_{0}}{2\epsilon_{K}}},v_{K}=\sqrt{\frac{\epsilon_{K}-\epsilon_{0}}{2\epsilon_{K}}}$
(18)
$\displaystyle\epsilon_{K}=\sqrt{\epsilon_{0}^{2}+|\Delta_{0}|^{2}e^{-K^{2}}},\epsilon_{0}=K^{2}-\mu$
(19)
We introduce the dimensionless energy
$\mathcal{F}\equiv(2\sqrt{2}\pi\tilde{\ell}/L{\cal E})\langle GS|H|GS\rangle$,
where the ground state $|GS\rangle$ is annihilated by quasi-particle operators
$\xi_{K},\chi_{K}$,
$\displaystyle\mathcal{F}=\int(\epsilon_{0}-\epsilon_{K})dK+|\Delta_{0}|^{2}/\alpha,$
(20)
where we have taken the continuum limit
$\sum_{K}\rightarrow(\sqrt{2}L/4\pi\tilde{\ell})\int dK$.
Making $(1/|\Delta_{0}|)\partial\mathcal{F}/\partial|\Delta_{0}|=0$ yields the
gap equation (7). Letting
$N=-(\sqrt{2}L/4\pi\tilde{\ell})\partial\mathcal{F}/\partial\mu$ yields the
number equation (8). These equations are further explored in the main text.
### I.2 B – Numerical approach
We here describe our numerical approach to solving the BdG equations in the
general case where $\xi>1$. Formally, Eq.(6) can be expressed in terms of non-
interacting Bogoliubov quasi-particles by a canonical transformation
$\displaystyle\left(\begin{array}[]{c}a_{K_{n}\uparrow}\\\
a^{\dagger}_{K_{n}\downarrow}\\\
\end{array}\right)=\sum_{n^{\prime}}\left(\begin{array}[]{cc}u_{n^{\prime}n}&-v_{n^{\prime}n}^{*}\\\
v_{n^{\prime}n}&u_{n^{\prime}n}^{*}\\\
\end{array}\right)\left(\begin{array}[]{c}\xi_{K_{n^{\prime}}}\\\
\chi^{\dagger}_{K_{n^{\prime}}}\\\ \end{array}\right),$ (27)
where we have defined $K_{n}\equiv K-nK_{0}$, and $u_{n^{\prime}n}\equiv
u_{n^{\prime}}(K_{n}),v_{n^{\prime}n}\equiv v_{n^{\prime}}(K_{n})$. The matrix
elements $u_{n^{\prime}n},v_{n^{\prime}n}$ are governed by the following BdG
equations,
$\displaystyle\epsilon_{K_{n}}\left(\begin{array}[]{c}v_{nn}\\\ u_{nn}\\\
\end{array}\right)=\sum_{n^{\prime}}\left(\begin{array}[]{cc}-\varepsilon_{n^{\prime}}\delta_{nn^{\prime}}&(\Delta_{n^{\prime}}^{n})^{*}\\\
\Delta_{n^{\prime}}^{n}&\varepsilon_{n^{\prime}}\delta_{nn^{\prime}}\\\
\end{array}\right)\left(\begin{array}[]{c}v_{nn^{\prime}}\\\
u_{nn^{\prime}}\\\ \end{array}\right)$ (34)
where $\epsilon_{K_{n}}$ is the dimensionless excitation energy of Bogoliubov
quasi-particles, and $\varepsilon_{n}=K_{n}^{2}-\mu$,
$\Delta_{n^{\prime}}^{n}=\Delta_{|n|K_{0}}e^{-\frac{1}{8}(2K_{n^{\prime}}-nK_{0})^{2}}$,
and $\delta_{nn^{\prime}}$ is the $\delta$-function. In terms of the
Bogoliubov operators the Hamiltonian is diagonal,
$\displaystyle H/{\cal E}$ $\displaystyle=$
$\displaystyle\sum_{n}\biggl{[}\sum_{K=-K_{0}/2}^{K_{0}/2}(\varepsilon_{n}-\epsilon_{K_{n}})-|\Delta_{|n|K_{0}}|^{2}/\beta$
(35) $\displaystyle+$
$\displaystyle\sum_{K=-K_{0}/2}^{K_{0}/2}\biggl{(}\epsilon_{K_{n}}(\xi_{K_{n}}^{\dagger}\xi_{K_{n}}+\chi_{K_{n}}^{\dagger}\chi_{K_{n}})\biggr{)}\biggr{]}.$
The dimensionless ground state energy
$\mathcal{F}=(2\sqrt{2}\pi\tilde{\ell}/L{\cal E})\langle GS|H|GS\rangle$ can
be written as
$\displaystyle\mathcal{F}=\sum_{n}\biggl{(}\int_{-K_{0}/2}^{K_{0}/2}(\varepsilon_{n}-\epsilon_{K_{n}})dK+|\Delta_{|n|K_{0}}|^{2}/\alpha\biggr{)}.$
(36)
For a given $\\{\mu,K_{0},\Delta_{|n|K_{0}}\\}$, we truncate Eq.(34), and use
standard linear algebra packages to extract $\epsilon_{K_{n}}$. This
effectively gives us $\mathcal{F}$ as a function of
$\\{\mu,\alpha,K_{0},\Delta_{|n|K_{0}}\\}$. This $\mathcal{F}$ is a
variational upperbound on the true ground state energy. We fix
$\\{\mu,\alpha\\}$ and numerically minimize $\mathcal{F}$, varying
$\\{K_{0},\Delta_{|n|K_{0}}\\}$, using a quasi-Newton algorithm. We restrict
the sum over $n$ in Eq.(36) to $-\zeta\leq n\leq\zeta$. We find for the
parameters studied, our results are independent of $\zeta$ if $\zeta\geq 6$.
### I.3 C – Linear stability analysis
Here we find the $\xi=1$ to $\xi=3$ phase boundary through a linear stability
analysis. We take $\Delta_{0}>0$, and assume
$\Delta_{K_{0}}=\Delta_{-K_{0}}=i\delta$ is small. We have chosen this factor
of $i$, as the unstable direction will then yield real $\delta$. We will
calculate $D=\partial^{2}\mathcal{F}/\partial\delta^{2}|_{\delta=0}$. For
small $\alpha$ the curvature $D$ is positive and the state with $\delta=0$ is
stable. We find the instability by seeking the point with when $D=0$.
Within our ansatz for $\Delta_{|n|K_{0}}$, the mean field Hamiltonian is
$\displaystyle H/{\cal E}=\mathcal{H}_{0}+i\delta\Lambda-2\delta^{2}/\beta,$
(37)
where
$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
f^{\dagger}(K_{0})+f^{\dagger}(-K_{0})-f(K_{0})-f(-K_{0}).$ (38)
Making use of the Hellmann-Feynman theorem, the second derivative of
$\mathcal{F}$ can be expressed as
$\displaystyle\frac{\partial^{2}\mathcal{F}}{\partial\delta^{2}}$
$\displaystyle=$
$\displaystyle\frac{2\sqrt{2}\pi\tilde{\ell}}{L}\frac{\partial^{2}}{\partial\delta^{2}}\langle
GS|i\Lambda\delta-\frac{2\delta^{2}}{\beta}|GS\rangle$ (39) $\displaystyle=$
$\displaystyle\frac{2\sqrt{2}\pi\tilde{\ell}}{L}\frac{\partial}{\partial\delta}\langle
GS|i\Lambda-\frac{4\delta}{\beta}|GS\rangle$ $\displaystyle=$
$\displaystyle-i\frac{4}{\alpha}\frac{\partial}{\partial\delta}\langle
GS|\beta f^{\dagger}(K_{0})|GS\rangle+\frac{4}{\alpha}.$
Setting $D=\partial^{2}\mathcal{F}/\partial\delta^{2}|_{\delta=0}=0$, one
finds that the points of instability is given by
$\displaystyle-i=\beta\frac{\partial}{\partial\delta}\langle
GS|f^{\dagger}(K_{0})|GS\rangle.$ (40)
Since the formal manipulations of perturbation theory are more transparent of
finite temperature, it is convenient to rewrite Eq.(40) as
$\displaystyle-i=\lim_{\mathcal{T}\rightarrow
0}\beta\frac{\partial}{\partial\delta}\frac{Tr(e^{-\mathcal{H}/\mathcal{T}}f^{\dagger}(K_{0}))}{Tr(e^{-\mathcal{H}/\mathcal{T}})}$
$\displaystyle=-i\beta\lim_{\mathcal{T}\rightarrow
0}\frac{\int_{0}^{1/\mathcal{T}}Tr(e^{-\tau\mathcal{H}_{0}}\Lambda
e^{(-1/\mathcal{T}+\tau)\mathcal{H}_{0}}f^{\dagger}(K_{0}))d\tau}{Tr(e^{-\mathcal{H}_{0}/\mathcal{T}})}$
(41)
where $\mathcal{T}$ is a formal parameter.
Substituting the results of Eq.(17)-(18) to Eq.(I.3), we obtain
$\displaystyle\alpha\int\frac{(u_{K}u_{K-K_{0}}+v_{K}v_{K-K_{0}})^{2}e^{-\frac{1}{4}(K_{0}-2K)^{2}}}{\epsilon_{K-K_{0}}+\epsilon_{K}}dK=1.$
(42)
This integral must be performed numerically, giving the second (right) black
solid curve in Fig.1(a).
|
arxiv-papers
| 2012-03-22T20:33:47 |
2024-09-04T02:49:28.961995
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ran Wei, Erich J. Mueller",
"submitter": "Ran Wei",
"url": "https://arxiv.org/abs/1203.5120"
}
|
1203.5167
|
# Study of wave chaos in a randomly-inhomogeneous oceanic acoustic waveguide:
spectral analysis of the finite-range evolution operator
D.V. Makarov makarov@poi.dvo.ru L.E. Kon’kov M.Yu. Uleysky V.I.Il’ichev
Pacific Oceanological Institute
of the Far-Eastern Branch of the Russian Academy of Sciences,
43 Baltiyskaya St., 690041, Vladivostok, Russia P.S. Petrov V.I.Il’ichev
Pacific Oceanological Institute
of the Far-Eastern Branch of the Russian Academy of Sciences,
43 Baltiyskaya St., 690041, Vladivostok, Russia Far-East Federal University,
8 Sukhanova St., 690950, Vladivostok, Russia
###### Abstract
The proplem of sound propagation in an oceanic waveguide is considered.
Scattering on random inhomogeneity of the waveguide leads to wave chaos. Chaos
reveals itself in spectral properties of the finite-range evolution operator
(FREO). FREO describes transformation of a wavefield in course of propagation
along a finite segment of a waveguide. We study transition to chaos by
tracking variations in spectral statistics with increasing length of the
segment. Analysis of the FREO is accompanied with ray calculations using the
one-step Poincaré map which is the classical counterpart of the FREO.
Underwater sound channel in the Sea of Japan is taken for an example. Several
methods of spectral analysis are utilized. In particular, we approximate level
spacing statistics by means of the Berry-Robnik and Brody distributions,
explore the spectrum using the procedure elaborated by A. Relano with
coworkers (Relano et al, Phys. Rev. Lett., 2002; Relano, Phys. Rev. Lett.,
2008), and analyze modal expansions of the eigenfunctions. We show that the
analysis of FREO eigenfunctions is more informative than the analysis of
eigenvalue statistics. It is found that near-axial sound propagation in the
Sea of Japan preserves stability even over distances of hundreds kilometers.
This phenomenon is associated with the presence of a shearless torus in the
classical phase space. Increasing of acoustic wavelength degrades scattering,
resulting in recovery of localization near periodic orbits of the one-step
Poincaré map. Relying upon the formal analogy between wave and quantum chaos,
we suggest that the concept of FREO, supported by classical calculations via
the one-step Poincaré map, can be efficiently applied for studying chaos-
induced decoherence in quantum systems.
###### pacs:
05.45.Mt, 43.30.Cq, 03.65.Yz, 05.45.Ac, 43.30.Ft
## I Introduction
Sound speed in the deep ocean typically has a minimum at some depth. This
results in formation of a refractive waveguide, the so-called underwater sound
channel, which prevents sound waves from contact with the absorbing bottom. As
sound absorption within water column is fairly weak, an underwater sound
channel enables sound propagation over distances of thousands kilometers. The
largest distance had been achieved using explosive charges in the seminal
experiment on sound transmission from Perth to Bermuda in 1960 Shockley _et
al._ (1982); Munk _et al._ (1988).
It is realized that small sound-speed variations induced by oceanic internal
waves lead to Lyapunov instability and chaos of sound rays. In the
mathematical sense, ray chaos is an analogue of classical chaos in Hamiltonian
systems. Indeed, ray motion in a waveguide is equivalent to motion of a point
particle in a potential well, and sound-speed variations along a waveguide
play the role of a nonstationary perturbation. Reciprocal Lyapunov exponent
for chaotic rays typically is of about several tens kilometers Makarov _et
al._ (2009), therefore, the problem of ray chaos is mainly important for long-
range sound propagation. During the last two decades ray chaos in ocean
acoustics was an object of intense research, both theoretical and experimental
Smith _et al._ (1992); Smirnov _et al._ (2001); Brown _et al._ (2003);
Makarov _et al._ (2004); Beron-Vera and Brown (2004). Considerable attention
was paid to wave chaos Makarov _et al._ (2009); Virovlyansky and Zaslavsky
(1999); Smirnov _et al._ (2004, 2005); Kon’kov _et al._ (2007); Makarov _et
al._ (2008); Virovlyansky _et al._ (2012). The term wave chaos relates to
wavefield manifestations of ray chaos. It was found that interference makes
wave refraction more regular than it is anticipated from ray modeling, albeit
influence of ray chaos persist even for very low sound frequencies Hegewisch
_et al._ (2005). This problem becomes especially important due to the growing
interest to hydroacoustical tomography, i. e. monitoring of environment using
sound signals. The classical scheme of tomography developed by Munk and Wunsch
Munk and Wunsch (1979) is based on computation of eigenrays connecting the
source and the receiver. It was shown in Tappert and Tang (1996) that ray
chaos leads to exponential proliferation of eigenrays with increasing
distance. As a result, the inverse problem becomes ill-posed, impeding
environment reconstruction. However, wave-based corrections “stabilize” wave
refraction, i. e. the standard semiclassical approximation typically
overestimates ray chaos. Thus, one needs either an improved version of the
semiclassical approximation for proper computation of eigenrays, or some
approach for making implications about eigenray stability relying upon wave
modeling, whereby a priori taking into account the wave-based suppression of
ray chaos.
Theory of ray and wave chaos extensively exploits methods borrowed from the
theory of Hamiltonian dynamical systems Makarov _et al._ (2009); Zaslavsky
(2007). In particular, ray motion is often studied by means of the phase space
representation. This provides the clear geometric interpretation of ray
dynamics for toy models of deterministic range-periodic waveguides. For
instance, one can easily separate domains of initial conditions corresponding
to stable and chaotic rays from each other. However, realistic underwater
sound channels are not range-periodic, and their range inhomogeneity should be
rather described as a stochastic process. On the other hand, statistical
methods Dozier and Tappert (1978a); Colosi and Morozov (2009) are implicitly
based on the assumption of ergodic chaos and ignore the existence of phase
space domains of finite-range stability Wolfson and Tomsovic (2001); Makarov
_et al._ (2006). Therefore, it is reasonable to elaborate some theoretical
approach combining advantages of statistical and deterministic methods.
Recently it had been shown that a bridge between the deterministic and
statistical descriptions of wave propagation in random media can be built up
by introducing the finite-range evolution operator (hereafter FREO) Makarov
_et al._ (2010); Virovlyansky _et al._ (2012). FREO acts as a propagator
determining wavefield transformation between two vertical sections of a
waveguide. Spectral analysis of the FREO in terms of the random matrix theory
allows one to estimate contribution of ray chaos to wave dynamics and track
transition to chaos with increasing distance between the sections. A somewhat
different approach had been offered in Hegewisch and Tomsovic (2012). These
approaches allow one to generalize the well-developed spectral theory of
quantum chaos (see, for instance, Stöckmann (2007)) on one-way wave
propagation in random media. On the other hand, concept of the FREO can serve
as a promising method for studying chaos-induced decoherence in randomly-
driven quantum systems Kolovsky (1997).
In the present paper we study spectral statistics of the FREO for the Sea of
Japan. Attention is concentrated on the track connecting the Gamov peninsula
and Kita-Yamato bank. The length of the track is about 350 km. Our interest to
this waveguide is motivated by results of the experiment conducted there in
2006 Bezotvetnykh _et al._ (2009), indicating on high stability of near-axial
propagation. Similar behavior was observed in a earlier experiment with a
slightly different propagation track Spindel _et al._ (2003). It should be
noted that stability of near-axial propagation is atypical for the deep ocean
Makarov _et al._ (2009); Virovlyansky _et al._ (2012).
The paper is organized as follows. The next section represents basic equations
describing long-range sound propagation in the ocean. In Section III, we
describe the waveguide used in the paper. Section IV contains description of
the FREO and methods of its analysis. Section V is devoted to the classical
counterpart of the FREO, namely the one-step Poincaré map. In Section VI, we
perform statistical analysis of the FREO for the underwater sound channel in
the Sea of Japan. In Conclusion we summarize and discuss the results obtained.
## II General equations
Ocean is a layered media, and its horizontal variability is much weaker than
vertical one. This allows one to reduce the initial three-dimensional problem
of wave propagation in an underwater sound channel to a two-dimensional one by
assuming cylindrical symmetry and neglecting azimuthal coupling. Sound
refraction is governed by spatial variability of sound speed
$c(z,\,r)=c_{0}+\Delta c(z)+\delta c(z,\,r),$ (1)
where $z$ is depth, $r$ is range coordinate, $c_{0}$ is a reference sound
speed. Sound-speed variations obey the double inequality
$\lvert\delta c\rvert_{\text{max}}\ll\lvert\Delta c\rvert_{\text{max}}\ll
c_{0}.$ (2)
Left inequality implies that the range-dependent term can be treated as a weak
perturbation of the background sound-speed profile. This term is mainly
contributed from oceanic internal waves. Right inequality means that
variations of the refractive index are weak, and only those waves which
propagate with small angles with respect to the horizontal plane can avoid
contact with the absorbing ocean bottom. Thus, one can invoke the small-angle
approximation, in which an acoustic wavefield is governed by the standard
parabolic equation
$\frac{i}{k_{0}}\frac{\partial\Phi}{\partial
r}=-\frac{1}{2k_{0}^{2}}\frac{\partial^{2}\Phi}{\partial
z^{2}}+\left[U(z)+\varepsilon V(z,\,r)\right]\Phi,$ (3)
where wave function $\Phi$ is linked to acoustic pressure $u$ by means of the
formula $u=\Phi\exp(ik_{0}r)/\sqrt{r}$. Here the denominator $\sqrt{r}$
responds for the cylindrical spreading of sound. Quantity $k_{0}$ is the
reference wavenumber related to sound frequency $f$ as $k_{0}=2\pi f/c_{0}$.
Functions $U(z)$ and $V(z,r)$ are determined by spatial sound-speed
variations. In the the small-angle approximation they can be expressed as
$U(z)=\frac{\Delta c(z)}{c_{0}},\quad V(z,\,r)=\frac{\delta c(z,\,r)}{c_{0}}.$
(4)
According to (2), $\lvert V\rvert_{\text{max}}\ll\lvert U\rvert_{\text{max}}$,
that is, function $V(z,r)$ can be treated as a small perturbation. One can
easily see that the replacement
$k_{0}^{-1}\to\hbar,\quad r\to t$ (5)
transforms the parabolic equation (3) into the Schrödinger equation for a
particle with unit mass. This circumstance enables study of wave propagation
using the approaches developed in quantum mechanics. In this relationship,
function $U(z)$ serves as an unperturbed potential. As $r$ is a timelike
variable, $V(z,\,r)$ plays the role of a nonstationary perturbation.
In the short-wavelength limit $k_{0}\to\infty$, solution of the parabolic
equation (3) can be expressed as a sum of rays whose trajectories are governed
by the Hamiltonian
$H=\frac{p^{2}}{2}+U(z)+V(z,\,r),$ (6)
where $p=\tan\chi$, $\chi$ is ray grazing angle (i. e. angle with respect to
the horizontal plane). The respective Hamiltonian equations read
$\frac{dz}{dr}=\frac{\partial H}{\partial
p}=p,\quad\frac{dp}{dr}=-\frac{\partial H}{\partial z}=-\frac{\partial
U}{\partial z}-\frac{\partial V}{\partial z}.$ (7)
Owing to the analogy with classical mechanics, $p$ is referred to as ray
momentum.
## III Model of a waveguide
### III.1 Background sound-speed profile
Figure 1: (a) Unperturbed sound-speed profile, (b) the first empirical
orthogonal function of the sound-speed perturbation (solid) and its smoothed
approximation (dotted).
Model of the underwater sound channel in the Sea of Japan was elaborated using
the hydrological data from the database Dat . Function $U(z)$ corresponding to
the background sound-speed profile was approximated by the expression
$U(z)=\left\\{\begin{aligned} &U_{1}(z),&z\leqslant z_{0},\\\
&U_{2}(z),&z>z_{0}.\end{aligned}\right.$ (8)
where
$\displaystyle U_{1}(z)$ $\displaystyle=\frac{c_{1}}{c_{0}}e^{-z/z_{1}},$ (9)
$\displaystyle U_{2}(z)$
$\displaystyle=\frac{c_{1}}{c_{0}}e^{-z_{0}/z_{1}}+\frac{g}{c_{0}}(z-z_{0}),$
$c_{0}=1455$ m/s, $c_{1}=70$ m/s, $z_{0}=250$ m is the depth of the channel
axis, i. e. the depth with the minimal sound speed, $z_{1}=30$ m, $g=0.017$
s-1. (see Fig. 1). The ocean bottom is assumed to be flat and placed at the
depth $h=3$ km. We consider only the deep-water propagation, albeit the source
in the aforementioned experiments Bezotvetnykh _et al._ (2009); Spindel _et
al._ (2003) was mounted into the bottom in the coastal zone near the Gamov
peninsula. The shallow-water part of the waveguide was relatively short, less
than 30 km, and didn’t have significant bottom features which could remarkably
alter ray stability.
Figure 2: Ray cycle length vs initial ray momentum for the source located at
the channel axis.
Expressions (8) and (9) permit analytical derivation of the basic model
characteritics in the absence of horizontal inhomogeneity. One can introduce
the ray action
$I=\frac{1}{\pi}\int\limits_{z_{\text{min}}}^{z_{\text{max}}}\sqrt{2[E-U(z)]}\,dz,$
(10)
where $z_{\text{min}}$ and $z_{\text{max}}$ are the upper and lower ray
turning points, respectively, and $E=0.5p^{2}+U(z)$. Ray action describes
steepness of a ray trajectory and enters into the Einstein-Brillouin-Keller
quantization rule.
$k_{0}I_{m}=m-1/2,\quad m=1,2,\dots$ (11)
establishing the link between normal modes of the unperturbed waveguide and
modal rays. Here $I_{m}$ is the action of a modal ray, and both ray turning
points are assumed to be inside the water column, that is, rays undergo total
internal reflection due to smooth vertical gradient of the refractive index
$n=c_{0}/c(z,\,r)$. Normal modes are the solutions of the Sturm-Liouville
equation
$-\frac{1}{2k_{0}^{2}}\frac{\partial^{2}\phi_{n}(z)}{\partial
z^{2}}+U(z)\phi_{n}(z)=E_{n}\phi_{n}(z),$ (12)
with appropriate boundary conditions at the ocean surface and bottom. The main
contribution to the mode $m$ is given from the rays with $I\simeq I_{m}$
Virovlyansky and Zaslavsky (1999). Integration of (10) yields
$I=2z_{1}\sqrt{2E_{\text{min}}}\left[\sqrt{\varepsilon}\ln(\sqrt{\varepsilon}+\sqrt{\varepsilon-1})-\sqrt{\varepsilon-1}\vphantom{\sqrt{\mathstrut}}\right]+\\\
\frac{2\sqrt{2}c_{0}}{3g}(E-E_{\text{min}})^{3/2},$ (13)
where $\varepsilon=E/E_{\text{min}}$,
$E_{\text{min}}=(c_{1}/c_{0})\exp(-z_{0}/z_{1})$. Ray cycle length, i. e.
horizontal distance between two successive upper (or lower) ray turning
points, can be determined as
$D(E)=2\pi\frac{dI}{dE}=\\\
2z_{1}\sqrt{\frac{2}{E}}\ln\left(\sqrt{\varepsilon}+\sqrt{\varepsilon-1}\right)+\frac{2c_{0}\sqrt{2E_{\text{min}}}}{g}\sqrt{\varepsilon-1}.$
(14)
Fig. 2 represents dependence of ray cycle length on initial ray momentum for
the source located at the channel axis $z=250$ m. Initial momentum $p_{0}$
depends on $E$ as $p_{0}=\sqrt{2(E-E_{\text{min}})}$. Function $D(p_{0})$ is
nonmonotonic and has two extrema, the sharp maximum and the smooth minimum.
The latter one can give rise to a so-called weakly-divergent beam Brekhovskikh
_et al._ (1990); Smirnov _et al._ (1999); Morozov and Colosi (2005); Petukhov
(2009). Its low divergence is associated with approximate equality of cycle
length values for rays forming the beam. It will be demonstrated in Section V
that the local minimum of $D(p_{0})$ plays a significant role in ray stabilty.
### III.2 Sound-speed perturbation
Model of the internal-wave-induced sound-speed perturbation was built up in
several steps. Firstly, we computed the range-averaged profile of buoyancy
frequency, using the hydrological data Dat . Then, we calculated realizations
of the sound-speed perturbation using the method proposed in Colosi and Brown
(1998). In order to facilitate numerical simulation, the perturbation was
expanded over empirical orthogonal functions LeBlanc and Middleton (1980)
$\delta c(z,\,r)=\left<\delta c(z)\right>+\sum\limits_{n}b_{n}(r)Y_{n}(z).$
(15)
Empirical orthogonal functions $Y_{n}(z)$ are the eigenvectors of the
covariance matrix $\hat{K}$ with elements
$K_{ij}=\frac{1}{L}\sum\limits_{l=1}^{L}[\delta c_{l}(z_{i})-\left<{\delta
c}(z_{i})\right>][\delta c_{l}(z_{j})-\left<{\delta c}(z_{j})\right>],$ (16)
where index $l$ numbers $L$ statistically independent realizations of $\delta
c(z)$, $\\{z_{i}\\}$ is a vector of depth values, and angular brackets mean
ensemble average. As $\delta c$ is caused by internal waves, one can set
$\left<\delta c\right>=0$. Eigenvalues of the matrix $\hat{K}$ quantify
contributions from the corresponding eigenvectors in the expansion (15). It
was found that the contribution of the first orthogonal function prevails, and
one can fairly represent the sound-speed perturbation as the product
$V(z,\,r)=b_{1}(r)Y_{1}(z)$ (17)
where $b_{1}(r)$ is a random function. For simplicity, it is assumed that
$b_{1}(r)$ is a stochastic process with the exponentially-decaying
autocorrelation function
$\left<b_{1}(r)b_{1}(r^{\prime})\right>=\exp(-\lvert
r-r^{\prime}\rvert/\bar{r}),$ (18)
where the correlation length $\bar{r}$ is taken of 10 km, that is typical for
the deep ocean Dozier and Tappert (1978b). Then the realizations of $b_{1}(r)$
can be computed via the formula $b_{1}=\sqrt{2\bar{r}}\eta$, where $\eta$ is a
solution of the Ornstein-Uhlenbeck stochastic differential equation
$\frac{d\eta}{dr}=-\frac{1}{\bar{r}}\eta(r)+\frac{1}{\bar{r}}\xi(r).$ (19)
The procedure of solving this equation is described, for instance, in Mallick
and Marcq (2002). Here $\xi$ is a Gaussian white noise satisfying
$\left<\xi(r)\right>=0,\quad\left<\xi(r)\xi(r^{\prime})\right>=\delta(r-r^{\prime}).$
(20)
The resulting function $b_{1}(r)$ satisfies $\left<b_{1}^{2}(r)\right>\simeq
1$.
The calculated function $Y_{1}(z)$ is depicted in Fig. 1(b). It involves step-
like changes in the depth interval from 100 to 300 meters. These changes are
caused by the depth discretization of the hydrological data and physically
irrelevant. Wave modeling is insensitive to them, but ray-based calculations
can be significantly affected. Therefore, in ray calculations we use a smooth
analytical approximation
$Y_{1}=Ay_{\mathrm{a}}\exp(-y_{\mathrm{a}}^{n})+B\exp(-y_{\mathrm{b}}^{2}),$
(21)
where $A=0.0027$, $y_{\mathrm{a}}=z/z_{\mathrm{a}}$, $n=1.1$,
$z_{\mathrm{a}}=24$ m, $B=2\cdot 10^{-5}$,
$y_{\mathrm{b}}=(z-z_{\mathrm{c}})/z_{\mathrm{b}}$, $z_{\mathrm{b}}=50$ m,
$z_{\mathrm{c}}=200$ m.
## IV Finite-range evolution operator
Finite-range evolution operator (FREO) had been introduced for studying wave
propagation in a randomly-inhomogeneous waveguide in Virovlyansky _et al._
(2012); Makarov _et al._ (2010). Its quantum-mechanical analogue was earlier
considered in Kolovsky (1997). Basically, a finite-range evolution operator
(FREO) is an element of one-parameter group generated by the operator in the
right-hand side of the equation (3). Consider a solution $\Phi(z,\,r)$ of the
parabolic equation (3) complemented with the standard boundary conditions
$\left.\Phi\right|_{z=0}=0,\quad\left.\frac{d\Phi}{dz}\right|_{z=h}=0$ (22)
and the initial condition $\Phi(z,\,r=0)=\bar{\Phi}(z)$, where $\bar{\Phi}(z)$
belongs to $L^{2}[0,h]$ and satisfies (22). Then FREO $\hat{G}(\tau)$ is
defined on the subspace of $L^{2}[0,h]$ (restricted by (22)) as
$\hat{G}(\tau)\bar{\Phi}(z)\equiv\left.\Phi(z,\,r)\right|_{r=\tau}.$ (23)
By definition, the FREO describes transformation of a wavefield in course of
propagation along a finite waveguide segment of length $\tau$. Each
realization of inhomogeneity produces its own realization of the FREO. Our
interest is concerned with statistical properties of the FREO and their
connection to classical ray stability.
Note that the choice of the hard wall boundary condition at the bottom (22) is
typical for the deep ocean acoustics problems when the attention is restricted
to the trapped modes, whose propagation is not affected by the bottom
interaction. Under these conditions no energy is absorbed by the bottom.
Hence, if the attenuation in the sea water is negligible (this is true for the
sound frequencies of our interest) and refraction index in (3) has no
imaginary part, then the FREO is a unitary operator.
FREO can be represented as a matrix in the basis of normal modes $\phi_{j}(z)$
satisfying the Sturm-Liouville problem (12,22). Matrix elements of the FREO
are given by
$G_{mn}(\tau)=\int\limits_{0}^{h}\phi_{m}(z)\hat{G}(\tau)\phi_{n}(z)\,dz.$
(24)
Thus, the matrix elements $G_{mn}$ are complex-valued amplitudes of modal
transitions. For the range-independent waveguide, the matrix of FREO is
diagonal with $|G_{mm}|=1$.
Eigenvalues and eigenvectors of the FREO obey the equation
$\hat{G}\Psi_{m}(z,\,r)=g_{m}\Psi_{m}(z,\,r).$ (25)
Owing to unitarity, eigenvalues $g_{m}$ can be recast as
$g_{m}=e^{-ik_{0}\epsilon_{m}},\quad\epsilon_{m}\in\Re.$ (26)
Since eigenvalues of the FREO belong to the unit circle in the complex plane,
the FREO corresponds to the circular ensemble Stöckmann (2007). The FREO has
much in common with the Floquet operator governing wave propagation in a
range-periodic waveguide Smirnov _et al._ (2005); Kon’kov _et al._ (2007)
and quantum dynamics in time-periodic systems. For instance, quantity
$\epsilon_{m}$ is the analogue of quasienergy in quantum mechanics.
Note that eigenvalues of $\hat{G}(\tau)$ may be easily computed using its
matrix representation $G_{mn}(\tau)$. To accomplish this one has to clip a
finite block of this (infinite) matrix corresponding to the trapped modes,
neglecting their interaction with high-order modes. This simplification is
reasonable and does not affect accuracy of the eigenvalues computation (and
the numerics confirms that) since the prevailing small-angle propagation
corresponds to the low-order modes.
### IV.1 Level spacing statistics
Wave chaos reveals itself in the spectrum of the FREO, in particular, in the
statistics of level spacings. A level spacing is defined as
$\begin{gathered}s=\frac{k_{0}M(\epsilon_{m+1}-\epsilon_{m})}{2\pi},\quad
m=1,2,\dots,M,\\\
\epsilon_{M+1}=\epsilon_{1}+\frac{2\pi}{k_{0}}.\end{gathered}$ (27)
where $\epsilon_{m}$ increases with increasing $m$, $M$ is the total number of
eigenvalues for a single realization of the FREO, equal to the number of
trapped modes.
Level spacing statistics can be studied in terms of the random matrix theory
Stöckmann (2007). Regular dynamics implies that the matrix of the FREO
consists of separate independent blocks. Then level sequences contributed from
different blocks are statistically independent, therefore, the resulting level
spacing distribution obeys the Poisson law
$\rho(s)\sim\exp(-s).$ (28)
Under conditions of ergodic chaos, all normal modes are coupled to each other.
This results in repulsion of neighbouring levels, the phenomenon closely
related to spectral splittings induced by tunneling Landau and Lifshitz
(1977). In this case level spacing statistics is described by the Wigner-Dyson
distribution
$\rho(s)\sim s^{\zeta}\exp\left(-Cs^{2}\right),$ (29)
where constants $\zeta$ and $C$ depend on symmetries of the FREO. As the FREO
doesn’t possess the symmetry $r\to-r$, it corresponds to the circular unitary
ensemble (CUE) with $\zeta=2$ and $C=4/\pi$ Kolovsky (1997).
Figure 3: Berry-Robnik distribution with $v_{\mathrm{r}}=0.9$,
$v_{\mathrm{r}}=0.5$ and $v_{\mathrm{r}}=0.1$.
The most interesting case is mixed phase space, with the coexistence of
regular and chaotic domains. Then level spacing statistics should be described
by some combination of Poisson and Wigner-Dyson laws. In the short-wavelength
limit one can use the Berry-Robnik distribution Berry and Robnik (1984)
$\rho(s)=\left[v_{\mathrm{r}}^{2}\operatorname{erfc}\left(\frac{\sqrt{\pi}}{2}v_{\mathrm{c}}s\right)+\right.\\\
\left.+\left(2v_{\mathrm{r}}v_{\mathrm{c}}+\frac{\pi}{2}v_{\mathrm{c}}^{3}s\right)\exp\left(-\frac{\pi}{4}v_{\mathrm{c}}^{2}s^{2}\right)\vphantom{\frac{\sqrt{\pi}}{2}}\right]\exp(-v_{\mathrm{r}}s),$
(30)
where $v_{\mathrm{r}}$ and $v_{\mathrm{c}}$ are relative phase space volumes
corresponding to regular and chaotic ray motion, respectively,
$v_{\mathrm{r}}+v_{\mathrm{c}}=1$. In (30), it is assumed that phase space
consists of only two distinct domains: one regular and one chaotic.
Taking into account the correspondence between rays and modes established by
the WKB representation (11), one can regard $v_{\mathrm{r}}$ as a fraction of
modes whose refraction is regular, and $v_{\mathrm{c}}$ as a fraction of modes
exhibiting wave chaos. In the limiting cases $v_{\mathrm{r}}=1$ and
$v_{\mathrm{c}}=1$, Berry-Robnik formula (30) reduces to the Poisson
distribution and the Wigner-Dyson distribution for the orthogonal ensemble
($\zeta=1$), respectively. As is shown in Fig. 3, Berry-Robnik distribution
undergoes smooth transition from the Poisson to the Wigner-Dyson law as
$v_{\mathrm{r}}$ decreases from 1 to 0. Wigner-Dyson distributions for
orthogonal ($\zeta=1$) and unitary ($\zeta=2$) ensembles are close to each
other, therefore, one can use (30) as an approximation for the circular
unitary ensemble. Berry-Robnik distribution is based on the assumption of the
total statistical independence of the matrix blocks corresponding to regular
and chaotic dynamics. This assumption is completely fulfilled only in the
semiclassical limit. As wave corrections grow, independence degrades due to
regular-to-chaotic tunneling Bäcker _et al._ (2005). Hence, Berry-Robnik
formula cannot work perfectly for low-frequency sound propagation. In this
case, a better fit is expected with the Brody distribution Prosen and Robnik
(1994)
$\rho(s)=(\beta+1)A_{\beta}s^{\beta}\exp(-A_{\beta}s^{\beta+1}),$ (31)
where $A_{\beta}=[\Gamma(\frac{\beta+2}{\beta+1})]^{\beta+1}$, $\Gamma$ is the
Euler gamma function. $\beta=0$ corresponds to the Poisson distribution,
$\beta=1$ yields the Wigner-Dyson distribution with $\zeta=1$. Unfortunately,
the Brody distribution is semiempirical and doesn’t have an explicit physical
interpretation in the intermediate regime $0<\beta<1$.
Functions (30) and (31) should describe level spacing statistics for single
realizations of the FREO. Very unfortunately, this is problematic because one
encounters insufficiency of the statistical ensemble. Indeed, long-range sound
propagation is feasible only with low acoustic frequencies of tens or hundreds
Hz. In this frequency range, number of trapped modes doesn’t exceed several
hundreds. To resolve this problem, we consider ensemble-averaged level spacing
distribution
$\rho(s,\,\tau)=\lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{n=1}^{N}P_{n}(s,\,\tau),$
(32)
where $P_{n}(s,\,\tau)$ is a level spacing distribution corresponding to
$n$-th realization of FREO. Fitting the function $\rho(s,\,\tau)$ with the
Berry-Robnik distribution (30), one can estimate number regularly propagating
modes for various values of $\tau$ and, whereby, track the transtion to chaos
with increasing $\tau$. However, it should be noted that formula (32) enables
accurate estimate of $v_{\mathrm{r}}$ only if fluctuations of $v_{\mathrm{r}}$
are weak, otherwise one should take into account nonlinearity of $\rho$ as a
function of $v_{\mathrm{r}}$ in (30).
Likewise we can observe transformation of level spacing statistics by fitting
(32) with the Brody distribution (31). Then the value of $\beta$ corresponding
to the best fit should increase with increasing $\tau$ from 0 to 1, reflecting
wavefield stochastization. Such calculations were performed in Makarov _et
al._ (2010); Virovlyansky _et al._ (2012) for a waveguide with a perturbed
biexponential sound-speed profile.
### IV.2 Eigenfunction statistics
Wave chaos is also reflected in eigenfunction statistics of the FREO. Each
eigenfunction can be expressed as a superposition of normal modes:
$\Phi_{m}(z)=\sum\limits_{n}c_{mn}\phi_{n}(z),$ (33)
where $c_{mn}$ is the $m$-th component of $n$-th eigenvector of the matrix
$\hat{G}$, $\phi_{n}(z)$ is the $n$-th normal mode.
There are many methods for identification of “chaotic” eigenfunctions. In the
present work we use only one of them. Ray chaos can lead to intense energy
exchange between normal modes Virovlyansky and Zaslavsky (1999); Makarov _et
al._ (2008); Hegewisch and Tomsovic (2012), therefore, a “chaotic”
eigenfunction is a compound of many modes. The stronger chaos, the larger
number of contributing modes. Hence, we can characterize “chaoticity” by
estimating number of principal components Varga and Pipek (2003) in the
expansion (33). Number of principal components is calculated as
$\nu(n)=\left(\sum\limits_{m=1}^{M}\lvert c_{mn}\rvert^{4}\right)^{-1}.$ (34)
Number of principal components $\nu$ is equal to 1 in an unperturbed waveguide
and grows as scattering intesifies, tending asymptotically to $M$.
## V One-step Poincaré map
In the geometrical acoustics approximation, dynamics of a wavepacket is
represented as motion of some ray bundle. The bundle remains compact in course
of propagation if ray dynamics is regular. Under conditions of ray chaos the
bundle rapidly diverges. Initial conditions for rays contributing to the
bundle can be found by means of the Husimi function
$W_{\mathrm{h}}(p,\,z)=\biggl{|}\frac{1}{\sqrt[4]{2\pi\Delta_{z}^{2}}}\int
dz^{\prime}\Phi_{0}^{*}(z^{\prime})\times\\\
\exp\left[ikp(z^{\prime}-z)-\frac{(z^{\prime}-z)}{4\Delta_{z}^{2}}\right]\biggr{|}^{2},$
(35)
projecting an initial wavepacket $\Phi_{0}$ onto phase space of ray equations
(7). In the limit $k_{0}\to\infty$, the transformation (35) turns
eigenfunctions $\Phi_{m}$ of the FREO into some phase space sets which are
invariant under the shift $r=0\to r=\tau$. The procedure proposed in Makarov
_et al._ (2006) allows one to find out these sets in a randomly-inhomogeneous
waveguide. This is the one-step Poincaré map (or the specific Poincaré map).
The ray analogue of the map looks as follows:
$p_{i+1}=p(r=\tau|\,p_{i},z_{i}),\quad z_{i+1}=q(r=\tau|\,p_{i},z_{i}),$ (36)
where $p(r=\tau|\,p_{i},z_{i})$ and $z(r=\tau|\,p_{i},z_{i})$ are the
solutions of ray equations (7) with initial conditions $p(r=0)=p_{i}$,
$z(r=0)=z_{i}$. Values of $p$ and $z$, calculated at the $i$-th step of
mapping, become the initial conditions for the $(i+1)$-th step. This procedure
is equivalent to the usual Poincaré map Zaslavsky (2007) for a range-periodic
waveguide with the ray Hamiltonian
$\bar{H}=\frac{p^{2}}{2}+U(z)+\tilde{V}(z,\,r).$ (37)
Here $\tilde{V}(z,\,r)$ is periodic in $r$ function
$\tilde{V}(z,\,r^{\prime}+n\tau)=V(z,\,r^{\prime}),\quad 0\leqslant
r^{\prime}\leqslant\tau,$ (38)
where $n$ is an integer. As it follows from (38), $\tilde{V}(z,\,r)$ is a
sequence of identical pieces of $V(z,\,r)$, each of them has the length
$\tau$. Thus we replace the original randomly-perturbed Hamiltonian system by
an equivalent periodically-perturbed one. This replacement is valid as long as
we restrict ourselves by considering dynamics within the range interval
$[0:\tau]$. One-step Poincaré map can be considered as the classical
counterpart of the FREO.
Owing to analogy with the usual Poincaré map, the main property of the one-
step Poincaré map can be formulated as follows: each point of a continuous
closed ray trajectory of the map (36) corresponds to a starting point of the
solution of (7) which remains stable by Lyapunov till the range $r=\tau$. The
inverse statement is not, in general, true. Hence, the one-step Poincaré map
provides a sufficient but not necessary criterion of stability.
Map (36) was studied in Makarov _et al._ (2009, 2006); Makarov and Uleysky
(2006); Gan _et al._ (2010); Gan and Lei (2011). Here we shall give its brief
description. It is reasonable to make canonical transformation of ray
variables from momentum-depth $(p-z)$ to the action–angle $(I,\vartheta)$.
This procedure provides more appropriate representation of ray equations. The
angle variable $\vartheta$ canonically conjugated to the action (10) is given
by
$\vartheta=\frac{\partial}{\partial I}\int\limits_{z_{0}}^{z}p\,dz.$ (39)
The transformed ray Hamiltonian is written as
$\bar{H}=H_{0}(I)+\tilde{V}(I,\vartheta,r).$ (40)
Ray equations in terms of the new variables:
$\begin{gathered}\frac{dI}{dr}=-\frac{\partial
H}{\partial\vartheta}=-\frac{\partial V}{\partial\vartheta},\\\
\frac{d\vartheta}{dr}=\frac{\partial H}{\partial I}=\omega(I)+\frac{\partial
V}{\partial I},\end{gathered}$ (41)
where $\omega=2\pi/D$ is spatial frequency of a ray trajectory in a waveguide.
Perturbation $\tilde{V}(I,\vartheta)$ can be expanded into a double Fourier
series
$\tilde{V}=\frac{1}{2}\sum\limits_{k,k^{\prime}=1}^{\infty}V_{k,k^{\prime}}e^{i(k\vartheta-k^{\prime}\Omega
r)}+\text{c.~{}c.},$ (42)
where $\Omega=2\pi/\tau$. Inserting (42) into (41), we obtain
$\displaystyle\frac{dI}{dr}$
$\displaystyle=-\frac{i}{2}\sum\limits_{k,k^{\prime}=1}^{\infty}kV_{k,k^{\prime}}e^{i(k\vartheta-k^{\prime}\Omega
r)}+\text{c.~{}c.},$ (43) $\displaystyle\frac{d\vartheta}{dr}$
$\displaystyle=\omega+\frac{1}{2}\sum\limits_{k,k^{\prime}=1}^{\infty}\frac{\partial
V_{k,k^{\prime}}}{\partial I}e^{i(k\vartheta-k^{\prime}\Omega
r)}+\text{c.~{}c.}$
If the condition
$k^{\prime}D(I)=k\tau$ (44)
is fulfilled, there occurs resonance. The pair of integers $k^{\prime}$ and
$k$ determines multiplicity of resonance $k:k^{\prime}$. Resonances occur at
certain values of the action, which correspond to the so-called resonant
torus. Ray dynamics in a small vicinity of a resonant torus with ray action
$I_{0}$ can be described using the so-called resonance approximation Zaslavsky
(2007), when one leaves only resonant terms in the r.h.s. of (43). It should
be mentioned that one and the same resonant torus corresponds to an infinite
number of resonances with multiplicities $(jk):(jk^{\prime})$, where $j$ is an
integer. However, resonance Fourier amplitudes $V_{k,k^{\prime}}$ rapidly
decrease with increasing $k$ and $k^{\prime}$, therefore, only few low-order
resonances influence significantly ray dynamics. Consequently, we can take
into account only some finite number of dominant resonances. For further
simplification, we make the following procedures:
1. 1.
As $\tilde{V}$ is a smooth function of $z$ in the underwater sound channel
considered, the derivative $d\tilde{V}/dI$ is small compared with $\omega$,
and the sum in the second equation of (43) can be dropped out.
2. 2.
Near resonance, spatial frequency $\omega$ can be expanded as
$\omega=\Omega+\omega_{I}^{\prime}\Delta I$, where
$\omega_{I}^{\prime}=d\omega/dI$.
Then introducing new variables
$\Delta I=I-I_{0},\quad\psi=k\vartheta-k^{\prime}\Omega r,$ (45)
and expressing $V_{k,k^{\prime}}$ as $\lvert
V_{k,k^{\prime}}\rvert\exp(i\zeta_{k,k^{\prime}})$, we can rewrite (43) as
$\begin{gathered}\frac{d(\Delta I)}{dr}=\sum\limits_{l=1}^{L}lk\lvert
V_{lk,lk^{\prime}}\rvert\sin(l\psi+\zeta_{lk,lk^{\prime}})=-\frac{\partial\tilde{H}}{\partial\psi},\\\
\frac{d\psi}{dr}=\omega_{I}\Delta I=\frac{\partial\tilde{H}}{\partial(\Delta
I)},\end{gathered}$ (46)
where $L$ is the number of dominant resonances, and
$\tilde{H}=\omega_{I}^{\prime}\frac{(\Delta
I)^{2}}{2}+\sum\limits_{l=1}^{L}lk\lvert
V_{lk,lk^{\prime}}\rvert\cos(l\psi+\zeta_{lk,lk^{\prime}}).$ (47)
If $L=1$, (47) turns into the universal Hamiltonian of nonlinear resonance
Makarov _et al._ (2009); Zaslavsky (2007), and a phase space portrait of Eqs.
(46) contains the domain of finite motion enclosed by the separatrix and
corresponding to trapping into resonance. Then maximal value $\Delta I$ on the
separatrix is determined as
$\Delta I_{\text{max}}=4\sqrt{\frac{k\lvert
V_{k,k^{\prime}}\rvert}{\omega_{I}^{\prime}}}$ (48)
The terms with $l>1$ may deform the pendulum-like phase space portrait and,
moreover, result in the presence of additional separatrices inside the domain
of finite motion. The latter phenomenon can occur when the perturbation
oscillates with depth Makarov _et al._ (2006).
Transition to global chaos in the one-step Poincaré map happens when
neighbouring dominant resonances overlap. The criterion of overlapping is the
well-known Chirikov criterion
$\frac{\Delta I_{\text{max}}}{\delta I}\geqslant 1.$ (49)
Here $\delta I$ is the distance between neighbouring dominant resonances in
the action space. Its variability with $\tau$ for $\tau>D$ is described by
equation
$\delta I=\frac{2\pi}{\omega_{I}^{\prime}\tau},$ (50)
that is, increasing of $\tau$ enhances resonance overlapping. Differences in
phase space patterns corresponding to different realizations of the
perturbation are associated with phase and amplitude fluctuations of Fourier-
amplitudes $V_{k,k^{\prime}}$. However, the contribution of these fluctuations
is limited, therefore, the ratio of the phase space volumes corresponding to
regular and chaotic motion is mainly controlled by $\tau$ and weakly varies
from one realization to another (see below for an illustration). This property
allows one to consider the one-step Poincaré map as an useful tool for
studying randomly-driven dynamical systems of various physical origins.
Resonance approximation presented above fails under violation of the
nondegeneracy condition $\omega_{I}^{\prime}\neq 0$. In this case one should
use more sophisticated approaches, one of them is presented in Rypina _et
al._ (2007). We shall address this issue in the end of this Section.
Figure 4: Ray phase space portraits constructed via the one-step Poincaré map
(36). Each column corresponds a single realization of the sound-speed
perturbation. Value of $\tau$ is indicated in the left lower corner.
Fig. 4 illustrates phase space portraits calculated using the one-step
Poincaré map with three different realizations of the sound-speed
perturbation. The jump of the derivative $dU/dz$ at $z=z_{0}$ leads to fast
growth of numerical error in ray calculations. Therefore, we replaced the
expression (8) for $U(z)$ by the smoothed function
$U(z)=U_{1}(z)+\frac{1}{2}\left[1+\tanh\frac{z-z_{0}}{\Delta}\right]U_{2}(z),$
(51)
where $\Delta=1$ m. Each of the phase space portraits represents a mixed phase
space structure consisted of regular and chaotic domains. Phase portraits with
the same $\tau$ mainly differ only in angular locations of regular islands,
whereas their overall structure is very similar. The main regular domain is
placed near the point $z=z_{0}$, $p=0$ and corresponds to flat rays
intersecting the horizontal plane with the smallest angles. This circumstance
deserves especial attention because stability of flat rays is not typical for
sound propagation in the deep ocean. Numerous experiments on long-range sound
propagation in the North-Eastern Pacific Ocean (see, for instance, Spiesberger
and Tappert (1996); Worcester _et al._ (1994, 1999); Wage _et al._ (2005);
Grigorieva _et al._ (2009)) indicate on strong irregularity of flat near-
axial rays, associated with ray chaos Simmen _et al._ (1997); Beron-Vera _et
al._ (2003). The “deterministic” mechanism of near-axial chaos is ray
scattering on vertical resonances caused by small-scale depth oscillations of
the sound-speed perturbation Hegewisch _et al._ (2005); Kon’kov _et al._
(2007); Makarov _et al._ (2008). These oscillations are contributed from
high-number modes of an internal-wave field. In the Sea of Japan, the effect
of the high-number modes is weak, therefore, the sound-speed perturbation can
be fairly described by equation (17), where depth dependence is given by a
smooth function $Y_{1}(z)$. Weakness of high-number internal-wave modes is a
peculiar feature of the Sea of Japan. It is caused by the specific form of the
buoyancy frequency profile Spindel _et al._ (2003). In particular, the
waveguide for internal waves, determined by the buoyancy frequency profile, is
too narrow. Consequently, it doesn’t efficiently focus high-number modes of
low-frequency internal waves. As low-frequency internal waves give the
dominant contribution into a total internal-wave field, depth oscillations of
the perturbation are suppressed.
Resonance overlapping is enhancing as $\tau$ grows, and stable islands
eventually submerge into the chaotic sea. However, there is a small region of
stability that survives for distances of hundreds kilometers, transforming
into a chain of islands around the region near $z=z_{0}$, $p=0$. This chain
corresponds to the smooth minimum of the function $D(p_{0})$ depicted in Fig.
2. In the absence of inhomogeneity, this minimum gives rise to the so-called
shearless torus in phase space, producing a weakly-divergent beam. It is
recognized that shearless tori can possess extraordinary persistence to chaos
Rypina _et al._ (2007); del Castillo-Negrete _et al._ (1996); Budyansky _et
al._ (2009); Uleysky _et al._ (2010a, b), therefore, a weakly-divergent beam
can survive in the presence of a sound-speed perturbation. Hence, formation of
a weakly-divergent beam can be considered as a possible mechanism responsible
for unusual stability of near-axial rays, observed in experiments Bezotvetnykh
_et al._ (2009); Spindel _et al._ (2003).
## VI Spectral statistics of FREO for the underwater sound channel in the Sea
of Japan
### VI.1 General remarks
Figure 5: Fraction of the phase space volume corresponding to regular motion
vs distance $\tau$ for various frequencies.
This section is devoted to numerical modeling of the FREO in the Sea of Japan.
Each realization of the FREO was represented as a matrix in the basis of
normal modes being solutions of the Sturm-Liouville problem (12) with the
boundary conditions (22). Only purely-water modes which propagate without
reaching the bottom were taken into account. They were selected using the
criterion $E_{m}<U(z=h)$, where $E_{m}$ is the $m$-th eigenvalue of the Sturm-
Liouville problem (12). This criterion ensues from the WKB approximation for
normal modes Virovlyansky and Zaslavsky (1999). Number of trapped modes $M$
depends on sound frequency. It is equal to 72 for $f=100$ Hz, 179 for $f=250$
Hz, 259 for $f=360$ Hz, and 361 for $f=500$ Hz. Statistical ensembles of the
FREOs, corresponding to the frequencies of 250, 360 and 500 Hz, were
calculated with 100 realizations of the perturbation. The ensemble
corresponding to 100 Hz was calculated with 500 realizations. For each
realization, we constructed a family of the FREOs $\hat{G}(\tau)$, where
$\tau=5,10,15,\dots,350$ km.
Figure 6: The Brody parameter $\beta$ vs distance $\tau$.
### VI.2 Eigenvalue statistics
We calculated the ensemble-averaged level spacing distribution $\rho(s,\tau)$
using the formula (32) and fitted it, for each value of $\tau$, by means of
the Berry-Robnik (30) and Brody (31) distributions. Thus, we obtained
dependencies of the regular phase space volume $v_{\mathrm{r}}$ and Brody
parameter $\beta$ on distance $\tau$. As is shown in Fig. 5, $v_{\mathrm{r}}$
rapidly decreases in the first 100–150 km. Then, it becomes almost constant.
This may indicate on the influence of the long-living stable islands in the
vicinity of the weakly-divergent beam. Notably, the curves corresponding to
250, 360 and 500 Hz are very close to each other, whereas the curve
corresponding to 100 Hz lies above them and undergoes strong fluctuations
which persist even with increasing number of realizations. It should be
mentioned that the above estimates of $v_{\mathrm{r}}$ have limited accuracy
because the assumptions underlying the formula (30) are satisfied only
approximately. Therefore, the results obtained using the Berry-Robnik
distribution are rather qualitative than quantitative, especially for low
sound frequencies.
Figure 7: Relano parameter $\alpha$ vs distance $\tau$.
Approximation of level spacing statistics by means of the Brody distribution
leads to qualitatively similar results. They are presented in Fig. 6. For
frequencies 250, 360 and 500 Hz, the Brody parameter $\beta$ gradually grows
from 0 to 1, reflecting transformation from Poissonian to Wigner-like
statistics. In the case of $f=100$ Hz the growth is remarkably slower and
affected by strong fluctuations. The fluctuations of $v_{\mathrm{r}}$ and
$\beta$ may be induced by regular-to-chaotic tunneling and dynamical
localization, the phenomena whose influence on spectral statistics is still
not well understood.
In addition, we used the method of spectral analysis, developed by A. Relano
with coworkers in Relaño _et al._ (2002). In this method, one firstly
constructs a series
$\delta_{n}=\sum\limits_{i=1}^{n}(s_{i}-\left<s\right>),$ (52)
where $n=1,2,\dots,N-1$, $N$ is the total number of eigenvalues. Then, making
discrete Fourier transform
$\bar{\delta}_{k}=\frac{1}{\sqrt{N}}\sum\limits_{n}\delta_{n}\exp\left(\frac{2\pi
ikn}{N}\right),$ (53)
one finds the power spectrum
$S(k)=\lvert\bar{\delta}_{k}\rvert^{2}.$ (54)
Generally, ensemble-averaged spectrum obeys a power low
$\left<S(k)\right>\sim k^{-\alpha}.$ (55)
Relano with coworkers found that regular dynamics corresponds to $\alpha=2$,
and global chaos results in $\alpha=1$. In the mixed regime, $\alpha$ takes on
an intermediate value between 1 and 2 Relaño (2008). Fig. 7 demonstrates that
$\alpha$ decreases with increasing $\tau$ for the frequencies of 360 and 500
Hz, indicating gradual transition to chaos. However, $\alpha$ varies
relatively slowly and remains near the middle value 1.5 for all distances
considered, despite of the marked changes in the classical phase space
portrait (see Fig. 4). Analogous dependencies for the frequencies of 100 and
250 Hz exhibit strong fluctuations and, therefore, are not presented in the
figure. This implies that the method developed in Relaño _et al._ (2002);
Relaño (2008) provides good agreement only for relatively short wavelengths.
### VI.3 Eigenfunction analysis
Analysis of eigenfunctions possesses some advantages as compared with analysis
of eigenvalues. The main advantage is the possibility to associate each
eigenfunction with some set of normal modes and, whereby, associate it with a
certain geometry of propagation. We can properly classify eigenfunctions,
taking into account the interplay with normal modes. Such classification can
be used for finding wavepacket configurations whose dynamics is expected to be
less or more regular. Proper classification can be obtained using the
parameter $\mu$ Smirnov _et al._ (2005). It is defined as
$\mu=\sum\limits_{m=1}^{M}\lvert c_{m}\rvert^{2}m.$ (56)
In an unperturbed waveguide, only one normal mode contributes to each
eigenfunction, and $\mu$ coincides with the number of this mode. Taking into
account the quantization rule (11), we obtain the formula
$\left<I\right>=\frac{\mu}{k_{0}}+\frac{1}{2k_{0}}$ (57)
that gives mean action corresponding to an eigenfunction. According to (57),
the parameter $\mu$ determines phase space location of the eigenfunction and
can serve as its identificator.
Figure 8: Distribution of eigenfunctions in the $\mu$–$\nu$ plane, where the
parameter $\mu$ is given by (56), and $\nu$ is a number of principal
components (34). Distance values: (a) $\tau=10$ km, (b) $\tau=35$ km, (c)
$\tau=100$ km, (d) $\tau=350$ km. The sound frequency is 500 Hz.
Figures 8 and 9 illustrate eigenfunction distributions in the $\mu$–$\nu$
plane, where $\nu$ is number of principal components (34), for the frequencies
of 500 and 100 Hz, respectively.
Let us firstly consider Fig. 8 corresponding to the frequency of 500 Hz. Mode
coupling is relatively weak for small values of $\tau$, therefore, the
distribution is mainly concentrated near $\nu=1$. We want to focus attention
on the dense spots elongated in the $\nu$-direction. They correspond to
eigenfunctions reflecting mode-medium resonance Virovlyansky and Zaslavsky
(1999) being the wave counterpart of ray-medium resonance (44). Indeed, an
isolated ray-medium resonance produces oscillations of ray action inside the
interval $I_{0}-\Delta I_{\text{max}}\leqslant I\leqslant I_{0}+\Delta
I_{\text{max}}$, where $I_{0}$ is a resonance action, $\Delta I_{\text{max}}$
is resonance width in the action space. According to the principle of ray-mode
duality, these oscillations of action correspond to coherent transitions
between the normal modes whose numbers satisfy the inequality
$m_{0}-\Delta m\leqslant m\leqslant m_{0}+\Delta m,$ (58)
where $m_{0}=k_{0}I_{0}+1/2$, $\Delta m=k_{0}\Delta I_{\text{max}}$.
Resonance-induced modal transtions give rise to eigenfunctions with $\mu\simeq
m_{0}$ and $\nu$ varying from 1 to $\Delta m$. As long as resonance values of
the action, being determined by $\tau$, are the same for all realizations of
the perturbation, these eigenfunctions form vertically-elongated
concentrations of points in the $\mu$–$\nu$ plane. Location of mode-medium
resonances along the $\mu$-axis can be found using the formula
$k^{\prime}D(I=\left<I\right>)=k\tau,$ (59)
where $\left<I\right>$ is linked to $\mu$ by (57). Concentrations induced by
mode-medium resonance disappear with increasing $\tau$ due to overlapping of
mode-medium resonances and delocalization Berman and Kolovskiĭ (1992).
Delocalization leads to abrupt growth of number of principal components. It
eventually subjects all eigenfunctions in the interval between $\mu\simeq 100$
and $\mu\simeq 300$, resulting in the “boomerang” pattern in the $\mu$–$\nu$
plane, as illustrated in Fig. 8(d). Left and right ends of the “boomerang” are
formed by weakly scattered eigenfunctions. The left end corresponds to the
almost horizontal near-axial propagation, that is, its regularity can be
associated with long-living stable islands in the one-step Poincaré map.
Figure 9: The same as in Fig. 8, but for the frequency of 100 Hz. (a)
$\tau=10$ km, (b) $\tau=35$ km, (c) $\tau=100$ km, (d) $\tau=350$ km.
Eigenfunction distribution in the $\mu$–$\nu$ plane for the frequency of 100
Hz possesses a more complicated structure. It is exceptionally regular for
$\tau=10$ km and $\tau=35$ km, as shown in Figs. 9(a) and 9(b). Mode-medium
resonances reveal themselves as “stalagmites”. Each stalagmite is drawn by a
family of distinct weakly biased lines. Contours of the most pronounced
stalagmite for $\tau=35$ km are somewhat disordered and smeared. Some traces
of stalagmite-like patterns survive even for distances of hundreds kilometers,
despite of global overlapping of ray-medium resonances. Persistence of
stalagmites for large $\tau$ indicates on the presence of eigenstates
localized near periodic orbits of the one-step Poincaré map. As long as these
eigenstates correspond to nonspeading wavepackets, we can associate such
localization with suppresion of ray chaos and recovery of regular refraction
in the vicinities of the periodic orbits. An analogous phenomenon had been
reported in Kon’kov _et al._ (2007) for range-periodic waveguides.
Besides of stalagmites, Figs. 9(a) and 9(b) illustrate the pattens in the form
of “bridges”. For instance, a pronounced “bridge” in the left part of Fig.
9(a) connects the points $\mu=5$, $\nu=1$ and $\mu=30$, $\nu=1$. The
eigenfunctions producing the “bridges” are consisted of normal modes
satisfying the condition
$k_{0}(E_{m}-E_{n})=\frac{2\pi l}{\tau},\quad m>n.$ (60)
The aforementioned “bridge” in the Fig. 9(a) satisfies (60) with $m=30$,
$n=5$, and $l=9$. Condition (60) is equivalent to quantum resonance between
two energy levels. In the ray limit
$k_{0}(E_{m}-E_{n})\to(m-n)\frac{dE}{dI}\equiv\frac{2\pi(m-n)}{D},$ (61)
and condition (60) reduces to (44). Each realization of the FREO can yield
eigenfunctions manifesting resonance (60). If resonance (60) corresponding to
some numbers ($l$, $m$, $n$) is localized, i. e. the modes $m$ and $n$ are not
affected by other resonances, then the respective eigenfunction is a
superposition of normal modes $m$ and $n$,
$\Phi_{\text{res}}(z)\simeq c_{m}\phi_{m}+c_{n}\phi_{n},\quad\lvert
c_{m}\rvert^{2}+\lvert c_{n}\rvert^{2}\simeq 1.$ (62)
As this takes place, the ratio of amplitudes $\lvert c_{m}\rvert/\lvert
c_{n}\rvert$ is determined by the phase of the resonance harmonics of the
perturbation. For the perturbation (17), amplitude of the $l$-th resonance
harmonics reads
$B_{l}=\frac{1}{2\pi}\int\limits_{0}^{2\pi/\tau}b_{1}(r)\exp\left(-i\frac{2\pi
lr}{\tau}\right)dr.$ (63)
Phase of the resonance harmonics is a random quantity with uniform probability
density in the range $[0:2\pi]$. Each value of the phase uniquely determines
the values of $\mu$ and $\nu$ via the formulae
$\mu=\lvert c_{m}\rvert^{2}m+\lvert c_{n}\rvert^{2}n,$ (64) $\nu=\left(\lvert
c_{m}\rvert^{4}+\lvert c_{n}\rvert^{4}\right)^{-1}.$ (65)
It turnes out that quantities $\mu$ and $\nu$ are correlated for each
eigenfunction corresponding to localized resonance (60). This results in
formation of “bridges” in the $\mu$–$\nu$ plane. Localization becomes violated
with increasing $\tau$, therefore, correlation between $\nu$ and $\mu$ ceases,
and ordered “bridges” transform into disordered clouds of points.
Figure 10: Mean number of principal components as function of distance.
The above analysis shows that there are qualitative differences in sound
scattering for different frequencies. To estimate these differences
quantitatively, we need a suitable parameter characterizing scattering
strength. For example, chaos-induced phase space delocalization can be
measured by the ensemble-averaged number of principal components, divided by
the total number of trapped modes. As it follows from Fig. 10, the rate of
delocalization increases with increasing sound frequency. This implies that
chaotic diffusion associated with ray chaos degrades with increasing sound
wavelength. It can be thought of as a manifestation of dynamical localization
Stöckmann (2007), an analogue of Anderson localization, when destructive
interference supresses wavepacket spreading.
Figure 11: Fraction of strongly-localized eigenfunctions as function of
distance. The criterion of strong localization is the inequality $\nu\leqslant
2$.
In practice, it is useful to know fraction of the eigenfunction ensemble
corresponding to regular propagation. This quantity can be regarded as an
analogue of the parameter $v_{\mathrm{r}}$ in the Berry-Robnik distribution.
It can be estimated by means of the cumulative distribution function
$F(\nu)=\int\limits_{1}^{\nu}\rho(\nu^{\prime})\,d\nu^{\prime},$ (66)
where $\rho(\nu^{\prime})$ is the probability density function of $\nu$. We
can conditionally distinguish two regimes of localization: strong localization
and moderate localization. Strong localization implies that eigenfunction of
FREO is close to one of normal modes of the unperturbed waveguide. To select
strongly-localized eigenfunctions, we can use the inequality $\nu\leqslant 2$.
Dependence of $F(2)$ on the horizontal distance $\tau$ is depicted in Fig. 11.
Evidently, fraction of strongly-localized eigenfunctions is much larger for
$f=100$ Hz than for higher frequencies. The curves corresponding to the higher
frequencies are close to each other. They drop down in the range of small
$\tau$, then decreasing of $F(2)$ becomes very slow. Slow decreasing can be
linked to the presence of the long-living islands of stability in the
neibourhood of the weakly-divergent beam.
Figure 12: Fraction of moderately localized eigenfunctions vs distance. The
criterion of moderate localization is the inequality $\nu\leqslant 0.1M$,
where $M$ is the number of trapped modes.
In the regime of moderate localization, mode coupling can be sufficiently
strong, but an eigenfunction occupies relatively small phase space volume,
that is, number of principal components is limited. We use the inequality
$\nu\leqslant 0.1M$ as the criterion of moderate localization. As is
demonstrated in Fig. 12, almost all eigenfunctions corresponding to the
frequency of 100 Hz are moderately-localized. This is not the case of higher
frequencies, when fraction of moderately-localized eigenfunctions
significantly decreases with $\tau$. We can conclude that decreasing of sound
frequency can result in remarkable suppression of wavepacket diffusion in
phase space.
## VII Conclusion
Spectral analysis of the finite-range evolution operator prompts a way to
explore wave dynamics in a randomly-inhomogeneous waveguide by means of the
quasideterministic approach, involving resonances, periodic orbits, phase
space portraits, e.t.c. This approach was originally proposed in Virovlyansky
_et al._ (2012); Makarov _et al._ (2010). The present work is devoted to its
further development. We demonstrate various methods of spectral analysis. For
instance, fitting of level spacing statistics by means of the Berry-Robnik
distribution yields approximate estimate for fraction of regularly propagating
normal modes of a waveguide. Also we use the method developed by A. Relano
with coworkers, and consider distribution of eigenfunctions in the $\mu$–$\nu$
space. In our opinion, the latter approach is the most robust, albeit its
analytical description is lacking. The most important advantage of the
eigenfunction analysis is the possibility to study separately scattering of
different modes of a waveguide and find out the modes corresponding to regular
propagation. A detailed view of the eigenfunction distribution in the
$\mu$–$\nu$ space suggests that the mechanism of the chaos onset with
increasing distance can be associated with overlapping and delocalization of
mode-medium resonances. The approach based on the statistical analysis of
level spacings by means of the Berry-Robnik formula gives qualitative
description of the transition but doesn’t ensure quantitative agreement. The
Relano method and the method based on the Brody distribution also give the
qualitative description but cannot make any quantitative estimates due to
their semiempirical nature.
We consider the underwater sound channel in the Sea of Japan as an example.
Our analysis shows that almost horizontal near-axial sound propagation
preserves regularity over distances of hundreds kilometers. There are two
factors responsible for near-axial stability. The first factor is the peculiar
hydrological structure in the region considered, resulting in the absence of
vertical oscillations of the sound-speed perturbation. This circumstance leads
to a qualitatively different scenario of ray chaos, as compared with the well-
known acoustic experiments in the North-Eastern Pacific Ocean. The second
factor is the formation of the weakly-divergent beam supported by long-living
stable islands in classical phase space. Weakly-divergent beam occurs in the
vicinity of the shearless torus. Non-dispersive motion of quantum wavepackets
near shearless tori was earlier observed in Kudo and Monteiro (2008). Also, we
should emphasize that sound propagation with the frequency of 100 Hz reveals
exceptionally high degree of stability, indicating on different physics of
scattering for low sound frequencies. This phenomenon can be associated with
strong dynamical localization.
We suppose that applicability of the approach presented in this paper is not
limited to the problems of wave propagation in random media. An analogue of
the FREO can be readily used for studying of noise-induced quantum transport
and related phenomena. We suggest that combination of deterministic and
statistical approaches should provide an insightful view on details of
dynamics, especially in the presence of intermittency or synchronization.
This work was supported by the Russian Foundation of Basic Research (projects
09-05-98608 and 09-02-01258-a), the Federal Program “World Ocean” and the
“Dynasty” Foundation. Authors are grateful to A.R. Kolovsky, S. Tomsovic, O.A.
Godin, K.V. Koshel and V.V. Novotryasov for helpful comments concerning the
subject of the research.
## References
* Shockley _et al._ (1982) R. C. Shockley, J. Northrop, P. G. Hansen, and C. Hartdegen, J. Acoust. Soc. Am. 71, 51 (1982).
* Munk _et al._ (1988) W. H. Munk, W. C. O’Reilly, and J. L. Reid, J. Phys. Oceanography 18, 1876 (1988).
* Makarov _et al._ (2009) D. Makarov, S. Prants, A. Virovlyansky, and G. Zaslavsky, _Ray and wave chaos in ocean acoustics: chaos in waveguides_ , Series on complexity, nonlinearity and chaos (World Scientific, Singapore, 2009) p. 388.
* Smith _et al._ (1992) K. B. Smith, M. G. Brown, and F. D. Tappert, J. Acoust. Soc. Am. 91, 1939 (1992).
* Smirnov _et al._ (2001) I. P. Smirnov, A. L. Virovlyansky, and G. M. Zaslavsky, Phys. Rev. E 64, 036221 (2001).
* Brown _et al._ (2003) M. G. Brown, J. A. Colosi, S. Tomsovic, A. L. Virovlyansky, M. A. Wolfson, and G. M. Zaslavsky, J. Acoust. Soc. Am. 113, 2533 (2003).
* Makarov _et al._ (2004) D. V. Makarov, M. Y. Uleysky, and S. V. Prants, Chaos 14, 79 (2004).
* Beron-Vera and Brown (2004) F. J. Beron-Vera and M. G. Brown, J. Acoust. Soc. Am. 115, 1068 (2004).
* Virovlyansky and Zaslavsky (1999) A. L. Virovlyansky and G. M. Zaslavsky, Phys. Rev. E 59, 1656 (1999).
* Smirnov _et al._ (2004) I. P. Smirnov, A. L. Virovlyansky, and G. M. Zaslavsky, Chaos 14, 317 (2004).
* Smirnov _et al._ (2005) I. P. Smirnov, A. L. Virovlyansky, M. Edelman, and G. M. Zaslavsky, Phys. Rev. E 72, 026206 (2005).
* Kon’kov _et al._ (2007) L. E. Kon’kov, D. V. Makarov, E. V. Sosedko, and M. Y. Uleysky, Phys. Rev. E 76, 056212 (2007).
* Makarov _et al._ (2008) D. V. Makarov, L. E. Kon’kov, and M. Y. Uleysky, Acoust. Phys. 54, 382 (2008).
* Virovlyansky _et al._ (2012) A. L. Virovlyansky, D. V. Makarov, and S. V. Prants, Phys. Usp. 55 (2012).
* Hegewisch _et al._ (2005) K. C. Hegewisch, N. R. Cerruti, and S. Tomsovic, J. Acoust. Soc. Am. 117, 1582 (2005).
* Munk and Wunsch (1979) W. Munk and C. Wunsch, Deep Sea Res. Part A. 26, 123 (1979).
* Tappert and Tang (1996) F. D. Tappert and X. Tang, J. Acoust. Soc. Am. 99, 185 (1996).
* Zaslavsky (2007) G. M. Zaslavsky, _The physics of chaos in Hamiltonian systems_ (Imperial College Press, London, 2007) p. 328.
* Dozier and Tappert (1978a) L. B. Dozier and F. D. Tappert, J. Acoust. Soc. Am. 63, 353 (1978a).
* Colosi and Morozov (2009) J. A. Colosi and A. K. Morozov, J. Acoust. Soc. Am. 126, 1026 (2009).
* Wolfson and Tomsovic (2001) M. A. Wolfson and S. Tomsovic, J. Acoust. Soc. Am. 109, 2693 (2001).
* Makarov _et al._ (2006) D. V. Makarov, M. Y. Uleysky, M. V. Budyansky, and S. V. Prants, Phys. Rev. E 73, 066210 (2006).
* Makarov _et al._ (2010) D. V. Makarov, L. E. Kon’kov, and M. Y. Uleysky, ArXiv e-prints (2010), arXiv:1008.3037 [nlin.CD] .
* Hegewisch and Tomsovic (2012) K. C. Hegewisch and S. Tomsovic, Europhys. Lett. 97, 34002 (2012).
* Stöckmann (2007) H. J. Stöckmann, _Quantum Chaos: An Introduction_ (Cambridge University Press, Cambridge, 2007) p. 384.
* Kolovsky (1997) A. R. Kolovsky, Phys. Rev. E 56, 2261 (1997).
* Bezotvetnykh _et al._ (2009) V. Bezotvetnykh, A. Burenin, Y. Morgunov, and Y. Polovinka, Acoust. Phys. 55, 376 (2009).
* Spindel _et al._ (2003) R. C. Spindel, J. Na, P. H. Dahl, S. Oh, C. Eggen, Y. G. Kim, V. A. Akulichev, and Y. N. Morgunov, IEEE J. Ocean. Engin. 28, 297 (2003).
* (29) “Oceanography and marine environment of the Far Eastern Region of Russia (proj. leader Rostov I. D.),” http://www.pacificinfo.ru/en.
* Brekhovskikh _et al._ (1990) L. M. Brekhovskikh, V. V. Goncharov, S. A. Dremuchev, V. M. Kurtepov, V. G. Selivanov, and Y. A. Chepurin, Sov. Phys. Acoust. 36, 461 (1990).
* Smirnov _et al._ (1999) I. Smirnov, J. Caruthers, and A. Khil’ko, Radiophys. Quantum Electron. 42, 864 (1999).
* Morozov and Colosi (2005) A. K. Morozov and J. A. Colosi, J. Acoust. Soc. Am. 117, 1611 (2005).
* Petukhov (2009) Y. Petukhov, Acoust. Phys. 55, 785 (2009).
* Colosi and Brown (1998) J. A. Colosi and M. G. Brown, J. Acoust. Soc. Am. 103, 2232 (1998).
* LeBlanc and Middleton (1980) L. R. LeBlanc and F. H. Middleton, J. Acoust. Soc. Am. 67, 2055 (1980).
* Dozier and Tappert (1978b) L. B. Dozier and F. D. Tappert, J. Acoust. Soc. Am. 64, 533 (1978b).
* Mallick and Marcq (2002) K. Mallick and P. Marcq, Phys. Rev. E 66, 041113 (2002).
* Landau and Lifshitz (1977) L. D. Landau and E. M. Lifshitz, _Course of theoretical physics._ , Vol. 3: Quantum mechanics. Nonrelativistic theory (Pergamon Press, Oxford, 1977).
* Berry and Robnik (1984) M. V. Berry and M. Robnik, J. Phys. A: Math. Gen. 17, 2413 (1984).
* Bäcker _et al._ (2005) A. Bäcker, R. Ketzmerick, and A. G. Monastra, Phys. Rev. Lett. 94, 054102 (2005).
* Prosen and Robnik (1994) T. Prosen and M. Robnik, J. Phys. A: Math. Gen. 27, 8059 (1994).
* Varga and Pipek (2003) I. Varga and J. Pipek, Phys. Rev. E 68, 026202 (2003).
* Makarov and Uleysky (2006) D. Makarov and M. Uleysky, J. Phys. A: Math. Gen. 39, 489 (2006).
* Gan _et al._ (2010) C. Gan, Q. Wang, and M. Perc, J. Physics A: Math. Theor. 43, 125102 (2010).
* Gan and Lei (2011) C. Gan and H. Lei, J. Sound Vibr. 330, 2174 (2011).
* Rypina _et al._ (2007) I. I. Rypina, M. G. Brown, F. J. Beron-Vera, H. Koçak, M. J. Olascoaga, and I. A. Udovydchenkov, Phys. Rev. Lett. 98, 104102 (2007).
* Spiesberger and Tappert (1996) J. L. Spiesberger and F. D. Tappert, J. Acoust. Soc. Am. 99, 173 (1996).
* Worcester _et al._ (1994) P. F. Worcester, B. D. Cornuelle, J. A. Hildebrand, J. William S. Hodgkiss, T. F. Duda, J. Boyd, B. M. Howe, J. A. Mercer, and R. C. Spindel, J. Acoust. Soc. Am. 95, 3118 (1994).
* Worcester _et al._ (1999) P. F. Worcester, B. D. Cornuelle, M. A. Dzieciuch, W. H. Munk, B. M. Howe, J. A. Mercer, R. C. Spindel, J. A. Colosi, K. Metzger, T. G. Birdsall, and A. B. Baggeroer, J. Acoust. Soc. Am. 105, 3185 (1999).
* Wage _et al._ (2005) K. E. Wage, M. A. Dzieciuch, P. F. Worcester, B. M. Howe, and J. A. Mercer, J. Acoust. Soc. Am. 117, 1565 (2005).
* Grigorieva _et al._ (2009) N. S. Grigorieva, G. M. Fridman, J. A. Mercer, R. K. Andrew, M. A. Wolfson, B. M. Howe, and J. A. Colosi, J. Acoust. Soc. Am. 125, 1919 (2009).
* Simmen _et al._ (1997) J. Simmen, S. M. Flatte, and G.-Y. Wang, J. Acoust. Soc. Am. 102, 239 (1997).
* Beron-Vera _et al._ (2003) F. J. Beron-Vera, M. G. Brown, J. A. Colosi, S. Tomsovic, A. L. Virovlyansky, M. A. Wolfson, and G. M. Zaslavsky, J. Acoust. Soc. Am. 114, 1226 (2003).
* del Castillo-Negrete _et al._ (1996) D. del Castillo-Negrete, J. M. Greene, and P. J. Morrison, Physica D 91, 1 (1996).
* Budyansky _et al._ (2009) M. V. Budyansky, M. Y. Uleysky, and S. V. Prants, Phys. Rev. E 79, 056215 (2009).
* Uleysky _et al._ (2010a) M. Y. Uleysky, M. V. Budyansky, and S. V. Prants, Phys. Rev. E 81, 017202 (2010a).
* Uleysky _et al._ (2010b) M. Uleysky, M. Budyansky, and S. Prants, J. of Exp. and Theor. Phys. 111, 1039 (2010b).
* Relaño _et al._ (2002) A. Relaño, J. M. G. Gómez, R. A. Molina, J. Retamosa, and E. Faleiro, Phys. Rev. Lett. 89, 244102 (2002).
* Relaño (2008) A. Relaño, Phys. Rev. Lett. 100, 224101 (2008).
* Berman and Kolovskiĭ (1992) G. P. Berman and A. R. Kolovskiĭ, Sov. Phys. Usp. 35, 303 (1992).
* Kudo and Monteiro (2008) K. Kudo and T. S. Monteiro, Phys. Rev. E 77, 055203 (2008).
|
arxiv-papers
| 2012-03-23T03:21:37 |
2024-09-04T02:49:28.970928
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. V. Makarov, L. E. Kon'kov, M. Yu. Uleysky, P. S. Petrov",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1203.5167"
}
|
1203.5281
|
# Impact of New $\beta$-decay Half-lives on $r$-process Nucleosynthesis
Nobuya Nishimura${}^{1,\;2\;}$ nobuya.nishimura@unibas.ch Toshitaka
Kajino${}^{3,\;4\;}$ Grant J. Mathews5
Shunji Nishimura6 Toshio Suzuki7 1 Department of Physics, University of Basel,
4056 Basel, Switzerland 2 GSI, Helmholtzzentrum für Schwerionenforschung
GmbH, 64291 Darmstadt, Germany 3 Division of Theoretical Astronomy, NAOJ,
181-8588 Mitaka, Japan 4 Department of Astronomy, The University of Tokyo,
113-033 Tokyo, Japan 5 Center for Astrophysics, University of Notre Dame,
Notre Dame, IN 46556, USA 6 RIKEN Nishina Center, Wako, Saitama 351-0198,
Japan 7 Department of Physics, Nihon University, 156-8550 Tokyo, Japan
###### Abstract
We investigate the effects of newly measured $\beta$-decay half-lives on
$r$-process nucleosynthesis. These new rates were determined by recent
experiments at the radioactive isotope beam factory facility in the RIKEN
Nishina Center. We adopt an $r$-process nucleosynthesis environment based on a
magnetohydrodynamic supernova explosion model that includes strong magnetic
fields and rapid rotation of the progenitor. A number of the new $\beta$-decay
rates are for nuclei on or near the $r$-process path, and hence, they affect
the nucleosynthesis yields and timescale of the $r$-process. The main effect
of the newly measured $\beta$-decay half-lives is an enhancement in the
calculated abundance of isotopes with mass number $A=110$ – $120$ relative to
calculated abundances based upon $\beta$-decay rates estimated with the
finite-range droplet mass model. This effect slightly alleviates, but does not
fully explain, the tendency of $r$-process models to underproduce isotopes
with $A=110$ – $120$ compared to the solar-system $r$-process abundances.
PACS numbers
23.40.-s, 25.30.-c, 26.30.Hj, 26.50.+x, 97.60.Bw
††preprint: APS/123-QED
Introduction. Rapid neutron-capture ($r$)-process nucleosynthesis is
responsible for the origin of approximately half of the elements heavier than
iron and is the only means of producing the naturally occurring radioactive
heavy actinide elements such as Th and U. In spite of more than a half century
of study and observational progress (e.g., Sneden:etal:2008, ), however, the
astrophysical sites for $r$-process nucleosynthesis have still not been
unambiguously identified (for recent reviews see, e.g., Arnould:etal:2007 ;
Thielemann:etal:2011 ). Although many candidate sites have been proposed and
supernovae appear to be well suited as the $r$-process site
(MathewsCowan:1990, ), up till now there has been no consensus as to the
correct astrophysical model.
Notwithstanding the difficulties in finding a suitable astronomical
environment, the physical conditions for the $r$-process are well constrained
Burbidge:etal:1957 . It is evident that the $r$-process occurs via a sequence
of near equilibrium rapid neutron captures and photo-neutron emission
reactions far on the neutron-rich side of stability. This equilibrium is
established with a maximum abundance strongly peaked on one or two isotopes
far from stability. The relative abundance of $r$-process elements is then
determined by the relative $\beta$-decay rates along this $r$-process path.,
i.e., slower $\beta$-decay lifetimes result in higher abundances. At least
part of the reason for the difficulty in finding the astrophysical site for
the $r$-process stems from the fact that it lies so far from the region of
stable isotopes where there is little experimental data on $\beta$-decay
rates.
In this context, it is of particular interest that $\beta$-decay half-lives of
38 neutron-rich isotopes including 100Kr, 103-105Sr, 106-108Y, 108-110Zr,
111,112Nb, 112-115Mo, and 116,117Tc have been measured (Nishimura:etal:2011, )
at the recently commissioned radioactive isotope beam factory (RIBF) facility
at the RIKEN Nishina Center. A secondary beam, containing the neutron-rich
nuclei of interest, was produced by inflight fission of a $345$ MeV/nucleon
238U beam in a Be target. The nuclei in the secondary beam were identified on
an event-by-event basis. Their atomic numbers were determined from the energy
loss in an ionization chamber, and the charge to mass ratio was determined by
combining the projectile time-of-flight and magnetic rigidity measurements.
The nuclei were then implanted in a silicon-strip detector where the
$\beta$-decay lifetimes could be measured. Most of the measured lifetimes are
an improvement on existing measurements and a number of them
(Nishimura:etal:2011, ) were measured for the first time.
Many of these isotopes are near or directly on the $r$-process path. As such,
they are of particular interest as they determine the $\beta$-flow toward the
important $r$-process peak at $A=130$. Thus, they regulate the ability of
models for the $r$-process to form heavier elements (Otsuki:etal:2003, ). It
is of particular interest, therefore, to examine the impact of these new rates
on $r$-process models.
The magnetohydrodynamic (MHD) supernova explosion model is of particular
interest in the present study as it provides a good fit to the observed
$r$-process abundances and avoids some of the problems (Otsuki:etal:2003, ,
and references therein) associated with other paradigms such as the neutrino
driven supernova wind scenario (Woosley:etal:1994, ). Robust $r$-process
nucleosynthesis appears to be a natural consequence of MHD mechanisms for
core-collapse supernova explosions involving strong magnetic fields and rapid
rotation. These models are also consistent with the observed jets and
asymmetry of core-collapse supernovae. The detailed hydrodynamic properties of
the MHD supernova model and the input nuclear physics are described below.
Magnetohydrodynamic Models. The main features of the MHD supernova model
employed here are described in Ref. Nishimura:etal:2006 and need not be
repeated here. However, we emphasize that although this calculation is
performed in the context of a particular $r$-process paradigm, this model is
selected because it matches the required conditions of the $r$-process
environment (timescale, neutron density, temperature, entropy, electron
fraction, etc.). Thus, the results presented here should be similar to those
obtained from more generic $r$-process models (e.g., Otsuki:etal:2003, ).
Figure 1: Thermodynamic properties for three representative tracer particles
are shown as A (red solid lines), B (green dotted lines), and C (blue dashed
lines). From top to bottom, each figure shows the time evolution of
temperature $T$ in GK, density $\rho$ in g $\rm{cm}^{-3}$, entropy per baryon
$S/k_{\rm{B}}$ in Boltzmann constant $k_{\rm{B}}$, and electron fraction
$Y_{\rm{e}}$, respectively. The evolution of each variable is plotted for
$0.5$ s from the time of the switch from NSE to a network calculation at $T=9$
GK.
We utilize the jet-like explosion (model 4 in Nishimura:etal:2006 ) based upon
a two-dimensional magnetohydrodynamic simulation. The ejecta were evolved with
23 tracers to describe the evolution of thermodynamic state variables. These
were then post-processed to obtain the nucleosynthesis yields. Figure 1 shows
the evolution of entropy, measured in units of the Boltzmann constant $k_{B}$,
and electron fraction, $Y_{\rm{e}}$, after nuclear statistical equilibrium
(NSE) is achieved for three representative tracer particles together with
temperature and density. Hydrodynamic quantities (e.g., entropy, temperature,
and density, etc.) are obtained from the magnetohydrodynamic simulation,
whereas the electron fractions are deduced from the nucleosynthesis
calculation. We switch from the NSE abundances to a nucleosynthesis network
calculation at $T=9\ \rm{GK}$ ($\sim 0.78\ \rm{MeV}$).
Nuclear Reaction Network. The nuclear reaction network utilized for the
nucleosynthesis simulations has been described in detail elsewhere
Nishimura:etal:2006 ; Nishimura:etal:2011b and need not be described in more
detail here. The network consists of more than $4000$ isotopes, including
neutrons, protons, and heavy isotopes with atomic number $Z\leq 100$ (for
detail, see Table 1 in Ref. Nishimura:etal:2006 ). We also consider possible
reactions related to the $r$-process involving two- and three-body reactions
or decay channels, and we include electron capture as well as positron capture
and screening effects for all of the relevant charged particle reactions.
Experimentally determined masses AudiWapstra:1995 and reaction rates are
adopted if available. Otherwise, the theoretical predictions for nuclear
masses, reaction rates, and $\beta$-decays are obtained from the finite-range
droplet model (FRDM) (Moller:etal:1995, ).
Table 1: Reaction rates used in nuclear reaction networks Network | $\beta$-decay half-lives | $(n,\gamma)$, $(\gamma,n)$ reaction
---|---|---
FRDM | REACLIB111REACLIB : the REACLIB compilation from Ref. RauscherThielemann:2000 . | REACLIB
RIBF | REACLIB + RIKEN222RIKEN : experimental data given in Ref. Nishimura:etal:2011 . | REACLIB
RIBF+ | REACLIB + RIKEN | REACLIB + $Q_{\rm{+}}$ modified33footnotemark: 3 22footnotetext: $Q_{+}$ modified : defined in Eq. (1).
In the present study, we perform $r$-process calculations based upon three
different nuclear reaction networks as summarized in Table 1. These are
extensions of the basic network described above. One of the networks utilizes
only the FRDM theoretical rates from the REACLIB compilation
RauscherThielemann:2000 . The other two (RIBF and RIBF+) utilize the new
experimental $\beta$-decay half-lives of 38 neutron-rich isotopes from Kr to
Tc and two versions of the theoretical FRDM $\beta$-decay rates for the other
isotopes. The RIBF network replaces the FRDM rates with the new measured ones
where possible. The network (RIBF$+$) is based on the RIBF network and FRDM
rates with modified $Q$ values for $(n,\gamma)$ and reverse reactions given
by:
$Q_{\rm{+}}=\begin{cases}Q-0.3~{}\rm{[MeV]}&(\;\;97\leq A\leq 103)\\\
Q+0.5~{}\rm{[MeV]}&(104\leq A\leq 107)\\\ Q+1.0~{}\rm{[MeV]}&(108\leq A\leq
115)\end{cases}$ (1)
where $Q$ is the theoretical $Q$ value (in MeV) obtained from the FRDM
(Moller:etal:1995, ). The motivation for these modified $Q$ values is that
they lead to a better fit to the observed $\beta$-decay lifetimes away from
stability.
Results. The nucleosynthesis calculations have been performed for $23$ tracer
particles from the MHD supernova model (e.g. Figure 1) with the three
different reaction networks listed in Table 1. In particular, the
underproduction of isotopes near $A=120$ becomes slightly less pronounced
relative to predictions based upon the FRDM rates when the new measured rates
are employed. Figure 2 illustrates the final abundances obtained from the
three representative trajectories (A, B, and C) of Figure 1 and the three
reaction networks employed. Note that as the $A\sim 130$ $r$-process peak is
approached, the new rates begin to make a difference in isotopes with $A\sim
90$ – $120$.
Figure 2: Upper, middle, and lower panels show final abundance distributions
for the illustrative tracer particles A, B, and C from Figure 1. Each panel
shows three different lines corresponding to the different reaction networks,
i.e., FRDM (solid red lines), RIBF (dashed green lines), and RIBF+ (dotted
blue lines). In the case of tracer A, the three lines are indistinguishable
because the abundance distribution does not reach mass numbers $A\sim 100$ –
$130$ where differences in the networks appear.
Figure 3 shows the final integrated abundance distribution from all of the
trajectories. These are compared to the solar system $r$-process abundance
distribution Arlandini:etal:1999 . Here the effect of new rates becomes more
apparent. There has been a recurrent conundrum in the $r$-process models in
that they tend to underproduce nuclei with $A\sim 120$ (Woosley:etal:1994, ).
One hope has been that the newly measured $\beta$-decay rates in this mass
region might shift the $\beta$-flow equilibrium thereby filling in the low
abundances near $A\sim 120$. Here, however, we see on Figure 3 that the
abundances in the $A=110$ – $120$ region are only slightly enhanced. Thus,
although the new rates provide a little assistance in enhancing the abundances
near the valley, they do not alleviate this problem.
Figure 3: Integrated mass averaged total final abundance distributions of
$r$-process elements from the adopted MHD supernova model (jet model 4 in Ref.
(Nishimura:etal:2006, )). Red solid, green dotted, and blue dashed lines
correspond to results from using the FRDM (standard), RIBF, and RIBF+ rates,
respectively. Abundances of solar system $r$-elements (Arlandini:etal:1999, )
are represented by black dots with error bars.
This suggests that a further modification of the $r$-process paradigm is
required. One possibility is suggested by this study. We note that the
$r$-process flow moved farther away from stability and proceeded faster in the
RIBF+ network in which the ($n,\gamma$) Q values were systematically enhanced
for isotopes with $A=104$ – $115$. This helped to fill in the abundances in
the higher mass region with $A\geq 115$ (see Figure 3). Thus, it is important
to measure the masses and/or neutron separation energies in this region. For
example, if the strength of the nuclear closed shell near the neutron magic
number $82$, and $A=120$ – $140$ was systematically diminished, this would
prevent the $r$-process path from bypassing the A $\sim 120$ nuclei as the
$(n,\gamma)$ equilibrium shifts toward the more strongly bound nuclei along
$N=82$ the neutron closed shell. Whatever the explanation for the filling of
$A\sim 110$ – $120$, however, it is evident that the new $\beta$-decay rates
have provided new insight into the nucleosynthesis of the heavy nuclei in the
$r$-process.
At the very least, these results confirm that the final abundances in the mass
region of $A\sim 110-120$ are quite sensitive to the $\beta$-decay half-lives
of isotopes along the $r$-process path. Indeed, the astrophysics of the
production of nuclei in the mass region of $A\sim 110-120$ is currently of
considerable interest. It is presently thought that the lighter heavy elements
with $A\leq 120$ observed in some ultra-metal-poor stars, may be the result of
a new light-element primary process (LEPP) Travaglio:etal:2014 . Our results
suggest that nucleosynthesis in the LEPP should also be sensitive to the
nuclear physics uncertainty from $\beta$-decay rates in this region. Hence,
further studies on both the nuclear physics and astrophysics of the synthesis
of elements with $A\sim 110-120$ are warranted.
###### Acknowledgements.
This study was supported in part by Grants-in-Aid for Scientific Research of
JSPS (19340074, 20244035, and 22540290), JSPS Fellows (21.6817), Scientific
Research on Innovative Area of MEXT (20105004), the U.S. National Science
Foundation Grant No. PHY-0855082, and U.S. Department of Energy under Nuclear
Theory Grant DE-FG02-95- ER40934.
## References
* (1) C. Sneden, J. J. Cowan, and R. Gallino, Annu. Rev. Astron. Astrophys., 46, 241 (2008)
* (2) M. Arnould, S. Goriely, and K. Takahashi, Phys. Rep., 450, 97 (2007)
* (3) F.-K. Thielemann et al., Prog. Part. Nucl. Phys., 66, 346 (2011)
* (4) G. J. Mathews and J. J. Cowan, Nature (London), 345, 491 (1990)
* (5) E. M. Burbidge et al., Rev. Mod. Phys., 29, 547 (1957)
* (6) S. Nishimura, et al., Phys. Rev. Lett. , 106, 052502 (2011)
* (7) K. Otsuki, G. J. Mathews, and T. Kajino, New Astronomy, 8, 767 (2003)
* (8) S. E. Woosley et al., Astrophys. J. , 433, 229, (1994)
* (9) S. Nishimura, et al., Astrophys. J. , 642, 410 (2006)
* (10) N. Nishimura, et al., arXiv:1112.5684
* (11) G. Audi and A. H. Wapstra, Nucl. Phys. A, 595, 409 (1995)
* (12) P. Möller, et al., At. Data Nucl. Data Tables, 59, 185 (1995)
* (13) T. Rauscher and F.-K. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000)
* (14) C. Arlandini et al., Astrophys. J. , 525, 886 (1999)
* (15) C. Travaglio et al., Astrophys. J. , 601, 864 (2004)
|
arxiv-papers
| 2012-03-23T16:20:53 |
2024-09-04T02:49:28.986021
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nobuya Nishimura, Toshitaka Kajino, Grant J. Mathews, Shunji\n Nishimura, and Toshio Suzuki",
"submitter": "Nobuya Nishimura",
"url": "https://arxiv.org/abs/1203.5281"
}
|
1203.5425
|
# CODATA Recommended Values of the Fundamental Physical Constants: 2010111
This report was prepared by the authors under the auspices of the CODATA Task
Group on Fundamental Constants. The members of the task group are:
F. Cabiati, Istituto Nazionale di Ricerca Metrologica, Italy
J. Fischer, Physikalisch-Technische Bundesanstalt, Germany
J. Flowers, National Physical Laboratory, United Kingdom
K. Fujii, National Metrology Institute of Japan, Japan
S. G. Karshenboim, Pulkovo Observatory, Russian Federation
P. J. Mohr, National Institute of Standards and Technology, United States of
America
D. B. Newell, National Institute of Standards and Technology, United States of
America
F. Nez, Laboratoire Kastler-Brossel, France
K. Pachucki, University of Warsaw, Poland
T. J. Quinn, Bureau international des poids et mesures
B. N. Taylor, National Institute of Standards and Technology, United States of
America
B. M. Wood, National Research Council, Canada
Z. Zhang, National Institute of Metrology, China (People’s Republic of)
Peter J. Mohr222Electronic address: mohr@nist.gov, Barry N.
Taylor333Electronic address: barry.taylor@nist.gov, and David B.
Newell444Electronic address: dnewell@nist.gov National Institute of Standards
and Technology, Gaithersburg, Maryland 20899-8420, USA
###### Abstract
This paper gives the 2010 self-consistent set of values of the basic constants
and conversion factors of physics and chemistry recommended by the Committee
on Data for Science and Technology (CODATA) for international use. The 2010
adjustment takes into account the data considered in the 2006 adjustment as
well as the data that became available from 1 January 2007, after the closing
date of that adjustment, until 31 December 2010, the closing date of the new
adjustment. Further, it describes in detail the adjustment of the values of
the constants, including the selection of the final set of input data based on
the results of least-squares analyses. The 2010 set replaces the previously
recommended 2006 CODATA set and may also be found on the World Wide Web at
physics.nist.gov/constants.
###### Contents
1. I Introduction
1. I.1 Background
2. I.2 Brief overview of CODATA 2010 adjustment
1. I.2.1 Fine-structure constant $\alpha$
2. I.2.2 Planck constant $h$
3. I.2.3 Molar gas constant $R$
4. I.2.4 Newtonian constant of gravitation $G$
5. I.2.5 Rydberg constant $R_{\infty}$ and proton radius $r_{\rm p}$
3. I.3 Outline of the paper
2. II Special quantities and units
3. III Relative atomic masses
1. III.1 Relative atomic masses of atoms
2. III.2 Relative atomic masses of ions and nuclei
3. III.3 Relative atomic masses of the proton, triton, and helion
4. III.4 Cyclotron resonance measurement of the electron relative atomic mass
4. IV Atomic transition frequencies
1. IV.1 Hydrogen and deuterium transition frequencies, the Rydberg constant $\bm{R_{\infty}}$, and the proton and deuteron charge radii $\bm{r_{\rm p},r_{\rm d}}$
1. IV.1.1 Theory of hydrogen and deuterium energy levels
2. IV.1.2 Experiments on hydrogen and deuterium
3. IV.1.3 Nuclear radii
2. IV.2 Antiprotonic helium transition frequencies and $\bm{A_{\rm r}({\rm e})}$
1. IV.2.1 Theory relevant to antiprotonic helium
2. IV.2.2 Experiments on antiprotonic helium
3. IV.2.3 Inferred value of $A_{\rm r}(\rm e)$ from antiprotonic helium
3. IV.3 Hyperfine structure and fine structure
5. V Magnetic moment anomalies and $\bm{g}$-factors
1. V.1 Electron magnetic moment anomaly $\bm{a_{\rm e}}$ and the fine-structure constant $\bm{\alpha}$
1. V.1.1 Theory of $a_{\rm e}$
2. V.1.2 Measurements of $a_{\rm e}$
3. V.1.3 Values of $\alpha$ inferred from $a_{\rm e}$
2. V.2 Muon magnetic moment anomaly $\bm{a_{\mbox{\scriptsize{{m}}}}}$
1. V.2.1 Theory of ${a_{\mbox{\scriptsize{{m}}}}}$
2. V.2.2 Measurement of $a_{\mbox{\scriptsize{{m}}}}$: Brookhaven
3. V.3 Bound electron $\bm{g}$-factor in $\bm{{}^{12}{\rm C}^{5+}}$ and in $\bm{{}^{16}{\rm O}^{7+}}$ and $\bm{A_{\rm r}({\rm e})}$
1. V.3.1 Theory of the bound electron $g$-factor
2. V.3.2 Measurements of $g_{\rm e}(^{12}{\rm C}^{5+})$ and $g_{\rm e}(^{16}{\rm O}^{7+})$
6. VI Magnetic moment ratios and the muon-electron mass ratio
1. VI.1 Magnetic moment ratios
1. VI.1.1 Theoretical ratios of atomic bound-particle to free-particle $g$-factors
2. VI.1.2 Bound helion to free helion magnetic moment ratio $\mu_{\rm h}^{\prime}/\mu_{\rm h}$
3. VI.1.3 Ratio measurements
2. VI.2 Muonium transition frequencies, the muon-proton magnetic moment ratio $\bm{\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}}$, and muon-electron mass ratio $\bm{m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}}$
1. VI.2.1 Theory of the muonium ground-state hyperfine splitting
2. VI.2.2 Measurements of muonium transition frequencies and values of $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ and $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$
7. VII Quotient of Planck constant and particle mass $\bm{h/m(X)}$ and $\bm{\alpha}$
1. VII.1 Quotient $h/m({\rm{}^{133}Cs})$
2. VII.2 Quotient $h/m({\rm{}^{87}Rb})$
3. VII.3 Other data
8. VIII Electrical measurements
1. VIII.1 Types of electrical quantities
2. VIII.2 Electrical data
1. VIII.2.1 $K^{2}_{\rm J}R_{\rm K}$ and $h$: NPL watt balance
2. VIII.2.2 $K^{2}_{\rm J}R_{\rm K}$ and $h$: METAS watt balance
3. VIII.2.3 Inferred value of $K_{\rm J}$
3. VIII.3 Josephson and quantum Hall effect relations
9. IX Measurements involving silicon crystals
1. IX.1 Measurements of $\bm{d_{220}({\scriptstyle X})}$ of natural silicon
2. IX.2 $\bm{d_{220}}$ difference measurements of natural silicon crystals
3. IX.3 Gamma-ray determination of the neutron relative atomic mass $\bm{A_{\rm r}({\rm n})}$
4. IX.4 Historic X-ray units
5. IX.5 Other data involving natural silicon crystals
6. IX.6 Determination of $\bm{N_{\rm A}}$ with enriched silicon
10. X Thermal physical quantities
1. X.1 Acoustic gas thermometry
1. X.1.1 NPL 1979 and NIST 1988 values of $\bm{R}$
2. X.1.2 LNE 2009 and 2011 values of $\bm{R}$
3. X.1.3 NPL 2010 value of R
4. X.1.4 INRIM 2010 value of R
2. X.2 Boltzmann constant $\bm{k}$ and quotient $\bm{k/h}$
1. X.2.1 NIST 2007 value of k
2. X.2.2 NIST 2011 value of k/h
3. X.3 Other data
4. X.4 Stefan-Boltzmann constant $\sigma$
11. XI Newtonian constant of gravitation $\bm{G}$
1. XI.1 Updated values
1. XI.1.1 National Institute of Standards and Technology and University of Virginia
2. XI.1.2 Los Alamos National Laboratory
2. XI.2 New values
1. XI.2.1 Huazhong University of Science and Technology
2. XI.2.2 JILA
12. XII Electroweak quantities
13. XIII Analysis of Data
1. XIII.1 Comparison of data through inferred values of $\bm{\alpha}$, $\bm{h}$, $\bm{k}$ and $\bm{A_{\rm r}({\rm e})}$
2. XIII.2 Multivariate analysis of data
1. XIII.2.1 Data related to the Newtonian constant of gravitation $G$
2. XIII.2.2 Data related to all other constants
3. XIII.2.3 Test of the Josephson and quantum Hall effect relations
14. XIV The 2010 CODATA recommended values
1. XIV.1 Calculational details
2. XIV.2 Tables of values
15. XV Summary and Conclusion
1. XV.1 Comparison of 2010 and 2006 CODATA recommended values
2. XV.2 Some implications of the 2010 CODATA recommended values and adjustment for metrology and physics
3. XV.3 Suggestions for future work
16. XVI Acknowledgments
## I Introduction
### I.1 Background
This article reports work carried out under the auspices of the Committee on
Data for Science and Technology (CODATA) Task Group on Fundamental
Constants.555CODATA was established in 1966 as an interdisciplinary committee
of the International Council of Science. The Task Group was founded 3 years
later. It describes in detail the CODATA 2010 least-squares adjustment of the
values of the constants, for which the closing date for new data was 31
December 2010. Equally important, it gives the 2010 self-consistent set of
over 300 CODATA recommended values of the fundamental physical constants based
on the 2010 adjustment. The 2010 set, which replaces its immediate predecessor
resulting from the CODATA 2006 adjustment Mohr _et al._ (2008), first became
available on 2 June 2011 at physics.nist.gov/constants, a Web site of the NIST
Fundamental Constants Data Center (FCDC).
The World Wide Web has engendered a sea change in expectations regarding the
availability of timely information. Further, in recent years new data that
influence our knowledge of the values of the constants seem to appear almost
continuously. As a consequence, the Task Group decided at the time of the 1998
CODATA adjustment to take advantage of the extensive computerization that had
been incorporated in that effort to issue a new set of recommended values
every 4 years; in the era of the Web, the 12-13 years between the first CODATA
set of 1973 Cohen and Taylor (1973) and the second CODATA set of 1986 Cohen
and Taylor (1987), and between this second set and the third set of 1998 Mohr
and Taylor (2000), could no longer be tolerated. Thus, if the 1998 set is
counted as the first of the new 4-year cycle, the 2010 set is the 4th of that
cycle.
Throughout this article we refer to the detailed reports describing the 1998,
2002, and 2006 adjustments as CODATA-98, CODATA-02, and CODATA-06,
respectively Mohr and Taylor (2000, 2005); Mohr _et al._ (2008). To keep the
paper to a reasonable length, our data review focuses on the new results that
became available between the 31 December 2006 and 31 December 2010 closing
dates of the 2006 and 2010 adjustments; the reader should consult these past
reports for detailed discussions of the older data. These past reports should
also be consulted for discussions of motivation, philosophy, the treatment of
numerical calculations and uncertainties, etc. A rather complete list of
acronyms and symbols may be found in the nomenclature section near the end of
the paper.
To further achieve a reduction in the length of this report compared to the
lengths of its three most recent predecessors, it has been decided to omit
extensive descriptions of new experiments and calculations and to comment only
on their most pertinent features; the original references should be consulted
for details. For the same reason, sometimes the older data used in the 2010
adjustment are not given in the portion of the paper that discusses the data
by category, but are given in the portion of the paper devoted to data
analysis. For example, the actual values of the 16 older items of input data
recalled in Sec. VIII are given only in Sec. XIII, rather than in both
sections as done in previous adjustment reports.
As in all previous CODATA adjustments, as a working principle, the validity of
the physical theory underlying the 2010 adjustment is assumed. This includes
special relativity, quantum mechanics, quantum electrodynamics (QED), the
standard model of particle physics, including $CPT$ invariance, and the
exactness (for all practical purposes–see Sec. VIII) of the relationships
between the Josephson and von Klitzing constants $K_{\rm J}$ and $R_{\rm K}$
and the elementary charge $e$ and Planck constant $h$, namely, $K_{\rm J} =
2e/h$ and $R_{\rm K} = h/e^{2}$.
Although the possible time variation of the constants continues to be an
active field of both experimental and theoretical research, there is no
observed variation relevant to the data on which the 2010 recommended values
are based; see, for example, the recent reviews by Uzan (2011) and Chiba
(2011). Other references may be found in the FCDC bibliographic database at
physics.nist.gov/constantsbib using, for example, the keywords “time
variation” or “constants.”
With regard to the 31 December closing date for new data, a datum was
considered to have met this date if the Task Group received a preprint
describing the work by that date and the preprint had already been, or shortly
would be, submitted for publication. Although results are identified by the
year in which they were published in an archival journal, it can be safely
assumed that any input datum labeled with an “11” or “12” identifier was in
fact available by the closing date. However, the 31 December 2010 closing date
does not apply to clarifying information requested from authors; indeed, such
information was received up to shortly before 2 June 2011, the date the new
values were posted on the FCDC Web site. This is the reason that some “private
communications” have 2011 dates.
### I.2 Brief overview of CODATA 2010 adjustment
The 2010 set of recommended values is the result of applying the same
procedures as in previous adjustments and is based on a least-squares
adjustment with, in this case, $N = 160$ items of input data, $M = 83$
variables called adjusted constants, and $\nu = N-M = 77$ degrees of freedom.
The statistic “chi-squared” is $\chi^{2} = 59.1$ with probability
$p(\chi^{2}|\nu) = 0.94$ and Birge ratio $R_{\rm B}=0.88$.
A significant number of new results became available for consideration, both
experimental and theoretical, from 1 January 2007, after the closing date of
the 2006 adjustment, to 31 December 2010, the closing date of the current
adjustment. Data that affect the determination of the fine-structure constant
$\alpha$, Planck constant $h$, molar gas constant $R$, Newtonian constant of
gravitation $G$, Rydberg constant $R_{\infty}$, and rms proton charge radius
$r_{\rm p}$ are the focus of this brief overview, because of their inherent
importance and, in the case of $\alpha$, $h$, and $R$, their impact on the
determination of the values of many other constants. (Constants that are not
among the directly adjusted constants are calculated from appropriate
combinations of those that are directly adjusted.)
#### I.2.1 Fine-structure constant $\alpha$
An improved measurement of the electron magnetic moment anomaly $a_{\rm e}$,
the discovery and correction of an error in its theoretical expression, and an
improved measurement of the quotient $h/m(^{87}{\rm Rb})$ have led to a 2010
value of $\alpha$ with a relative standard uncertainty of $3.2 \times
10^{-10}$ compared to $6.8 \times 10^{-10}$ for the 2006 value. Of more
significance, because of the correction of the error in the theory, the 2010
value of $\alpha$ shifted significantly and now is larger than the 2006 value
by $6.5$ times the uncertainty of that value. This change has rather profound
consequences, because many constants depend on $\alpha$, for example, the
molar Planck constant $N_{\rm A}h$.
#### I.2.2 Planck constant $h$
A new value of the Avogadro constant $N_{\rm A}$ with a relative uncertainty
of $3.0 \times 10^{-8}$ obtained from highly enriched silicon with amount of
substance fraction $x(^{28}{\rm Si}) \approx 0.999\,96$ replaces the 2006
value based on natural silicon and provides an inferred value of $h$ with
essentially the same uncertainty. This uncertainty is somewhat smaller than
$3.6 \times 10^{-8}$, the uncertainty of the most accurate directly measured
watt-balance value of $h$. Because the two values disagree, the uncertainties
used for them in the adjustment were increased by a factor of two to reduce
the inconsistency to an acceptable level; hence the relative uncertainties of
the recommended values of $h$ and $N_{\rm A}$ are $4.4 \times 10^{-8}$, only
slightly smaller than the uncertainties of the corresponding 2006 values. The
2010 value of $h$ is larger than the 2006 value by the fractional amount $9.2
\times 10^{-8}$ while the 2010 value of $N_{\rm A}$ is smaller than the 2006
value by the fractional amount $8.3 \times 10^{-8}$. A number of other
constants depend on $h$, for example, the first radiation constant $c_{1}$,
and consequently the 2010 recommended values of these constants reflect the
change in $h$.
#### I.2.3 Molar gas constant $R$
Four consistent new values of the molar gas constant together with the two
previous consistent values, with which the new values also agree, have led to
a new 2010 recommended value of $R$ with an uncertainty of $9.1\times 10^{-7}$
compared to $1.7 \times 10^{-6}$ for the 2006 value. The 2010 value is smaller
than the 2006 value by the fractional amount $1.2 \times 10^{-6}$ and the
relative uncertainty of the 2010 value is a little over half that of the 2006
value. This shift and uncertainty reduction is reflected in a number of
constants that depend on $R$, for example, the Boltzmann constant $k$ and the
Stefan-Boltzmann constant $\sigma$.
#### I.2.4 Newtonian constant of gravitation $G$
Two new values of $G$ resulting from two new experiments each with
comparatively small uncertainties but in disagreement with each other and with
earlier measurements with comparable uncertainties led to an even larger
expansion of the a priori assigned uncertainties of the data for $G$ than was
necessary in 2006. In both cases the expansion was necessary to reduce the
inconsistencies to an acceptable level. This increase has resulted in a 20 %
increase in uncertainty of the 2010 recommended value compared to that of the
2006 value: $12$ parts in $10^{5}$ vs. $10$ parts in $10^{5}$. Furthermore,
the 2010 recommended value of $G$ is smaller than the 2006 value by the
fractional amount $6.6 \times 10^{-5}$.
#### I.2.5 Rydberg constant $R_{\infty}$ and proton radius $r_{\rm p}$
New experimental and theoretical results that have become available in the
past 4 years have led to the reduction in the relative uncertainty of the
recommended value of the Rydberg constant from $6.6 \times 10^{-12}$ to $5.0
\times 10^{-12}$, and the reduction in uncertainty from $0.0069$ fm to
$0.0051$ fm of the proton rms charge radius based on spectroscopic and
scattering data but not muonic hydrogen data. Data from muonic hydrogen, with
the assumption that the muon and electron interact with the proton at short
distances in exactly the same way, are so inconsistent with the other data
that they have not been included in the determination of $r_{\rm p}$ and thus
do not have an influence on $R_{\infty}$. The 2010 value of $R_{\infty}$
exceeds the 2006 value by the fractional amount $1.1 \times 10^{-12}$ and the
2010 value of $r_{\rm p}$ exceeds the 2006 value by $0.0007$ fm.
### I.3 Outline of the paper
Section II briefly recalls some constants that have exact values in the
International System of Units (SI) BIPM (2006), the unit system used in all
CODATA adjustments. Sections III-XII discuss the input data with a strong
focus on those results that became available between the 31 December 2006 and
31 December 2010 closing dates of the 2006 and 2010 adjustments. It should be
recalled (see especially Appendix E of CODATA-98) that in a least-squares
analysis of the constants, both the experimental and theoretical numerical
data, also called observational data or input data, are expressed as functions
of a set of independent variables called directly adjusted constants (or
sometimes simply adjusted constants). The functions themselves are called
observational equations, and the least-squares procedure provides best
estimates, in the least-squares sense, of the adjusted constants. In essence,
the procedure determines the best estimate of a particular adjusted constant
by automatically taking into account all possible ways of determining its
value from the input data. As already noted, the recommended values of those
constants not directly adjusted are calculated from the adjusted constants.
Section XIII describes the analysis of the data. The analysis includes
comparison of measured values of the same quantity, measured values of
different quantities through inferred values of another quantity such as
$\alpha$ or $h$, and by the method of least-squares. The final input data used
to determine the adjusted constants, and hence the entire 2010 CODATA set of
recommended values, are based on these investigations.
Section XIV provides, in several tables, the set of over 300 recommended
values of the basic constants and conversion factors of physics and chemistry,
including the covariance matrix of a selected group of constants. Section XV
concludes the report with a comparison of a small representative subset of
2010 recommended values with their 2006 counterparts, comments on some of the
more important implications of the 2010 adjustment for metrology and physics,
and suggestions for future experimental and theoretical work that will improve
our knowledge of the values of the constants. Also touched upon is the
potential importance of this work and that of the next CODATA constants
adjustment (expected 31 December 2014 closing date) for the redefinition of
the kilogram, ampere, kelvin, and mole currently under discussion
internationally Mills _et al._ (2011).
## II Special quantities and units
As a consequence of the SI definitions of the meter, the ampere, and the mole,
$c$, $\mu_{0}$ and $\epsilon_{0}$, and $M(^{12}$C) and $M_{\rm u}$, have exact
values; see Table 1. Since the relative atomic mass $A_{\rm r}(X)$ of an
entity $X$ is defined by $A_{\rm r}(X)=m(X)/m_{\rm u}$, where $m(X)$ is the
mass of $X$, and the (unified) atomic mass constant $m_{u}$ is defined
according to $m_{\rm u}$ = $m(^{12}{\rm C})/12$, $A_{\rm r}(^{12}{\rm C})=12$
exactly, as shown in the table. Since the number of specified entities in one
mole is equal to the numerical value of the Avogadro constant $N_{\rm
A}\approx 6.022\times 10^{23}$/mol, it follows that the molar mass of an
entity $X$, $M(X)$, is given by $M(X)=N_{\rm A}m(X)=A_{\rm r}(X)M_{\rm u}$ and
$M_{\rm u}=N_{\rm A}m_{\rm u}$. The (unified) atomic mass unit u (also called
the dalton, Da), is defined as 1 u $=m_{\rm u}\approx 1.66\times 10^{-27}$ kg.
The last two entries in Table 1, $K_{\rm J-90}$ and $R_{\rm K-90}$, are the
conventional values of the Josephson and von Klitzing constants introduced on
1 January 1990 by the International Committee for Weights and Measures (CIPM)
to foster worldwide uniformity in the measurement of electrical quantities. In
this paper, those electrical quantities measured in terms of the Josephson and
quantum Hall effects with the assumption that $K_{\rm J}$ and $R_{\rm K}$ have
these conventional values are labeled with a subscript 90.
Measurements of the quantity $K_{\rm J}^{2}R_{\rm K}=4/h$ using a moving coil
watt balance (see Sec. VIII) require the determination of the local
acceleration of free fall $g$ at the site of the balance with a relative
uncertainty of a few parts in $10^{9}$. That currently available absolute
gravimeters can achieve such an uncertainty if properly used has been
demonstrated by comparing different instruments at essentially the same
location. An important example is the periodic international comparison of
absolute gravimeters (ICAG) carried out at the International Bureau of Weights
and Measures (BIPM), Sèvres, France Jiang _et al._ (2011). The good agreement
obtained between a commercial optical interferometer-based gravimeter that is
in wide use and a cold atom, atomic interferometer-based instrument also
provides evidence that the claimed uncertainties of determinations of $g$ are
realistic Merlet _et al._ (2010). However, not all gravimeter comparisons have
obtained such satisfactory results Louchet-Chauvet _et al._ (2011). Additional
work in this area may be needed when the relative uncertainties of watt-
balance experiments reach the level of 1 part in $10^{8}$.
Table 1: Some exact quantities relevant to the 2010 adjustment. Quantity | Symbol | Value
---|---|---
speed of light in vacuum | $c$, $c_{0}$ | $299\,792\,458\ {\rm m\ s^{-1}}$
magnetic constant | $\mu_{0}$ | $4\mbox{{p}}\times 10^{-7}\ {\rm N\ A^{-2}}$ $=12.566\,370\,614...\times 10^{-7}\ {\rm N\ A^{-2}}$
electric constant | $\epsilon_{0}$ | $(\mu_{0}c^{2})^{-1}$ $=8.854\,187\,817...\ \times 10^{-12}\ {\rm F\ m^{-1}}$
molar mass of 12C | $M(^{12}{\rm C})$ | $12\times 10^{-3}\ {\rm kg\ mol^{-1}}$
molar mass constant | $M_{\rm u}$ | $10^{-3}\ {\rm kg\ mol^{-1}}$
relative atomic mass of 12C | $A_{\rm r}(^{12}{\rm C})$ | $12$
conventional value of Josephson constant | $K_{{\rm J}-90}$ | $483\,597.9\ {\rm GHz\ V^{-1}}$
conventional value of von Klitzing constant | $R_{{\rm K}-90}$ | $25\,812.807\ {\rm\Omega}$
## III Relative atomic masses
The directly adjusted constants include the relative atomic masses $A_{\rm
r}(X)$ of a number of particles, atoms, and ions. Further, values of $A_{\rm
r}(X)$ of various atoms enter the calculations of several potential input
data. The following sections and Tables 2 to 4 summarize the relevant
information.
Table 2: Values of the relative atomic masses of the neutron and various atoms as given in the 2003 atomic mass evaluation together with the defined value for 12C. Atom | Relative atomic | Relative standard
---|---|---
| mass $A_{\rm r}({\rm X})$ | uncertainty $u_{\rm r}$
n | $1.008\,664\,915\,74(56)$ | $5.6\times 10^{-10}$
1H | $1.007\,825\,032\,07(10)$ | $1.0\times 10^{-10}$
2H | $2.014\,101\,777\,85(36)$ | $1.8\times 10^{-10}$
3H | $3.016\,049\,2777(25)$ | $8.2\times 10^{-10}$
3He | $3.016\,029\,3191(26)$ | $8.6\times 10^{-10}$
4He | $4.002\,603\,254\,153(63)$ | $1.6\times 10^{-11}$
12C | $12$ | (exact)
16O | $15.994\,914\,619\,56(16)$ | $1.0\times 10^{-11}$
28Si | $27.976\,926\,5325(19)$ | $6.9\times 10^{-11}$
29Si | $28.976\,494\,700(22)$ | $7.6\times 10^{-10}$
30Si | $29.973\,770\,171(32)$ | $1.1\times 10^{-9}$
36Ar | $35.967\,545\,105(28)$ | $7.8\times 10^{-10}$
38Ar | $37.962\,732\,39(36)$ | $9.5\times 10^{-9}$
40Ar | $39.962\,383\,1225(29)$ | $7.2\times 10^{-11}$
87Rb | $86.909\,180\,526(12)$ | $1.4\times 10^{-10}$
107Ag | $106.905\,0968(46)$ | $4.3\times 10^{-8}$
109Ag | $108.904\,7523(31)$ | $2.9\times 10^{-8}$
133Cs | $132.905\,451\,932(24)$ | $1.8\times 10^{-10}$
Table 3: Values of the relative atomic masses of various atoms that have become available since the 2003 atomic mass evaluation. Atom | Relative atomic | Relative standard
---|---|---
| mass $A_{\rm r}(X)$ | uncertainty $u_{\rm r}$
2H | $2.014\,101\,778\,040(80)$ | $4.0\times 10^{-11}$
4He | $4.002\,603\,254\,131(62)$ | $1.5\times 10^{-11}$
16O | $15.994\,914\,619\,57(18)$ | $1.1\times 10^{-11}$
28Si | $27.976\,926\,534\,96(62)$ | $2.2\times 10^{-11}$
29Si | $28.976\,494\,6625(20)$ | $6.9\times 10^{-11}$
87Rb | $86.909\,180\,535(10)$ | $1.2\times 10^{-10}$
133Cs | $132.905\,451\,963(13)$ | $9.8\times 10^{-11}$
### III.1 Relative atomic masses of atoms
Table 2, which is identical to Table II in CODATA-06, gives values of $A_{\rm
r}(X)$ taken from the 2003 atomic mass evaluation (AME2003) carried out by the
Atomic Mass Data Center (AMDC), Centre de Spectrométrie Nucléaire et de
Spectrométrie de Masse (CSNMS), Orsay, France Wapstra _et al._ (2003); Audi
_et al._ (2003); AMDC (2006). However, not all of these values are actually
used in the adjustment; some are given for comparison purposes only. Although
these values are correlated to a certain extent, the only correlation that
needs to be taken into account in the current adjustment is that between
$A_{\rm r}(^{1}{\rm H})$ and $A_{\rm r}(^{2}{\rm H})$; their correlation
coefficient is 0.0735 AMDC (2003).
Table 3 lists seven values of $A_{\rm r}(X)$ relevant to the 2010 adjustment
obtained since the publication of ASME2003. It is the updated version of Table
IV discussed in CODATA-06. The changes made are the deletion of the
${}^{3}{\rm H}$ and ${}^{3}{\rm He}$ values obtained by the SMILETRAP group at
Stockholm University (StockU), Sweden; and the inclusion of values for
${}^{28}{\rm Si}$, ${}^{87}{\rm Rb}$, and ${}^{133}{\rm Cs}$ obtained by the
group at Florida State University (FSU), Tallahassee, FL, USA Redshaw _et al._
(2008); Mount _et al._ (2010). This group uses the method initially developed
at the Massachusetts Institute of Technology, Cambridge, MA, USA Rainville _et
al._ (2005). In the MIT approach, which eliminates or reduces a number of
systematic effects and their associated uncertainties, mass ratios are
determined by directly comparing the cyclotron frequencies of two different
ions simultaneously confined in a Penning trap. (The value of $A_{\rm
r}(^{29}{\rm Si})$ in Table 3 is given in the supplementary information of the
last cited reference. The MIT atomic mass work was transferred to FSU a number
of years ago.)
The deleted SMILETRAP results are not discarded but are included in the
adjustment in a more fundamental way, as described in Sec. III.3. The values
of $A_{\rm r}(^{2}{\rm H})$, $A_{\rm r}(^{4}{\rm He})$, and $A_{\rm
r}(^{16}{\rm O})$ in Table 3 were obtained by the University of Washington
(UWash) group, Seattle, WA, USA and were used in the 2006 adjustment. The
three values are correlated and their variances, covariances, and correlation
coefficients are given in Table 4, which is identical to Table IV in CODATA-06
Table 4: The variances, covariances, and correlation coefficients of the University of Washington values of the relative atomic masses of deuterium, helium 4, and oxygen 16. The numbers in bold above the main diagonal are $10^{20}$ times the numerical values of the covariances; the numbers in bold on the main diagonal are $10^{20}$ times the numerical values of the variances; and the numbers in italics below the main diagonal are the correlation coefficients. | $A_{\rm r}(^{2}{\rm H})$ | $A_{\rm r}(^{4}{\rm He})$ | $A_{\rm r}(^{16}{\rm O})$
---|---|---|---
$A_{\rm r}(^{2}{\rm H})$ | ${\bf 0.6400}$ | ${\bf 0.0631}$ | ${\bf 0.1276}$
$A_{\rm r}(^{4}{\rm He})$ | ${\it 0.1271}$ | ${\bf 0.3844}$ | ${\bf 0.2023}$
$A_{\rm r}(^{16}{\rm O})$ | ${\it 0.0886}$ | ${\it 0.1813}$ | ${\bf 3.2400}$
The values of $A_{\rm r}(X)$ from Table 2 initially used as input data for the
2010 adjustment are $A_{\rm r}(^{1}{\rm H})$, $A_{\rm r}(^{2}{\rm H})$,
$A_{\rm r}(^{87}{\rm Rb})$, and $A_{\rm r}(^{133}{\rm Cs})$; and from Table 3,
$A_{\rm r}(^{2}{\rm H})$, $A_{\rm r}(^{4}{\rm He})$, $A_{\rm r}(^{16}{\rm
O})$, $A_{\rm r}(^{87}{\rm Rb})$, and $A_{\rm r}(^{133}{\rm Cs})$. These
values are items $B1$, $B2.1$, $B2.2$, and $B7$ to $B10.2$ in Table
LABEL:tab:pdata, Sec. XIII. As in the 2006 adjustment, the ASME2003 values for
$A_{\rm r}(^{3}{\rm H})$, and $A_{\rm r}(^{3}{\rm He})$ in Table 2 are not
used because they were influenced by an earlier ${}^{3}{\rm He}$ result of the
UWash group that disagrees with their newer, more accurate result Van Dyck
(2010). Although not yet published, it can be said that it agrees well with
the value from the SMILETRAP group; see Sec. III.3.
Also as in the 2006 adjustment, the UWash group’s values for $A_{\rm
r}(^{4}{\rm He})$ and $A_{\rm r}(^{16}{\rm O})$ in Table 3 are used in place
of the corresponding ASME2003 values in Table 2 because the latter are based
on a preliminary analysis of the data while those in Table 3 are based on a
thorough reanalysis of the data Van Dyck _et al._ (2006).
Finally, we note that the $A_{\rm r}(^{2}{\rm H})$ value of the UWash group in
Table 3 is the same as used in the 2006 adjustment. As discussed in CODATA-06,
it is a near-final result with a conservatively assigned uncertainty based on
the analysis of 10 runs taken over a 4-year period privately communicated to
the Task Group in 2006 by R. S. Van Dyck. A final result completely consistent
with it based on the analysis of 11 runs but with an uncertainty of about half
that given in the table should be published in due course together with the
final result for $A_{\rm r}(^{3}{\rm He})$ Van Dyck (2010).
### III.2 Relative atomic masses of ions and nuclei
For a neutral atom $X$, $A_{\rm r}(X)$ can be expressed in terms of $A_{\rm
r}$ of an ion of the atom formed by the removal of $n$ electrons according to
$\displaystyle A_{\rm\rm r}(X)$ $\displaystyle=$ $\displaystyle A_{\rm
r}(X^{n+})+nA_{\rm r}({\rm e})$ (1) $\displaystyle-\frac{E_{\rm b}(X)-E_{\rm
b}(X^{n+})}{m_{\rm u}c^{2}}\ .$
In this expression, $E_{\rm b}(X)/m_{\rm u}c^{2}$ is the relative-atomic-mass
equivalent of the total binding energy of the $Z$ electrons of the atom and
$Z$ is the atom’s atomic number (proton number). Similarly, $E_{\rm
b}(X^{n+})/m_{\rm u}c^{2}$ is the relative-atomic-mass equivalent of the
binding energy of the $Z-n$ electrons of the $X^{n+}$ ion. For an ion that is
fully stripped $n=Z$ and $X^{Z+}$ is simply $N$, the nucleus of the atom. In
this case $E_{\rm b}(X^{Z+})/m_{\rm u}c^{2}=0$ and Eq. (1) becomes of the form
of the first two equations of Table 33, Sec. XIII.
The binding energies $E_{\rm b}$ employed in the 2010 adjustment are the same
as those used in that of 2002 and 2006; see Table IV of CODATA-02. As noted in
CODATA-06, the binding energy for tritium, ${}^{3}{\rm H}$, is not included in
that table. We employ the value used in the 2006 adjustment,
$1.097\,185\,439\times 10^{7}~{}{\rm m^{-1}}$, due to Kotochigova (2006). For
our purposes here, the uncertainties of the binding energies are negligible.
### III.3 Relative atomic masses of the proton, triton, and helion
The focus of this section is the cyclotron frequency ratio measurements of the
SMILETRAP group that lead to values of $A_{\rm r}({\rm p})$, $A_{\rm r}({\rm
t})$, and $A_{\rm r}({\rm h})$, where the triton t and helion h are the nuclei
of ${}^{3}{\rm H}$ and ${}^{3}{\rm He}$. As noted in Sec. III.1 above, the
reported values of Nagy _et al._ (2006) for $A_{\rm r}(^{3}{\rm H})$ and
$A_{\rm r}(^{3}{\rm He})$ were used as input data in the 2006 adjustment but
are not used in this adjustment. Instead, the actual cyclotron frequency ratio
results underlying those values are used as input data. This more fundamental
way of handling the SMILETRAP group’s results is motivated by the similar but
more recent work of the group related to the proton, which we discuss before
considering the earlier work.
Solders _et al._ (2008) used the Penning-trap mass spectrometer SMILETRAP,
described in detail by Bergström _et al._ (2002), to measure the ratio of the
cyclotron frequency $f_{\rm c}$ of the ${\rm H_{2}}^{+*}$ molecular ion to
that of the deuteron d, the nucleus of the ${}^{2}{\rm H}$ atom. (The
cyclotron frequency of an ion of charge $q$ and mass $m$ in a magnetic flux
density $B$ is given by $f_{\rm c}=qB/2\mbox{{p}}{m}$.) Here the asterisk
indicates that the singly ionized ${\rm H_{2}}$ molecules are in excited
vibrational states as a result of the 3.4 keV electrons used to bombard
neutral ${\rm H_{2}}$ molecules in their vibrational ground state in order to
ionize them. The reported result is
$\displaystyle\frac{f_{\rm c}({\rm H}_{2}^{+*})}{f_{\rm c}({\rm d})}$
$\displaystyle=$ $\displaystyle 0.999\,231\,659\,33(17)\qquad[1.7\times
10^{-10}]\,.\qquad$ (2)
This value was obtained using a two-pulse Ramsey technique to excite the
cyclotron frequencies, thereby enabling a more precise determination of the
cyclotron resonance frequency line-center than was possible with the one-pulse
excitation used in earlier work George _et al._ (2007); Suhonen _et al._
(2007). The uncertainty is essentially all statistical; components of
uncertainty from systematic effects such as “$q/A$ asymmetry” (difference of
charge-to-mass ratio of the two ions), time variation of the 4.7 T applied
magnetic flux density, relativistic mass increase, and ion-ion interactions
were deemed negligible by comparison.
The frequency ratio $f_{\rm c}({\rm H_{2}}^{+*})/{f_{\rm c}(\rm d)}$ can be
expressed in terms of adjusted constants and ionization and binding energies
that have negligible uncertainties in this context. Based on Sec. III.2 we can
write
$\displaystyle A_{\rm r}({\rm H}_{2})$ $\displaystyle=$ $\displaystyle 2A_{\rm
r}({\rm H})-E_{\rm B}({\rm H}_{2})/m_{\rm u}c^{2}\,,$ (3) $\displaystyle
A_{\rm r}({\rm H})$ $\displaystyle=$ $\displaystyle A_{\rm r}({\rm p})+A_{\rm
r}({\rm e})-E_{\rm I}({\rm H})/m_{\rm u}c^{2}\,,$ (4) $\displaystyle A_{\rm
r}({\rm H}_{2})$ $\displaystyle=$ $\displaystyle A_{\rm r}({\rm
H}_{2}^{+})+A_{\rm r}({\rm e})-E_{\rm I}({\rm H}_{2})/m_{\rm u}c^{2}\,,$ (5)
$\displaystyle A_{\rm r}({\rm H}_{2}^{+*})$ $\displaystyle=$ $\displaystyle
A_{\rm r}({\rm H}_{2}^{+})+E_{\rm av}/m_{\rm u}c^{2}\,,$ (6)
which yields
$\displaystyle A_{\rm r}({\rm H}_{2}^{+*})$ $\displaystyle=$ $\displaystyle
2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-E_{\rm B}({\rm H}_{2}^{+*})/m_{\rm
u}c^{2}\,,$ (7)
where
$\displaystyle E_{\rm B}({\rm H}_{2}^{+*})$ $\displaystyle=$ $\displaystyle
2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}$
(8)
is the binding energy of the ${\rm H}_{2}^{+*}$ excited molecule. Here $E_{\rm
I}({\rm H})$is the ionization energy of hydrogen, $E_{\rm B}({\rm H_{2}})$ is
the disassociation energy of the ${\rm H_{2}}$ molecule, $E_{\rm I}({\rm
H}_{2})$ is the single electron ionization energy of ${\rm H}_{2}$, and
$E_{\rm av}$ is the average vibrational excitation energy of an ${\rm
H}_{2}^{+}$ molecule as a result of the ionization of ${\rm H}_{2}$ by 3.4 keV
electron impact.
The observational equation for the frequency ratio is thus
$\displaystyle\frac{f_{\rm c}({\rm H}_{2}^{+*})}{f_{\rm c}({\rm d})}$
$\displaystyle=$ $\displaystyle\frac{A_{\rm r}({\rm d})}{2A_{\rm r}({\rm
p})+A_{\rm r}({\rm e})-E_{\rm B}({\rm H}_{2}^{+*})/m_{\rm u}c^{2}}\,.$ (9)
We treat $E_{\rm av}$ as an adjusted constant in addition to $A_{\rm r}({\rm
e})$, $A_{\rm r}({\rm p})$, and $A_{\rm r}({\rm d})$ in order to take its
uncertainty into account in a consistent way, especially since it enters into
the observational equations for the frequency ratios to be discussed below.
The required ionization and binding energies as well as $E_{\rm av}$ that we
use are as given by Solders _et al._ (2008) and except for $E_{\rm av}$, have
negligible uncertainties:
$\displaystyle E_{\rm I}({\rm H})$ $\displaystyle=$ $\displaystyle
13.5984\mbox{ eV}=14.5985\times 10^{-9}\,m_{\rm u}c^{2}\,,$ (10)
$\displaystyle E_{\rm B}({\rm H}_{2})$ $\displaystyle=$ $\displaystyle
4.4781\mbox{ eV}=4.8074\times 10^{-9}\,m_{\rm u}c^{2}\,,$ (11) $\displaystyle
E_{\rm I}({\rm H}_{2})$ $\displaystyle=$ $\displaystyle 15.4258\mbox{
eV}=16.5602\times 10^{-9}\,m_{\rm u}c^{2}\,,$ (12) $\displaystyle E_{\rm av}$
$\displaystyle=$ $\displaystyle 0.740(74)\mbox{ eV}=0.794(79)\times
10^{-9}\,m_{\rm u}c^{2}\,.\qquad$ (13)
We now consider the SMILETRAP results of Nagy _et al._ (2006) for the ratio of
the cyclotron frequency of the triton t and of the ${}^{3}{\rm He}^{+}$ ion to
that of the ${\rm H_{2}}^{+*}$ molecular ion. These authors report for the
triton
$\displaystyle\frac{f_{\rm c}({\rm t})}{f_{\rm c}({\rm H}_{2}^{+*})}$
$\displaystyle=$ $\displaystyle 0.668\,247\,726\,86(55)\quad[8.2\times
10^{-10}]\qquad$ (14)
and for the ${}^{3}{\rm He}^{+}$ ion
$\displaystyle\frac{f_{\rm c}(^{3}{\rm He}^{+})}{f_{\rm c}({\rm H}_{2}^{+*})}$
$\displaystyle=$ $\displaystyle 0.668\,252\,146\,82(55)\quad[8.2\times
10^{-10}]\,.\qquad$ (15)
The relative uncertainty of the triton ratio consists of the following
uncertainty components in parts in 109: 0.22 statistical, and 0.1, 0.1, 0.77,
and 0.1 due to relativistic mass shift, ion number dependence, $q/A$
asymmetry, and contaminant ions, respectively. The components for the
${}^{3}{\rm He}^{+}$ ion ratio are the same except the statistical uncertainty
is 0.24. All of these components are independent except the 0.77$\times
10^{-9}$ component due to $q/A$ asymmetry; it leads to a correlation
coefficient between the two frequency ratios of 0.876.
Observational equations for these frequency ratios are
$\displaystyle\frac{f_{\rm c}({\rm t})}{f_{\rm c}({\rm H}_{2}^{+*})}$
$\displaystyle=$ $\displaystyle\frac{2A_{\rm r}({\rm p})+A_{\rm r}({\rm
e})-E_{\rm B}({\rm H}_{2}^{+*})/m_{\rm u}c^{2}}{A_{\rm r}({\rm t})}\qquad$
(16)
and
$\displaystyle\frac{f_{\rm c}(^{3}{\rm He}^{+})}{f_{\rm c}({\rm H}_{2}^{+*})}$
$\displaystyle=$ $\displaystyle\frac{2A_{\rm r}({\rm p})+A_{\rm r}({\rm
e})-E_{\rm B}({\rm H}_{2}^{+*})/m_{\rm u}c^{2}}{A_{\rm r}({\rm h})+A_{\rm
r}({\rm e})-E_{\rm I}(^{3}{\rm He}^{+})/m_{\rm u}c^{2}}\,,\qquad$ (17)
where
$\displaystyle A_{\rm r}(^{3}{\rm He}^{+})=A_{\rm r}({\rm h})+A_{\rm r}({\rm
e})-E_{\rm I}(^{3}{\rm He}^{+})/m_{\rm u}c^{2}$ (18)
and
$\displaystyle E_{\rm I}(^{3}{\rm He}^{+})=51.4153\mbox{ eV}=58.4173\times
10^{-9}\,m_{\rm u}c^{2}$ (19)
is the ionization energy of the ${}^{3}{\rm He}^{+}$ ion, based on Table IV of
CODATA-02.
The energy $E_{\rm av}$ and the three frequency ratios given in Eqs. (2),
(14), and (15) are items $B3$ to $B6$ in Table LABEL:tab:pdata.
### III.4 Cyclotron resonance measurement of the electron relative atomic
mass
As in the 2002 and 2006 CODATA adjustments, we take as an input datum the
Penning-trap result for the electron relative atomic mass $A_{\rm r}({\rm e})$
obtained by the University of Washington group Farnham _et al._ (1995):
$\displaystyle A_{\rm r}({\rm e})=0.000\,548\,579\,9111(12)\quad[2.1\times
10^{-9}]\,.$ (20)
This is item $B11$ of Table LABEL:tab:pdata.
## IV Atomic transition frequencies
Measurements and theory of transition frequencies in hydrogen, deuterium,
anti-protonic helium, and muonic hydrogen provide information on the Rydberg
constant, the proton and deuteron charge radii, and the relative atomic mass
of the electron. These topics as well as hyperfine and fine-structure
splittings are considered in this section.
### IV.1 Hydrogen and deuterium transition frequencies, the Rydberg constant
$\bm{R_{\infty}}$, and the proton and deuteron charge radii $\bm{r_{\rm
p},r_{\rm d}}$
Transition frequencies between states $a$ and $b$ in hydrogen and deuterium
are given by
$\displaystyle\nu_{ab}$ $\displaystyle=$ $\displaystyle\frac{E_{b}-E_{a}}{h}\
,$ (21)
where $E_{a}$ and $E_{b}$ are the energy levels of the states. The energy
levels divided by $h$ are given by
$\displaystyle\frac{E_{a}}{h}$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{2}m_{\rm
e}c^{2}}{2n_{a}^{2}h}\left(1+\delta_{a}\right)=-\frac{R_{\infty}c}{n_{a}^{2}}\left(1+\delta_{a}\right),$
(22)
where $R_{\infty}c$ is the Rydberg constant in frequency units, $n_{a}$ is the
principle quantum number of state $a$, and $\delta_{a}$ is a small correction
factor ( $|\delta_{a}|\ll 1$ ) that contains the details of the theory of the
energy level, including the effect of the finite size of the nucleus as a
function of the rms charge radius $r_{\rm p}$ for hydrogen or $r_{\rm d}$ for
deuterium. In the following summary, corrections are given in terms of the
contribution to the energy level, but in the numerical evaluation for the
least-squares adjustment, $R_{\infty}$ is factored out of the expressions and
is an adjusted constant.
#### IV.1.1 Theory of hydrogen and deuterium energy levels
Here we provide the information necessary to determine theoretical values of
the relevant energy levels, with the emphasis of the discussion on results
that have become available since the 2006 adjustment. For brevity, most
references to earlier work, which can be found in Eides _et al._ (2001b);
Eides _et al._ (2007), for example, are not included here.
Theoretical values of the energy levels of different states are highly
correlated. In particular, uncalculated terms for S states are primarily of
the form of an unknown common constant divided by $n^{3}$. We take this fact
into account by calculating covariances between energy levels in addition to
the uncertainties of the individual levels (see Sec. IV.1.1). The correlated
uncertainties are denoted by $u_{0}$, while the uncorrelated uncertainties are
denoted by $u_{n}$.
##### Dirac eigenvalue
The Dirac eigenvalue for an electron in a Coulomb field is
$\displaystyle E_{\rm D}=f(n,j)\,m_{\rm e}c^{2}\ ,$ (23)
where
$\displaystyle
f(n,j)=\left[1+{(Z\alpha)^{2}\over(n-\delta)^{2}}\right]^{-1/2}\ ,$ (24)
$n$ and $j$ are the principal quantum number and total angular momentum of the
state, respectively, and
$\displaystyle\delta=j+{\textstyle{1\over 2}}-\left[(j+{\textstyle{1\over
2}})^{2}-(Z\alpha)^{2}\right]^{1/2}\ .$ (25)
In Eqs. (24) and (25), $Z$ is the charge number of the nucleus, which for
hydrogen and deuterium is 1. However, we shall retain $Z$ as a parameter to
classify the various contributions.
Equation (23) is only valid for an infinitely heavy nucleus. For a nucleus
with a finite mass $m_{\rm N}$ that expression is replaced by Barker and
Glover (1955); Sapirstein and Yennie (1990):
$\displaystyle E_{M}(H)$ $\displaystyle=$ $\displaystyle
Mc^{2}+[f(n,j)-1]m_{\rm r}c^{2}-[f(n,j)-1]^{2}{m_{\rm r}^{2}c^{2}\over 2M}$
(26) $\displaystyle+\,{1-\delta_{\ell
0}\over\kappa(2\ell+1)}{(Z\alpha)^{4}m_{\rm r}^{3}c^{2}\over 2n^{3}m_{\rm
N}^{2}}+\cdots$
for hydrogen or by Pachucki and Karshenboim (1995)
$\displaystyle E_{M}(D)$ $\displaystyle=$ $\displaystyle
Mc^{2}+[f(n,j)-1]m_{\rm r}c^{2}-[f(n,j)-1]^{2}{m_{\rm r}^{2}c^{2}\over 2M}$
(27) $\displaystyle+\,{1\over\kappa(2\ell+1)}{(Z\alpha)^{4}m_{\rm
r}^{3}c^{2}\over 2n^{3}m_{\rm N}^{2}}+\cdots$
for deuterium. In Eqs. (26) and (27) $\ell$ is the nonrelativistic orbital
angular momentum quantum number,
$\kappa=(-1)^{j-\ell+1/2}(j+{\textstyle{1\over 2}})$ is the angular-momentum-
parity quantum number, $M=m_{\rm e}+m_{\rm N}$, and $m_{\rm r}=m_{\rm e}m_{\rm
N}/(m_{\rm e}+m_{\rm N})$ is the reduced mass.
Equations (26) and (27) differ in that the Darwin-Foldy term proportional to
$\delta_{\ell 0}$ is absent in Eq. (27), because it does not occur for a spin-
one nucleus such as the deuteron Pachucki and Karshenboim (1995). In the three
previous adjustments, Eq. (26) was used for both hydrogen and deuterium and
the absence of the Darwin-Foldy term in the case of deuterium was accounted
for by defining an effective deuteron radius given by Eq. (A56) of CODATA-98
and using it to calculate the finite nuclear size correction given by Eq.
(A43) and the related equations. The extra term in the size correction
canceled the Darwin-Foldy term in Eq. (26). See also Sec. IV.1.1.
##### Relativistic recoil
The leading relativistic-recoil correction, to lowest order in $Z\alpha$ and
all orders in $m_{\rm e}/m_{\rm N}$, is Erickson (1977); Sapirstein and Yennie
(1990)
$\displaystyle E_{\rm S}$ $\displaystyle=$ $\displaystyle{m_{\rm r}^{3}\over
m_{\rm e}^{2}m_{\rm N}}{(Z\alpha)^{5}\over\mbox{{p}}n^{3}}m_{\rm e}c^{2}$
$\displaystyle\times\bigg{\\{}{\textstyle{1\over 3}}\delta_{\ell
0}\ln(Z\alpha)^{-2}-{\textstyle{8\over 3}}\ln k_{0}(n,\ell)-{\textstyle{1\over
9}}\delta_{\ell 0}-{\textstyle{7\over 3}}a_{n}$ $\displaystyle-\,{2\over
m_{\rm N}^{2}-m_{\rm e}^{2}}\delta_{\ell 0}\left[m_{\rm
N}^{2}\ln\Big{(}{m_{\rm e}\over m_{\rm r}}\Big{)}-m_{\rm
e}^{2}\ln\Big{(}{m_{\rm N}\over m_{\rm r}}\Big{)}\right]\bigg{\\}},$
where
$\displaystyle a_{n}$ $\displaystyle=$ $\displaystyle-2\left[\ln\Big{(}{2\over
n}\Big{)}+\sum_{i=1}^{n}{1\over i}+1-\frac{1}{2n}\right]\delta_{\ell 0}$ (29)
$\displaystyle+\,{1-\delta_{\ell 0}\over\ell(\ell+1)(2\ell+1)}\,.$
To lowest order in the mass ratio, the next two orders in $Z\alpha$ are
$\displaystyle E_{\rm R}$ $\displaystyle=$ $\displaystyle{m_{\rm e}\over
m_{\rm N}}{(Z\alpha)^{6}\over n^{3}}m_{\rm e}c^{2}$ (30)
$\displaystyle\times\left[D_{60}+D_{72}Z\alpha\ln^{2}{(Z\alpha)^{-2}}+\cdots\right]\
,$
where for $n{\rm S}_{1/2}$ states Pachucki and Grotch (1995); Eides and Grotch
(1997c); Pachucki and Karshenboim (1999); Melnikov and Yelkhovsky (1999)
$\displaystyle D_{60}$ $\displaystyle=$ $\displaystyle 4\ln 2-{7\over 2}\ ,$
(31) $\displaystyle D_{72}$ $\displaystyle=$ $\displaystyle-{11\over
60\mbox{{p}}}\ ,$ (32)
and for states with $\ell\geq 1$ Golosov _et al._ (1995); Elkhovskiĭ (1996);
Jentschura and Pachucki (1996)
$\displaystyle D_{60}$ $\displaystyle=$
$\displaystyle\left[3-{\ell(\ell+1)\over
n^{2}}\right]{2\over(4\ell^{2}-1)(2\ell+3)}\ .$ (33)
Based on the general pattern of the magnitudes of higher-order coefficients,
the uncertainty for S states is taken to be 10 % of Eq. (30), and for states
with $\ell\geq 1$, it is taken to be 1 %. Numerical values for Eq. (30) to all
orders in $Z\alpha$ have been obtained by Shabaev _et al._ (1998), and
although they disagree somewhat with the analytic result, they are consistent
within the uncertainty assigned here. We employ the analytic equations in the
adjustment. The covariances of the theoretical values are calculated by
assuming that the uncertainties are predominately due to uncalculated terms
proportional to $(m_{\rm e}/m_{\rm N})/n^{3}$.
##### Nuclear polarizability
For hydrogen, we use the result Khriplovich and Sen’kov (2000)
$\displaystyle E_{\rm P}({\rm H})$ $\displaystyle=$
$\displaystyle-0.070(13)h{\delta_{l0}\over n^{3}}\ {\rm kHz}\ .$ (34)
More recent results are a model calculation by Nevado and Pineda (2008) and a
slightly different result than Eq. (34) calculated by Martynenko (2006).
For deuterium, the sum of the proton polarizability, the neutron
polarizibility Khriplovich and Sen’kov (1998), and the dominant nuclear
structure polarizibility Friar and Payne (1997a), gives
$\displaystyle E_{\rm P}({\rm D})$ $\displaystyle=$
$\displaystyle-21.37(8)h{\delta_{l0}\over n^{3}}\ {\rm kHz}\ .$ (35)
Presumably the polarization effect is negligible for states of higher $\ell$
in either hydrogen or deuterium.
##### Self energy
The one-photon self energy of the bound electron is
$\displaystyle E_{\rm SE}^{(2)}={\alpha\over\mbox{{p}}}{(Z\alpha)^{4}\over
n^{3}}F(Z\alpha)\,m_{\rm e}c^{2}\ ,$ (36)
where
$\displaystyle F(Z\alpha)$ $\displaystyle=$ $\displaystyle
A_{41}\ln(Z\alpha)^{-2}+A_{40}+A_{50}\,(Z\alpha)$ (37)
$\displaystyle+A_{62}\,(Z\alpha)^{2}\ln^{2}(Z\alpha)^{-2}+A_{61}\,(Z\alpha)^{2}\ln(Z\alpha)^{-2}$
$\displaystyle+G_{\rm SE}(Z\alpha)\,(Z\alpha)^{2}\ .$
From Erickson and Yennie (1965) and earlier papers cited therein,
$\displaystyle A_{41}$ $\displaystyle=$ $\displaystyle{\textstyle{4\over
3}}\,\delta_{\ell 0}\,,$ $\displaystyle A_{40}$ $\displaystyle=$
$\displaystyle-{\textstyle{4\over 3}}\ln k_{0}(n,\ell)+{\textstyle{10\over
9}}\,\delta_{\ell 0}-{1\over 2\kappa(2\ell+1)}(1-\delta_{\ell 0})\,,$
$\displaystyle A_{50}$ $\displaystyle=$
$\displaystyle\left({\textstyle{139\over 32}}-2\ln
2\right)\mbox{{p}}\,\delta_{\ell 0}\,,$ (38) $\displaystyle A_{62}$
$\displaystyle=$ $\displaystyle-\delta_{\ell 0}\,,$ $\displaystyle A_{61}$
$\displaystyle=$
$\displaystyle\left[4\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)+{28\over
3}\ln{2}-4\ln{n}\right.$ $\displaystyle-$ $\displaystyle\left.{601\over
180}-{77\over 45n^{2}}\right]\delta_{\ell 0}+\left(1-{1\over
n^{2}}\right)\left({2\over 15}+{1\over 3}\delta_{j{1\over
2}}\right)\delta_{\ell 1}\,,$ $\displaystyle+$
$\displaystyle{\left[96n^{2}-32\ell(\ell+1)\right]\left(1-\delta_{\ell
0}\right)\over 3n^{2}(2\ell-1)(2\ell)(2\ell+1)(2\ell+2)(2\ell+3)})\ .$
The Bethe logarithms $\ln k_{0}(n,\ell)$ in Eq. (38) are given in Table 5
Drake and Swainson (1990).
Table 5: Relevant values of the Bethe logarithms $\ln k_{0}(n,l)$. $n$ | S | P | D
---|---|---|---
1 | $2.984\,128\,556$ | |
2 | $2.811\,769\,893$ | $-0.030\,016\,709$ |
3 | $2.767\,663\,612$ | |
4 | $2.749\,811\,840$ | $-0.041\,954\,895$ | $-0.006\,740\,939$
6 | $2.735\,664\,207$ | | $-0.008\,147\,204$
8 | $2.730\,267\,261$ | | $-0.008\,785\,043$
12 | | | $-0.009\,342\,954$
For S and P states with $n\leq 4$, the values we use here for $G_{\rm
SE}(Z\alpha)$ in Eq. (37) are listed in Table 6 and are based on direct
numerical evaluations by Jentschura _et al._ (1999, 2001); Jentschura and Mohr
(2004, 2005). The values of $G_{\rm SE}(\alpha)$ for the 6S and 8S states are
based on the low-$Z$ limit $G_{\rm SE}(0)=A_{60}$ Jentschura _et al._ (2005a)
together with extrapolations of the results of complete numerical calculations
of $F(Z\alpha)$ in Eq. (36) at higher $Z$ Kotochigova and Mohr (2006). A
calculation of the constant $A_{60}$ for various D states, including 12D
states, has been done by Wundt and Jentschura (2008). In CODATA-06 this
constant was obtained by extrapolation from lower-$n$ states. The more recent
calculated values are
$\displaystyle A_{60}(12\mbox{D}_{3/2})$ $\displaystyle=$ $\displaystyle
0.008\,909\,60(5)\,,$ (39) $\displaystyle A_{60}(12\mbox{D}_{5/2})$
$\displaystyle=$ $\displaystyle 0.034\,896\,67(5)\,.$ (40)
To estimate the corresponding value of $G_{\rm SE}(\alpha)$, we use the data
from Jentschura _et al._ (2005b) given in Table 7. It is evident from the
table that
$\displaystyle G_{\rm SE}(\alpha)-A_{60}\approx 0.000\,22$ (41)
for the $n\mbox{D}_{3/2}$ and $n\mbox{D}_{5/2}$ states for $n=4,5,6,7,8$, so
we make the approximation
$\displaystyle G_{\rm SE}(\alpha)=A_{60}+0.000\,22\,,$ (42)
with an uncertainty given by $0.000\,09$ and $0.000\,22$ for the
$12\mbox{D}_{3/2}$ and $12\mbox{D}_{5/2}$ states, respectively. This yields
$\displaystyle G_{\rm SE}(\alpha)$ $\displaystyle=$ $\displaystyle
0.009\,13(9)\qquad\ \,\mbox{ for }12\mbox{D}_{3/2}\,,$ (43) $\displaystyle
G_{\rm SE}(\alpha)$ $\displaystyle=$ $\displaystyle 0.035\,12(22)\qquad\mbox{
for }12\mbox{D}_{5/2}\,.$ (44)
All values for $G_{\rm SE}(\alpha)$ that we use here are listed in Table 6.
The uncertainty of the self energy contribution to a given level arises
entirely from the uncertainty of $G_{\rm SE}(\alpha)$ listed in that table and
is taken to be type $u_{n}$.
Table 6: Values of the function $G_{\rm SE}(\alpha)$. $n$ | S1/2 | P1/2 | P3/2 | D3/2 | D5/2
---|---|---|---|---|---
$1$ | $-30.290\,240(20)$ | | | |
$2$ | $-31.185\,150(90)$ | $-0.973\,50(20)$ | $-0.486\,50(20)$ | |
$3$ | $-31.047\,70(90)$ | | | |
$4$ | $-30.9120(40)$ | $-1.1640(20)$ | $-0.6090(20)$ | | $0.031\,63(22)$
$6$ | $-30.711(47)$ | | | | $0.034\,17(26)$
$8$ | $-30.606(47)$ | | | $0.007\,940(90)$ | $0.034\,84(22)$
$12$ | | | | $0.009\,130(90)$ | $0.035\,12(22)$
Table 7: Data from Jentschura _et al._ (2005b) and the deduced values of $G_{\rm SE}(\alpha)$ for $n=12$. | $A_{60}\ \ $ | $G_{\rm SE}(\alpha)\ \ \ $ | $G_{\rm SE}(\alpha)-A_{60}$
---|---|---|---
$n$ | $\qquad\rm D_{3/2}$ | $\qquad\rm D_{5/2}$ | $\quad\rm D_{3/2}$ | $\quad\rm D_{5/2}$ | $\quad\rm D_{3/2}$ | $\quad\rm D_{5/2}$
$3$ | $0.005\,551\,575(1)$ | $0.027\,609\,989(1)$ | $0.005\,73(15)$ | $0.027\,79(18)$ | $0.000\,18(15)$ | $0.000\,18(18)$
$4$ | $0.005\,585\,985(1)$ | $0.031\,411\,862(1)$ | $0.005\,80(9)$ | $0.031\,63(22)$ | $0.000\,21(9)$ | $0.000\,22(22)$
$5$ | $0.006\,152\,175(1)$ | $0.033\,077\,571(1)$ | $0.006\,37(9)$ | $0.033\,32(25)$ | $0.000\,22(9)$ | $0.000\,24(25)$
$6$ | $0.006\,749\,745(1)$ | $0.033\,908\,493(1)$ | $0.006\,97(9)$ | $0.034\,17(26)$ | $0.000\,22(9)$ | $0.000\,26(26)$
$7$ | $0.007\,277\,403(1)$ | $0.034\,355\,926(1)$ | $0.007\,50(9)$ | $0.034\,57(22)$ | $0.000\,22(9)$ | $0.000\,21(22)$
$8$ | $0.007\,723\,850(1)$ | $0.034\,607\,492(1)$ | $0.007\,94(9)$ | $0.034\,84(22)$ | $0.000\,22(9)$ | $0.000\,23(22)$
$12$ | $0.008\,909\,60(5)$ | $0.034\,896\,67(5)$ | $0.009\,13(9)$ | $0.035\,12(22)$ | $0.000\,22(9)$ | $0.000\,22(22)$
The dominant effect of the finite mass of the nucleus on the self energy
correction is taken into account by multiplying each term of $F(Z\alpha)$ by
the reduced-mass factor $(m_{\rm r}/m_{\rm e})^{3}$, except that the magnetic
moment term $-1/[2\kappa(2\ell+1)]$ in $A_{40}$ is instead multiplied by the
factor $(m_{\rm r}/m_{\rm e})^{2}$. In addition, the argument $(Z\alpha)^{-2}$
of the logarithms is replaced by $(m_{\rm e}/m_{\rm r})(Z\alpha)^{-2}$
Sapirstein and Yennie (1990).
##### Vacuum polarization
The second-order vacuum-polarization level shift is
$\displaystyle E_{\rm VP}^{(2)}={\alpha\over\mbox{{p}}}{(Z\alpha)^{4}\over
n^{3}}H\\!(Z\alpha)\,m_{\rm e}c^{2}\ ,$ (45)
where the function $H\\!(Z\alpha)$ consists of the Uehling potential
contribution $H^{(1)}\\!(Z\alpha)$ and a higher-order remainder $H^{({\rm
R})}\\!(Z\alpha)$:
$\displaystyle H^{(1)}\\!(Z\alpha)$ $\displaystyle=$ $\displaystyle
V_{40}+V_{50}\,(Z\alpha)+V_{61}\,(Z\alpha)^{2}\ln(Z\alpha)^{-2}$ (46)
$\displaystyle+\,G_{\rm VP}^{(1)}(Z\alpha)\,(Z\alpha)^{2}\,,$ $\displaystyle
H^{({\rm R})}\\!(Z\alpha)$ $\displaystyle=$ $\displaystyle G_{\rm VP}^{({\rm
R})}(Z\alpha)\,(Z\alpha)^{2}\ ,$ (47)
with
$\displaystyle V_{40}$ $\displaystyle=$
$\displaystyle-\frac{4}{15}\,\delta_{\ell 0}\,,$ $\displaystyle V_{50}$
$\displaystyle=$ $\displaystyle\frac{5}{48}\mbox{{p}}\,\delta_{\ell 0}\,,$
(48) $\displaystyle V_{61}$ $\displaystyle=$
$\displaystyle-\frac{2}{15}\,\delta_{\ell 0}\,.$
Values of $G_{\rm VP}^{(1)}(Z\alpha)$ are given in Table 8 Mohr (1982);
Kotochigova _et al._ (2002). The Wichmann-Kroll contribution $G_{\rm
VP}^{({\rm R})}(Z\alpha)$ has the leading powers in $Z\alpha$ given by
Wichmann and Kroll (1956); Mohr (1975, 1983)
$\displaystyle G_{\rm VP}^{\rm(R)}(Z\alpha)$ $\displaystyle=$
$\displaystyle\left(\frac{19}{45}-\frac{\mbox{{p}}^{2}}{27}\right)\delta_{\ell
0}$ (49)
$\displaystyle+\left(\frac{1}{16}-\frac{31\mbox{{p}}^{2}}{2880}\right)\mbox{{p}}(Z\alpha)\delta_{\ell
0}+\cdots\ .\qquad$
Higher-order terms are negligible.
Table 8: Values of the function $G_{\rm VP}^{(1)}(\alpha)$. (The minus signs on the zeros in the last two columns indicate that the values are nonzero negative numbers smaller than the digits shown.) $n$ | S1/2 | P1/2 | P3/2 | D3/2 | D5/2
---|---|---|---|---|---
$1$ | $-0.618\,724$ | | | |
$2$ | $-0.808\,872$ | $-0.064\,006$ | $-0.014\,132$ | |
$3$ | $-0.814\,530$ | | | |
$4$ | $-0.806\,579$ | $-0.080\,007$ | $-0.017\,666$ | | $-0.000\,000$
$6$ | $-0.791\,450$ | | | | $-0.000\,000$
$8$ | $-0.781\,197$ | | | $-0.000\,000$ | $-0.000\,000$
$12$ | | | | $-0.000\,000$ | $-0.000\,000$
The finite mass of the nucleus is taken into account by multiplying Eq. (45)
by $(m_{\rm r}/m_{\rm e})^{3}$ and including a factor of $(m_{\rm e}/m_{\rm
r})$ in the argument of the logarithm in Eq. (46).
Vacuum polarization from ${\mbox{{m}}}^{+}{\mbox{{m}}}^{-}$ pairs is Eides and
Shelyuto (1995); Karshenboim (1995)
$\displaystyle E_{{\mbox{\scriptsize{{m}}}}{\rm
VP}}^{(2)}={\alpha\over\mbox{{p}}}{(Z\alpha)^{4}\over
n^{3}}\left(-\frac{4}{15}\,\delta_{\ell 0}\right)\left({m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\right)^{2}\left({m_{\rm r}\over m_{\rm
e}}\right)^{3}m_{\rm e}c^{2}\ ,$ (50)
and the effect of ${\mbox{{t}}}^{+}{\mbox{{t}}}^{-}$ pairs is negligible.
Hadronic vacuum polarization gives Friar _et al._ (1999)
$\displaystyle E_{\rm had\,VP}^{(2)}=0.671(15)E_{{\mbox{\scriptsize{{m}}}}{\rm
VP}}^{(2)}\ ,$ (51)
where the uncertainty is of type $u_{0}$.
The muonic and hadronic vacuum polarization contributions are negligible for
higher-$\ell$ states.
##### Two-photon corrections
The two-photon correction, in powers of $Z\alpha$, is
$\displaystyle E^{(4)}$ $\displaystyle=$
$\displaystyle\left({\alpha\over\mbox{{p}}}\right)^{2}{(Z\alpha)^{4}\over
n^{3}}m_{\rm e}c^{2}F^{(4)}(Z\alpha)\ ,$ (52)
where
$\displaystyle F^{(4)}(Z\alpha)$ $\displaystyle=$ $\displaystyle
B_{40}+B_{50}\,(Z\alpha)+B_{63}\,(Z\alpha)^{2}\ln^{3}(Z\alpha)^{-2}$ (53)
$\displaystyle+B_{62}\,(Z\alpha)^{2}\ln^{2}(Z\alpha)^{-2}$
$\displaystyle+B_{61}\,(Z\alpha)^{2}\ln(Z\alpha)^{-2}+B_{60}\,(Z\alpha)^{2}$
$\displaystyle+\cdots\ .$
The leading term $B_{40}$ is
$\displaystyle B_{40}$ $\displaystyle=$
$\displaystyle\left[\frac{3\mbox{{p}}^{2}}{2}\ln
2-\frac{10\mbox{{p}}^{2}}{27}-\frac{2179}{648}-\frac{9}{4}\zeta(3)\right]\delta_{\ell
0}$ $\displaystyle+\left[\frac{\mbox{{p}}^{2}\ln
2}{2}-\frac{\mbox{{p}}^{2}}{12}-\frac{197}{144}-\frac{3\zeta(3)}{4}\right]{1-\delta_{\ell
0}\over\kappa(2\ell+1)}\ ,$
where $\zeta$ is the Riemann zeta function Olver _et al._ (2010), and the next
term is Pachucki (1993a); Eides _et al._ (1997); Eides and Shelyuto (1995);
Pachucki (1994); Dowling _et al._ (2010)
$\displaystyle B_{50}$ $\displaystyle=$
$\displaystyle-21.554\,47(13)\delta_{\ell 0}\ .$ (55)
The leading sixth-order coefficient is Karshenboĭm (1993); Yerokhin (2000);
Manohar and Stewart (2000); Pachucki (2001)
$\displaystyle B_{63}=-{8\over 27}\delta_{\ell 0}\ .$ (56)
For S states $B_{62}$ is Karshenboim (1996); Pachucki (2001)
$\displaystyle B_{62}$ $\displaystyle=$ $\displaystyle{16\over
9}\left[{71\over 60}-\ln{2}+\mbox{{g}}+\psi(n)-\ln n-{1\over n}+{1\over
4n^{2}}\right]\ ,$
where $\mbox{{g}}=0.577...$ is Euler’s constant and $\psi$ is the psi function
Olver _et al._ (2010). For P states Karshenboim (1996); Jentschura and Nándori
(2002)
$\displaystyle B_{62}={4\over 27}{n^{2}-1\over n^{2}}\ ,$ (58)
and $B_{62}=0$ for $\ell\geq 2$.
For S states $B_{61}$ is Pachucki (2001); Jentschura _et al._ (2005a)
$\displaystyle B_{61}$ $\displaystyle=$ $\displaystyle{413\,581\over
64\,800}+{4N(n{\rm S})\over 3}+{2027\mbox{{p}}^{2}\over 864}-{616\,\ln{2}\over
135}$ (59) $\displaystyle-{2\mbox{{p}}^{2}\ln{2}\over 3}+{40\ln^{2}{2}\over
9}+\zeta(3)+\left(\frac{304}{135}-\frac{32\,\ln{2}}{9}\right)$
$\displaystyle\times\left[{3\over 4}+\mbox{{g}}+\psi(n)-\ln n-{1\over
n}+{1\over 4n^{2}}\right]\ .$
For P states Jentschura _et al._ (2005a); Jentschura (2003)
$\displaystyle B_{61}(n{\rm P}_{1/2})$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,N(n{\rm
P})+\frac{n^{2}-1}{n^{2}}\left(\frac{166}{405}-\frac{8}{27}\,\ln{2}\right),\qquad$
(60) $\displaystyle B_{61}(n{\rm P}_{3/2})$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,N(n{\rm
P})+\frac{n^{2}-1}{n^{2}}\left(\frac{31}{405}-\frac{8}{27}\,\ln{2}\right),\qquad$
(61)
and $B_{61}=0$ for $\ell\geq 2$. Values for $B_{61}$ used in the adjustment
are listed in Table 9
Table 9: Values of $B_{61}$ used in the 2010 adjustment.
$n$ | $B_{61}(n$S${}_{1/2})$ | ${B}_{61}(n$P${}_{1/2})$ | ${B}_{61}(n$P${}_{3/2})$ | ${B}_{61}(n$D${}_{3/2})$ | ${B}_{61}(n$D${}_{5/2})$
---|---|---|---|---|---
1 | $48.958\,590\,24(1)$ | | | |
2 | $41.062\,164\,31(1)$ | $0.004\,400\,847(1)$ | $0.004\,400\,847(1)$ | |
3 | $38.904\,222(1)$ | | | |
4 | $37.909\,514(1)$ | $-0.000\,525\,776(1)$ | $-0.000\,525\,776(1)$ | | $0.0(0)$
6 | $36.963\,391(1)$ | | | | $0.0(0)$
8 | $36.504\,940(1)$ | | | $0.0(0)$ | $0.0(0)$
12 | | | | $0.0(0)$ | $0.0(0)$
For the 1S state, the result of a perturbation theory estimate for the term
$B_{60}$ is Pachucki (2001); Pachucki and Jentschura (2003)
$\displaystyle B_{60}(1{\rm S})=-61.6(9.2)\,.$ (62)
All-order numerical calculations of the two-photon correction have also been
carried out. The diagrams with closed electron loops have been evaluated by
Yerokhin _et al._ (2008). They obtained results for the 1S, 2S, and 2P states
at $Z=1$ and higher $Z$, and obtained a value for the contribution of the
terms of order $(Z\alpha)^{6}$ and higher. The remaining contributions to
$B_{60}$ are from the self-energy diagrams. These have been evaluated by
Yerokhin _et al._ (2005c, b); Yerokhin _et al._ (2003, 2007) for the 1S state
for $Z=10$ and higher $Z$, and more recently, Yerokhin (2010) has done an all-
order calculation of the 1S-state no-electron-loop two-loop self energy
correction for $Z\geq 10$. His extrapolation of the higher-$Z$ values to
obtain a value for $Z=1$ yields a contribution to $B_{60}$, including higher-
order terms, given by $-86(15)$. This result combined with the result for the
electron-loop two-photon diagrams, reported by Yerokhin _et al._ (2008), gives
a total of $B_{60}+\dots=-101(15)$, where the dots represent the contribution
of the higher-order terms. This may be compared to the earlier evaluation
which gave $-127(39)$ Yerokhin _et al._ (2005c, b); Yerokhin _et al._ (2003,
2007). The new value also differs somewhat from the result in Eq. (62). In
view of this difference between the two calculations, to estimate $B_{60}$ for
the 2010 adjustment, we use the average of the analytic value of $B_{60}$ and
the numerical result for $B_{60}$ with higher-order terms included, with an
uncertainty that is half the difference. The higher-order contribution is
small compared to the difference between the results of the two methods of
calculation. The average result is
$\displaystyle B_{60}(1{\rm S})=-81.3(0.3)(19.7)\ .$ (63)
In Eq. (63), the first number in parentheses is the state-dependent
uncertainty $u_{n}(B_{60})$ associated with the two-loop Bethe logarithm, and
the second number in parentheses is the state-independent uncertainty
$u_{0}(B_{60})$ that is common to all S-state values of $B_{60}$. Two-loop
Bethe logarithms needed to evaluate $B_{60}(n{\rm S})$ have been given for
$n=1$ to 6 Pachucki and Jentschura (2003); Jentschura (2004), and a value at
$n=8$ may be obtained by a simple extrapolation from the calculated values
[see Eq. (43) of CODATA-06]. The complete state dependence of $B_{60}(nS)$ in
terms of the two-loop Bethe logarithms has been calculated by Czarnecki _et
al._ (2005); Jentschura _et al._ (2005a). Values of $B_{60}$ for all relevant
S-states are given in Table 10.
Table 10: Values of $B_{60}$, $\overline{B}_{60}$, or $\Delta B_{71}$ used in
the 2010 adjustment
$n$ | $B_{60}(n$S${}_{1/2})$ | $\overline{B}_{60}(n$P${}_{1/2})$ | $\overline{B}_{60}(n$P${}_{3/2})$ | $\overline{B}_{60}(n$D${}_{3/2})$ | $\overline{B}_{60}(n$D${}_{5/2})$ | $\Delta B_{71}(n$S${}_{1/2})$
---|---|---|---|---|---|---
1 | $-81.3(0.3)(19.7)$ | | | | |
2 | $-66.2(0.3)(19.7)$ | $-1.6(3)$ | $-1.7(3)$ | | | $16(8)$
3 | $-63.0(0.6)(19.7)$ | | | | | $22(11)$
4 | $-61.3(0.8)(19.7)$ | $-2.1(3)$ | $-2.2(3)$ | | $-0.005(2)$ | $25(12)$
6 | $-59.3(0.8)(19.7)$ | | | | $-0.008(4)$ | $28(14)$
8 | $-58.3(2.0)(19.7)$ | | | $0.015(5)$ | $-0.009(5)$ | $29(15)$
12 | | | | $0.014(7)$ | $-0.010(7)$ |
For higher-$\ell$ states, an additional consideration is necessary. The
radiative level shift includes contributions associated with decay to lower
levels. At the one-loop level, this is the imaginary part of the level shift
corresponding to the resonance scattering width of the level. At the two-loop
level there is an imaginary contribution corresponding to two-photon decays
and radiative corrections to the one-photon decays, but in addition there is a
real contribution from the square of one-photon decay width. This can be
thought of as the second-order term that arises in the expansion of the
resonance denominator for scattering of photons from the atom in its ground
state in powers of the level width Jentschura _et al._ (2002). As such, this
term should not be included in the calculation of the resonant line center
shift of the scattering cross section, which is the quantity of interest for
the least-squares adjustment. The leading contribution of the square of the
one-photon width is of order $\alpha(Z\alpha)^{6}m_{\rm e}c^{2}/\hbar$. This
correction vanishes for the 1S and 2S states, because the 1S level has no
width and the 2S level can only decay with transition rates that are higher
order in $\alpha$ and/or $Z\alpha$. The higher-$n$ S states have a
contribution from the square of the one-photon width from decays to lower P
states, but for the 3S and 4S states for which it has been separately
identified, this correction is negligible compared to the uncertainty in
$B_{60}$ Jentschura (2004, 2006). We assume the correction for higher S states
is also negligible compared to the numerical uncertainty in $B_{60}$. However,
the correction is taken into account in the 2010 adjustment for P and D states
for which it is relatively larger Jentschura _et al._ (2002); Jentschura
(2006).
Calculations of $B_{60}$ for higher-$\ell$ states have been made by Jentschura
(2006). The results can be expressed as
$\displaystyle B_{60}(nL_{j})$ $\displaystyle=$ $\displaystyle
a(nL_{j})+b_{\rm L}(nL)\,,$ (64)
where $a(nL_{j})$ is a precisely calculated term that depends on $j$, and the
two-loop Bethe logarithm $b_{\rm L}(nL)$ has a larger numerical uncertainty
but does not depend on $j$. Jentschura (2006) gives semianalytic formulas for
$a(nL_{j})$ that include numerically calculated terms. The information needed
for the 2010 adjustment is in Eqs. (22a), (22b), (23a), (23b), Tables VII,
VIII, XI, and X of Jentschura (2006) and Eq. (17) of Jentschura (2003). Two
corrections to Eq. (22b) are
$\displaystyle-\frac{73321}{103680}+\frac{185}{1152n}+\frac{8111}{25920n^{2}}$
$\displaystyle\qquad\qquad\rightarrow-\frac{14405}{20736}+\frac{185}{1152n}+\frac{1579}{5184n^{2}}\qquad\qquad$
(65)
on the first line and
$\displaystyle-\frac{3187}{3600n^{2}}\rightarrow+\frac{3187}{3600n^{2}}$ (66)
on the fourth line Jentschura (2011a).
Values of the two-photon Bethe logarithm $b_{\rm L}(nL)$ may be divided into a
contribution of the “squared level width” term $\delta^{2}B_{60}$ and the rest
$\overline{b}_{\rm L}(nL)$, so that
$\displaystyle b_{\rm L}(nL)$ $\displaystyle=$
$\displaystyle\delta^{2}B_{60}+\overline{b}_{\rm L}(nL)\,.$ (67)
The corresponding value $\overline{B}_{60}$ that represents the shift of the
level center is given by
$\displaystyle\overline{B}_{60}(nL_{j})$ $\displaystyle=$ $\displaystyle
a(nL_{j})+\overline{b}_{\rm L}(nL)\,.$ (68)
Here we give the numerical values for $\overline{B}(nL_{j})$ in Table 10 and
refer the reader to Jentschura (2006) for the separate values for $a(nL_{j})$
and $\overline{b}_{\rm L}(nL)$. The D-state values for $n=6,8$ are
extrapolated from the corresponding values at $n=5,6$ with a function of the
form $a+b/n$. The values in Table 10 for S states may be regarded as being
either $B_{60}$ or $\overline{B}_{60}$, since the difference is expected to be
smaller than the uncertainty. The uncertainties listed for the P- and D-state
values of $\overline{B}(nL_{j})$ in that table are predominately from the two-
photon Bethe logarithm which depends on $n$ and $L$, but not on $j$ for a
given $n,L$. Therefore there is a large covariance between the corresponding
two values of $\overline{B}(nL_{j})$. However, we do not take this into
consideration when calculating the uncertainty in the fine structure
splitting, because the uncertainty of higher-order coefficients dominates over
any improvement in accuracy the covariance would provide. We assume that the
uncertainties in the two-photon Bethe logarithms are sufficiently large to
account for higher-order P and D state two-photon uncertainties as well.
For S states, higher-order terms have been estimated by Jentschura _et al._
(2005a) with an effective potential model. They find that the next term has a
coefficient of $B_{72}$ and is state independent. We thus assume that the
uncertainty $u_{0}[B_{60}(n{\rm S})]$ is sufficient to account for the
uncertainty due to omitting such a term and higher-order state-independent
terms. In addition, they find an estimate for the state dependence of the next
term, given by
$\displaystyle\Delta B_{71}(n{\rm S})$ $\displaystyle=$ $\displaystyle
B_{71}(n{\rm S})-B_{71}(1{\rm
S})=\mbox{{p}}\left(\frac{427}{36}-\frac{16}{3}\,\ln{2}\right)$ (69)
$\displaystyle\times$
$\displaystyle\left[\frac{3}{4}-\frac{1}{n}+\frac{1}{4n^{2}}+\mbox{{g}}+\psi(n)-\ln{n}\right],$
with a relative uncertainty of 50 %. We include this additional term, which is
listed in Table 10, along with the estimated uncertainty
$u_{n}(B_{71})=B_{71}/2$.
##### Three-photon corrections
The three-photon contribution in powers of $Z\alpha$ is
$\displaystyle E^{(6)}$ $\displaystyle=$
$\displaystyle\left({\alpha\over\mbox{{p}}}\right)^{3}{(Z\alpha)^{4}\over
n^{3}}m_{\rm e}c^{2}\left[C_{40}+C_{50}(Z\alpha)+\cdots\right]\ .$
The leading term $C_{40}$ is Melnikov and van Ritbergen (2000); Laporta and
Remiddi (1996); Baikov and Broadhurst (1995); Eides and Grotch (1995a)
$\displaystyle C_{40}$ $\displaystyle=$ $\displaystyle\bigg{[}-{{568\,{\rm
a_{4}}}\over{9}}+{{85\,\zeta(5)}\over{24}}$
$\displaystyle-{{121\,\mbox{{p}}^{2}\,\zeta(3)}\over{72}}-{{84\,071\,\zeta(3)}\over{2304}}-{{71\,\ln^{4}2}\over{27}}$
$\displaystyle-{{239\,\mbox{{p}}^{2}\,\ln^{2}2}\over{135}}+{{4787\,\mbox{{p}}^{2}\,\ln
2}\over{108}}+{{1591\,\mbox{{p}}^{4}}\over{3240}}$
$\displaystyle-{{252\,251\,\mbox{{p}}^{2}}\over{9720}}+{679\,441\over
93\,312}\bigg{]}\delta_{\ell 0}$ $\displaystyle+\bigg{[}-{{100\,{\rm
a_{4}}}\over{3}}+{{215\,\zeta(5)}\over{24}}$
$\displaystyle-{{83\,\mbox{{p}}^{2}\,\zeta(3)}\over{72}}-{{139\,\zeta(3)}\over{18}}-{{25\,\ln^{4}2}\over{18}}$
$\displaystyle+{{25\,\mbox{{p}}^{2}\,\ln^{2}2}\over{18}}+{{298\,\mbox{{p}}^{2}\,\ln
2}\over{9}}+{{239\,\mbox{{p}}^{4}}\over{2160}}$
$\displaystyle-{{17\,101\,\mbox{{p}}^{2}}\over{810}}-{28\,259\over
5184}\bigg{]}{1-\delta_{\ell 0}\over\kappa(2\ell+1)}\ ,$
where $a_{4}=\sum_{n=1}^{\infty}1/(2^{n}\,n^{4})=0.517\,479\,061\dots$ .
Partial results for $C_{50}$ have been calculated by Eides and Shelyuto (2004,
2007). The uncertainty is taken to be $u_{0}(C_{50})=30\delta_{\ell 0}$ and
$u_{n}(C_{63})=1$, where $C_{63}$ would be the coefficient of
$(Z\alpha)^{2}\ln^{3}{(Z\alpha)^{-2}}$ in the square brackets in Eq.
(LABEL:eq:total6). The dominant effect of the finite mass of the nucleus is
taken into account by multiplying the term proportional to $\delta_{\ell 0}$
by the reduced-mass factor $(m_{\rm r}/m_{\rm e})^{3}$ and the term
proportional to $1/[\kappa(2\ell+1)]$, the magnetic moment term, by the factor
$(m_{\rm r}/m_{\rm e})^{2}$.
The contribution from four photons would be of order
$\displaystyle\left({\alpha\over\mbox{{p}}}\right)^{4}{(Z\alpha)^{4}\over
n^{3}}m_{\rm e}c^{2}\ ,$ (72)
which is about 10 Hz for the 1S state and is negligible at the level of
uncertainty of current interest.
##### Finite nuclear size
In the nonrelativistic limit, the level shift due to the finite size of the
nucleus is
$\displaystyle E^{(0)}_{\rm NS}={\cal E}_{\rm NS}\delta_{\ell 0}\ ,$ (73)
where
$\displaystyle{\cal E}_{\rm NS}={2\over 3}\left({m_{\rm r}\over m_{\rm
e}}\right)^{3}{(Z\alpha)^{2}\over n^{3}}\ m_{\rm e}c^{2}\left({Z\alpha r_{\rm
N}\over\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm
C}}\right)^{2}\ ,$ (74)
$r_{\rm N}$ is the bound-state root-mean-square (rms) charge radius of the
nucleus, and $\lambda\hskip-4.5pt\vrule
height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ is the Compton wavelength of
the electron divided by $2\mbox{{p}}$.
Higher-order contributions have been examined by Friar (1979b); Friar and
Payne (1997b); Karshenboim (1997) [see also Borisoglebsky and Trofimenko
(1979); Mohr (1983)]. For S states the leading and next-order corrections are
given by
$\displaystyle E_{\rm NS}$ $\displaystyle=$ $\displaystyle{\cal E}_{\rm
NS}\Bigg{\\{}1-C_{\eta}{m_{\rm r}\over m_{\rm e}}{r_{\rm
N}\over\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm
C}}Z\alpha-\bigg{[}\ln{\left({m_{\rm r}\over m_{\rm e}}{r_{\rm
N}\over\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm
C}}{Z\alpha\over n}\right)}$
$\displaystyle+\psi(n)+\mbox{{g}}-{(5n+9)(n-1)\over
4n^{2}}-C_{\theta}\bigg{]}(Z\alpha)^{2}\Bigg{\\}}\ ,$
where $C_{\eta}$ and $C_{\theta}$ are constants that depend on the charge
distribution in the nucleus with values $C_{\eta}=1.7(1)$ and
$C_{\theta}=0.47(4)$ for hydrogen or $C_{\eta}=2.0(1)$ and
$C_{\theta}=0.38(4)$ for deuterium.
For the P1/2 states in hydrogen the leading term is
$\displaystyle E_{\rm NS}={\cal E}_{\rm NS}{(Z\alpha)^{2}(n^{2}-1)\over
4n^{2}}\ .$ (76)
For P3/2 states and higher-$\ell$ states the nuclear-size contribution is
negligible.
As mentioned in Sec. IV.1.1, in the 2010 adjustment, we do not use an
effective radius for the deuteron, but rather simply $r_{\rm d}$ which is
defined by Eq. (74). In CODATA-02, and CODATA-06, the adjustment code used
$r_{\rm d}$ as an adjusted variable and that value was reported for the rms
radius, rather than the value for $R_{\rm d}$ defined by Eq. (56) of
CODATA-98, which differs from $r_{\rm d}$ by less than 0.1 %.
##### Nuclear-size correction to self energy and vacuum polarization
There is a correction from the finite size of the nucleus to the self energy
Pachucki (1993b); Eides and Grotch (1997b); Milstein _et al._ (2002); Milstein
_et al._ (2003a),
$\displaystyle E_{\rm
NSE}=\left(4\ln{2}-\frac{23}{4}\right)\alpha(Z\alpha){\cal E}_{\rm
NS}\delta_{\ell 0}\,,$ (77)
and to the vacuum polarization Friar (1979a); Hylton (1985); Eides and Grotch
(1997b),
$\displaystyle E_{\rm NVP}={3\over 4}\alpha(Z\alpha){\cal E}_{\rm
NS}\delta_{\ell 0}\,.$ (78)
For the self-energy, higher-order size corrections have been calculated for S
states by Milstein _et al._ (2002) and for P states by Jentschura (2003);
Milstein _et al._ (2003b); Milstein _et al._ (2004). Yerokhin (2011)
calculated the finite nuclear size corrections to the self energy and vacuum
polarization nonperturbatively in $Z\alpha$ and has extrapolated the values
for the 1S state to $Z=1$. The results are consistent with the higher-order
analytic results. Pachucki in a private communication quoted by Yerokhin
(2011) notes that the coefficients of the leading log terms are the same for
the nuclear size correction to the self energy as they are for the self-energy
correction to the hyperfine splitting. The latter terms have been calculated
by Jentschura and Yerokhin (2010). However, these higher-order terms are
negligible at the level of accuracy under consideration. Corrections for
higher-$\ell$ states are also expected to be negligible.
##### Radiative-recoil corrections
Corrections to the self energy and vacuum polarization for the finite mass of
the nucleus, beyond the reduced-mass corrections already included, are
radiative-recoil effects given by Eides _et al._ (2001a); Czarnecki and
Melnikov (2001); Eides and Grotch (1995b); Pachucki (1995); Pachucki and
Karshenboim (1999); Melnikov and Yelkhovsky (1999):
$\displaystyle E_{\rm RR}$ $\displaystyle=$ $\displaystyle{m_{\rm r}^{3}\over
m_{\rm e}^{2}m_{\rm N}}{\alpha(Z\alpha)^{5}\over\mbox{{p}}^{2}\,n^{3}}m_{\rm
e}c^{2}\delta_{\ell 0}$ (79)
$\displaystyle\times\bigg{[}6\,\zeta(3)-2\,\mbox{{p}}^{2}\ln{2}+{35\,\mbox{{p}}^{2}\over
36}-{448\over 27}$ $\displaystyle\qquad+{2\over
3}\mbox{{p}}(Z\alpha)\,\ln^{2}{(Z\alpha)^{-2}}+\cdots\bigg{]}\ .\qquad$
The uncertainty is taken to be the term $(Z\alpha)\ln(Z\alpha)^{-2}$ relative
to the square brackets with numerical coefficients 10 for $u_{0}$ and 1 for
$u_{n}$. Corrections for higher-$\ell$ states are expected to be negligible.
##### Nucleus self energy
A correction due to the self energy of the nucleus is Pachucki (1995); Eides
_et al._ (2001b)
$\displaystyle E_{\rm SEN}$ $\displaystyle=$
$\displaystyle{4Z^{2}\alpha(Z\alpha)^{4}\over 3\mbox{{p}}n^{3}}{m_{\rm
r}^{3}\over m_{\rm N}^{2}}c^{2}$ (80)
$\displaystyle\times\left[\ln{\left({m_{\rm N}\over m_{\rm
r}(Z\alpha)^{2}}\right)}\delta_{\ell 0}-\ln k_{0}(n,\ell)\right]\,.\qquad$
For the uncertainty, we assign a value to $u_{0}$ corresponding to an additive
constant of 0.5 in the square brackets in Eq. (80) for S states. For
higher-$\ell$ states, the correction is not included.
##### Total energy and uncertainty
The energy $E_{X}(n{\rm L}_{j})$ of a level (where L = S, P, … and $X$ = H, D)
is the sum of the various contributions listed in the preceding sections plus
an additive correction $\delta_{X}(n{\rm L}_{j})$ that is zero with an
uncertainty that is the rms sum of the uncertainties of the individual
contributions
$\displaystyle u^{2}[\delta_{X}(n{\rm L}_{j})]=\sum_{i}{u_{0i}^{2}(X{\rm
L}_{j})+u_{ni}^{2}(X{\rm L}_{j})\over n^{6}}\ ,$ (81)
where $u_{0i}(X{\rm L}_{j})/n^{3}$ and $u_{ni}(X{\rm L}_{j})/n^{3}$ are the
components of uncertainty $u_{0}$ and $u_{n}$ of contribution $i$.
Uncertainties from the fundamental constants are not explicitly included here,
because they are taken into account through the least-squares adjustment.
The covariance of any two $\delta$’s follows from Eq. (F7) of Appendix F of
CODATA-98. For a given isotope
$\displaystyle u\left[\delta_{X}(n_{1}{\rm L}_{j}),\delta_{X}(n_{2}{\rm
L}_{j})\right]=\sum_{i}{u^{2}_{0i}(X{\rm L}_{j})\over(n_{1}n_{2})^{3}}\ ,$
(82)
which follows from the fact that $u(u_{0i},u_{ni})=0$ and
$u(u_{n_{1}i},u_{n_{2}i})=0$ for $n_{1}\neq n_{2}$. We also assume that
$\displaystyle u\left[\delta_{X}(n_{1}{\rm
L_{1}}_{j_{1}}),\delta_{X}(n_{2}{\rm L_{2}}_{j_{2}})\right]=0\ ,$ (83)
if ${\rm L}_{1}\neq{\rm L}_{2}$ or $j_{1}\neq j_{2}$.
For covariances between $\delta$’s for hydrogen and deuterium, we have for
states of the same $n$
$\displaystyle u\left[\delta_{\rm H}(n{\rm L}_{j}),\delta_{\rm D}(n{\rm
L}_{j})\right]$ $\displaystyle=\sum_{i=\\{i_{\rm c}\\}}{u_{0i}({\rm
HL}_{j})u_{0i}({\rm DL}_{j})+u_{ni}({\rm HL}_{j})u_{ni}({\rm DL}_{j})\over
n^{6}}\,,\qquad\quad$ (84)
and for $n_{1}\neq n_{2}$
$\displaystyle u\left[\delta_{\rm H}(n_{1}{\rm L}_{j}),\delta_{\rm
D}(n_{2}{\rm L}_{j})\right]=\sum_{i=i_{\rm c}}{u_{0i}({\rm HL}_{j})u_{0i}({\rm
DL}_{j})\over(n_{1}n_{2})^{3}}\,,$ (85)
where the summation is over the uncertainties common to hydrogen and
deuterium. We assume
$\displaystyle u\left[\delta_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}}),\delta_{\rm
D}(n_{2}{\rm L_{2}}_{j_{2}})\right]=0\,,$ (86)
if ${\rm L}_{1}\neq{\rm L}_{2}$ or $j_{1}\neq j_{2}$.
The values of $u\left[\delta_{X}(n{\rm L}_{j})\right]$ of interest for the
2010 adjustment are given in Table 18 of Sec. XIII, and the non negligible
covariances of the $\delta$’s are given as correlation coefficients in Table
19 of that section. These coefficients are as large as 0.9999.
##### Transition frequencies between levels with $n=2$ and the fine-structure
constant $\alpha$
To test the QED predictions, we calculate the values of the transition
frequencies between levels with $n=2$ in hydrogen. This is done by running the
least-squares adjustment with the hydrogen and deuterium spectroscopic data
included, but excluding experimental values for the transitions being
calculated (items $A39$, $A40.1$, and $A40.2$ in Table 18). The necessary
constants $A_{\rm r}$(e), $A_{\rm r}$(p), $A_{\rm r}$(d), and $\alpha$, are
assigned their 2010 adjusted values. The results are
$\displaystyle\nu_{\rm H}(2{\rm P}_{1/2}-2{\rm S}_{1/2})$ $\displaystyle=$
$\displaystyle 1\,057\,844.4(1.8)\ {\rm kHz}\ [1.7\times 10^{-6}],$
$\displaystyle\nu_{\rm H}(2{\rm S}_{1/2}-2{\rm P}_{3/2})$ $\displaystyle=$
$\displaystyle 9\,911\,197.1(1.8)\ {\rm kHz}\ [1.8\times 10^{-7}],$
$\displaystyle\nu_{\rm H}(2{\rm P}_{1/2}-2{\rm P}_{3/2})$ (87)
$\displaystyle\hbox to-65.0pt{}=10\,969\,041.571(41)\ {\rm kHz}\ [3.7\times
10^{-9}],$
which are consistent with the relevant experimental results given in Table 18.
There is more than a factor of two reduction in uncertainty in the first two
frequencies compared to the corresponding 2006 theoretical values.
We obtain a value for the fine-structure constant $\alpha$ from the data on
the hydrogen and deuterium transitions. This is done by running a variation of
the 2010 least-squares adjustment that includes all the transition frequency
data in Table 18 and the 2010 adjusted values of $A_{\rm r}$(e), $A_{\rm
r}$(p), and $A_{\rm r}$(d). This yields
$\displaystyle\alpha^{-1}$ $\displaystyle=$ $\displaystyle
137.036\,003(41)\qquad[3.0\times 10^{-7}]\ ,$ (88)
which is in excellent agreement with, but substantially less accurate than,
the 2010 recommended value, and is included in Table 25.
##### Isotope shift and the deuteron-proton radius difference
A new experimental result for the hydrogen-deuterium isotope shift is included
in Table 11 Parthey _et al._ (2010); Jentschura _et al._ (2011a). In
Jentschura _et al._ (2011a) there is a discussion of the theory of the isotope
shift, with the objective of extracting the difference of the squares of the
charge radii for the deuteron and proton. The analysis in Jentschura _et al._
(2011a) is in general agreement with the review given in the preceding
sections of the present work, with a few differences in the estimates of
uncertainties.
As pointed out by Jentschura _et al._ (2011a), the isotope shift is roughly
given by
$\displaystyle\Delta f_{\rm 1S-2S,d}-\Delta f_{\rm 1S-2S,p}$
$\displaystyle\approx$
$\displaystyle-\frac{3}{4}\,R_{\infty}c\left(\frac{m_{\rm e}}{m_{\rm
d}}-\frac{m_{\rm e}}{m_{\rm p}}\right)$ (89) $\displaystyle=$
$\displaystyle\frac{3}{4}\,R_{\infty}c\,\frac{m_{\rm e}\left({m_{\rm
d}}-{m_{\rm p}}\right)}{{m_{\rm d}}{m_{\rm p}}}\,,\qquad$
and from a comparison of experiment and theory, they obtain
$\displaystyle r_{\rm d}^{2}-r_{\rm p}^{2}$ $\displaystyle=$ $\displaystyle
3.820\,07(65)\mbox{ fm}^{2}$ (90)
for the difference of the squares of the radii. This can be compared to the
result of the 2010 adjustment given by
$\displaystyle r_{\rm d}^{2}-r_{\rm p}^{2}$ $\displaystyle=$ $\displaystyle
3.819\,89(42)\mbox{ fm}^{2}\,,$ (91)
which is in good agreement. (The difference of the squares of the quoted 2010
recommended values of the radii gives 87 in the last two digits of the
difference, rather than 89, due to rounding.) The uncertainty follows from
Eqs. (F11) and (F12) of CODATA-98. Here there is a significant reduction in
the uncertainty compared to the uncertainties of the individual radii because
of the large correlation coefficient (physics.nist.gov/constants)
$\displaystyle r(r_{\rm d},r_{\rm p})=0.9989\,.$ (92)
Part of the reduction in uncertainty in Eq. (91) compared to Eq. (90) is due
to the fact that the correlation coefficient takes into account the covariance
of the electron-nucleon mass ratios in Eq. (89).
#### IV.1.2 Experiments on hydrogen and deuterium
The hydrogen and deuterium transition frequencies used in the 2010 adjustment
for the determination of the Rydberg constant $R_{\infty}$ are given in Table
11. These are items $A26$ to $A48$ in Table 18, Sec. XIII. There are only
three differences between Table 11 and its counterpart, Table XII, in
CODATA-06.
First, the last two digits of the $1{\rm S}_{1/2}-2{\rm S}_{1/2}$ transition
frequency obtained by the group at the Max-Planck-Institute für Quantenoptik
(MPQ), Garching, Germany have changed from $74$ to $80$ as a result of the
group’s improved measurement of the $2{\rm S}$ hydrogen hyperfine splitting
frequency (HFS). Their result is Kolachevsky _et al._ (2009)
$\displaystyle\nu_{\rm HFS}({\rm H};2\rm S)$ $\displaystyle=$ $\displaystyle
177\,556\,834.3(6.7)\mbox{ Hz}\quad[3.8\times 10^{-8}].$ (93)
The reduction in the uncertainty of their previous value for this frequency
Kolachevsky _et al._ (2004) by a factor of $2.4$ was mainly due to the use of
a new ultra stable optical reference Alnis _et al._ (2008) and a reanalysis of
the shift with pressure of the $2{\rm S}$ HFS frequency that showed it was
negligible in their apparatus. The $2{\rm S}$ HFS enters the determination of
the $1{\rm S}_{1/2}-2{\rm S}_{1/2}$ transition frequency because the
transition actually measured is $(1{\rm S},F=1,m_{F}={\pm 1})\rightarrow(2{\rm
S},F^{\prime}=1,m_{F}^{\prime}={\pm 1})$ and the well known $1{\rm S}$ HFS
Ramsey (1990) and the $2{\rm S}$ HFS are required to convert the measured
frequency to the frequency of the hyperfine centroid.
For completeness, we note that the MPQ group has very recently reported a new
value for the $1{\rm S}_{1/2}-2{\rm S}_{1/2}$ transition frequency that has an
uncertainty of 10 Hz, corresponding to a relative standard uncertainty of
$4.2\times 10^{-15}$, or about 30 % of the uncertainty of the value in the
table Parthey _et al._ (2011).
Second, the previous MPQ value Huber _et al._ (1998) for the hydrogen-
deuterium $1{\rm S}-2{\rm S}$ isotope shift, that is, the frequency difference
$\nu_{\rm D}(1{\rm S}_{1/2}-2{\rm S}_{1/2})-\nu_{\rm H}(1{\rm S}_{1/2}-2{\rm
S}_{1/2})$, has been replaced by their recent, much more accurate value
Parthey _et al._ (2010); its uncertainty of $15~{}{\rm Hz}$, corresponding to
a relative uncertainty of $2.2\times 10^{-11}$, is a factor of 10 smaller than
the uncertainty of their previous result. Many experimental advances enabled
this significant uncertainty reduction, not the least of which was the use of
a fiber frequency comb referenced to an active hydrogen maser steered by the
Global Positioning System (GPS) to measure laser frequencies. The principal
uncertainty components in the measurement are $11~{}{\rm Hz}$ due to density
effects in the atomic beam, $6~{}{\rm Hz}$ from second-order Doppler shift,
and $5.1~{}{\rm Hz}$ statistical.
Third, Table 11 includes a new result from the group at the Laboratoire
Kastler-Brossel (LKB), École Normale Supérieure et Université Pierre et Marie
Curie, Paris, France. These researchers have extended their previous work and
determined the $1{\rm S}_{1/2}-3{\rm S}_{1/2}$ transition frequency in
hydrogen using Doppler-free two-photon spectroscopy with a relative
uncertainty of $4.4\times 10^{-12}$ Arnoult _et al._ (2010), the second
smallest uncertainty for a hydrogen or deuterium optical transition frequency
ever obtained. The transition occurs at a wavelength of $205~{}{\rm nm}$, and
light at this wavelength was obtained by twice doubling the frequency of light
emitted by a titanium-sapphire laser of wavelength $820~{}{\rm nm}$ whose
frequency was measured using an optical frequency comb.
A significant problem in the experiment was the second-order Doppler effect
due to the velocity $v$ of the $1{\rm S}$ atomic beam which causes an apparent
shift of the transition frequency. The velocity was measured by having the
beam pass through a transverse magnetic field, thereby inducing a motional
electric field and hence a quadratic Stark shift that varies as $v^{2}$. The
variation of this Stark shift with field was used to determine $v$ and thus
the correction for the second-order Doppler effect. The dominant $12.0~{}{\rm
kHz}$ uncertainty component in the LKB experiment is statistical,
corresponding to a relative uncertainty of $4.1\times 10^{-12}$; the remaining
components together contribute an additional uncertainty of only $4.8~{}{\rm
kHz}$.
As discussed in CODATA-98, some of the transition frequencies measured in the
same laboratory are correlated. Table 19, Sec XIII, gives the relevant
correlation coefficients.
Table 11: Summary of measured transition frequencies $\nu$ considered in the
present work for the determination of the Rydberg constant $R_{\infty}$ (H is
hydrogen and D is deuterium).
Authors | Laboratory1 | Frequency interval(s) | Reported value | Rel. stand.
---|---|---|---|---
| | | $\nu$/kHz | uncert. $u_{\rm r}$
Fischer _et al._ (2004) | MPQ | $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $2\,466\,061\,413\,187.080(34)$ | $1.4\times 10^{-14}$
Weitz _et al._ (1995) | MPQ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,797\,338(10)$ | $2.1\times 10^{-6}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-{1\over 4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,490\,144(24)$ | $3.7\times 10^{-6}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-{1\over 4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,801\,693(20)$ | $4.2\times 10^{-6}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-{1\over 4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,494\,841(41)$ | $6.3\times 10^{-6}$
Parthey _et al._ (2010) | MPQ | $\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})-\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $670\,994\,334.606(15)$ | $2.2\times 10^{-11}$
de Beauvoir _et al._ (1997) | LKB/SYRTE | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ | $770\,649\,350\,012.0(8.6)$ | $1.1\times 10^{-11}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ | $770\,649\,504\,450.0(8.3)$ | $1.1\times 10^{-11}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ | $770\,649\,561\,584.2(6.4)$ | $8.3\times 10^{-12}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ | $770\,859\,041\,245.7(6.9)$ | $8.9\times 10^{-12}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ | $770\,859\,195\,701.8(6.3)$ | $8.2\times 10^{-12}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ | $770\,859\,252\,849.5(5.9)$ | $7.7\times 10^{-12}$
Schwob _et al._ (1999) | LKB/SYRTE | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ | $799\,191\,710\,472.7(9.4)$ | $1.2\times 10^{-11}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ | $799\,191\,727\,403.7(7.0)$ | $8.7\times 10^{-12}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ | $799\,409\,168\,038.0(8.6)$ | $1.1\times 10^{-11}$
| | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ | $799\,409\,184\,966.8(6.8)$ | $8.5\times 10^{-12}$
Arnoult _et al._ (2010) | LKB | $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $2\,922\,743\,278\,678(13)$ | $4.4\times 10^{-12}$
Bourzeix _et al._ (1996) | LKB | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6S_{1/2}})-{1\over 4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $4\,197\,604(21)$ | $4.9\times 10^{-6}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6D_{5/2}})-{1\over 4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $4\,699\,099(10)$ | $2.2\times 10^{-6}$
Berkeland _et al._ (1995) | Yale | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{1/2}})-{1\over 4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,664\,269(15)$ | $3.2\times 10^{-6}$
| | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{3/2}})-{1\over 4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,035\,373(10)$ | $1.7\times 10^{-6}$
Hagley and Pipkin (1994) | Harvard | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 2P_{3/2}})$ | $9\,911\,200(12)$ | $1.2\times 10^{-6}$
Lundeen and Pipkin (1986) | Harvard | $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ | $1\,057\,845.0(9.0)$ | $8.5\times 10^{-6}$
Newton _et al._ (1979) | U. Sussex | $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ | $1\,057\,862(20)$ | $1.9\times 10^{-5}$
1MPQ: Max-Planck-Institut für Quantenoptik, Garching. LKB: Laboratoire
Kastler-Brossel, Paris. SYRTE: Systèmes de référence Temps Espace, Paris,
formerly Laboratoire Primaire du Temps et des Fréquences (LPTF).
#### IV.1.3 Nuclear radii
Transition frequencies in hydrogen and deuterium depend on the rms charge
radius of the nucleus, denoted by $r_{\rm p}$ and $r_{\rm d}$ respectively.
The main difference between energy levels for a point charge nucleus and for a
nucleus with a finite charge radius is given by Eq. (74). These radii are
treated as adjusted constants, so the H and D experimental transition-
frequency input data, together with theory, provide adjusted values for them.
##### Electron scattering
The radii can also be determined from elastic electron-proton (e-p) scattering
data in the case of $r_{\rm p}$, and from elastic electron-deuteron (e-d)
scattering data in the case of $r_{\rm d}$. These independently determined
values are used as additional input data which, together with the H and D
spectroscopic data and the theory, determine the 2010 recommended values of
the radii. The experimental electron-scattering values of $r_{\rm p}$ and
$r_{\rm d}$ that we take as input data in the 2010 adjustment are
$\displaystyle r_{\rm p}$ $\displaystyle=$ $\displaystyle 0.895(18)\ {\rm
fm}\,,$ (94) $\displaystyle r_{\rm p}$ $\displaystyle=$ $\displaystyle
0.8791(79)\ {\rm fm}\,,$ (95) $\displaystyle r_{\rm d}$ $\displaystyle=$
$\displaystyle 2.130(10)\ {\rm fm}\,.$ (96)
The first result for $r_{\rm p}$, which was also used in the 2002 and 2006
adjustments, is due to Sick (2003, 2007, 2008) and is based on a reanalysis of
the world e-p cross section and polarization transfer data. The value in Eq.
(94) is consistent with the more accurate result $r_{\rm p}=0.894(8)$ reported
after the closing date of the 2010 adjustment by Sick (2011) using an improved
method to treat the proton’s charge density at large radii. It is also
consistent with the very recent result $r_{\rm p}=0.886(8)$ calculated by Sick
(2012) that extends this method and is based in part on the data obtained by
Bernauer _et al._ (2010) in the experiment that yields the second result for
$r_{\rm p}$, which we now discuss. [Note that the recent paper of Sick (2012)
gives an overview of the problems associated with determining a reliable value
of $r_{\rm p}$ from e-p scattering data. Indeed, Adamuscin _et al._ (2012)
find $r_{\rm p}=0.844(7)$ based on a reanalysis of selected nucleon form-
factor data; see also Arrington _et al._ (2007).]
The value of $r_{\rm p}$ given in Eq. (95) was obtained at the Mainz
University, Germany, with the Mainz linear electron accelerator MAMI. About
1400 elastic e-p scattering cross sections were measured at six beam energies
from 180 MeV to 855 MeV, covering the range of four-momentum transfers squared
from $Q^{2}=0.004$ (GeV/c)2 to 1 (GeV/c)2. The value of $r_{\rm p}$ was
extracted from the data using spline fits or polynomial fits, and because the
reason for the comparatively small difference between the resulting values
could not be identified, Bernauer _et al._ (2010) give as their final result
the average of the two values with an added uncertainty equal to half the
difference. [Note that the value in Eq. (95) contains extra digits provided by
Bernauer (2010). See also the exchange of comments of Arrington (2011);
Bernauer _et al._ (2011).]
The result for $r_{\rm d}$ is that given by Sick (2008) and is based on an
analysis of the world data on e-d scattering similar to that used to determine
the value of $r_{\rm p}$ in Eq. (94).
For completeness we note the recent e-p scattering result for $r_{\rm p}$
based in part on new data obtained in the range $Q^{2}=0.3~{}({\rm
GeV}/c)^{2}$ to $0.7~{}({\rm GeV}/c)^{2}$ at the Thomas Jefferson National
Accelerator Facility, Newport News, Virginia, USA, often referred to as simply
JLab. The new data, acquired using a technique called polarization transfer or
recoil polarimetry, were combined with previous cross section and polarization
measurements to produce the result $r_{\rm p}=0.875(10)$ fm from an updated
global fit in this range of $Q^{2}$ Zhan _et al._ (2011); Ron _et al._ (2011).
It is independent of and agrees with the Mainz result in Eq. (95), and it also
agrees with the result in Eq. (94) but the two are not independent since the
data used to obtain the latter result were included in the JLab fit. This
result became available after the 31 December 2010 closing date of the 2010
adjustment.
##### Muonic hydrogen
A muonic hydrogen atom, $\mbox{{m}}^{-}$p, consists of a negative muon and a
proton. Since $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}\approx 207$, the Bohr
radius of the muon is about 200 times smaller than the electron Bohr radius,
so the muon is more sensitive to the size of the nucleus. Indeed, the finite-
size effect for the 2S state in ${\mbox{{m}}^{-}}$p is about 2 % of the total
Lamb shift, that is, the energy difference between the 2S and 2P states, which
should make it an ideal system for measuring the size of the proton. (Because
of the large electron vacuum polarization effect in muonic hydrogen, the
$2{\rm S}_{1/2}$ level is well below both the $2{\rm P}_{3/2}$ and $2{\rm
P}_{1/2}$ levels.)
In a seminal experiment carried out using pulsed laser spectroscopy at a
specially built muon beam line at the proton accelerator of the Paul Scherrer
Institute (PSI), Villigen, Switzerland, Pohl _et al._ (2010, 2011) have
measured the 206 meV (50 THz or 6 ${\mbox{{m}}\rm m}$) ${\mbox{{m}}^{-}}$p
Lamb shift, in particular, the $2{\rm S}_{1/2}(F=1)-2{\rm P}_{3/2}(F=2)$
transition, with an impressive relative standard uncertainty of 15 parts in
$10^{6}$. The result, when combined with the theoretical expression for the
transition, leads to Jentschura (2011b)
$\displaystyle r_{\rm p}$ $\displaystyle=$ $\displaystyle 0.84169(66)\ {\rm
fm}\,.$ (97)
The value given in Eq. (97) is based on a review and reanalysis of the theory
by Jentschura (2011b, c) but is not significantly different from the value
first given by Pohl _et al._ (2010). Because the muonic hydrogen value of
$r_{\rm p}$ differs markedly from the 2006 CODATA recommended value given in
CODATA-06, its publication in 2010 has led to a significant number of papers
that reexamine various aspects of the theory or propose possible reasons for
the disagreement; see, for example, the recent review of Borie (2012). If Eq.
(97) is compared to the 2010 recommended value of $0.8775(51)$ fm, the
disagreement is $7\,\sigma$. If it is compared to the value $0.8758(77)$ fm
based on only H and D spectroscopic data (see Table 38), the disagreement is
$4.4\,\sigma$.
The impact of including Eq. (97) on the 2010 adjustment and the reasons the
Task Group decided not to include it are discussed in Sec. XIII.2.2. We also
note the following fact. If the least-squares adjustment that leads to the
value of $\alpha$ given in Eq. (88) is carried out with the value in Eq. (97)
added as an input datum, the result is
$\alpha^{-1}=137.035\,881(35)~{}[2.6\times 10^{-7}]$, which differs from the
2010 recommended value by $3.4\,\sigma$. The value of $R_{\infty}$ from this
adjustment is $10\,973\,731.568\,016(49)$ m-1.
### IV.2 Antiprotonic helium transition frequencies and $\bm{A_{\rm r}({\rm
e})}$
Consisting of a ${}^{4}{\rm He}$ or a ${}^{3}{\rm He}$ nucleus, an antiproton,
and an electron, the antiprotonic helium atom is a three-body system denoted
by $\bar{\rm p}{\rm He}^{+}$. Because it is assumed that $CPT$ is a valid
symmetry, determination of the antiproton-electron mass ratio from
antiprotonic helium experiments can be interpreted as determination of the
proton-electron mass ratio. Further, because the relative atomic mass of the
proton $A_{\rm r}(\rm p)$ is known with a significantly smaller relative
uncertainty from other data than is $A_{\rm r}(\rm e)$, a value of the
antiproton-electron mass ratio with a sufficiently small uncertainty can
provide a competitive value of $A_{\rm r}(\rm e)$.
Theoretical and experimental values of frequencies corresponding to
transitions between atomic levels of the antiprotons with large principal
quantum number $n$ and angular momentum quantum number $l$, such that
$n\approx{l+1}\approx 38$, were used to obtain a value of $A_{\rm r}(\rm e)$
in the 2006 adjustment. Table 12 summarizes the relevant experimental and
theoretical data. The first column indicates the mass number of the helium
nucleus of the antiprotonic atom and the principal and angular momentum
quantum numbers of the energy levels involved in the transitions. The second
column gives the experimentally measured values of the transition frequencies
while the third gives the theoretically calculated values. The last two
columns give the values in the unit $2cR_{\infty}$ of quantities $a$ and $b$
used in the observational equations that relate the experimental values of the
transition frequencies to their calculated values and relevant adjusted
constants, as discussed in the next section. Besides a few comparatively minor
changes in some of the calculated frequencies and their uncertainties, the
only significant difference between Table 12 and the corresponding Table XIII
in CODATA-06 is the addition of recently acquired data on three two-photon
transitions: $(33,32)\rightarrow(31,30)$ and $(36,34)\rightarrow(34,32)$ for
$\bar{\rm p}^{4}{\rm He}^{+}$, and $(35,33)\rightarrow(33,31)$ for $\bar{\rm
p}^{3}{\rm He}^{+}$.
It is noteworthy that Hori _et al._ (2011), who determined the experimental
values of these three frequencies (discussed further in Sec. IV.2.2 below),
have used the new experimental and theoretical data to obtain an important new
limit. With the aid of the long-known result that the absolute value of the
charge-to-mass ratio of $\rm p$ and $\bar{\rm p}$ are the same within at least
$9$ parts in $10^{11}$ Gabrielse (2006), they showed that the charge and mass
of $\rm p$ and $\bar{\rm p}$ are the same within $7$ parts in $10^{10}$ at the
$90\%$ confidence level.
#### IV.2.1 Theory relevant to antiprotonic helium
The calculated transition frequencies in Table 12 are due to Korobov (2008,
2010) and are based on the 2002 recommended values of the required fundamental
constants with no uncertainties. Korobov’s publication updates some of the
values and uncertainties of the calculated transition frequencies used in the
2006 adjustment that he provided directly to the Task Group Korobov (2006),
but it also includes results for the $\bar{\rm p}^{4}{\rm He}^{+}$ and
$\bar{\rm p}^{3}{\rm He}^{+}$ two-photon transition frequencies
$(36,34)\rightarrow(34,32)$ and $(35,33)\rightarrow(33,31)$. The calculated
value for the $\bar{\rm p}^{4}{\rm He}^{+}$ two-photon frequency
$(33,32)\rightarrow(31,30)$ was again provided directly to the Task Group by
Korobov (2010), as were slightly updated values for the two other two-photon
frequencies. The same calculated values of the three two-photon frequencies
are also given in the paper by Hori _et al._ (2011) cited above.
The quantities $a\equiv a_{\bar{\rm p}\rm He}(n,l:{n^{\prime}},{l^{\prime}})$
and $b\equiv b_{\bar{\rm p}\rm He}(n,l:{n^{\prime}},{l^{\prime}})$ in Table
12, also directly provided to the Task Group by Korobov (2010, 2006), are
actually the numerical values of derivatives defined and used as follows (in
these and other similar expressions in this section, $\rm He$ is ${}^{3}{\rm
He}$ or ${}^{4}{\rm He}$).
The theoretical values of the transition frequencies are functions of the mass
ratios $A_{\rm r}(\bar{\rm p})/A_{\rm r}(\rm e)$ and $A_{\rm r}(N)/A_{\rm
r}(\bar{\rm p})$, where $N$ is either ${}^{4}{\rm He}^{2+}$ or ${}^{3}{\rm
He}^{2+}$, that is, the alpha particle $\rm\alpha$ or helion $\rm h$. If the
transition frequencies as a function of these mass ratios are denoted by
$\nu_{\bar{\rm p}\rm He}(n,l:{n^{\prime}},{l^{\prime}})$, and the calculated
values in Table 12 by $\nu^{(0)}_{\bar{\rm p}\rm
He}(n,l:{n^{\prime}},{l^{\prime}})$, we have
$\displaystyle a_{\rm\bar{p}\,{\rm He}}(n,l:n^{\prime},l^{\prime})$
$\displaystyle=$ $\displaystyle\left(\frac{A_{\rm r}({\rm\bar{p}})}{A_{\rm
r}({\rm e})}\right)^{(0)}\frac{\partial\Delta\nu_{\rm\bar{p}\,{\rm
He}}(n,l:n^{\prime},l^{\prime})}{\partial\left(\frac{A_{\rm
r}({\rm\bar{p}})}{A_{\rm r}({\rm e})}\right)}\,,$ $\displaystyle
b_{\rm\bar{p}\,{\rm He}}(n,l:n^{\prime},l^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{A_{\rm r}({\rm He})}{A_{\rm
r}({\rm\bar{p}})}\right)^{(0)}\frac{\partial\Delta\nu_{\rm\bar{p}\,{\rm
He}}(n,l:n^{\prime},l^{\prime})}{\partial\left(\frac{A_{\rm r}({N})}{A_{\rm
r}({\rm\bar{p}})}\right)}\,,$
where the superscript $(0)$ denotes the fact that the 2002 CODATA values of
the relative atomic mass ratios were used by Korobov in his calculations. The
zero-order frequencies, mass ratios, and the derivatives $a$ and $b$ provide a
first-order approximation to the transition frequencies as a function of
changes in the mass ratios:
$\displaystyle\hbox to-10.0pt{}\nu_{\rm\bar{p}\,{\rm
He}}\,(n,l:n^{\prime},l^{\prime})=\nu_{\rm\bar{p}\,{\rm
He}}^{(0)}(n,l:n^{\prime},l^{\prime})$ $\displaystyle+a_{\rm\bar{p}\,{\rm
He}}(n,l:n^{\prime},l^{\prime})\left[\left(\frac{A_{\rm r}({\rm e})}{A_{\rm
r}({\rm\bar{p})}}\right)^{\\!(0)}\\!\\!\left(\frac{A_{\rm
r}({\rm\bar{p}}\,)}{A_{\rm r}({\rm e})}\right)-1\right]$
$\displaystyle+b_{\bar{\rm p}\,{\rm
He}}(n,l:n^{\prime},l^{\prime})\left[\left(\frac{A_{\rm
r}({\rm\bar{p}})}{A_{\rm r}({N)}}\right)^{\\!(0)}\\!\\!\left(\frac{A_{\rm
r}({N})}{A_{\rm r}({\rm\bar{p}})}\right)-1\right]$ $\displaystyle+\dots\,.$
(100)
This expression is the basis for the observational equations for the measured
and calculated transition frequencies as a function of the mass ratios in the
least-squares adjustment; see Table 35, Sec. XIII. Although $A_{\rm r}(\rm
e)$, $A_{\rm r}(\rm p)$ and $A_{\rm r}(N)$ are adjusted constants, the
principal effect of including the antiprotonic helium transition frequencies
in the adjustment is to provide information about $A_{\rm r}(\rm e)$. This is
because independent data in the adjustment provide values of $A_{\rm r}(\rm
p)$ and $A_{\rm r}(N)$ with significantly smaller relative uncertainties than
the uncertainty of $A_{\rm r}(\rm e)$.
The uncertainties of the calculated transition frequencies are taken into
account by including an additive constant $\delta_{\bar{\rm p}\rm
He}(n,l:{n^{\prime}},{l^{\prime}})$ in the observational equation for each
measured frequency; see Table 34 and $C13-C24$ in Table 35, Sec. XIII. The
additive constants are adjusted constants and their assigned values are zero
with the uncertainties of the theoretical values. They are data items $C1$ to
$C15$ in Table 22. Moreover, the input data for the additive constants are
correlated; their correlation coefficients, calculated from information
provided by Korobov (2010), are given in Table 23. (In the 2006 adjustment,
the correlations between the ${}^{4}{\rm He}$ and ${}^{3}{\rm He}$ calculated
frequencies were omitted.)
#### IV.2.2 Experiments on antiprotonic helium
Recent reviews of the experimental work, which is carried out at CERN, have
been given by Hori (2011) and by Hayano (2010). The first seven ${}^{4}{\rm
He}$ and the first five ${}^{3}{\rm He}$ experimental transition frequencies
in Table 12, obtained by Hori _et al._ (2006), were used in the 2006
adjustment and are discussed in CODATA-06. The measurements were carried out
with antiprotons from the CERN Antiproton Decelerator and employed the
technique of single-photon precision laser-spectroscopy. The transition
frequencies and their uncertainties include an extra digit beyond those
reported by Hori _et al._ (2006) that were provided to the Task Group by Hori
(2006) to reduce rounding errors.
During the past 4 years the CERN group has been able to improve their
experiment and, as noted above, Hori _et al._ (2011) have recently reported
results for three transitions based on two-photon laser spectroscopy. In this
work $\bar{\rm p}^{4}{\rm He}^{+}$ or $\bar{\rm p}^{3}{\rm He}^{+}$ atoms are
irradiated by two counter-propagating laser beams that excite deep
ultraviolet, nonlinear, two-photon transitions of the type
$(n,l)\rightarrow(n-2,,l-2)$. This technique reduces thermal Doppler
broadening of the resonances of the antiprotonic atoms, thereby producing
narrower spectral lines and reducing the uncertainties of the measured
transition frequencies.
In normal two-photon spectroscopy the frequencies of the two counter
propagating laser beams are the same and equal to one-half the resonance
frequency. In consequence, to first order in the atom’s velocity, Doppler
broadening is reduced to zero. However, normal two-photon spectroscopy is
difficult to do in antiprotonic helium because of the small transition
probabilities of the nonlinear two-photon transitions. The CERN group was able
to mitigate this problem by using the fact that the probability can be
increased some five orders of magnitude if the two beams have different
frequencies $\nu_{1}$ and $\nu_{2}$ such that the virtual state of the two-
photon transition is within approximately $10~{}{\rm GHz}$ of a real state
with quantum numbers $(n-1,l-1)$ Hori and Korobov (2010). In this case the
first-order Doppler width of the resonance is reduced by the factor
$|\nu_{1}-\nu_{2}|/(\nu_{1}+\nu_{2})$.
As for the earlier data, an extra digit, provided to the Task Group by Hori
(2010), has been added to the three new two-photon frequencies and their
uncertainties. Further, as for the one-photon transitions used in 2006, Hori
(2010) has provided the Task Group with a detailed uncertainty budget for each
of the new frequencies so that their correlation coefficients could be
properly evaluated. (There are no correlations between the 12 older one-photon
frequencies and the 3 new two-photon frequencies.) As for the one-photon
frequencies, the dominant uncertainty component for the two-photon frequencies
is statistical; it varies from $3.0~{}{\rm MHz}$ to $6.6~{}{\rm MHz}$ compared
to $3.2~{}{\rm MHz}$ to $13.8~{}{\rm MHz}$ for the one-photon frequencies. The
15 transition frequencies are data items $16$ to $30$ in Table 22; all
relevant correlation coefficients are given in Table 23.
#### IV.2.3 Inferred value of $A_{\rm r}(\rm e)$ from antiprotonic helium
Use of the 2010 recommended values of $A_{\rm r}(\rm p)$, $A_{\rm
r}(\rm\alpha)$, and $A_{\rm r}(\rm h)$, the experimental and theoretical
values of the $15$ transition frequencies in Table 12, the correlation
coefficients in Table 23, and the observational equations in Table 35 derived
as discussed above, yields the following inferred value of the electron
relative atomic mass:
$\displaystyle A_{\rm r}({\rm e})$ $\displaystyle=$ $\displaystyle
0.000\,548\,579\,909\,14(75)\quad[1.4\times 10^{-9}]\,.\qquad$ (101)
The $\bar{\rm p}^{3}{\rm He}$ data alone give a value of $A_{\rm r}({\rm e})$
that has an uncertainty that is $1.7$ times as large as the uncertainty of the
value in Eq. (101); and it is smaller by a factor $1.2$ times its uncertainty.
The combined result is consistent and competitive with other values, as
discussed in Sec. XIII.
Table 12: Summary of data related to the determination of $A_{\rm r}(\rm e)$ from measurements of antiprotonic helium. The uncertainties of the 15 calculated values are the root-sum-square (rss) of the following 15 pairs of uncertainty components in MHz, where the first component reflects the possible size of uncalculated terms of order $R_{\infty}\alpha^{5}\ln\alpha$ and higher, and the second component reflects the uncertainty of the numerical calculations: (0.8, 0.2); (1.0, 0.3); (1.1, 0.3); (1.1, 0.3); (1.1, 0.4); (1.0, 0.8); (1.8, 0.4); (1.6, 0.3); (2.1, 0.3); (0.9, 0.1); (1.1, 0.2); (1.1, 0.4); (1.1, 0.3); (1.8, 0.3); (2.2, 0.2). Transition | Experimental | Calculated | $a\hbox to-10.0pt{}$ | $b\hbox to-10.0pt{}$
---|---|---|---|---
$(n,l)\rightarrow(n^{\prime},l^{\prime})$ | Value (MHz) | Value (MHz) | $(2cR_{\infty})\hbox to-25.0pt{}$ | $(2cR_{\infty})\hbox to-25.0pt{}$
$\bar{\rm p}^{4}$He+: $(32,31)\rightarrow(31,30)$ | $1\,132\,609\,209(15)\hbox to-16.0pt{}$ | $1\,132\,609\,223.50(82)$ | $0.2179$ | $0.0437$
$\bar{\rm p}^{4}$He+: $(35,33)\rightarrow(34,32)$ | $804\,633\,059.0(8.2)$ | $804\,633\,058.0(1.0)$ | $0.1792$ | $0.0360$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(35,33)$ | $717\,474\,004(10)\hbox to-16.0pt{}$ | $717\,474\,001.1(1.1)$ | $0.1691$ | $0.0340$
$\bar{\rm p}^{4}$He+: $(37,34)\rightarrow(36,33)$ | $636\,878\,139.4(7.7)$ | $636\,878\,151.7(1.1)$ | $0.1581$ | $0.0317$
$\bar{\rm p}^{4}$He+: $(39,35)\rightarrow(38,34)$ | $501\,948\,751.6(4.4)$ | $501\,948\,755.6(1.2)$ | $0.1376$ | $0.0276$
$\bar{\rm p}^{4}$He+: $(40,35)\rightarrow(39,34)$ | $445\,608\,557.6(6.3)$ | $445\,608\,569.3(1.3)$ | $0.1261$ | $0.0253$
$\bar{\rm p}^{4}$He+: $(37,35)\rightarrow(38,34)$ | $412\,885\,132.2(3.9)$ | $412\,885\,132.8(1.8)$ | $-0.1640$ | $-0.0329$
$\bar{\rm p}^{4}$He+: $(33,32)\rightarrow(31,30)$ | $2\,145\,054\,858.2(5.1)$ | $2\,145\,054\,857.9(1.6)$ | $0.4213$ | $0.0846$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(34,32)$ | $1\,522\,107\,061.8(3.5)$ | $1\,522\,107\,058.9(2.1)$ | $0.3483$ | $0.0699$
$\bar{\rm p}^{3}$He+: $(32,31)\rightarrow(31,30)$ | $1\,043\,128\,608(13)\hbox to-16.0pt{}$ | $1\,043\,128\,579.70(91)$ | $0.2098$ | $0.0524$
$\bar{\rm p}^{3}$He+: $(34,32)\rightarrow(33,31)$ | $822\,809\,190(12)\hbox to-16.0pt{}$ | $822\,809\,170.9(1.1)$ | $0.1841$ | $0.0460$
$\bar{\rm p}^{3}$He+: $(36,33)\rightarrow(35,32)$ | $646\,180\,434(12)\hbox to-16.0pt{}$ | $646\,180\,408.2(1.2)$ | $0.1618$ | $0.0405$
$\bar{\rm p}^{3}$He+: $(38,34)\rightarrow(37,33)$ | $505\,222\,295.7(8.2)$ | $505\,222\,280.9(1.1)$ | $0.1398$ | $0.0350$
$\bar{\rm p}^{3}$He+: $(36,34)\rightarrow(37,33)$ | $414\,147\,507.8(4.0)$ | $414\,147\,507.8(1.8)$ | $-0.1664$ | $-0.0416$
$\bar{\rm p}^{3}$He+: $(35,33)\rightarrow(33,31)$ | $1\,553\,643\,099.6(7.1)$ | $1\,553\,643\,100.7(2.2)$ | $0.3575$ | $0.0894$
### IV.3 Hyperfine structure and fine structure
During the past 4 years two highly accurate values of the fine-structure
constant $\rm\alpha$ from dramatically different experiments have become
available, one from the electron magnetic-moment anomaly $a_{\rm e}$ and the
other from $h/m(^{87}{\rm Rb})$ obtained by atom recoil. They are consistent
and have relative standard uncertainties of $3.7\times 10^{-10}$ and
$6.6\times 10^{-10}$, respectively; see Table 25. These uncertainties imply
that for another value of $\rm\alpha$ to be competitive, its relative
uncertainty should be no more than about a factor of 10 larger.
By equating the experimentally measured ground-state hyperfine transition
frequency of a simple atom such as hydrogen, muonium (${\rm\mu}^{+}{\rm
e}^{-}$ atom), or positronium (${\rm e}^{+}{\rm e}^{-}$ atom) to its
theoretical expression, one could in principle obtain a value of $\rm\alpha$,
since this frequency is proportional to ${\rm\alpha}^{2}{R_{\infty}}c$.
Muonium is, however, still the only atom for which both the measured value of
the hyperfine frequency and its theoretical expression have sufficiently small
uncertainties to be of possible interest, and even for this atom with a
structureless nucleus the resulting value of ${\rm\alpha}$ is no longer
competitive; instead, muonium provides the most accurate value of the
electron-muon mass ratio, as discussed in Sec. VI.2.
Also proportional to ${\rm\alpha}^{2}{R_{\infty}}c$ are fine-structure
transition frequencies, and thus in principal these could provide a useful
value of $\rm\alpha$. However, even the most accurate measurements of such
frequencies in the relatively simple one-electron atoms hydrogen and deuterium
do not provide a competitive value; see Table 11 and Sec. IV.1.1, especially
Eq. (88). Rather, the experimental hydrogen fine-structure transition
frequencies given in that table are included in the 2010 adjustment, as in
past adjustments, because of their influence on the adjusted constant
$R_{\infty}$.
The large natural line widths of the $2{\rm P}$ levels in H and D limit the
accuracy with which the fine-structure frequencies in these atoms can be
measured. By comparison, the $2^{3}{\rm P}_{J}$ states of ${}^{4}{\rm He}$ are
narrow ($1.6~{}{\rm MHz}$ vs. $100~{}{\rm MHz}$) because they cannot decay to
the ground $1^{1}{\rm S}_{0}$ state by allowed electric dipole transitions.
Since the energy differences between the three $2^{3}{\rm P}$ levels and the
corresponding transition frequencies can be calculated and measured with
reasonably small uncertainties, it has long been hoped that the fine structure
of ${}^{4}{\rm He}$ could one day provide a competitive value of $\rm\alpha$.
Although the past 4 years has seen considerable progress toward this goal, it
has not yet been reached. In brief, the situation is as follows.
The fine structure of the $2^{3}{\rm P}_{J}$ triplet state of ${}^{4}{\rm He}$
consists of three levels; they are, from highest to lowest, $2^{3}{\rm
P}_{0}$, $2^{3}{\rm P}_{1}$, and $2^{3}{\rm P}_{2}$. The three transition
frequencies of interest are ${\rm\nu}_{01}\approx 29.6~{}{\rm GHz}$,
${\rm\nu}_{12}\approx 2.29~{}{\rm GHz}$, and ${\rm\nu}_{02}\approx 31.9~{}{\rm
GHz}$. In a series of papers Pachucki (2006) and Pachucki and Yerokhin (2009,
2010a, 2011b, 2011a), but see also Pachucki and Sapirstein (2010) and
Sapirstein (2010), have significantly advanced the theory of these transitions
in both helium and light helium-like ions. Based on this work, the theory is
now complete to orders $m\alpha^{7}$ and $m(m/M)\alpha^{6}$ ($m$ the electron
mass and $m/M$ the electron-alpha particle mass ratio), previous disagreements
among calculations have been resolved, and an estimate of uncertainty due to
the uncalculated $m\alpha^{8}$ term has been made. Indeed, the uncertainty of
the theoretical expression for the ${\rm\nu}_{02}$ transition, which is the
most accurately known both theoretically and experimentally, is estimated to
be $1.7~{}{\rm kHz}$, corresponding to a relative uncertainty of $5.3\times
10^{-8}$ or $2.7\times 10^{-8}$ for ${\alpha}$. Nevertheless, even if an
experimental value of ${\rm\nu}_{02}$ with an uncertainty of just a few hertz
were available, the uncertainty in the value of $\alpha$ from helium fine
structure would still be too large to be included in the 2010 adjustment
In fact, the most accurate experimental value of ${\rm\nu}_{02}$ is that
measured by Smiciklas and Shiner (2010) with an uncertainty of $300~{}\rm Hz$,
corresponding to a relative uncertainty of $9.4\times 10^{-9}$ or $4.7\times
10^{-9}$ for ${\rm\alpha}$. As given by Pachucki and Yerokhin (2011b), the
value of ${\rm\alpha}$ obtained by equating this experimental result and the
theoretical result is ${\alpha}^{-1}=137.035\,9996(37)$ $[2.7\times 10^{-8}]$,
which agrees well with the two most accurate values mentioned at the start of
this section but is not competitive with them.
Another issue is that the agreement among different experimental values of the
various helium fine-structure transitions and their agreement with theory is
not completely satisfactory. Besides the result of Smiciklas and Shiner (2010)
for ${\rm\nu}_{02}$, there is the measurement of ${\rm\nu}_{12}$ by Borbely
_et al._ (2009), all three frequencies by Zelevinsky _et al._ (2005),
${\rm\nu}_{01}$ by Giusfredi _et al._ (2005), ${\rm\nu}_{01}$ by George _et
al._ (2001), ${\rm\nu}_{12}$ by Castillega _et al._ (2000), and
${\rm\nu}_{02}$ by Shiner and Dixson (1995). Graphical comparisons of these
data among themselves and with theory may be found in the paper by Smiciklas
and Shiner (2010).
In summary, no ${}^{4}{\rm He}$ fine-structure datum is included in the 2010
adjustment, because the resulting value of $\rm\alpha$ has too large an
uncertainty compared to the uncertainties of the values from $a_{\rm e}$ and
$h/m(^{87}{\rm Rb})$.
## V Magnetic moment anomalies and $\bm{g}$-factors
As discussed in CODATA-06, the magnetic moment of any of the three charged
leptons $\ell={\rm e},\,\mbox{{m}},\,\mbox{{t}}$ is
$\displaystyle\bm{\mu}_{\ell}=g_{\ell}{e\over 2m_{\ell}}\bm{s}\ ,$ (102)
where $g_{\ell}$ is the $g$-factor of the particle, $m_{\ell}$ is its mass,
and $\bm{s}$ is its spin. In Eq. (102), $e$ is the (positive) elementary
charge. For the negatively charged leptons $\ell^{\,-}$, $g_{\ell}$ is
negative. These leptons have eigenvalues of spin projection
$s_{z}=\pm\hbar/2$, so that
$\displaystyle\mu_{\rm\ell}={g_{\ell}\over 2}\,\frac{e\hbar}{2m_{\ell}}\ ,$
(103)
and $\hbar/2m_{\rm e}=\mu_{\rm B}$, the Bohr magneton. The magnetic moment
anomaly $a_{\ell}$ is defined by
$\displaystyle|g_{\ell}|$ $\displaystyle=$ $\displaystyle 2(1+a_{\ell})\ ,$
(104)
where the free-electron Dirac equation gives $a_{\ell}=0$. In fact, the
anomaly is not zero, but is given by
$\displaystyle a_{\ell}({\rm th})=a_{\ell}({\rm QED})+a_{\ell}({\rm
weak})+a_{\ell}({\rm had})\ ,$ (105)
where the terms denoted by QED, weak, and had account for the purely quantum
electrodynamic, predominantly electroweak, and predominantly hadronic (that
is, strong interaction) contributions to $a_{\ell}$, respectively.
For a comprehensive review of the theory of $a_{\rm e}$, but particularly of
$a_{\mbox{\scriptsize{{m}}}}$, see Jegerlehner and Nyffeler (2009). It has
long been recognized, as these authors duly note, that the comparison of
experimental and theoretical values of the electron and muon $g$-factors can
test our description of nature, in particular, the Standard Model of particle
physics, which is the theory of the electromagnetic, weak, and strong
interactions. Nevertheless, our main purpose here is not to test physical
theory critically, but to obtain “best” values of the fundamental constants.
### V.1 Electron magnetic moment anomaly $\bm{a_{\rm e}}$ and the fine-
structure constant $\bm{\alpha}$
Comparison of theory and experiment for the electron magnetic moment anomaly
gives the value for the fine-structure constant $\alpha$ with the smallest
estimated uncertainty in the 2010 adjustment.
#### V.1.1 Theory of $a_{\rm e}$
The QED contribution for the electron may be written as Kinoshita _et al._
(1990)
$\displaystyle a_{\rm e}({\rm QED})$ $\displaystyle=$ $\displaystyle
A_{1}+A_{2}(m_{\rm e}/m_{\mbox{\scriptsize{{m}}}})+A_{2}(m_{\rm
e}/m_{\mbox{\scriptsize{{t}}}})$ (106) $\displaystyle+A_{3}(m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}},m_{\rm e}/m_{\mbox{\scriptsize{{t}}}})\ .$
The leading term $A_{1}$ is mass independent and the masses in the
denominators of the ratios in $A_{2}$ and $A_{3}$ correspond to particles in
vacuum polarization loops.
Each of the four terms on the right-hand side of Eq. (106) is expressed as a
power series in the fine-structure constant $\alpha$:
$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle
A_{i}^{(2)}\left({\alpha\over\mbox{{p}}}\right)+A_{i}^{(4)}\left({\alpha\over\mbox{{p}}}\right)^{2}+A_{i}^{(6)}\left({\alpha\over\mbox{{p}}}\right)^{3}$
(107)
$\displaystyle+A_{i}^{(8)}\left({\alpha\over\mbox{{p}}}\right)^{4}+A_{i}^{(10)}\left({\alpha\over\mbox{{p}}}\right)^{5}+\cdots\
,$
where $A_{2}^{(2)}=A_{3}^{(2)}=A_{3}^{(4)}=0$. Coefficients proportional to
$(\alpha/\mbox{{p}})^{n}$ are of order $e^{2n}$ and are referred to as 2nth-
order coefficients. The second-order coefficient is known exactly, and the
fourth- and sixth-order coefficients are known analytically in terms of
readily evaluated functions:
$\displaystyle A_{1}^{(2)}$ $\displaystyle=$ $\displaystyle{\textstyle{1\over
2}}\,,$ (108) $\displaystyle A_{1}^{(4)}$ $\displaystyle=$
$\displaystyle-0.328\,478\,965\,579\ldots\,,$ (109) $\displaystyle
A_{1}^{(6)}$ $\displaystyle=$ $\displaystyle 1.181\,241\,456\ldots\ .$ (110)
The eighth-order coefficient $A_{1}^{(8)}$ arises from $891$ Feynman diagrams
of which only a few are known analytically. Evaluation of this coefficient
numerically by Kinoshita and co-workers has been underway for many years
Kinoshita (2010). The value used in the 2006 adjustment is
$A_{1}^{(8)}=-1.7283(35)$ as reported by Kinoshita and Nio (2006). However,
and as discussed in CODATA-06, well after the 31 December 2006 closing date of
the 2006 adjustment, as well as the date when the 2006 CODATA recommended
values of the constants were made public, it was discovered by Aoyama _et al._
(2007) that a significant error had been made in the calculation. In
particular, 2 of the 47 integrals representing $518$ diagrams that had not
been confirmed independently required a corrected treatment of infrared
divergences. The error was identified by using FORTRAN code generated by an
automatic code generator. The new value is Aoyama _et al._ (2007)
$\displaystyle A_{1}^{(8)}$ $\displaystyle=$ $\displaystyle-1.9144(35)\,;$
(111)
details of the calculation are given by Aoyama _et al._ (2008). In view of the
extensive effort made by these workers to ensure that the result in Eq. (111)
is reliable, the Task Group adopts both its value and quoted uncertainty for
use in the 2010 adjustment.
Independent work is in progress on analytic calculations of eighth-order
integrals. See, for example, Laporta (2001); Mastrolia and Remiddi (2001);
Laporta _et al._ (2004); Laporta (2008). Work is also in progress on numerical
calculations of the 12 672 Feynman diagrams for the tenth-order coefficient.
See Aoyama _et al._ (2011) and references cited therein.
The evaluation of the contribution to the uncertainty of $a_{\rm e}({\rm th})$
from the fact that $A_{1}^{(10)}$ is unknown follows the procedure in
CODATA-98 and yields $A_{1}^{(10)}=0.0(4.6)$, which contributes a standard
uncertainty component to $a_{\rm e}({\rm th})$ of $2.7\times 10^{-10}\,a_{\rm
e}$. This uncertainty is larger than the uncertainty attributed to
$A_{1}^{(10)}$ in CODATA-06, because the absolute value of $A_{1}^{(8)}$ has
increased. All higher-order coefficients are assumed to be negligible.
The mass-dependent coefficients for the electron based on the 2010 recommended
values of the mass ratios are
$\displaystyle A_{2}^{(4)}\\!(m_{\rm e}/m_{\mbox{\scriptsize{{m}}}})$
$\displaystyle=$ $\displaystyle 5.197\,386\,68(26)\times 10^{-7}$ (112)
$\displaystyle\quad\rightarrow 24.182\times 10^{-10}a_{\rm e}\,,$
$\displaystyle A_{2}^{(4)}\\!(m_{\rm e}/m_{\mbox{\scriptsize{{t}}}})$
$\displaystyle=$ $\displaystyle 1.837\,98(33)\times 10^{-9}$ (113)
$\displaystyle\quad\rightarrow 0.086\times 10^{-10}a_{\rm e}\,,$
$\displaystyle A^{(6)}_{2}\\!(m_{\rm e}/m_{\mbox{\scriptsize{{m}}}})$
$\displaystyle=$ $\displaystyle-7.373\,941\,62(27)\times 10^{-6}$ (114)
$\displaystyle\quad\rightarrow-0.797\times 10^{-10}a_{\rm e}\,,$
$\displaystyle A^{(6)}_{2}\\!(m_{\rm e}/m_{\mbox{\scriptsize{{t}}}})$
$\displaystyle=$ $\displaystyle-6.5830(11)\times 10^{-8}$ (115)
$\displaystyle\quad\rightarrow-0.007\times 10^{-10}a_{\rm e}\,,$
where the standard uncertainties of the coefficients are due to the
uncertainties of the mass ratios and are negligible. The contributions from
$A_{3}^{(6)}\\!(m_{\rm e}/m_{\mbox{\scriptsize{{m}}}},m_{\rm
e}/m_{\mbox{\scriptsize{{t}}}})$ and all higher-order mass-dependent terms are
also negligible.
The dependence on $\alpha$ of any contribution other than $a_{\rm e}({\rm
QED})$ is negligible, hence the anomaly as a function of $\alpha$ is given by
combining QED terms that have like powers of $\alpha/\mbox{{p}}$:
$\displaystyle a_{\rm e}({\rm QED})$ $\displaystyle=$ $\displaystyle C_{\rm
e}^{(2)}\left({\alpha\over\mbox{{p}}}\right)+C_{\rm
e}^{(4)}\left({\alpha\over\mbox{{p}}}\right)^{2}+C_{\rm
e}^{(6)}\left({\alpha\over\mbox{{p}}}\right)^{3}$ (116) $\displaystyle+C_{\rm
e}^{(8)}\left({\alpha\over\mbox{{p}}}\right)^{4}+C_{\rm
e}^{(10)}\left({\alpha\over\mbox{{p}}}\right)^{5}+\cdots,$
with
$\displaystyle C_{\rm e}^{(2)}$ $\displaystyle=$ $\displaystyle 0.5\,,$
$\displaystyle C_{\rm e}^{(4)}$ $\displaystyle=$
$\displaystyle-0.328\,478\,444\,00\,,$ $\displaystyle C_{\rm e}^{(6)}$
$\displaystyle=$ $\displaystyle 1.181\,234\,017\,,$ $\displaystyle C_{\rm
e}^{(8)}$ $\displaystyle=$ $\displaystyle-1.9144(35)\,,$ $\displaystyle C_{\rm
e}^{(10)}$ $\displaystyle=$ $\displaystyle 0.0(4.6)\,.$ (117)
The electroweak contribution, calculated as in CODATA-98 but with the 2010
values of $G_{\rm F}$ and ${\rm sin}^{2}\theta_{\rm W}$, is
$\displaystyle a_{\rm e}({\rm weak})$ $\displaystyle=$ $\displaystyle
0.029\,73(52)\times 10^{-12}$ (118) $\displaystyle=$ $\displaystyle
0.2564(45)\times 10^{-10}a_{\rm e}\,.$
The hadronic contribution can be written as
$\displaystyle a_{\rm e}({\rm had})=a_{\rm e}^{(4)}({\rm had})+a_{\rm
e}^{(6a)}({\rm had})+a_{\rm
e}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}({\rm had})+\cdots,$
(119)
where $a_{\rm e}^{(4)}{({\rm had})}$ and $a_{\rm e}^{(6a)}{({\rm had})}$ are
due to hadronic vacuum polarization and are of order
$({\alpha/\mbox{{p}}})^{2}$ and $({\alpha/\mbox{{p}}})^{3}$, respectively;
also of order $({\alpha/\mbox{{p}}})^{3}$ is
$a_{\mbox{\scriptsize{{m}}}}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}$,
which is due to light-by-light vacuum polarization. Its value,
$\displaystyle a_{\rm e}({\rm had})$ $\displaystyle=$ $\displaystyle
1.685(22)\times 10^{-12}$ (120) $\displaystyle=$ $\displaystyle
1.453(19)\times 10^{-9}a_{\rm e}\,,$
is the sum of the following three contributions: $a_{\rm e}^{(4)}({\rm
had})=1.875(18)\times 10^{-12}$ obtained by Davier and Höcker (1998); $a_{\rm
e}^{(6a)}({\rm had})=-0.225(5)\times 10^{-12}$ given by Krause (1997); and
$a_{\rm e}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}(\rm
had)=0.035(10)\times 10^{-12}$ as given by Prades _et al._ (2010). In past
adjustments this contribution was calculated by assuming that $a_{\rm
e}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}=(m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}})^{2}\,a_{\mbox{\scriptsize{{m}}}}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}(\rm
had)$. However, Prades _et al._ (2010) have shown that such scaling is not
adequate for the neutral pion exchange contribution to
$a_{\mbox{\scriptsize{{m}}}}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}(\rm
had)$ and have taken this into account in obtaining their above result for
$a_{\rm e}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}(\rm had)$ from
their muon value
$a_{\mbox{\scriptsize{{m}}}}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}(\rm
had)=105(26)\times 10^{-11}$.
The theoretical prediction is
$\displaystyle a_{\rm e}({\rm th})=a_{\rm e}({\rm QED})+a_{\rm e}({\rm
weak})+a_{\rm e}({\rm had})\ .$ (121)
The various contributions can be put into context by comparing them to the
most accurate experimental value of $a_{\rm e}$ currently available, which has
an uncertainty of $2.8\times 10^{-10}a_{\rm e}$; see Eq. (125) below.
The standard uncertainty of $a_{\rm e}({\rm th})$ from the uncertainties of
the terms listed above is
$\displaystyle u[a_{\rm e}({\rm th)}]=0.33\times 10^{-12}=2.8\times
10^{-10}\,a_{\rm e},$ (122)
and is dominated by the uncertainty of the coefficient $C_{\rm e}^{(10)}$.
For the purpose of the least-squares calculations carried out in Sec. XIII, we
include an additive correction $\delta_{\rm e}$ to $a_{\rm e}({\rm th)}$ to
account for the uncertainty of $a_{\rm e}({\rm th)}$ other than that due to
$\alpha$, and hence the complete theoretical expression in the observational
equation for the electron anomaly ($B13$ in Table 33) is
$\displaystyle a_{\rm e}(\alpha,\delta_{\rm e})=a_{\rm e}({\rm
th)}+\delta_{\rm e}\,.$ (123)
The input datum for $\delta_{\rm e}$ is zero with standard uncertainty
$u[a_{\rm e}({\rm th})]$, or $0.00(33)\times 10^{-12}$, which is data item
$B12$ in Table LABEL:tab:pdata.
#### V.1.2 Measurements of $a_{\rm e}$
##### University of Washington.
The classic series of measurements of the electron and positron anomalies
carried out at the University of Washington by Van Dyck _et al._ (1987) yield
the value
$\displaystyle a_{\rm e}=1.159\,652\,1883(42)\times 10^{-3}~{}~{}[3.7\times
10^{-9}]\ ,$ (124)
as discussed in CODATA-98. This result, which assumes that $CPT$ invariance
holds for the electron-positron system, is data item $B13.1$ in Table
LABEL:tab:pdata.
##### Harvard University.
In both the University of Washington and Harvard University, Cambridge MA, USA
experiments, the electron magnetic moment anomaly is essentially determined
from the relation $a_{\rm e}=f_{\rm a}/f_{\rm c}$ by measuring in the same
magnetic flux density $B\approx 5~{}{\rm T}$ the anomaly difference frequency
$f_{\rm a}=f_{\rm s}-f_{\rm c}$ and cyclotron frequency $f_{\rm
c}=eB/2{\mbox{{p}}}m_{\rm e}$, where $f_{\rm s}=|g_{\rm
e}|{\mbox{\scriptsize{{m}}}}_{\rm B}B/h$ is the electron spin-flip (or
precession) frequency.
Because of its small relative standard uncertainty of $7.6\times 10^{-10}$,
the then new result for $a_{e}$ obtained by Odom _et al._ (2006) at Harvard
using a cylindrical rather than a hyperbolic Penning trap played the dominant
role in determining the 2006 recommended value of $\rm\alpha$. This work
continued with a number of significant improvements and a new value of $a_{e}$
consistent with the earlier one but with an uncertainty nearly a factor of
three smaller was reported by Hanneke _et al._ (2008):
$\displaystyle a_{\rm e}$ $\displaystyle=$ $\displaystyle
1.159\,652\,180\,73(28)\times 10^{-3}\,.$ (125)
A paper that describes this measurement in detail was subsequently published
by Hanneke _et al._ (2011) (see also the review by Gabrielse (2010)). As
discussed by Hanneke _et al._ (2011), the improvement that contributed most to
the reduction in uncertainty is a better understanding of the Penning trap
cavity frequency shifts of the radiation used to measure $f_{\rm c}$. A
smaller reduction resulted from narrower linewidths of the anomaly and
cyclotron resonant frequencies. Consequently, Hanneke _et al._ (2011) state
that their 2008 result should be viewed as superseding the earlier Harvard
result. Therefore, only the value of $a_{e}$ in Eq. (125) is included as an
input datum in the 2010 adjustment; it is data item $B13.2$ in Table
LABEL:tab:pdata.
#### V.1.3 Values of $\alpha$ inferred from $a_{\rm e}$
Equating the theoretical expression with the two experimental values of
$a_{\rm e}$ given in Eqs. (124) and (125) yields
$\displaystyle\alpha^{-1}(a_{\rm e})=137.035\,998\,19(50)~{}~{}[3.7\times
10^{-9}]$ (126)
from the University of Washington result and
$\displaystyle\alpha^{-1}(a_{\rm e})=137.035\,999\,084(51)~{}~{}[3.7\times
10^{-10}]$ (127)
from the Harvard University result. The contribution of the uncertainty in
$a_{\rm e}({\rm th})$ to the relative uncertainty of either of these results
is $2.8\times 10^{-10}$. The value in Eq. (127) has the smallest uncertainty
of any value of alpha currently available. The fact that the next most
accurate value of ${\rm\alpha}$, which has a relative standard uncertainty of
$6.6\times 10^{-10}$ and is obtained from the quotient $h/m(^{87}{\rm Rb})$
measured by atom recoil, is consistent with this value suggests that the
theory of $a_{\rm e}$ is well in hand; see Sec. XIII.
### V.2 Muon magnetic moment anomaly $\bm{a_{\mbox{\scriptsize{{m}}}}}$
The 2006 adjustment included data that provided both an experimental value and
a theoretical value for $a_{\mbox{\scriptsize{{m}}}}$. Because of problems
with the theory, the uncertainty assigned to the theoretical value was over
three times larger than that of the experimental value. Nevertheless, the
theoretical value with its increased uncertainty was included in the
adjustment, even if with a comparatively small weight.
For the 2010 adjustment, the Task Group decided not to include the theoretical
value for $a_{\mbox{\scriptsize{{m}}}}$, with the result that the 2010
recommended value is based mainly on experiment. This is consistent with the
fact that the value of $a_{\mbox{\scriptsize{{m}}}}$ recommended by the
Particle Data Group in their biennial 2010 _Review of Particle Physics_
Nakamura _et al._ (2010) is the experimental value. The current situation is
briefly summarized in the following sections.
#### V.2.1 Theory of ${a_{\mbox{\scriptsize{{m}}}}}$
The mass-independent coefficients $A_{1}^{(n)}$ for the muon are the same as
for the electron. Based on the 2010 recommended values of the mass ratios, the
relevant mass-dependent terms are
$\displaystyle A_{2}^{(4)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm e})$
$\displaystyle=$ $\displaystyle 1.094\,258\,3118(81)$ (128)
$\displaystyle\rightarrow$ $\displaystyle 506\,386.4620(38)\times
10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A_{2}^{(4)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}})$
$\displaystyle=$ $\displaystyle 0.000\,078\,079(14)$ (129)
$\displaystyle\rightarrow$ $\displaystyle 36.1325(65)\times
10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A^{(6)}_{2}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm e})$ $\displaystyle=$
$\displaystyle 22.868\,380\,04(19)$ (130) $\displaystyle\rightarrow$
$\displaystyle 24\,581.766\,56(20)\times
10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A^{(6)}_{2}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}})$
$\displaystyle=$ $\displaystyle 0.000\,360\,63(11)$ (131)
$\displaystyle\rightarrow$ $\displaystyle 0.387\,65(12)\times
10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A_{2}^{(8)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm e})$ $\displaystyle=$
$\displaystyle 132.6823(72)$ (132) $\displaystyle\rightarrow$ $\displaystyle
331.288(18)\times 10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A_{2}^{(10)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm e})$ $\displaystyle=$
$\displaystyle 663(20)$ (133) $\displaystyle\rightarrow$ $\displaystyle
3.85(12)\times 10^{-8}a_{\mbox{\scriptsize{{m}}}}\,,$ $\displaystyle
A_{3}^{(6)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm
e},m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}})$ $\displaystyle=$
$\displaystyle 0.000\,527\,762(94)$ (134) $\displaystyle\rightarrow$
$\displaystyle 0.567\,30(10)\times 10^{-8}a_{\mbox{\scriptsize{{m}}}\,,}$
$\displaystyle A_{3}^{(8)}\\!(m_{\mbox{\scriptsize{{m}}}}/m_{\rm
e},m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}})$ $\displaystyle=$
$\displaystyle 0.037\,594(83)$ (135) $\displaystyle\rightarrow$ $\displaystyle
0.093\,87(21)\times 10^{-8}a_{\mbox{\scriptsize{{m}}}}\,.\qquad$
The QED contribution to the theory of $a_{\mbox{\scriptsize{{m}}}}$, where
terms that have like powers of $\alpha/\mbox{{p}}$ are combined, is
$\displaystyle a_{\mbox{\scriptsize{{m}}}}({\rm QED})$ $\displaystyle=$
$\displaystyle
C_{\mbox{\scriptsize{{m}}}}^{(2)}\left({\alpha\over\mbox{{p}}}\right)+C_{\mbox{\scriptsize{{m}}}}^{(4)}\left({\alpha\over\mbox{{p}}}\right)^{2}+C_{\mbox{\scriptsize{{m}}}}^{(6)}\left({\alpha\over\mbox{{p}}}\right)^{3}$
(136)
$\displaystyle+C_{\mbox{\scriptsize{{m}}}}^{(8)}\left({\alpha\over\mbox{{p}}}\right)^{4}+C_{\mbox{\scriptsize{{m}}}}^{(10)}\left({\alpha\over\mbox{{p}}}\right)^{5}+\cdots,$
with
$\displaystyle C_{\mbox{\scriptsize{{m}}}}^{(2)}$ $\displaystyle=$
$\displaystyle 0.5\,,$ $\displaystyle C_{\mbox{\scriptsize{{m}}}}^{(4)}$
$\displaystyle=$ $\displaystyle 0.765\,857\,426(16)\,,$ $\displaystyle
C_{\mbox{\scriptsize{{m}}}}^{(6)}$ $\displaystyle=$ $\displaystyle
24.050\,509\,88(28)\,,$ $\displaystyle C_{\mbox{\scriptsize{{m}}}}^{(8)}$
$\displaystyle=$ $\displaystyle 130.8055(80)\,,$ $\displaystyle
C_{\mbox{\scriptsize{{m}}}}^{(10)}$ $\displaystyle=$ $\displaystyle
663(21)\,,$ (137)
which yields, using the 2010 recommended value of $\alpha$,
$\displaystyle a_{\mbox{\scriptsize{{m}}}}(\rm
QED)=0.001\,165\,847\,1810(15)\quad[1.3\times 10^{-9}]\,.\quad$ (138)
In absolute terms, the uncertainty in $a_{\mbox{\scriptsize{{m}}}}({\rm QED})$
is $0.15\times 10^{-11}$.
The current theoretical expression for the muon anomaly is of the same form as
for the electron:
$\displaystyle a_{\mbox{\scriptsize{{m}}}}({\rm
th})=a_{\mbox{\scriptsize{{m}}}}({\rm QED})+a_{\mbox{\scriptsize{{m}}}}({\rm
weak})+a_{\mbox{\scriptsize{{m}}}}({\rm had})\,.$ (139)
The electroweak contribution, calculated by Czarnecki _et al._ (2003), is
$a_{\mbox{\scriptsize{{m}}}}{({\rm weak})}=154(2)\times 10^{-11}$. In contrast
to the case of the electron, $a_{\mbox{\scriptsize{{m}}}}({\rm weak})$ is a
significant contribution compared to $a_{\mbox{\scriptsize{{m}}}}({\rm QED})$.
In a manner similar to that for the electron, the hadronic contribution can be
written as
$\displaystyle a_{\mbox{\scriptsize{{m}}}}{({\rm
had})}=a_{\mbox{\scriptsize{{m}}}}^{(4)}{({\rm
had})}+a_{\mbox{\scriptsize{{m}}}}^{(6a)}({\rm
had})+a_{\mbox{\scriptsize{{m}}}}^{(\mbox{\scriptsize{{g}}}\mbox{\scriptsize{{g}}})}({\rm
had})+\cdots\ .$ (140)
It is also of much greater importance for the muon than for the electron.
Indeed, $a_{\mbox{\scriptsize{{m}}}}(\rm had)$ is roughly $7000(50)\times
10^{-11}$, which should be compared with the $63\times 10^{-11}$ uncertainty
of the experimental value $a_{\mbox{\scriptsize{{m}}}}(\rm exp)$ discussed in
the next section.
For well over a decade a great deal of effort has been devoted by many
researchers to the improved evaluation of $a_{\mbox{\scriptsize{{m}}}}({\rm
had})$. The standard method of calculating
$a_{\mbox{\scriptsize{{m}}}}^{(4)}{({\rm had})}$ and
$a_{\mbox{\scriptsize{{m}}}}^{(6a)}({\rm had})$ is to evaluate dispersion
integrals over experimentally measured cross sections for the scattering of
${\rm e}^{+}{\rm e}^{-}$ into hadrons. However, in some calculations data on
decays of the t into hadrons are used to replace the ${\rm e}^{+}{\rm e}^{-}$
data in certain energy regions. The results of three evaluations which include
the most recent data can be concisely summarized as follows.
Davier _et al._ (2011) find that $a_{\mbox{\scriptsize{{m}}}}({\rm exp})$
exceeds their theoretically predicted value $a_{\mbox{\scriptsize{{m}}}}({\rm
th})$ by $3.6$ times the combined standard uncertainty of the difference, or
$3.6{\sigma}$, using only ${\rm e}^{+}{\rm e}^{-}$ data, and by $2.4\sigma$ if
t data are included. On the other hand, Jegerlehner and Szafron (2011) find
that by correcting the t data for the effect they term r \- g mixing, the
values of $a_{\mbox{\scriptsize{{m}}}}^{(4)}({\rm had})$ obtained from only
${\rm e}^{+}{\rm e}^{-}$ data, and from ${\rm e}^{+}{\rm e}^{-}$ and t data
together, are nearly identical and that the difference between experiment and
theory is $3.3{\sigma}$. And Hagiwara _et al._ (2011) find the same
$3.3{\sigma}$ difference using ${\rm e}^{+}{\rm e}^{-}$ data alone. Finally,
we note that in a very recent paper, Benayoun _et al._ (2012) obtain a
difference in the range $4.07\sigma$ to $4.65\sigma$, depending on the
assumptions made, using a “hidden local symmetry” model.
The disagreement between experiment and theory has long been known and
numerous theoretical papers have been published that attempt to explain the
discrepancy in terms of New Physics; see the review by Stöckinger (2010).
Although a contribution to $a_{\mbox{\scriptsize{{m}}}}({\rm th})$ large
enough to bring it into agreement with $a_{\mbox{\scriptsize{{m}}}}({\rm
exp})$ from physics beyond the Standard Model is possible, no outside
experimental evidence currently exists for such physics. Thus, because of the
persistence of the discrepancy and its confirmation by the most recent
calculations, and because no known physics has yet been able to eliminate it,
the Task Group has decided to omit the theory of $a_{\mbox{\scriptsize{{m}}}}$
from the 2010 adjustment.
#### V.2.2 Measurement of $a_{\mbox{\scriptsize{{m}}}}$: Brookhaven
Experiment E821 at BNL has been discussed in the past three CODATA reports. It
involves the direct measurement of the anomaly difference frequency $f_{\rm
a}=f_{\rm s}-f_{\rm c}$, where $f_{\rm
s}=|g_{\mbox{\scriptsize{{m}}}}|(e\hbar/2m_{\mbox{\scriptsize{{m}}}})B/h$ is
the muon spin-flip (or precession) frequency in the applied magnetic flux
density $B$ and $f_{\rm c}=eB/2{\mbox{{p}}}m_{\mbox{\scriptsize{{m}}}}$ is the
corresponding muon cyclotron frequency. However, in contrast to the case of
the electron where both $f_{\rm a}$ and $f_{\rm c}$ are measured directly and
the electron anomaly is calculated from $a_{\rm e}=f_{\rm a}/f_{\rm c}$, for
the muon $B$ is eliminated by determining its value from proton nuclear
magnetic resonance (NMR) measurements. This means that the muon anomaly is
calculated from
$\displaystyle a_{\mbox{\scriptsize{{m}}}}({\rm
exp})=\frac{\overline{R}}{|\mu_{\mbox{\scriptsize{{m}}}}/{\mu_{p}}|-\overline{R}}\,,$
(141)
where $\overline{R}=f_{\rm a}/\overline{f}_{\rm p}$ and $\overline{f}_{\rm p}$
is the free proton NMR frequency corresponding to the average flux density $B$
seen by the muons in their orbits in the muon storage ring.
The final value of $\overline{R}$ obtained in the E821 experiment is Bennett
_et al._ (2006)
$\displaystyle\overline{R}=0.003\,707\,2063(20)\,,$ (142)
which is used as an input datum in the 2010 adjustment and is data item $B14$
in Table LABEL:tab:pdata. [The last digit of this value is one less than that
of the value used in 2006, because the 2006 value was taken from Eq. (57) in
the paper by Bennett _et al._ (2006) but the correct value is that given in
Table XV Roberts (2009).] Based on this value of $\overline{R}$, Eq. (141),
and the 2010 recommended value of $\mu_{\mbox{\scriptsize{{m}}}}/{\mu_{\rm
p}}$, whose uncertainty is negligible in this context, the experimental value
of the muon anomaly is
$\displaystyle a_{\mbox{\scriptsize{{m}}}}({\rm exp})=1.165\,920\,91(63)\times
10^{-3}\,.$ (143)
Further, with the aid of Eq. (229), the equation for $\overline{R}$ can be
written as
$\displaystyle\overline{R}=-\frac{a_{\mbox{\scriptsize{{m}}}}}{1+a_{\rm
e}(\alpha,\delta_{\rm e})}\frac{m_{\rm
e}}{m_{\mbox{\scriptsize{{m}}}}}\frac{\mu_{{\rm e}^{-}}}{\mu_{\rm p}}\ ,$
(144)
where use has been made of the relations $g_{\rm e}=-2(1+a_{\rm e})$,
$g_{\mbox{\scriptsize{{m}}}}=-2(1+a_{\mbox{\scriptsize{{m}}}})$, and $a_{\rm
e}$ is replaced by the theoretical expression $a_{\rm e}(\alpha,\delta_{\rm
e})$ given in Eq. (105). However, since the theory of
$a_{\mbox{\scriptsize{{m}}}}$ is omitted from the 2010 adjustment,
$a_{\mbox{\scriptsize{{m}}}}$ is not replaced in Eq. (144) by a theoretical
expression, rather it is made to be an adjusted constant.
### V.3 Bound electron $\bm{g}$-factor in $\bm{{}^{12}{\rm C}^{5+}}$ and in
$\bm{{}^{16}{\rm O}^{7+}}$ and $\bm{A_{\rm r}({\rm e})}$
Competitive values of $A_{\rm r}$(e) can be obtained from precise measurements
and theoretical calculations of the $g$-factor of the electron in hydrogenic
12C and 16O.
For a ground-state hydrogenic ion ${}^{A}X^{(Z-1)+}$ with mass number $A$,
atomic number (proton number) $Z$, nuclear spin quantum number $i$ = 0, and
$g$-factor $g_{\rm e^{-}}(^{A}{X}^{(Z-1)+})$ in an applied magnetic flux
density $B$, the ratio of the electron’s spin-flip (or precession) frequency
$f_{\rm s}=|g_{{\rm e}^{-}}(^{A}{X}^{(Z-1)+})|(e\hbar/2m_{\rm e})B/h$ to the
cyclotron frequency of the ion $f_{\rm
c}=(Z-1)eB/2\mbox{{p}}m(^{A}{X}^{(Z-1)+})$ in the same magnetic flux density
is
$\displaystyle\frac{f_{\rm s}(^{A}{X}^{(Z-1)+})}{f_{\rm c}(^{A}{X}^{(Z-1)+})}$
$\displaystyle=$ $\displaystyle-\frac{g_{{\rm
e}^{-}}(^{A}{X}^{(Z-1)+})}{2(Z-1)}\frac{A_{\rm r}(^{A}{X}^{(Z-1)+})}{A_{\rm
r}({\rm e})}\ ,$
where $A_{\rm r}(X)$ is the relative atomic mass of particle $X$.
This expression can be used to obtain a competitive result for $A_{\rm r}(\rm
e)$ if for a particular ion the quotient $f_{\rm s}/f_{\rm c}$, its bound
state $g$-factor, and the relative atomic mass of the ion can be obtained with
sufficiently small uncertainties. In fact, work underway since the mid-1990s
has been so successful that Eq. (LABEL:eq:fsfcgx) now provides the most
accurate values of $A_{\rm r}(\rm e)$. Measurements of $f_{\rm s}/f_{\rm c}$
for ${}^{12}{\rm C}^{5+}$ and ${}^{16}{\rm O}^{7+}$, performed at the
Gesellschaft für Schwerionenforschung, Darmstadt, Germany (GSI) by GSI and
University of Mainz researchers, are discussed in CODATA-06 and the results
were included in the 2006 adjustment. These data are recalled in Sec. V.3.2
below, and the present status of the theoretical expressions for the bound-
state $g$-factors of the two ions are discussed in the following section.
For completeness, we note that well after the closing date of the 2010
adjustment Sturm _et al._ (2011) reported a value of $f_{\rm s}/f_{\rm c}$ for
the hydrogenic ion ${}^{28}{\rm Si}^{13+}$. Using the 2006 recommended value
of $A_{\rm r}(\rm e)$ and the applicable version of Eq. (LABEL:eq:fsfcgx),
they found good agreement between the theoretical and experimental values of
the $g$-factor of this ion, thereby strengthening confidence in our
understanding of bound-state QED theory.
#### V.3.1 Theory of the bound electron $g$-factor
The energy of a free electron with spin projection $s_{z}$ in a magnetic flux
density $B$ in the $z$ direction is
$\displaystyle E$ $\displaystyle=$ $\displaystyle-\bm{\mu}\cdot\bm{B}=-g_{\rm
e^{-}}{e\over 2m_{\rm e}}s_{z}B\ ,$ (146)
and hence the spin-flip energy difference is
$\displaystyle\Delta E=-g_{\rm e^{-}}\mu_{\rm B}B\ .$ (147)
(In keeping with the definition of the $g$-factor in Sec. V, the quantity
$g_{\rm e^{-}}$ is negative.) The analogous expression for ions with no
nuclear spin is
$\displaystyle\Delta E_{\rm b}(X)$ $\displaystyle=$ $\displaystyle-g_{\rm
e^{-}}(X)\mu_{\rm B}B\ ,$ (148)
which defines the bound-state electron $g$-factor, and where $X$ is either
${}^{12}{\rm C}^{5+}$ or 16O7+.
The theoretical expression for $g_{\rm e^{-}}(X)$ is written as
$\displaystyle g_{\rm e^{-}}(X)=g_{\rm D}+\Delta g_{\rm rad}+\Delta g_{\rm
rec}+\Delta g_{\rm ns}+\cdots\ ,$ (149)
where the individual terms are the Dirac value, the radiative corrections, the
recoil corrections, and the nuclear size corrections, respectively. Numerical
results are summarized in Tables 13 and 14.
Table 13: Theoretical contributions and total for the $g$-factor of the electron in hydrogenic carbon 12 based on the 2010 recommended values of the constants. Contribution | Value | Source
---|---|---
Dirac $g_{\rm D}$ | $-1.998\,721\,354\,390\,9(8)$ | Eq. (LABEL:eq:diracg)
$\Delta g^{(2)}_{\rm SE}$ | $-0.002\,323\,672\,436(4)$ | Eq. (158)
$\Delta g^{(2)}_{\rm VP}$ | $\phantom{-}0.000\,000\,008\,512(1)$ | Eq. (161)
$\Delta g^{(4)}$ | $\phantom{-}0.000\,003\,545\,677(25)$ | Eq. (165)
$\Delta g^{(6)}$ | $-0.000\,000\,029\,618$ | Eq. (167)
$\Delta g^{(8)}$ | $\phantom{-}0.000\,000\,000\,111$ | Eq. (168)
$\Delta g^{(10)}$ | $\phantom{-}0.000\,000\,000\,000(1)$ | Eq. (169)
$\Delta g_{\rm rec}$ | $-0.000\,000\,087\,629$ | Eqs. (170)-(172)
$\Delta g_{\rm ns}$ | $-0.000\,000\,000\,408(1)$ | Eq. (174)
$g_{\rm e^{-}}(^{12}{\rm C}^{5+})$ | $-2.001\,041\,590\,181(26)\vbox to11.38092pt{}$ | Eq. (LABEL:eq:gco)
Table 14: Theoretical contributions and total for the $g$-factor of the electron in hydrogenic oxygen 16 based on the 2010 recommended values of the constants. Contribution | Value | Source
---|---|---
Dirac $g_{\rm D}$ | $-1.997\,726\,003\,06$ | Eq. (LABEL:eq:diracg)
$\Delta g^{(2)}_{\rm SE}$ | $-0.002\,324\,442\,14(1)$ | Eq. (158)
$\Delta g^{(2)}_{\rm VP}$ | $\phantom{-}0.000\,000\,026\,38$ | Eq. (161)
$\Delta g^{(4)}$ | $\phantom{-}0.000\,003\,546\,54(11)$ | Eq. (165)
$\Delta g^{(6)}$ | $-0.000\,000\,029\,63$ | Eq. (167)
$\Delta g^{(8)}$ | $\phantom{-}0.000\,000\,000\,11$ | Eq. (168)
$\Delta g^{(10)}$ | $\phantom{-}0.000\,000\,000\,00$ | Eq. (169)
$\Delta g_{\rm rec}$ | $-0.000\,000\,117\,00$ | Eqs. (170)-(172)
$\Delta g_{\rm ns}$ | $-0.000\,000\,001\,56(1)$ | Eq. (174)
$g_{\rm e^{-}}(^{16}{\rm O}^{7+})$ | $-2.000\,047\,020\,35(11)\vbox to11.38092pt{}$ | Eq. (LABEL:eq:gco)
Breit (1928) obtained the exact value
$\displaystyle g_{\rm D}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\left[1+2\sqrt{1-(Z\alpha)^{2}}\right]$
$\displaystyle=$
$\displaystyle-2\left[1-\frac{1}{3}(Z\alpha)^{2}-\frac{1}{12}(Z\alpha)^{4}-\frac{1}{24}(Z\alpha)^{6}+\cdots\right]$
from the Dirac equation for an electron in the field of a fixed point charge
of magnitude $Ze$, where the only uncertainty is that due to the uncertainty
in $\alpha$.
For the radiative corrections we have
$\displaystyle\Delta g_{\rm rad}$ $\displaystyle=$
$\displaystyle-2\left[C_{\rm
e}^{(2)}(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)+C_{\rm
e}^{(4)}(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)^{2}+\cdots\right]\ ,$
where
$\displaystyle\lim_{Z\alpha\rightarrow 0}C_{\rm e}^{(2n)}(Z\alpha)=C_{\rm
e}^{(2n)}\,,$ (152)
and where the $C_{\rm e}^{(2n)}$ are given in Eq. (117).
For the coefficient $C_{\rm e}^{(2)}(Z\alpha)$, we have Grotch (1970); Faustov
(1970); Close and Osborn (1971); Pachucki _et al._ (2004); Pachucki _et al._
(2005a)
$\displaystyle C_{\rm e,SE}^{(2)}(Z\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{\\{}1+\frac{(Z\alpha)^{2}}{6}+(Z\alpha)^{4}\left[\frac{32}{9}\,\ln{(Z\alpha)^{-2}}\right.$
(153)
$\displaystyle\left.\quad+\frac{247}{216}-\frac{8}{9}\,\ln{k_{0}}-\frac{8}{3}\,\ln{k_{3}}\right]$
$\displaystyle\quad+(Z\alpha)^{5}\,R_{\rm SE}(Z\alpha)\bigg{\\}}\ ,$
where
$\displaystyle\ln{k_{0}}$ $\displaystyle=$ $\displaystyle 2.984\,128\,556\,,$
(154) $\displaystyle\ln{k_{3}}$ $\displaystyle=$ $\displaystyle
3.272\,806\,545\,,$ (155) $\displaystyle R_{\rm SE}(6\alpha)$ $\displaystyle=$
$\displaystyle 22.160(10)\,,$ (156) $\displaystyle R_{\rm SE}(8\alpha)$
$\displaystyle=$ $\displaystyle 21.859(4)\,.$ (157)
The quantity $\ln{k_{0}}$ is the Bethe logarithm for the 1S state (see Table
5), $\ln{k_{3}}$ is a generalization of the Bethe logarithm, and $R_{\rm
SE}(Z\alpha)$ was obtained by extrapolation of the results of numerical
calculations at higher $Z$ Yerokhin _et al._ (2002); Pachucki _et al._ (2004).
Equation (153) yields
$\displaystyle C_{\rm e,SE}^{(2)}(6\alpha)$ $\displaystyle=$ $\displaystyle
0.500\,183\,606\,65(80)\,,$ $\displaystyle C_{\rm e,SE}^{(2)}(8\alpha)$
$\displaystyle=$ $\displaystyle 0.500\,349\,2887(14)\,.$ (158)
The one loop self energy has been calculated directly at $Z=6$ and $Z=8$ by
Yerokhin and Jentschura (2008, 2010). The results are in agreement with, but
less accurate than the extrapolation from higher-$Z$.
The lowest-order vacuum-polarization correction consists of a wave-function
correction and a potential correction, each of which can be separated into a
lowest-order Uehling potential contribution and a Wichmann-Kroll higher-
contribution. The wave-function correction is Beier _et al._ (2000);
Karshenboim (2000); Karshenboim _et al._ (2001a, b)
$\displaystyle C_{\rm e,VPwf}^{(2)}(6\alpha)$ $\displaystyle=$
$\displaystyle-0.000\,001\,840\,3431(43)\,,$ $\displaystyle C_{\rm
e,VPwf}^{(2)}(8\alpha)$ $\displaystyle=$
$\displaystyle-0.000\,005\,712\,028(26)\,.$ (159)
For the potential correction, we have Beier _et al._ (2000); Beier (2000);
Karshenboim and Milstein (2002); Mohr and Taylor (2005); Lee _et al._ (2005)
$\displaystyle C_{\rm e,VPp}^{(2)}(6\alpha)$ $\displaystyle=$ $\displaystyle
0.000\,000\,008\,08(12)\,,$ $\displaystyle C_{\rm e,VPp}^{(2)}(8\alpha)$
$\displaystyle=$ $\displaystyle 0.000\,000\,033\,73(50)\,,$ (160)
which is the unweighted average of two slightly inconsistent results with an
uncertainty of half their difference. The total one-photon vacuum polarization
coefficients are given by the sum of Eqs. (159) and (160):
$\displaystyle C_{\rm e,VP}^{(2)}(6\alpha)$ $\displaystyle=$ $\displaystyle
C_{\rm e,VPwf}^{(2)}(6\alpha)+C_{\rm e,VPp}^{(2)}(6\alpha)$ $\displaystyle=$
$\displaystyle-0.000\,001\,832\,26(12)\,,$ $\displaystyle C_{\rm
e,VP}^{(2)}(8\alpha)$ $\displaystyle=$ $\displaystyle C_{\rm
e,VPwf}^{(2)}(8\alpha)+C_{\rm e,VPp}^{(2)}(8\alpha)$ (161) $\displaystyle=$
$\displaystyle-0.000\,005\,678\,30(50)\,.$
The total one-photon coefficient is the sum of Eqs. (158) and (161):
$\displaystyle C_{\rm e}^{(2)}(6\alpha)$ $\displaystyle=$ $\displaystyle
C_{\rm e,SE}^{(2)}(6\alpha)+C_{\rm e,VP}^{(2)}(6\alpha)$ $\displaystyle=$
$\displaystyle 0.500\,181\,774\,39(81)\,,$ $\displaystyle C_{\rm
e}^{(2)}(8\alpha)$ $\displaystyle=$ $\displaystyle C_{\rm
e,SE}^{(2)}(8\alpha)+C_{\rm e,VP}^{(2)}(8\alpha)$ (162) $\displaystyle=$
$\displaystyle 0.500\,343\,6104(14)\,,$
and the total one-photon contribution is
$\displaystyle\Delta g^{(2)}$ $\displaystyle=$ $\displaystyle-2\,C_{\rm
e}^{(2)}(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)$ $\displaystyle=$
$\displaystyle-0.002\,323\,663\,924(4)\quad{\rm for}~{}Z=6$ $\displaystyle=$
$\displaystyle-0.002\,324\,415\,756(7)\quad{\rm for}~{}Z=8\ .$
Separate one-photon self energy and vacuum polarization contributions to the
$g$-factor are given in Tables 13 and 14.
The leading binding correction to the higher-order coefficients is Eides and
Grotch (1997a); Czarnecki _et al._ (2001)
$\displaystyle C_{\rm e}^{(2n)}(Z\alpha)=C_{\rm
e}^{(2n)}\left(1+{(Z\alpha)^{2}\over 6}+\cdots\right)\,.$ (164)
The two-loop contribution of relative order $(Z\alpha)^{4}$ for the ground S
state is Pachucki _et al._ (2005a); Jentschura _et al._ (2006)
$\displaystyle C_{\rm e}^{(4)}(Z\alpha)$ $\displaystyle=$ $\displaystyle
C_{\rm e}^{(4)}\left(1+{(Z\alpha)^{2}\over 6}\right)$ (165)
$\displaystyle\hbox
to-56.9055pt{}+(Z\alpha)^{4}\,\bigg{[}\frac{14}{9}\,\ln{(Z\alpha)^{-2}}+\frac{991343}{155520}-\frac{2}{9}\,\ln{k_{0}}-\frac{4}{3}\,\ln{k_{3}}$
$\displaystyle\hbox
to-56.9055pt{}+\frac{679\,\mbox{{p}}^{2}}{12960}-\frac{1441\,\mbox{{p}}^{2}}{720}\,\ln{2}+\frac{1441}{480}\,\zeta({3})\bigg{]}+{\cal
O}(Z\alpha)^{5}$ $\displaystyle=$ $\displaystyle-0.328\,5778(23)\quad{\rm
for}~{}Z=6$ $\displaystyle=$ $\displaystyle-0.328\,6578(97)\quad{\rm
for}~{}Z=8\,,$
where $\ln{k_{0}}$ and $\ln{k_{3}}$ are given in Eqs. (154) and (155). As in
CODATA-06, the uncertainty due to uncalculated terms is taken to be Pachucki
_et al._ (2005a)
$\displaystyle u\left[C_{\rm e}^{(4)}(Z\alpha)\right]$ $\displaystyle=$
$\displaystyle 2\,\left|(Z\alpha)^{5}\,C_{\rm e}^{(4)}\,R_{\rm
SE}(Z\alpha)\right|.$ (166)
Jentschura (2009) has calculated a two-loop gauge-invariant set of vacuum
polarization diagrams to obtain a contribution of the same order in $Z\alpha$
as the above uncertainty. However, in general we do not include partial
results of a given order. Jentschura also speculates that the complete term of
that order could be somewhat larger than our uncertainty.
The three- and four-photon terms are calculated with the leading binding
correction included:
$\displaystyle C_{\rm e}^{(6)}(Z\alpha)$ $\displaystyle=$ $\displaystyle
C_{\rm e}^{(6)}\left(1+{(Z\alpha)^{2}\over 6}+\cdots\right)$ (167)
$\displaystyle=$ $\displaystyle 1.181\,611\dots\quad{\rm for}~{}Z=6$
$\displaystyle=$ $\displaystyle 1.181\,905\dots\quad{\rm for}~{}Z=8\,,$
where $C_{\rm e}^{(6)}=1.181\,234\dots$ , and
$\displaystyle C_{\rm e}^{(8)}(Z\alpha)$ $\displaystyle=$ $\displaystyle
C_{\rm e}^{(8)}\left(1+{(Z\alpha)^{2}\over 6}+\cdots\right)$ (168)
$\displaystyle=$ $\displaystyle-1.9150(35)\dots\quad{\rm for}~{}Z=6$
$\displaystyle=$ $\displaystyle-1.9155(35)\dots\quad{\rm for}~{}Z=8\,,$
where $C_{\rm e}^{(8)}=-1.9144(35)$. An uncertainty estimate
$\displaystyle C_{\rm e}^{(10)}(Z\alpha)$ $\displaystyle\approx$
$\displaystyle C_{\rm e}^{(10)}=0.0(4.6)$ (169)
is included for the five-loop correction.
The recoil correction to the bound-state $g$-factor is $\Delta g_{\rm
rec}=\Delta g_{\rm rec}^{(0)}+\Delta g_{\rm rec}^{(2)}+\dots$ where the terms
on the right are zero- and first-order in $\alpha/\mbox{{p}}$, respectively.
We have
$\displaystyle\Delta g_{\rm rec}^{(0)}$ $\displaystyle=$
$\displaystyle\bigg{\\{}-(Z\alpha)^{2}+{(Z\alpha)^{4}\over
3[1+\sqrt{1-(Z\alpha)^{2}}]^{2}}$ (170)
$\displaystyle-(Z\alpha)^{5}\,P(Z\alpha)\bigg{\\}}{m_{\rm e}\over m_{\rm
N}}+{\cal O}\left({m_{\rm e}\over m_{\rm N}}\right)^{\\!2}$ $\displaystyle=$
$\displaystyle-0.000\,000\,087\,70\dots~{}{\rm for}~{}Z=6$ $\displaystyle=$
$\displaystyle-0.000\,000\,117\,09\dots~{}{\rm for}~{}Z=8\,,$
where $m_{\rm N}$ is the mass of the nucleus. The mass ratios, obtained from
the 2010 adjustment, are ${m_{\rm e}/m(^{12}{\rm
C}^{6+})}=0.000\,045\,727\,5\ldots$ and ${m_{\rm e}/m(^{16}{\rm
O}^{8+})}=0.000\,034\,306\,5\ldots$. The recoil terms are the same as in
CODATA-02 and references to the original calculations are given there. An
additional term of the order of the mass ratio squared Eides and Grotch
(1997a); Eides (2002)
$\displaystyle(1+Z)(Z\alpha)^{2}\left({m_{\rm e}\over m_{\rm
N}}\right)^{\\!2}$ (171)
should also be included in the theory. The validity of this term for a nucleus
of any spin has been reconfirmed by Pachucki (2008); Eides and Martin (2010,
2011).
For $\Delta g_{\rm rec}^{(2)}$, we have
$\displaystyle\Delta g_{\rm rec}^{(2)}$ $\displaystyle=$
$\displaystyle{\alpha\over\mbox{{p}}}{(Z\alpha)^{2}\over 3}{m_{\rm e}\over
m_{\rm N}}+\cdots$ (172) $\displaystyle=$ $\displaystyle
0.000\,000\,000\,06\ldots~{}{\rm for}~{}Z=6$ $\displaystyle=$ $\displaystyle
0.000\,000\,000\,09\ldots~{}{\rm for}~{}Z=8\,.$
There is a small correction to the bound-state $g$-factor due to the finite
size of the nucleus, of order Karshenboim (2000)
$\displaystyle\Delta g_{\rm ns}=-{8\over 3}(Z\alpha)^{4}\left({R_{\rm
N}\over\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm
C}}\right)^{2}+\cdots\,,$ (173)
where $R_{\rm N}$ is the bound-state nuclear rms charge radius and
$\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ is
the Compton wavelength of the electron divided by $2\mbox{{p}}$. This term is
calculated by scaling the results of Glazov and Shabaev (2002) with the
squares of updated values for the nuclear radii $R_{\rm N}=2.4703(22)$ fm and
$R_{\rm N}=2.7013(55)$ from the compilation of Angeli (2004) for 12C and 16O,
respectively. This yields the correction
$\displaystyle\Delta g_{\rm ns}$ $\displaystyle=$
$\displaystyle-0.000\,000\,000\,408(1)\quad{\rm for}~{}^{12}{\rm C}\,,$
$\displaystyle\Delta g_{\rm ns}$ $\displaystyle=$
$\displaystyle-0.000\,000\,001\,56(1)\quad{\rm for}~{}^{16}{\rm O}\,.$ (174)
The theoretical value for the $g$-factor of the electron in hydrogenic carbon
12 or oxygen 16 is the sum of the individual contributions discussed above and
summarized in Tables 13 and 14:
$\displaystyle g_{\rm e^{-}}(^{12}{\rm C}^{5+})$ $\displaystyle=$
$\displaystyle-2.001\,041\,590\,181(26)\,,$ $\displaystyle g_{\rm
e^{-}}(^{16}{\rm O}^{7+})$ $\displaystyle=$
$\displaystyle-2.000\,047\,020\,35(11)\,.$
For the purpose of the least-squares calculations carried out in Sec. XIII, we
define $g_{\rm C}({\rm th})$ to be the sum of $g_{\rm D}$ as given in Eq.
(LABEL:eq:diracg), the term $-2(\alpha/\mbox{{p}})C_{\rm e}^{(2)}$, and the
numerical values of the remaining terms in Eq. (149) as given in Table 13,
where the standard uncertainty of these latter terms is
$\displaystyle u[g_{\rm C}({\rm th})]$ $\displaystyle=$ $\displaystyle
0.3\times 10^{-10}=1.3\times 10^{-11}|g_{\rm C}({\rm th})|\,.$
The uncertainty in $g_{\rm C}({\rm th})$ due to the uncertainty in $\alpha$
enters the adjustment primarily through the functional dependence of $g_{\rm
D}$ and the term $-2(\alpha/\mbox{{p}})C_{\rm e}^{(2)}$ on $\alpha$. Therefore
this particular component of uncertainty is not explicitly included in
$u[g_{\rm C}({\rm th})]$. To take the uncertainty $u[g_{\rm C}({\rm th})]$
into account we employ as the theoretical expression for the $g$-factor ($B17$
in Table 33)
$\displaystyle g_{\rm C}(\alpha,\delta_{\rm C})$ $\displaystyle=$
$\displaystyle g_{\rm C}({\rm th})+\delta_{\rm C}\,,$ (177)
where the input value of the additive correction $\delta_{\rm C}$ is taken to
be zero with standard uncertainty $u[g_{\rm C}({\rm th})]$, or $0.00(26)\times
10^{-10}$, which is data item $B15$ in Table LABEL:tab:pdata. Analogous
considerations apply for the $g$-factor in oxygen, where
$\displaystyle u[g_{\rm O}({\rm th})]$ $\displaystyle=$ $\displaystyle
1.1\times 10^{-10}=5.3\times 10^{-11}|g_{\rm O}({\rm th})|$
and ($B18$ in Table 33)
$\displaystyle g_{\rm O}(\alpha,\delta_{\rm O})$ $\displaystyle=$
$\displaystyle g_{\rm O}({\rm th})+\delta_{\rm O}\,.$ (179)
The input value for $\delta_{\rm O}$ is $0.0(1.1)\times 10^{-10}$, which is
data item $B16$ in Table LABEL:tab:pdata.
The covariance of the quantities $\delta_{\rm C}$ and $\delta_{\rm O}$ is
$\displaystyle u(\delta_{\rm C},\delta_{\rm O})=27\times 10^{-22}\ ,$ (180)
which corresponds to a correlation coefficient of $r(\delta_{\rm
C},\delta_{\rm O})=0.994$.
The theoretical value of the ratio of the two $g$-factors is
$\displaystyle{g_{\rm e^{-}}(^{12}{\rm C}^{5+})\over g_{\rm e^{-}}(^{16}{\rm
O}^{7+})}$ $\displaystyle=$ $\displaystyle 1.000\,497\,273\,224(40)\ ,$ (181)
where the covariance of the two values is taken into account.
#### V.3.2 Measurements of $g_{\rm e}(^{12}{\rm C}^{5+})$ and $g_{\rm
e}(^{16}{\rm O}^{7+})$
The experimental values of $f_{\rm s}/f_{\rm c}$ for ${}^{12}{\rm C}^{5+}$ and
${}^{16}{\rm O}^{7+}$ obtained at GSI using the double Penning trap method are
discussed in CODATA-02 and the slightly updated result for the oxygen ion is
discussed in CODATA-06. For ${}^{12}{\rm C}^{5+}$ we have Beier _et al._
(2002); Häffner _et al._ (2003); Werth (2003)
$\displaystyle{f_{\rm s}\left({}^{12}{\rm C}^{5+}\right)\over f_{\rm
c}\left({}^{12}{\rm C}^{5+}\right)}=4376.210\,4989(23)\ ,$ (182)
while for ${}^{16}{\rm O}^{7+}$ we have Tomaselli _et al._ (2002); Verdú
(2006)
$\displaystyle{f_{\rm s}\left({}^{16}{\rm O}^{7+}\right)\over f_{\rm
c}\left({}^{16}{\rm O}^{7+}\right)}=4164.376\,1837(32)\,.$ (183)
The correlation coefficient of these two frequency ratios, which are data
items $B17$ and $B18$ in Table LABEL:tab:pdata, is $0.082$.
Equations (1) and (LABEL:eq:fsfcgx) together yield
$\displaystyle{f_{\rm s}\left({}^{12}{\rm C}^{5+}\right)\over f_{\rm
c}\left({}^{12}{\rm C}^{5+}\right)}$ $\displaystyle=$ $\displaystyle-{g_{\rm
e^{-}}\left({}^{12}{\rm C}^{5+}\right)\over 10A_{\rm r}({\rm e})}$ (184)
$\displaystyle\hbox to-56.9055pt{}\times\left[12-5A_{\rm r}({\rm e})+{E_{\rm
b}\left({}^{12}{\rm C}\right)-E_{\rm b}\left({}^{12}{\rm C}^{5+}\right)\over
m_{\rm u}c^{2}}\right]\,,\qquad$
which is the basis of the observational equation for the ${}^{12}{\rm C}^{5+}$
frequency ratio input datum, Eq. (182); see $B17$ in Table 33. In a similar
manner we may write
$\displaystyle{f_{\rm s}\left({}^{16}{\rm O}^{7+}\right)\over f_{\rm
c}\left({}^{16}{\rm O}^{7+}\right)}=-{g_{\rm e^{-}}\left({}^{16}{\rm
O}^{7+}\right)\over 14A_{\rm r}({\rm e})}\,A_{\rm r}\\!\left({}^{16}{\rm
O}^{7+}\right),\qquad$ (185)
with
$\displaystyle A_{\rm r}\left({}^{16}{\rm O}\right)$ $\displaystyle=$
$\displaystyle A_{\rm r}\left({}^{16}{\rm O}^{7+}\right)+7A_{\rm r}({\rm e})$
(186) $\displaystyle\qquad-{E_{\rm b}\left({}^{16}{\rm O}\right)-E_{\rm
b}\left({}^{16}{\rm O}^{7+}\right)\over m_{\rm u}c^{2}}\,,\qquad$
which are the basis for the observational equations for the oxygen frequency
ratio and $A_{\rm r}(^{16}{\rm O})$, respectively; see $B18$ and $B8$ in Table
33.
Evaluation of Eq. (184) using the result for the carbon frequency ratio in Eq.
(182), the theoretical result for $g_{{\rm e}^{-}}(^{12}$C${}^{5+})$ in Table
13, and the relevant binding energies in Table IV of CODATA-02, yields
$\displaystyle A_{\rm r}({\rm e})=0.000\,548\,579\,909\,32(29)\quad[5.2\times
10^{-10}]\,.\quad$ (187)
A similar calculation for oxygen using the value of $A_{\rm r}(^{16}$O) in
Table 3 yields
$\displaystyle A_{\rm r}({\rm e})=0.000\,548\,579\,909\,57(42)\quad[7.6\times
10^{-10}]\,.\quad$ (188)
These values of $A_{\rm r}({\rm e})$ are consistent with each other.
Finally, as a further consistency test, the experimental and theoretical
values of the ratio of $g_{{\rm e}^{-}}(^{12}$C${}^{5+})$ to $g_{{\rm
e}^{-}}(^{16}$O${}^{7+})$ can be compared Karshenboim and Ivanov (2002). The
theoretical value of the ratio is given in Eq. (181) and the experimental
value is
$\displaystyle\frac{g_{{\rm e}^{-}}(^{12}{\rm C}^{5+})}{g_{{\rm
e}^{-}}(^{16}{\rm O}^{7+})}$ $\displaystyle=$ $\displaystyle
1.000\,497\,273\,68(89)~{}[8.9\times 10^{-10}]\,,\ \ $
in agreement with the theoretical value.
## VI Magnetic moment ratios and the muon-electron mass ratio
Magnetic moment ratios and the muon-electron mass ratio are determined by
experiments on bound states of the relevant particles and must be corrected to
determine the free particle moments.
For nucleons or nuclei with spin $\bm{I}$, the magnetic moment can be written
as
$\displaystyle\bm{\mu}=g{e\over 2m_{\rm p}}\bm{I}\ ,$ (190)
or
$\displaystyle\mu=g\mu_{\rm N}i\ .$ (191)
In Eq. (191), $\mu_{\rm N}=e\hbar/2m_{\rm p}$ is the nuclear magneton, defined
in analogy with the Bohr magneton, and $i$ is the spin quantum number of the
nucleus defined by $\bm{I}^{2}=i(i+1)\hbar^{2}$ and
$I_{z}=-i\hbar,...,(i-1)\hbar,i\hbar$, where $I_{z}$ is the spin projection.
Bound state $g$-factors for atoms with a non-zero nuclear spin are defined by
considering their interactions in an applied magnetic flux density $\bm{B}$.
For hydrogen, in the Pauli approximation, we have
$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\beta({\rm
H})\bm{\mu}_{\rm e^{-}}\cdot\bm{\mu}_{\rm p}-\bm{\mu}_{\rm e^{-}}({\rm
H})\cdot\bm{B}-\bm{\mu}_{\rm p}({\rm H})\cdot\bm{B}$ $\displaystyle=$
$\displaystyle{2\mbox{{p}}\over\hbar}\Delta\nu_{\rm H}\bm{s}\cdot\bm{I}-g_{\rm
e^{-}}({\rm H})\,{\mu_{\rm B}\over\hbar}\ \bm{s}\cdot\bm{B}-g_{\rm p}({\rm
H})\,{\mu_{\rm N}\over\hbar}\ \bm{I}\cdot\bm{B}\ ,$
where $\beta({\rm H})$ characterizes the strength of the hyperfine
interaction, $\Delta\nu_{\rm H}$ is the ground-state hyperfine frequency,
$\bm{s}$ is the spin of the electron, and $\bm{I}$ is the spin of the nucleus.
Equation (LABEL:eq:gdefs) defines the corresponding bound-state $g$-factors
$g_{\rm e^{-}}({\rm H})$ and $g_{\rm p}({\rm H})$.
### VI.1 Magnetic moment ratios
Theoretical binding corrections relate $g$-factors measured in the bound state
to the corresponding free-particle $g$-factors. The corrections are
sufficiently small that the adjusted constants used to calculate them are
taken as exactly known. These corrections and the references for the relevant
calculations are discussed in CODATA-98 and CODATA-02.
#### VI.1.1 Theoretical ratios of atomic bound-particle to free-particle
$g$-factors
For the electron in hydrogen, we have
$\displaystyle{g_{\rm e^{-}}({\rm H})\over g_{\rm e^{-}}}$ $\displaystyle=$
$\displaystyle 1-{\textstyle{1\over 3}}(Z\alpha)^{2}-{\textstyle{1\over
12}}(Z\alpha)^{4}+{\textstyle{1\over
4}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right)$ (193)
$\displaystyle+{\textstyle{1\over 2}}(Z\alpha)^{2}{m_{\rm e}\over m_{\rm
p}}+{\textstyle{1\over 2}}\left(A_{1}^{(4)}-{\textstyle{1\over
4}}\right)(Z\alpha)^{2}\left(\alpha\over\mbox{{p}}\right)^{2}$
$\displaystyle-{\textstyle{5\over
12}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right){m_{\rm e}\over m_{\rm
p}}+\cdots\ ,$
where $A_{1}^{(4)}$ is given in Eq. (109). For the proton in hydrogen, we have
$\displaystyle{g_{\rm p}({\rm H})\over g_{\rm p}}$ $\displaystyle=$
$\displaystyle 1-{\textstyle{1\over 3}}\alpha(Z\alpha)-{\textstyle{97\over
108}}\alpha(Z\alpha)^{3}$ (194) $\displaystyle+{\textstyle{1\over
6}}\alpha(Z\alpha){m_{\rm e}\over m_{\rm p}}{3+4a_{\rm p}\over 1+a_{\rm
p}}+\cdots\ ,$
where the proton magnetic moment anomaly $a_{\rm p}$ is defined by
$\displaystyle a_{\rm p}$ $\displaystyle=$ $\displaystyle{\mu_{\rm
p}\over\left(e\hbar/2m_{\rm p}\right)}-1\approx 1.793\ .$ (195)
For deuterium, similar expressions apply for the electron
$\displaystyle{g_{\rm e^{-}}({\rm D})\over g_{\rm e^{-}}}$ $\displaystyle=$
$\displaystyle 1-{\textstyle{1\over 3}}(Z\alpha)^{2}-{\textstyle{1\over
12}}(Z\alpha)^{4}+{\textstyle{1\over
4}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right)$ (196)
$\displaystyle+{\textstyle{1\over 2}}(Z\alpha)^{2}{m_{\rm e}\over m_{\rm
d}}+{\textstyle{1\over 2}}\left(A_{1}^{(4)}-{\textstyle{1\over
4}}\right)(Z\alpha)^{2}\left(\alpha\over\mbox{{p}}\right)^{2}$
$\displaystyle-{\textstyle{5\over
12}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right){m_{\rm e}\over m_{\rm
d}}+\cdots\ ,$
and deuteron
$\displaystyle{g_{\rm d}({\rm D})\over g_{\rm d}}$ $\displaystyle=$
$\displaystyle 1-{\textstyle{1\over 3}}\alpha(Z\alpha)-{\textstyle{97\over
108}}\alpha(Z\alpha)^{3}$ (197) $\displaystyle+{\textstyle{1\over
6}}\alpha(Z\alpha){m_{\rm e}\over m_{\rm d}}{3+4a_{\rm d}\over 1+a_{\rm
d}}+\cdots\ ,$
where the deuteron magnetic moment anomaly $a_{\rm d}$ is defined by
$\displaystyle a_{\rm d}={\mu_{\rm d}\over\left(e\hbar/m_{\rm
d}\right)}-1\approx-0.143\ .$ (198)
In the case of muonium Mu, some additional higher-order terms are included.
For the electron in muonium, we have
$\displaystyle{g_{\rm e^{-}}({\rm Mu})\over g_{\rm e^{-}}}$ $\displaystyle=$
$\displaystyle 1-{\textstyle{1\over 3}}(Z\alpha)^{2}-{\textstyle{1\over
12}}(Z\alpha)^{4}+{\textstyle{1\over
4}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right)$ (199)
$\displaystyle+{\textstyle{1\over 2}}(Z\alpha)^{2}{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}+{\textstyle{1\over
2}}\left(A_{1}^{(4)}-{\textstyle{1\over
4}}\right)(Z\alpha)^{2}\left(\alpha\over\mbox{{p}}\right)^{2}$
$\displaystyle\hbox to-14.22636pt{}-{\textstyle{5\over
12}}(Z\alpha)^{2}\left({\alpha\over\mbox{{p}}}\right){m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}-{\textstyle{1\over
2}}(1+Z)(Z\alpha)^{2}\left({m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\right)^{\\!2}$ $\displaystyle\hbox
to-14.22636pt{}+\cdots\ ,$
and for the muon in muonium, the ratio is
$\displaystyle{g_{{\mbox{\scriptsize{{m}}}}^{+}}({\rm Mu})\over
g_{{\mbox{\scriptsize{{m}}}}^{+}}}$ $\displaystyle=$ $\displaystyle
1-{\textstyle{1\over 3}}\alpha(Z\alpha)-{\textstyle{97\over
108}}\alpha(Z\alpha)^{3}$ $\displaystyle+{\textstyle{1\over
2}}\alpha(Z\alpha){m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}+{\textstyle{1\over
12}}\alpha(Z\alpha)\left({\alpha\over\mbox{{p}}}\right){m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}$ $\displaystyle-{\textstyle{1\over
2}}(1+Z)\alpha(Z\alpha)\left({m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\right)^{2}+\cdots\ .$
The numerical values of the corrections in Eqs. (193) to (LABEL:eq:mumugrat),
based on the 2010 adjusted values of the relevant constants, are listed in
Table 15; uncertainties are negligible here. An additional term of order
$\alpha(Z\alpha)^{5}$, relevant to Eqs. (194), (197), and (LABEL:eq:mumugrat)
has been calculated by Ivanov _et al._ (2009), but it is negligible at the
present level of uncertainty.
Table 15: Theoretical values for various bound-particle to free-particle $g$-factor ratios relevant to the 2010 adjustment based on the 2010 recommended values of the constants. Ratio | Value
---|---
$g_{\rm e^{-}}({\rm H})/g_{\rm e^{-}}$ | $1-17.7054\times 10^{-6}$
$g_{\rm p}({\rm H})/g_{\rm p}$ | $1-17.7354\times 10^{-6}$
$g_{\rm e^{-}}({\rm D})/g_{\rm e^{-}}$ | $1-17.7126\times 10^{-6}$
$g_{\rm d}({\rm D})/g_{\rm d}$ | $1-17.7461\times 10^{-6}$
$g_{\rm e^{-}}({\rm Mu})/g_{\rm e^{-}}$ | $1-17.5926\times 10^{-6}$
$g_{\mbox{\scriptsize{{m}}}^{+}}({\rm Mu})/g_{\mbox{\scriptsize{{m}}}^{+}}$ | $1-17.6254\times 10^{-6}$
#### VI.1.2 Bound helion to free helion magnetic moment ratio $\mu_{\rm
h}^{\prime}/\mu_{\rm h}$
The bound helion to free helion magnetic moment ratio correction $\sigma_{\rm
h}$, defined by
$\displaystyle\frac{\mu_{\rm h}^{\prime}}{\mu_{\rm h}}$ $\displaystyle=$
$\displaystyle 1-\sigma_{\rm h}\,,$ (201)
has been calculated by Rudziński _et al._ (2009), who obtain
$\displaystyle\sigma_{\rm h}$ $\displaystyle=$ $\displaystyle
59.967\,43(10)\times 10^{-6}\quad[1.7\times 10^{-6}]\,.$ (202)
This provides a recommended value for the unshielded helion magnetic moment,
along with other related quantities.
#### VI.1.3 Ratio measurements
Since all of the experimental bound-state magnetic-moment ratios of interest
for the 2010 adjustment are discussed in one or more of the previous three
CODATA reports, only minimal information is given here. The relevant input
data are items $B19$-$B27$ of Table LABEL:tab:pdata and their respective
observational equations are $B19$-$B27$ in Table 33. The adjusted constants in
those equations may be identified using Table 32, and theoretical bound-
particle to free-particle $g$-factor ratios, which are taken to be exact, are
given in Table 15. The symbol $\mu_{\rm p}^{\,\prime}$ denotes the magnetic
moment of a proton in a spherical sample of pure $\rm H_{2}{\rm O}$ at 25
${}^{\circ}{\rm C}$ surrounded by vacuum; and the symbol $\mu_{\rm
h}^{\,\prime}$ denotes the magnetic moment of a helion bound in a ${}^{3}{\rm
He}$ atom. Although the exact shape and temperature of the gaseous ${}^{3}{\rm
He}$ sample is unimportant, we assume that it is spherical, at 25
${}^{\circ}{\rm C}$, and surrounded by vacuum.
Item $B19$, labeled MIT-72, is the ratio $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm
p}({\rm H})$ in the 1S state of hydrogen obtained at MIT by Winkler _et al._
(1972); Kleppner (1997); and $B20$, labeled MIT-84, is the ratio $\mu_{\rm
d}({\rm D})/\mu_{\rm e^{-}}({\rm D})$ in the 1S state of deuterium also
obtained at MIT Phillips _et al._ (1984).
Item $B21$ with identification StPtrsb-03 is the magnetic moment ratio
$\mu_{\rm p}(\rm HD)/\mu_{\rm d}(\rm HD)$, and $B23$ with the same
identification is the ratio $\mu_{\rm t}(\rm HT)/\mu_{\rm p}(\rm HT)$, both of
which were determined from NMR measurements on the HD and HT molecules (bound
state of hydrogen and deuterium and of hydrogen and tritium, respectively) by
researchers working at institutes in St. Petersburg, Russian Federation
Neronov and Karshenboim (2003); Karshenboim _et al._ (2005). Here $\mu_{\rm
p}(\rm HD)$ and $\mu_{\rm d}(\rm HD)$ are the proton and the deuteron magnetic
moments in HD and $\mu_{\rm t}(\rm HT)$ and $\mu_{\rm p}(\rm HT)$ are the
triton and the proton magnetic moments in HT. Item $B22$ and $B24$, also with
the identifications StPtrsb-03 and due to Neronov and Karshenboim (2003) and
Karshenboim _et al._ (2005), are defined according to $\sigma_{\rm
dp}\equiv\sigma_{\rm d}(\rm HD)-\sigma_{\rm p}(\rm HD)$ and $\sigma_{\rm
tp}\equiv\sigma_{\rm t}(\rm HT)-\sigma_{\rm p}(\rm HT)$, where $\sigma_{\rm
p}(\rm HD)$, $\sigma_{\rm d}(\rm HD)$, $\sigma_{\rm t}(\rm HT)$, and
$\sigma_{\rm p}(\rm HT)$ are the corresponding nuclear magnetic shielding
corrections, which are small: $\mu({\rm bound})=(1-\sigma){\mu}({\rm free})$.
We note that after the 31 December 2010 closing date of the 2010 adjustment,
Neronov and Aleksandrov (2011) reported a result for the ratio $\mu_{\rm
t}(\rm HT)/\mu_{\rm p}(\rm HT)$ with a relative standard uncertainty of
$7\times 10^{-10}$ and which is consistent with data item $B23$.
Item $B25$, labeled MIT-77, is the ratio $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm
p}^{\,\prime}$ obtained at MIT by Phillips _et al._ (1977), where the electron
is in the 1S state of hydrogen. The results of Petley and Donaldson (1984) are
used to correct the measured value of the ratio based on a spherical $\rm
H_{2}{\rm O}$ NMR sample at 34.7 ${}^{\circ}{\rm C}$ to the reference
temperature 25 ${}^{\circ}{\rm C}$.
Item $B26$ with identification NPL-93 is the ratio $\mu_{\rm
h}^{\,\prime}/\mu_{\rm p}^{\,\prime}$ determined at the National Physical
Laboratory (NPL), Teddington, UK, by Flowers _et al._ (1993). And $B27$,
labeled ILL-79, is the neutron to shielded proton magnetic-moment ratio
$\mu_{\rm n}/\mu_{\rm p}^{\,\prime}$ determined at the Institut Max von Laue-
Paul Langevin (ILL) in Grenoble, France Greene _et al._ (1979, 1977).
### VI.2 Muonium transition frequencies, the muon-proton magnetic moment
ratio $\bm{\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}}$, and muon-electron mass
ratio $\bm{m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}}$
Experimental frequencies for transitions between Zeeman energy levels in
muonium ($\mbox{{m}}^{+}{\rm e}^{-}$ atom) provide measured values of
$\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ and the muonium ground-state
hyperfine splitting $\Delta\nu_{\rm Mu}$ that depend only on the commonly used
Breit-Rabi equation Breit and Rabi (1931).
The theoretical expression for the hyperfine splitting $\Delta\nu_{\rm
Mu}({\rm th})$ is discussed in the following section and may be written as
$\displaystyle\Delta\nu_{\rm Mu}({\rm th})$ $\displaystyle=$
$\displaystyle{16\over 3}cR_{\infty}\alpha^{2}{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\left(1+{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\right)^{-3}{\cal F}\hbox
to-2.0pt{}\left(\alpha,{m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}}\right)$ (203)
$\displaystyle=$ $\displaystyle\Delta\nu_{\rm F}{\cal F}\hbox
to-2.0pt{}\left(\alpha,{m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}}\right)\,,$
where the function ${\cal F}$ depends weakly on $\alpha$ and $m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}$.
#### VI.2.1 Theory of the muonium ground-state hyperfine splitting
Presented here is a brief summary of the present theory of $\Delta\nu_{\rm
Mu}$. Complete results of the relevant calculations are given along with
references to new work; references to the original literature included in
earlier CODATA reports are not repeated.
The hyperfine splitting is given mainly by the Fermi formula:
$\displaystyle\Delta\nu_{\rm F}={16\over 3}cR_{\infty}Z^{3}\alpha^{2}{m_{\rm
e}\over m_{\mbox{\scriptsize{{m}}}}}\left[1+{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\right]^{-3}\,.$ (204)
In order to identify the source of the terms, some of the theoretical
expressions are for a muon with charge $Ze$ rather than $e$.
The general expression for the hyperfine splitting is
$\displaystyle\Delta\nu_{\rm Mu}(\hbox{th})$ $\displaystyle=$
$\displaystyle\Delta\nu_{\rm D}+\Delta\nu_{\rm rad}+\Delta\nu_{\rm rec}$ (205)
$\displaystyle+\Delta\nu_{\rm r\hbox{-}r}+\Delta\nu_{\rm weak}+\Delta\nu_{\rm
had}\,,$
where the terms labeled D, rad, rec, r-r, weak, and had account for the Dirac,
radiative, recoil, radiative-recoil, electroweak, and hadronic contributions
to the hyperfine splitting, respectively.
The Dirac equation yields
$\displaystyle\Delta\nu_{\rm D}$ $\displaystyle=$ $\displaystyle\Delta\nu_{\rm
F}(1+a_{\mbox{\scriptsize{{m}}}})\left[1+{\textstyle{3\over
2}}(Z\alpha)^{2}+{\textstyle{17\over 8}}(Z\alpha)^{4}+\cdots\ \right],$
where $a_{\mbox{\scriptsize{{m}}}}$ is the muon magnetic moment anomaly.
The radiative corrections are
$\displaystyle\Delta\nu_{\rm rad}$ $\displaystyle=$
$\displaystyle\Delta\nu_{\rm
F}(1+a_{\mbox{\scriptsize{{m}}}})\Big{[}D^{(2)}\\!(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)$
$\displaystyle+D^{(4)}\\!(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)^{2}+D^{(6)}\\!(Z\alpha)\left({\alpha\over\mbox{{p}}}\right)^{3}+\cdots\Big{]}\
,$
where the functions $D^{(2n)}\\!(Z\alpha)$ are contributions from $n$ virtual
photons. The leading term is
$\displaystyle D^{(2)}\\!(Z\alpha)$ $\displaystyle=$ $\displaystyle
A_{1}^{(2)}+\left(\ln 2-{\textstyle{5\over 2}}\right)\mbox{{p}}Z\alpha$ (208)
$\displaystyle+\Big{[}-{\textstyle{2\over
3}}\ln^{2}(Z\alpha)^{-2}+\left({\textstyle{281\over 360}}-{\textstyle{8\over
3}}\ln 2\right)\ln(Z\alpha)^{-2}$
$\displaystyle+16.9037\ldots\Big{]}(Z\alpha)^{2}$
$\displaystyle+\Big{[}\left({\textstyle{5\over 2}}\ln 2-{\textstyle{547\over
96}}\right)\ln(Z\alpha)^{-2}\Big{]}\mbox{{p}}(Z\alpha)^{3}$
$\displaystyle+G(Z\alpha)(Z\alpha)^{3}\ ,$
where $A_{1}^{(2)}={\textstyle{1\over 2}}$, as in Eq. (108). The function
$G(Z\alpha)$ accounts for all higher-order contributions in powers of
$Z\alpha$; it can be divided into self-energy and vacuum polarization
contributions, $G(Z\alpha)=G_{\rm SE}(Z\alpha)+G_{\rm VP}(Z\alpha)$. Yerokhin
and Jentschura (2008, 2010) have calculated the one-loop self energy for the
muonium HFS with the result
$\displaystyle G_{\rm SE}(\alpha)=-13.8308(43)\,,$ (209)
which agrees with the value $G_{\rm SE}(\alpha)=-13.8(3)$ from an earlier
calculation by Yerokhin _et al._ (2005a), as well as with other previous
estimates. The vacuum polarization part is
$\displaystyle G_{\rm VP}(\alpha)=7.227(9)\ .$ (210)
For $D^{(4)}\\!(Z\alpha)$, we have
$\displaystyle D^{(4)}\\!(Z\alpha)$ $\displaystyle=$ $\displaystyle
A_{1}^{(4)}+0.770\,99(2)\mbox{{p}}Z\alpha+\Big{[}-{\textstyle{1\over
3}}\ln^{2}(Z\alpha)^{-2}$ (211)
$\displaystyle-0.6390\ldots\times\ln(Z\alpha)^{-2}+10(2.5)\Big{]}(Z\alpha)^{2}$
$\displaystyle+\cdots\ ,$
where $A_{1}^{(4)}$ is given in Eq. (109), and the coefficient of
$\mbox{{p}}Z\alpha$ has been calculated by Mondéjar _et al._ (2010).
The next term is
$\displaystyle D^{(6)}\\!(Z\alpha)=A_{1}^{(6)}+\cdots\ ,$ (212)
where the leading contribution $A_{1}^{(6)}$ is given in Eq. (110), but only
partial results of relative order $Z\alpha$ have been calculated Eides and
Shelyuto (2007). Higher-order functions $D^{(2n)}\\!(Z\alpha)$ with $n>3$ are
expected to be negligible.
The recoil contribution is
$\displaystyle\Delta\nu_{\rm rec}$ $\displaystyle=$
$\displaystyle\Delta\nu_{\rm F}{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}}\Bigg{(}-{3\over 1-\left(m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}\right)^{2}}\ln\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over
m_{\rm e}}\Big{)}{Z\alpha\over\mbox{{p}}}$
$\displaystyle+\,{1\over\left(1+m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}\right)^{2}}\bigg{\\{}\ln{(Z\alpha)^{-2}}-8\ln
2+{65\over 18}$ $\displaystyle+\bigg{[}{9\over
2\mbox{{p}}^{2}}\ln^{2}\left({m_{\mbox{\scriptsize{{m}}}}\over m_{\rm
e}}\right)+\left({27\over
2\mbox{{p}}^{2}}-1\right)\ln\left({m_{\mbox{\scriptsize{{m}}}}\over m_{\rm
e}}\right)$ $\displaystyle+{93\over
4\mbox{{p}}^{2}}+{33\zeta(3)\over\mbox{{p}}^{2}}-{13\over 12}-12\ln
2\bigg{]}{m_{\rm e}\over m_{\mbox{\scriptsize{{m}}}}}\bigg{\\}}(Z\alpha)^{2}$
$\displaystyle+\,\bigg{\\{}-{3\over
2}\ln{\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over m_{\rm
e}}\Big{)}}\ln(Z\alpha)^{-2}-{1\over 6}\ln^{2}{(Z\alpha)^{-2}}$
$\displaystyle+\left({101\over 18}-10\ln 2\right)\ln(Z\alpha)^{-2}$
$\displaystyle\qquad\qquad+40(10)\bigg{\\}}{(Z\alpha)^{3}\over\mbox{{p}}}\Bigg{)}+\cdots\,,$
as discussed in CODATA-02
The radiative-recoil contribution is
$\displaystyle\Delta\nu_{\rm r\hbox{-}r}$ $\displaystyle=$
$\displaystyle\Delta\nu_{\rm F}\left({\alpha\over\mbox{{p}}}\right)^{2}{m_{\rm
e}\over
m_{\mbox{\scriptsize{{m}}}}}\bigg{\\{}\bigg{[}-2\ln^{2}\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over
m_{\rm e}}\Big{)}+{13\over 12}\ln{\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over
m_{\rm e}}\Big{)}}$ (214) $\displaystyle+\,{21\over
2}\zeta(3)+{\mbox{{p}}^{2}\over 6}+{35\over 9}\bigg{]}+\bigg{[}\,{4\over
3}\ln^{2}\alpha^{-2}$ $\displaystyle+\left({16\over 3}\ln 2-{341\over
180}\right)\ln\alpha^{-2}-40(10)\bigg{]}\mbox{{p}}\alpha$
$\displaystyle+\bigg{[}-{4\over
3}\ln^{3}\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over m_{\rm e}}\Big{)}+{4\over
3}\ln^{2}\Big{(}{m_{\mbox{\scriptsize{{m}}}}\over m_{\rm
e}}\Big{)}\bigg{]}{\alpha\over\mbox{{p}}}\bigg{\\}}$ $\displaystyle-\nu_{\rm
F}\alpha^{2}\\!\left(m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}\right)^{\\!2}\left(6\ln{2}+{13\over
6}\right)+\cdots\ ,$
where, for simplicity, the explicit dependence on $Z$ is not shown. Partial
radiative recoil results are given by Eides and Shelyuto (2009b, a, 2010), and
are summarized as
$\displaystyle\Delta\nu_{\rm ES}$ $\displaystyle=$
$\displaystyle\Delta\nu_{\rm
F}\left(\frac{\alpha}{\mbox{{p}}}\right)^{3}\frac{m_{\rm
e}}{m_{\mbox{\scriptsize{{m}}}}}\bigg{\\{}\left[3\zeta(3)-6\mbox{{p}}^{2}\ln{2}+\mbox{{p}}^{2}-8\right]\ln{\frac{m_{\mbox{\scriptsize{{m}}}}}{m_{\rm
e}}}$ (215) $\displaystyle\qquad+63.127(2)\bigg{\\}}=-34.7\mbox{ Hz}\,.$
The electroweak contribution due to the exchange of a Z0 boson is Eides (1996)
$\displaystyle\Delta\nu_{\rm weak}$ $\displaystyle=$ $\displaystyle-65\ {\rm
Hz}\ ,$ (216)
while for the hadronic vacuum polarization contribution we have Eidelman _et
al._ (2002)
$\displaystyle\Delta\nu_{\rm had}$ $\displaystyle=$ $\displaystyle 236(4)\
{\rm Hz}\,,$ (217)
as in CODATA-06. A negligible contribution ($\approx 0.0065$ Hz) from the
hadronic light-by-light correction has been given by Karshenboim _et al._
(2008). Tau vacuum polarization contributes 3 Hz, which is also negligible at
the present level of uncertainty Sapirstein _et al._ (1984).
The four principle sources of uncertainty in $\Delta\nu_{\rm Mu}(\rm th)$ are
$\Delta\nu_{\rm rad}$, $\Delta\nu_{\rm rec}$, $\Delta\nu_{\rm r-r}$, and
$\Delta\nu_{\rm had}$ in Eq. (205). Based on the discussion in CODATA-02,
CODATA-06, and the new results above, the current uncertainties from these
contributions are 7 Hz, 74 Hz, 63 Hz, and 4 Hz, respectively, for a total of
98 Hz. Since this is only 3 % less than the value 101 Hz used in the 2006
adjustment, and in view of the incomplete nature of the calculations, the Task
Group has retained the 101 Hz standard uncertainty of that adjustment:
$\displaystyle u[\Delta\nu_{\rm Mu}{\rm(th)}]=101\ {\rm Hz}\quad[2.3\times
10^{-8}]\ .$ (218)
For the least-squares calculations, we use as the theoretical expression for
the hyperfine splitting
$\displaystyle\Delta\nu_{\rm Mu}\\!\\!\left(R_{\infty},\alpha,{m_{\rm e}\over
m_{\mbox{\scriptsize{{m}}}}},\delta_{\mbox{\scriptsize{{m}}}},\delta_{\rm
Mu}\right)=\Delta\nu_{\rm Mu}\rm(th)+\delta_{\rm Mu}\ ,$
where the input datum for the additive correction $\delta_{\rm Mu}$, which
accounts for the uncertainty of the theoretical expression and is data item
$B28$ in Table LABEL:tab:pdata, is 0(101) Hz.
The above theory yields
$\displaystyle\Delta\nu_{\rm Mu}=4\,463\,302\,891(272)\ {\rm
Hz}\quad[6.1\times 10^{-8}]$ (220)
using values of the constants obtained from the 2010 adjustment without the
two LAMPF measured values of $\Delta\nu_{\rm Mu}$ discussed in the following
section. The main source of uncertainty in this value is the mass ratio
$m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$.
#### VI.2.2 Measurements of muonium transition frequencies and values of
$\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ and
$m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$
The two most precise determinations of muonium Zeeman transition frequencies
were carried out at the Clinton P. Anderson Meson Physics Facility at Los
Alamos (LAMPF), USA, and were reviewed in detail in CODATA-98. The results are
as follows.
Data reported in 1982 by Mariam _et al._ (1982); Mariam (1981) are
$\displaystyle\Delta\nu_{\rm Mu}=4\,463\,302.88(16)\ {\rm kHz}\quad[3.6\times
10^{-8}]\,,$ (221) $\displaystyle\nu(f_{\rm p})=627\,994.77(14)\ {\rm
kHz}\quad[2.2\times 10^{-7}]\,,$ (222) $\displaystyle r[\Delta\nu_{\rm
Mu},\nu(f_{\rm p})]=0.227\,,$ (223)
where $f_{\rm p}$ is $57.972\,993$ MHz, corresponding to the magnetic flux
density of about $1.3616$ T used in the experiment, and $r[\Delta\nu_{\rm
Mu},\nu(f_{\rm p})]$ is the correlation coefficient of $\Delta\nu_{\rm Mu}$
and $\nu(f_{\rm p})$. The data reported in 1999 by Liu _et al._ (1999) are
$\displaystyle\Delta\nu_{\rm Mu}=4\,463\,302\,765(53)\ {\rm Hz}\quad[1.2\times
10^{-8}]\,,$ (224) $\displaystyle\nu(f_{\rm p})=668\,223\,166(57)\ {\rm
Hz}\quad[8.6\times 10^{-8}]\,,$ (225) $\displaystyle r[\Delta\nu_{\rm
Mu},\nu(f_{\rm p})]=0.195\,,$ (226)
where $f_{\rm p}$ is $72.320\,000$ MHz, corresponding to the flux density of
approximately $1.7$ T used in the experiment, and $r[\Delta\nu_{\rm
Mu},\nu(f_{\rm p})]$ is the correlation coefficient of $\Delta\nu_{\rm Mu}$
and $\nu(f_{\rm p})$. The data in Eqs. (221), (222), (224), and (225) are data
items $B29.1$, $B30$, $B29.2$, and $B31$, respectively, in Table
LABEL:tab:pdata.
The expression for the magnetic moment ratio is
$\displaystyle{\mu_{{\mbox{\scriptsize{{m}}}}^{+}}\over\mu_{\rm p}}$
$\displaystyle=$ $\displaystyle{\Delta\nu_{\rm Mu}^{2}-\nu^{2}(f_{\rm
p})+2s_{\rm e}f_{\rm p}\,\nu(f_{\rm p})\over 4s_{\rm e}f_{\rm p}^{2}-2f_{\rm
p}\,\nu(f_{\rm p})}\left({g_{{\mbox{\scriptsize{{m}}}}^{+}}(\rm Mu})\over
g_{{\mbox{\scriptsize{{m}}}}^{+}}\right)^{-1},$
where $\Delta\nu_{\rm Mu}$ and $\nu(f_{\rm p})$ are the sum and difference of
two measured transition frequencies, $f_{\rm p}$ is the free proton NMR
reference frequency corresponding to the flux density used in the experiment,
$g_{{\mbox{\scriptsize{{m}}}}^{+}}({\rm
Mu})/g_{{\mbox{\scriptsize{{m}}}}^{+}}$ is the bound-state correction for the
muon in muonium given in Table 15, and
$\displaystyle s_{\rm e}={\mu_{\rm e^{-}}\over\mu_{\rm p}}{g_{\rm e^{-}}({\rm
Mu})\over g_{\rm e^{-}}}\,,$ (228)
where $g_{\rm e^{-}}({\rm Mu})/g_{\rm e^{-}}$ is the bound-state correction
for the electron in muonium given in the same table.
The muon to electron mass ratio $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$ and
the muon to proton magnetic moment ratio
$\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ are related by
$\displaystyle{m_{\mbox{\scriptsize{{m}}}}\over m_{\rm e}}=\left({\mu_{\rm
e}\over\mu_{\rm p}}\right)\left({\mu_{\mbox{\scriptsize{{m}}}}\over\mu_{\rm
p}}\right)^{-1}\left({g_{\mbox{\scriptsize{{m}}}}\over g_{\rm e}}\right).$
(229)
A least-squares adjustment using the LAMPF data, the 2010 recommended values
of $R_{\infty}$, $\mu_{\rm e}/\mu_{\rm p}$, $g_{\rm e}$, and
$g_{\mbox{\scriptsize{{m}}}}$, together with Eq. (203) and Eqs.
(LABEL:eq:murat) to (229), yields
$\displaystyle{\mu_{{\mbox{\scriptsize{{m}}}}^{+}}\over\mu_{\rm p}}$
$\displaystyle=$ $\displaystyle 3.183\,345\,24(37)\quad[1.2\times 10^{-7}]\,,$
(230) $\displaystyle{m_{\mbox{\scriptsize{{m}}}}\over m_{\rm e}}$
$\displaystyle=$ $\displaystyle 206.768\,276(24)\quad[1.2\times 10^{-7}]\,,$
(231) $\displaystyle\alpha^{-1}$ $\displaystyle=$ $\displaystyle
137.036\,0018(80)\quad[5.8\times 10^{-8}]\,,$ (232)
where this value of $\alpha$ is denoted as $\alpha^{-1}(\Delta\nu_{\rm Mu})$.
The uncertainty of $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$ in Eq. (231) is
nearly five times the uncertainty of the 2010 recommended value. In Eq. (231),
the value follows from Eqs. (LABEL:eq:murat) to (229) with almost the same
uncertainty as the moment ratio in Eq. (230). Taken together, the experimental
value of and theoretical expression for the hyperfine splitting essentially
determine the value of the product $\alpha^{2}m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}$, as is evident from Eq. (203), with an
uncertainty dominated by the $2.3\times 10^{-8}$ relative uncertainty in the
theory, and in this limited least-squares adjustment $\alpha$ is otherwise
unconstrained. However, in the full adjustment the value of $\alpha$ is
determined by other data which in turn determines the value of
$m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$ with a significantly smaller
uncertainty than that of Eq. (231).
## VII Quotient of Planck constant and particle mass $\bm{h/m(X)}$ and
$\bm{\alpha}$
Measurements of $h/m(X)$ are of potential importance because the relation
$R_{\infty}=\alpha^{2}m_{\rm e}c/2h$ implies
$\displaystyle\alpha=\left[{2R_{\infty}\over c}{A_{\rm r}(X)\over A_{\rm
r}(\rm e)}{h\over m(X)}\right]^{1/2},$ (233)
where $A_{\rm r}(X)$ is the relative atomic mass of particle $X$ with mass
$m(X)$ and $A_{\rm r}({\rm e})$ is the relative atomic mass of the electron.
Because $c$ is exactly known, the relative standard uncertainties of
$R_{\infty}$ and $A_{\rm r}({\rm e})$ are $5.0\times 10^{-12}$ and $4.0\times
10^{-10}$, respectively, and the uncertainty of $A_{\rm r}(X)$ for many
particles and atoms is less than that of $A_{\rm r}({\rm e})$, Eq. (233) can
provide a competitive value of $\alpha$ if $h/m(X)$ is determined with a
sufficiently small uncertainty. This section discusses measurements of
$h/m({\rm{}^{133}Cs})$ and $h/m({\rm{}^{87}Rb})$.
### VII.1 Quotient $h/m({\rm{}^{133}Cs})$
Wicht _et al._ (2002) determined $h/m({\rm{}^{133}Cs})$ by measuring the
atomic recoil frequency shift of photons absorbed and emitted by
${\rm{}^{133}Cs}$ atoms using atom interferometry. Carried out at Stanford
University, Stanford, California, USA, the experiment is discussed in
CODATA-06 and CODATA-02. Consequently, only the final result is given here:
$\displaystyle{h\over m(^{133}{\rm Cs})}$ $\displaystyle=$ $\displaystyle
3.002\,369\,432(46)\times 10^{-9}~{}{\rm m}^{2}~{}{\rm s}^{-1}$ (234)
$\displaystyle\qquad\qquad[1.5\times 10^{-8}]\ .$
The observational equation for this datum is, from Eq. (233),
$\displaystyle{h\over m(^{133}{\rm Cs})}={A_{\rm r}({\rm e})\over A_{\rm
r}({\rm{}^{133}Cs})}{c\,\alpha^{2}\over 2R_{\infty}}\ .$ (235)
The value of $\alpha$ inferred from this expression and Eq. (234) is given in
Table 25, Sec. XIII.1.
The Stanford result for $h/m({\rm{}^{133}Cs})$ was not included as an input
datum in the final adjustment on which the 2006 recommended values are based
because of its low weight, and is omitted from the 2010 final adjustment for
the same reason. Nevertheless, it is included as an initial input datum to
provide a complete picture of the available data that provide values of
$\alpha$.
### VII.2 Quotient $h/m({\rm{}^{87}Rb})$
A value of $h/m({\rm{}^{87}Rb})$ with a relative standard uncertainty of
$1.3\times 10^{-8}$ obtained at LKB in Paris was taken as an input datum in
the 2006 adjustment and its uncertainty was sufficiently small for it to be
included in the 2006 final adjustment. Reported by Cladé _et al._ (2006) and
discussed in CODATA-06, $h/m({\rm{}^{87}Rb})$ was determined by measuring the
rubidium recoil velocity $v_{\rm r}=\hbar k/m({\rm{}^{87}Rb})$ when a rubidium
atom absorbs or emits a photon of wave vector $k=2{\mbox{{p}}}/\lambda$, where
$\lambda$ is the wavelength of the photon and $\nu=c/\lambda$ is its
frequency. The measurements were based on Bloch oscillations in a moving
standing wave.
A value of $h/m({\rm{}^{87}Rb})$ with a relative uncertainty of $9.2\times
10^{-9}$ and in agreement with the earlier result, obtained from a new LKB
experiment using combined Bloch oscillations and atom interferometry, was
subsequently reported by Cadoret _et al._ (2008). In this approach Bloch
oscillations are employed to transfer a large number of photon momenta to
rubidium atoms and an atom interferometer is used to accurately determine the
resulting variation in the velocity of the atoms. Significant improvements
incorporated into this version of the experiment have now provided a newer
value of $h/m({\rm{}^{87}Rb})$ that not only agrees with the two previous
values, but has an uncertainty over 10 and 7 times smaller, respectively. As
given by Bouchendira _et al._ (2011), the new LKB result is
$\displaystyle{h\over m(^{87}{\rm Rb})}$ $\displaystyle=$ $\displaystyle
4.591\,359\,2729(57)\times 10^{-9}~{}{\rm m}^{2}~{}{\rm s}^{-1}$ (236)
$\displaystyle\qquad\qquad[1.2\times 10^{-9}]\ .$
Because the LKB researchers informed the Task Group that this result should be
viewed as superseding the two earlier results Biraben (2011), it is the only
value of $h/m({\rm{}^{87}Rb})$ included as an input datum in the 2010
adjustment . The observational equation for this datum is, from Eq. (233),
$\displaystyle{h\over m(^{87}{\rm Rb})}={A_{\rm r}({\rm e})\over A_{\rm
r}({\rm{}^{87}Rb})}{c\,\alpha^{2}\over 2R_{\infty}}\ .$ (237)
The value of $\alpha$ inferred from this expression and Eq. (236) is given in
Table 25, Sec. XIII.1.
The experiment of the LKB group from which the result given in Eq. (236) was
obtained is described in the paper by Bouchendira _et al._ (2011), the
references cited therein; see also Cadoret _et al._ (2011); Cladé _et al._
(2010); Cadoret _et al._ (2009, 2008). It is worth noting, however, that the
reduction in uncertainty of the 2008 result by over a factor of 7 was achieved
by reducing the uncertainties of a number of individual components, especially
those due to the alignment of beams, wave front curvature and Gouy phase, and
the second order Zeeman effect. The total fractional correction for systematic
effects is $-26.4(5.9)\times 10^{-10}$ and the statistical or Type A
uncertainty is 2 parts in $10^{10}$.
### VII.3 Other data
A result for the quotient $h/m_{\rm n}d_{220}({\rm\scriptstyle W04})$ with a
relative standard uncertainty of $4.1\times 10^{-8}$, where $m_{\rm n}$ is the
neutron mass and $d_{220}({\rm{\scriptstyle W04}})$ is the {220} lattice
spacing of the crystal WASO 04, was included in the past three CODATA
adjustments, although its uncertainty was increased by the multiplicative
factor 1.5 in the 2006 final adjustment. It was obtained by PTB researchers
working at the ILL high-neutron-flux reactor in Grenoble Krüger _et al._
(1999).
Since the result has a relative uncertainty of $4.1\times 10^{-8}$, the value
of $\alpha$ that can be inferred from it, even assuming that
$d_{220}({\rm\scriptstyle W04})$ is exactly known, has an uncertainty of about
$2\times 10^{-8}$. This is over 50 times larger than that of $\alpha$ from
$a_{\rm e}$ and is not competitive. Further, the inferred value disagrees with
the $a_{\rm e}$ value.
On the other hand, the very small uncertainty of the $a_{\rm e}$ value of
$\alpha$ means that the PTB result for $h/m_{\rm n}d_{220}({\rm\scriptstyle
W04})$ can provide an inferred value of $d_{220}({\rm\scriptstyle W04})$ with
the competitive relative uncertainty of about 4 parts in $10^{8}$. However,
this inferred lattice-spacing value, reflecting the disagreement of the
inferred value of alpha, is inconsistent with the directly determined XROI
value. This discrepancy could well be the result of the different effective
lattice parameters for the different experiments. In the PTB measurement of
$h/m_{\rm n}d_{220}({\rm\scriptstyle W04})$, the de Broglie wavelength,
$\lambda\approx 0.25$ nm, of slow neutrons was determined using back
reflection from the surface of a silicon crystal. As pointed out to the Task
Group by Peter Becker (2011) of the PTB, the lattice spacings near the surface
of the crystal, which play a more critical role than in the XROI measurements
carried out using x-ray transmission, may be strained and not the same as the
spacings in the bulk of the crystal.
For these reasons, the Task Group decided not to consider this result for
inclusion in the 2010 adjustment.
## VIII Electrical measurements
This section focuses on 18 input data resulting from high-accuracy electrical
measurements, 16 of which were also available for the 2006 adjustment. The
remaining two became available in the intervening 4 years. Of the 16, 13 were
not included in the final adjustment on which the 2006 recommended values are
based because of their low weight. These same data and one of the two new
values are omitted in the final 2010 adjustment for the same reason.
Nevertheless, all are initially included as input data because of their
usefulness in providing an overall picture of the consistency of the data and
in testing the exactness of the Josephson and quantum Hall effect relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$. As an aid, we begin with a concise
overview of the seven different types of electrical quantities of which the 18
input data are particular examples.
### VIII.1 Types of electrical quantities
If microwave radiation of frequency $f$ is applied to a Josephson effect
device, quantized voltages $U_{\rm J}(n)=nf/K_{\rm J}$ are induced across the
device, where $n$, an integer, is the step number of the voltage and $K_{\rm
J}=2e/h$ is the Josephson constant. Similarly, the quantized Hall resistance
of the $i$th resistance plateau of a quantum Hall effect device carrying a
current and in a magnetic field, $i$ an integer, is given by $R_{\rm
H}(i)=R_{\rm K}/i$, where $R_{\rm K}=h/e^{2}=\mu_{0}c/2\alpha$ is the von
Klitzing constant. Thus, measurement of $K_{\rm J}$ in its SI unit Hz/V
determines the quotient $2e/h$, and since in the SI $c$ and $\mu_{0}$ are
exactly known constants, measurement of $R_{\rm K}$ in its SI unit $\Omega$
determines $\alpha$. Further, since $K_{\rm J}^{2}R_{\rm K}=4/h$, a
measurement of this product in its SI unit $({\rm J}~{}{\rm s})^{-1}$
determines $h$.
The gyromagnetic ratio $\gamma_{x}$ of a bound particle $x$ of spin quantum
number $i$ and magnetic moment $\mu_{x}$ is given by
$\displaystyle\gamma_{x}=\frac{2\mbox{{p}}{f}}{B}=\frac{\omega}{B}=\frac{|\mu_{x}|}{i\hbar},$
(238)
where $f$ is the spin-flip (or precession) frequency and $\omega$ is the
angular precession frequency of the particle in the magnetic flux density $B$.
For a bound and shielded proton p and helion h Eq. (238) gives
$\displaystyle\gamma^{\prime}_{\rm p}=\frac{2\mu^{\prime}_{\rm
p}}{\hbar},\qquad\gamma^{\prime}_{\rm h}=\frac{2\mu^{\prime}_{\rm h}}{\hbar},$
(239)
where the protons are in a spherical sample of pure $\rm H_{2}{\rm O}$ at 25
${}^{\circ}{\rm C}$ surrounded by vacuum; and the helions are in a spherical
sample of low-pressure, pure ${}^{3}{\rm He}$ gas at 25 ${}^{\circ}{\rm C}$
surrounded by vacuum.
The shielded gyromagnetic ratio of a particle can be determined by two methods
but the quantities actually measured are different: the low-field method
determines $\gamma_{x}^{\prime}/K_{\rm J}R_{\rm K}$ while the high-field
method determines $\gamma_{x}^{\prime}K_{\rm J}R_{\rm K}$. In both cases an
electric current $I$ is measured using the Josephson and quantum Hall effects
with the conventional values of the Josephson and von Klitzing constants. We
have for the two methods
$\displaystyle\gamma_{x}^{\,\prime}$ $\displaystyle=$
$\displaystyle{\it\Gamma}_{x-90}^{\,\prime}({\rm lo}){K_{\rm J}\,R_{\rm
K}\over K_{{\rm J}-90}\,R_{{\rm K}-90}}\ ,$ (240)
$\displaystyle\gamma_{x}^{\,\prime}$ $\displaystyle=$
$\displaystyle{\it\Gamma}_{x-90}^{\,\prime}({\rm hi}){K_{{\rm J}-90}\,R_{{\rm
K}-90}\over K_{\rm J}\,R_{\rm K}}\ ,$ (241)
where ${\it\Gamma}^{\prime}_{x-90}{(\rm lo)}$ and
${\it{\Gamma}}^{\prime}_{x-90}{(\rm hi)}$ are the experimental values of
$\gamma_{x}^{\prime}$ in SI units that would result from low- and hi- field
experiments, respectively, if $K_{\rm J}$ and $R_{\rm K}$ had the exactly
known conventional values $K_{\rm J-90}$ and $R_{\rm K-90}$. The actual input
data used in the adjustment are ${\it{\Gamma}}^{\prime}_{x-90}{(\rm lo)}$ and
${\it{\Gamma}}^{\prime}_{x-90}{(\rm hi)}$ since these are the quantities
actually measured in the experiments, but their observational equations (see
Table 33) account for the fact that $K_{\rm J-90}\neq K_{\rm J}$ and $R_{\rm
K-90}\neq R_{\rm K}$.
Finally, for the Faraday constant $F$ we have
$\displaystyle F$ $\displaystyle=$ $\displaystyle{\cal F}_{90}{K_{{\rm
J}-90}\,R_{{\rm K}-90}\over K_{\rm J}\,R_{\rm K}}\ ,$ (242)
where ${\cal F}_{90}$ is the actual quantity experimentally measured. Equation
(242) is similar to Eq. (241) because ${\cal F}_{90}$ depends on current in
the same way as ${\it\Gamma}_{x-90}^{\,\prime}({\rm hi})$, and the same
comments apply.
### VIII.2 Electrical data
The 18 electrical input data are data items $B32.1$ through $B38$ in Table
LABEL:tab:pdata, Sec. XIII. Data items $B37.4$ and $B37.5$, the two new input
data mentioned above and which, like the other three data in this category,
are moving-coil watt balance results for the product $K_{\rm J}^{2}R_{\rm K}$,
are discussed in the next two sections. Since the other 16 input data have
been discussed in one or more of the three previous CODATA reports, we provide
only limited information here.
$B32.1$ and $B32.2$, labeled NIST-89 and NIM-95, are values of
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ obtained at the National
Institute of Standards and Technology (NIST), Gaithersburg, MD, USA Williams
_et al._ (1989), and at the National Institute of Metrology (NIM), Beijing,
PRC Liu _et al._ (1995), respectively. $B33$, identified as KR/VN-98, is a
similar value of ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ obtained at the
Korea Research Institute of Standards and Science (KRISS), Taedok Science
Town, Republic of Korea in a collaborative effort with researchers from the
Mendeleyev All-Russian Research Institute for Metrology (VNIIM), St.
Petersburg, Russian Federation Park _et al._ (1999); Shifrin _et al._ (1999);
Shifrin _et al._ (1998b, a). $B34.1$ and $B34.2$ are values of
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ from NIM Liu _et al._ (1995) and
NPL Kibble and Hunt (1979), respectively, with identifications NIM-95 and
NPL-79.
$B35.1$-$B35.5$ are five calculable-capacitor determinations of $R_{\rm K}$
from NIST Jeffery _et al._ (1997, 1998), the National Metrology Institute
(NMI), Lindfield, Australia Small _et al._ (1997), NPL Hartland _et al._
(1988), NIM Zhang _et al._ (1995), and Laboratoire national de métrologie et
d’essais (LNE), Trappes, France Trapon _et al._ (2001, 2003), respectively,
and are labeled NIST-97, NMI-97, NPL-88, NIM-95, and LNE-01.
$B36.1$ with identification NMI-89 is the mercury electrometer result for
$K_{\rm J}$ from NMI K. Clothier _et al._ (1989); and $B36.2$, labeled PTB-91,
is the capacitor voltage balance result for $K_{\rm J}$ from the Physikalisch-
Technische Bundesanstalt (PTB), Braunschweig, Germany Funck and Sienknecht
(1991); Sienknecht and Funck (1986, 1985).
$B37.1$-$B37.3$, with identifications NPL-90, NIST-98, and NIST-07,
respectively, are moving-coil watt-balance results for $K_{\rm J}^{2}R_{\rm
K}$ from NPL Kibble _et al._ (1990) and from NIST Williams _et al._ (1998);
Steiner _et al._ (2007).
The last electrical input datum, $B38$ and labeled NIST-80, is the silver
dissolution coulometer result for ${\cal F}_{90}$ from NIST Bower and Davis
(1980).
The correlation coefficients of these data, as appropriate, are given in Table
21, Sec. XIII; the observational equations for the seven different types of
electrical data of which the 18 input data are particular examples are given
in Table 33 in the same section and are $B32$-$B38$. Recalling that the
relative standard uncertainties of $R_{\infty}$, $\alpha$, $\mu_{\rm
e^{-}}/{\mu^{\prime}_{\rm p}}$, ${\mu^{\prime}_{\rm h}}/{\mu^{\prime}_{\rm
p}}$, and $A_{\rm r}(\rm e)$ are significantly smaller that those of the
electrical input data, inspection of these equations shows that measured
values of ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$, ${\it\Gamma}_{\rm
h-90}^{\,\prime}({\rm lo})$, ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$,
$R_{\rm K}$, $K_{\rm J}$, $K_{\rm J}^{2}R_{\rm K}$, and ${\cal F}_{90}$
principally determine $\alpha$, $\alpha$, $h$, $\alpha$, $h$, $h$, and $h$,
respectively.
#### VIII.2.1 $K^{2}_{\rm J}R_{\rm K}$ and $h$: NPL watt balance
We consider here and in the following section the two new watt-balance
measurements of $K_{\rm J}^{2}R_{\rm K}=4/h$. For reviews of such experiments,
see, for example, the papers of Li _et al._ (2012); Stock (2011); Eichenberger
_et al._ (2009). The basic idea is to compare electrical power measured in
terms of the Josephson and quantum Hall effects to the equivalent mechanical
power measured in the SI unit W = m2 kg s-3. The comparison employs an
apparatus now called a moving-coil watt balance, or simply a watt balance,
first proposed by Kibble (1975) at NPL. A watt balance experiment can be
described by the simple equation $m_{\rm s}gv=UI$, where, for example, $I$ is
the current in a circular coil in a radial magnetic flux density $B$ and the
force on the coil due to $I$ and $B$ is balanced by the weight $m_{\rm s}g$ of
a standard of mass $m_{\rm s}$; and $U$ is the voltage induced across the
terminals of the coil when it is moved vertically with a velocity $v$ in the
same flux density $B$. Thus, a watt balance is operated in two different
modes: the weighing mode and the velocity mode.
The NPL Mark II watt balance and its early history were briefly discussed in
CODATA-06, including the initial result obtained with it by Robinson and
Kibble (2007). Based on measurements carried out from October 2006 to March
2007 and having a relative standard uncertainty of 66 parts in $10^{9}$, this
result became available only after the closing date of the 2006 adjustment.
Moreover, the NPL value of $K_{\rm J}^{2}R_{\rm K}$ was 308 parts in $10^{9}$
smaller than the NIST-07 value with a relative uncertainty of 36 parts in
$10^{9}$.
Significant modifications were subsequently made to the NPL apparatus in order
to identify previously unknown sources of error as well as to reduce
previously identified sources. The modifications were completed in November
2008, the apparatus was realigned in December 2008, and measurements and error
investigations were continued until June 2009. From then to August 2009 the
apparatus was dismantled, packed, and shipped to the National Research Council
(NRC), Ottawa, Canada. A lengthy, highly detailed preprint reporting the final
Mark II result was provided to the Task Group by I. A. Robinson of NPL prior
to the 31 December 2010 closing date of the 2010 adjustment. This paper has
now been published and the reported value is Robinson (2012)
$\displaystyle h=6.626\,07123(133)\times 10^{-34}\ {\rm J\ s}\quad[2.0\times
10^{-7}]\,.\qquad$ (243)
This corresponds to
$\displaystyle K_{\rm J}^{2}R_{\rm K}$ $\displaystyle=$ $\displaystyle
6.036\,7597(12)\times 10^{33}~{}{\rm J}^{-1}\ {\rm s}^{-1}$ (244)
$\displaystyle\qquad\qquad\qquad\qquad[2.0\times 10^{-7}]$
identified as NPL-12 and which is included as an input datum in the current
adjustment, data item $B37.4$.
The NPL final result is based on the initial data obtained from October 2006
to March 2007, data obtained during the first half of 2008, and data obtained
during the first half of 2009, the final period. Many variables were
investigated to determine their possible influence on the measured values of
$K_{\rm J}^{2}R_{\rm K}$. For example, several mass standards with different
masses and fabricated from different materials were used during the course of
the data taking. A comparison of the uncertainty budgets for the 2007 data and
the 2009 data shows significant reductions in all categories, with the
exception of the calibration of the mass standards, resulting in the reduction
of the overall uncertainty from 66 parts in $10^{9}$ to 36 parts in $10^{9}$.
Nevertheless, during the week before the balance was to be dismantled, a
previously unrecognized possible systematic error in the weighing mode of the
experiment came to light. Although there was insufficient time to derive a
correction for the effect, Robinson obtained an uncertainty estimate for it.
This additional uncertainty component, 197 parts in $10^{9}$, when combined
with the initially estimated overall uncertainty, leads to the 200 parts in
$10^{9}$ final uncertainty in Eqs. (243) and (244). Since the same component
applies to the initial Mark II result, its uncertainty is increased from 66
parts in $10^{9}$ to $208$ parts in $10^{9}$.
Finally, there is a slight correlation between the final Mark II value of
$K_{\rm J}^{2}R_{\rm K}$, NPL-12, item $B37.4$ in Table LABEL:tab:pdata, and
its 1990 predecessor, NPL-90, item $B37.1$ in the same table. Based on the
paper by Robinson (2012), the correlation coefficient is 0.0025.
#### VIII.2.2 $K^{2}_{\rm J}R_{\rm K}$ and $h$: METAS watt balance
The watt-balance experiment at the Federal Office of Metrology (METAS), Bern-
Wabern, Switzerland, was initiated in 1997, and progress reports describing
more than a decade of improvements and investigations of possible systematic
errors have been published and presented at conferences Beer _et al._ (1999,
2001, 2003). A detailed preprint giving the final result of this effort,which
is being continued with a new apparatus, was provided to the Task Group by A.
Eichenberger of METAS prior to the 31 December 2010 closing date of the 2010
adjustment, and was subsequently published by Eichenberger _et al._ (2011).
The METAS value for $h$ and the corresponding value for $K_{\rm J}^{2}R_{\rm
K}$, identified as METAS-11, input datum $B37.5$, are
$\displaystyle h$ $\displaystyle=$ $\displaystyle 6.626\,0691(20)\times
10^{-34}\ {\rm J\ s}\quad[2.9\times 10^{-7}]\,,\qquad$ (245)
and
$\displaystyle K_{\rm J}^{2}R_{\rm K}$ $\displaystyle=$ $\displaystyle
6.036\,7617(18)\times 10^{33}~{}{\rm J}^{-1}\ {\rm s}^{-1}\quad[2.9\times
10^{-7}]\,.\qquad$ (246)
The METAS watt balance differs in a number of respects from those of NIST and
NPL. For example, the METAS apparatus was designed to use a 100 g mass
standard and a commercial mass comparator rather than a 1 kg standard and a
specially designed and constructed balance in order to reduce the size and
complexity of the apparatus. Also, the velocity mode was designed to be
completely independent of the weighing mode. The use of two separated
measuring systems for the two modes in the same apparatus make it possible to
optimize each, but does require the transfer of the coil between the two
systems during the course of the measurements. Improvements in the apparatus
over the last several years of its operation focused on alignment, control of
the coil position, and reducing magnet hysteresis.
The METAS result is based on six sets of data acquired in 2010, each
containing at least 500 individual measurements which together represent over
3400 hours of operation of the apparatus. The $7\times 10^{-8}$ relative
standard uncertainty of the mean of the means of the six data sets is
considered by Eichenberger _et al._ (2011) to be a measure of the
reproducibility of the apparatus. The uncertainty budget from which the
$29\times 10^{-8}$ relative uncertainty of the METAS value of $K_{\rm
J}^{2}R_{\rm K}$ is obtained contains nine components, but the dominant
contributions, totaling 20 parts in $10^{8}$, are associated with the
alignment of the apparatus. Eichenberger _et al._ (2011) point out that
because of the mechanical design of the current METAS watt balance, it is not
possible to reduce this source of uncertainty in a significant way.
#### VIII.2.3 Inferred value of $K_{\rm J}$
As indicated in CODATA-06, a value of $K_{\rm J}$ with an uncertainty
significantly smaller than those of the two directly measured values $B36.1$
and $B36.2$ can be obtained without assuming the validity of the relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$. Dividing the weighted mean of the
five directly measured watt-balance values of $K^{2}_{\rm J}R_{\rm K}$,
$B37.1$-$B37.5$, by the weighted mean of the five directly measured
calculable-capacitor values $R_{\rm K}$, $B35.1$-$B35.5$, we have
$\displaystyle K_{\rm J}$ $\displaystyle=$ $\displaystyle K_{\rm
J-90}[1-3.0(1.9)\times 10^{-8}]$ (247) $\displaystyle=$ $\displaystyle
483\,597.8853(92)\ {\rm GHz/V}\quad[1.9\times 10^{-8}]\,.\qquad$
This result is consistent with the two directly measured values but has an
uncertainty that is smaller by more than an order of magnitude.
### VIII.3 Josephson and quantum Hall effect relations
The theoretical and experimental evidence accumulated over the past 50 years
for the Josephson effect and 30 years for the quantum Hall effect that
supports the exactness of the relations $K_{\rm J}=2e/h$ and $R_{\rm
K}=h/e^{2}$ has been discussed in the three previous CODATA reports and
references cited therein. The vast majority of the experimental evidence for
both effects over the years comes from tests of the universality of these
relations; that is, their invariance with experimental variables such as the
material of which the Josephson effect and quantum Hall effect devices are
fabricated. However, in both the 2002 and 2006 adjustments, the input data
were used to test these relations experimentally in an “absolute” sense, that
is by comparing the values of $2e/h$ and $h/e^{2}=\mu_{0}c/2\alpha$ implied by
the data assuming the relations are exact with those implied by the data under
the assumption that they are not exact. Indeed, such an analysis is given in
this report in Sec. XIII.2.3. Also briefly discussed there is the “metrology
triangle.” Here we discuss other developments of interest that have occurred
between the closing dates of the 2006 and 2010 adjustments.
Noteworthy for the Josephson effect is the publication by Wood and Solve
(2009) of “A review of Josephson comparison results.” These authors examined a
vast number of Josephson junction voltage comparisons conducted over the past
30 years involving many different laboratories, junction materials, types of
junctions, operating frequencies, step numbers, number of junctions in series,
voltage level, and operating temperature with some comparisons achieving a
precision of a few parts in $10^{11}$. They find no evidence that the relation
$K_{\rm J}=2e/h$ is not universal.
There are three noteworthy developments for the quantum Hall effect. First is
the recent publication of a _C. R. Physique_ special issue on the quantum Hall
effect and metrology with a number of theoretical as well as experimental
papers that support the exactness of the relation $R_{\rm K}=h/e^{2}$; see the
Foreword to this issue by Glattli (2011) and the papers contained therein, as
well as the recent review article by Weis and von Klitzing (2011).
The second is the agreement found between the value of $R_{\rm K}$ in a normal
GaAs/AlGaAs heterostructure quantum Hall effect device and a graphene (two
dimensional graphite) device to within the 8.6 parts in $10^{11}$ uncertainty
of the experiment Janssen _et al._ (2011). This is an extremely important
result in support of the universality of the above relation, because of the
significant difference in the charge carriers in graphene and the usual two
dimensional semiconductor systems; see Goerbig (2011); Peres (2010); Kramer
_et al._ (2010).
The third is the theoretical paper by Penin (2009). This author’s calculations
appear to show that the relation $R_{\rm K}=h/e^{2}$ is not exact but should
be written as $R_{\rm K}=(h/e^{2})[1+C]$, where the correction $C$ is due to
vacuum polarization and is given by
$C=-(2/45)(\alpha/\mbox{{p}})(B/B_{0})^{2}$. Here $B$ is the magnetic flux
density applied to the quantum Hall effect device and
$B_{0}=2{\mbox{{p}}}{c^{2}}{m^{2}_{\rm e}}/he\approx 4.4\times 10^{9}$ T.
However, since $B$ is generally no larger than 20 T, the correction,
approximately $-2\times 10^{-21}$, is vanishingly small and can be completely
ignored. Further, Penin (2009) argues that because of the topological nature
of the quantum Hall effect, there can be no other type of correction including
finite size effects.
## IX Measurements involving silicon crystals
Experimental results obtained using nearly perfect single crystals of natural
silicon are discussed here, along with a new result for $N_{\rm A}$ with a
relative standard uncertainty of $3.0\times 10^{-8}$ obtained using highly-
enriched silicon. For this material, ${x}(^{28}{\rm Si})\approx 0.999\,96$,
compared to ${x}(^{28}{\rm Si})\approx 0.92$, for natural silicon, where
$x(^{A}{\rm Si})$ is the amount-of-substance fraction of the indicated
isotope.
The new $N_{\rm A}$ result (see Sec. IX.6 below), as well as much of the
natural silicon data used in the current and previous CODATA adjustments, were
obtained as part of an extensive international effort under way since the
early 1990s to determine $N_{\rm A}$ with the smallest possible uncertainty.
This worldwide enterprise, which has many participating laboratories and is
called the International Avogadro Coordination (IAC), carries out its work
under the auspices of the Consultative Committee for Mass and Related
Quantities (CCM) of the CIPM.
The eight natural silicon crystal samples of interest here are denoted WASO
4.2a, WASO 04, WASO 17, NRLM3, NRLM4, MO*, ILL, and N, and the {220} crystal
lattice spacing of each, $d_{220}({\scriptstyle X})$, is taken as an adjusted
constant. For simplicity the shortened forms W4.2a, W04, W17, NR3, and NR4 are
used in quantity symbols for the first five crystals. Note also that crystal
labels actually denote the single crystal ingot from which the crystal samples
are taken, since no distinction is made between different samples taken from
the same ingot.
Silicon is a cubic crystal with $n=8$ atoms per face-centered cubic unit cell
of edge length (or lattice parameter) $a\approx 543~{}{\rm pm}$ with {220}
crystal lattice spacing $d_{220}=a/\sqrt{8}\approx 192~{}{\rm pm}$. For
practical purposes, it can be assumed that $a$, and thus $d_{220}$, of an
impurity free, crystallographically perfect or “ideal” silicon crystal at
specified conditions of temperature $t$, pressure $p$, and isotopic
composition is an invariant of nature. The currently adopted reference
conditions for natural silicon are $t_{90}=\,\,$22.5 ${}^{\circ}{\rm C}$ and
$p=0$ (vacuum), where $t_{90}$ is Celsius temperature on the International
Temperature Scale of 1990 (ITS-90). Reference values for $x(^{A}{\rm Si})$
have not been adopted, because any variation of $d_{220}({\scriptstyle X})$
with the typical isotopic composition variation observed for the natural
silicon crystals used is deemed negligible. To convert the lattice spacing
$d_{220}({\scriptstyle X})$ of a real crystal to the lattice spacing $d_{220}$
of an ideal crystal requires the application of corrections for impurities,
mainly carbon, oxygen, and nitrogen.
Typical variation in the lattice spacing of different samples from the same
ingot is taken into account by including an additional relative standard
uncertainty component of $\sqrt{2}\times 10^{-8}$ for each crystal in the
uncertainty budget of any measurement result involving one or more silicon
lattice spacings. However, the component is $(3/2)\sqrt{2}\times 10^{-8}$ in
the case of crystal MO* because it is known to contain a comparatively large
amount of carbon. For simplicity, we do not explicitly mention the inclusion
of such components in the following discussion.
### IX.1 Measurements of $\bm{d_{220}({\scriptstyle X})}$ of natural silicon
Measurements of $d_{220}({\scriptstyle X})$ are performed using a combined
x-ray and optical interferometer (XROI). The interferometer has three lamenae
from a single crystal, one of which can be displaced and is called the
analyzer; see CODATA-98. Also discussed there is the measurement at PTB using
an XROI with WASO 4.2a Becker _et al._ (1981). This result, which was taken as
an input datum in the past three adjustments, is also used in the current
adjustment; its value is
$\displaystyle d_{220}({\scriptstyle{\rm W4.2a}})=192\,015.563(12)\ {\rm
fm}\quad[6.2\times 10^{-8}]\,,\qquad$ (248)
which is data item $B41.1$, labeled PTB-81, in Table LABEL:tab:pdata.
The three other {220} natural silicon lattice spacings taken as input data in
the 2010 adjustment, determined at the Istituto Nazionale di Ricerca
Metrologica, (INRIM) Torino, Italy, using XROIs fabricated from MO*, WASO 04,
and WASO 4.2a, are much more recent results. Ferroglio _et al._ (2008) report
$\displaystyle d_{220}({\scriptstyle{\rm MO^{*}}})$ $\displaystyle=$
$\displaystyle 192\,015.5508(42)\mbox{ fm}\quad[2.2\times 10^{-8}]\,,\qquad$
(249)
which is data item $B39$, labeled INRIM-08; Massa _et al._ (2009b) find
$\displaystyle d_{220}({\scriptstyle{\rm WO4}})$ $\displaystyle=$
$\displaystyle 192\,015.5702(29)\mbox{ fm}\quad[1.5\times 10^{-8}]\,,\qquad$
(250)
which is data item $B40$, labeled INRIM-09; and Massa _et al._ (2009a) give
$\displaystyle d_{220}({\scriptstyle{\rm W4.2a}})$ $\displaystyle=$
$\displaystyle 192\,015.5691(29)\mbox{ fm}\quad[1.5\times 10^{-8}]\,,\qquad$
(251)
which is data item $B41.2$, labeled INRIM-09.
The XROI used to obtain these three results is a new design with many special
features. The most significant advance over previous designs is the capability
to displace the analyzer by up to 5 cm. In the new apparatus, laser
interferometers and capacitive transducers sense crystal displacement,
parasitic rotations, and transverse motions, and feedback loops provide
positioning with picometer resolution, alignment with nanometer resolution,
and movement of the analyzer with nanometer straightness. A number of
fractional corrections for different effects, such as laser wavelength, laser
beam diffraction, laser beam alignment, and temperature of the crystal, are
applied in each determination; the total correction for each of the three
results, in parts in $10^{9}$, is 6.5, $-4.0$, and 3.7, respectively. The
relative standard uncertainties of the three lattice spacing measurements
without the additional uncertainty component for possible variation in the
lattice spacing of different samples from the same ingot, again in parts in
$10^{9}$, are 6.1, 5.2, and 5.2.
The three INRIM lattice spacing values are correlated with one another, as
well as with the enriched silicon value of $N_{\rm A}$ discussed in Sec. IX.6
below. The latter correlation arises because the {220} lattice spacing of the
enriched silicon was determined at INRIM by Massa _et al._ (2011b) using the
same XROI apparatus (relative standard uncertainty of 3.5 parts in $10^{9}$
achieved). The relevant correlation coefficients for these data are given in
Table 21 and are calculated using information provided to the Task Group by
Mana (2011).
The many successful cross-checks of the performance of the new INRIM combined
x-ray and optical interferometer lend support to the reliability of the
results obtained with it. Indeed, Massa _et al._ (2011a) describe a highly
successful test based on the comparison of the lattice spacings of enriched
and natural silicon determined using the new XROI. Consequently, the IAC Mana
(2011). and the Task Group view the new INRIM values for
$d_{220}({\scriptstyle{\rm MO^{*}}})$ and $d_{220}({\scriptstyle{\rm W04}})$
as superseding the earlier INRIM values of these lattice spacings used in the
2006 adjustment.
### IX.2 $\bm{d_{220}}$ difference measurements of natural silicon crystals
Measurements of the fractional difference $\left[d_{220}({\scriptstyle
X})-d_{220}({\rm ref})\right]/d_{220}({\rm ref})$ of the {220} lattice spacing
of a sample of a single crystal ingot $X$ and that of a reference crystal
“ref” enable the lattice spacings of crystals used in various experiments to
be related to one another. Both NIST and PTB have carried out such
measurements, and the fractional differences from these two laboratories that
we take as input data in the 2010 adjustment are data items $B42$ to $B53$ in
Table LABEL:tab:pdata, labeled NIST-97, NIST-99, NIST-06, PTB-98, and PTB-03.
Their relevant correlation coefficients can be found in Table 21. For details
concerning the NIST and PTB difference measurements, see the three previous
CODATA reports. A discussion of item $B53$, the fractional difference between
the {220} lattice spacing of an ideal natural silicon crystal $d_{220}$ and
$d_{220}({\scriptstyle{\rm W04}})$, is given in CODATA-06 following Eq. (312).
### IX.3 Gamma-ray determination of the neutron relative atomic mass
$\bm{A_{\rm r}({\rm n})}$
The value of $A_{\rm r}({\rm n})$ listed in Table 2 from AME2003 is not used
in the 2010 adjustment. Rather, $A_{\rm r}({\rm n})$ is obtained as described
here so that the 2010 recommended value is consistent with the current data on
the {220} lattice spacing of silicon.
The value of $A_{\rm r}$(n) is obtained from measurement of the wavelength of
the 2.2 MeV g ray in the reaction n + p $\rightarrow$ d + g. The result
obtained from Bragg-angle measurements carried out at the high-flux reactor of
ILL in a NIST and ILL collaboration, is Kessler _et al._ (1999)
$\displaystyle\frac{\lambda_{\rm meas}}{d_{220}({\rm{\scriptstyle
ILL}})}=0.002\,904\,302\,46(50)\qquad[1.7\times 10^{-7}]\ .\quad$ (252)
Here $d_{220}({\rm{\scriptstyle ILL}})$ is the {220} lattice spacing of the
silicon crystals of the ILL GAMS4 spectrometer at $t_{90}=22.5~{}^{\circ}$C
and $p$ = 0 used in the measurements. Relativistic kinematics of the reaction
yields the observational equation
$\displaystyle{\lambda_{\rm meas}\over d_{220}({\rm{\scriptstyle ILL}})}$
$\displaystyle=$ $\displaystyle{\alpha^{2}A_{\rm r}({\rm e})\over
R_{\infty}d_{220}({\rm{\scriptstyle ILL}})}{A_{\rm r}({\rm n})+A_{\rm r}({\rm
p})\over\left[A_{\rm r}({\rm n})+A_{\rm r}({\rm p})\right]^{2}-A_{\rm
r}^{2}({\rm d})}\ ,$
where the quantities on the right-hand side are adjusted constants.
### IX.4 Historic X-ray units
Units used in the past to express the wavelengths of x-ray lines are the
copper ${\rm K\mbox{{a}}_{1}}$ x unit, symbol ${\rm xu(CuK\mbox{{a}}_{1})}$,
the molybdenum K${\rm\mbox{{a}}}_{1}$ x unit, symbol ${\rm
xu(MoK\mbox{{a}}_{1})}$, and the ångstrom star, symbol Å∗. They are defined by
assigning an exact, conventional value to the wavelength of the ${\rm
CuK\mbox{{a}}_{1}}$, ${\rm MoK\mbox{{a}}_{1}}$, and ${\rm WK\mbox{{a}}_{1}}$
x-ray lines when each is expressed in its corresponding unit:
$\displaystyle\lambda({\rm CuK\mbox{{a}}_{1}})$ $\displaystyle=$
$\displaystyle 1\,537.400\ {\rm xu(CuK\mbox{{a}}_{1})}\,,\qquad$ (254)
$\displaystyle\lambda({\rm MoK\mbox{{a}}_{1}})$ $\displaystyle=$
$\displaystyle 707.831\ {\rm xu(MoK\mbox{{a}}_{1})}\,,$ (255)
$\displaystyle\lambda({\rm WK\mbox{{a}}_{1}})$ $\displaystyle=$ $\displaystyle
0.209\,010\,0\ {\rm\AA^{*}}\,.$ (256)
The data relevant to these units are (see CODATA-98)
$\displaystyle{\lambda({\rm CuK\mbox{{a}}_{1}})\over d_{220}({\rm{\scriptstyle
W4.2a}})}$ $\displaystyle=$ $\displaystyle 0.802\,327\,11(24)\quad[3.0\times
10^{-7}]\,,$ (257) $\displaystyle{\lambda({\rm WK\mbox{{a}}_{1}})\over
d_{220}({\rm{\scriptstyle N}})}$ $\displaystyle=$ $\displaystyle
0.108\,852\,175(98)\quad[9.0\times 10^{-7}]\,,$ (258)
$\displaystyle{\lambda({\rm MoK\mbox{{a}}_{1}})\over d_{220}({\rm{\scriptstyle
N}})}$ $\displaystyle=$ $\displaystyle 0.369\,406\,04(19)\quad[5.3\times
10^{-7}]\,,$ (259) $\displaystyle{\lambda({\rm CuK\mbox{{a}}_{1}})\over
d_{220}({\rm{\scriptstyle N}})}$ $\displaystyle=$ $\displaystyle
0.802\,328\,04(77)\quad[9.6\times 10^{-7}]\,,\qquad$ (260)
where ${d_{220}({\rm{\scriptstyle W4.2a}})}$ and ${d_{220}({\rm{\scriptstyle
N}})}$ denote the {220} lattice spacings, at the standard reference conditions
$p=0$ and $t_{90}=22.5~{}^{\circ}{\rm C}$, of particular silicon crystals used
in the measurements. The result in Eq. (257) is from a collaboration between
researchers from Friedrich-Schiller University (FSUJ), Jena, Germany and the
PTB Härtwig _et al._ (1991).
To obtain recommended values for ${\rm xu(CuK\mbox{{a}}_{1})}$, ${\rm
xu(MoK\mbox{{a}}_{1})}$, and ${\rm\AA^{*}}$, we take these units to be
adjusted constants. The observational equations for the data of Eqs. (257) to
(260) are
$\displaystyle\frac{\lambda({\rm
CuK\mbox{{a}}_{1}})}{d_{220}({\rm{\scriptstyle{W4.2a}})}}$ $\displaystyle=$
$\displaystyle\frac{\rm
1\,537.400~{}xu(CuK\mbox{{a}}_{1})}{d_{220}({\rm{\scriptstyle{W4.2a}})}}\,,$
(261) $\displaystyle\frac{\lambda({\rm
WK\mbox{{a}}_{1}})}{d_{220}({\rm{\scriptstyle N}})}$ $\displaystyle=$
$\displaystyle\frac{\rm 0.209\,010\,0~{}\AA^{*}}{d_{220}({\rm{\scriptstyle
N}})}\,,$ (262) $\displaystyle\frac{\lambda({\rm
MoK\mbox{{a}}_{1}})}{d_{220}({\rm{\scriptstyle N}})}$ $\displaystyle=$
$\displaystyle\frac{\rm
707.831~{}xu(MoK\mbox{{a}}_{1})}{d_{220}({\rm{\scriptstyle N}})}\,,$ (263)
$\displaystyle\frac{\lambda({\rm
CuK\mbox{{a}}_{1}})}{d_{220}({\rm{\scriptstyle N}})}$ $\displaystyle=$
$\displaystyle\frac{\rm
1\,537.400~{}xu(CuK\mbox{{a}}_{1})}{d_{220}({\rm{\scriptstyle N}})}\,,\qquad$
(264)
where $d_{220}({\rm{\scriptstyle N}})$ is taken to be an adjusted constant and
$d_{220}({\rm{\scriptstyle W17}})$ and $d_{220}({\rm{\scriptstyle{W4.2a}}})$
are adjusted constants as well.
### IX.5 Other data involving natural silicon crystals
Two input data used in the 2006 adjustment but not used in the 2010 adjustment
at the request of the IAC Fujii (2010) are discussed in this section.
The first is the NMIJ value of $d_{220}({\scriptstyle{\rm NR3}})$, the {220}
lattice spacing reported by Cavagnero _et al._ (2004a). The IAC formally
requested that the Task Group not consider this result for the 2010
adjustment, because of its questionable reliability due to the problems
discussed in Sec. VIII.A.1._b_ of CODATA-06.
The second is the molar volume of natural silicon $V_{\rm m}({\rm Si})$ from
which $N_{\rm A}$ can be determined. The value used in the 2006 adjustment is
Fujii _et al._ (2005) $12.058\,8254(34)\times 10^{-6}~{}{\rm
m^{3}~{}mol^{-1}}~{}~{}[2.8\times 10^{-7}]$. The IAC requested that the Task
Group no longer consider this result, because of problems uncovered with the
molar mass measurements of natural silicon $M({\rm Si})$ at the Institute for
Reference Materials and Measurements (IRMM), Geel, Belgium.
One problem is associated with the experimental determination of the
calibration factors of the mass spectrometer used to measure the amount-of-
substance ratios (see following section) of the silicon isotopes ${}^{28}{\rm
Si}$, ${}^{29}{\rm Si}$, and ${}^{30}{\rm Si}$ in various silicon crystals, as
discussed by Valkiers _et al._ (2011). The factors are critical, because molar
masses are calculated from these ratios and the comparatively well-known
relative atomic masses of the isotopes. Another problem is the unexplained
large scatter of $\pm 7$ parts in $10^{7}$ in molar mass values among crystals
taken from the same ingot, as discussed by Fujii _et al._ (2005) in connection
with their result for $V_{\rm m}({\rm Si})$ given above.
More specifically, from 1994 to 2005 IRMM measured the molar masses of natural
silicon in terms of the molar mass of WASO17.2, which was determined using the
now suspect calibration factors Valkiers _et al._ (2011). Based on a new
determination of the calibration factors, Valkiers _et al._ (2011) report a
value for the molar mass of WASO17.2 that has a relative standard uncertainty
of $2.4\times 10^{-7}$, compared to the $1.3\times 10^{-7}$ uncertainty of the
value used since 1994, and which is fractionally larger by $1.34\times
10^{-6}$ than the earlier value. (The recent paper by Yi _et al._ (2012) also
points to a correction of the same general magnitude.) This new result and the
data and calculations in Fujii _et al._ (2005) yield the following revised
value for the molar volume of natural silicon:
$\displaystyle V_{\rm m}({\rm Si})$ $\displaystyle=$ $\displaystyle
12.058\,8416(45)\times 10^{-6}~{}{\rm m^{3}~{}mol^{-1}}$ (265)
$\displaystyle\qquad\qquad\qquad[3.7\times 10^{-7}]\,.\qquad$
Although the IAC does not consider this result to be sufficiently reliable for
the Task Group to consider it for inclusion in the 2010 adjustment, we note
that based on the 2010 recommended values of $d_{220}$ and the molar Planck
constant $N_{\rm A}h$, Eq. (265) implies
$\displaystyle N_{\rm A}$ $\displaystyle=$ $\displaystyle
6.022\,1456(23)\times 10^{23}~{}{\rm mol^{-1}}\quad[3.8\times 10^{-7}]\,,$
$\displaystyle h$ $\displaystyle=$ $\displaystyle 6.626\,0649(25)\times
10^{-34}~{}{\rm J~{}s}~{}~{}[3.8\times 10^{-7}]\,.\quad\qquad$ (266)
The difference between this value of $N_{\rm A}$ and the value with relative
standard uncertainty $3.0\times 10^{-8}$ obtained from enriched silicon
discussed in the next section is $7.9(3.8)$ parts in $10^{7}$, while the
difference between the NIST 2007 watt-balance value of $h$ with uncertainty
$3.6\times 10^{-8}$ and this value of $h$ is $6.1(3.8)$ parts in $10^{7}$.
### IX.6 Determination of $\bm{N_{\rm A}}$ with enriched silicon
The IAC project to determine $N_{\rm A}$ using the XRCD method and silicon
crystals highly enriched with ${}^{28}{\rm Si}$ was formally initiated in
2004, but its origin dates back two decades earlier. Its initial result is
discussed in detail in a _Metrologia_ special issue; see the Foreword by Massa
and Nicolaus (2011), the 14 technical papers in the issue, and the references
cited therein. The first paper, by Andreas _et al._ (2011a), provides an
extensive overview of the entire project. The value of the Avogadro constant
obtained from this unique international collaborative effort, identified as
IAC-11, input datum $B54$, is Andreas _et al._ (2011a)
$\displaystyle N_{\rm A}$ $\displaystyle=$ $\displaystyle
6.022\,140\,82(18)\times 10^{23}~{}{\rm mol^{-1}}\quad[3.0\times
10^{-8}]\,.\qquad$ (267)
Note that this result differs slightly from the somewhat earlier result
reported by Andreas _et al._ (2011b) but is the preferred value Bettin (2011).
The basic equation for the XRCD determination of $N_{\rm A}$ has been
discussed in previous CODATA reports. In brief,
$\displaystyle N_{\rm A}$ $\displaystyle=$ $\displaystyle{A_{\rm r}({\rm
Si})M_{\rm u}\over\sqrt{8}\,d_{220}^{\,3}\,\rho{\rm(Si)}}\ ,$ (268)
which would apply to an impurity free, crystallographically perfect, “ideal”
silicon crystal. Here $A_{\rm r}$(Si) is the mean relative atomic mass of the
silicon atoms in such a crystal, and $\rho$(Si) is the crystal’s macroscopic
mass density. Thus, to determine $N_{\rm A}$ from Eq. (268) requires
determining the density $\rho$(Si), the {220} lattice spacing $d_{220}$, and
the amount-of-substance ratios $R_{29/28}=n(^{29}\rm Si)$/$n(^{28}\rm Si)$ and
$R_{30/28}=n(^{30}\rm Si)$/$n(^{28}\rm Si)$ so that $A_{\rm r}(\rm Si)$ can be
calculated using the well-known values of $A_{\rm r}(^{A}\rm Si)$. Equally
important is the characterization of the material properties of the crystals
used, for example, impurity content, non-impurity point defects, dislocations,
and microscopic voids must be considered.
The international effort to determine the Avogadro constant, as described in
the _Metrologia_ special issue, involved many tasks including the following:
enrichment and poly-crystal growth of silicon in the Russian Federation;
growth and purification of a 5 kg single silicon crystal ingot in Germany;
measurement of the isotopic composition of the crystals at PTB; measurement of
the lattice spacing with the newly developed XROI described above at INRIM;
grinding and polishing of two spheres cut from the ingot to nearly perfect
spherical shape at NMI; optical interferometric measurement of the diameters
of the spheres at PTB and NMIJ; measurement of the masses of the spheres in
vacuum at PTB, NMIJ, and BIPM; and characterization of and correction for the
effect of the contaminants on the surfaces of the spheres at various
laboratories.
The uncertainty budget for the IAC value of $N_{\rm A}$ is dominated by
components associated with determining the volumes and the surface properties
of the spheres, followed by those related to measuring their lattice spacings
and their molar masses. These four components, in parts in $10^{9}$, are 29,
15, 11, and 8 for the sphere designated AVO28-S5.
How this result compares with other data and its role in the 2010 adjustment
is discussed in Sec. XIII.
## X Thermal physical quantities
Table 16 summarizes the eight results for the thermal physical quantities $R$,
$k$, and $k/h$, the molar gas constant, the Boltzmann constant, and the
quotient of the Boltzmann and Planck constants, respectively, that are taken
as input data in the 2010 adjustment. They are data items $B58.1$ to $B60$ in
Table LABEL:tab:pdata with correlation coefficients as given in Table 21 and
observational equations as given in Table 33. Values of $k$ that can be
inferred from these data are given in Table 27 and are graphically compared in
Fig. 4. The first two results, the NPL 1979 and NIST 1988 values of $R$, were
included in the three previous CODATA adjustments, but the other six became
available during the 4 years between the closing dates of the 2006 and 2010
adjustments. (Note that not every result in Table 16 appears in the cited
reference. For some, additional digits have been provided to the Task Group to
reduce rounding errors; for others, the value of $R$ or $k$ actually
determined in the experiment is recovered from the reported result using the
relation $R=kN_{\rm A}$ and the value of $N_{\rm A}$ used by the researchers
to obtain that result.)
### X.1 Acoustic gas thermometry
As discussed in CODATA-98 and the references cited therein, measurement of $R$
by the method of acoustic gas thermometry (AGT) is based on the following
expressions for the square of the speed of sound in a real gas of atoms or
molecules in thermal equilibrium at thermodynamic temperature $T$ and pressure
$p$ and occupying a volume $V$:
$\displaystyle c_{\rm a}^{2}{(T,p)}$ $\displaystyle=$ $\displaystyle
A_{0}(T)+A_{1}(T)p$ (269)
$\displaystyle+A_{2}(T)p^{2}+A_{3}(T)p^{3}+\cdots\,.\qquad$
Here $A_{1}(T)$ is the first acoustic virial coefficient, $A_{2}(T)$ is the
second, etc. In the limit $p\rightarrow 0$, this becomes
$\displaystyle c_{\rm a}^{2}{(T,0)}=A_{0}(T)={\gamma_{0}RT\over A_{\rm
r}(X)M_{\rm u}}\,,$ (270)
where $\gamma_{0}=c_{p}/c_{V}$ is the ratio of the specific heat capacity of
the gas at constant pressure to that at constant volume and is 5/3 for an
ideal monotonic gas. The basic experimental approach to determining the speed
of sound of a gas, usually argon or helium, is to measure the acoustic
resonant frequencies of a cavity at or near the triple point of water, $T_{\rm
TPW}=273.16~{}{\rm K}$, and at various pressures and extrapolating to $p=0$.
The cavities are either cylindrical of fixed or variable length, or spherical,
but most commonly quasispherical in the form of a triaxial ellipsoid. This
shape removes the degeneracy of the microwave resonances used to measure the
volume of the resonator in order to calculate $c_{\rm a}^{2}(T,p)$ from the
measured acoustic frequencies and the corresponding acoustic resonator
eigenvalues known from theory. The cavities are formed by carefully joining
hemispherical cavities.
In practice, the determination of $R$ by AGT with a relative standard
uncertainty of order one part in $10^{6}$ is complex; the application of
numerous corrections is required as well as the investigation of many possible
sources of error. For a review of the advances made in AGT in the past 20
years, see Moldover (2009).
#### X.1.1 NPL 1979 and NIST 1988 values of $\bm{R}$
Both the NPL and NIST experiments are discussed in detail in CODATA-98. We
only note here that the NPL measurement used argon in a vertical, variable-
path-length, 30 mm inner diameter cylindrical acoustic resonator operated at a
fixed frequency, and the displacement of the acoustic reflector that formed
the top of the resonator was measured using optical interferometry. The NIST
experiment also used argon, and the volume of the stainless steel spherical
acoustic resonator, of approximate inside diameter 180 mm, was determined from
the mass of mercury of known density required to fill it. The 1986 CODATA
recommended value of $R$ is the NPL result while the 1998, 2002, and 2006
CODATA recommended values are the weighted means of the NPL and NIST results.
#### X.1.2 LNE 2009 and 2011 values of $\bm{R}$
Pitre _et al._ (2009); Pitre (2011) obtained the LNE 2009 result using a
copper quasisphere of about 100 mm inner diameter and helium gas. The
principal advantage of helium is that its thermophysical properties are well-
known based on ab initio theoretical calculations; the principal disadvantage
is that because of its comparatively low mass, impurities have a larger effect
on the speed of sound. This problem is mitigated by passing the helium gas
through a liquid helium trap and having a continuous flow of helium through
the resonator, thereby reducing the effect of outgassing from the walls of the
resonator. In calculating the molar mass of the helium Pitre _et al._ (2009)
assumed that the only remaining impurity is 3He and that the ratio of 3He to
4He is less than $1.3\times 10^{-6}$.
The critically important volume of the resonator was determined from
measurements of its electromagnetic (EM) resonances together with relevant
theory of the eigenvalues. The dimensions of the quasihemispheres were also
measured using a coordinate measuring machine (CMM). The volumes so obtained
agreed, but the $17\times 10^{-6}$ relative standard uncertainty of the CMM
determination far exceeded the $0.85\times 10^{-6}$ relative uncertainty of
the EM determination. The principal uncertainty components that contribute to
the 2.7 parts in $10^{6}$ uncertainty of the final result are, in parts in
$10^{6}$, 1.8, 1.0, 1.5, and 0.8 due, respectively, to measurement of the
volume of the quasisphere (including various corrections), its temperature
relative to $T_{\rm TPW}$, extrapolation of $c_{\rm a}^{2}(T_{\rm TPW},p)$ to
$p=0$, and the reproducibility of the result, based on two runs using
different purities of helium and different acoustic transducers Pitre (2011).
The 2011 LNE result for $R$, which has the smallest uncertainty of any
reported to date, is described in great detail by Pitre _et al._ (2011). It
was obtained using the same quasispherical resonator employed in the 2009
experiment, but with argon in place of helium. The reduction in uncertainty by
more than a factor of two was achieved by improving all aspects of the
experiment Pitre _et al._ (2011). The volume of the resonator was again
determined from measurements of its EM resonances and cross checked with CMM
dimensional measurements of the quasispheres carried out at NPL de Podesta _et
al._ (2010). As usual in AGT, the square of the speed-of-sound was determined
from measurements of the quasisphere’s acoustic resonant frequencies at
different pressures (50 kPa to 700 kPa in this case) and extrapolation to
$p=0$. The isotopic composition of the argon and its impurity content was
determined at IRMM Valkiers _et al._ (2010).
The five uncertainty components of the final 1.24 parts in $10^{6}$
uncertainty of the result, with each component itself being composed of a
number of subcomponents, are, in parts in $10^{6}$, the following: 0.30 from
temperature measurements (the nominal temperature of the quasisphere was
$T_{\rm TPW})$; 0.57 from the EM measurement of the quasisphere’s volume; 0.84
from the determination of $c_{\rm a}^{2}(T_{\rm TPW},0)$; 0.60 associated with
the argon molar mass and its impurities; and 0.25 for experimental
repeatability based on the results from two series of measurements carried out
in May and July of 2009.
Because the LNE 2009 and 2011 results are from experiments in which some of
the equipment and measuring techniques are the same or similar, they are
correlated. Indeed, for the same reason, there are non-negligible correlations
among the four recent AGT determinations of $R$, that is, LNE-09, NPL-10,
INRIM-10, and LNE-11. These correlations are given in Table 21 and have been
calculated using information provided to the Task Group by researchers
involved in the experiments Gavioso _et al._ (2011).
#### X.1.3 NPL 2010 value of R
This result was obtained at NPL by Sutton _et al._ (2010); de Podesta (2011)
at $T_{\rm TPW}$ using a thin-walled copper quasispherical resonator of about
100 mm inner diameter on loan from LNE and argon as the working gas. The
internal surfaces of the quasihemispheres were machined using diamond turning
techniques. The 5 mm wall thickness of the quasisphere, about one-half that of
the usual AGT resonators, was specially chosen to allow improved study of the
effect of resonator shell vibrations on acoustic resonances. The volume of the
quasisphere was determined from measurements of EM resonances and checked with
CMM dimensional measurements of the quasihemispheres before assembly de
Podesta _et al._ (2010). Two series of measurements were carried out, each
lasting several days: one with the quasisphere rigidly attached to a fixed
stainless steel post and one with it freely suspended by three wires attached
to its equator. Pressures ranged from 50 kPa to 650 kPa and were measured with
commercial pressure meters. The isotopic composition of the argon and its
impurity content were again determined at IRMM Valkiers _et al._ (2010).
The final result is the average of the value obtained from each run. The 3.78
parts in $10^{6}$ difference between the molar mass of the argon used in the
fixed and hanging quasisphere runs is to a large extent canceled by the
$-2.77$ parts in $10^{6}$ difference between the values of $c_{\rm
a}^{2}(T_{\rm TPW},0)$ for the two runs, so the two values of $R$ agree within
1.01 parts in $10^{6}$. The largest uncertainty components in parts in
$10^{6}$ contributing to the final uncertainty of 3.1 parts in $10^{6}$ are,
respectively Sutton _et al._ (2010); de Podesta _et al._ (2010), 2, 1.1, 0.9,
1, and 1.4 arising from the difference between the acoustic and microwave
volumes of the resonator, temperature calibration, temperature measurement,
argon gas impurities, and correction for the layer of gas near the wall of the
resonator (thermal boundary layer correction).
#### X.1.4 INRIM 2010 value of R
The INRIM determination of $R$ by Gavioso _et al._ (2010); Gavioso (2011)
employed a stainless steel spherical resonator of about 182 mm inner diameter
and non-flowing helium gas. Although the measurements were performed with the
resonator very near $T_{\rm TPW}$ as in the other AGT molar-gas-constant
determinations, two important aspects of the INRIM experiment are quite
different. First, the speed of sound was measured at only one pressure,
namely, 410 kPa, and the extrapolation to $p=0$ was implemented using the
comparatively well-known theoretical values of the required ${}^{4}{\rm He}$
equation-of-state and acoustic virial coefficients. Second, the radius of the
resonator was determined using the theoretical value of the ${}^{4}{\rm He}$
index of refraction together with eight measured EM resonance frequencies and
the corresponding predicted eigenvalues. The speed of sound was then
calculated from this value of the radius and measured acoustic resonant
frequencies. Gavioso _et al._ (2010) calculated the molar mass of their He
sample assuming the known atmospheric abundance of 3He represents an upper
limit.
The two uncertainty components that are by far the largest contributors to the
7.5 parts in $10^{6}$ final uncertainty of the experiment are, in parts in
$10^{6}$, 4.2 from fitting the shape of the eight measured microwave modes and
4.8 from the scatter of the squared frequencies of the six measured radial
acoustic modes used to determine $c_{\rm a}^{2}(T_{\rm TPW},p=410~{}{\rm
kPa})$.
### X.2 Boltzmann constant $\bm{k}$ and quotient $\bm{k/h}$
The following two sections discuss the two NIST experiments that have yielded
the last two entries of Table 16.
#### X.2.1 NIST 2007 value of k
This result was obtained by Schmidt _et al._ (2007) using the technique of
refractive index gas thermometry (RIGT), an approach similar to that of
dielectric constant gas thermometry (DCGT) discussed in CODATA-98, and to a
lesser extent in CODATA-02 and CODATA-06. The starting point of both DCGT and
RIGT is the virial expansion of the equation of state for a real gas of amount
of substance $n$ in a volume $V$ Schmidt _et al._ (2007),
$\displaystyle p=\rho RT\left[1+\rho
b(T)+\rho^{2}c(T)+\rho^{3}d(T)+\cdots\right],$ (271)
where $\rho=n/V$ is the amount of substance density of the gas at
thermodynamic temperature $T$, and $b(T)$ is the first virial coefficient,
$c(T)$ is the second, etc.; and the Clausius-Mossotti equation
$\displaystyle{\epsilon_{\rm r}-1\over\epsilon_{\rm r}+2}$ $\displaystyle=$
$\displaystyle\rho A_{\epsilon}\bigg{[}1+\rho B_{\epsilon}(T)$ (272)
$\displaystyle\qquad+\rho^{2}C_{\epsilon}(T)+\rho^{3}D_{\epsilon}(T)+\cdots\bigg{]},\qquad$
where $\epsilon_{\rm r}=\epsilon/\epsilon_{0}$ is the relative dielectric
constant (relative permittivity) of the gas, $\epsilon$ is its dielectric
constant, $\epsilon_{0}$ is the exactly known electric constant,
$A_{\epsilon}$ is the molar polarizability of the atoms, and
$B_{\epsilon}(T)$, $C_{\epsilon}(T)$, etc. are the dielectric virial
coefficients. The static electric polarizability of a gas atom $\alpha_{0}$,
$A_{\epsilon}$, $R$, and $k$ are related by
$A_{\epsilon}/R=\alpha_{0}/3{\epsilon_{0}}k$, which shows that if $\alpha_{0}$
is known sufficiently well from theory, which it currently is for ${}^{4}{\rm
He}$ Łach _et al._ (2004); Jentschura _et al._ (2011b); Puchalski _et al._
(2011), then a competitive value of $k$ can be obtained if the quotient
$A_{\epsilon}/R$ can be measured with a sufficiently small uncertainty.
In fact, by appropriately combining Eqs. (271) and (272), an expression is
obtained from which $A_{\epsilon}/R$ can be experimentally determined by
measuring $\epsilon_{\rm r}$ at a known constant temperature such as $T_{\rm
TPW}$ and at different pressures and extrapolating to zero pressure. This is
done in practice by measuring the fractional change in capacitance of a
specially constructed capacitor, first without helium gas and then with helium
gas at a known pressure. This is the DCGT technique.
In the RIGT technique of Schmidt _et al._ (2007), $A_{\epsilon}/R$ is
determined, and hence $k$, from measurements of $n^{2}(T,p)\equiv\epsilon_{\rm
r}\mu_{\rm r}$ of a gas of helium, where $n(T,p)$ is the index of refraction
of the gas, $\mu_{\rm r}=\mu/\mu_{0}$ is the relative magnetic permeability of
the gas, $\mu$ is its magnetic permeability, and $\mu_{0}$ is the exactly
known magnetic constant. Because ${}^{4}{\rm He}$ is slightly diamagnetic, the
quantity actually determined is $(A_{\epsilon}+A_{\mu})/R$, where
$A_{\mu}=4\mbox{{p}}\chi_{0}/3$ and $\chi_{0}$ is the diamagnetic
susceptibility of a ${}^{4}{\rm He}$ atom. The latter quantity is known from
theory and the theoretical value of $A_{\mu}$ was used to obtain
$A_{\epsilon}/R$ from the determined quantity.
Schmidt _et al._ (2007) obtained $n(T,p)$ by measuring the microwave resonant
frequencies from 2.7 GHz to 7.6 GHz of a quasispherical copper plated
resonator, either evacuated or filled with He at pressures of 0.1 MPa to 6.3
MPa. The temperature of the resonator was within a few millikelvin of $T_{\rm
TPW}$. A network analyzer was used to measure the resonant frequencies and a
calibrated pressure balance to measure $p$. The extrapolation to $p=0$
employed both theoretical and experimental values of the virial coefficients
$B,C,D,b$, and $c$ taken from the literature. The uncertainties of these
coefficients and of the pressure and temperature measurements, and the
uncertainty of the isothermal compressibility of the resonator, are the
largest components in the uncertainty budget.
#### X.2.2 NIST 2011 value of k/h
As discussed in CODATA-98, the Nyquist theorem predicts, with a fractional
error of less than one part in $10^{6}$ at frequencies less than 10 MHz and
temperatures greater than 250 K, that
$\displaystyle\langle U^{2}\rangle=4kTR_{\rm s}\Delta f\,.$ (273)
Here $\langle U^{2}\rangle$ is the mean-square-voltage, or Johnson noise
voltage, in a measurement bandwidth of frequency $\Delta f$ across the
terminals of a resistor of resistance $R_{\rm s}$ in thermal equilibrium at
thermodynamic temperature $T$. If $\langle U^{2}\rangle$ is measured in terms
of the Josephson constant $K_{\rm J}=2e/h$ and $R_{\rm s}$ in terms of the von
Klitzing constant $R_{\rm K}=h/e^{2}$, then this experiment yields a value of
$k/h$.
Such an experiment has been carried out at NIST, yielding the result in Table
16; see the paper by Benz _et al._ (2011) and references therein. In that
work, digitally synthesized pseudo-noise voltages are generated by means of a
pulse-biased Josephson junction array. These known voltages are compared to
the unknown thermal-noise voltages generated by a specially designed 100
$\Omega$ resistor in a well regulated thermal cell at or near $T_{\rm TPW}$.
Since the spectral density of the noise voltage of a 100 $\Omega$ resistor at
273.16 K is only 1.23 nV$\sqrt{\rm Hz}\,$, it is measured using a low-noise,
two-channel, cross-correlation technique that enables the resistor signal to
be extracted from uncorrelated amplifier noise of comparable amplitude and
spectral density. The bandwidths range from 10 kHz to 650 kHz. The final
result is based on two data runs, each of about 117 hours duration, separated
in time by about three months.
The dominant uncertainty component of the 12.1 parts in $10^{6}$ total
uncertainty is the 12.0 parts in $10^{6}$ component due to the measurement of
the ratio $\langle V^{2}_{\rm R}/V^{2}_{\rm Q}\rangle$, where $V_{\rm R}$ is
the resistor noise voltage and $V_{\rm Q}$ is the synthesized voltage. The
main uncertainty component contributing to the uncertainty of the ratio is
10.4 parts in $10^{6}$ due to spectral aberrations, that is, effects that lead
to variations of the ratio with bandwidth.
### X.3 Other data
We note for completeness the following three results, each of which agrees
with its corresponding 2010 recommended value. The first has a non-competitive
uncertainty but is of interest because it is obtained from a relatively new
method that could yield a value with a competitive uncertainty in the future.
The other two became available only after the 31 December 2010 closing date of
the 2010 adjustment.
Lemarchand _et al._ (2011) find $R=8.314\,80(42)~{}{\rm
J~{}m^{-1}~{}K^{-1}}~{}[50\times 10^{-6}]$ determined by the method of Doppler
spectroscopy, in particular, by measuring near the ice point $T=273.15~{}{\rm
K}$ the absorption profile of a rovibrational line at $\nu=30~{}{\rm THz}$ of
ammonia molecules in an ammonium gas in thermal equilibrium. The width of the
line is mainly determined by the Doppler width due to the velocity
distribution of the ${}^{4}{\rm NH}_{3}$ molecules along the direction of the
incident laser beam. The relevant expression is
$\displaystyle{\Delta\omega_{\rm D}\over\omega_{0}}=\left({2kT\over m(^{4}{\rm
NH}_{3})c^{2}}\right)^{1/2}=\left({2RT\over A_{\rm r}(^{4}{\rm NH}_{3})M_{\rm
u}c^{2}}\right)^{1/2},$ (274)
where $\Delta\omega_{\rm D}$ is the e-fold angular frequency half-width of the
Doppler profile of the ammonium line at temperature $T$, $\omega_{0}$ is its
angular frequency, and $m(^{4}{\rm NH}_{3})$ and $A_{\rm r}(^{4}{\rm NH}_{3})$
are the mass and relative atomic mass of the ammonium molecule.
Zhang _et al._ (2011); Zhang (2011) obtain $R=8.314\,474(66)~{}{\rm
J~{}m^{-1}~{}K^{-1}}~{}[7.9\times 10^{-6}]$ using acoustic gas thermometry
with argon gas, more specifically, by measuring resonant frequencies of a
fixed-path-length cylindrical acoustic resonator at $T_{\rm TPW}$; its
approximate 129 mm length is measured by two-color optical interferometry.
Gaiser and Fellmuth (2012); Fellmuth _et al._ (2011) give
$k=1.380\,655(11)\times 10^{-23}~{}{\rm J/K}~{}[7.9\times 10^{-6}]$ measured
using dielectric gas thermometry (see Sec. X.2.1 above) and helium gas at
$T_{\rm TPW}$ and also at temperatures in the range 21 K to 27 K surrounding
the triple point of neon at $T\approx 25~{}{\rm K}$.
### X.4 Stefan-Boltzmann constant $\sigma$
The Stefan-Boltzmann constant is related to $c$, $h$, and $k$ by
$\sigma=2\mbox{{p}}^{5}k^{4}/{15h^{3}c^{2}}$, which, with the aid of the
relations $k=R/N_{\rm A}$ and $N_{\rm A}h=cA_{\rm r}({\rm e})M_{\rm
u}\alpha^{2}/2R_{\infty}$, can be expressed in terms of the molar gas constant
and other adjusted constants as
$\displaystyle\sigma=\frac{32\mbox{{p}}^{5}h}{15c^{6}}\left(\frac{R_{\infty}R}{A_{\rm
r}({\rm e})M_{\rm u}\alpha^{2}}\right)^{4}.$ (275)
Since no competitive directly measured value of $\sigma$ is available for the
2010 adjustment, the 2010 recommended value is obtained from this equation.
Table 16: Summary of thermal physical measurements relevant to the 2010
adjustment (see text for details). AGT: acoustic gas thermometry; RIGT:
refractive index gas thermometry; JNT: Johnson noise thermometry; cylindrical,
spherical, quasispherical: shape of resonator used; JE and QHE: Josephson
effect voltage and quantum Hall effect resistance standards.
Source | Ident.${}^{\text{a}}$ | Quant. | Method | Value | Rel. stand.
---|---|---|---|---|---
| | | | | uncert $u_{\rm r}$
Colclough _et al._ (1979) | NPL-79 | $R$ | AGT, cylindrical, argon | $8.314\,504(70)$ J mol-1 K-1 | $8.4\times 10^{-6}$
Moldover _et al._ (1988) | NIST-88 | $R$ | AGT, spherical, argon | $8.314\,471(15)$ J mol-1 K-1 | $1.8\times 10^{-6}$
Pitre _et al._ (2009) | LNE-09 | $R$ | AGT, quasispherical, helium | $8.314\,467(22)$ J mol-1 K-1 | $2.7\times 10^{-6}$
Sutton _et al._ (2010) | NPL-10 | $R$ | AGT, quasispherical, argon | $8.314\,468(26)$ J mol-1 K-1 | $3.1\times 10^{-6}$
Gavioso _et al._ (2010) | INRIM-10 | $R$ | AGT, spherical, helium | $8.314\,412(63)$ J mol-1 K-1 | $7.5\times 10^{-6}$
Pitre _et al._ (2011) | LNE-11 | $R$ | AGT, quasispherical, argon | $8.314\,456(10)$ J mol-1 K-1 | $1.2\times 10^{-6}$
Schmidt _et al._ (2007) | NIST-07 | $k$ | RIGT, quasispherical, helium | $1.380\,653(13)\times 10^{-23}$ J K-1 | $9.1\times 10^{-6}$
Benz _et al._ (2011) | NIST-11 | $k/h$ | JNT, JE and QHE | $2.083\,666(25)\times 10^{10}$ Hz K-1 | $1.2\times 10^{-5}$
${}^{\text{a}}$NPL: National Physical Laboratory, Teddington, UK; NIST:
National Institute of Standards and Technology, Gaithersburg, MD, and Boulder,
CO, USA; LNE: Laboritoire commun de métrologie (LCM), Saint-Denis, France, of
the Laboratoire national de métrologie et d’essais (LNE); INRIM: Istituto
Nazionale di Ricerca Metrologica, Torino, Italy.
## XI Newtonian constant of gravitation $\bm{G}$
Table 17: Summary of the results of measurements of the Newtonian constant of
gravitation relevant to the 2010 adjustment.
Source | Identification${}^{\text{a}}$ | Method | $10^{11}\,G$ | Rel. stand.
---|---|---|---|---
| | | $\overline{{\rm m}^{3}\ {\rm kg}^{-1}\ {\rm s}^{-2}}$ | uncert $u_{\rm r}$
Luther and Towler (1982) | NIST-82 | Fiber torsion balance, | $6.672\,48(43)$ | $6.4\times 10^{-5}$
| | dynamic mode | |
Karagioz and Izmailov (1996) | TR&D-96 | Fiber torsion balance, | $6.672\,9(5)$ | $7.5\times 10^{-5}$
| | dynamic mode | |
Bagley and Luther (1997) | LANL-97 | Fiber torsion balance, | $6.673\,98(70)$ | $1.0\times 10^{-4}$
| | dynamic mode | |
Gundlach and Merkowitz (2000, 2002) | UWash-00 | Fiber torsion balance, | $6.674\,255(92)$ | $1.4\times 10^{-5}$
| | dynamic compensation | |
Quinn _et al._ (2001) | BIPM-01 | Strip torsion balance, | $6.675\,59(27)$ | $4.0\times 10^{-5}$
| | compensation mode, static deflection | |
Kleinevoß (2002); Kleinvoß _et al._ (2002) | UWup-02 | Suspended body, | $6.674\,22(98)$ | $1.5\times 10^{-4}$
| | displacement | |
Armstrong and Fitzgerald (2003) | MSL-03 | Strip torsion balance, | $6.673\,87(27)$ | $4.0\times 10^{-5}$
| | compensation mode | |
Hu _et al._ (2005) | HUST-05 | Fiber torsion balance, | $6.672\,28(87)$ | $1.3\times 10^{-4}$
| | dynamic mode | |
Schlamminger _et al._ (2006) | UZur-06 | Stationary body, | $6.674\,25(12)$ | $1.9\times 10^{-5}$
| | weight change | |
Luo _et al._ (2009); Tu _et al._ (2010) | HUST-09 | Fiber torsion balance, | $6.673\,49(18)$ | $2.7\times 10^{-5}$
| | dynamic mode | |
Parks and Faller (2010) | JILA-10 | Suspended body, | $6.672\,34(14)$ | $2.1\times 10^{-5}$
| | displacement | |
${}^{\text{a}}$NIST: National Institute of Standards and Technology,
Gaithersburg, MD, USA; TR&D: Tribotech Research and Development Company,
Moscow, Russian Federation; LANL: Los Alamos National Laboratory, Los Alamos,
New Mexico, USA; UWash: University of Washington, Seattle, Washington, USA;
BIPM: International Bureau of Weights and Measures, Sèvres, France; UWup:
University of Wuppertal, Wuppertal, Germany; MSL: Measurement Standards
Laboratory, Lower Hutt, New Zeland; HUST: Huazhong University of Science and
Technology, Wuhan, PRC; UZur: University of Zurich, Zurich, Switzerland; JILA:
JILA, University of Colorado and National Institute of Standards and
Technology, Boulder, Colorado, USA.
Table 17 summarizes the 11 values of the Newtonian constant of gravitation $G$
of interest in the 2010 adjustment. Because they are independent of the other
data relevant to the current adjustment, and because there is no known
quantitative theoretical relationship between $G$ and other fundamental
constants, they contribute only to the determination of the 2010 recommended
value of $G$. The calculation of this value is discussed in Sec. XIII.2.1.
The inconsistencies between different measurements of $G$ as discussed in the
reports of previous CODATA adjustments demonstrate the historic difficulty of
determining this most important constant. Unfortunately, this difficulty has
been demonstrated anew with the publication of two new competitive results for
$G$ during the past 4 years. The first is an improved value from the group at
the Huazhong University of Science and Technology (HUST), PRC, identified as
HUST-09 Luo _et al._ (2009); Tu _et al._ (2010); the second is a completely
new value from researchers at JILA, Boulder, Colorado, USA, identified as
JILA-10 Parks and Faller (2010). (JILA, formerly known as the Joint Institute
for Laboratory Astrophysics, is a joint institute of NIST and the University
of Colorado and is located on the University of Colorado campus, Boulder,
Colorado.)
The publication of the JILA value has led the Task Group to re-examine and
modify two earlier results. The first is that obtained at NIST (then known as
the National Bureau of Standards) by Luther and Towler (1982) in a NIST-
University of Virginia (UVa) collaboration, labeled NIST-82. This value was
the basis for the CODATA 1986 recommended value Cohen and Taylor (1987) and
was taken into account in determining the CODATA 1998 value Mohr and Taylor
(2000), but played no role in either the 2002 or 2006 adjustments. The second
is the Los Alamos National Laboratory (LANL), Los Alamos, USA, result of
Bagley and Luther (1997), labeled LANL-97; it was first included in the 1998
CODATA adjustment and in all subsequent adjustments. Details of the
modifications to NIST-82 and LANL-97 (quite minor for the latter), the reasons
for including NIST-82 in the 2010 adjustment, and discussions of the new
values HUST-09 and JILA-10 are given below. The 11 available values of $G$,
which are data items $G1$-$G11$ in Table 24, Sec. XIII, are the same as in
2006 with the exception of NIST-82, slightly modified LANL-97, and the two new
values. Thus, in keeping with our approach in this report, there is no
discussion of the other seven values since they have been covered in one or
more of the previous reports.
For simplicity, in the following text, we write $G$ as a numerical factor
multiplying $G_{0}$, where
$\displaystyle G_{0}=10^{-11}~{}{\rm m^{3}~{}kg^{-1}~{}s^{-2}}\ .$ (276)
### XI.1 Updated values
#### XI.1.1 National Institute of Standards and Technology and University of
Virginia
As discussed in CODATA-98, the experiment of Luther and Towler (1982) used a
fiber-based torsion balance operated in the dynamic mode and the time-of-swing
method, thereby requiring measurement of a small change in the long
oscillation period of the balance. Ideally, the torsional spring constant of
the fiber should be independent of frequency at very low frequencies, for
example, at 3 mHz.
Long after the publication of the NIST-UVa result, Kuroda (1995) [see also
Matsumura _et al._ (1998) and Kuroda (1999)] pointed out that the anelasticity
of such fibers is sufficiently large to cause the value of $G$ determined in
this way to be biased. If $Q$ is the quality factor of the main torsional mode
of the fiber and it is assumed that the damping of the torsion balance is
solely due to the losses in the fiber, then the unbiased value of $G$ is
related to the experimentally observed value $G({\rm obs})$ by Kuroda (1995)
$\displaystyle G={G({\rm obs})\over 1+\mbox{{p}}Q\,}\,.$ (277)
Although the exact value of the $Q$ of the fiber used in the NIST-UVa
experiment is unknown, one of the researchers Luther (2010) has provided an
estimate, based on data obtained during the course of the experiment, of no
less than 10 000 and no greater than 30 000. Assuming a rectangular
probability density function for $Q$ with these lower and upper limits then
leads to $Q=2\times 10^{4}$ with a relative standard uncertainty of $4.6\times
10^{-6}$. Using these values, the result $G({\rm obs})=6.672~{}59(43)G_{0}$
$[64\times 10^{-6}]$ Luther and Towler (1982), Luther (1986), and Eq. (277) we
obtain
$\displaystyle G=6.672\,48(43)G_{0}\quad[6.4\times 10^{-5}]\,.$ (278)
In this case the correction $1/(1+\mbox{{p}}Q)$ reduced $G({\rm obs})$ by the
fractional amount $15.9(4.6)\times 10^{-6}$, but increased its $64\times
10^{-6}$ relative standard uncertainty by a negligible amount.
The Task Group decided to include the value given in Eq. (278) as an input
datum in the 2010 adjustment even though it was not included in the 2002 and
2006 adjustments, because information provided by Luther (2010) allows the
original result to be corrected for the Kuroda effect. Further, although there
were plans to continue the NIST-UVa experiment Luther and Towler (1982),
recent conversations with Luther (2010) made clear that the measurements on
which the result is based were thorough and complete.
#### XI.1.2 Los Alamos National Laboratory
The experiment of Bagley and Luther (1997), also described in detail in
CODATA-98, is similar to the NIST-UVa experiment of Luther and Towler (1982),
and in fact used some of the same components including the tungsten source
masses. Its purpose was not only to determine $G$, but also to test the Kuroda
hypothesis by using two different fibers, one with $Q=950$ and the other with
$Q=490$. Because the value of $G$ resulting from this experiment is correlated
with the NIST-UVa value and both values are now being included in the
adjustment, we evaluated the correlation coefficient of the two results. This
was done with information from Bagley (2010), Luther (2010), and the Ph.D.
thesis of Bagley (1996). We take into account the uncertainties of the two $Q$
values (2 %) and the correlation coefficient of the two values of $G$ obtained
from the two fibers (0.147) when computing their weighted mean. The final
result is
$\displaystyle G=6.673\,98(70)\quad[1.0\times 10^{-4}]\,,$ (279)
which in fact is essentially the same as the value used in the 2002 and 2006
adjustments. The correlation coefficient of the NIST-UVa and LANL values of
$G$ is 0.351.
### XI.2 New values
#### XI.2.1 Huazhong University of Science and Technology
The improved HUST-09 result for $G$ was first reported by Luo _et al._ (2009)
and subsequently described in detail by Tu _et al._ (2010); it represents a
reduction in uncertainty, compared to the previous Huazhong University result
HUST-05, of about a factor of five. As pointed out by Tu _et al._ (2010), a
number of changes in the earlier experiment contributed to this uncertainty
reduction, including (i) replacement of the two stainless steel cylindrical
source masses by spherical source masses with a more homogeneous density; (ii)
use of a rectangular quartz block as the principal portion of the torsion
balance’s pendulum, thereby improving the stability of the period of the
balance and reducing the uncertainty of the pendulum’s moment of inertia;
(iii) a single vacuum chamber for the source masses and pendulum leading to a
reduction of the uncertainty of their relative positions; (iv) a remotely
operated stepper motor to change the positions of the source masses, thereby
reducing environmental changes; and (v) measurement of the anelasticity of the
torsion fiber with the aid of a high-$Q$ quartz fiber.
The final result is the average of two values of $G$ that differ by 9 parts in
$10^{6}$ obtained from two partially correlated determinations using the same
apparatus. The dominant components of uncertainty, in parts in $10^{6}$, are
19 from the measurement of the fiber’s anelasticity, 14 (statistical) from the
measurement of the change in the square of the angular frequency of the
pendulum when the source masses are in their near and far positions, and 10
from the measured distance between the geometric centers of the source masses.
Although the uncertainty of HUST-05 is five times larger than that of HUST-09,
Luo (2010) and co-workers do not believe that HUST-09 supersedes HUST-05.
Thus, both are considered for inclusion in the 2010 adjustment. Based on
information provided to the Task Group by the researchers Luo (2010), their
correlation coefficient is estimated to be 0.234 and is used in the
calculations of Sec. XIII. The extra digits for the value and uncertainty of
HUST-05 were also provided by Luo (2011).
#### XI.2.2 JILA
As can be seen from Table 17, the $21\times 10^{-6}$ relative standard
uncertainty of the value of $G$ identified as JILA-10 and obtained at JILA by
Parks and Faller (2010) has the third smallest estimated uncertainty of the
values listed and is the second smallest of those values. It differs from the
value with the smallest uncertainty, identified as UWash-00, by $287(25)$
parts in $10^{6}$, which is $11$ times the standard uncertainty of their
difference $u_{\rm diff}$, or “$11\sigma$.” This disagreement is an example of
the “historic difficulty” referred to at the very beginning of this section.
The data on which the JILA researchers based their result was taken in 2004,
but being well aware of this inconsistency they hesitated to publish it until
they checked and rechecked their work Parks and Faller (2010). With this done,
they decided it was time to report their value for $G$.
The apparatus used in the JILA experiment of Parks and Faller (2010) consisted
of two $780~{}{\rm g}$ copper test masses (or “pendulum bobs”) separated by
$34~{}{\rm cm}$, each of which was suspended from a supporting bar by four
wires and together they formed a Fabry-Perot cavity. When the four $120~{}{\rm
kg}$ cylindrical tungsten source masses, two pairs with each member of the
pair on either side of the laser beam traversing the cavity, were periodically
moved parallel to the laser beam from their inner and outer positions (they
remained stationary for $80~{}{\rm s}$ in each position), the separation
between the bobs changed by about $90~{}{\rm nm}$. This change was observed as
a $125~{}{\rm MHz}$ beat frequency between the laser locked to the pendulum
cavity and the laser locked to a reference cavity that was part of the
supporting bar. The geometry of the experiment reduces the most difficult
aspect of determining the gravitational field of the source masses to six one
dimensional measurements: the distance between opposite source mass pairs in
the inner and outer positions and the distances between adjacent source masses
in the inner position. The most important relative standard uncertainty
components contributing to the uncertainty of $G$ are, in parts in $10^{6}$
Parks and Faller (2010), the six critical dimension measurements, 14; all
other dimension measurements and source mass density inhomogeneities, 8 each;
pendulum spring constants, 7; and total mass measurement and interferometer
misalignment, 6 each.
As already noted, we leave the calculation of the 2010 recommended value of
$G$ to Sec. XIII.2.1.
## XII Electroweak quantities
As in previous adjustments, there are a few cases in the 2010 adjustment where
an inexact constant that is used in the analysis of input data is not treated
as an adjusted quantity, because the adjustment has a negligible effect on its
value. Three such constants, used in the calculation of the theoretical
expression for the electron magnetic moment anomaly $a_{\rm e}$, are the mass
of the tau lepton $m_{\mbox{\scriptsize{{t}}}}$, the Fermi coupling constant
$G_{\rm F}$, and sine squared of the weak mixing angle sin${}^{2}{\theta}_{\rm
W}$; they are obtained from the most recent report of the Particle Data Group
Nakamura _et al._ (2010):
$\displaystyle m_{\mbox{\scriptsize{{t}}}}c^{2}$ $\displaystyle=$
$\displaystyle 1776.82(16)\ {\rm MeV}\qquad[9.0\times 10^{-5}]\,,$ (280)
$\displaystyle{G_{\rm F}\over(\hbar c)^{3}}$ $\displaystyle=$ $\displaystyle
1.166\,364(5)\times 10^{-5}\ {\rm GeV}^{-2}\quad[4.3\times 10^{-6}]\,,$ (281)
$\displaystyle{\rm sin}^{2}{\theta}_{\rm W}$ $\displaystyle=$ $\displaystyle
0.2223(21)\qquad[9.5\times 10^{-3}]\,.$ (282)
The value for $G_{\rm F}/(\hbar c)^{3}$ is taken from p. 127 of Nakamura _et
al._ (2010). We use the definition sin${}^{2}{\theta}_{\rm W}=1-(m_{\rm
W}/m_{\rm Z})^{2}$, where $m_{\rm W}$ and $m_{\rm Z}$ are, respectively, the
masses of the ${\rm W}^{\pm}$ and ${\rm Z}^{0}$ bosons, because it is employed
in the calculation of the electroweak contributions to $a_{\rm e}$ Czarnecki
_et al._ (1996). The Particle Data Group’s recommended value for the mass
ratio of these bosons is $m_{\rm W}/m_{\rm Z}=0.8819(12)$, which leads to the
value of sin${}^{2}{\theta}_{\rm W}$ given above.
## XIII Analysis of Data
We examine in this section the input data discussed in the previous sections
and, based upon that examination, select the data to be used in the least-
squares adjustment that determines the 2010 CODATA recommended values of the
constants. Tables 18, LABEL:tab:pdata, 22, and 24 give the input data,
including the $\delta$’s, which are corrections added to theoretical
expressions to account for the uncertainties of those expressions. The
covariances of the data are given as correlation coefficients in Tables 19,
21, 23, and 24. There are 14 types of input data for which there are two or
more experiments, and the data of the same type generally agree.
Table 18: Summary of principal input data for the determination of the 2010
recommended value of the Rydberg constant $R_{\infty}$.
Item | Input datum | Value | Relative standard | Identification | Sec.
---|---|---|---|---|---
number | | | uncertainty1 $u_{\rm r}$ | |
$A1$ | $\delta_{\rm H}({\rm 1S_{1/2}})$ | $0.0(2.5)$ kHz | $[7.5\times 10^{-13}]$ | theory | IV.1.1
$A2$ | $\delta_{\rm H}({\rm 2S_{1/2}})$ | $0.00(31)$ kHz | $[3.8\times 10^{-13}]$ | theory | IV.1.1
$A3$ | $\delta_{\rm H}({\rm 3S_{1/2}})$ | $0.000(91)$ kHz | $[2.5\times 10^{-13}]$ | theory | IV.1.1
$A4$ | $\delta_{\rm H}({\rm 4S_{1/2}})$ | $0.000(39)$ kHz | $[1.9\times 10^{-13}]$ | theory | IV.1.1
$A5$ | $\delta_{\rm H}({\rm 6S_{1/2}})$ | $0.000(15)$ kHz | $[1.6\times 10^{-13}]$ | theory | IV.1.1
$A6$ | $\delta_{\rm H}({\rm 8S_{1/2}})$ | $0.0000(63)$ kHz | $[1.2\times 10^{-13}]$ | theory | IV.1.1
$A7$ | $\delta_{\rm H}({\rm 2P_{1/2}})$ | $0.000(28)$ kHz | $[3.5\times 10^{-14}]$ | theory | IV.1.1
$A8$ | $\delta_{\rm H}({\rm 4P_{1/2}})$ | $0.0000(38)$ kHz | $[1.9\times 10^{-14}]$ | theory | IV.1.1
$A9$ | $\delta_{\rm H}({\rm 2P_{3/2}})$ | $0.000(28)$ kHz | $[3.5\times 10^{-14}]$ | theory | IV.1.1
$A10$ | $\delta_{\rm H}({\rm 4P_{3/2}})$ | $0.0000(38)$ kHz | $[1.9\times 10^{-14}]$ | theory | IV.1.1
$A11$ | $\delta_{\rm H}({\rm 8D_{3/2}})$ | $0.000\,00(44)$ kHz | $[8.5\times 10^{-15}]$ | theory | IV.1.1
$A12$ | $\delta_{\rm H}({\rm 12D_{3/2}})$ | $0.000\,00(13)$ kHz | $[5.7\times 10^{-15}]$ | theory | IV.1.1
$A13$ | $\delta_{\rm H}({\rm 4D_{5/2}})$ | $0.0000(35)$ kHz | $[1.7\times 10^{-14}]$ | theory | IV.1.1
$A14$ | $\delta_{\rm H}({\rm 6D_{5/2}})$ | $0.0000(10)$ kHz | $[1.1\times 10^{-14}]$ | theory | IV.1.1
$A15$ | $\delta_{\rm H}({\rm 8D_{5/2}})$ | $0.000\,00(44)$ kHz | $[8.5\times 10^{-15}]$ | theory | IV.1.1
$A16$ | $\delta_{\rm H}({\rm 12D_{5/2}})$ | $0.000\,00(13)$ kHz | $[5.7\times 10^{-15}]$ | theory | IV.1.1
$A17$ | $\delta_{\rm D}({\rm 1S_{1/2}})$ | $0.0(2.3)$ kHz | $[6.9\times 10^{-13}]$ | theory | IV.1.1
$A18$ | $\delta_{\rm D}({\rm 2S_{1/2}})$ | $0.00(29)$ kHz | $[3.5\times 10^{-13}]$ | theory | IV.1.1
$A19$ | $\delta_{\rm D}({\rm 4S_{1/2}})$ | $0.000(36)$ kHz | $[1.7\times 10^{-13}]$ | theory | IV.1.1
$A20$ | $\delta_{\rm D}({\rm 8S_{1/2}})$ | $0.0000(60)$ kHz | $[1.2\times 10^{-13}]$ | theory | IV.1.1
$A21$ | $\delta_{\rm D}({\rm 8D_{3/2}})$ | $0.000\,00(44)$ kHz | $[8.5\times 10^{-15}]$ | theory | IV.1.1
$A22$ | $\delta_{\rm D}({\rm 12D_{3/2}})$ | $0.000\,00(13)$ kHz | $[5.6\times 10^{-15}]$ | theory | IV.1.1
$A23$ | $\delta_{\rm D}({\rm 4D_{5/2}})$ | $0.0000(35)$ kHz | $[1.7\times 10^{-14}]$ | theory | IV.1.1
$A24$ | $\delta_{\rm D}({\rm 8D_{5/2}})$ | $0.000\,00(44)$ kHz | $[8.5\times 10^{-15}]$ | theory | IV.1.1
$A25$ | $\delta_{\rm D}({\rm 12D_{5/2}})$ | $0.000\,00(13)$ kHz | $[5.7\times 10^{-15}]$ | theory | IV.1.1
$A26$ | $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $2\,466\,061\,413\,187.080(34)$ kHz | $1.4\times 10^{-14}$ | MPQ-04 | IV.1.2
$A27$ | $\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $2\,922\,743\,278\,678(13)$ kHz | $4.4\times 10^{-12}$ | LKB-10 | IV.1.2
$A28$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ | $770\,649\,350\,012.0(8.6)$ kHz | $1.1\times 10^{-11}$ | LK/SY-97 | IV.1.2
$A29$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ | $770\,649\,504\,450.0(8.3)$ kHz | $1.1\times 10^{-11}$ | LK/SY-97 | IV.1.2
$A30$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ | $770\,649\,561\,584.2(6.4)$ kHz | $8.3\times 10^{-12}$ | LK/SY-97 | IV.1.2
$A31$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ | $799\,191\,710\,472.7(9.4)$ kHz | $1.2\times 10^{-11}$ | LK/SY-98 | IV.1.2
$A32$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ | $799\,191\,727\,403.7(7.0)$ kHz | $8.7\times 10^{-12}$ | LK/SY-98 | IV.1.2
$A33$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,797\,338(10)$ kHz | $2.1\times 10^{-6}$ | MPQ-95 | IV.1.2
$A34$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,490\,144(24)$ kHz | $3.7\times 10^{-6}$ | MPQ-95 | IV.1.2
$A35$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6S_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $4\,197\,604(21)$ kHz | $4.9\times 10^{-6}$ | LKB-96 | IV.1.2
$A36$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 6D_{5/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 3S_{1/2}})$ | $4\,699\,099(10)$ kHz | $2.2\times 10^{-6}$ | LKB-96 | IV.1.2
$A37$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{1/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,664\,269(15)$ kHz | $3.2\times 10^{-6}$ | YaleU-95 | IV.1.2
$A38$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 4P_{3/2}})-\frac{1}{4}\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,035\,373(10)$ kHz | $1.7\times 10^{-6}$ | YaleU-95 | IV.1.2
$A39$ | $\nu_{\rm H}({\rm 2S_{1/2}}-{\rm 2P_{3/2}})$ | $9\,911\,200(12)$ kHz | $1.2\times 10^{-6}$ | HarvU-94 | IV.1.2
$A40.1$ | $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ | $1\,057\,845.0(9.0)$ kHz | $8.5\times 10^{-6}$ | HarvU-86 | IV.1.2
$A40.2$ | $\nu_{\rm H}({\rm 2P_{1/2}}-{\rm 2S_{1/2}})$ | $1\,057\,862(20)$ kHz | $1.9\times 10^{-5}$ | USus-79 | IV.1.2
$A41$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8S_{1/2}})$ | $770\,859\,041\,245.7(6.9)$ kHz | $8.9\times 10^{-12}$ | LK/SY-97 | IV.1.2
$A42$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{3/2}})$ | $770\,859\,195\,701.8(6.3)$ kHz | $8.2\times 10^{-12}$ | LK/SY-97 | IV.1.2
$A43$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 8D_{5/2}})$ | $770\,859\,252\,849.5(5.9)$ kHz | $7.7\times 10^{-12}$ | LK/SY-97 | IV.1.2
$A44$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{3/2}})$ | $799\,409\,168\,038.0(8.6)$ kHz | $1.1\times 10^{-11}$ | LK/SY-98 | IV.1.2
$A45$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 12D_{5/2}})$ | $799\,409\,184\,966.8(6.8)$ kHz | $8.5\times 10^{-12}$ | LK/SY-98 | IV.1.2
$A46$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4S_{1/2}})-\frac{1}{4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $4\,801\,693(20)$ kHz | $4.2\times 10^{-6}$ | MPQ-95 | IV.1.2
$A47$ | $\nu_{\rm D}({\rm 2S_{1/2}}-{\rm 4D_{5/2}})-\frac{1}{4}\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $6\,494\,841(41)$ kHz | $6.3\times 10^{-6}$ | MPQ-95 | IV.1.2
$A48$ | $\nu_{\rm D}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})-\nu_{\rm H}({\rm 1S_{1/2}}-{\rm 2S_{1/2}})$ | $670\,994\,334.606(15)$ kHz | $2.2\times 10^{-11}$ | MPQ-10 | IV.1.2
$A49.1$ | $r_{\rm p}$ | $0.895(18)$ fm | $2.0\times 10^{-2}$ | rp-03 | IV.1.3
$A49.2$ | $r_{\rm p}$ | $0.8791(79)$ fm | $9.0\times 10^{-3}$ | rp-10 | IV.1.3
$A50$ | $r_{\rm d}$ | $2.130(10)$ fm | $4.7\times 10^{-3}$ | rd-98 | IV.1.3
1 The values in brackets are relative to the frequency equivalent of the
binding energy of the indicated level.
Table 19: Correlation coefficients $r(x_{i},x_{j})\geq 0.0001$ of the input data related to $R_{\infty}$ in Table 18. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table 18. $r$($A1$, $A2$) | = $0.9905$ | $r$($A6$, $A19$) | = $0.7404$ | $r$($A28$, $A29$) | = $0.3478$ | $r$($A31$, $A45$) | = $0.1136$
---|---|---|---|---|---|---|---
$r$($A1$, $A3$) | = $0.9900$ | $r$($A6$, $A20$) | = $0.9851$ | $r$($A28$, $A30$) | = $0.4532$ | $r$($A32$, $A35$) | = $0.0278$
$r$($A1$, $A4$) | = $0.9873$ | $r$($A7$, $A8$) | = $0.0237$ | $r$($A28$, $A31$) | = $0.0899$ | $r$($A32$, $A36$) | = $0.0553$
$r$($A1$, $A5$) | = $0.7640$ | $r$($A9$, $A10$) | = $0.0237$ | $r$($A28$, $A32$) | = $0.1206$ | $r$($A32$, $A41$) | = $0.1512$
$r$($A1$, $A6$) | = $0.7627$ | $r$($A11$, $A12$) | = $0.0006$ | $r$($A28$, $A35$) | = $0.0225$ | $r$($A32$, $A42$) | = $0.1647$
$r$($A1$, $A17$) | = $0.9754$ | $r$($A11$, $A21$) | = $0.9999$ | $r$($A28$, $A36$) | = $0.0448$ | $r$($A32$, $A43$) | = $0.1750$
$r$($A1$, $A18$) | = $0.9656$ | $r$($A11$, $A22$) | = $0.0003$ | $r$($A28$, $A41$) | = $0.1225$ | $r$($A32$, $A44$) | = $0.1209$
$r$($A1$, $A19$) | = $0.9619$ | $r$($A12$, $A21$) | = $0.0003$ | $r$($A28$, $A42$) | = $0.1335$ | $r$($A32$, $A45$) | = $0.1524$
$r$($A1$, $A20$) | = $0.7189$ | $r$($A12$, $A22$) | = $0.9999$ | $r$($A28$, $A43$) | = $0.1419$ | $r$($A33$, $A34$) | = $0.1049$
$r$($A2$, $A3$) | = $0.9897$ | $r$($A13$, $A14$) | = $0.0006$ | $r$($A28$, $A44$) | = $0.0980$ | $r$($A33$, $A46$) | = $0.2095$
$r$($A2$, $A4$) | = $0.9870$ | $r$($A13$, $A15$) | = $0.0006$ | $r$($A28$, $A45$) | = $0.1235$ | $r$($A33$, $A47$) | = $0.0404$
$r$($A2$, $A5$) | = $0.7638$ | $r$($A13$, $A16$) | = $0.0006$ | $r$($A29$, $A30$) | = $0.4696$ | $r$($A34$, $A46$) | = $0.0271$
$r$($A2$, $A6$) | = $0.7625$ | $r$($A13$, $A23$) | = $0.9999$ | $r$($A29$, $A31$) | = $0.0934$ | $r$($A34$, $A47$) | = $0.0467$
$r$($A2$, $A17$) | = $0.9656$ | $r$($A13$, $A24$) | = $0.0003$ | $r$($A29$, $A32$) | = $0.1253$ | $r$($A35$, $A36$) | = $0.1412$
$r$($A2$, $A18$) | = $0.9754$ | $r$($A13$, $A25$) | = $0.0003$ | $r$($A29$, $A35$) | = $0.0234$ | $r$($A35$, $A41$) | = $0.0282$
$r$($A2$, $A19$) | = $0.9616$ | $r$($A14$, $A15$) | = $0.0006$ | $r$($A29$, $A36$) | = $0.0466$ | $r$($A35$, $A42$) | = $0.0307$
$r$($A2$, $A20$) | = $0.7187$ | $r$($A14$, $A16$) | = $0.0006$ | $r$($A29$, $A41$) | = $0.1273$ | $r$($A35$, $A43$) | = $0.0327$
$r$($A3$, $A4$) | = $0.9864$ | $r$($A14$, $A23$) | = $0.0003$ | $r$($A29$, $A42$) | = $0.1387$ | $r$($A35$, $A44$) | = $0.0226$
$r$($A3$, $A5$) | = $0.7633$ | $r$($A14$, $A24$) | = $0.0003$ | $r$($A29$, $A43$) | = $0.1475$ | $r$($A35$, $A45$) | = $0.0284$
$r$($A3$, $A6$) | = $0.7620$ | $r$($A14$, $A25$) | = $0.0003$ | $r$($A29$, $A44$) | = $0.1019$ | $r$($A36$, $A41$) | = $0.0561$
$r$($A3$, $A17$) | = $0.9651$ | $r$($A15$, $A16$) | = $0.0006$ | $r$($A29$, $A45$) | = $0.1284$ | $r$($A36$, $A42$) | = $0.0612$
$r$($A3$, $A18$) | = $0.9648$ | $r$($A15$, $A23$) | = $0.0003$ | $r$($A30$, $A31$) | = $0.1209$ | $r$($A36$, $A43$) | = $0.0650$
$r$($A3$, $A19$) | = $0.9611$ | $r$($A15$, $A24$) | = $0.9999$ | $r$($A30$, $A32$) | = $0.1622$ | $r$($A36$, $A44$) | = $0.0449$
$r$($A3$, $A20$) | = $0.7183$ | $r$($A15$, $A25$) | = $0.0003$ | $r$($A30$, $A35$) | = $0.0303$ | $r$($A36$, $A45$) | = $0.0566$
$r$($A4$, $A5$) | = $0.7613$ | $r$($A16$, $A23$) | = $0.0003$ | $r$($A30$, $A36$) | = $0.0602$ | $r$($A37$, $A38$) | = $0.0834$
$r$($A4$, $A6$) | = $0.7600$ | $r$($A16$, $A24$) | = $0.0003$ | $r$($A30$, $A41$) | = $0.1648$ | $r$($A41$, $A42$) | = $0.5699$
$r$($A4$, $A17$) | = $0.9625$ | $r$($A16$, $A25$) | = $0.9999$ | $r$($A30$, $A42$) | = $0.1795$ | $r$($A41$, $A43$) | = $0.6117$
$r$($A4$, $A18$) | = $0.9622$ | $r$($A17$, $A18$) | = $0.9897$ | $r$($A30$, $A43$) | = $0.1908$ | $r$($A41$, $A44$) | = $0.1229$
$r$($A4$, $A19$) | = $0.9755$ | $r$($A17$, $A19$) | = $0.9859$ | $r$($A30$, $A44$) | = $0.1319$ | $r$($A41$, $A45$) | = $0.1548$
$r$($A4$, $A20$) | = $0.7163$ | $r$($A17$, $A20$) | = $0.7368$ | $r$($A30$, $A45$) | = $0.1662$ | $r$($A42$, $A43$) | = $0.6667$
$r$($A5$, $A6$) | = $0.5881$ | $r$($A18$, $A19$) | = $0.9856$ | $r$($A31$, $A32$) | = $0.4750$ | $r$($A42$, $A44$) | = $0.1339$
$r$($A5$, $A17$) | = $0.7448$ | $r$($A18$, $A20$) | = $0.7366$ | $r$($A31$, $A35$) | = $0.0207$ | $r$($A42$, $A45$) | = $0.1687$
$r$($A5$, $A18$) | = $0.7445$ | $r$($A19$, $A20$) | = $0.7338$ | $r$($A31$, $A36$) | = $0.0412$ | $r$($A43$, $A44$) | = $0.1423$
$r$($A5$, $A19$) | = $0.7417$ | $r$($A21$, $A22$) | = $0.0002$ | $r$($A31$, $A41$) | = $0.1127$ | $r$($A43$, $A45$) | = $0.1793$
$r$($A5$, $A20$) | = $0.5543$ | $r$($A23$, $A24$) | = $0.0001$ | $r$($A31$, $A42$) | = $0.1228$ | $r$($A44$, $A45$) | = $0.5224$
$r$($A6$, $A17$) | = $0.7435$ | $r$($A23$, $A25$) | = $0.0001$ | $r$($A31$, $A43$) | = $0.1305$ | $r$($A46$, $A47$) | = $0.0110$
$r$($A6$, $A18$) | = $0.7433$ | $r$($A24$, $A25$) | = $0.0002$ | $r$($A31$, $A44$) | = $0.0901$ | |
Table 20: Summary of principal input data for the determination of the 2010 recommended values of the fundamental constants ($R_{\infty}$ and $G$ excepted). Item | Input datum | Value | Relative standard | Identification | Sec. and Eq.
---|---|---|---|---|---
number | | | uncertainty1 $u_{\rm r}$ | |
$B1$ | $A_{\rm r}(^{1}{\rm H})$ | $\phantom{-}1.007\,825\,032\,07(10)$ | $1.0\times 10^{-10}$ | AMDC-03 | III.1
$B2.1$ | $A_{\rm r}(^{2}{\rm H})$ | $\phantom{-}2.014\,101\,777\,85(36)$ | $1.8\times 10^{-10}$ | AMDC-03 | III.1
$B2.2$ | $A_{\rm r}(^{2}{\rm H})$ | $\phantom{-}2.014\,101\,778\,040(80)$ | $4.0\times 10^{-11}$ | UWash-06 | III.1
$B3$ | $A_{\rm r}(E_{\rm av})$ | $\phantom{-}0.794(79)$ | $1.0\times 10^{-1}$ | StockU-08 | III.3 (13)
$B4$ | $f_{\rm c}({\rm H}_{2}^{+*})/f_{\rm c}({\rm d})$ | $\phantom{-}0.999\,231\,659\,33(17)$ | $1.7\times 10^{-10}$ | StockU-08 | III.3 (2)
$B5$ | $f_{\rm c}({\rm t})/f_{\rm c}({\rm H}_{2}^{+*})$ | $\phantom{-}0.668\,247\,726\,86(55)$ | $8.2\times 10^{-10}$ | StockU-06 | III.3 (14)
$B6$ | $f_{\rm c}(^{3}{\rm He}^{+})/f_{\rm c}({\rm H}_{2}^{+*})$ | $\phantom{-}0.668\,252\,146\,82(55)$ | $8.2\times 10^{-10}$ | StockU-06 | III.3 (15)
$B7$ | $A_{\rm r}(^{4}{\rm He})$ | $\phantom{-}4.002\,603\,254\,131(62)$ | $1.5\times 10^{-11}$ | UWash-06 | III.1
$B8$ | $A_{\rm r}(^{16}{\rm O})$ | $\phantom{-}15.994\,914\,619\,57(18)$ | $1.1\times 10^{-11}$ | UWash-06 | III.1
$B9.1$ | $A_{\rm r}(^{87}{\rm Rb})$ | $\phantom{-}86.909\,180\,526(12)$ | $1.4\times 10^{-10}$ | AMDC-03 | III.1
$B9.2$ | $A_{\rm r}(^{87}{\rm Rb})$ | $\phantom{-}86.909\,180\,535(10)$ | $1.2\times 10^{-10}$ | FSU-10 | III.1
$B10.1^{2}$ | $A_{\rm r}(^{133}{\rm Cs})$ | $\phantom{-}132.905\,451\,932(24)$ | $1.8\times 10^{-10}$ | AMDC-03 | III.1
$B10.2^{2}$ | $A_{\rm r}(^{133}{\rm Cs})$ | $\phantom{-}132.905\,451\,963(13)$ | $9.8\times 10^{-11}$ | FSU-10 | III.1
$B11$ | $A_{\rm r}({\rm e})$ | $\phantom{-}0.000\,548\,579\,9111(12)$ | $2.1\times 10^{-9}$ | UWash-95 | III.4 (20)
$B12$ | $\delta_{\rm e}$ | $\phantom{-}0.00(33)\times 10^{-12}$ | [$2.8\times 10^{-10}$] | theory | V.1.1
$B13.1^{2}$ | $a_{\rm e}$ | $\phantom{-}1.159\,652\,1883(42)\times 10^{-3}$ | $3.7\times 10^{-9}$ | UWash-87 | V.1.2 (124)
$B13.2$ | $a_{\rm e}$ | $\phantom{-}1.159\,652\,180\,73(28)\times 10^{-3}$ | $2.4\times 10^{-10}$ | HarvU-08 | V.1.2 (125)
$B14$ | $\overline{R}$ | $\phantom{-}0.003\,707\,2063(20)$ | $5.4\times 10^{-7}$ | BNL-06 | V.2.2 (142)
$B15$ | $\delta_{\rm C}$ | $\phantom{-}0.00(26)\times 10^{-10}$ | [$1.3\times 10^{-11}$] | theory | V.3.1
$B16$ | $\delta_{\rm O}$ | $\phantom{-}0.0(1.1)\times 10^{-10}$ | [$5.3\times 10^{-11}$] | theory | V.3.1
$B17$ | $f_{\rm s}({\rm{}^{12}C^{5+}})/f_{\rm c}({\rm{}^{12}C^{5+}})$ | $\phantom{-}4376.210\,4989(23)$ | $5.2\times 10^{-10}$ | GSI-02 | V.3.2 (182)
$B18$ | $f_{\rm s}({\rm{}^{16}O^{7+}})/f_{\rm c}({\rm{}^{16}O^{7+}})$ | $\phantom{-}4164.376\,1837(32)$ | $7.6\times 10^{-10}$ | GSI-02 | V.3.2 (183)
$B19$ | $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm p}({\rm H})$ | $-658.210\,7058(66)$ | $1.0\times 10^{-8}$ | MIT-72 | VI.1.3
$B20$ | $\mu_{\rm d}({\rm D})/\mu_{\rm e^{-}}({\rm D})$ | $-4.664\,345\,392(50)\times 10^{-4}$ | $1.1\times 10^{-8}$ | MIT-84 | VI.1.3
$B21$ | $\mu_{\rm p}({\rm HD})/\mu_{\rm d}({\rm HD})$ | $\phantom{-}3.257\,199\,531(29)$ | $8.9\times 10^{-9}$ | StPtrsb-03 | VI.1.3
$B22$ | $\sigma_{\rm dp}$ | $\phantom{-}15(2)\times 10^{-9}$ | | StPtrsb-03 | VI.1.3
$B23$ | $\mu_{\rm t}({\rm HT})/\mu_{\rm p}({\rm HT})$ | $\phantom{-}1.066\,639\,887(10)$ | $9.4\times 10^{-9}$ | StPtrsb-03 | VI.1.3
$B24$ | $\sigma_{\rm tp}$ | $\phantom{-}20(3)\times 10^{-9}$ | | StPtrsb-03 | VI.1.3
$B25$ | $\mu_{\rm e^{-}}({\rm H})/\mu_{\rm p}^{\prime}$ | $-658.215\,9430(72)$ | $1.1\times 10^{-8}$ | MIT-77 | VI.1.3
$B26$ | $\mu_{\rm h}^{\prime}/\mu_{\rm p}^{\prime}$ | $-0.761\,786\,1313(33)$ | $4.3\times 10^{-9}$ | NPL-93 | VI.1.3
$B27$ | $\mu_{\rm n}/\mu_{\rm p}^{\prime}$ | $-0.684\,996\,94(16)$ | $2.4\times 10^{-7}$ | ILL-79 | VI.1.3
$B28$ | $\delta_{\rm Mu}$ | $\phantom{-}0(101)$ Hz | [$2.3\times 10^{-8}$] | theory | VI.2.1
$B29.1$ | $\Delta\nu_{\rm Mu}$ | $\phantom{-}4\,463\,302.88(16)$ kHz | $3.6\times 10^{-8}$ | LAMPF-82 | VI.2.2 (221)
$B29.2$ | $\Delta\nu_{\rm Mu}$ | $\phantom{-}4\,463\,302\,765(53)$ Hz | $1.2\times 10^{-8}$ | LAMPF-99 | VI.2.2 (224)
$B30$ | $\nu(58~{}{\rm MHz})$ | $\phantom{-}627\,994.77(14)$ kHz | $2.2\times 10^{-7}$ | LAMPF-82 | VI.2.2 (222)
$B31$ | $\nu(72~{}{\rm MHz})$ | $\phantom{-}668\,223\,166(57)$ Hz | $8.6\times 10^{-8}$ | LAMPF-99 | VI.2.2 (225)
$B32.1^{2}$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\phantom{-}2.675\,154\,05(30)\times 10^{8}$ s-1 T-1 | $1.1\times 10^{-7}$ | NIST-89 | VIII.2
$B32.2^{2}$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\phantom{-}2.675\,1530(18)\times 10^{8}$ s-1 T-1 | $6.6\times 10^{-7}$ | NIM-95 | VIII.2
$B33^{2}$ | ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $\phantom{-}2.037\,895\,37(37)\times 10^{8}$ s-1 T-1 | $1.8\times 10^{-7}$ | KR/VN-98 | VIII.2
$B34.1^{2}$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\phantom{-}2.675\,1525(43)\times 10^{8}$ s-1 T-1 | $1.6\times 10^{-6}$ | NIM-95 | VIII.2
$B34.2^{2}$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\phantom{-}2.675\,1518(27)\times 10^{8}$ s-1 T-1 | $1.0\times 10^{-6}$ | NPL-79 | VIII.2
$B35.1^{2}$ | $R_{\rm K}$ | $\phantom{-}25\,812.808\,31(62)\ {\rm\Omega}$ | $2.4\times 10^{-8}$ | NIST-97 | VIII.2
$B35.2^{2}$ | $R_{\rm K}$ | $\phantom{-}25\,812.8071(11)\ {\rm\Omega}$ | $4.4\times 10^{-8}$ | NMI-97 | VIII.2
$B35.3^{2}$ | $R_{\rm K}$ | $\phantom{-}25\,812.8092(14)\ {\rm\Omega}$ | $5.4\times 10^{-8}$ | NPL-88 | VIII.2
$B35.4^{2}$ | $R_{\rm K}$ | $\phantom{-}25\,812.8084(34)\ {\rm\Omega}$ | $1.3\times 10^{-7}$ | NIM-95 | VIII.2
$B35.5^{2}$ | $R_{\rm K}$ | $\phantom{-}25\,812.8081(14)\ {\rm\Omega}$ | $5.3\times 10^{-8}$ | LNE-01 | VIII.2
$B36.1^{2}$ | $K_{\rm J}$ | $\phantom{-}483\,597.91(13)\ {\rm GHz\ V^{-1}}$ | $2.7\times 10^{-7}$ | NMI-89 | VIII.2
$B36.2^{2}$ | $K_{\rm J}$ | $\phantom{-}483\,597.96(15)\ {\rm GHz\ V^{-1}}$ | $3.1\times 10^{-7}$ | PTB-91 | VIII.2
$B37.1^{3}$ | $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7625(12)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.0\times 10^{-7}$ | NPL-90 | VIII.2
$B37.2^{3}$ | $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,761\,85(53)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $8.7\times 10^{-8}$ | NIST-98 | VIII.2
$B37.3^{3}$ | $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,761\,85(22)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $3.6\times 10^{-8}$ | NIST-07 | VIII.2
$B37.4^{3}$ | $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7597(12)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.0\times 10^{-7}$ | NPL-12 | VIII.2.1 (244)
$B37.5^{2}$ | $K_{\rm J}^{2}R_{\rm K}$ | $\phantom{-}6.036\,7617(18)\times 10^{33}\ {\rm J^{-1}\ s^{-1}}$ | $2.9\times 10^{-7}$ | METAS-11 | VIII.2.2 (246)
$B38^{2}$ | ${\cal F}_{90}$ | $\phantom{-}96\,485.39(13)\ {\rm C\ mol^{-1}}$ | $1.3\times 10^{-6}$ | NIST-80 | VIII.2
$B39$ | $d_{220}({\rm{\scriptstyle{MO^{*}}}})$ | $\phantom{-}192\,015.5508(42)$ fm | $2.2\times 10^{-8}$ | INRIM-08 | IX.1 (249)
$B40$ | $d_{220}({\rm{\scriptstyle{W04}}})$ | $\phantom{-}192\,015.5702(29)$ fm | $1.5\times 10^{-8}$ | INRIM-09 | IX.1 (250)
$B41.1$ | $d_{220}({\rm{\scriptstyle{W4.2a}}})$ | $\phantom{-}192\,015.563(12)$ fm | $6.2\times 10^{-8}$ | PTB-81 | IX.1 (248)
$B41.2$ | $d_{220}({\rm{\scriptstyle{W4.2a}}})$ | $\phantom{-}192\,015.5691(29)$ fm | $1.5\times 10^{-8}$ | INRIM-09 | IX.1 (251)
$B42$ | $1-d_{220}({\rm{\scriptstyle{N}}})/d_{220}({\rm{\scriptstyle W17}})$ | $\phantom{-}7(22)\times 10^{-9}$ | | NIST-97 | IX.2
$B43$ | $1-d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $-8(22)\times 10^{-9}$ | | NIST-99 | IX.2
$B44$ | $1-d_{220}({\rm{\scriptstyle{MO^{*}}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $\phantom{-}86(27)\times 10^{-9}$ | | NIST-99 | IX.2
$B45$ | $1-d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle ILL}})$ | $\phantom{-}33(22)\times 10^{-9}$ | | NIST-99 | IX.2
$B46$ | $d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-11(21)\times 10^{-9}$ | | NIST-06 | IX.2
$B47$ | $d_{220}({\rm{\scriptstyle{NR4}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}25(21)\times 10^{-9}$ | | NIST-06 | IX.2
$B48$ | $d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}11(21)\times 10^{-9}$ | | NIST-06 | IX.2
$B49$ | $d_{220}({\rm{\scriptstyle{W4.2a}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-1(21)\times 10^{-9}$ | | PTB-98 | IX.2
$B50$ | $d_{220}({\rm{\scriptstyle{W17}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}22(22)\times 10^{-9}$ | | PTB-98 | IX.2
$B51$ | $d_{220}({\rm{\scriptstyle{MO^{*}4}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-103(28)\times 10^{-9}$ | | PTB-98 | IX.2
$B52$ | $d_{220}({\rm{\scriptstyle{NR3}}})/d_{220}({\rm{\scriptstyle W04}})-1$ | $-23(21)\times 10^{-9}$ | | PTB-98 | IX.2
$B53$ | $d_{220}/d_{220}({\rm{\scriptstyle W04}})-1$ | $\phantom{-}10(11)\times 10^{-9}$ | | PTB-03 | IX.2
$B54^{3}$ | $N_{\rm A}$ | $\phantom{-}6.022\,140\,82(18)\times 10^{23}$ m3 mol-1 | $3.0\times 10^{-8}$ | IAC-11 | IX.6 (267)
$B55$ | $\lambda_{\rm meas}/d_{220}({\rm{\scriptstyle ILL}})$ | $\phantom{-}0.002\,904\,302\,46(50)$ m s-1 | $1.7\times 10^{-7}$ | NIST-99 | IX.3 (252)
$B56^{2}$ | $h/m({\rm{}^{133}Cs})$ | $\phantom{-}3.002\,369\,432(46)\times 10^{-9}$ m2 s-1 | $1.5\times 10^{-8}$ | StanfU-02 | VII.1 (234)
$B57$ | $h/m({\rm{}^{87}Rb})$ | $\phantom{-}4.591\,359\,2729(57)\times 10^{-9}$ m2 s-1 | $1.2\times 10^{-9}$ | LKB-11 | VII.2 (236)
$B58.1$ | $R$ | $\phantom{-}8.314\,504(70)$ J mol-1 K-1 | $8.4\times 10^{-6}$ | NPL-79 | X.1.1
$B58.2$ | $R$ | $\phantom{-}8.314\,471(15)$ J mol-1 K-1 | $1.8\times 10^{-6}$ | NIST-88 | X.1.1
$B58.3$ | $R$ | $\phantom{-}8.314\,467(22)$ J mol-1 K-1 | $2.7\times 10^{-6}$ | LNE-09 | X.1.2
$B58.4$ | $R$ | $\phantom{-}8.314\,468(26)$ J mol-1 K-1 | $3.1\times 10^{-6}$ | NPL-10 | X.1.3
$B58.5$ | $R$ | $\phantom{-}8.314\,412(63)$ J mol-1 K-1 | $7.5\times 10^{-6}$ | INRIM-10 | X.1.4
$B58.6$ | $R$ | $\phantom{-}8.314\,456(10)$ J mol-1 K-1 | $1.2\times 10^{-6}$ | LNE-11 | X.1.2
$B59^{2}$ | $k$ | $\phantom{-}1.380\,653(13)\times 10^{-23}$ J K-1 | $9.1\times 10^{-6}$ | NIST-07 | X.2.1
$B60^{2}$ | $k/h$ | $\phantom{-}2.083\,666(25)\times 10^{10}$ Hz K-1 | $1.2\times 10^{-5}$ | NIST-11 | X.2.2
$B61$ | $\lambda({\rm CuK\alpha_{1}})/d_{220}({\rm{\scriptstyle{W4.2a}}})$ | $\phantom{-}0.802\,327\,11(24)$ | $3.0\times 10^{-7}$ | FSUJ/PTB-91 | IX.4 (257)
$B62$ | $\lambda({\rm WK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.108\,852\,175(98)$ | $9.0\times 10^{-7}$ | NIST-79 | IX.4 (258)
$B63$ | $\lambda({\rm MoK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.369\,406\,04(19)$ | $5.3\times 10^{-7}$ | NIST-73 | IX.4 (259)
$B64$ | $\lambda({\rm CuK\alpha_{1}})/d_{220}({\rm{\scriptstyle{N}}})$ | $\phantom{-}0.802\,328\,04(77)$ | $9.6\times 10^{-7}$ | NIST-73 | IX.4 (260)
1 The values in brackets are relative to the quantities $a_{\rm e}$, $g_{\rm
e^{-}}(^{12}{\rm C}^{5+})$, $g_{\rm e^{-}}(^{16}{\rm O}^{7+})$, or
$\Delta\nu_{\rm Mu}$ as appropriate.
2 Datum not included in the final least-squares adjustment that provides the
recommended values of the constants.
3 Datum included in the final least-squares adjustment with an expanded
uncertainty.
Table 20: (Continued). Summary of principal input data for the determination of the 2006 recommended values of the fundamental constants ($R_{\infty}$ and $G$ excepted). Table 21: Non-negligible correlation coefficients $r(x_{i},x_{j})$ of the input data in Table LABEL:tab:pdata. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table LABEL:tab:pdata. $r$($B1$, $B2.1$) | = $0.073$ | $r$($B39$, $B40$) | = $0.023$ | $r$($B43$, $B45$) | = $0.516$ | $r$($B47$, $B48$) | = $0.509$
---|---|---|---|---|---|---|---
$r$($B2.2$, $B7$) | = $0.127$ | $r$($B39$, $B41.2$) | = $0.023$ | $r$($B43$, $B46$) | = $0.065$ | $r$($B49$, $B50$) | = $0.469$
$r$($B2.2$, $B8$) | = $0.089$ | $r$($B39$, $B54$) | = $-0.026$ | $r$($B43$, $B47$) | = $0.065$ | $r$($B49$, $B51$) | = $0.372$
$r$($B5$, $B6$) | = $0.876$ | $r$($B40$, $B41.2$) | = $0.027$ | $r$($B43$, $B48$) | = $-0.367$ | $r$($B49$, $B52$) | = $0.502$
$r$($B7$, $B8$) | = $0.181$ | $r$($B40$, $B54$) | = $-0.029$ | $r$($B44$, $B45$) | = $0.421$ | $r$($B50$, $B51$) | = $0.347$
$r$($B15$, $B16$) | = $0.994$ | $r$($B41.2$, $B54$) | = $-0.029$ | $r$($B44$, $B46$) | = $0.053$ | $r$($B50$, $B52$) | = $0.469$
$r$($B17$, $B18$) | = $0.082$ | $r$($B42$, $B43$) | = $-0.288$ | $r$($B44$, $B47$) | = $0.053$ | $r$($B51$, $B52$) | = $0.372$
$r$($B29.1$, $B30$) | = $0.227$ | $r$($B42$, $B44$) | = $0.096$ | $r$($B44$, $B48$) | = $0.053$ | $r$($B58.3$, $B58.4$) | = $0.002$
$r$($B29.2$, $B31$) | = $0.195$ | $r$($B42$, $B45$) | = $0.117$ | $r$($B45$, $B46$) | = $-0.367$ | $r$($B58.3$, $B58.5$) | = $0.001$
$r$($B32.2$, $B34.1$) | = $-0.014$ | $r$($B42$, $B46$) | = $0.066$ | $r$($B45$, $B47$) | = $0.065$ | $r$($B58.3$, $B58.6$) | = $0.032$
$r$($B36.1$, $B58.2$) | = $0.068$ | $r$($B42$, $B47$) | = $0.066$ | $r$($B45$, $B48$) | = $0.065$ | $r$($B58.4$, $B58.6$) | = $0.012$
$r$($B37.1$, $B37.4$) | = $0.003$ | $r$($B42$, $B48$) | = $0.504$ | $r$($B46$, $B47$) | = $0.509$ | |
$r$($B37.2$, $B37.3$) | = $0.140$ | $r$($B43$, $B44$) | = $0.421$ | $r$($B46$, $B48$) | = $0.509$ | |
Table 22: Summary of principal input data for the determination of the
relative atomic mass of the electron from antiprotonic helium transitions. The
numbers in parentheses $(n,l:n^{\prime},l^{\prime})$ denote the transition
$(n,l)\rightarrow(n^{\prime},l^{\prime})$.
Item | Input Datum | Value | Relative standard | Identification | Sec.
---|---|---|---|---|---
number | | | uncertainty1 $u_{\rm r}$ | |
$C1$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(32,31:31,30)$ | $0.00(82)$ MHz | $[7.3\times 10^{-10}]$ | JINR-06 | IV.2.1
$C2$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(35,33:34,32)$ | $0.0(1.0)$ MHz | $[1.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C3$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34:35,33)$ | $0.0(1.1)$ MHz | $[1.6\times 10^{-9}]$ | JINR-06 | IV.2.1
$C4$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,34:36,33)$ | $0.0(1.1)$ MHz | $[1.8\times 10^{-9}]$ | JINR-06 | IV.2.1
$C5$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(39,35:38,34)$ | $0.0(1.2)$ MHz | $[2.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C6$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(40,35:39,34)$ | $0.0(1.3)$ MHz | $[2.9\times 10^{-9}]$ | JINR-06 | IV.2.1
$C7$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,35:38,34)$ | $0.0(1.8)$ MHz | $[4.5\times 10^{-9}]$ | JINR-06 | IV.2.1
$C8$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(33,32:31,30)$ | $0.0(1.6)$ MHz | $[7.6\times 10^{-10}]$ | JINR-10 | IV.2.1
$C9$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34:34,32)$ | $0.0(2.1)$ MHz | $[1.4\times 10^{-9}]$ | JINR-10 | IV.2.1
$C10$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(32,31:31,30)$ | $0.00(91)$ MHz | $[8.7\times 10^{-10}]$ | JINR-06 | IV.2.1
$C11$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(34,32:33,31)$ | $0.0(1.1)$ MHz | $[1.4\times 10^{-9}]$ | JINR-06 | IV.2.1
$C12$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,33:35,32)$ | $0.0(1.2)$ MHz | $[1.8\times 10^{-9}]$ | JINR-06 | IV.2.1
$C13$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(38,34:37,33)$ | $0.0(1.1)$ MHz | $[2.3\times 10^{-9}]$ | JINR-06 | IV.2.1
$C14$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,34:37,33)$ | $0.0(1.8)$ MHz | $[4.4\times 10^{-9}]$ | JINR-06 | IV.2.1
$C15$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(35,33:33,31)$ | $0.0(2.2)$ MHz | $[1.4\times 10^{-9}]$ | JINR-10 | IV.2.1
$C16$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(32,31:31,30)$ | $1\,132\,609\,209(15)$ MHz | $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C17$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(35,33:34,32)$ | $804\,633\,059.0(8.2)$ MHz | $1.0\times 10^{-8}$ | CERN-06 | IV.2.2
$C18$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(36,34:35,33)$ | $717\,474\,004(10)$ MHz | $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C19$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(37,34:36,33)$ | $636\,878\,139.4(7.7)$ MHz | $1.2\times 10^{-8}$ | CERN-06 | IV.2.2
$C20$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(39,35:38,34)$ | $501\,948\,751.6(4.4)$ MHz | $8.8\times 10^{-9}$ | CERN-06 | IV.2.2
$C21$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(40,35:39,34)$ | $445\,608\,557.6(6.3)$ MHz | $1.4\times 10^{-8}$ | CERN-06 | IV.2.2
$C22$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(37,35:38,34)$ | $412\,885\,132.2(3.9)$ MHz | $9.4\times 10^{-9}$ | CERN-06 | IV.2.2
$C23$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(33,32:31,30)$ | $2\,145\,054\,858.2(5.1)$ MHz | $2.4\times 10^{-9}$ | CERN-10 | IV.2.2
$C24$ | $\nu_{\bar{\rm p}{\rm{}^{4}He}^{+}}(36,34:34,32)$ | $1\,522\,107\,061.8(3.5)$ MHz | $2.3\times 10^{-9}$ | CERN-10 | IV.2.2
$C25$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(32,31:31,30)$ | $1\,043\,128\,608(13)$ MHz | $1.3\times 10^{-8}$ | CERN-06 | IV.2.2
$C26$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(34,32:33,31)$ | $822\,809\,190(12)$ MHz | $1.5\times 10^{-8}$ | CERN-06 | IV.2.2
$C27$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(36,33:35,32)$ | $646\,180\,434(12)$ MHz | $1.9\times 10^{-8}$ | CERN-06 | IV.2.2
$C28$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(38,34:37,33)$ | $505\,222\,295.7(8.2)$ MHz | $1.6\times 10^{-8}$ | CERN-06 | IV.2.2
$C29$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(36,34:37,33)$ | $414\,147\,507.8(4.0)$ MHz | $9.7\times 10^{-9}$ | CERN-06 | IV.2.2
$C30$ | $\nu_{\bar{\rm p}{\rm{}^{3}He}^{+}}(35,33:33,31)$ | $1\,553\,643\,099.6(7.1)$ MHz | $4.6\times 10^{-9}$ | CERN-10 | IV.2.2
1 The values in brackets are relative to the corresponding transition
frequency.
Table 23: Non-negligible correlation coefficients $r(x_{i},x_{j})$ of the input data in Table 22. For simplicity, the two items of data to which a particular correlation coefficient corresponds are identified by their item numbers in Table 22. $r$($C1$, $C2$) | = $0.929$ | $r$($C4$, $C10$) | = $0.959$ | $r$($C9$, $C14$) | = $-0.976$ | $r$($C18$, $C27$) | = $0.141$
---|---|---|---|---|---|---|---
$r$($C1$, $C3$) | = $0.936$ | $r$($C4$, $C11$) | = $0.949$ | $r$($C9$, $C15$) | = $0.986$ | $r$($C18$, $C28$) | = $0.106$
$r$($C1$, $C4$) | = $0.936$ | $r$($C4$, $C12$) | = $0.907$ | $r$($C10$, $C11$) | = $0.978$ | $r$($C18$, $C29$) | = $0.217$
$r$($C1$, $C5$) | = $0.912$ | $r$($C4$, $C13$) | = $0.931$ | $r$($C10$, $C12$) | = $0.934$ | $r$($C19$, $C20$) | = $0.268$
$r$($C1$, $C6$) | = $0.758$ | $r$($C4$, $C14$) | = $-0.952$ | $r$($C10$, $C13$) | = $0.959$ | $r$($C19$, $C21$) | = $0.193$
$r$($C1$, $C7$) | = $-0.947$ | $r$($C4$, $C15$) | = $0.961$ | $r$($C10$, $C14$) | = $-0.980$ | $r$($C19$, $C22$) | = $0.302$
$r$($C1$, $C8$) | = $0.954$ | $r$($C5$, $C6$) | = $0.734$ | $r$($C10$, $C15$) | = $0.990$ | $r$($C19$, $C25$) | = $0.172$
$r$($C1$, $C9$) | = $0.960$ | $r$($C5$, $C7$) | = $-0.917$ | $r$($C11$, $C12$) | = $0.925$ | $r$($C19$, $C26$) | = $0.190$
$r$($C1$, $C10$) | = $0.964$ | $r$($C5$, $C8$) | = $0.924$ | $r$($C11$, $C13$) | = $0.949$ | $r$($C19$, $C27$) | = $0.189$
$r$($C1$, $C11$) | = $0.954$ | $r$($C5$, $C9$) | = $0.930$ | $r$($C11$, $C14$) | = $-0.970$ | $r$($C19$, $C28$) | = $0.144$
$r$($C1$, $C12$) | = $0.912$ | $r$($C5$, $C10$) | = $0.934$ | $r$($C11$, $C15$) | = $0.980$ | $r$($C19$, $C29$) | = $0.294$
$r$($C1$, $C13$) | = $0.936$ | $r$($C5$, $C11$) | = $0.925$ | $r$($C12$, $C13$) | = $0.907$ | $r$($C20$, $C21$) | = $0.210$
$r$($C1$, $C14$) | = $-0.957$ | $r$($C5$, $C12$) | = $0.883$ | $r$($C12$, $C14$) | = $-0.927$ | $r$($C20$, $C22$) | = $0.295$
$r$($C1$, $C15$) | = $0.966$ | $r$($C5$, $C13$) | = $0.907$ | $r$($C12$, $C15$) | = $0.936$ | $r$($C20$, $C25$) | = $0.152$
$r$($C2$, $C3$) | = $0.924$ | $r$($C5$, $C14$) | = $-0.927$ | $r$($C13$, $C14$) | = $-0.952$ | $r$($C20$, $C26$) | = $0.167$
$r$($C2$, $C4$) | = $0.924$ | $r$($C5$, $C15$) | = $0.936$ | $r$($C13$, $C15$) | = $0.961$ | $r$($C20$, $C27$) | = $0.169$
$r$($C2$, $C5$) | = $0.900$ | $r$($C6$, $C7$) | = $-0.762$ | $r$($C14$, $C15$) | = $-0.982$ | $r$($C20$, $C28$) | = $0.141$
$r$($C2$, $C6$) | = $0.748$ | $r$($C6$, $C8$) | = $0.767$ | $r$($C16$, $C17$) | = $0.210$ | $r$($C20$, $C29$) | = $0.287$
$r$($C2$, $C7$) | = $-0.935$ | $r$($C6$, $C9$) | = $0.773$ | $r$($C16$, $C18$) | = $0.167$ | $r$($C21$, $C22$) | = $0.235$
$r$($C2$, $C8$) | = $0.941$ | $r$($C6$, $C10$) | = $0.776$ | $r$($C16$, $C19$) | = $0.224$ | $r$($C21$, $C25$) | = $0.107$
$r$($C2$, $C9$) | = $0.948$ | $r$($C6$, $C11$) | = $0.768$ | $r$($C16$, $C20$) | = $0.197$ | $r$($C21$, $C26$) | = $0.118$
$r$($C2$, $C10$) | = $0.952$ | $r$($C6$, $C12$) | = $0.734$ | $r$($C16$, $C21$) | = $0.138$ | $r$($C21$, $C27$) | = $0.122$
$r$($C2$, $C11$) | = $0.942$ | $r$($C6$, $C13$) | = $0.753$ | $r$($C16$, $C22$) | = $0.222$ | $r$($C21$, $C28$) | = $0.112$
$r$($C2$, $C12$) | = $0.900$ | $r$($C6$, $C14$) | = $-0.770$ | $r$($C16$, $C25$) | = $0.129$ | $r$($C21$, $C29$) | = $0.229$
$r$($C2$, $C13$) | = $0.924$ | $r$($C6$, $C15$) | = $0.778$ | $r$($C16$, $C26$) | = $0.142$ | $r$($C22$, $C25$) | = $0.170$
$r$($C2$, $C14$) | = $-0.945$ | $r$($C7$, $C8$) | = $-0.959$ | $r$($C16$, $C27$) | = $0.141$ | $r$($C22$, $C26$) | = $0.188$
$r$($C2$, $C15$) | = $0.954$ | $r$($C7$, $C9$) | = $-0.966$ | $r$($C16$, $C28$) | = $0.106$ | $r$($C22$, $C27$) | = $0.191$
$r$($C3$, $C4$) | = $0.931$ | $r$($C7$, $C10$) | = $-0.970$ | $r$($C16$, $C29$) | = $0.216$ | $r$($C22$, $C28$) | = $0.158$
$r$($C3$, $C5$) | = $0.907$ | $r$($C7$, $C11$) | = $-0.960$ | $r$($C17$, $C18$) | = $0.209$ | $r$($C22$, $C29$) | = $0.324$
$r$($C3$, $C6$) | = $0.753$ | $r$($C7$, $C12$) | = $-0.917$ | $r$($C17$, $C19$) | = $0.280$ | $r$($C23$, $C24$) | = $0.155$
$r$($C3$, $C7$) | = $-0.942$ | $r$($C7$, $C13$) | = $-0.942$ | $r$($C17$, $C20$) | = $0.247$ | $r$($C23$, $C30$) | = $0.104$
$r$($C3$, $C8$) | = $0.948$ | $r$($C7$, $C14$) | = $0.963$ | $r$($C17$, $C21$) | = $0.174$ | $r$($C24$, $C30$) | = $0.167$
$r$($C3$, $C9$) | = $0.955$ | $r$($C7$, $C15$) | = $-0.972$ | $r$($C17$, $C22$) | = $0.278$ | $r$($C25$, $C26$) | = $0.109$
$r$($C3$, $C10$) | = $0.959$ | $r$($C8$, $C9$) | = $0.973$ | $r$($C17$, $C25$) | = $0.161$ | $r$($C25$, $C27$) | = $0.108$
$r$($C3$, $C11$) | = $0.949$ | $r$($C8$, $C10$) | = $0.977$ | $r$($C17$, $C26$) | = $0.178$ | $r$($C25$, $C28$) | = $0.081$
$r$($C3$, $C12$) | = $0.907$ | $r$($C8$, $C11$) | = $0.967$ | $r$($C17$, $C27$) | = $0.177$ | $r$($C25$, $C29$) | = $0.166$
$r$($C3$, $C13$) | = $0.931$ | $r$($C8$, $C12$) | = $0.924$ | $r$($C17$, $C28$) | = $0.132$ | $r$($C26$, $C27$) | = $0.120$
$r$($C3$, $C14$) | = $-0.952$ | $r$($C8$, $C13$) | = $0.948$ | $r$($C17$, $C29$) | = $0.271$ | $r$($C26$, $C28$) | = $0.090$
$r$($C3$, $C15$) | = $0.961$ | $r$($C8$, $C14$) | = $-0.969$ | $r$($C18$, $C19$) | = $0.223$ | $r$($C26$, $C29$) | = $0.184$
$r$($C4$, $C5$) | = $0.907$ | $r$($C8$, $C15$) | = $0.979$ | $r$($C18$, $C20$) | = $0.198$ | $r$($C27$, $C28$) | = $0.091$
$r$($C4$, $C6$) | = $0.753$ | $r$($C9$, $C10$) | = $0.984$ | $r$($C18$, $C21$) | = $0.140$ | $r$($C27$, $C29$) | = $0.186$
$r$($C4$, $C7$) | = $-0.942$ | $r$($C9$, $C11$) | = $0.974$ | $r$($C18$, $C22$) | = $0.223$ | $r$($C28$, $C29$) | = $0.154$
$r$($C4$, $C8$) | = $0.948$ | $r$($C9$, $C12$) | = $0.930$ | $r$($C18$, $C25$) | = $0.128$ | |
$r$($C4$, $C9$) | = $0.955$ | $r$($C9$, $C13$) | = $0.955$ | $r$($C18$, $C26$) | = $0.142$ | |
Table 24: Summary of values of $G$ used to determine the 2010 recommended
value (see also Table 17, Sec. XI).
| | Relative |
---|---|---|---
Item | Value1 | standard |
number | ($10^{-11}$ m3 kg-1 s-2) | uncertainty $u_{\rm r}$ | Identification
$G1$ | $6.672\,48(43)$ | $6.4\times 10^{-5}$ | NIST-82
$G2$ | $6.672\,9(5)$ | $7.5\times 10^{-5}$ | TR&D-96
$G3$ | $6.673\,98(70)$ | $1.0\times 10^{-4}$ | LANL-97
$G4$ | $6.674\,255(92)$ | $1.4\times 10^{-5}$ | UWash-00
$G5$ | $6.675\,59(27)$ | $4.0\times 10^{-5}$ | BIPM-01
$G6$ | $6.674\,22(98)$ | $1.5\times 10^{-4}$ | UWup-02
$G7$ | $6.673\,87(27)$ | $4.0\times 10^{-5}$ | MSL-03
$G8$ | $6.672\,28(87)$ | $1.3\times 10^{-4}$ | HUST-05
$G9$ | $6.674\,25(12)$ | $1.9\times 10^{-5}$ | UZur-06
$G10$ | $6.673\,49(18)$ | $2.7\times 10^{-5}$ | HUST-09
$G11$ | $6.672\,34(14)$ | $2.1\times 10^{-5}$ | JILA-10
1Correlation coefficients: $r(G1,G3)=0.351$; $r(G8,G10)=0.234$.
### XIII.1 Comparison of data through inferred values of $\bm{\alpha}$,
$\bm{h}$, $\bm{k}$ and $\bm{A_{\rm r}({\rm e})}$
Here the level of consistency of the data is shown by comparing values of
$\alpha$, $h$, $k$ and $A_{\rm r}({\rm e})$ that can be inferred from
different types of experiments. Note, however, that the inferred value is for
comparison purposes only; the datum from which it is obtained, not the
inferred value, is used as the input datum in the least-squares calculations.
Table 25: Inferred values of the fine-structure constant $\alpha$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary | Item | Identification | Sec. and Eq. | $\alpha^{-1}$ | Relative standard
---|---|---|---|---|---
source | number | | | | uncertainty $u_{\rm r}$
$a_{\rm e}$ | $B13.2$ | HarvU-08 | V.1.3 (127) | $137.035\,999\,084(51)$ | $3.7\times 10^{-10}$
$h/m(^{87}{\rm Rb})$ | $B57$ | LKB-11 | VII.2 | $137.035\,999\,049(90)$ | $6.6\times 10^{-10}$
$a_{\rm e}$ | $B11$ | UWash-87 | V.1.3 (126) | $137.035\,998\,19(50)$ | $3.7\times 10^{-9}$
$h/m(^{133}{\rm Cs})$ | $B56$ | StanfU-02 | VII.1 | $137.036\,0000(11)$ | $7.7\times 10^{-9}$
$R_{\rm K}$ | $B35.1$ | NIST-97 | VIII.2 | $137.036\,0037(33)$ | $2.4\times 10^{-8}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $B32.1$ | NIST-89 | VIII.2 | $137.035\,9879(51)$ | $3.7\times 10^{-8}$
$R_{\rm K}$ | $B35.2$ | NMI-97 | VIII.2 | $137.035\,9973(61)$ | $4.4\times 10^{-8}$
$R_{\rm K}$ | $B35.5$ | LNE-01 | VIII.2 | $137.036\,0023(73)$ | $5.3\times 10^{-8}$
$R_{\rm K}$ | $B35.3$ | NPL-88 | VIII.2 | $137.036\,0083(73)$ | $5.4\times 10^{-8}$
$\Delta\nu_{\rm Mu}$ | $B29.1,B29.2$ | LAMPF | VI.2.2 (232) | $137.036\,0018(80)$ | $5.8\times 10^{-8}$
${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $B33$ | KR/VN-98 | VIII.2 | $137.035\,9852(82)$ | $6.0\times 10^{-8}$
$R_{\rm K}$ | $B35.4$ | NIM-95 | VIII.2 | $137.036\,004(18)$ | $1.3\times 10^{-7}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $B32.2$ | NIM-95 | VIII.2 | $137.036\,006(30)$ | $2.2\times 10^{-7}$
$\nu_{\rm H},\nu_{\rm D}$ | | | IV.1.1 (88) | $137.036\,003(41)$ | $3.0\times 10^{-7}$
Figure 1: Values of the fine-structure constant $\alpha$ with $u_{\rm
r}<10^{-7}$ implied by the input data in Table LABEL:tab:pdata, in order of
decreasing uncertainty from top to bottom (see Table 25).
Figure 2: Values of the fine-structure constant $\alpha$ with $u_{\rm
r}<10^{-8}$ implied by the input data in Table LABEL:tab:pdata and the 2006
and 2010 CODATA recommended values in chronological order from top to bottom
(see Table 25).
Table 25 and Figs. 1 and 2 compare values of $\alpha$ obtained from the
indicated input data. These values are calculated using the appropriate
observational equation for each input datum as given in Table 33 and the 2010
recommended values of the constants other than $\alpha$ that enter that
equation. (Some inferred values have also been given in the portion of the
paper where the relevant datum is discussed.) Inspection of the Table and
figures shows that there is agreement among the vast majority of the various
values of $\alpha$, and hence the data from which they are obtained, to the
extent that the difference between any two values of $\alpha$ is less than
$2u_{\rm diff}$, the standard uncertainty of the difference.
The two exceptions are the values of $\alpha$ from the NIST-89 result for
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ and, to a lesser extent, the
KR/VN-98 result for ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$; of the 91
differences, six involving $\alpha$ from NIST-89 and two involving $\alpha$
from KR/VN-98 are greater than $2u_{\rm diff}$. The inconsistency of these
data has in fact been discussed in previous CODATA reports but, as in 2006,
because their self-sensitivity coefficients $S_{\rm c}$ (see Sec. XIII.2
below) are less than $0.01$, they are not included in the final adjustment on
which the 2010 recommended values are based. Hence, their disagreement is not
a serious issue. Examination of the table and figures also shows that even if
all of the data from which these values of $\alpha$ have been inferred were to
be included in the final adjustment, the recommended value of $\alpha$ would
still be determined mainly by the HarvU-08 $a_{\rm e}$ and LKB-10
$h/m(^{87}{\rm Rb})$ data. Indeed, the comparatively large uncertainties of
some of the values of $\alpha$ means that the data from which they are
obtained will have values of $S_{\rm c}<0.01$ and will not be included in the
final adjustment.
Table 26: Inferred values of the Planck constant $h$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary | Item | Identification | Sec. and Eq. | $h/({\rm J\ s})$ | Relative standard
---|---|---|---|---|---
source | number | | | | uncertainty $u_{\rm r}$
$N_{\rm A}$(28Si) | $B54$ | IAC-11 | IX.6 (267) | $6.626\,070\,09(20)\times 10^{-34}$ | $3.0\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ | $B37.3$ | NIST-07 | VIII.2 | $6.626\,068\,91(24)\times 10^{-34}$ | $3.6\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ | $B37.2$ | NIST-98 | VIII.2 | $6.626\,068\,91(58)\times 10^{-34}$ | $8.7\times 10^{-8}$
$K_{\rm J}^{2}R_{\rm K}$ | $B37.1$ | NPL-90 | VIII.2 | $6.626\,0682(13)\times 10^{-34}$ | $2.0\times 10^{-7}$
$K_{\rm J}^{2}R_{\rm K}$ | $B37.4$ | NPL-12 | VIII.2.1 (243) | $6.626\,0712(13)\times 10^{-34}$ | $2.0\times 10^{-7}$
$K_{\rm J}^{2}R_{\rm K}$ | $B37.5$ | METAS-11 | VIII.2.2 (245) | $6.626\,0691(20)\times 10^{-34}$ | $2.9\times 10^{-7}$
$K_{\rm J}$ | $B36.1$ | NMI-89 | VIII.2 | $6.626\,0684(36)\times 10^{-34}$ | $5.4\times 10^{-7}$
$K_{\rm J}$ | $B36.2$ | PTB-91 | VIII.2 | $6.626\,0670(42)\times 10^{-34}$ | $6.3\times 10^{-7}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $B34.2$ | NPL-79 | VIII.2 | $6.626\,0730(67)\times 10^{-34}$ | $1.0\times 10^{-6}$
${\cal F}_{90}$ | $B38$ | NIST-80 | VIII.2 | $6.626\,0657(88)\times 10^{-34}$ | $1.3\times 10^{-6}$
${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $B34.1$ | NIM-95 | VIII.2 | $6.626\,071(11)\times 10^{-34}$ | $1.6\times 10^{-6}$
Figure 3: Values of the Planck constant $h$ with $u_{{\rm r}}<10^{-6}$
implied by the input data in Table LABEL:tab:pdata and the 2006 and 2010
CODATA recommended values in chronological order from top to bottom (see Table
26).
Table 26 and Fig. 3 compare values of $h$ obtained from the indicated input
data. The various values of $h$, and hence the data from which they are
calculated, agree to the extent that the 55 differences between any two values
of $h$ is less than $2u_{\rm diff}$, except for the difference between the
NIST-07 and IAC-11 values. In this case, the difference is $3.8u_{\rm diff}$.
Because the uncertainties of these two values of $h$ are smaller than other
values and are comparable, they play the dominant role in the determination of
the recommended value of $h$. This discrepancy is dealt with before carrying
out the final adjustment. The relatively large uncertainties of many of the
other values of $h$ means that the data from which they are calculated will
not be included in the final adjustment.
Table 27: Inferred values of the Boltzmann constant $k$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary | Item | Identification | Section | $k/({\rm J\ K^{-1}})$ | Relative standard
---|---|---|---|---|---
source | number | | | | uncertainty $u_{\rm r}$
$R$ | $B58.6$ | LNE-11 | X.1.2 | $1.380\,6477(17)\times 10^{-23}$ | $1.2\times 10^{-6}$
$R$ | $B58.2$ | NIST-88 | X.1.1 | $1.380\,6503(25)\times 10^{-23}$ | $1.8\times 10^{-6}$
$R$ | $B58.3$ | LNE-09 | X.1.2 | $1.380\,6495(37)\times 10^{-23}$ | $2.7\times 10^{-6}$
$R$ | $B58.4$ | NPL-10 | X.1.3 | $1.380\,6496(43)\times 10^{-23}$ | $3.1\times 10^{-6}$
$R$ | $B58.5$ | INRIM-10 | X.1.4 | $1.380\,640(10)\times 10^{-23}$ | $7.5\times 10^{-6}$
$R$ | $B58.1$ | NPL-79 | X.1.1 | $1.380\,656(12)\times 10^{-23}$ | $8.4\times 10^{-6}$
$k$ | $B59$ | NIST-07 | X.2.1 | $1.380\,653(13)\times 10^{-23}$ | $9.1\times 10^{-6}$
$k/h$ | $B60$ | NIST-11 | X.2.2 | $1.380\,652(17)\times 10^{-23}$ | $1.2\times 10^{-5}$
Figure 4: Values of the Boltzmann constant $k$ implied by the input data in
Table LABEL:tab:pdata and the 2006 and 2010 CODATA recommended values in
chronological order from top to bottom (see Table 27). AGT: acoustic gas
thermometry; RIGT: refractive index gas thermometry; JNT: Johnson noise
thermometry.
Table 27 and Fig. 4 compare values of $k$ obtained from the indicated input
data. Although most of the source data are values of $R$, values of
$k=R/N_{\rm A}$ are compared, because that is the constant used to define the
kelvin in the “New” SI; see, for example, Mills _et al._ (2011). All of these
values are in general agreement, with none of the 28 differences exceeding
$2u_{\rm diff}$. However, some of the input data from which they are
calculated have uncertainties so large that they will not be included in the
final adjustment.
Table 28: Inferred values of the electron relative atomic mass $A_{\rm r}({\rm e})$ in order of increasing standard uncertainty obtained from the indicated experimental data in Table LABEL:tab:pdata. Primary | Item | Identification | Sec. and Eq. | $A_{\rm r}({\rm e})$ | Relative standard
---|---|---|---|---|---
source | number | | | | uncertainty $u_{\rm r}$
$f_{\rm s}({\rm C})/f_{\rm c}({\rm C})$ | $B17$ | GSI-02 | V.3.2 (187) | $0.000\,548\,579\,909\,32(29)$ | $5.2\times 10^{-10}$
$f_{\rm s}({\rm O})/f_{\rm c}({\rm O})$ | $B18$ | GSI-02 | V.3.2 (188) | $0.000\,548\,579\,909\,57(42)$ | $7.6\times 10^{-10}$
$\Delta\nu_{\bar{\rm p}\,{\rm He^{+}}}$ | $C16-C30$ | CERN-06/10 | IV.2.3 (101) | $0.000\,548\,579\,909\,14(75)$ | $1.4\times 10^{-9}$
$A_{\rm r}({\rm e})$ | $B11$ | UWash-95 | III.4 (20) | $0.000\,548\,579\,9111(12)$ | $2.1\times 10^{-9}$
Figure 5: Values of the electron relative atomic mass $A_{\rm r}({\rm e})$
implied by the input data in Tables LABEL:tab:pdata and 22 and the 2006 and
2010 CODATA recommended values in chronological order from top to bottom (see
Table 28).
Finally, in Table 28 and Fig. 5 we compare four values of $A_{\rm r}({\rm e})$
calculated from different input data as indicated. They are in agreement, with
all six differences less than $2u_{\rm diff}$. Further, since the four
uncertainties are comparable, all four of the source data are included in the
final adjustment.
### XIII.2 Multivariate analysis of data
Our multivariate analysis of the data employs a well known least-squares
method that takes correlations among the input data into account. Used in the
three previous adjustments, it is described in Appendix E of CODATA-98 and
references cited therein. We recall from that appendix that a least-squares
adjustment is characterized by the number of input data $N$, number of
variables or adjusted constants $M$, degrees of freedom $\nu=N-M$, measure
$\chi^{2}$, probability $p\,({\chi^{2}|\nu})$ of obtaining an observed value
of $\chi^{2}$ that large or larger for the given value of $\nu$, Birge ratio
$R_{\rm B}=\sqrt{\chi^{2}/{\nu}}$, and normalized residual of the $i$th input
datum $r_{i}=(x_{i}-\langle x_{i}\rangle)/u_{i}$, where $x_{i}$ is the input
datum, $\langle x_{i}\rangle$ its adjusted value, and $u_{i}$ its standard
uncertainty.
The observational equations for the input data are given in Tables 31, 33, and
35. These equations are written in terms of a particular independent subset of
constants (broadly interpreted) called, as already noted, _adjusted
constants_. These are the variables (or unknowns) of the adjustment. The
least-squares calculation yields values of the adjusted constants that predict
values of the input data through their observational equations that best agree
with the data themselves in the least squares sense. The adjusted constants
used in the 2010 calculations are given in Tables 30, 32, and 34.
The symbol $\doteq$ in an observational equation indicates that an input datum
of the type on the left-hand side is ideally given by the expression on the
right-hand side containing adjusted constants. But because the equation is one
of an overdetermined set that relates a datum to adjusted constants, the two
sides are not necessarily equal. The best estimate of the value of an input
datum is its observational equation evaluated with the least-squares adjusted
values of the adjusted constants on which its observational equation depends.
For some input data such as $\delta_{\rm e}$ and $R$, the observational
equation is simply $\delta_{\rm e}\doteq\delta_{\rm e}$ and $R\doteq R$.
The binding energies $E_{\rm b}(X)/m_{\rm u}c^{2}$ in the observational
equations of Table 33 are treated as fixed quantities with negligible
uncertainties, as are the bound-state $g$-factor ratios. The frequency $f_{\rm
p}$ is not an adjusted constant but is included in the equation for data items
$B30$ and $B31$ to indicate that they are functions of $f_{\rm p}$. Finally,
the observational equations for items $B30$ and $B31$, which are based on Eqs.
(LABEL:eq:murat)-(229) of Sec. VI.2.2, include the function $a_{\rm
e}(\alpha,\delta_{\rm e})$, as well as the theoretical expression for input
data of type $B29$, $\Delta\nu_{\rm Mu}$. The latter expression is discussed
in Sec. VI.2.1 and is a function of $R_{\infty}$, $\alpha$, $m_{\rm
e}/m_{\mbox{\scriptsize{{m}}}}$, and $a_{\mbox{\scriptsize{{m}}}}$.
The self-sensitivity coefficient $S_{\rm c}$ for an input datum is a measure
of the influence of a particular item of data on its corresponding adjusted
value. As in previous adjustments, in general, for an input datum to be
included in the final adjustment on which the 2010 recommended values are
based, its value of $S_{\rm c}$ must be greater than 0.01, or $1\,\%$, which
means that its uncertainty must be no more than about a factor of 10 larger
than the uncertainty of the adjusted value of that quantity; see Sec. I.D of
CODATA-98 for the justification of this $1\,\%$ cutoff. However, the exclusion
of a datum is not followed if, for example, a datum with $S_{\rm c}<0.01$ is
part of a group of data obtained in a given experiment, or series of
experiments, where most of the other data have self-sensitivity coefficients
greater than $0.01$. It is also not followed for $G$, because in this case
there is substantial disagreement of some of the data with the smallest
uncertainties and hence relatively greater significance of the data with
larger uncertainties.
In summary, there is one major discrepancy among the data discussed in this
section: the disagreement of the NIST-07 watt balance value of $K_{\rm
J}^{2}R_{\rm K}$ and the IAC-11 enriched 28Si XRCD value of $N_{\rm A}$, items
$B37.3$ and $B54$ of Table LABEL:tab:pdata.
#### XIII.2.1 Data related to the Newtonian constant of gravitation $G$
Figure 6: Values of the Newtonian constant of gravitation $G$ in Table 24 and
the 2006 and 2010 CODATA recommended values in chronological order from top to
bottom
Our least-squares analysis of the input data begins with the 11 values of $G$
in Table 24, which are graphically compared in Fig. 6. (Because the $G$ data
are independent of all other data, they can be treated separately.) As
discussed in Secs. XI.1.2 and XI.2.1, there are two correlation coefficients
associated with these data: $r(G1,G3)=0.351$ and $r(G8,G10)=0.234$. It is
clear from both the table and figure that the data are highly inconsistent. Of
the 55 differences among the 11 values, the three largest, $11.4u_{\rm diff}$,
$10.7u_{\rm diff}$, and $10.2u_{\rm diff}$ are between JILA-10 and UWash-00,
BIPM-01, and UZur-06, respectively. Further, eight range from $4u_{\rm diff}$
to $7u_{\rm diff}$. The weighted mean of the 11 values has a relative standard
uncertainty of $8.6\times 10^{-6}$. For this calculation, with $\nu=11-1=10$,
we have $\chi^{2}=209.6$, $p\,(209.6|10)\approx 0$, and $R_{\rm B}=4.58$.
(Recall that a multivariate least-squares calculation with only one variable
is a weighted mean with covariances.) Five data have normalized residuals
$r_{i}>2.0$: JILA-10, BIPM-01, UWash-00, NIST-82, and UZur-06; their
respective values are $-10.8$, $6.4$, $4.4$, $-3.2$ and $3.2$.
Repeating the calculation using only the six values of $G$ with relative
uncertainties $\leq 4.0\times 10^{-5}$, namely, UWash-00, BIPM-01, MSL-03,
UZur-06, HUST-09, and JILA-10, has little impact: the value of $G$ increases
by the fractional amount $5.0\times 10^{-6}$ and the relative uncertainty
increases to $8.8\times 10^{-6}$; for this calculation $\nu=6-1=5$,
$\chi^{2}=191.4$, $p\,(191.4|5)\approx 0$, and $R_{\rm B}=6.19$; the values of
$r_{i}$ are $4.0$, $6.3$, $-0.05$, $3.0$, $-2.2$, and $-11.0$, respectively.
Taking into account the historic difficulty in measuring $G$ and the fact that
all $11$ values of $G$ have no apparent issue besides the disagreement among
them, the Task Group decided to take as the 2010 recommended value the
weighted mean of the 11 values in Table 24 after each of their uncertainties
is multiplied by the factor 14. This yields
$\displaystyle
G=6.673\,84(80)~{}\mbox{m}^{3}~{}\mbox{kg}^{-1}~{}\mbox{s}^{-2}\quad[1.2\times
10^{-4}]\,.$ (283)
The largest normalized residual, that of JILA-10, is now 0.77, and the largest
difference between values of $G$, that between JILA-10 and UWash-00, is
$0.82u_{\rm diff}$. For the calculation yielding the recommended value,
$\nu=11-1=10$, $\chi^{2}=1.07$, $p\,(1.07|10)=1.00$, and $R_{\rm B}=0.33$. In
view of the significant scatter of the measured values of $G$, the factor of
14 was chosen so that the smallest and largest values would differ from the
recommended value by about twice its uncertainty; see Fig. 6. The 2010
recommended value represents a fractional decrease in the 2006 value of
$0.66\times 10^{-4}$ and an increase in uncertainty of $20\,\%$.
#### XIII.2.2 Data related to all other constants
Tables 36, 37, and 38 summarize 12 least-squares analyses, discussed in the
following paragraphs, of the input data and correlation coefficients in Tables
18 to 23. Because the adjusted value of $R_{\infty}$ is essentially the same
for all five adjustments summarized in Table 36 and equal to that of
adjustment 3 of Table 38, the values are not listed in Table 36. (Note that
adjustment 3 in Tables 36 and 38 is the same adjustment.)
_Adjustment 1_. The initial adjustment includes all of the input data, three
of which have normalized residuals whose absolute magnitudes are
problematically greater than 2; see Table 37. They are the 2007 NIST watt-
balance result for $K^{2}_{\rm J}R_{\rm K}$, the 2011 IAC enriched silicon
XRCD result for $N_{\rm A}$, and the 1989 NIST result for
${\it\Gamma}^{\prime}_{\rm p-90}$(lo). All other input data have values of
$|r_{i}|$ less than 2, except those for two antiprotonic ${}^{3}{\rm He}$
transitions, data items $C25$ and $C27$ in Table 22, for which $r_{25}=2.12$
and $r_{27}=2.10$. However, the fact that their normalized residuals are
somewhat greater than 2 is not a major concern, because their self-sensitivity
coefficients $S_{\rm c}$ are considerably less than 0.01. In this regard, we
see from Table 37 that two of the three inconsistent data have values of
$S_{\rm c}$ considerably larger than 0.01; the exception is
${\it\Gamma}^{\prime}_{\rm p-90}$(lo) with $S_{\rm c}=0.0096$, which is
rounded to 0.010 in the table.
_Adjustment 2_. The difference in the IAC-11 and NIST-07 values of $h$ (see
first two lines of Table 26) is 3.8$u_{\rm diff}$, where as before $u_{\rm
diff}$ is the standard uncertainty of the difference. To reduce the difference
between these two highly credible results to an acceptable level, that is, to
2$u_{\rm diff}$ or slightly below, the Task Group decided that the
uncertainties used in the adjustment for these data would be those in the
Table LABEL:tab:pdata multiplied by a factor of two. It was also decided to
apply the same factor to the uncertainties of all the data that contribute in
a significant way to the determination of $h$, so that the relative weights of
this set of data are unchanged. (Recall that if the difference between two
values of the same quantity is $au_{\rm diff}$ and the uncertainty of each is
increased by a factor $b$, the difference is reduced to $(a/b)u_{\rm diff}$.)
Thus, adjustment 2 differs from adjustment 1 in that the uncertainties of data
items $B36.1$, $B36.2$, $B37.1$ to $B37.5$, and $B54$ in Table
LABEL:tab:pdata, which are the two values of $K_{\rm J}$, the five values of
$K^{2}_{\rm J}R_{\rm K}$, and the value of $N_{\rm A}$, are increased by a
factor of 2. (Although items $B31.1$, $B31.2$, and $B38$, the two values of
${\it\Gamma}^{\prime}_{\rm p-90}$(hi) and ${\cal F}_{90}$, also contribute to
the determination of $h$, their contribution is small and no multiplicative
factor is applied.)
From Tables 36 and 37 we see that the values of $\alpha$ and $h$ from
adjustment 2 are very nearly the same as from adjustment 1, that $|r_{i}|$ for
both $B37.3$ and $B54$ have been reduced to below 1.4, and that the residual
for ${\it\Gamma}^{\prime}_{\rm p-90}$(lo) is unchanged.
_Adjustment 3_. Adjustment 3 is the adjustment on which the 2010 CODATA
recommended values are based, and as such it is referred to as the “final
adjustment.”. It differs from adjustment 2 in that, following the prescription
described above, 18 input data with values of $S_{\rm c}$ less than 0.01 are
deleted. These are data items $B13.1$, $B32.1$ to $B36.2$, $B37.5$, $B38$,
$B56$, $B59$, and $B60$ in Table LABEL:tab:pdata. (The range in values of
$S_{\rm c}$ for the deleted data is 0.0003 to 0.0097, and no datum with a
value of $S_{\rm c}>1$ was “converted” to a value with $S_{\rm c}<1$ due to
the multiplicative factor.) Further, because $h/m(^{133}{\rm Cs})$, item
$B56$, is deleted as an input datum due to its low weight, the two values of
$A_{\rm r}(^{133}{\rm Cs})$, items $B10.1$ and $10.2$, which are not relevant
to any other input datum, are also deleted and $A_{\rm r}(^{133}{\rm Cs})$ is
omitted as an adjusted constant. This brings the total number of omitted data
items to 20. Table 36 shows that deleting them has virtually no impact on the
values of $\alpha$ and $h$ and Birge ratio $R_{\rm B}$. The data for the final
adjustment are quite consistent, as demonstrated by the value of $\chi^{2}$:
$p\,(58.1|67)=0.77$.
_Adjustments 4 and 5_. The purpose of these adjustments is to test the
robustness of the 2010 recommended values of $\alpha$ and $h$ by omitting the
most accurate data relevant to these constants. Adjustment 4 differs from
adjustment 2 in that the four data that provide values of $\alpha$ with the
smallest uncertainties are deleted, namely, items $B13.1$, $B13.2$, $B56$ and
$B57$, the two values of $a_{\rm e}$ and the values of $h/m(^{133}{\rm Cs})$
and $h/m(^{87}{\rm Rb})$; see the first four entries of Table 25. (For the
same reason as in adjustment 3, in adjustment 4 the two values of $A_{\rm
r}(^{133}{\rm Cs})$ are also deleted as input data and $A_{\rm r}(^{133}{\rm
Cs})$ is omitted as an adjusted constant; the same applies to $A_{\rm
r}(^{87}{\rm Rb})$.) Adjustment 5 differs from adjustment 1 in that the three
data that provide values of $h$ with the smallest uncertainties are deleted,
namely, items $B37.2$, $B37.3$, and $B54$, the two NIST values of $K^{2}_{\rm
J}R_{\rm K}$ and the IAC value of $N_{\rm A}$; see the first three entries of
Table 26. Also deleted are the data with $S_{\rm c}<0.01$ that contribute in a
minimal way to the determination of $\alpha$ and are deleted in the final
adjustment. Table 36 shows that the value of $\alpha$ from the less accurate
$\alpha$-related data used in adjustment 4, and the value of $h$ from the less
accurate $h$-related data used in adjustment 5, agree with the corresponding
recommended values from adjustment 3. This agreement provides a consistency
check on the 2010 recommended values.
_Adjustments 6 to 12_. The aim of the seven adjustments summarized in Table 38
is to investigate the data that determine the recommended values of
$R_{\infty}$, $r_{\rm p}$, and $r_{\rm d}$. Results from adjustment 3, the
final adjustment, are included in the table for reference purposes. We begin
with a discussion of adjustments 6 to 10, which are derived from adjustment 3
by deleting selected input data. We then discuss adjustments 11 and 12, which
examine the impact of the value of the proton rms charge radius derived from
the measurement of the Lamb shift in muonic hydrogen discussed in Sec. IV.1.3
and given in Eq. (97). Note that the value of $R_{\infty}$ depends only weakly
on the data in Tables LABEL:tab:pdata and 22.
In adjustment 6, the electron scattering values of $r_{\rm p}$ and $r_{\rm
d}$, data items $A49.1$, $A49.2$, and $A50$ in Table 18, are not included.
Thus, the values of these two quantities from adjustment 6 are based solely on
H and D spectroscopic data. It is evident from a comparison of the results of
this adjustment and adjustment 3 that the scattering values of the radii play
a smaller role than the spectroscopic data in determining the 2010 recommended
values of $R_{\infty}$, $r_{\rm p}$ and $r_{\rm d}$.
Adjustment 7 is based on only hydrogen data, including the two scattering
values of $r_{\rm p}$ but not the difference between the $1{\rm S_{1/2}}-2{\rm
S_{1/2}}$ transition frequencies in H and D, item $A48$ in Table 18, hereafter
referred to as the “isotope shift.” Adjustment 8 differs from adjustment 7 in
that the two scattering values of $r_{\rm p}$ are deleted. Adjustments 9 and
10 are similar to 7 and 8 but are based on only deuterium data; that is,
adjustment 9 includes the scattering value of $r_{\rm d}$ but not the isotope
shift, while for adjustment 10 the scattering value is deleted. The results of
these four adjustments show the dominant role of the hydrogen data and the
importance of the isotope shift in determining the recommended value of
$r_{\rm d}$. Further, the four values of $R_{\infty}$ from these adjustments
agree with the 2010 recommended value, and the two values of $r_{\rm p}$ and
of $r_{\rm d}$ also agree with their respective recommended values: the
largest difference from the recommended value for the eight results is
$1.4u_{\rm diff}$.
Adjustment 11 differs from adjustment 3 in that it includes the muonic
hydrogen value $r_{\rm p}=0.841\,69(66)~{}{\rm fm}$, and adjustment 12 differs
from adjustment 11 in that the three scattering values of the nuclear radii
are deleted. Because the muonic hydrogen value is significantly smaller and
has a significantly smaller uncertainty than the purely spectroscopic value of
adjustment 6 and the two scattering values, it has a major impact on the
results of adjustments 11 and 12, as can be seen from Table 38: for both
adjustments the value of $R_{\infty}$ shifts down by over 6 standard
deviations and its uncertainty is reduced by a factor of 4.6. Moreover, and
not surprisingly, the values of $r_{\rm p}$ and of $r_{\rm d}$ from both
adjustments are significantly smaller than the recommended values and have
significantly smaller uncertainties. The inconsistencies between the muonic
hydrogen result for $r_{\rm p}$ and the spectroscopic and scattering results
is demonstrated by the large value and low probability of $\chi^{2}$ for
adjustment 11; $p\,(104.9|68)=0.0027$.
The impact of the muonic hydrogen value of $r_{\rm p}$ can also be seen by
examining for adjustments 3, 11, and 12 the normalized residuals and self-
sensitivity coefficients of the principal experimental data that determine
$R_{\infty}$, namely, items $A26$ to $A50$ of Table 18. In brief, $|r_{i}|$
for these data in the final adjustment range from near 0 to 1.24 for item
$A50$, the $r_{\rm d}$ scattering result, with the vast majority being less
than 1. For the three greater than 1, $|r_{i}|$ is 1.03, 1.08, and 1.04. The
value of $S_{\rm c}$ is 1.00 for items $A26$ and $A48$, the hydrogen $1{\rm
S}_{1/2}-2{\rm S}_{1/2}$ transition frequency and the H-D isotope shift; and
0.42 for item $A49.2$, which is the more accurate of the two scattering values
of $r_{\rm p}$. Most others are a few percent, although some values of $S_{\rm
c}$ are near 0. The situation is markedly different for adjustment 12. First,
$|r_{i}|$ for item $A30$, the hydrogen transition frequency involving the
$8{\rm D}_{5/2}$ state, is 3.06 compared to 0.87 in adjustment 3; and items
$A41$, $A42$, and $A43$, deuterium transitions involving the $8{\rm S}_{1/2}$,
$8{\rm D}_{3/2}$, and $8{\rm D}_{5/2}$ states, are now 2.5, 2.4, and 3.0,
respectively, compared to 0.40, 0.17, and 0.68. Further, ten other transitions
have residuals in the range 1.02 to 1.76. As a result, with this proton
radius, the predictions of the theory for hydrogen and deuterium transition
frequencies are not generally consistent with the experiments. Equally
noteworthy is the fact that although $S_{\rm c}$ for items $A26$ and $A48$
remain equal to 1.00, for all other transition frequencies $S_{\rm c}$ is less
than 0.01, which means that they play an inconsequential role in determining
$R_{\infty}$. The results for adjustment 11, which includes the scattering
values of the nuclear radii as well as the muonic hydrogen value, are similar.
In view of the impact of the latter value on the internal consistency of the
$R_{\infty}$ data and its disagreement with the spectroscopic and scattering
values, the Task Group decided that it was premature to include it as an input
datum in the 2010 CODATA adjustment; it was deemed more prudent to wait to see
if further research can resolve the discrepancy. See Sec. IV.1.3 for
additional discussion.
#### XIII.2.3 Test of the Josephson and quantum Hall effect relations
As in CODATA-02 and CODATA-06, the exactness of the relations $K_{\rm J}=2e/h$
and $R_{\rm K}=h/e^{2}$ is investigated by writing
$\displaystyle K_{\rm J}$ $\displaystyle=$
$\displaystyle\frac{2e}{h}\left({1+\varepsilon_{\rm
J}}\right)=\left({\frac{8\alpha}{\mu_{0}ch}}\right)^{1/2}\left({1+\varepsilon_{\rm
J}}\right),$ (284) $\displaystyle R_{\rm K}$ $\displaystyle=$
$\displaystyle\frac{h}{e^{2}}\left({1+\varepsilon_{\rm
K}}\right)=\frac{\mu_{0}c}{2\alpha}\left({1+\varepsilon_{\rm K}}\right),$
(285)
where $\varepsilon_{\rm J}$ and $\varepsilon_{\rm K}$ are unknown correction
factors taken to be additional adjusted constants. Replacing the relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$ in the analysis leading to the
observational equations in Table 33 with the generalizations in Eqs. (284) and
(285) leads to the modified observational equations given in Table 39.
Although the NIST value of $k/h$, item $B60$, was obtained using the Josephson
and quantum Hall effects, it is not included in the tests of the relations
$K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$, because of its large uncertainty.
The results of seven different adjustments are summarized in Table 29. An
entry of 0 in the $\varepsilon_{\rm K}$ column means that it is assumed that
$R_{\rm K}=h/e^{2}$ in the corresponding adjustment; similarly, an entry of 0
in the $\varepsilon_{\rm J}$ column means that it is assumed that $K_{\rm
J}=2e/h$ in the corresponding adjustment. The following comments apply to the
adjustments of Table 29.
Adjustment (i) uses all of the data and thus differs from adjustment 1 of
Table 36 discussed in the previous section only in that the assumption $K_{\rm
J}=2e/h$ and $R_{\rm K}=h/e^{2}$ is relaxed. For this adjustment, $\nu=86$,
$\chi^{2}=78.1$, and $R_{\rm B}=1.02$. The normalized residuals $r_{i}$ for
the three inconsistent data items in Table 37, the companion table to Table
36, are $0.75$, $-0.56$, and $2.88$. Examination of Table 29 shows that
$\epsilon_{\rm K}$ is consistent with 0 within 1.2 times its uncertainty of
$1.8\times 10^{-8}$, while $\epsilon_{\rm J}$ is consistent with 0 within 2.4
times its uncertainty of $5.7\times 10^{-8}$.
It is important to recognize that any conclusions that can be drawn from the
values of $\varepsilon_{\rm K}$ and $\varepsilon_{\rm J}$ of adjustment (i)
must be tempered, because not all of the individual values of
$\varepsilon_{\rm K}$ and $\varepsilon_{\rm J}$ that contribute to their
determination are consistent. This is demonstrated by adjustments (ii) to
(vii) and Figs. 7 and 8. (Because of their comparatively small uncertainties,
it is possible in these adjustments to take the 2010 recommended values for
the constants $a_{\rm e}$, $\alpha$, $R_{\infty}$, and $A_{\rm r}(\rm e)$,
which appear in the observational equations of Table 39, and assume that they
are exactly known.)
Adjustments (ii) and (iii) focus on $\epsilon_{\rm K}$: $\epsilon_{\rm J}$ is
set equal to 0 and values of $\epsilon_{\rm K}$ are obtained from data whose
observational equations are independent of $h$. These data are the five values
of $R_{\rm K}$, items $B35.1$ to $B35.5$; and the three low-field gyromagnetic
ratios, items $B32.1$, $B32.2$, and $B33$. We see from Table 29 that the two
values of $\epsilon_{\rm K}$ resulting from the two adjustments not only have
opposite signs but their difference is $3.0u_{\rm diff}$. Figure 7 compares
the combined value of $\epsilon_{\rm K}$ obtained from the five values of
$R_{\rm K}$ with the five individual values, while Fig. 8 does the same for
the results obtained from the three gyromagnetic ratios.
Figure 7: Comparison of the five individual values of $\epsilon_{\rm K}$
obtained from the five values of $R_{\rm K}$, data items $B35.1$ to $B35.5$,
and the combined value (open circle) from adjustment (ii) given in Table 29.
The applicable observational equation in Table 39 is $B35^{*}$.
Figure 8: Comparison of the three individual values of $\epsilon_{\rm K}$
obtained from the three low-field gyromagnetic ratios, data items $B32.1$,
$B32.2$, and $B33$, and the combined value (open circle) from adjustment (iii)
given in Table 29. The applicable observational equations in Table 39 are
$B32^{*}$ and $B33^{*}$. Because of the form of these equations, the value of
$\epsilon_{\rm K}$ when $\epsilon_{\rm J}=0$ is identical to the value of
$\epsilon_{\rm J}$ when $\epsilon_{\rm K}=0$, hence the label at the bottom of
the figure.
Adjustments (iv) to (vii) focus on $\epsilon_{\rm J}$: $\epsilon_{\rm K}$ is
set equal to 0 and values of $\epsilon_{\rm J}$ are, with the exception of
adjustment (iv), obtained from data whose observational equations are
dependent on $h$. Examination of Table 29 shows that although the values of
$\epsilon_{\rm J}$ from adjustments (iv) and (v) are of opposite sign, their
difference of $49.1\times 10^{-8}$ is less than the $72.0\times 10^{-8}$
uncertainty of the difference. However, the difference between the values of
$\epsilon_{\rm J}$ from adjustments (iv) and (vi) is $3.6u_{\rm diff}$, and is
$3.3u_{\rm diff}$ even for the value of $\epsilon_{\rm J}$ from adjustment
(vii), in which the uncertainties of the most accurate data have been
increased by the factor 2. (The multiplicative factor 2 is that used in
adjustment 2 and the final adjustment; see Tables 37, 36, and their associated
text.) On the other hand, we see that the value of $\epsilon_{\rm J}$ from
adjustment (vi) is consistent with 0 only to within 3.9 times its uncertainty,
but that this is reduced to 2.0 for the value of $\epsilon_{\rm J}$ from
adjustment (vii) which uses expanded uncertainties.
The results of the adjustments discussed above reflect the disagreement of the
NIST-07 watt-balance value for $K_{\rm J}^{2}R_{\rm K}$, and to a lesser
extent that of the similar NIST-98 value, items $B37.2$ and $B37.3$, with the
IAC-11 enriched silicon value of $N_{\rm A}$, item $B54$; and the disagreement
of the NIST-89 result for ${\it\Gamma}^{\prime}_{p-90}$(lo), and to a lesser
extent the KR/VN-98 result for ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$,
items $B32.1$ and $B33$, with the highly accurate values of $\alpha$. If
adjustment 1 is repeated with these five data deleted, we find
$\varepsilon_{\rm K}=2.8(1.8)\times 10^{-8}$ and $\varepsilon_{\rm
J}=15(49)\times 10^{-8}$. These values can be interpreted as confirming that
$\varepsilon_{\rm K}$ is consistent with 0 to within 1.6 times its uncertainty
of $1.8\times 10^{-8}$ and that $\varepsilon_{\rm J}$ is consistent with 0
well within its uncertainty of $49\times 10^{-8}$.
We conclude this section by briefly discussing recent efforts to close what is
called the “metrology triangle.” Although there are variants, the basic idea
is to use a single electron tunneling (SET) device that generates a quantized
current $I=ef$ when an alternating voltage of frequency $f$ is applied to it,
where as usual $e$ is the elementary charge. The current $I$ is then compared
to a current derived from Josephson and quantum Hall effect devices. In view
of quantization of charge in units of $e$ and conservation of charge, the
equality of the currents shows that $K_{\rm J}R_{\rm K}e=2$, as expected,
within the uncertainty of the measurements Keller (2008); Keller _et al._
(2008); Feltin and Piquemal (2009). Although there is no indication from the
results reported to date that this relation is not valid, the uncertainties of
the results are at best at the 1 to 2 parts in $10^{6}$ level Keller _et al._
(2007, 2008); Feltin _et al._ (2011); Camarota _et al._ (2012).
Table 29: Summary of the results of several least-squares adjustments to
investigate the relations $K_{\rm J}=(2e/h)(1+\epsilon_{\rm J})$ and $R_{\rm
K}=(h/e^{2})(1+\epsilon_{\rm K})$. See the text for an explanation and
discussion of each adjustment, but in brief, adjustment (i) uses all the data,
(ii) assumes $K_{\rm J}=2e/h$ (that is, $\epsilon_{\rm J}=0$) and obtains
$\epsilon_{\rm K}$ from the five measured values of $R_{\rm K}$, (iii) is
based on the same assumption and obtains $\epsilon_{\rm K}$ from the two
values of the proton gyromagnetic ratio and one value of the helion
gyromagnetic ratio, (iv) is (iii) but assumes $R_{\rm K}=h/e^{2}$ (that is,
$\epsilon_{\rm K}=0$) and obtains $\epsilon_{\rm J}$ in place of
$\epsilon_{\rm K}$, (v) to (vii) are based on the same assumption and obtain
$\epsilon_{\rm J}$ from all the measured values given in Table LABEL:tab:pdata
for the quantities indicated.
Adj. | Data included1 | $10^{8}\varepsilon_{\rm K}$ | $10^{8}\varepsilon_{\rm J}$
---|---|---|---
(i) | All | $2.2(1.8)$ | $5.7(2.4)$
(ii) | $R_{\rm K}$ | $2.6(1.8)$ | $0$
(iii) | ${\it\Gamma}^{\prime}_{\rm p,h-90}({\rm lo})$ | $-25.4(9.3)$ | $0$
(iv) | ${\it\Gamma}^{\prime}_{\rm p,h-90}({\rm lo})$ | $0$ | $-25.4(9.3)$
(v) | ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),K_{\rm J},K_{\rm J}^{\rm 2}\\!R_{\rm K},{\cal F}_{{\rm 90}}$ | $0$ | $23.7(72.0)$
(vi) | ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),K_{\rm J},K_{\rm J}^{\rm 2}\\!R_{\rm K},{\cal F}_{{\rm 90}},\\!N_{\rm A}$ | $0$ | $8.6(2.2)$
(vii) | ${\it\Gamma}^{\prime}_{\rm p-90}({\rm hi}),[K_{\rm J}],[K_{\rm J}^{\rm 2}\\!R_{\rm K}],{\cal F}_{{\rm 90}},\\![N_{\rm A}]$ | $0$ | $8.6(4.4)$
1The data items in brackets have their uncertainties expanded by a factor of
two.
Table 30: The 28 adjusted constants (variables) used in the least-squares multivariate analysis of the Rydberg-constant data given in Table 18. These adjusted constants appear as arguments of the functions on the right-hand side of the observational equations of Table 31. Adjusted constant | Symbol
---|---
Rydberg constant | $R_{\infty}$
bound-state proton rms charge radius | $r_{\rm p}$
bound-state deuteron rms charge radius | $r_{\rm d}$
additive correction to $E_{\rm H}(1{\rm S}_{1/2})/h$ | $\delta_{\rm H}(1{\rm S}_{1/2})$
additive correction to $E_{\rm H}(2{\rm S}_{1/2})/h$ | $\delta_{\rm H}(2{\rm S}_{1/2})$
additive correction to $E_{\rm H}(3{\rm S}_{1/2})/h$ | $\delta_{\rm H}(3{\rm S}_{1/2})$
additive correction to $E_{\rm H}(4{\rm S}_{1/2})/h$ | $\delta_{\rm H}(4{\rm S}_{1/2})$
additive correction to $E_{\rm H}(6{\rm S}_{1/2})/h$ | $\delta_{\rm H}(6{\rm S}_{1/2})$
additive correction to $E_{\rm H}(8{\rm S}_{1/2})/h$ | $\delta_{\rm H}(8{\rm S}_{1/2})$
additive correction to $E_{\rm H}(2{\rm P}_{1/2})/h$ | $\delta_{\rm H}(2{\rm P}_{1/2})$
additive correction to $E_{\rm H}(4{\rm P}_{1/2})/h$ | $\delta_{\rm H}(4{\rm P}_{1/2})$
additive correction to $E_{\rm H}(2{\rm P}_{3/2})/h$ | $\delta_{\rm H}(2{\rm P}_{3/2})$
additive correction to $E_{\rm H}(4{\rm P}_{3/2})/h$ | $\delta_{\rm H}(4{\rm P}_{3/2})$
additive correction to $E_{\rm H}(8{\rm D}_{3/2})/h$ | $\delta_{\rm H}(8{\rm D}_{3/2})$
additive correction to $E_{\rm H}(12{\rm D}_{3/2})/h$ | $\delta_{\rm H}(12{\rm D}_{3/2})$
additive correction to $E_{\rm H}(4{\rm D}_{5/2})/h$ | $\delta_{\rm H}(4{\rm D}_{5/2})$
additive correction to $E_{\rm H}(6{\rm D}_{5/2})/h$ | $\delta_{\rm H}(6{\rm D}_{5/2})$
additive correction to $E_{\rm H}(8{\rm D}_{5/2})/h$ | $\delta_{\rm H}(8{\rm D}_{5/2})$
additive correction to $E_{\rm H}(12{\rm D}_{5/2})/h$ | $\delta_{\rm H}(12{\rm D}_{5/2})$
additive correction to $E_{\rm D}(1{\rm S}_{1/2})/h$ | $\delta_{\rm D}(1{\rm S}_{1/2})$
additive correction to $E_{\rm D}(2{\rm S}_{1/2})/h$ | $\delta_{\rm D}(2{\rm S}_{1/2})$
additive correction to $E_{\rm D}(4{\rm S}_{1/2})/h$ | $\delta_{\rm D}(4{\rm S}_{1/2})$
additive correction to $E_{\rm D}(8{\rm S}_{1/2})/h$ | $\delta_{\rm D}(8{\rm S}_{1/2})$
additive correction to $E_{\rm D}(8{\rm D}_{3/2})/h$ | $\delta_{\rm D}(8{\rm D}_{3/2})$
additive correction to $E_{\rm D}(12{\rm D}_{3/2})/h$ | $\delta_{\rm D}(12{\rm D}_{3/2})$
additive correction to $E_{\rm D}(4{\rm D}_{5/2})/h$ | $\delta_{\rm D}(4{\rm D}_{5/2})$
additive correction to $E_{\rm D}(8{\rm D}_{5/2})/h$ | $\delta_{\rm D}(8{\rm D}_{5/2})$
additive correction to $E_{\rm D}(12{\rm D}_{5/2})/h$ | $\delta_{\rm D}(12{\rm D}_{5/2})$
Table 31: Observational equations that express the input data related to $R_{\infty}$ in Table 18 as functions of the adjusted constants in Table 30. The numbers in the first column correspond to the numbers in the first column of Table 18. Energy levels of hydrogenic atoms are discussed in Sec. IV.1. As pointed out at the beginning of that section, $E_{X}(n{\rm L}_{j})/h$ is in fact proportional to $cR_{\infty}$ and independent of $h$, hence $h$ is not an adjusted constant in these equations. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input | | Observational equation |
---|---|---|---
datum | | | |
$A1$–$A16\quad$ | $\delta_{\rm H}(n{\rm L}_{j})$ | $\doteq$ | $\delta_{\rm H}(n{\rm L}_{j})$ |
$A17$–$A25\quad$ | $\delta_{\rm D}(n{\rm L}_{j})$ | $\doteq$ | $\delta_{\rm D}(n{\rm L}_{j})$ |
$A26$–$A31\quad$ | $\nu_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})$ | $\doteq$ | $\big{[}E_{\rm H}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
$A38,A39$ | | | $-E_{\rm H}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}\big{]}/h$ |
$A32$–$A37\quad$ | $\nu_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})-{\textstyle{1\over 4}}\nu_{\rm H}(n_{3}{\rm L_{3}}_{j_{3}}-n_{4}{\rm L_{4}}_{j_{4}})$ | $\doteq$ | $\Big{\\{}E_{\rm H}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
| | | $\ -E_{\rm H}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}$ |
| | | $\ -{\textstyle{1\over 4}}\big{[}E_{\rm H}\big{(}n_{4}{\rm L_{4}}_{j_{4}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{4}{\rm L_{4}}_{j_{4}})\big{)}$ |
| | | $\quad-E_{\rm H}\big{(}n_{3}{\rm L_{3}}_{j_{3}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(n_{3}{\rm L_{3}}_{j_{3}})\big{)}\big{]}\Big{\\}}/h$ |
$A40$–$A44\quad$ | $\nu_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})$ | $\doteq$ | $\big{[}E_{\rm D}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
| | | $-E_{\rm D}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}\big{]}/h$ |
$A45$–$A46\quad$ | $\nu_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}}-n_{2}{\rm L_{2}}_{j_{2}})-{\textstyle{1\over 4}}\nu_{\rm D}(n_{3}{\rm L_{3}}_{j_{3}}-n_{4}{\rm L_{4}}_{j_{4}})$ | $\doteq$ | $\Big{\\{}E_{\rm D}\big{(}n_{2}{\rm L_{2}}_{j_{2}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{2}{\rm L_{2}}_{j_{2}})\big{)}$ |
| | | $\ -E_{\rm D}\big{(}n_{1}{\rm L_{1}}_{j_{1}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{1}{\rm L_{1}}_{j_{1}})\big{)}$ |
| | | $\ -{\textstyle{1\over 4}}\big{[}E_{\rm D}\big{(}n_{4}{\rm L_{4}}_{j_{4}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{4}{\rm L_{4}}_{j_{4}})\big{)}$ |
| | | $\quad-E_{\rm D}\big{(}n_{3}{\rm L_{3}}_{j_{3}};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(n_{3}{\rm L_{3}}_{j_{3}})\big{)}\big{]}\Big{\\}}/h$ |
$A47\quad$ | $\nu_{\rm D}(1{\rm S}_{1/2}-2{\rm S}_{1/2})-\nu_{\rm H}(1{\rm S}_{1/2}-2{\rm S}_{1/2})$ | $\doteq$ | $\Big{\\{}E_{\rm D}\big{(}2{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(2{\rm S}_{1/2})\big{)}$ |
| | | $\ -E_{\rm D}\big{(}1{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm d}),r_{\rm d},\delta_{\rm D}(1{\rm S}_{1/2})\big{)}$ |
| | | $\ -\big{[}E_{\rm H}\big{(}2{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(2{\rm S}_{1/2})\big{)}$ |
| | | $\quad-E_{\rm H}\big{(}1{\rm S}_{1/2};R_{\infty},\alpha,A_{\rm r}({\rm e}),A_{\rm r}({\rm p}),r_{\rm p},\delta_{\rm H}(1{\rm S}_{1/2})\big{)}\big{]}\Big{\\}}/h$ |
$A48\quad$ | $r_{\rm p}$ | $\doteq$ | $r_{\rm p}$ |
$A49\quad$ | $r_{\rm d}$ | $\doteq$ | $r_{\rm d}$ |
Table 32: The 39 adjusted constants (variables) used in the least-squares multivariate analysis of the input data in Table LABEL:tab:pdata. These adjusted constants appear as arguments of the functions on the right-hand side of the observational equations of Table 33. Adjusted constant | Symbol
---|---
electron relative atomic mass | $A_{\rm r}({\rm e})$
proton relative atomic mass | $A_{\rm r}({\rm p})$
neutron relative atomic mass | $A_{\rm r}({\rm n})$
deuteron relative atomic mass | $A_{\rm r}({\rm d})$
triton relative atomic mass | $A_{\rm r}({\rm t})$
helion relative atomic mass | $A_{\rm r}({\rm h})$
alpha particle relative atomic mass | $A_{\rm r}(\mbox{{a}})$
16O7+ relative atomic mass | $A_{\rm r}(^{16}{\rm O}^{7+})$
87Rb relative atomic mass | $A_{\rm r}(^{87}{\rm Rb})$
133Cs relative atomic mass | $A_{\rm r}(^{133}{\rm Cs})$
average vibrational excitation energy | $A_{\rm r}(E_{\rm av})$
fine-structure constant | $\alpha$
additive correction to $a_{\rm e}$(th) | $\delta_{\rm e}$
muon magnetic moment anomaly | $a_{\mbox{\scriptsize{{m}}}}$
additive correction to $g_{\rm C}$(th) | $\delta_{\rm C}$
additive correction to $g_{\rm O}$(th) | $\delta_{\rm O}$
electron-proton magnetic moment ratio | $\mu_{\rm e^{-}}/\mu_{\rm p}$
deuteron-electron magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm e^{-}}$
triton-proton magnetic moment ratio | $\mu_{\rm t}/\mu_{\rm p}$
shielding difference of d and p in HD | $\sigma_{\rm dp}$
shielding difference of t and p in HT | $\sigma_{\rm tp}$
electron to shielded proton |
magnetic moment ratio | $\mu_{\rm e^{-}}/\mu^{\prime}_{\rm p}$
shielded helion to shielded proton |
magnetic moment ratio | $\mu^{\prime}_{\rm h}/\mu^{\prime}_{\rm p}$
neutron to shielded proton |
magnetic moment ratio | $\mu_{\rm n}/\mu_{\rm p}^{\prime}$
electron-muon mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$
additive correction to $\Delta\nu_{\rm Mu}({\rm th})$ | $\delta_{\rm Mu}$
Planck constant | $h$
molar gas constant | $R$
copper K${\rm\alpha}_{1}$ x unit | xu(CuK${\rm\alpha}_{1})$
molybdenum K${\rm\alpha}_{1}$ x unit | xu(MoK${\rm\alpha}_{1})$
ångstrom star | Å∗
$d_{220}$ of Si crystal ILL | $d_{220}({\rm{\scriptstyle ILL}})$
$d_{220}$ of Si crystal N | $d_{220}({\rm{\scriptstyle N}})$
$d_{220}$ of Si crystal WASO 17 | $d_{220}({\rm{\scriptstyle W17}})$
$d_{220}$ of Si crystal WASO 04 | $d_{220}({\rm{\scriptstyle W04}})$
$d_{220}$ of Si crystal WASO 4.2a | $d_{220}({\rm{\scriptstyle W4.2a}})$
$d_{220}$ of Si crystal MO∗ | $d_{220}({\rm{\scriptstyle MO^{*}}})$
$d_{220}$ of Si crystal NR3 | $d_{220}({\rm{\scriptstyle NR3}})$
$d_{220}$ of Si crystal NR4 | $d_{220}({\rm{\scriptstyle NR4}})$
$d_{220}$ of an ideal Si crystal | $d_{220}$
Table 33: Observational equations that express the input data in Table LABEL:tab:pdata as functions of the adjusted constants in Table 32. The numbers in the first column correspond to the numbers in the first column of Table LABEL:tab:pdata. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input | | Observational equation | Sec.
---|---|---|---
datum | | | |
$B1\quad$ | $A_{\rm r}(^{1}{\rm H})$ | $\doteq$ | $A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-E_{\rm b}(^{1}{\rm H})/m_{\rm u}c^{2}$ | III.2
$B2\quad$ | $A_{\rm r}(^{2}{\rm H})$ | $\doteq$ | $A_{\rm r}({\rm d})+A_{\rm r}({\rm e})-E_{\rm b}(^{2}{\rm H})/m_{\rm u}c^{2}$ | III.2
$B3\quad$ | $A_{\rm r}(E_{\rm av})$ | $\doteq$ | $A_{\rm r}(E_{\rm av})$ | III.3
$B4\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm d})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm d})}\over\vbox to9.0pt{}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}}$ | III.3
$B5\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}({\rm t})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm t})}}$ | III.3
$B6\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm c}(^{3}{\rm He}^{+})}\over\vbox to9.0pt{}\textstyle{f_{\rm c}({\rm H}_{2}^{+*})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2A_{\rm r}({\rm p})+A_{\rm r}({\rm e})-\left[\,2E_{\rm I}({\rm H})+E_{\rm B}({\rm H}_{2})-E_{\rm I}({\rm H}_{2})-E_{\rm av}\,\right]/m_{\rm u}c^{2}}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm h})+A_{\rm r}({\rm e})-E_{\rm I}(^{3}{\rm He}^{+})/m_{\rm u}c^{2}}}$ | III.3
$B7\quad$ | $A_{\rm r}(^{4}{\rm He})$ | $\doteq$ | $A_{\rm r}({\mbox{{a}}})+2A_{\rm r}({\rm e})-E_{\rm b}(^{4}{\rm He})/m_{\rm u}c^{2}$ | III.2
$B8\quad$ | $A_{\rm r}(^{16}{\rm O})$ | $\doteq$ | $A_{\rm r}(^{16}{\rm O}^{7+})+7A_{\rm r}({\rm e})-\left[E_{\rm b}(^{16}{\rm O})-E_{\rm b}(^{16}{\rm O}^{7+})\right]\\!/m_{\rm u}c^{2}$ | III.2
$B9\quad$ | $A_{\rm r}(^{87}{\rm Rb})$ | $\doteq$ | $A_{\rm r}(^{87}{\rm Rb})$ |
$B10\quad$ | $A_{\rm r}(^{133}{\rm Cs})$ | $\doteq$ | $A_{\rm r}(^{133}{\rm Cs})$ |
$B11\quad$ | $A_{\rm r}({\rm e})$ | $\doteq$ | $A_{\rm r}({\rm e})$ |
$B12\quad$ | $\delta_{\rm e}$ | $\doteq$ | $\delta_{\rm e}$ |
$B13\quad$ | $a_{\rm e}$ | $\doteq$ | $a_{\rm e}(\alpha,\delta_{e})$ | V.1.1
$B14\quad$ | $\overline{R}$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{a_{\mbox{\scriptsize{{m}}}}}\over\vbox to9.0pt{}\textstyle{1+a_{\rm e}(\alpha,\delta_{\rm e})}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ | V.2.2
$B15\quad$ | $\delta_{\rm C}$ | $\doteq$ | $\delta_{\rm C}$ |
$B16\quad$ | $\delta_{\rm O}$ | $\doteq$ | $\delta_{\rm O}$ |
$B17\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm s}\left({}^{12}{\rm C}^{5+}\right)}\over\vbox to9.0pt{}\textstyle{f_{\rm c}\left({}^{12}{\rm C}^{5+}\right)}}$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm C}(\alpha,\delta_{\rm C})}\over\vbox to9.0pt{}\textstyle{10A_{\rm r}({\rm e})}}\left[12-5A_{\rm r}({\rm e})+{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{E_{\rm b}\left({}^{12}{\rm C}\right)-E_{\rm b}\left({}^{12}{\rm C}^{5+}\right)}\over\vbox to9.0pt{}\textstyle{m_{\rm u}c^{2}}}\right]$ | V.3.2
$B18\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{f_{\rm s}\left({}^{16}{\rm O}^{7+}\right)}\over\vbox to9.0pt{}\textstyle{f_{\rm c}\left({}^{16}{\rm O}^{7+}\right)}}$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm O}(\alpha,\delta_{\rm O})}\over\vbox to9.0pt{}\textstyle{14A_{\rm r}({\rm e})}}A_{\rm r}(^{16}{\rm O}^{7+})$ | V.3.2
$B19\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}({\rm H})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm p}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm p}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ |
$B20\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm d}({\rm D})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}({\rm D})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm d}({\rm D})}\over\vbox to9.0pt{}\textstyle{{g_{\rm d}}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm D})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm d}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}}}$ |
$B21\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm p}({\rm HD})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm d}({\rm HD})}}$ | $\doteq$ | $\left[1+\sigma_{\rm dp}\right]{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm p}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm e^{-}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm d}}}$ |
$B22\quad$ | $\sigma_{\rm dp}$ | $\doteq$ | $\sigma_{\rm dp}$ |
$B23\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm t}({\rm HT})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}({\rm HT})}}$ | $\doteq$ | $\left[1-\sigma_{\rm tp}\right]{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm t}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}}$ |
$B24\quad$ | $\sigma_{\rm tp}$ | $\doteq$ | $\sigma_{\rm tp}$ |
$B25\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{g_{\rm e^{-}}({\rm H})}\over\vbox to9.0pt{}\textstyle{g_{\rm e^{-}}}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B26\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
Table 33: (Continued). Observational equations that express the input data in Table LABEL:tab:pdata as functions of the adjusted constants in Table 32. The numbers in the first column correspond to the numbers in the first column of Table LABEL:tab:pdata. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input | | Observational equation |
---|---|---|---
datum | | | | Sec.
$B27\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm n}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm n}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B28\quad$ | $\delta_{\rm Mu}$ | $\doteq$ | $\delta_{\rm Mu}$ |
$B29\quad$ | $\Delta\nu_{\rm Mu}$ | $\doteq$ | $\Delta\nu_{\rm Mu}\\!\\!\left(R_{\infty},\alpha,{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}},\delta_{\rm Mu}\right)$ | VI.2.1
$B30,B31\quad$ | $\nu(f_{\rm p})$ | $\doteq$ | $\nu\\!\left(f_{\rm p};R_{\infty},\alpha,{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{m_{\rm e}}\over\vbox to9.0pt{}\textstyle{m_{\mbox{\scriptsize{{m}}}}}},{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}}},\delta_{\rm e},\delta_{\rm Mu}\right)$ | VI.2.2
$B32\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$ |
$B33\quad$ | ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$ |
$B34\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$ |
$B35\quad$ | $R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{0}c}\over\vbox to9.0pt{}\textstyle{2\alpha}}$ |
$B36\quad$ | $K_{\rm J}$ | $\doteq$ | $\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{8\alpha}\over\vbox to9.0pt{}\textstyle{\mu_{0}ch}}\right)^{1/2}$ |
$B37\quad$ | $K_{\rm J}^{2}R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{4}\over\vbox to9.0pt{}\textstyle{h}}$ |
$B38\quad$ | ${\cal F}_{90}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}$ |
$B39$-$B41\quad$ | $d_{220}({{\scriptstyle X}})$ | $\doteq$ | $d_{220}({{\scriptstyle X}})$ |
$B42$-$B53\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{d_{220}({{\scriptstyle X}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle Y}})}}-1$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{d_{220}({{\scriptstyle X}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle Y}})}}-1$ |
$B54\quad$ | $N_{\rm A}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{2R_{\infty}h}}$ |
$B55\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda_{\rm meas}}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle ILL}})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\alpha^{2}A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{R_{\infty}d_{220}({\rm{\scriptstyle ILL}})}}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm n})+A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{\left[A_{\rm r}({\rm n})+A_{\rm r}({\rm p})\right]^{2}-A_{\rm r}^{2}({\rm d})}}$ | IX.3
$B56,B57\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{h}\over\vbox to9.0pt{}\textstyle{m(X)}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}(X)}}$ ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c\alpha^{2}}\over\vbox to9.0pt{}\textstyle{2R_{\infty}}}$ | VII.1
$B58\quad$ | $R$ | $\doteq$ | $R$ |
$B59\quad$ | $k$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2R_{\infty}hR}\over\vbox to9.0pt{}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}}$ |
$B60\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{k}\over\vbox to9.0pt{}\textstyle{h}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{2R_{\infty}R}\over\vbox to9.0pt{}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}}$ |
$B61,B64\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm CuK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle X}})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 1\,537.400~{}xu(CuK\mbox{{a}}_{1})}\over\vbox to9.0pt{}\textstyle{d_{220}({{\scriptstyle X}})}}$ | IX.4
$B62\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm WK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 0.209\,010\,0~{}\AA^{*}}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | IX.4
$B63\quad$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\lambda({\rm MoK\mbox{{a}}_{1}})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\rm 707.831~{}xu(MoK\mbox{{a}}_{1})}\over\vbox to9.0pt{}\textstyle{d_{220}({\rm{\scriptstyle N}})}}$ | IX.4
Table 34: The 15 adjusted constants relevant to the antiprotonic helium data given in Table 22. These adjusted constants appear as arguments of the theoretical expressions on the right-hand side of the observational equations of Table 35. Transition | Adjusted constant
---|---
$\bar{\rm p}^{4}$He+: $(32,31)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(32,31\\!\\!:\\!31,30)$
$\bar{\rm p}^{4}$He+: $(35,33)\rightarrow(34,32)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(35,33\\!\\!:\\!34,32)$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(35,33)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34\\!\\!:\\!35,33)$
$\bar{\rm p}^{4}$He+: $(37,34)\rightarrow(36,33)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,34\\!\\!:\\!36,33)$
$\bar{\rm p}^{4}$He+: $(39,35)\rightarrow(38,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(39,35\\!\\!:\\!38,34)$
$\bar{\rm p}^{4}$He+: $(40,35)\rightarrow(39,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(40,35\\!\\!:\\!39,34)$
$\bar{\rm p}^{4}$He+: $(37,35)\rightarrow(38,34)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(37,35\\!\\!:\\!38,34)$
$\bar{\rm p}^{4}$He+: $(33,32)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(33,32\\!\\!:\\!31,30)$
$\bar{\rm p}^{4}$He+: $(36,34)\rightarrow(34,32)$ | $\delta_{\bar{\rm p}^{4}{\rm He}^{+}}(36,34\\!\\!:\\!34,32)$
$\bar{\rm p}^{3}$He+: $(32,31)\rightarrow(31,30)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(32,31\\!\\!:\\!31,30)$
$\bar{\rm p}^{3}$He+: $(34,32)\rightarrow(33,31)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(34,32\\!\\!:\\!33,31)$
$\bar{\rm p}^{3}$He+: $(36,33)\rightarrow(35,32)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,33\\!\\!:\\!35,32)$
$\bar{\rm p}^{3}$He+: $(38,34)\rightarrow(37,33)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(38,34\\!\\!:\\!37,33)$
$\bar{\rm p}^{3}$He+: $(36,34)\rightarrow(37,33)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(36,34\\!\\!:\\!37,33)$
$\bar{\rm p}^{3}$He+: $(35,33)\rightarrow(33,31)$ | $\delta_{\bar{\rm p}^{3}{\rm He}^{+}}(35,33\\!\\!:\\!33,31)$
Table 35: Observational equations that express the input data related to antiprotonic helium in Table 22 as functions of adjusted constants in Tables 32 and 34. The numbers in the first column correspond to the numbers in the first column of Table 22. Definitions of the symbols and values of the parameters in these equations are given in Sec. IV.2. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input | | Observational equation |
---|---|---|---
datum | | | |
$C1$–$C7\quad$ | $\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C8$–$C12\quad$ | $\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C13$–$C19\quad$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{4}He^{+}}}^{(0)}(n,l:n^{\prime},l^{\prime})+a_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm p)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]$ |
| | | $+b_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\mbox{{a}})}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\mbox{{a}}})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]+\delta_{\bar{\rm p}{\rm{}^{4}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
$C20$–$C24\quad$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ | $\doteq$ | $\Delta\nu_{\bar{\rm p}{\rm{}^{3}He^{+}}}^{(0)}(n,l:n^{\prime},l^{\prime})+a_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm p)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm p})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]$ |
| | | $+b_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})\left[\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm e})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm h)}}}\right)^{\\!(0)}\\!\\!\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{A_{\rm r}({\rm h})}\over\vbox to9.0pt{}\textstyle{A_{\rm r}({\rm e})}}\right)-1\right]+\delta_{\bar{\rm p}{\rm{}^{3}He^{+}}}(n,l:n^{\prime},l^{\prime})$ |
Table 36: Summary of the results of some of the least-squares adjustments used to analyze the input data given in Tables 18-23. The values of $\alpha$ and $h$ are those obtained in the adjustment, $N$ is the number of input data, $M$ is the number of adjusted constants, $\nu=N-M$ is the degrees of freedom, and $R_{\rm B}=\sqrt{{\it\chi}^{2}/\nu}$ is the Birge ratio. See the text for an explanation and discussion of each adjustment, but in brief, adjustment 1 is all the data; 2 is the same as 1 except with the uncertainties of the key data that determine $h$ multiplied by 2; 3 is 2 with the low-weight input data deleted and is the adjustment on which the 2010 recommended values are based; 4 is 2 with the input data that provide the most accurate values of alpha deleted; and 5 is 1 with the input data that provide the most accurate values of $h$ deleted. Adj. | $N$ | $M$ | $\nu$ | ${\it\chi}^{2}$ | $R_{\rm B}$ | $\alpha^{-1}$ | $u_{\rm r}(\alpha^{-1})$ | $h$/(J s) | $u_{\rm r}(h)$
---|---|---|---|---|---|---|---|---|---
1 | 169 | 83 | 86 | 89.3 | 1.02 | $137.035\,999\,075(44)$ | $3.2\times 10^{-10}$ | $6.626\,069\,58(15)\times 10^{-34}$ | $2.2\times 10^{-8}$
2 | 169 | 83 | 86 | 75.7 | 0.94 | $137.035\,999\,073(44)$ | $3.2\times 10^{-10}$ | $6.626\,069\,57(29)\times 10^{-34}$ | $4.4\times 10^{-8}$
3 | 149 | 82 | 67 | 58.1 | 0.93 | $137.035\,999\,074(44)$ | $3.2\times 10^{-10}$ | $6.626\,069\,57(29)\times 10^{-34}$ | $4.4\times 10^{-8}$
4 | 161 | 81 | 80 | 69.4 | 0.93 | $137.036\,0005(20)\phantom{\,44}$ | $1.4\times 10^{-8~{}}$ | $6.626\,069\,50(31)\times 10^{-34}$ | $4.7\times 10^{-8}$
5 | 154 | 82 | 72 | 57.2 | 0.89 | $137.035\,999\,074(44)$ | $3.2\times 10^{-10}$ | $6.626\,069\,48(80)\times 10^{-34}$ | $1.2\times 10^{-7}$
Table 37: Normalized residuals $r_{i}$ and self-sensitivity coefficients $S_{\rm c}$ that result from the five least-squares adjustments summarized in Table 36 for the three input data with the largest absolute values of $r_{i}$ in adjustment 1. $S_{\rm c}$ is a measure of how the least-squares estimated value of a given type of input datum depends on a particular measured or calculated value of that type of datum; see Appendix E of CODATA-98. See the text for an explanation and discussion of each adjustment; brief explanations are given at the end of the caption to the previous table. Item | Input | Identification | Adj. 1 | Adj. 2 | Adj. 3 | Adj. 4 | Adj. 5
---|---|---|---|---|---|---|---
number | quantity | | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$ | $~{}r_{i}~{}~{}~{}~{}S_{\rm c}$
$B37.3$ | $K_{\rm J}^{2}R_{\rm K}$ | NIST-07 | $\phantom{-}2.83~{}~{}0.367~{}$ | $\phantom{-}1.39~{}~{}0.367~{}$ | $\phantom{-}1.39~{}~{}0.371~{}$ | $\phantom{-}1.23~{}~{}0.413~{}$ | Deleted
$B54$ | $N_{\rm A}$ | IAC-11 | $-2.57~{}~{}0.555~{}$ | $-1.32~{}~{}0.539~{}$ | $-1.31~{}~{}0.546~{}$ | $-1.16~{}~{}0.587~{}$ | Deleted
$B32.1$ | ${\it\Gamma}^{\prime}_{\rm p-90}({\rm lo})$ | NIST-89 | $\phantom{-}2.19~{}~{}0.010~{}$ | $\phantom{-}2.19~{}~{}0.010~{}$ | Deleted | $\phantom{-}2.46~{}~{}0.158~{}$ | Deleted
Table 38: Summary of the results of some of the least-squares adjustments used
to analyze the input data related to $R_{\infty}$. The values of $R_{\infty}$,
$r_{\rm p}$, and $r_{\rm d}$ are those obtained in the indicated adjustment,
$N$ is the number of input data, $M$ is the number of adjusted constants,
$\nu=N-M$ is the degrees of freedom, and $R_{\rm B}=\sqrt{{\it\chi}^{2}/\nu}$
is the Birge ratio. See the text for an explanation and discussion of each
adjustment, but in brief, adjustment 6 is 3, but the scattering data for the
nuclear radii are omitted; 7 is 3, but with only the hydrogen data included
(but not the isotope shift); 8 is 7 with the $r_{\rm p}$ data deleted; 9 and
10 are similar to 7 and 8, but for the deuterium data; 11 is 3 with the muonic
Lamb-shift value of $r_{\rm p}$ included; and 12 is 11, but without the
scattering values of $r_{\rm p}$ and $r_{\rm d}$.
Adj. | $N$ | $M$ | $\nu$ | ${\it\chi}^{2}$ | $R_{\rm B}$ | $R_{\infty}/{\rm m}^{-1}$ | $u_{\rm r}(R_{\infty})$ | $r_{\rm p}$/fm | $r_{\rm d}$/fm
---|---|---|---|---|---|---|---|---|---
$\phantom{3}3$ | $149$ | $82$ | 67 | 58.1 | 0.93 | $10\,973\,731.568\,539(55)$ | $5.0\times 10^{-12}$ | $0.8775(51)$ | $2.1424(21)$
$\phantom{3}6$ | $146$ | $82$ | 64 | 55.5 | 0.93 | $10\,973\,731.568\,521(82)$ | $7.4\times 10^{-12}$ | $0.8758(77)$ | $2.1417(31)$
$\phantom{3}7$ | $131$ | $72$ | 59 | 53.4 | 0.95 | $10\,973\,731.568\,561(60)$ | $5.5\times 10^{-12}$ | $0.8796(56)$ |
$\phantom{3}8$ | $129$ | $72$ | 57 | 52.5 | 0.96 | $10\,973\,731.568\,528(94)$ | $8.6\times 10^{-12}$ | $0.8764(89)$ |
$\phantom{3}9$ | $114$ | $65$ | 49 | 46.9 | 0.98 | $10\,973\,731.568\,37(13)$ | $1.1\times 10^{-11}$ | | $2.1288(93)$
$10$ | $113$ | $65$ | 48 | 46.8 | 0.99 | $10\,973\,731.568\,28(30)$ | $2.7\times 10^{-11}$ | | $2.121(25)$
$11$ | $150$ | $82$ | 68 | 104.9 | 1.24 | $10\,973\,731.568\,175(12)$ | $1.1\times 10^{-12}$ | $0.842\,25(65)$ | $2.128\,24(28)$
$12$ | $147$ | $82$ | 65 | 74.3 | 1.07 | $10\,973\,731.568\,171(12)$ | $1.1\times 10^{-12}$ | $0.841\,93(66)$ | $2.128\,11(28)$
Table 39: Generalized observational equations that express input data $B32$-$B38$ in Table LABEL:tab:pdata as functions of the adjusted constants in Tables 32 and 30 with the additional adjusted constants $\varepsilon_{\rm J}$ and $\varepsilon_{\rm K}$ as given in Eqs. (284) and (285). The numbers in the first column correspond to the numbers in the first column of Table LABEL:tab:pdata. For simplicity, the lengthier functions are not explicitly given. See Sec. XIII.2 for an explanation of the symbol $\doteq$. Type of input | | Generalized observational equation
---|---|---
datum | | |
$B32^{*}\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm lo})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$
$B33^{*}\quad$ | ${\it\Gamma}_{\rm h-90}^{\,\prime}({\rm lo})$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{K_{\rm J-90}R_{\rm K-90}[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{3}}\over\vbox to9.0pt{}\textstyle{2\mu_{0}R_{\infty}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})}}\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm h}^{\prime}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}$
$B34^{*}\quad$ | ${\it\Gamma}_{\rm p-90}^{\,\prime}({\rm hi})$ | $\doteq$ | $-{\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{c[1+a_{\rm e}(\alpha,\delta_{\rm e})]\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{\rm e^{-}}}\over\vbox to9.0pt{}\textstyle{\mu_{\rm p}^{\prime}}}\right)^{-1}$
$B35^{*}\quad$ | $R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{\mu_{0}c}\over\vbox to9.0pt{}\textstyle{2\alpha}}(1+\varepsilon_{\rm K})$
$B36^{*}\quad$ | $K_{\rm J}$ | $\doteq$ | $\left({\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{8\alpha}\over\vbox to9.0pt{}\textstyle{\mu_{0}ch}}\right)^{1/2}(1+\varepsilon_{\rm J})$
$B37^{*}\quad$ | $K_{\rm J}^{2}R_{\rm K}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{4}\over\vbox to9.0pt{}\textstyle{h}}(1+\varepsilon_{\rm J})^{2}(1+\varepsilon_{\rm K})$
$B38^{*}\quad$ | ${\cal F}_{90}$ | $\doteq$ | ${\vbox to9.0pt{}\phantom{{}_{I}}\textstyle{cM_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}\over\vbox to9.0pt{}\textstyle{K_{\rm J-90}R_{\rm K-90}R_{\infty}h}}(1+\varepsilon_{\rm J})(1+\varepsilon_{\rm K})$
## XIV The 2010 CODATA recommended values
### XIV.1 Calculational details
The 168 input data and their correlation coefficients initially considered for
inclusion in the 2010 CODATA adjustment of the values of the constants are
given in Tables 18 to 23. The 2010 recommended values are based on adjustment
3, called the final adjustment, summarized in Tables 36 to 38 and discussed in
the associated text. Adjustment 3 omits 20 of the 168 initially considered
input data, namely, items $B10.1$, $B10.2$, $B13.1$, $B32.1$ to $B36.2$,
$B37.5$, $B38$, $B56$, $B59$, and $B56$, because of their low weight (self
sensitivity coefficient $S_{\rm c}<0.01$). However, because the observational
equation for $h/m(^{133}{\rm Cs})$, item $B56$, depends on $A_{\rm
r}(^{133}{\rm Cs})$ but item $B56$ is deleted because of its low weight, the
two values of $A_{\rm r}(^{133}{\rm Cs})$, items $B10.1$ and $B10.2$, are also
deleted and $A_{\rm r}(^{133}{\rm Cs})$ itself is deleted as an adjusted
constant. Further, the initial uncertainties of five input data, items $B37.1$
to $B37.4$ and $B56$, are multiplied by the factor 2, with the result that the
absolute values of the normalized residuals $|r_{i}|$ of the five data are
less than 1.4 and their disagreement is reduced to an acceptable level.
Each input datum in this final adjustment has a self sensitivity coefficient
$S_{c}$ greater than 0.01, or is a subset of the data of an experiment or
series of experiments that provide an input datum or input data with $S_{\rm
c}>0.01$. Not counting such input data with $S_{\rm c}<0.01$, the seven data
with $|r_{i}|>1.2$ are $A50$, $B11$, $B37.3$, $B54$, $C19$, $C21$, and $C28$;
their values of $r_{i}$ are $-1.24$, $1.43$, $1.39$, $-1.31$, $-1.60$,
$-1.83$, and $1.76$, respectively.
As discussed in Sec. XIII.2.1, the 2010 recommended value of $G$ is the
weighted mean of the 11 measured values in Table 24 after the uncertainty of
each is multiplied by the factor 14. Although these data can be treated
separately because they are independent of all of the other data, they could
have been included with the other data. For example, if the 11 values of $G$
with expanded uncertainties are added to the 148 input data of adjustment 3,
$G$ is taken as an additional adjusted constant so that these 11 values can be
included in a new adjustment using the observational equation $G\doteq G$, and
the so-modified adjustment 3 is repeated, then we find for this “grand final
adjustment” that $N=160$, $M=83$, $\nu=77$, $\chi^{2}=59.1$,
$p(59.1|77)=0.94$, and $R_{\rm B}=0.88$. Of course, the resulting values of
the adjusted constants, and of the normalized residuals and self sensitivity
coefficients of the input data, are exactly the same as those from adjustment
3 and the weighted mean of the 11 measured values of $G$ with expanded
uncertainties.
In any event, the 2010 recommended values are calculated from the set of best
estimated values, in the least-squares sense, of 82 adjusted constants,
including $G$, and their variances and covariances, together with (i) those
constants that have exact values such as $\mu_{0}$ and $c$; and (ii) the
values of $m_{\mbox{\scriptsize{{t}}}}$, $G_{\rm F}$, and $\sin^{2}\theta_{\rm
W}$ given in Sec. XII. See Sec. V.B of CODATA-98 for details.
### XIV.2 Tables of values
Tables 40 to 47 give the 2010 CODATA recommended values of the basic constants
and conversion factors of physics and chemistry and related quantities.
Although very similar in form and content to their 2006 counterparts, several
new recommended values have been included in the 2010 tables and a few have
been deleted. The values of the four new constants, $m_{\rm n}-m_{\rm p}$ in
kg and u, and $(m_{\rm n}-m_{\rm p})c^{2}$ in J and MeV, are given in Table
LABEL:tab:constants under the heading “Neutron, n”; and the values of the four
new constants $\mu_{\rm h}$, $\mu_{\rm h}/\mu_{\rm B}$, $\mu_{\rm h}/\mu_{\rm
N}$, and $g_{\rm h}$ are given in the same table under the heading “Helion,
h.” The three constants deleted, $\mu_{\rm t}/\mu_{\rm e}$, $\mu_{\rm
t}/\mu_{\rm p}$, and $\mu_{\rm t}/\mu_{\rm n}$, were in the 2006 version of
Table LABEL:tab:constants under the heading “Triton, t.” It was decided that
these constants were of limited interest and the values can be calculated from
other constants in the table.
The values of the four new helion-related constants are calculated from the
adjusted constant $\mu_{\rm h}^{\prime}/\mu_{\rm p}^{\prime}$ and the
theoretically predicted shielding correction $\sigma_{\rm
h}=59.967\,43(10)\times 10^{-6}$ due to Rudziński _et al._ (2009) using the
relation $\mu_{\rm h}^{\prime}=\mu_{\rm h}(1-\sigma_{\rm h})$; see Sec.
VI.1.2.
Table 40: An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chemistry based on the 2010 adjustment. | | | | Relative std.
---|---|---|---|---
Quantity | Symbol | Numerical value | Unit | uncert. $u_{\rm r}$
speed of light in vacuum | $c,c_{0}$ | 299 792 458 | m s-1 | exact
magnetic constant | $\mu_{0}$ | $4\mbox{{p}}\times 10^{-7}$ | N A-2 |
| | $=12.566\,370\,614...\times 10^{-7}$ | N A-2 | exact
electric constant 1/$\mu_{0}c^{2}$ | $\epsilon_{0}$ | $8.854\,187\,817...\times 10^{-12}$ | F m-1 | exact
Newtonian constant of gravitation | $G$ | $6.673\,84(80)\times 10^{-11}$ | m3 kg-1 s-2 | $1.2\times 10^{-4}$
Planck constant | $h$ | $6.626\,069\,57(29)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
$h/2\mbox{{p}}$ | $\hbar$ | $1.054\,571\,726(47)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
elementary charge | $e$ | $1.602\,176\,565(35)\times 10^{-19}$ | C | $2.2\times 10^{-8}$
magnetic flux quantum $h$/2$e$ | ${\it\Phi}_{0}$ | $2.067\,833\,758(46)\times 10^{-15}$ | Wb | $2.2\times 10^{-8}$
conductance quantum $2e^{2}\\!/h$ | $G_{0}$ | $7.748\,091\,7346(25)\times 10^{-5}$ | S | $3.2\times 10^{-10}$
electron mass | $m_{\rm e}$ | $9.109\,382\,91(40)\times 10^{-31}$ | kg | $4.4\times 10^{-8}$
proton mass | $m_{\rm p}$ | $1.672\,621\,777(74)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
proton-electron mass ratio | $m_{\rm p}$/$m_{\rm e}$ | $1836.152\,672\,45(75)$ | | $4.1\times 10^{-10}$
fine-structure constant $e^{2}\\!/4\mbox{{p}}\epsilon_{0}\hbar c$ | $\alpha$ | $7.297\,352\,5698(24)\times 10^{-3}$ | | $3.2\times 10^{-10}$
inverse fine-structure constant | $\alpha^{-1}$ | $137.035\,999\,074(44)$ | | $3.2\times 10^{-10}$
Rydberg constant $\alpha^{2}m_{\rm e}c/2h$ | $R_{\infty}$ | $10\,973\,731.568\,539(55)$ | m-1 | $5.0\times 10^{-12}$
Avogadro constant | $N_{\rm A},L$ | $6.022\,141\,29(27)\times 10^{23}$ | mol-1 | $4.4\times 10^{-8}$
Faraday constant $N_{\rm A}e$ | $F$ | $96\,485.3365(21)$ | C mol-1 | $2.2\times 10^{-8}$
molar gas constant | $R$ | $8.314\,4621(75)$ | J mol-1 K-1 | $9.1\times 10^{-7}$
Boltzmann constant $R$/$N_{\rm A}$ | $k$ | $1.380\,6488(13)\times 10^{-23}$ | J K-1 | $9.1\times 10^{-7}$
Stefan-Boltzmann constant | | | |
($\mbox{{p}}^{2}$/60)$k^{4}\\!/\hbar^{3}c^{2}$ | $\sigma$ | $5.670\,373(21)\times 10^{-8}$ | W m-2 K-4 | $3.6\times 10^{-6}$
Non-SI units accepted for use with the SI
electron volt ($e$/C) J | eV | $1.602\,176\,565(35)\times 10^{-19}$ | J | $2.2\times 10^{-8}$
(unified) atomic mass unit ${1\over 12}m(^{12}$C) | u | $1.660\,538\,921(73)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$ —
Table 40 is a highly-abbreviated list of the values of the constants and
conversion factors most commonly used. Table LABEL:tab:constants is a much
more extensive list of values categorized as follows: UNIVERSAL;
ELECTROMAGNETIC; ATOMIC AND NUCLEAR; and PHYSICOCHEMICAL. The ATOMIC AND
NUCLEAR category is subdivided into 11 subcategories: General; Electroweak;
Electron, ${\rm e}^{-}$; Muon, ${\mbox{{m}}}^{-}$; Tau, ${\mbox{{t}}}^{-}$;
Proton, ${\rm p}$; Neutron, ${\rm n}$; Deuteron, ${\rm d}$; Triton, ${\rm t}$;
Helion, ${\rm h}$; and Alpha particle, a. Table 42 gives the variances,
covariances, and correlation coefficients of a selected group of constants.
(Use of the covariance matrix is discussed in Appendix E of CODATA-98.) Table
43 gives the internationally adopted values of various quantities; Table 44
lists the values of a number of x-ray related quantities; Table 45 lists the
values of various non-SI units; and Tables 46 and 47 give the values of
various energy equivalents.
All of the values given in Tables 40 to 47 are available on the Web pages of
the Fundamental Constants Data Center of the NIST Physical Measurement
Laboratory at physics.nist.gov/constants. This electronic version of the 2010
CODATA recommended values of the constants also includes a much more extensive
correlation coefficient matrix. In fact, the correlation coefficient of any
two constants listed in the tables is accessible on the Web site, as well as
the automatic conversion of the value of an energy-related quantity expressed
in one unit to the corresponding value expressed in another unit (in essence,
an automated version of Tables 46 and 47).
Table 41: The CODATA recommended values of the fundamental constants of physics and chemistry based on the 2010 adjustment. | | | | Relative std.
---|---|---|---|---
Quantity | Symbol | Numerical value | Unit | uncert. $u_{\rm r}$
UNIVERSAL
speed of light in vacuum | $c,c_{0}$ | $299\,792\,$458 | m s-1 | exact
magnetic constant | $\mu_{0}$ | 4$\mbox{{p}}\times 10^{-7}$ | N A-2 |
| | $=12.566\,370\,614...\times 10^{-7}$ | N A-2 | exact
electric constant 1/$\mu_{0}c^{2}$ | $\epsilon_{0}$ | $8.854\,187\,817...\times 10^{-12}$ | F m-1 | exact
characteristic impedance of vacuum $\mu_{0}c$ | $Z_{0}$ | $376.730\,313\,461...$ | ${\rm\Omega}$ | exact
Newtonian constant of gravitation | $G$ | $6.673\,84(80)\times 10^{-11}$ | m3 kg-1 s-2 | $1.2\times 10^{-4}$
| $G/\hbar c$ | $6.708\,37(80)\times 10^{-39}$ | $({\rm GeV}/c^{2})^{-2}$ | $1.2\times 10^{-4}$
Planck constant | $h$ | $6.626\,069\,57(29)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
| | $4.135\,667\,516(91)\times 10^{-15}$ | eV s | $2.2\times 10^{-8}$
$h/2\mbox{{p}}$ | $\hbar$ | $1.054\,571\,726(47)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
| | $6.582\,119\,28(15)\times 10^{-16}$ | eV s | $2.2\times 10^{-8}$
| $\hbar c$ | $197.326\,9718(44)$ | MeV fm | $2.2\times 10^{-8}$
Planck mass $(\hbar c/G)^{1/2}$ | $m_{\rm P}$ | $2.176\,51(13)\times 10^{-8}$ | kg | $6.0\times 10^{-5}$
energy equivalent | $m_{\rm P}c^{2}$ | $1.220\,932(73)\times 10^{19}$ | GeV | $6.0\times 10^{-5}$
Planck temperature $(\hbar c^{5}/G)^{1/2}/k$ | $T_{\rm P}$ | $1.416\,833(85)\times 10^{32}$ | K | $6.0\times 10^{-5}$
Planck length $\hbar/m_{\rm P}c=(\hbar G/c^{3})^{1/2}$ | $l_{\rm P}$ | $1.616\,199(97)\times 10^{-35}$ | m | $6.0\times 10^{-5}$
Planck time $l_{\rm P}/c=(\hbar G/c^{5})^{1/2}$ | $t_{\rm P}$ | $5.391\,06(32)\times 10^{-44}$ | s | $6.0\times 10^{-5}$
ELECTROMAGNETIC
elementary charge | $e$ | $1.602\,176\,565(35)\times 10^{-19}$ | C | $2.2\times 10^{-8}$
| $e/h$ | $2.417\,989\,348(53)\times 10^{14}$ | A J-1 | $2.2\times 10^{-8}$
magnetic flux quantum $h/2e$ | ${\it\Phi}_{0}$ | $2.067\,833\,758(46)\times 10^{-15}$ | Wb | $2.2\times 10^{-8}$
conductance quantum $2e^{2}\\!/h$ | $G_{0}$ | $7.748\,091\,7346(25)\times 10^{-5}$ | S | $3.2\times 10^{-10}$
inverse of conductance quantum | $G_{0}^{-1}$ | $12\,906.403\,7217(42)$ | ${\rm\Omega}$ | $3.2\times 10^{-10}$
Josephson constant666See Table 43 for the conventional value adopted internationally for realizing representations of the volt using the Josephson effect. 2$e/h$ | $K_{\rm J}$ | $483\,597.870(11)\times 10^{9}$ | Hz V-1 | $2.2\times 10^{-8}$
von Klitzing constant777See Table 43 for the conventional value adopted internationally for realizing representations of the ohm using the quantum Hall effect. $h/e^{2}=\mu_{0}c/2\alpha$ | $R_{\rm K}$ | $25\,812.807\,4434(84)$ | ${\rm\Omega}$ | $3.2\times 10^{-10}$
Bohr magneton $e\hbar/2m_{\rm e}$ | $\mu_{\rm B}$ | $927.400\,968(20)\times 10^{-26}$ | J T-1 | $2.2\times 10^{-8}$
| | $5.788\,381\,8066(38)\times 10^{-5}$ | eV T-1 | $6.5\times 10^{-10}$
| $\mu_{\rm B}/h$ | $13.996\,245\,55(31)\times 10^{9}$ | Hz T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm B}/hc$ | $46.686\,4498(10)$ | m${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm B}/k$ | $0.671\,713\,88(61)$ | K T-1 | $9.1\times 10^{-7}$
nuclear magneton $e\hbar/2m_{\rm p}$ | $\mu_{\rm N}$ | $5.050\,783\,53(11)\times 10^{-27}$ | J T-1 | $2.2\times 10^{-8}$
| | $3.152\,451\,2605(22)\times 10^{-8}$ | eV T-1 | $7.1\times 10^{-10}$
| $\mu_{\rm N}/h$ | $7.622\,593\,57(17)$ | MHz T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm N}/hc$ | $2.542\,623\,527(56)\times 10^{-2}$ | m${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\mu_{\rm N}/k$ | $3.658\,2682(33)\times 10^{-4}$ | K T-1 | $9.1\times 10^{-7}$
ATOMIC AND NUCLEAR
General
fine-structure constant $e^{2}\\!/4\mbox{{p}}\epsilon_{0}\hbar c$ | $\alpha$ | $7.297\,352\,5698(24)\times 10^{-3}$ | | $3.2\times 10^{-10}$
inverse fine-structure constant | $\alpha^{-1}$ | $137.035\,999\,074(44)$ | | $3.2\times 10^{-10}$
Rydberg constant $\alpha^{2}m_{\rm e}c/2h$ | $R_{\infty}$ | $10\,973\,731.568\,539(55)$ | m-1 | $5.0\times 10^{-12}$
| $R_{\infty}c$ | $3.289\,841\,960\,364(17)\times 10^{15}$ | Hz | $5.0\times 10^{-12}$
| $R_{\infty}hc$ | $2.179\,872\,171(96)\times 10^{-18}$ | J | $4.4\times 10^{-8}$
| | $13.605\,692\,53(30)$ | eV | $2.2\times 10^{-8}$
Bohr radius $\alpha/4\mbox{{p}}R_{\infty}=4\mbox{{p}}\epsilon_{0}\hbar^{2}\\!/m_{\rm e}e^{2}$ | $a_{\rm 0}$ | $0.529\,177\,210\,92(17)\times 10^{-10}$ | m | $3.2\times 10^{-10}$
Hartree energy $e^{2}\\!/4\mbox{{p}}\epsilon_{\rm 0}a_{\rm 0}=2R_{\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ | $E_{\rm h}$ | $4.359\,744\,34(19)\times 10^{-18}$ | J | $4.4\times 10^{-8}$
| | $27.211\,385\,05(60)$ | eV | $2.2\times 10^{-8}$
quantum of circulation | $h/2m_{\rm e}$ | $3.636\,947\,5520(24)\times 10^{-4}$ | m${}^{2}~{}$s-1 | $6.5\times 10^{-10}$
| $h/m_{\rm e}$ | $7.273\,895\,1040(47)\times 10^{-4}$ | m${}^{2}~{}$s-1 | $6.5\times 10^{-10}$
Electroweak
Fermi coupling constant888Value recommended by the Particle Data Group Nakamura _et al._ (2010). | $G_{\rm F}/(\hbar c)^{3}$ | $1.166\,364(5)\times 10^{-5}$ | GeV-2 | $4.3\times 10^{-6}$
weak mixing angle999Based on the ratio of the masses of the W and Z bosons $m_{\rm W}/m_{\rm Z}$ recommended by the Particle Data Group Nakamura _et al._ (2010). The value for ${\rm sin}^{2}{\theta}_{\rm W}$ they recommend, which is based on a particular variant of the modified minimal subtraction $({\scriptstyle{\rm\overline{MS}}})$ scheme, is ${\rm sin}^{2}\hat{\theta}_{\rm W}(M_{\rm Z})=0.231\,16(13)$. $\theta_{\rm W}$ (on-shell scheme) | | | |
$\sin^{2}\theta_{\rm W}=s^{2}_{\rm W}\equiv 1-(m_{\rm W}/m_{\rm Z})^{2}$ | $\sin^{2}\theta_{\rm W}$ | $0.2223(21)$ | | $9.5\times 10^{-3}$
Electron, e-
electron mass | $m_{\rm e}$ | $9.109\,382\,91(40)\times 10^{-31}$ | kg | $4.4\times 10^{-8}$
| | $5.485\,799\,0946(22)\times 10^{-4}$ | u | $4.0\times 10^{-10}$
energy equivalent | $m_{\rm e}c^{2}$ | $8.187\,105\,06(36)\times 10^{-14}$ | J | $4.4\times 10^{-8}$
| | $0.510\,998\,928(11)$ | MeV | $2.2\times 10^{-8}$
electron-muon mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$ | $4.836\,331\,66(12)\times 10^{-3}$ | | $2.5\times 10^{-8}$
electron-tau mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{t}}}}$ | $2.875\,92(26)\times 10^{-4}$ | | $9.0\times 10^{-5}$
electron-proton mass ratio | $m_{\rm e}/m_{\rm p}$ | $5.446\,170\,2178(22)\times 10^{-4}$ | | $4.1\times 10^{-10}$
electron-neutron mass ratio | $m_{\rm e}/m_{\rm n}$ | $5.438\,673\,4461(32)\times 10^{-4}$ | | $5.8\times 10^{-10}$
electron-deuteron mass ratio | $m_{\rm e}/m_{\rm d}$ | $2.724\,437\,1095(11)\times 10^{-4}$ | | $4.0\times 10^{-10}$
electron-triton mass ratio | $m_{\rm e}/m_{\rm t}$ | $1.819\,200\,0653(17)\times 10^{-4}$ | | $9.1\times 10^{-10}$
electron-helion mass ratio | $m_{\rm e}/m_{\rm h}$ | $1.819\,543\,0761(17)\times 10^{-4}$ | | $9.2\times 10^{-10}$
electron to alpha particle mass ratio | $m_{\rm e}/m_{\mbox{\scriptsize{{a}}}}$ | $1.370\,933\,555\,78(55)\times 10^{-4}$ | | $4.0\times 10^{-10}$
electron charge to mass quotient | $-e/m_{\rm e}$ | $-1.758\,820\,088(39)\times 10^{11}$ | C kg-1 | $2.2\times 10^{-8}$
electron molar mass $N_{\rm A}m_{\rm e}$ | $M({\rm e}),M_{\rm e}$ | $5.485\,799\,0946(22)\times 10^{-7}$ | kg mol-1 | $4.0\times 10^{-10}$
Compton wavelength $h/m_{\rm e}c$ | $\lambda_{\rm C}$ | $2.426\,310\,2389(16)\times 10^{-12}$ | m | $6.5\times 10^{-10}$
$\lambda_{\rm C}/2\mbox{{p}}=\alpha a_{\rm 0}=\alpha^{2}\\!/4\mbox{{p}}R_{\infty}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ | $386.159\,268\,00(25)\times 10^{-15}$ | m | $6.5\times 10^{-10}$
classical electron radius $\alpha^{2}a_{\rm 0}$ | $r_{\rm e}$ | $2.817\,940\,3267(27)\times 10^{-15}$ | m | $9.7\times 10^{-10}$
Thomson cross section (8$\mbox{{p}}/3)r^{2}_{\rm e}$ | $\sigma_{\rm e}$ | $0.665\,245\,8734(13)\times 10^{-28}$ | m2 | $1.9\times 10^{-9}$
electron magnetic moment | $\mu_{\rm e}$ | $-928.476\,430(21)\times 10^{-26}$ | J T-1 | $2.2\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm e}/\mu_{\rm B}$ | $-1.001\,159\,652\,180\,76(27)$ | | $2.6\times 10^{-13}$
to nuclear magneton ratio | $\mu_{\rm e}/\mu_{\rm N}$ | $-1838.281\,970\,90(75)$ | | $4.1\times 10^{-10}$
electron magnetic moment | | | |
anomaly $|\mu_{\rm e}|/\mu_{\rm B}-1$ | $a_{\rm e}$ | $1.159\,652\,180\,76(27)\times 10^{-3}$ | | $2.3\times 10^{-10}$
electron $g$-factor $-2(1+a_{\rm e})$ | $g_{\rm e}$ | $-2.002\,319\,304\,361\,53(53)$ | | $2.6\times 10^{-13}$
electron-muon magnetic moment ratio | $\mu_{\rm e}/\mu_{\mbox{\scriptsize{{m}}}}$ | $206.766\,9896(52)$ | | $2.5\times 10^{-8}$
electron-proton magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm p}$ | $-658.210\,6848(54)$ | | $8.1\times 10^{-9}$
electron to shielded proton magnetic | | | |
moment ratio (H2O, sphere, 25 ∘C) | $\mu_{\rm e}/\mu^{\prime}_{\rm p}$ | $-658.227\,5971(72)$ | | $1.1\times 10^{-8}$
electron-neutron magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm n}$ | $960.920\,50(23)$ | | $2.4\times 10^{-7}$
electron-deuteron magnetic moment ratio | $\mu_{\rm e}/\mu_{\rm d}$ | $-2143.923\,498(18)$ | | $8.4\times 10^{-9}$
electron to shielded helion magnetic | | | |
moment ratio (gas, sphere, 25 ∘C) | $\mu_{\rm e}/\mu^{\prime}_{\rm h}$ | $864.058\,257(10)$ | | $1.2\times 10^{-8}$
electron gyromagnetic ratio $2|\mu_{\rm e}|/\hbar$ | $\gamma_{\rm e}$ | $1.760\,859\,708(39)\times 10^{11}$ | s${}^{-1}~{}$T-1 | $2.2\times 10^{-8}$
| $\gamma_{\rm e}/2\mbox{{p}}$ | $28\,024.952\,66(62)$ | MHz T-1 | $2.2\times 10^{-8}$
Muon, ${\mbox{{m}}}^{-}$
muon mass | $m_{\mbox{\scriptsize{{m}}}}$ | $1.883\,531\,475(96)\times 10^{-28}$ | kg | $5.1\times 10^{-8}$
| | $0.113\,428\,9267(29)$ | u | $2.5\times 10^{-8}$
energy equivalent | $m_{\mbox{\scriptsize{{m}}}}c^{2}$ | $1.692\,833\,667(86)\times 10^{-11}$ | J | $5.1\times 10^{-8}$
| | $105.658\,3715(35)$ | MeV | $3.4\times 10^{-8}$
muon-electron mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm e}$ | $206.768\,2843(52)$ | | $2.5\times 10^{-8}$
muon-tau mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\mbox{\scriptsize{{t}}}}$ | $5.946\,49(54)\times 10^{-2}$ | | $9.0\times 10^{-5}$
muon-proton mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm p}$ | $0.112\,609\,5272(28)$ | | $2.5\times 10^{-8}$
muon-neutron mass ratio | $m_{\mbox{\scriptsize{{m}}}}/m_{\rm n}$ | $0.112\,454\,5177(28)$ | | $2.5\times 10^{-8}$
muon molar mass $N_{\rm A}m_{\mbox{\scriptsize{{m}}}}$ | $M({\mbox{{m}}}),M_{\mbox{\scriptsize{{m}}}}$ | $0.113\,428\,9267(29)\times 10^{-3}$ | kg mol-1 | $2.5\times 10^{-8}$
muon Compton wavelength $h/m_{\mbox{\scriptsize{{m}}}}c$ | $\lambda_{{\rm C},{\mbox{\scriptsize{{m}}}}}$ | $11.734\,441\,03(30)\times 10^{-15}$ | m | $2.5\times 10^{-8}$
$\lambda_{{\rm C},{\mbox{\scriptsize{{m}}}}}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{{\rm C},{\mbox{\scriptsize{{m}}}}}$ | $1.867\,594\,294(47)\times 10^{-15}$ | m | $2.5\times 10^{-8}$
muon magnetic moment | $\mu_{\mbox{\scriptsize{{m}}}}$ | $-4.490\,448\,07(15)\times 10^{-26}$ | J T-1 | $3.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm B}$ | $-4.841\,970\,44(12)\times 10^{-3}$ | | $2.5\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm N}$ | $-8.890\,596\,97(22)$ | | $2.5\times 10^{-8}$
muon magnetic moment anomaly | | | |
$|\mu_{\mbox{\scriptsize{{m}}}}|/(e\hbar/2m_{\mbox{\scriptsize{{m}}}})-1$ | $a_{\mbox{\scriptsize{{m}}}}$ | $1.165\,920\,91(63)\times 10^{-3}$ | | $5.4\times 10^{-7}$
muon $g$-factor $-2(1+a_{\mbox{\scriptsize{{m}}}}$) | $g_{\mbox{\scriptsize{{m}}}}$ | $-2.002\,331\,8418(13)$ | | $6.3\times 10^{-10}$
muon-proton magnetic moment ratio | $\mu_{\mbox{\scriptsize{{m}}}}/\mu_{\rm p}$ | $-3.183\,345\,107(84)$ | | $2.6\times 10^{-8}$
Tau, ${\mbox{{t}}}^{-}$
tau mass101010This and all other values involving $m_{\mbox{\scriptsize{{t}}}}$ are based on the value of $m_{\mbox{\scriptsize{{t}}}}c^{2}$ in MeV recommended by the Particle Data Group Nakamura _et al._ (2010). | $m_{\mbox{\scriptsize{{t}}}}$ | $3.167\,47(29)\times 10^{-27}$ | kg | $9.0\times 10^{-5}$
| | $1.907\,49(17)$ | u | $9.0\times 10^{-5}$
energy equivalent | $m_{\mbox{\scriptsize{{t}}}}c^{2}$ | $2.846\,78(26)\times 10^{-10}$ | J | $9.0\times 10^{-5}$
| | $1776.82(16)$ | MeV | $9.0\times 10^{-5}$
tau-electron mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm e}$ | $3477.15(31)$ | | $9.0\times 10^{-5}$
tau-muon mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\mbox{\scriptsize{{m}}}}$ | $16.8167(15)$ | | $9.0\times 10^{-5}$
tau-proton mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm p}$ | $1.893\,72(17)$ | | $9.0\times 10^{-5}$
tau-neutron mass ratio | $m_{\mbox{\scriptsize{{t}}}}/m_{\rm n}$ | $1.891\,11(17)$ | | $9.0\times 10^{-5}$
tau molar mass $N_{\rm A}m_{\mbox{\scriptsize{{t}}}}$ | $M({\mbox{{t}}}),M_{\mbox{\scriptsize{{t}}}}$ | $1.907\,49(17)\times 10^{-3}$ | kg mol-1 | $9.0\times 10^{-5}$
tau Compton wavelength $h/m_{\mbox{\scriptsize{{t}}}}c$ | $\lambda_{{\rm C},{\mbox{\scriptsize{{t}}}}}$ | $0.697\,787(63)\times 10^{-15}$ | m | $9.0\times 10^{-5}$
$\lambda_{{\rm C},{\mbox{\scriptsize{{t}}}}}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{{\rm C},{\mbox{\scriptsize{{t}}}}}$ | $0.111\,056(10)\times 10^{-15}$ | m | $9.0\times 10^{-5}$
Proton, p
proton mass | $m_{\rm p}$ | $1.672\,621\,777(74)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $1.007\,276\,466\,812(90)$ | u | $8.9\times 10^{-11}$
energy equivalent | $m_{\rm p}c^{2}$ | $1.503\,277\,484(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $938.272\,046(21)$ | MeV | $2.2\times 10^{-8}$
proton-electron mass ratio | $m_{\rm p}/m_{\rm e}$ | $1836.152\,672\,45(75)$ | | $4.1\times 10^{-10}$
proton-muon mass ratio | $m_{\rm p}/m_{\mbox{\scriptsize{{m}}}}$ | $8.880\,243\,31(22)$ | | $2.5\times 10^{-8}$
proton-tau mass ratio | $m_{\rm p}/m_{\mbox{\scriptsize{{t}}}}$ | $0.528\,063(48)$ | | $9.0\times 10^{-5}$
proton-neutron mass ratio | $m_{\rm p}/m_{\rm n}$ | $0.998\,623\,478\,26(45)$ | | $4.5\times 10^{-10}$
proton charge to mass quotient | $e/m_{\rm p}$ | $9.578\,833\,58(21)\times 10^{7}$ | C kg-1 | $2.2\times 10^{-8}$
proton molar mass $N_{\rm A}m_{\rm p}$ | $M$(p), $M_{\rm p}$ | $1.007\,276\,466\,812(90)\times 10^{-3}$ | kg mol-1 | $8.9\times 10^{-11}$
proton Compton wavelength $h/m_{\rm p}c$ | $\lambda_{\rm C,p}$ | $1.321\,409\,856\,23(94)\times 10^{-15}$ | m | $7.1\times 10^{-10}$
$\lambda_{\rm C,p}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C,p}$ | $0.210\,308\,910\,47(15)\times 10^{-15}$ | m | $7.1\times 10^{-10}$
proton rms charge radius | $r_{\rm p}$ | $0.8775(51)\times 10^{-15}$ | m | $5.9\times 10^{-3}$
proton magnetic moment | $\mu_{\rm p}$ | $1.410\,606\,743(33)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm p}/\mu_{\rm B}$ | $1.521\,032\,210(12)\times 10^{-3}$ | | $8.1\times 10^{-9}$
to nuclear magneton ratio | $\mu_{\rm p}/\mu_{\rm N}$ | $2.792\,847\,356(23)$ | | $8.2\times 10^{-9}$
proton $g$-factor $2\mu_{\rm p}/\mu_{\rm N}$ | $g_{\rm p}$ | $5.585\,694\,713(46)$ | | $8.2\times 10^{-9}$
proton-neutron magnetic moment ratio | $\mu_{\rm p}/\mu_{\rm n}$ | $-1.459\,898\,06(34)$ | | $2.4\times 10^{-7}$
shielded proton magnetic moment | $\mu^{\prime}_{\rm p}$ | $1.410\,570\,499(35)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
(H2O, sphere, 25 ∘C) | | | |
to Bohr magneton ratio | $\mu^{\prime}_{\rm p}/\mu_{\rm B}$ | $1.520\,993\,128(17)\times 10^{-3}$ | | $1.1\times 10^{-8}$
to nuclear magneton ratio | $\mu^{\prime}_{\rm p}/\mu_{\rm N}$ | $2.792\,775\,598(30)$ | | $1.1\times 10^{-8}$
proton magnetic shielding correction | | | |
$1-\mu^{\prime}_{\rm p}/\mu_{\rm p}$ (H2O, sphere, 25 ∘C) | $\sigma^{\prime}_{\rm p}$ | $25.694(14)\times 10^{-6}$ | | $5.3\times 10^{-4}$
proton gyromagnetic ratio $2\mu_{\rm p}/\hbar$ | $\gamma_{\rm p}$ | $2.675\,222\,005(63)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.4\times 10^{-8}$
| $\gamma_{\rm p}/2\mbox{{p}}$ | $42.577\,4806(10)$ | MHz T-1 | $2.4\times 10^{-8}$
shielded proton gyromagnetic ratio | | | |
$2\mu^{\prime}_{\rm p}/\hbar$ (H2O, sphere, 25 ∘C) | $\gamma^{\prime}_{\rm p}$ | $2.675\,153\,268(66)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.5\times 10^{-8}$
| $\gamma^{\prime}_{\rm p}/2\mbox{{p}}$ | $42.576\,3866(10)$ | MHz T-1 | $2.5\times 10^{-8}$
Neutron, n
neutron mass | $m_{\rm n}$ | $1.674\,927\,351(74)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $1.008\,664\,916\,00(43)$ | u | $4.2\times 10^{-10}$
energy equivalent | $m_{\rm n}c^{2}$ | $1.505\,349\,631(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $939.565\,379(21)$ | MeV | $2.2\times 10^{-8}$
neutron-electron mass ratio | $m_{\rm n}/m_{\rm e}$ | $1838.683\,6605(11)$ | | $5.8\times 10^{-10}$
neutron-muon mass ratio | $m_{\rm n}/m_{\mbox{\scriptsize{{m}}}}$ | $8.892\,484\,00(22)$ | | $2.5\times 10^{-8}$
neutron-tau mass ratio | $m_{\rm n}/m_{\mbox{\scriptsize{{t}}}}$ | $0.528\,790(48)$ | | $9.0\times 10^{-5}$
neutron-proton mass ratio | $m_{\rm n}/m_{\rm p}$ | $1.001\,378\,419\,17(45)$ | | $4.5\times 10^{-10}$
neutron-proton mass difference | $m_{\rm n}-m_{\rm p}$ | $2.305\,573\,92(76)\times 10^{-30}$ | kg | $3.3\times 10^{-7}$
| | $0.001\,388\,449\,19(45)$ | u | $3.3\times 10^{-7}$
energy equivalent | ($m_{\rm n}-m_{\rm p})c^{2}$ | $2.072\,146\,50(68)\times 10^{-13}$ | J | $3.3\times 10^{-7}$
| | $1.293\,332\,17(42)$ | MeV | $3.3\times 10^{-7}$
neutron molar mass $N_{\rm A}m_{\rm n}$ | $M({\rm n}),M_{\rm n}$ | $1.008\,664\,916\,00(43)\times 10^{-3}$ | kg mol-1 | $4.2\times 10^{-10}$
neutron Compton wavelength $h/m_{\rm n}c$ | $\lambda_{\rm C,n}$ | $1.319\,590\,9068(11)\times 10^{-15}$ | m | $8.2\times 10^{-10}$
$\lambda_{\rm C,n}/2\mbox{{p}}$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C,n}$ | $0.210\,019\,415\,68(17)\times 10^{-15}$ | m | $8.2\times 10^{-10}$
neutron magnetic moment | $\mu_{\rm n}$ | $-0.966\,236\,47(23)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-7}$
to Bohr magneton ratio | $\mu_{\rm n}/\mu_{\rm B}$ | $-1.041\,875\,63(25)\times 10^{-3}$ | | $2.4\times 10^{-7}$
to nuclear magneton ratio | $\mu_{\rm n}/\mu_{\rm N}$ | $-1.913\,042\,72(45)$ | | $2.4\times 10^{-7}$
neutron $g$-factor $2\mu_{\rm n}/\mu_{\rm N}$ | $g_{\rm n}$ | $-3.826\,085\,45(90)$ | | $2.4\times 10^{-7}$
neutron-electron magnetic moment ratio | $\mu_{\rm n}/\mu_{\rm e}$ | $1.040\,668\,82(25)\times 10^{-3}$ | | $2.4\times 10^{-7}$
neutron-proton magnetic moment ratio | $\mu_{\rm n}/\mu_{\rm p}$ | $-0.684\,979\,34(16)$ | | $2.4\times 10^{-7}$
neutron to shielded proton magnetic | | | |
moment ratio (H2O, sphere, 25 ∘C) | $\mu_{\rm n}/\mu_{\rm p}^{\prime}$ | $-0.684\,996\,94(16)$ | | $2.4\times 10^{-7}$
neutron gyromagnetic ratio $2|\mu_{\rm n}|/\hbar$ | $\gamma_{\rm n}$ | $1.832\,471\,79(43)\times 10^{8}$ | s${}^{-1}~{}$T-1 | $2.4\times 10^{-7}$
| $\gamma_{\rm n}/2\mbox{{p}}$ | $29.164\,6943(69)$ | MHz T-1 | $2.4\times 10^{-7}$
Deuteron, d
deuteron mass | $m_{\rm d}$ | $3.343\,583\,48(15)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $2.013\,553\,212\,712(77)$ | u | $3.8\times 10^{-11}$
energy equivalent | $m_{\rm d}c^{2}$ | $3.005\,062\,97(13)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $1875.612\,859(41)$ | MeV | $2.2\times 10^{-8}$
deuteron-electron mass ratio | $m_{\rm d}/m_{\rm e}$ | $3670.482\,9652(15)$ | | $4.0\times 10^{-10}$
deuteron-proton mass ratio | $m_{\rm d}/m_{\rm p}$ | $1.999\,007\,500\,97(18)$ | | $9.2\times 10^{-11}$
deuteron molar mass $N_{\rm A}m_{\rm d}$ | $M({\rm d}),M_{\rm d}$ | $2.013\,553\,212\,712(77)\times 10^{-3}$ | kg mol-1 | $3.8\times 10^{-11}$
deuteron rms charge radius | $r_{\rm d}$ | $2.1424(21)\times 10^{-15}$ | m | $9.8\times 10^{-4}$
deuteron magnetic moment | $\mu_{\rm d}$ | $0.433\,073\,489(10)\times 10^{-26}$ | J T-1 | $2.4\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm d}/\mu_{\rm B}$ | $0.466\,975\,4556(39)\times 10^{-3}$ | | $8.4\times 10^{-9}$
to nuclear magneton ratio | $\mu_{\rm d}/\mu_{\rm N}$ | $0.857\,438\,2308(72)$ | | $8.4\times 10^{-9}$
deuteron $g$-factor $\mu_{\rm d}/\mu_{\rm N}$ | $g_{\rm d}$ | $0.857\,438\,2308(72)$ | | $8.4\times 10^{-9}$
deuteron-electron magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm e}$ | $-4.664\,345\,537(39)\times 10^{-4}$ | | $8.4\times 10^{-9}$
deuteron-proton magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm p}$ | $0.307\,012\,2070(24)$ | | $7.7\times 10^{-9}$
deuteron-neutron magnetic moment ratio | $\mu_{\rm d}/\mu_{\rm n}$ | $-0.448\,206\,52(11)$ | | $2.4\times 10^{-7}$
Triton, t
triton mass | $m_{\rm t}$ | $5.007\,356\,30(22)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $3.015\,500\,7134(25)$ | u | $8.2\times 10^{-10}$
energy equivalent | $m_{\rm t}c^{2}$ | $4.500\,387\,41(20)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $2808.921\,005(62)$ | MeV | $2.2\times 10^{-8}$
triton-electron mass ratio | $m_{\rm t}/m_{\rm e}$ | $5496.921\,5267(50)$ | | $9.1\times 10^{-10}$
triton-proton mass ratio | $m_{\rm t}/m_{\rm p}$ | $2.993\,717\,0308(25)$ | | $8.2\times 10^{-10}$
triton molar mass $N_{\rm A}m_{\rm t}$ | $M({\rm t}),M_{\rm t}$ | $3.015\,500\,7134(25)\times 10^{-3}$ | kg mol-1 | $8.2\times 10^{-10}$
triton magnetic moment | $\mu_{\rm t}$ | $1.504\,609\,447(38)\times 10^{-26}$ | J T-1 | $2.6\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm t}/\mu_{\rm B}$ | $1.622\,393\,657(21)\times 10^{-3}$ | | $1.3\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\rm t}/\mu_{\rm N}$ | $2.978\,962\,448(38)$ | | $1.3\times 10^{-8}$
triton $g$-factor $2\mu_{\rm t}/\mu_{\rm N}$ | $g_{\rm t}$ | $5.957\,924\,896(76)$ | | $1.3\times 10^{-8}$
Helion, h
helion mass | $m_{\rm h}$ | $5.006\,412\,34(22)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $3.014\,932\,2468(25)$ | u | $8.3\times 10^{-10}$
energy equivalent | $m_{\rm h}c^{2}$ | $4.499\,539\,02(20)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $2808.391\,482(62)$ | MeV | $2.2\times 10^{-8}$
helion-electron mass ratio | $m_{\rm h}/m_{\rm e}$ | $5495.885\,2754(50)$ | | $9.2\times 10^{-10}$
helion-proton mass ratio | $m_{\rm h}/m_{\rm p}$ | $2.993\,152\,6707(25)$ | | $8.2\times 10^{-10}$
helion molar mass $N_{\rm A}m_{\rm h}$ | $M({\rm h}),M_{\rm h}$ | $3.014\,932\,2468(25)\times 10^{-3}$ | kg mol-1 | $8.3\times 10^{-10}$
helion magnetic moment | $\mu_{\rm h}$ | $-1.074\,617\,486(27)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
to Bohr magneton ratio | $\mu_{\rm h}/\mu_{\rm B}$ | $-1.158\,740\,958(14)\times 10^{-3}$ | | $1.2\times 10^{-8}$
to nuclear magneton ratio | $\mu_{\rm h}/\mu_{\rm N}$ | $-2.127\,625\,306(25)$ | | $1.2\times 10^{-8}$
helion $g$-factor $2\mu_{\rm h}/\mu_{\rm N}$ | $g_{\rm h}$ | $-4.255\,250\,613(50)$ | | $1.2\times 10^{-8}$
shielded helion magnetic moment | $\mu^{\prime}_{\rm h}$ | $-1.074\,553\,044(27)\times 10^{-26}$ | J T-1 | $2.5\times 10^{-8}$
(gas, sphere, 25 ∘C) | | | |
to Bohr magneton ratio | $\mu^{\prime}_{\rm h}/\mu_{\rm B}$ | $-1.158\,671\,471(14)\times 10^{-3}$ | | $1.2\times 10^{-8}$
to nuclear magneton ratio | $\mu^{\prime}_{\rm h}/\mu_{\rm N}$ | $-2.127\,497\,718(25)$ | | $1.2\times 10^{-8}$
shielded helion to proton magnetic | | | |
moment ratio (gas, sphere, 25 ∘C) | $\mu^{\prime}_{\rm h}/\mu_{\rm p}$ | $-0.761\,766\,558(11)$ | | $1.4\times 10^{-8}$
shielded helion to shielded proton magnetic | | | |
moment ratio (gas/H2O, spheres, 25 ∘C) | $\mu^{\prime}_{\rm h}/\mu^{\prime}_{\rm p}$ | $-0.761\,786\,1313(33)$ | | $4.3\times 10^{-9}$
shielded helion gyromagnetic ratio | | | |
$2|\mu^{\prime}_{\rm h}|/\hbar$ (gas, sphere, 25 ∘C) | $\gamma^{\prime}_{\rm h}$ | $2.037\,894\,659(51)\times 10^{8}$ | s-1 T-1 | $2.5\times 10^{-8}$
| $\gamma^{\prime}_{\rm h}/2\mbox{{p}}$ | $32.434\,100\,84(81)$ | MHz T-1 | $2.5\times 10^{-8}$
Alpha particle, a
alpha particle mass | $m_{\mbox{\scriptsize{{a}}}}$ | $6.644\,656\,75(29)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
| | $4.001\,506\,179\,125(62)$ | u | $1.5\times 10^{-11}$
energy equivalent | $m_{\mbox{\scriptsize{{a}}}}c^{2}$ | $5.971\,919\,67(26)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $3727.379\,240(82)$ | MeV | $2.2\times 10^{-8}$
alpha particle to electron mass ratio | $m_{\mbox{\scriptsize{{a}}}}/m_{\rm e}$ | $7294.299\,5361(29)$ | | $4.0\times 10^{-10}$
alpha particle to proton mass ratio | $m_{\mbox{\scriptsize{{a}}}}/m_{\rm p}$ | $3.972\,599\,689\,33(36)$ | | $9.0\times 10^{-11}$
alpha particle molar mass $N_{\rm A}m_{\mbox{\scriptsize{{a}}}}$ | $M({\mbox{{a}}}),M_{\mbox{\scriptsize{{a}}}}$ | $4.001\,506\,179\,125(62)\times 10^{-3}$ | kg mol-1 | $1.5\times 10^{-11}$
PHYSICOCHEMICAL
Avogadro constant | $N_{\rm A},L$ | $6.022\,141\,29(27)\times 10^{23}$ | mol-1 | $4.4\times 10^{-8}$
atomic mass constant | | | |
$m_{\rm u}=\frac{1}{12}m(^{12}{\rm C})=1$ u | $m_{\rm u}$ | $1.660\,538\,921(73)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
energy equivalent | $m_{\rm u}c^{2}$ | $1.492\,417\,954(66)\times 10^{-10}$ | J | $4.4\times 10^{-8}$
| | $931.494\,061(21)$ | MeV | $2.2\times 10^{-8}$
Faraday constant111111The numerical value of $F$ to be used in coulometric chemical measurements is $96\,485.3321(43)$ [$4.4\times 10^{-8}$] when the relevant current is measured in terms of representations of the volt and ohm based on the Josephson and quantum Hall effects and the internationally adopted conventional values of the Josephson and von Klitzing constants $K_{\rm J-90}$ and $R_{\rm K-90}$ given in Table 43. $N_{\rm A}e$ | $F$ | $96\,485.3365(21)$ | C mol-1 | $2.2\times 10^{-8}$
molar Planck constant | $N_{\rm A}h$ | $3.990\,312\,7176(28)\times 10^{-10}$ | J s mol-1 | $7.0\times 10^{-10}$
| $N_{\rm A}hc$ | $0.119\,626\,565\,779(84)$ | J m mol-1 | $7.0\times 10^{-10}$
molar gas constant | $R$ | $8.314\,4621(75)$ | J mol${}^{-1}~{}$K-1 | $9.1\times 10^{-7}$
Boltzmann constant $R/N_{\rm A}$ | $k$ | $1.380\,6488(13)\times 10^{-23}$ | J K-1 | $9.1\times 10^{-7}$
| | $8.617\,3324(78)\times 10^{-5}$ | eV K-1 | $9.1\times 10^{-7}$
| $k/h$ | $2.083\,6618(19)\times 10^{10}$ | Hz K-1 | $9.1\times 10^{-7}$
| $k/hc$ | $69.503\,476(63)$ | m${}^{-1}~{}$K-1 | $9.1\times 10^{-7}$
molar volume of ideal gas $RT/p$ | | | |
$T=273.15\ {\rm K},\,p=100\ {\rm kPa}$ | $V_{\rm m}$ | $22.710\,953(21)\times 10^{-3}$ | m${}^{3}~{}$mol-1 | $9.1\times 10^{-7}$
Loschmidt constant $N_{\rm A}/V_{\rm m}$ | $n_{0}$ | $2.651\,6462(24)\times 10^{25}$ | m-3 | $9.1\times 10^{-7}$
molar volume of ideal gas $RT/p$ | | | |
$T=273.15\ {\rm K},\,p=101.325\ {\rm kPa}$ | $V_{\rm m}$ | $22.413\,968(20)\times 10^{-3}$ | m${}^{3}~{}$mol-1 | $9.1\times 10^{-7}$
Loschmidt constant $N_{\rm A}/V_{\rm m}$ | $n_{0}$ | $2.686\,7805(24)\times 10^{25}$ | m-3 | $9.1\times 10^{-7}$
Sackur-Tetrode (absolute entropy) constant121212The entropy of an ideal monatomic gas of relative atomic mass $A_{\rm r}$ is given by $S=S_{0}+{3\over 2}R\,\ln A_{\rm r}-R\,\ln(p/p_{0})+{5\over 2}R\,\ln(T/{\rm K}).$ | | | |
$\frac{5}{2}+\ln[(2\mbox{{p}}m_{\rm u}kT_{1}/h^{2})^{3/2}kT_{1}/p_{0}]$ | | | |
$T_{1}=1\ {\rm K},\,p_{0}\,=\,100\ {\rm kPa}$ | $S_{0}/R$ | $-1.151\,7078(23)$ | | $2.0\times 10^{-6}$
$T_{1}=1\ {\rm K},\,p_{0}\,=\,101.325\ {\rm kPa}$ | | $-1.164\,8708(23)$ | | $1.9\times 10^{-6}$
Stefan-Boltzmann constant | | | |
($\mbox{{p}}^{2}/60)k^{4}\\!/\hbar^{3}c^{2}$ | $\sigma$ | $5.670\,373(21)\times 10^{-8}$ | W m${}^{-2}~{}$K-4 | $3.6\times 10^{-6}$
first radiation constant 2$\mbox{{p}}hc^{2}$ | $c_{1}$ | $3.741\,771\,53(17)\times 10^{-16}$ | W m2 | $4.4\times 10^{-8}$
first radiation constant for spectral radiance 2$hc^{2}$ | $c_{\rm 1L}$ | $1.191\,042\,869(53)\times 10^{-16}$ | W m2 sr-1 | $4.4\times 10^{-8}$
second radiation constant $hc/k$ | $c_{2}$ | $1.438\,7770(13)\times 10^{-2}$ | m K | $9.1\times 10^{-7}$
Wien displacement law constants | | | |
$b=\lambda_{\rm max}T=c_{2}/4.965\,114\,231...$ | $b$ | $2.897\,7721(26)\times 10^{-3}$ | m K | $9.1\times 10^{-7}$
$b^{\prime}=\nu_{\rm max}/T=2.821\,439\,372...\,c/c_{2}$ | $b^{\prime}$ | $5.878\,9254(53)\times 10^{10}$ | Hz K-1 | $9.1\times 10^{-7}$
Table 41: (Continued).
Table 42: The variances, covariances, and correlation coefficients of the
values of a selected group of constants based on the 2010 CODATA adjustment.
The numbers in bold above the main diagonal are $10^{16}$ times the numerical
values of the relative covariances; the numbers in bold on the main diagonal
are $10^{16}$ times the numerical values of the relative variances; and the
numbers in italics below the main diagonal are the correlation coefficients.1
$\alpha$ $h$ $e$ $m_{\rm e}$ $N_{\rm A}$ $m_{\rm e}/m_{\rm\mu}$ $F$ $\alpha$
${\bf 0.0010}$ ${\bf 0.0010}$ ${\bf 0.0010}$ ${\bf-0.0011}$ ${\bf 0.0009}$
${\bf-0.0021}$ ${\bf 0.0019}$ $h$ ${\it 0.0072}$ ${\bf 19.4939}$ ${\bf
9.7475}$ ${\bf 19.4918}$ ${\bf-19.4912}$ ${\bf-0.0020}$ ${\bf-9.7437}$ $e$
${\it 0.0145}$ ${\it 1.0000}$ ${\bf 4.8742}$ ${\bf 9.7454}$ ${\bf-9.7452}$
${\bf-0.0020}$ ${\bf-4.8709}$ $m_{\rm e}$ ${\it-0.0075}$ ${\it 0.9999}$ ${\it
0.9998}$ ${\bf 19.4940}$ ${\bf-19.4929}$ ${\bf 0.0021}$ ${\bf-9.7475}$ $N_{\rm
A}$ ${\it 0.0060}$ ${\it-0.9999}$ ${\it-0.9997}$ ${\it-1.0000}$ ${\bf
19.4934}$ ${\bf-0.0017}$ ${\bf 9.7483}$ $m_{\rm e}/m_{\rm\mu}$ ${\it-0.0251}$
${\it-0.0002}$ ${\it-0.0004}$ ${\it 0.0002}$ ${\it-0.0002}$ ${\bf 6.3872}$
${\bf-0.0037}$ $F$ ${\it 0.0265}$ ${\it-0.9993}$ ${\it-0.9990}$ ${\it-0.9997}$
${\it 0.9997}$ ${\it-0.0007}$ ${\bf 4.8774}$ 1 The relative covariance is
$u_{\rm r}(x_{i},x_{j})=u(x_{i},x_{j})/(x_{i}x_{j})$, where $u(x_{i},x_{j})$
is the covariance of $x_{i}$ and $x_{j}$; the relative variance is $u_{\rm
r}^{2}(x_{i})=u_{\rm r}(x_{i},x_{i})$: and the correlation coefficient is
$r(x_{i},x_{j})=u(x_{i},x_{j})/[u(x_{i})u(x_{j})]$.
Table 43: Internationally adopted values of various quantities.
| | | | Relative std.
---|---|---|---|---
Quantity | Symbol | Numerical value | Unit | uncert. $u_{\rm r}$
relative atomic mass1 of 12C | $A_{\rm r}(^{12}$C) | $12$ | | exact
molar mass constant | $M_{\rm u}$ | $1\times 10^{-3}$ | kg mol-1 | exact
molar mass2 of 12C | $M(^{12}$C) | $12\times 10^{-3}$ | kg mol-1 | exact
conventional value of Josephson constant3 | $K_{\rm J-90}$ | 483 597.9 | GHz V-1 | exact
conventional value of von Klitzing constant4 | $R_{\rm K-90}$ | 25 812.807 | ${\rm\Omega}$ | exact
standard-state pressure | | $100$ | kPa | exact
standard atmosphere | | $101.325$ | kPa | exact
1 The relative atomic mass $A_{\rm r}(X)$ of particle $X$ with mass $m(X)$ is
defined by $A_{\rm r}(X)=m(X)/m_{\rm u}$, where $m_{\rm u}=m(^{12}{\rm
C})/12=M_{\rm u}/N_{\rm A}=1~{}{\rm u}$ is the atomic mass constant, $M_{\rm
u}$ is the molar mass constant, $N_{\rm A}$ is the Avogadro constant, and u is
the unified atomic mass unit. Thus the mass of particle $X$ is $m(X)=A_{\rm
r}(X)$ u and the molar mass of $X$ is $M(X)=A_{\rm r}(X)M_{\rm u}$.
2 Value fixed by the SI definition of the mole.
3 This is the value adopted internationally for realizing representations of
the volt using the Josephson effect.
4 This is the value adopted internationally for realizing representations of
the ohm using the quantum Hall effect.
Table 44: Values of some x-ray-related quantities based on the 2010 CODATA
adjustment of the values of the constants.
Relative std. Quantity Symbol Numerical value Unit uncert. $u_{\rm r}$ Cu x
unit: $\lambda({\rm CuK}{\rm\alpha}_{\rm 1})/1\,537.400$ ${\rm xu}({\rm
CuK}{\rm\alpha}_{\rm 1})$ $1.002\,076\,97(28)\times 10^{-13}$ m $2.8\times
10^{-7}$ Mo x unit: $\lambda({\rm MoK}{\rm\alpha}_{\rm 1})/707.831$ ${\rm
xu}({\rm MoK}{\rm\alpha}_{\rm 1})$ $1.002\,099\,52(53)\times 10^{-13}$ m
$5.3\times 10^{-7}$ ångstrom star$:\lambda({\rm WK}{\rm\alpha}_{\rm
1})/0.209\,010\,0$ Å∗ $1.000\,014\,95(90)\times 10^{-10}$ m $9.0\times
10^{-7}$ lattice parameter1 of Si (in vacuum, 22.5 ∘C) $a$
$543.102\,0504(89)\times 10^{-12}$ m $1.6\times 10^{-8}$ {220} lattice spacing
of Si $a/\sqrt{8}$ $d_{\rm 220}$ $192.015\,5714(32)\times 10^{-12}$ m
$1.6\times 10^{-8}$ (in vacuum, 22.5 ∘C) molar volume of Si $M({\rm
Si})/\rho({\rm Si})=N_{\rm A}a^{3}\\!/8$ $V_{\rm m}$(Si)
$12.058\,833\,01(80)\times 10^{-6}$ m3 mol-1 $6.6\times 10^{-8}$ (in vacuum,
22.5 ∘C) 1 This is the lattice parameter (unit cell edge length) of an ideal
single crystal of naturally occurring Si free of impurities and imperfections,
and is deduced from measurements on extremely pure and nearly perfect single
crystals of Si by correcting for the effects of impurities.
Table 45: The values in SI units of some non-SI units based on the 2010 CODATA adjustment of the values of the constants. | | | | Relative std.
---|---|---|---|---
Quantity | Symbol | Numerical value | Unit | uncert. $u_{\rm r}$
Non-SI units accepted for use with the SI
electron volt: ($e/{\rm C}$) J | eV | $1.602\,176\,565(35)\times 10^{-19}$ | J | $2.2\times 10^{-8}$
(unified) atomic mass unit: ${1\over 12}m(^{12}$C) | u | $1.660\,538\,921(73)\times 10^{-27}$ | kg | $4.4\times 10^{-8}$
Natural units (n.u.)
n.u. of velocity | $c,c_{0}$ | 299 792 458 | m s-1 | exact
n.u. of action: $h/2\mbox{{p}}$ | $\hbar$ | $1.054\,571\,726(47)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
| | $6.582\,119\,28(15)\times 10^{-16}$ | eV s | $2.2\times 10^{-8}$
| $\hbar c$ | $197.326\,9718(44)$ | MeV fm | $2.2\times 10^{-8}$
n.u. of mass | $m_{\rm e}$ | $9.109\,382\,91(40)\times 10^{-31}$ | kg | $4.4\times 10^{-8}$
n.u. of energy | $m_{\rm e}c^{2}$ | $8.187\,105\,06(36)\times 10^{-14}$ | J | $4.4\times 10^{-8}$
| | $0.510\,998\,928(11)$ | MeV | $2.2\times 10^{-8}$
n.u. of momentum | $m_{\rm e}c$ | $2.730\,924\,29(12)\times 10^{-22}$ | kg m s-1 | $4.4\times 10^{-8}$
| | $0.510\,998\,928(11)$ | MeV/$c$ | $2.2\times 10^{-8}$
n.u. of length: $\hbar/m_{\rm e}c$ | $\lambda\hskip-4.5pt\vrule height=4.6pt,depth=-4.3pt,width=4.0pt_{\rm C}$ | $386.159\,268\,00(25)\times 10^{-15}$ | m | $6.5\times 10^{-10}$
n.u. of time | $\hbar/m_{\rm e}c^{2}$ | $1.288\,088\,668\,33(83)\times 10^{-21}$ | s | $6.5\times 10^{-10}$
Atomic units (a.u.)
a.u. of charge | $e$ | $1.602\,176\,565(35)\times 10^{-19}$ | C | $2.2\times 10^{-8}$
a.u. of mass | $m_{\rm e}$ | $9.109\,382\,91(40)\times 10^{-31}$ | kg | $4.4\times 10^{-8}$
a.u. of action: $h/2\mbox{{p}}$ | $\hbar$ | $1.054\,571\,726(47)\times 10^{-34}$ | J s | $4.4\times 10^{-8}$
a.u. of length: Bohr radius (bohr) | | | |
$\alpha/4\mbox{{p}}R_{\infty}$ | $a_{0}$ | $0.529\,177\,210\,92(17)\times 10^{-10}$ | m | $3.2\times 10^{-10}$
a.u. of energy: Hartree energy (hartree) | | | |
$e^{2}\\!/4\mbox{{p}}\epsilon_{0}a_{0}=2R_{\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ | $E_{\rm h}$ | $4.359\,744\,34(19)\times 10^{-18}$ | J | $4.4\times 10^{-8}$
a.u. of time | $\hbar/E_{\rm h}$ | $2.418\,884\,326\,502(12)\times 10^{-17}$ | s | $5.0\times 10^{-12}$
a.u. of force | $E_{\rm h}/a_{0}$ | $8.238\,722\,78(36)\times 10^{-8}$ | N | $4.4\times 10^{-8}$
a.u. of velocity: $\alpha c$ | $a_{0}E_{\rm h}/\hbar$ | $2.187\,691\,263\,79(71)\times 10^{6}$ | m s-1 | $3.2\times 10^{-10}$
a.u. of momentum | $\hbar/a_{0}$ | $1.992\,851\,740(88)\times 10^{-24}$ | kg m s-1 | $4.4\times 10^{-8}$
a.u. of current | $eE_{\rm h}/\hbar$ | $6.623\,617\,95(15)\times 10^{-3}$ | A | $2.2\times 10^{-8}$
a.u. of charge density | $e/a_{0}^{3}$ | $1.081\,202\,338(24)\times 10^{12}$ | C m-3 | $2.2\times 10^{-8}$
a.u. of electric potential | $E_{\rm h}/e$ | $27.211\,385\,05(60)$ | V | $2.2\times 10^{-8}$
a.u. of electric field | $E_{\rm h}/ea_{0}$ | $5.142\,206\,52(11)\times 10^{11}$ | V m-1 | $2.2\times 10^{-8}$
a.u. of electric field gradient | $E_{\rm h}/ea_{0}^{2}$ | $9.717\,362\,00(21)\times 10^{21}$ | V m-2 | $2.2\times 10^{-8}$
a.u. of electric dipole moment | $ea_{0}$ | $8.478\,353\,26(19)\times 10^{-30}$ | C m | $2.2\times 10^{-8}$
a.u. of electric quadrupole moment | $ea_{0}^{2}$ | $4.486\,551\,331(99)\times 10^{-40}$ | C m2 | $2.2\times 10^{-8}$
a.u. of electric polarizability | $e^{2}a_{0}^{2}/E_{\rm h}$ | $1.648\,777\,2754(16)\times 10^{-41}$ | C2 m2 J-1 | $9.7\times 10^{-10}$
a.u. of 1st hyperpolarizability | $e^{3}a_{0}^{3}/E_{\rm h}^{2}$ | $3.206\,361\,449(71)\times 10^{-53}$ | C3 m3 J-2 | $2.2\times 10^{-8}$
a.u. of 2nd hyperpolarizability | $e^{4}a_{0}^{4}/E_{\rm h}^{3}$ | $6.235\,380\,54(28)\times 10^{-65}$ | C4 m4 J-3 | $4.4\times 10^{-8}$
a.u. of magnetic flux density | $\hbar/ea_{0}^{2}$ | $2.350\,517\,464(52)\times 10^{5}$ | T | $2.2\times 10^{-8}$
a.u. of magnetic dipole moment: $2\mu_{\rm B}$ | $\hbar e/m_{\rm e}$ | $1.854\,801\,936(41)\times 10^{-23}$ | J T-1 | $2.2\times 10^{-8}$
a.u. of magnetizability | $e^{2}a_{0}^{2}/m_{\rm e}$ | $7.891\,036\,607(13)\times 10^{-29}$ | J T-2 | $1.6\times 10^{-9}$
a.u. of permittivity: $10^{7}/c^{2}$ | $e^{2}/a_{0}E_{\rm h}$ | $1.112\,650\,056\ldots\times 10^{-10}$ | F m-1 | exact
Table 46: The values of some energy equivalents derived from the relations
$E=mc^{2}=hc/\lambda=h\nu=kT$, and based on the 2010 CODATA adjustment of the
values of the constants; 1 eV $=(e/{\rm C})$ J, 1 u $=m_{\rm
u}=\textstyle{1\over 12}m(^{12}{\rm C})=10^{-3}$ kg mol${}^{-1}\\!/N_{\rm A}$,
and $E_{\rm h}=2R_{\rm\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ is the Hartree
energy (hartree). Relevant unit
---
| J | kg | m-1 | Hz
1 J | $(1\ {\rm J})=$ | (1 J)/$c^{2}=$ | (1 J)/$hc=$ | (1 J)/$h=$
| 1 J | $1.112\,650\,056\ldots\times 10^{-17}$ kg | $5.034\,117\,01(22)\times 10^{24}$ m-1 | $1.509\,190\,311(67)\times 10^{33}$ Hz
1 kg | (1 kg)$c^{2}=$ | $(1\ {\rm kg})=$ | (1 kg)$c/h=$ | (1 kg)$c^{2}/h=$
| $8.987\,551\,787\ldots\times 10^{16}$ J | 1 kg | $4.524\,438\,73(20)\times 10^{41}$ m-1 | $1.356\,392\,608(60)\times 10^{50}$ Hz
1 m-1 | (1 m${}^{-1})hc=$ | (1 m${}^{-1})h/c=$ | $(1$ m${}^{-1})=$ | (1 m${}^{-1})c=$
| $1.986\,445\,684(88)\times 10^{-25}$ J | $2.210\,218\,902(98)\times 10^{-42}$ kg | 1 m-1 | $299\,792\,458$ Hz
1 Hz | (1 Hz)$h=$ | (1 Hz)$h/c^{2}=$ | (1 Hz)/$c=$ | $(1$ Hz$)=$
| $6.626\,069\,57(29)\times 10^{-34}$ J | $7.372\,496\,68(33)\times 10^{-51}$ kg | $3.335\,640\,951\ldots\times 10^{-9}$ m-1 | 1 Hz
1 K | (1 K)$k=$ | (1 K)$k/c^{2}=$ | (1 K)$k/hc=$ | (1 K)$k/h=$
| $1.380\,6488(13)\times 10^{-23}$ J | $1.536\,1790(14)\times 10^{-40}$ kg | $69.503\,476(63)$ m-1 | $2.083\,6618(19)\times 10^{10}$ Hz
1 eV | (1 eV) = | $(1~{}{\rm eV})/c^{2}=$ | $(1~{}{\rm eV})/hc=$ | $(1~{}{\rm eV})/h=$
| $1.602\,176\,565(35)\times 10^{-19}$ J | $1.782\,661\,845(39)\times 10^{-36}$ kg | $8.065\,544\,29(18)\times 10^{5}$ m-1 | $2.417\,989\,348(53)\times 10^{14}$ Hz
1 u | $(1~{}{\rm u})c^{2}=$ | (1 u) = | $(1~{}{\rm u})c/h=$ | $(1~{}{\rm u})c^{2}/h=$
| $1.492\,417\,954(66)\times 10^{-10}$ J | $1.660\,538\,921(73)\times 10^{-27}$ kg | $7.513\,006\,6042(53)\times 10^{14}$ m-1 | $2.252\,342\,7168(16)\times 10^{23}$ Hz
1 $E_{\rm h}$ | $(1~{}E_{\rm h})=$ | $(1~{}E_{\rm h})/c^{2}=$ | $(1~{}E_{\rm h})/hc=$ | $(1~{}E_{\rm h})/h=$
| $4.359\,744\,34(19)\times 10^{-18}$ J | $4.850\,869\,79(21)\times 10^{-35}$ kg | $2.194\,746\,313\,708(11)\times 10^{7}$ m-1 | $6.579\,683\,920\,729(33)\times 10^{15}$ Hz
Table 47: The values of some energy equivalents derived from the relations
$E=mc^{2}=hc/\lambda=h\nu=kT$, and based on the 2010 CODATA adjustment of the
values of the constants; 1 eV $=(e/{\rm C})$ J, 1 u $=m_{\rm
u}=\textstyle{1\over 12}m(^{12}{\rm C})=10^{-3}$ kg mol${}^{-1}\\!/N_{\rm A}$,
and $E_{\rm h}=2R_{\rm\infty}hc=\alpha^{2}m_{\rm e}c^{2}$ is the Hartree
energy (hartree). Relevant unit
---
| K | eV | u | $E_{\rm h}$
1 J | (1 J)/$k=$ | (1 J) = | (1 J)/$c^{2}$ = | (1 J) =
| $7.242\,9716(66)\times 10^{22}$ K | $6.241\,509\,34(14)\times 10^{18}$ eV | $6.700\,535\,85(30)\times 10^{9}$ u | $2.293\,712\,48(10)\times 10^{17}$ $E_{\rm h}$
1 kg | (1 kg)$c^{2}/k=$ | (1 kg)$c^{2}$ = | (1 kg) = | (1 kg)$c^{2}=$
| $6.509\,6582(59)\times 10^{39}$ K | $5.609\,588\,85(12)\times 10^{35}$ eV | $6.022\,141\,29(27)\times 10^{26}$ u | $2.061\,485\,968(91)\times 10^{34}$ $E_{\rm h}$
1 m-1 | (1 m${}^{-1})hc/k=$ | (1 m${}^{-1})hc=$ | (1 m${}^{-1})h/c$ = | (1 m${}^{-1})hc=$
| $1.438\,7770(13)\times 10^{-2}$ K | $1.239\,841\,930(27)\times 10^{-6}$ eV | $1.331\,025\,051\,20(94)\times 10^{-15}$ u | $4.556\,335\,252\,755(23)\times 10^{-8}$ $E_{\rm h}$
1 Hz | (1 Hz)$h/k=$ | (1 Hz)$h=$ | (1 Hz)$h/c^{2}$ = | (1 Hz)$h=$
| $4.799\,2434(44)\times 10^{-11}$ K | $4.135\,667\,516(91)\times 10^{-15}$ eV | $4.439\,821\,6689(31)\times 10^{-24}$ u | $1.519\,829\,846\,0045(76)\times 10^{-16}$ $E_{\rm h}$
1 K | $(1$ K$)=$ | (1 K)$k=$ | (1 K)$k/c^{2}=$ | (1 K)$k=$
| 1 K | $8.617\,3324(78)\times 10^{-5}$ eV | $9.251\,0868(84)\times 10^{-14}$ u | $3.166\,8114(29)\times 10^{-6}$ $E_{\rm h}$
1 eV | (1 eV)/$k=$ | $(1$ eV$)=$ | $(1~{}{\rm eV})/c^{2}=$ | $(1~{}{\rm eV})=$
| $1.160\,4519(11)\times 10^{4}$ K | 1 eV | $1.073\,544\,150(24)\times 10^{-9}$ u | $3.674\,932\,379(81)\times 10^{-2}$ $E_{\rm h}$
1 u | $(1~{}{\rm u})c^{2}/k=$ | $(1~{}{\rm u})c^{2}=$ | $(1$ u$)=$ | $(1~{}{\rm u})c^{2}=$
| $1.080\,954\,08(98)\times 10^{13}$ K | $931.494\,061(21)\times 10^{6}$ eV | 1 u | $3.423\,177\,6845(24)\times 10^{7}$ $E_{\rm h}$
1 $E_{\rm h}$ | $(1~{}E_{\rm h})/k=$ | $(1~{}E_{\rm h})=$ | $(1~{}E_{\rm h})/c^{2}=$ | $(1~{}E_{\rm h})=$
| $3.157\,7504(29)\times 10^{5}$ K | $27.211\,385\,05(60)$ eV | $2.921\,262\,3246(21)\times 10^{-8}$ u | $1~{}E_{\rm h}$
## XV Summary and Conclusion
The focus of this section is (i) comparison of the 2010 and 2006 recommended
values of the constants and identification of those new results that have
contributed most to the changes in the 2006 values; (ii) presentation of
several conclusions that can be drawn from the 2010 recommended values and the
input data on which they are based; and (iii) identification of new
experimental and theoretical work that can advance our knowledge of the values
of the constants.
Topic (iii) is of special importance in light of the adoption by the 24th
General Conference on Weights and Measures (CGPM) at its meeting in Paris in
October 2011 of Resolution 1 entitled “On the possible future revision of the
International System of Units, the SI,” available on the BIPM Web site at
bipm.org/utils/common/pdf/24_CGPM_Resolutions.pdf.
In brief, this resolution notes the intention of the CIPM to propose, possibly
to the 25th CGPM in 2014, a revision of the SI. The “New SI,” as it is called
to distinguish it from the current SI, will be the system of units in which
seven reference constants, including the Planck constant $h$, elementary
charge $e$, Boltzmann constant $k$, and Avogadro constant $N_{\rm A}$, have
exact assigned values. Resolution 1 also looks to CODATA to provide the
necessary values of these four constants for the new definition. Details of
the proposed New SI may be found in Mills _et al._ (2011) and the references
cited therein; see also Taylor (2011); Mohr and Newell (2010).
### XV.1 Comparison of 2010 and 2006 CODATA recommended values
Table 48: Comparison of the 2010 and 2006 CODATA adjustments of the values of the constants by the comparison of the corresponding recommended values of a representative group of constants. Here $D_{\rm r}$ is the 2010 value minus the 2006 value divided by the standard uncertainty $u$ of the 2006 value (i.e., $D_{\rm r}$ is the change in the value of the constant from 2006 to 2010 relative to its 2006 standard uncertainty). Quantity | 2010 rel. std. | Ratio 2006 $u_{\rm r}$ | $D_{\rm r}$
---|---|---|---
| uncert. $u_{\rm r}$ | to 2010 $u_{\rm r}$ |
$\alpha$ | $3.2\times 10^{-10}$ | $2.1$ | $6.5$
$R_{\rm K}$ | $3.2\times 10^{-10}$ | $2.1$ | $-6.5$
$a_{\rm 0}$ | $3.2\times 10^{-10}$ | $2.1$ | $6.5$
$\lambda_{\rm C}$ | $6.5\times 10^{-10}$ | $2.1$ | $6.5$
$r_{\rm e}$ | $9.7\times 10^{-10}$ | $2.1$ | $6.5$
$\sigma_{\rm e}$ | $1.9\times 10^{-9}$ | $2.1$ | $6.5$
$h$ | $4.4\times 10^{-8}$ | $1.1$ | $1.9$
$m_{\rm e}$ | $4.4\times 10^{-8}$ | $1.1$ | $1.7$
$m_{\rm h}$ | $4.4\times 10^{-8}$ | $1.1$ | $1.7$
$m_{\mbox{\scriptsize{{a}}}}$ | $4.4\times 10^{-8}$ | $1.1$ | $1.7$
$N_{\rm A}$ | $4.4\times 10^{-8}$ | $1.1$ | $-1.7$
$E_{\rm h}$ | $4.4\times 10^{-8}$ | $1.1$ | $1.9$
$c_{1}$ | $4.4\times 10^{-8}$ | $1.1$ | $1.9$
$e$ | $2.2\times 10^{-8}$ | $1.1$ | $1.9$
$K_{\rm J}$ | $2.2\times 10^{-8}$ | $1.1$ | $-1.8$
$F$ | $2.2\times 10^{-8}$ | $1.1$ | $-1.4$
$\gamma^{\,\prime}_{\rm p}$ | $2.5\times 10^{-8}$ | $1.1$ | $-1.3$
$\mu_{\rm B}$ | $2.2\times 10^{-8}$ | $1.1$ | $2.3$
$\mu_{\rm N}$ | $2.2\times 10^{-8}$ | $1.1$ | $2.3$
$\mu_{\rm e}$ | $2.2\times 10^{-8}$ | $1.1$ | $-2.3$
$\mu_{\rm p}$ | $2.4\times 10^{-8}$ | $1.1$ | $2.2$
$R$ | $9.1\times 10^{-7}$ | $1.9$ | $-0.7$
$k$ | $9.1\times 10^{-7}$ | $1.9$ | $-0.7$
$V_{\rm m}$ | $9.1\times 10^{-7}$ | $1.9$ | $-0.7$
$c_{2}$ | $9.1\times 10^{-7}$ | $1.9$ | $0.7$
$\sigma$ | $3.6\times 10^{-6}$ | $1.9$ | $-0.7$
$G$ | $1.2\times 10^{-4}$ | $0.8$ | $-0.7$
$R_{\infty}$ | $5.0\times 10^{-12}$ | $1.3$ | $0.2$
$m_{\rm e}/m_{\rm p}$ | $4.1\times 10^{-10}$ | $1.1$ | $0.0$
$m_{\rm e}/m_{\mbox{\scriptsize{{m}}}}$ | $2.5\times 10^{-8}$ | $1.0$ | $-0.4$
$A_{\rm r}({\rm e})$ | $4.0\times 10^{-10}$ | $1.1$ | $0.1$
$A_{\rm r}({\rm p})$ | $8.9\times 10^{-11}$ | $1.2$ | $0.4$
$A_{\rm r}({\rm n})$ | $4.2\times 10^{-10}$ | $1.0$ | $0.1$
$A_{\rm r}({\rm d})$ | $3.8\times 10^{-11}$ | $1.0$ | $-0.2$
$A_{\rm r}({\rm t})$ | $8.2\times 10^{-10}$ | $1.0$ | $0.0$
$A_{\rm r}({\rm h})$ | $8.3\times 10^{-10}$ | $1.0$ | $-0.2$
$A_{\rm r}({\mbox{{a}}})$ | $1.5\times 10^{-11}$ | $1.0$ | $0.0$
$d_{\rm 220}$ | $1.6\times 10^{-8}$ | $1.6$ | $-1.0$
$g_{\rm e}$ | $2.6\times 10^{-13}$ | $2.8$ | $0.5$
$g_{\mbox{\scriptsize{{m}}}}$ | $6.3\times 10^{-10}$ | $1.0$ | $-0.3$
$\mu_{\rm p}/\mu_{\rm B}$ | $8.1\times 10^{-9}$ | $1.0$ | $0.0$
$\mu_{\rm p}/\mu_{\rm N}$ | $8.2\times 10^{-9}$ | $1.0$ | $0.0$
$\mu_{\rm n}/\mu_{\rm N}$ | $2.4\times 10^{-7}$ | $1.0$ | $0.0$
$\mu_{\rm d}/\mu_{\rm N}$ | $8.4\times 10^{-9}$ | $1.0$ | $0.0$
$\mu_{\rm e}/\mu_{\rm p}$ | $8.1\times 10^{-9}$ | $1.0$ | $0.0$
$\mu_{\rm n}/\mu_{\rm p}$ | $2.4\times 10^{-7}$ | $1.0$ | $0.0$
$\mu_{\rm d}/\mu_{\rm p}$ | $7.7\times 10^{-9}$ | $1.0$ | $0.0$
Table 48 compares the 2010 and 2006 recommended values of a representative
group of constants. The fact that the values of many constants are obtained
from expressions proportional to the fine-structure constant $\alpha$, Planck
constant $h$, or molar gas constant $R$ raised to various powers leads to the
regularities observed in the numbers in columns 2 to 4. For example, the first
six quantities are obtained from expressions proportional to $\alpha^{a}$,
where $|a|=1,~{}2,~{}3$, or $6$. The next 15 quantities, $h$ through the
magnetic moment of the proton $\mu_{\rm p}$, are calculated from expressions
containing the factor $h^{a}$, where $|a|=1$ or $1/2$. And the five quantities
$R$ through the Stefan Boltzmann constant $\sigma$ are proportional to
$R^{a}$, where $|a|=1$ or $4$.
Further comments on some of the entries in Table 48 are as follows.
(i) The large shift in the 2006 recommended value of $\alpha$ is mainly due to
the discovery and correction of an error in the numerically calculated value
of the eighth-order coefficient $A_{1}^{(8)}$ in the theoretical expression
for $a_{\rm e}$; see Sec. V.1.1. Its reduction in uncertainty is due to two
new results. The first is the 2008 improved value of $a_{\rm e}$ obtained at
Harvard University with a relative standard uncertainty of $2.4\times
10^{-10}$ compared to the $7.0\times 10^{-10}$ uncertainty of the earlier
Harvard result used in the 2006 adjustment. The second result is the 2011
improved LKB atom-recoil value of $h/m(^{87}{\rm Rb})$ with an uncertainty of
$1.2\times 10^{-9}$ compared to the $1.3\times 10^{-8}$ uncertainty of the
earlier LKB result used in 2006. The much reduced uncertainty of $g_{\rm e}$
is also due to the improved value of $\alpha$.
(ii) The change in the 2006 recommended value of $h$ is due to the 2011 IAC
result for $N_{\rm A}$ with a relative standard uncertainty of $3.0\times
10^{-8}$ obtained using ${}^{28}{\rm Si}$ enriched single crystals. It
provides a value of $h$ with the same uncertainty, which is smaller than the
$3.6\times 10^{-8}$ uncertainty of the value of $h$ from the 2007 NIST watt-
balance measurement of $K_{\rm J}^{2}R_{\rm K}$; the latter played the
dominant role in determining the 2006 recommended value. The two differ by
about 18 parts in $10^{8}$, resulting in a shift of the 2006 recommended value
by nearly twice its uncertainty. In the 2006 adjustment inconsistencies among
some of the electrical and silicon crystal data (all involving natural
silicon) led the Task Group to increase the uncertainties of these data by the
multiplicative factor 1.5 to reduce the inconsistencies to an acceptable
level. In the 2010 adjustment, inconsistencies among the data that determine
$h$ are reduced to an acceptable level by using a multiplicative factor of 2.
Consequently the uncertainties of the 2006 and 2010 recommended values of $h$
do not differ significantly.
(iii) The 2006 recommended value of the molar gas constant $R$ was determined
by the 1988 NIST speed-of-sound result with a relative standard uncertainty of
$1.8\times 10^{-6}$, and to a much lesser extent the 1979 NPL speed-of-sound
result with an uncertainty of $8.4\times 10^{-6}$ obtained with a rather
different type of apparatus. The six new data of potential interest related to
$R$ that became available during the 4 years between the 2006 and 2010
adjustments have uncertainties ranging from $1.2\times 10^{-6}$ to $12\times
10^{-6}$ and agree with each other as well as with the NIST and NPL values.
Further, the self-sensitivity coefficients of four of the six were
sufficiently large for them to be included in the 2010 final adjustment, and
they are responsible for the small shift in the 2006 recommended value and the
reduction of its uncertainty by nearly a factor of 2.
(iv) Other constants in Table 48 whose changes are worth noting are the
Rydberg constant $R_{\infty}$, proton relative atomic mass $A_{\rm r}({\rm
p})$, and {220} natural Si lattice spacing $d_{220}$. The reduction in
uncertainty of $R_{\infty}$ is due to improvements in the theory of H and D
energy levels and the 2010 LKB result for the $1{\rm S_{1/2}}-3{\rm S_{1/2}}$
transition frequency in hydrogen with a relative standard uncertainty of
$4.4\times 10^{-12}$. For $A_{\rm r}({\rm p})$, the reduction in uncertainty
is due to the 2008 Stockholm University (SMILETRAP) result for the ratio of
the cyclotron frequency of the excited hydrogen molecular ion to that of the
deuteron, $f_{\rm c}({\rm H_{2}^{+*}})/f_{\rm c}({\rm d})$, with a relative
uncertainty of $1.7\times 10^{-10}$. The changes in $d_{220}$ arise from the
omission of the 1999 PTB result for $h/m_{\rm n}d_{220}({\scriptstyle{\rm
W}04})$, the 2004 NMIJ result for $d_{220}({\scriptstyle{\rm NR3}})$, the 2007
INRIM results for $d_{220}({\scriptstyle{\rm W4.2a}})$, and
$d_{220}({\scriptstyle{\rm MO^{*}}})$, and the inclusion of the new 2008 INRIM
result for $d_{220}({\scriptstyle{\rm MO^{*}}})$ as well as the new 2009 INRIM
results for $d_{220}({\scriptstyle{\rm W}04})$ and $d_{220}({\scriptstyle{\rm
W4.2a}})$.
### XV.2 Some implications of the 2010 CODATA recommended values and
adjustment for metrology and physics
_Conventional electric units_. The adoption of the conventional values $K_{\rm
J-90}=483\,597.9~{}{\rm GHz/V}$ and $R_{\rm K-90}=25\,812.807~{}{\Omega}$ for
the Josephson and von Klitzing constants in 1990 can be viewed as establishing
conventional, practical units of voltage and resistance, $V_{90}$ and
${\it\Omega}_{90}$, given by $V_{90}=(K_{\rm J-90}/K_{\rm J})$ V and
${\it\Omega}_{90}=(R_{\rm K}/R_{\rm K-90})~{}{\rm\Omega}$. Other conventional
electric units follow from $V_{90}$ and ${\it\Omega}_{90}$, for example,
$A_{90}=V_{90}/{\it\Omega}_{90}$, $C_{90}=A_{90}$ s, $W_{90}=A_{90}V_{90}$,
$F_{90}=C_{90}/V_{90}$, and $H_{90}={\it\Omega}_{90}$ s, which are the
conventional, practical units of current, charge, power, capacitance, and
inductance, respectively Taylor and Mohr (2001). For the relations between
$K_{\rm J}$ and $K_{\rm J-90}$, and $R_{\rm K}$ and $R_{\rm K-90}$, the 2010
adjustment gives
$\displaystyle K_{\rm J}$ $\displaystyle=$ $\displaystyle K_{\rm
J-90}[1-6.3(2.2)\times 10^{-8}]\,,$ (286) $\displaystyle R_{\rm K}$
$\displaystyle=$ $\displaystyle R_{\rm K-90}[1+1.718(32)\times 10^{-8}]\,,$
(287)
which lead to
$\displaystyle V_{90}$ $\displaystyle=$ $\displaystyle[1+6.3(2.2)\times
10^{-8}]~{}{\rm V},$ (288) $\displaystyle{\it\Omega}_{90}$ $\displaystyle=$
$\displaystyle[1+1.718(32)\times 10^{-8}]~{}{\rm\Omega},$ (289) $\displaystyle
A_{90}$ $\displaystyle=$ $\displaystyle[1-4.6(2.2)\times 10^{-8}]~{}{\rm
A}\,,$ (290) $\displaystyle C_{90}$ $\displaystyle=$
$\displaystyle[1-4.6(2.2)\times 10^{-8}]~{}{\rm C}\,,$ (291) $\displaystyle
W_{90}$ $\displaystyle=$ $\displaystyle[1+10.8(5.0)\times 10^{-8}]~{}{\rm
W}\,,$ (292) $\displaystyle F_{90}$ $\displaystyle=$
$\displaystyle[1-1.718(32)\times 10^{-8}]~{}{\rm F}\,,$ (293) $\displaystyle
H_{90}$ $\displaystyle=$ $\displaystyle[1+1.718(32)\times 10^{-8}]~{}{\rm
H}\,.$ (294)
Equations (288) and (289) show that $V_{90}$ exceeds V and ${\it\Omega}_{90}$
exceeds $\Omega$ by $6.3(2.2)\times 10^{-8}$ and $1.718(32)\times 10^{-8}$,
respectively. This means that measured voltages and resistances traceable to
the Josephson effect and $K_{\rm J-90}$ and the quantum Hall effect and
$R_{\rm K-90}$, respectively, are too small relative to the SI by these same
fractional amounts. However, these differences are well within the $40\times
10^{-8}$ uncertainty assigned to $V_{90}/$V and the $10\times 10^{-8}$
uncertainty assigned to ${\it\Omega}_{90}/\Omega$ by the Consultative
Committee for Electricity and Magnetism (CCEM) of the CIPM Quinn (2001, 1989).
_Josephson and quantum Hall effects_. Although there is extensive theoretical
and experimental evidence for the exactness of the Josephson and quantum-Hall-
effect relations $K_{\rm J}=2e/h$ and $R_{\rm K}=h/e^{2}$, and some of the
input data available for the 2010 adjustment provide additional supportive
evidence for these expressions, some other data are not supportive. This
dichotomy reflects the rather significant inconsistencies among a few key
data, particularly the highly accurate IAC enriched silicon XRCD result for
$N_{\rm A}$, and the comparably accurate NIST watt-balance result for $K_{\rm
J}^{2}R_{\rm K}$, and will only be fully resolved when the inconsistencies are
reconciled.
_The New SI_. Implementation of the New SI requires that the four reference
constants $h$, $e$, $k$, and $N_{\rm A}$ must be known with sufficiently small
uncertainties to meet current and future measurement needs. However, of equal
if not greater importance, the causes of any inconsistencies among the data
that provide their values must be understood. Although the key data that
provide the 2010 recommended value of $k$ would appear to be close to meeting
both requirements, this is not the case for $h$, $e$, and $N_{\rm A}$, which
are in fact interrelated. We have
$\displaystyle N_{\rm A}h$ $\displaystyle=$ $\displaystyle{cA_{\rm r}({\rm
e})M_{\rm u}\alpha^{2}\over 2R_{\infty}}\,,$ (295) $\displaystyle e$
$\displaystyle=$ $\displaystyle\left({2\alpha h\over\mu_{0}c}\right)^{1/2}\,.$
(296)
Since the combined relative standard uncertainty of the 2010 recommended
values of the constants on the right-hand-side of Eq. (295) is only $7.0\times
10^{-10}$, a measurement of $h$ with a given relative uncertainty, even as
small as $5\times 10^{-9}$, determines $N_{\rm A}$ with essentially the same
relative uncertainty. Further, since the recommended value of $\alpha$ has a
relative uncertainty of only $3.2\times 10^{-10}$, based on Eq. (296) the
relative uncertainty of $e$ will be half that of $h$ or $N_{\rm A}$. For these
reasons, the 2010 recommended values of $h$ and $N_{\rm A}$ have the same
$4.4\times 10^{-8}$ relative uncertainty, and the uncertainty of the
recommended value of $e$ is $2.2\times 10^{-8}$. However, these uncertainties
are twice as large as they would have been if there were no disagreement
between the watt-balance values of $h$ and the enriched silicon XRCD value of
$N_{\rm A}$. This disagreement led to an increase in the uncertainties of the
relevant data by a factor of 2. More specifically, if the data had been
consistent the uncertainties of the recommended values of $h$ and $N_{\rm A}$
would be $2.2\times 10^{-8}$ and $1.1\times 10^{-8}$ for $e$. Because these
should be sufficiently small for the New SI to be implemented, the
significance of the disagreement and the importance of measurements of $h$ and
$N_{\rm A}$ are apparent.
_Proton radius_. The proton rms charge radius $r_{\rm p}$ determined from the
Lamb shift in muonic hydrogen disagrees significantly with values determined
from H and D transition frequencies as well as from electron-proton scattering
experiments. Although the uncertainty of the muonic hydrogen value is
significantly smaller than the uncertainties of these other values, its
negative impact on the internal consistency of the theoretically predicted and
experimentally measured frequencies, as well as on the value of the Rydberg
constant, was deemed so severe that the only recourse was to not include it in
the final least-squares adjustment on which the 2010 recommended values are
based.
_Muon magnetic moment anomaly_. Despite extensive new theoretical work, the
long-standing significant difference between the theoretically predicted,
standard-model value of $a_{\mbox{\scriptsize{{m}}}}$ and the experimentally
determined value remains unresolved. Because the difference is from 3.3 to
possibly 4.5 times the standard uncertainty of the difference, depending on
the way the all-important hadronic contribution to the theoretical expression
for $a_{\mbox{\scriptsize{{m}}}}$ is evaluated, the theory was not
incorporated in the 2010 adjustment. The recommended values of
$a_{\mbox{\scriptsize{{m}}}}$ and those of other constants that depend on it
are, therefore, based on experiment.
_Electron magnetic moment anomaly, fine-structure constant, and QED_. The most
accurate value of the fine-structure constant $\alpha$ currently available
from a single experiment has a relative standard uncertainty of $3.7\times
10^{-10}$; it is obtained by equating the QED theoretical expression for the
electron magnetic moment anomaly $a_{\rm e}$ and the most accurate
experimental value of $a_{\rm e}$, obtained from measurements on a single
electron in a Penning trap. This value of $\alpha$ is in excellent agreement
with a competitive experimental value with an uncertainty of $6.6\times
10^{-10}$. Because the latter is obtained from the atom-recoil determination
of the quotient $h/m(^{87}{\rm Rb})$ using atom-interferometry and is only
weakly dependent on QED theory, the agreement provides one of the most
significant confirmations of quantum electrodynamics.
_Newtonian constant of gravitation_. The situation regarding measurements of
$G$ continues to be problematic and has become more so in the past 4 years.
Two new results with comparatively small uncertainties have become available
for the 2010 adjustment, leading to an increase in the scatter among the now
11 values of $G$. This has resulted in a $20\,\%$ increase in the uncertainty
of the 2010 recommended value compared to that of its 2006 predecessor.
Clearly, there is a continuing problem for the determination of this
important, but poorly-known, fundamental constant; the uncertainty of the 2010
recommended value is now 120 parts in $10^{6}$.
### XV.3 Suggestions for future work
For evaluation of the fundamental constants, it is desirable not only to have
multiple results with competitive uncertainties for a given quantity, but also
to have one or more results obtained by a different method. If the term
“redundant” is used to describe such an ideal set of data, there is usually
only limited redundancy among the key data available for any given CODATA
adjustment.
With this in mind, based on the preceding discussion, our suggestions are as
follows.
(i) Resolution of the disagreement between the most accurate watt-balance
result for $K_{\rm J}^{2}R_{\rm K}$ and the XRCD result for $N_{\rm A}$.
Approaches to solving this problem might include new measurements of $K_{\rm
J}^{2}R_{\rm K}$ using watt balances of different design (or their equivalent)
with uncertainties at the 2 to 3 parts in $10^{8}$ level, a thorough review by
the researchers involved of their existing measurements of this quantity,
tests of the exactness of the relations $K_{\rm J}=2e/h$ and $R_{\rm
K}=h/e^{2}$, independent measurements of the isotopic composition of the
enriched silicon crystals and their $d_{220}$ lattice spacing used in the
determination of $N_{\rm A}$ (these are the two principal quantities for which
only one measurement exists), and a thorough review by the researchers
involved of the many corrections required to obtain $N_{\rm A}$ from the
principal quantities measured.
(ii) Measurements of $k$ (and related quantities such as $k/h$) with
uncertainties at the 1 to 3 parts in $10^{6}$ level using the techniques of
dielectric gas thermometry, refractive index gas thermometry, noise
thermometry, and Doppler broadening, because these methods are so very
different from acoustic gas thermometry, which is the dominant method used to
date.
(iii) Resolution of the discrepancy between the muonic hydrogen inferred value
of $r_{\rm p}$ and the spectroscopic value from H and D transition
frequencies. Work underway on frequency measurements in hydrogen as well as
the analysis of $\mbox{{m}}^{-}{\rm p}$ and $\mbox{{m}}^{-}{\rm d}$ data and
possible measurements in $\mbox{{m}}^{-}{\rm h}$ and
$\mbox{{m}}^{-}\mbox{{a}}$ should provide additional useful information.
Independent evaluations of electron scattering data to determine $r_{\rm p}$
are encouraged as well as verification of the theory of H, D, and muonic
hydrogen-like energy levels.
(iv) Independent calculation of the eighth- and tenth-order coefficients in
the QED expression for $a_{\rm e}$, in order to increase confidence in the
value of $\alpha$ from $a_{\rm e}$.
(v) Resolution of the disagreement between the theoretical expression for
$a_{\mbox{\scriptsize{{m}}}}$ and its experimental value. This discrepancy
along with the discrepancy between theory and experiment in muonic hydrogen
are two important problems in muon-related physics.
(vi) Determinations of $G$ with an uncertainty of one part in $10^{5}$ using
new and innovative approaches that might resolve the disagreements among the
measurements made within the past three decades.
## XVI Acknowledgments
We gratefully acknowledge the help of our many colleagues throughout the world
who provided the CODATA Task Group on Fundamental Constants with results prior
to formal publication and for promptly and patiently answering our many
questions about their work. We wish to thank Barry M. Wood, the current chair
of the Task Group, as well as our fellow Task-Group members, for their
invaluable guidance and suggestions during the course of the 2010 adjustment
effort.
## Nomenclature
AMDC | Atomic Mass Data Center, Centre de Spectrométrie Nucléaire et de Spectrométrie de Masse (CSNSM), Orsay, France
---|---
$A_{\rm r}(X)$ | Relative atomic mass of $X$: $A_{\rm r}(X)=m(X)/m_{\rm u}$
$A_{90}$ | Conventional unit of electric current:
$A_{90}=V_{90}/{\it\Omega}_{90}$
Å∗ | Ångström-star: ${\rm\lambda}({\rm WK}{\mbox{{a}}}_{1})=0.209\,010\,0\ {\rm\AA}^{\ast}$
$a_{\rm e}$ | Electron magnetic moment anomaly:
$a_{\rm e}=(|g_{\rm e}|-2$)/2
$a_{\mbox{\scriptsize{{m}}}}$ | Muon magnetic moment anomaly:
$a_{\mbox{\scriptsize{{m}}}}=(|g_{\mbox{\scriptsize{{m}}}}|-2$)/2
BIPM | International Bureau of Weights and Measures, Sèvres, France
BNL | Brookhaven National Laboratory, Upton, New York, USA
CERN | European Organization for Nuclear Research, Geneva, Switzerland
CIPM | International Committee for Weights and Measures
CODATA | Committee on Data for Science and Technology of the International Council for Science
$CPT$ | Combined charge conjugation, parity inversion, and time reversal
$c$ | Speed of light in vacuum
d | Deuteron (nucleus of deuterium D, or 2H)
$d_{220}$ | $\\{220\\}$ lattice spacing of an ideal crystal of naturally occurring silicon
$d_{220}({\scriptstyle X})$ | $\\{220\\}$ lattice spacing of crystal $X$ of naturally occurring silicon
$E_{\rm b}$ | Binding energy
e | Symbol for either member of the electron-positron pair; when necessary, e- or e+ is used to indicate the electron or positron
$e$ | Elementary charge: absolute value of the charge of the electron
$F$ | Faraday constant: $F$ = $N_{\rm A}e$
FSU | Florida State University, Tallahassee, Florida, USA
FSUJ | Friedrich-Schiller University, Jena, Germany
${\cal F}_{90}$ | ${\cal F}_{90}=(F/A_{90})$ A
$G$ | Newtonian constant of gravitation
$g$ | Local acceleration of free fall
$g_{\rm d}$ | Deuteron $g$-factor: $g_{\rm d}=\mu_{\rm d}/\mu_{\rm N}$
$g_{\rm e}$ | Electron $g$-factor: $g_{\rm e}=2\mu_{\rm e}/\mu_{\rm B}$
$g_{\rm p}$ | Proton $g$-factor: $g_{\rm p}=2\mu_{\rm p}/\mu_{\rm N}$
$g^{\prime}_{\rm p}$ | Shielded proton $g$-factor: $g_{\rm p}^{\prime}=2\mu_{\rm p}^{\prime}/\mu_{\rm N}$
$g_{\rm t}$ | Triton $g$-factor: $g_{\rm t}=2\mu_{\rm t}/\mu_{\rm N}$
$g_{X}(Y)$ | $g$-factor of particle $X$ in the ground (1S) state of hydrogenic atom $Y$
$g_{\mbox{\scriptsize{{m}}}}$ | Muon $g$-factor: $g_{\mbox{\scriptsize{{m}}}}=2\mu_{\mbox{\scriptsize{{m}}}}/(e\hbar/2m_{\mbox{\scriptsize{{m}}}})$
GSI | Gesellschaft für Schweironenforschung, Darmstadt, Germany
HD | HD molecule (bound state of hydrogen and deuterium atoms)
HT | HT molecule (bound state of hydrogen and tritium atoms)
h | Helion (nucleus of 3He)
$h$ | Planck constant; $\hbar=h/2\mbox{{p}}$
HarvU | Harvard University, Cambridge, Massachusetts, USA
IAC | International Avogadro Coordination
ILL | Institut Max von Laue-Paul Langevin, Grenoble, France
INRIM | Istituto Nazionale di Ricerca Metrologica, Torino, Italy
IRMM | Institute for Reference Materials and Measurements, Geel, Belgium
KRISS | Korea Research Institute of Standards and Science, Taedok Science Town, Republic of Korea
KR/VN | KRISS-VNIIM collaboration
$K_{\rm J}$ | Josephson constant: $K_{\rm J}=2e/h$
$K_{\rm J-90}$ | Conventional value of the Josephson constant $K_{\rm J}$: $K_{\rm J-90}=483\,597.9$ GHz V-1
$k$ | Boltzmann constant: $k=R/N_{\rm A}$
LAMPF | Clinton P. Anderson Meson Physics Facility at Los Alamos National Laboratory, Los Alamos, New Mexico, USA
LKB | Laboratoire Kastler-Brossel, Paris, France
LK/SY | LKB and SYRTE collaboration
LNE | Laboratoire national de métrologie et d’essais, Trappes, France
METAS | Federal Office of Metrology, Bern-Wabern, Switzerland
MIT | Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
MPQ | Max-Planck-Institut für Quantenoptik, Garching, Germany
$M(X)$ | Molar mass of $X$: $M(X)=A_{\rm r}(X)M_{\rm u}$
Mu | Muonium (${\mbox{{m}}}^{+}{\rm e}^{-}$ atom)
$M_{\rm u}$ | Molar mass constant: $M_{\rm u}=10^{-3}~{}{\rm kg~{}mol^{-1}}$
$m_{\rm u}$ | Unified atomic mass constant: $m_{\rm u}=m(^{12}{\rm C})/12$
$m_{X}$, $m(X)$ | Mass of $X$ (for the electron e, proton p, and other elementary particles, the first symbol is used, i.e., $m_{\rm e}$, $m_{\rm p}$, etc.)
$N_{\rm A}$ | Avogadro constant
NIM | National Institute of Metrology, Beijing, China (People’s Republic of)
NIST | National Institute of Standards and Technology, Gaithersburg, Maryland and Boulder, Colorado, USA
NMI | National Metrology Institute, Lindfield, Australia
NMIJ | National Metrology Institute of Japan, Tsukuba, Japan
NPL | National Physical Laboratory, Teddington, UK
n | Neutron
PTB | Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany
p | Proton
$\overline{\rm p}\,^{A}$He+ | Antiprotonic helium (AHe+ \+ $\overline{\rm p}$ atom, $A=3\mbox{ or }4$)
QED | Quantum electrodynamics
$p(\chi^{2}|\nu)$ | Probability that an observed value of chi-square for $\nu$ degrees of freedom would exceed $\chi^{2}$
$R$ | Molar gas constant
$\overline{R}$ | Ratio of muon anomaly difference frequency to free proton NMR frequency
$R_{\rm B}$ | Birge ratio: $R_{\rm B}=(\chi^{2}/\nu)^{\frac{1}{2}}$
$r_{\rm d}$ | Bound-state rms charge radius of the deuteron
$R_{\rm K}$ | von Klitzing constant: $R_{\rm K}=h/e^{2}$
$R_{\rm K-90}$ | Conventional value of the von Klitzing constant $R_{\rm K}$: $R_{\rm K-90}=25\,812.807~{}{\rm\Omega}$
$r_{\rm p}$ | Bound-state rms charge radius of the proton
$R_{\infty}$ | Rydberg constant: $R_{\infty}=m_{\rm e}c\alpha^{2}/2h$
$r(x_{i},x_{j})$ | Correlation coefficient of estimated values $x_{i}$ and $x_{j}$: $r(x_{i},x_{j})=u(x_{i},x_{j})/[u(x_{i})u(x_{j})]$
$S_{\rm c}$ | Self-sensitivity coefficient
SI | Système international d’unités (International System of Units)
StanfU | Stanford University, Stanford, California, USA
StockU | Stockholm University, Stockholm, Sweden
StPtrsb | St. Petersburg, Russian Federation
SYRTE | Systèmes de référence Temps Espace, Paris, France
$T$ | Thermodynamic temperature
Type A | Uncertainty evaluation by the statistical analysis of series of observations
Type B | Uncertainty evaluation by means other than the statistical analysis of series of observations
$t_{90}$ | Celsius temperature on the International Temperature Scale of 1990 (ITS-90)
t | Triton (nucleus of tritium T, or 3H)
USus | University of Sussex, Sussex, UK
UWash | University of Washington, Seattle, Washington, USA
u | Unified atomic mass unit (also called the dalton, Da): 1 u = $m_{\rm u}$ = $m(^{12}$C)/12
$u(x_{i})$ | Standard uncertainty (i.e., estimated standard deviation) of an estimated value $x_{i}$ of a quantity $X_{i}$ (also simply $u$)
$u_{\rm r}(x_{i})$ | Relative standard uncertainty of an estimated value $x_{i}$ of a quantity $X_{i}$:
$u_{\rm r}(x_{i})=u(x_{i})/|x_{i}|,\ x_{i}\neq 0$ (also simply $u_{\rm r}$)
$u(x_{i},x_{j})$ | Covariance of estimated values $x_{i}$ and $x_{j}$
$u_{\rm r}(x_{i},x_{j})$ | Relative covariance of estimated values $x_{i}$ and $x_{j}$: $u_{\rm r}(x_{i},x_{j})=u(x_{i},x_{j})/(x_{i}x_{j})$
$V_{\rm m}({\rm Si)}$ | Molar volume of naturally occurring silicon
VNIIM | D. I. Mendeleyev All-Russian Research Institute for Metrology, St. Petersburg, Russian Federation
$V_{90}$ | Conventional unit of voltage based on the Josephson effect and $K_{\rm J-90}$: $V_{90}=(K_{\rm J-90}/K_{\rm J}$) V
$W_{90}$ | Conventional unit of power: $W_{90}=V^{2}_{90}/{\it\Omega}_{90}$
XROI | Combined x-ray and optical interferometer
xu(CuK${\mbox{{a}}}_{1}$) | Cu x unit: ${\rm\lambda}$(CuK${\mbox{{a}}}_{1}$) = 1 537.400 xu(CuK${\mbox{{a}}}_{1}$)
xu(MoK${\mbox{{a}}}_{1}$) | Mo x unit: ${\rm\lambda}$(MoK${\mbox{{a}}}_{1}$) = 707.831 xu(MoK${\mbox{{a}}}_{1}$)
$x(X)$ | Amount-of-substance fraction of $X$
YaleU | Yale University, New Haven, Connecticut, USA
$\alpha$ | Fine-structure constant: $\alpha=e^{2}/4\mbox{{p}}\epsilon_{0}\hbar c\approx 1/137$
a | Alpha particle (nucleus of 4He)
$\it\Gamma^{\prime}_{X-{\rm 90}}$(lo) | $\it\Gamma^{\prime}_{X-{\rm 90}}({\rm lo})=(\gamma_{X}^{\prime}\,A_{\rm 90})$ A-1, $X$ = p or h
$\it\Gamma^{\prime}_{\rm p-90}$(hi) | $\it\Gamma^{\prime}_{\rm p-90}({\rm hi})=(\gamma_{\rm p}^{\prime}/A_{\rm 90})$ A
$\gamma_{\rm p}$ | Proton gyromagnetic ratio: $\gamma_{\rm p}=2\mu_{\rm p}/\hbar$
$\gamma_{\rm p}^{\prime}$ | Shielded proton gyromagnetic ratio: $\gamma_{\rm p}^{\prime}=2\mu^{\prime}_{\rm p}/\hbar$
$\gamma^{\prime}_{\rm h}$ | Shielded helion gyromagnetic ratio: $\gamma^{\prime}_{\rm h}=2|\mu^{\prime}_{\rm h}|/\hbar$
$\Delta\nu_{\rm Mu}$ | Muonium ground-state hyperfine splitting
$\delta_{\rm e}$ | Additive correction to the theoretical expression for the electron magnetic moment anomaly $a_{\rm e}$
$\delta_{\rm Mu}$ | Additive correction to the theoretical expression for the ground-state hyperfine splitting of muonium ${\rm\Delta\nu}_{\rm Mu}$
$\delta_{\overline{\rm p}\,{\rm He}}$ | Additive correction to the theoretical expression for a particular transition frequency of antiprotonic helium
$\delta_{X}(n{\rm L}_{j})$ | Additive correction to the theoretical expression for an energy level of either hydrogen H or deuterium D with quantum numbers $n$, L, and $j$
$\delta_{\mbox{\scriptsize{{m}}}}$ | Additive correction to the theoretical expression for the muon magnetic moment anomaly $a_{\mbox{\scriptsize{{m}}}}$
$\epsilon_{\rm 0}$ | Electric constant: $\epsilon_{\rm 0}=1/\mu_{\rm 0}c^{2}$
$\doteq$ | Symbol used to relate an input datum to its observational equation
${\rm\lambda}({X\,{\rm K}}{\mbox{{a}}}_{1})$ | Wavelength of K${\mbox{{a}}}_{1}$ x-ray line of element $X$
${\rm\lambda}_{\rm meas}$ | Measured wavelength of the 2.2 MeV capture g-ray emitted in the reaction n + p $\rightarrow$ d + g
m | Symbol for either member of the muon-antimuon pair; when necessary, ${\mbox{{m}}}^{-}$ or ${\mbox{{m}}}^{+}$ is used to indicate the negative muon or positive muon
$\mu_{\rm B}$ | Bohr magneton: $\mu_{\rm B}=e\hbar/2m_{\rm e}$
$\mu_{\rm N}$ | Nuclear magneton: $\mu_{\rm N}=e\hbar/2m_{\rm p}$
$\mu_{X}(Y)$ | Magnetic moment of particle $X$ in atom or molecule $Y$.
$\mu_{\rm 0}$ | Magnetic constant: $\mu_{\rm 0}=4{\mbox{{p}}}\times 10^{-7}$ N/A2
$\mu_{X}$, $\mu^{\prime}_{X}$ | Magnetic moment, or shielded magnetic moment, of particle $X$
$\nu$ | Degrees of freedom of a particular adjustment
$\nu(f_{\rm p})$ | Difference between muonium hyperfine splitting Zeeman transition frequencies $\nu_{34}$ and $\nu_{12}$ at a magnetic flux density $B$ corresponding to the free proton NMR frequency $f_{\rm p}$
$\sigma$ | Stefan-Boltzmann constant: $\sigma=2{\mbox{{p}}}^{5}k^{4}/(15h^{3}c^{2})$
t | Symbol for either member of the tau-antitau pair; when necessary, ${\mbox{{t}}}^{-}$ or ${\mbox{{t}}}^{+}$ is used to indicate the negative tau or positive tau
${\rm\chi}^{2}$ | The statistic “chi square”
${\it\Omega}_{90}$ | Conventional unit of resistance based on the quantum Hall effect and $R_{\rm K-90}:{\it\Omega}_{90}=(R_{\rm K}/R_{\rm K-90})~{}{\rm\Omega}$
## References
* Adamuscin _et al._ (2012) Adamuscin, C., S. Dubnicka, and A. Z. Dubnickova, 2012, to be published, Prog. Part. Nuc. Phys.
* Alnis _et al._ (2008) Alnis, J., A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, 2008, Phys. Rev. A 77, 053809.
* AMDC (2003) AMDC, 2003, amdc.in2p3.fr/masstables/Ame2003/
a0p4sqza.cov03.
* AMDC (2006) AMDC, 2006, amdc.in2p3.fr.
* Andreas _et al._ (2011a) Andreas, B., Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, _et al._ , 2011a, Metrologia 48(2), S1.
* Andreas _et al._ (2011b) Andreas, B., Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, _et al._ , 2011b, Phys. Rev. Lett. 106, 030801.
* Angeli (2004) Angeli, I., 2004, At. Data. Nucl. Data Tables 87(2), 185.
* Aoyama _et al._ (2007) Aoyama, T., M. Hayakawa, T. Kinoshita, and M. Nio, 2007, Phys. Rev. Lett. 99, 110406\.
* Aoyama _et al._ (2008) Aoyama, T., M. Hayakawa, T. Kinoshita, and M. Nio, 2008, Phys. Rev. D 77, 053012\.
* Aoyama _et al._ (2011) Aoyama, T., M. Hayakawa, T. Kinoshita, and M. Nio, 2011, Phys. Rev. D 84, 053003\.
* Armstrong and Fitzgerald (2003) Armstrong, T. R., and M. P. Fitzgerald, 2003, Phys. Rev. Lett. 91, 201101.
* Arnoult _et al._ (2010) Arnoult, O., F. Nez, L. Julien, and F. Biraben, 2010, Eur. Phys. J. D 60(2), 243.
* Arrington (2011) Arrington, J., 2011, Phys. Rev. Lett. 107, 119101.
* Arrington _et al._ (2007) Arrington, J., W. Melnitchouk, and J. A. Tjon, 2007, Phys. Rev. C 76, 035205.
* Audi _et al._ (2003) Audi, G., A. H. Wapstra, and C. Thibault, 2003, Nucl. Phys. A 729(1), 337.
* Bagley (1996) Bagley, C. H., 1996, Ph.D. thesis (University of Colorado, Boulder, CO).
* Bagley (2010) Bagley, C. H., 2010, private communication.
* Bagley and Luther (1997) Bagley, C. H., and G. G. Luther, 1997, Phys. Rev. Lett. 78(16), 3047.
* Baikov and Broadhurst (1995) Baikov, P. A., and D. J. Broadhurst, 1995, in _New Computing Techniques in Physics Research IV. International Workshop on Software Engineering, Artificial Intelligence, and Expert Systems for High Energy and Nuclear Physics_ , edited by B. Denby and D. Perret-Gallix (World Scientific, Singapore), pp. 167–172.
* Barker and Glover (1955) Barker, W. A., and F. N. Glover, 1955, Phys. Rev. 99(1), 317.
* de Beauvoir _et al._ (1997) de Beauvoir, B., F. Nez, L. Julien, B. Cagnac, F. Biraben, D. Touahri, L. Hilico, O. Acef, A. Clairon, and J. J. Zondy, 1997, Phys. Rev. Lett. 78(3), 440.
* Becker (2011) Becker, P., 2011, private communication.
* Becker _et al._ (1981) Becker, P., K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, 1981, Phys. Rev. Lett. 46(23), 1540.
* Beer _et al._ (2003) Beer, W., A. L. Eichenberger, B. Jeanneret, B. Jeckelmann, A. R. Pourzand, P. Richard, and J. P. Schwarz, 2003, IEEE Trans. Instrum. Meas. 52(2), 626.
* Beer _et al._ (2001) Beer, W., A. L. Eichenberger, B. Jeanneret, B. Jeckelmann, P. Richard, H. Schneiter, A. R. Pourzand, A. Courteville, and R. Dändliker, 2001, IEEE Trans. Instrum. Meas. 50(2), 583.
* Beer _et al._ (1999) Beer, W., B. Jeanneret, B. Jeckelmann, P. Richard, A. Courteville, Y. Salvadé, and R. Dändliker, 1999, IEEE Trans. Instrum. Meas. 48(2), 192.
* Beier (2000) Beier, T., 2000, Phys. Rep. 339(2,3), 79.
* Beier _et al._ (2002) Beier, T., H. Häffner, N. Hermanspahn, S. G. Karshenboim, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, and G. Werth, 2002, Phys. Rev. Lett. 88, 011603.
* Beier _et al._ (2000) Beier, T., I. Lindgren, H. Persson, S. Salomonson, P. Sunnergren, H. Häffner, and N. Hermanspahn, 2000, Phys. Rev. A 62, 032510.
* Benayoun _et al._ (2012) Benayoun, M., P. David, L. DelBuono, and F. Jegerlehner, 2012, Eur. Phys. J. C 72, 1848.
* Bennett _et al._ (2006) Bennett, G. W., B. Bousquet, H. N. Brown, G. Bunce, R. M. Carey, P. Cushman, G. T. Danby, P. T. Debevec, M. Deile, H. Deng, W. Deninger, S. K. Dhawan, _et al._ , 2006, Phys. Rev. D 73, 072003.
* Benz _et al._ (2011) Benz, S. P., A. Pollarolo, J. Qu, H. Rogalla, C. Urano, W. L. Tew, P. D. Dresselhaus, and D. R. White, 2011, Metrologia 48(3), 142.
* Bergström _et al._ (2002) Bergström, I., C. Carlberg, T. Fritioff, G. Douysset, J. Schönfelder, and R. Schuch, 2002, Nucl. Instrum. Methods Phys. Res. A 487(3), 618.
* Berkeland _et al._ (1995) Berkeland, D. J., E. A. Hinds, and M. G. Boshier, 1995, Phys. Rev. Lett. 75(13), 2470.
* Bernauer (2010) Bernauer, J., 2010, private communication.
* Bernauer _et al._ (2010) Bernauer, J. C., P. Achenbach, C. Ayerbe Gayoso, R. Böhm, D. Bosnar, L. Debenjak, M. O. Distler, L. Doria, A. Esser, H. Fonvieille, J. M. Friedrich, J. Friedrich, _et al._ , 2010, Phys. Rev. Lett. 105, 242001.
* Bernauer _et al._ (2011) Bernauer, J. C., P. Achenbach, C. Ayerbe Gayoso, R. Böhm, D. Bosnar, L. Debenjak, M. O. Distler, L. Doria, A. Esser, H. Fonvieille, J. M. Friedrich, J. Friedrich, _et al._ , 2011, Phys. Rev. Lett. 107, 119102.
* Bettin (2011) Bettin, H., 2011, private communication.
* BIPM (2006) BIPM, 2006, _International System of Units (SI)_ (Bureau international des poids et mesures, Sèvres, France), 8th edition.
* Biraben (2011) Biraben, F., 2011, private communication.
* Borbely _et al._ (2009) Borbely, J. S., M. C. George, L. D. Lombardi, M. Weel, D. W. Fitzakerley, and E. A. Hessels, 2009, Phys. Rev. A 79, 060503.
* Borie (2012) Borie, E., 2012, Ann. Phys. (N.Y.) 327(3), 733.
* Borisoglebsky and Trofimenko (1979) Borisoglebsky, L. A., and E. E. Trofimenko, 1979, Phys. Lett. B 81(2), 175.
* Bouchendira _et al._ (2011) Bouchendira, R., P. Cladé, S. Guellati-Khélifa, F. Nez, and F. Biraben, 2011, Phys. Rev. Lett. 106, 080801.
* Bourzeix _et al._ (1996) Bourzeix, S., B. de Beauvoir, F. Nez, M. D. Plimmer, F. de Tomasi, L. Julien, F. Biraben, and D. N. Stacey, 1996, Phys. Rev. Lett. 76(3), 384.
* Bower and Davis (1980) Bower, V. E., and R. S. Davis, 1980, J. Res. Natl. Bur. Stand. 85(3), 175.
* Breit (1928) Breit, G., 1928, Nature 122(3078), 649.
* Breit and Rabi (1931) Breit, G., and I. I. Rabi, 1931, Phys. Rev. 38(11), 2082.
* Cadoret _et al._ (2009) Cadoret, M., E. De Mirandes, P. Cladé, F. Nez, L. Julien, F. Biraben, and S. Guellati-Khélifa, 2009, Eur. Phys. J. Special Topics 172(1), 121.
* Cadoret _et al._ (2011) Cadoret, M., E. d. Mirandés, P. Cladé, S. Guellati-Khélifa, F. Nez, and F. Biraben, 2011, C. R. Physique 12(4), 379.
* Cadoret _et al._ (2008) Cadoret, M., E. d. Mirandes, P. Cladé, S. Guellati-Khélifa, C. Schwob, F. Nez, L. Julien, and F. Biraben, 2008, Eur. Phys. J. Special Topics 163, 101.
* Camarota _et al._ (2012) Camarota, B., H. Scherer, M. W. Keller, S. V. Lotkhov, G.-D. Willenberg, and F. J. Ahlers, 2012, Metrologia 49(1), 8.
* Castillega _et al._ (2000) Castillega, J., D. Livingston, A. Sanders, and D. Shiner, 2000, Phys. Rev. Lett. 84(19), 4321.
* Cavagnero _et al._ (2004a) Cavagnero, G., H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, 2004a, Metrologia 41(1), 56, erratum: see Cavagnero _et al._ (2004b).
* Cavagnero _et al._ (2004b) Cavagnero, G., H. Fujimoto, G. Mana, E. Massa, K. Nakayama, and G. Zosi, 2004b, Metrologia 41(6), 445.
* Chiba (2011) Chiba, T., 2011, Prog. Theor. Phys. 126(6), 993.
* Cladé _et al._ (2006) Cladé, P., E. de Mirandes, M. Cadoret, S. Guellati-Khélifa, C. Schwob, F. Nez, L. Julien, and F. Biraben, 2006, Phys. Rev. A 74, 052109.
* Cladé _et al._ (2010) Cladé, P., T. Plisson, S. Guellati-Khélifa, F. Nez, and F. Biraben, 2010, Eur. Phys. J. D 59(3), 349.
* Close and Osborn (1971) Close, F. E., and H. Osborn, 1971, Phys. Lett. B 34(5), 400.
* Cohen and Taylor (1973) Cohen, E. R., and B. N. Taylor, 1973, J. Phys. Chem. Ref. Data 2(4), 663.
* Cohen and Taylor (1987) Cohen, E. R., and B. N. Taylor, 1987, Rev. Mod. Phys. 59(4), 1121.
* Colclough _et al._ (1979) Colclough, A. R., T. J. Quinn, and T. R. D. Chandler, 1979, Proc. R. Soc. London, Ser. A 368(1732), 125.
* Czarnecki _et al._ (2005) Czarnecki, A., U. D. Jentschura, and K. Pachucki, 2005, Phys. Rev. Lett. 95, 180404.
* Czarnecki _et al._ (1996) Czarnecki, A., B. Krause, and W. J. Marciano, 1996, Phys. Rev. Lett. 76(18), 3267.
* Czarnecki _et al._ (2003) Czarnecki, A., W. J. Marciano, and A. Vainshtein, 2003, Phys. Rev. D 67, 073006, erratum: see Czarnecki _et al._ (2006).
* Czarnecki _et al._ (2006) Czarnecki, A., W. J. Marciano, and A. Vainshtein, 2006, Phys. Rev. D 73, 119901.
* Czarnecki and Melnikov (2001) Czarnecki, A., and K. Melnikov, 2001, Phys. Rev. Lett. 87, 013001.
* Czarnecki _et al._ (2001) Czarnecki, A., K. Melnikov, and A. Yelkhovsky, 2001, Phys. Rev. A 63, 012509.
* Davier and Höcker (1998) Davier, M., and A. Höcker, 1998, Phys. Lett. B 435(3-4), 427.
* Davier _et al._ (2011) Davier, M., A. Hoecker, B. Malaescu, and Z. Zhang, 2011, Eur. Phys. J. C 71, 1515.
* Dowling _et al._ (2010) Dowling, M., J. Mondéjar, J. H. Piclum, and A. Czarnecki, 2010, Phys. Rev. A 81, 022509.
* Drake and Swainson (1990) Drake, G. W. F., and R. A. Swainson, 1990, Phys. Rev. A 41(3), 1243.
* Eichenberger _et al._ (2011) Eichenberger, A., H. Baumann, B. Jeanneret, B. Jeckelmann, P. Richard, and W. Beer, 2011, Metrologia 48(3), 133.
* Eichenberger _et al._ (2009) Eichenberger, A., G. Genevès, and P. Gournay, 2009, Eur. Phys. J. Special Topics 172(1), 363.
* Eidelman _et al._ (2002) Eidelman, S. I., S. G. Karshenboim, and V. A. Shelyuto, 2002, Can. J. Phys. 80(11), 1297.
* Eides (1996) Eides, M. I., 1996, Phys. Rev. A 53(5), 2953.
* Eides (2002) Eides, M. I., 2002, private communication.
* Eides and Grotch (1995a) Eides, M. I., and H. Grotch, 1995a, Phys. Rev. A 52(4), 3360.
* Eides and Grotch (1995b) Eides, M. I., and H. Grotch, 1995b, Phys. Rev. A 52(2), 1757.
* Eides and Grotch (1997a) Eides, M. I., and H. Grotch, 1997a, Ann. Phys. (N.Y.) 260(1), 191.
* Eides and Grotch (1997b) Eides, M. I., and H. Grotch, 1997b, Phys. Rev. A 56(4), R2507.
* Eides and Grotch (1997c) Eides, M. I., and H. Grotch, 1997c, Phys. Rev. A 55(5), 3351.
* Eides _et al._ (1997) Eides, M. I., H. Grotch, and V. A. Shelyuto, 1997, Phys. Rev. A 55(3), 2447.
* Eides _et al._ (2001a) Eides, M. I., H. Grotch, and V. A. Shelyuto, 2001a, Phys. Rev. A 63, 052509.
* Eides _et al._ (2001b) Eides, M. I., H. Grotch, and V. A. Shelyuto, 2001b, Phys. Rep. 342(2-3), 63.
* Eides _et al._ (2007) Eides, M. I., H. Grotch, and V. A. Shelyuto, 2007, _Theory of Light Hydrogenic Bound States_ (Springer, Springer Tracts in Mod. Phys. 222, Berlin, Heidelberg, New York).
* Eides and Martin (2010) Eides, M. I., and T. J. S. Martin, 2010, Phys. Rev. Lett. 105, 100402.
* Eides and Martin (2011) Eides, M. I., and T. J. S. Martin, 2011, Can. J. Phys. 89(1), 117.
* Eides and Shelyuto (1995) Eides, M. I., and V. A. Shelyuto, 1995, Phys. Rev. A 52(2), 954.
* Eides and Shelyuto (2004) Eides, M. I., and V. A. Shelyuto, 2004, Phys. Rev. A 70, 022506.
* Eides and Shelyuto (2007) Eides, M. I., and V. A. Shelyuto, 2007, Can. J. Phys. 85(5), 509.
* Eides and Shelyuto (2009a) Eides, M. I., and V. A. Shelyuto, 2009a, Phys. Rev. D 80, 053008.
* Eides and Shelyuto (2009b) Eides, M. I., and V. A. Shelyuto, 2009b, Phys. Rev. Lett. 103, 133003.
* Eides and Shelyuto (2010) Eides, M. I., and V. A. Shelyuto, 2010, J. Exp. Theor. Phys. 110(1), 17.
* Elkhovskiĭ (1996) Elkhovskiĭ, A. S., 1996, Zh. Eksp. Teor. Fiz. 110(2), 431, [JETP 83(2), 230 (1996)].
* Erickson (1977) Erickson, G. W., 1977, J. Phys. Chem. Ref. Data 6(3), 831.
* Erickson and Yennie (1965) Erickson, G. W., and D. R. Yennie, 1965, Ann. Phys. (N.Y.) 35(2), 271.
* Farnham _et al._ (1995) Farnham, D. L., R. S. Van Dyck, Jr., and P. B. Schwinberg, 1995, Phys. Rev. Lett. 75(20), 3598\.
* Faustov (1970) Faustov, R., 1970, Phys. Lett. B 33(6), 422.
* Fellmuth _et al._ (2011) Fellmuth, B., J. Fischer, C. Gaiser, O. Jusko, T. Priruenrom, W. Sabuga, and T. Zandt, 2011, Metrologia 48(5), 382.
* Feltin and Piquemal (2009) Feltin, N., and F. Piquemal, 2009, Eur. Phys. J. Special Topics 172(1), 267.
* Feltin _et al._ (2011) Feltin, N., B. Steck, L. DeVoille, S. Sassine, B. Chenaud, W. Poirier, F. Schopfer, G. Sprengler, S. Djordjevic, O. Seron, and F. Piquemal, 2011, Revue Française de Métrologie 2011-1(25), 3.
* Ferroglio _et al._ (2008) Ferroglio, L., G. Mana, and E. Massa, 2008, Opt. Express 16(21), 16877.
* Fischer _et al._ (2004) Fischer, M., N. Kolachevsky, M. Zimmermann, R. Holzwarth, T. Udem, T. W. Hänsch, M. Abgrall, J. Grünert, I. Maksimovic, S. Bize, H. Marion, F. P. Dos Santos, _et al._ , 2004, Phys. Rev. Lett. 92, 230802.
* Flowers _et al._ (1993) Flowers, J. L., B. W. Petley, and M. G. Richards, 1993, Metrologia 30(2), 75.
* Friar (1979a) Friar, J. L., 1979a, Z. Phys. A 292(1), 1, erratum: see Friar (1981).
* Friar (1979b) Friar, J. L., 1979b, Ann. Phys. (N.Y.) 122(1), 151.
* Friar (1981) Friar, J. L., 1981, Z. Phys. A 303(1), 84.
* Friar _et al._ (1999) Friar, J. L., J. Martorell, and D. W. L. Sprung, 1999, Phys. Rev. A 59(5), 4061.
* Friar and Payne (1997a) Friar, J. L., and G. L. Payne, 1997a, Phys. Rev. C 56(2), 619.
* Friar and Payne (1997b) Friar, J. L., and G. L. Payne, 1997b, Phys. Rev. A 56(6), 5173.
* Fujii (2010) Fujii, K., 2010, private communication.
* Fujii _et al._ (2005) Fujii, K., A. Waseda, N. Kuramoto, S. Mizushima, P. Becker, H. Bettin, A. Nicolaus, U. Kuetgens, S. Valkiers, P. Taylor, P. De Bièvre, G. Mana, _et al._ , 2005, IEEE Trans. Instrum. Meas. 54(2), 854.
* Funck and Sienknecht (1991) Funck, T., and V. Sienknecht, 1991, IEEE Trans. Instrum. Meas. 40(2), 158.
* Gabrielse (2006) Gabrielse, G., 2006, Int. J. Mass Spectrom. 251(2-3), 273.
* Gabrielse (2010) Gabrielse, G., 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 5, pp. 157–194.
* Gaiser and Fellmuth (2012) Gaiser, C., and B. Fellmuth, 2012, Metrologia 49(1), L4.
* Gavioso (2011) Gavioso, R., 2011, private communication.
* Gavioso _et al._ (2011) Gavioso, R., M. de Podesta, and L. Pitre, 2011, private communication.
* Gavioso _et al._ (2010) Gavioso, R. M., G. Benedetto, P. A. Giuliano Albo, D. Madonna Ripa, A. Merlone, C. Guianvarc’h, F. Moro, and R. Cuccaro, 2010, Metrologia 47(4), 387.
* George _et al._ (2001) George, M. C., L. D. Lombardi, and E. A. Hessels, 2001, Phys. Rev. Lett. 87, 173002.
* George _et al._ (2007) George, S., K. Blaum, F. Herfurth, A. Herlert, M. Kretzschmar, S. Nagy, S. Schwarz, L. Schweikhard, and C. Yazidjian, 2007, Int. J. Mass Spectrom. 264(2-3), 110.
* Giusfredi _et al._ (2005) Giusfredi, G., P. C. Pastor, P. D. Natale, D. Mazzotti, C. D. Mauro, L. Fallani, G. Hagel, V. Krachmalnicoff, and M. Inguscio, 2005, Can. J. Phys. 83(4), 301.
* Glattli (2011) Glattli, C., 2011, C. R. Physique 12(4), 319.
* Glazov and Shabaev (2002) Glazov, D. A., and V. M. Shabaev, 2002, Phys. Lett. A 297(5-6), 408.
* Goerbig (2011) Goerbig, M. O., 2011, C. R. Physique 12(4), 369.
* Golosov _et al._ (1995) Golosov, E. A., A. S. Elkhovskiĭ, A. I. Mil’shteĭn, and I. B. Khriplovich, 1995, Zh. Eksp. Teor. Fiz. 107(2), 393, [JETP 80(2), 208 (1995)].
* Greene _et al._ (1979) Greene, G. L., N. F. Ramsey, W. Mampe, J. M. Pendlebury, K. Smith, W. B. Dress, P. D. Miller, and P. Perrin, 1979, Phys. Rev. D 20(9), 2139.
* Greene _et al._ (1977) Greene, G. L., N. F. Ramsey, W. Mampe, J. M. Pendlebury, K. Smith, W. D. Dress, P. D. Miller, and P. Perrin, 1977, Phys. Lett. B 71(2), 297.
* Grotch (1970) Grotch, H., 1970, Phys. Rev. Lett. 24(2), 39.
* Gundlach and Merkowitz (2000) Gundlach, J. H., and S. M. Merkowitz, 2000, Phys. Rev. Lett. 85(14), 2869.
* Gundlach and Merkowitz (2002) Gundlach, J. H., and S. M. Merkowitz, 2002, private communication.
* Häffner _et al._ (2003) Häffner, H., T. Beier, S. Djekić, N. Hermanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, T. Valenzuela, and G. Werth, 2003, Eur. Phys. J. D 22(2), 163.
* Hagiwara _et al._ (2011) Hagiwara, K., R. Liao, A. D. Martin, D. Nomura, and T. Teubner, 2011, J. Phys. G 38, 085003.
* Hagley and Pipkin (1994) Hagley, E. W., and F. M. Pipkin, 1994, Phys. Rev. Lett. 72(8), 1172.
* Hanneke _et al._ (2008) Hanneke, D., S. Fogwell, and G. Gabrielse, 2008, Phys. Rev. Lett. 100, 120801.
* Hanneke _et al._ (2011) Hanneke, D., S. Fogwell Hoogerheide, and G. Gabrielse, 2011, Phys. Rev. A 83, 052122.
* Hartland _et al._ (1988) Hartland, A., R. G. Jones, and D. J. Legg, 1988, Document CCE/88-9 submitted to the 18th meeting of the Comité Consultatif d’Électricité of the CIPM.
* Härtwig _et al._ (1991) Härtwig, J., S. Grosswig, P. Becker, and D. Windisch, 1991, Phys. Status Solidi A 125(1), 79.
* Hayano (2010) Hayano, R. S., 2010, Proc. Jpn. Acad., Ser. B 86(1), 1.
* Hori (2006) Hori, M., 2006, private communication.
* Hori (2010) Hori, M., 2010, private communication.
* Hori (2011) Hori, M., 2011, Can. J. Phys. 89(1), 1.
* Hori _et al._ (2006) Hori, M., A. Dax, J. Eades, K. Gomikawa, R. Hayano, N. Ono, W. Pirkl, E. Widmann, H. A. Torii, B. Juhász, D. Barna, and D. Horváth, 2006, Phys. Rev. Lett. 96, 243401.
* Hori and Korobov (2010) Hori, M., and V. I. Korobov, 2010, Phys. Rev. A 81, 062508.
* Hori _et al._ (2011) Hori, M., A. Sótér, D. Barna, A. Dax, R. Hayano, S. Friedreich, B. Juhász, T. Pask, E. Widmann, D. Horváth, L. Venturelli, and N. Zurlo, 2011, Nature 475(7357), 484.
* Hu _et al._ (2005) Hu, Z.-K., J.-Q. Guo, and J. Luo, 2005, Phys. Rev. D 71, 127505\.
* Huber _et al._ (1998) Huber, A., T. Udem, B. Gross, J. Reichert, M. Kourogi, K. Pachucki, M. Weitz, and T. W. Hänsch, 1998, Phys. Rev. Lett. 80(3), 468.
* Hylton (1985) Hylton, D. J., 1985, Phys. Rev. A 32(3), 1303.
* Ivanov _et al._ (2009) Ivanov, V. G., S. G. Karshenboim, and R. N. Lee, 2009, Phys. Rev. A 79, 012512.
* Janssen _et al._ (2011) Janssen, T. J. B. M., N. E. Fletcher, R. Goebel, J. M. Willaims, A. Tzalenchuk, R. Yakimova, S. Kubatkin, S. Lara-Avila, and V. I. Falko, 2011, New J. Phys. 13, 093026.
* Jeffery _et al._ (1998) Jeffery, A., R. E. Elmquist, J. Q. Shields, L. H. Lee, M. E. Cage, S. H. Shields, and R. F. Dziuba, 1998, Metrologia 35(2), 83.
* Jeffery _et al._ (1997) Jeffery, A.-M., R. E. Elmquist, L. H. Lee, J. Q. Shields, and R. F. Dziuba, 1997, IEEE Trans. Instrum. Meas. 46(2), 264.
* Jegerlehner and Nyffeler (2009) Jegerlehner, F., and A. Nyffeler, 2009, Phys. Rep. 477(1-3), 1.
* Jegerlehner and Szafron (2011) Jegerlehner, F., and R. Szafron, 2011, Eur. Phys. J. C 71, 1632.
* Jentschura and Pachucki (1996) Jentschura, U., and K. Pachucki, 1996, Phys. Rev. A 54(3), 1853.
* Jentschura (2003) Jentschura, U. D., 2003, J. Phys. A 36, L229.
* Jentschura (2004) Jentschura, U. D., 2004, Phys. Rev. A 70, 052108.
* Jentschura (2006) Jentschura, U. D., 2006, Phys. Rev. A 74, 062517.
* Jentschura (2009) Jentschura, U. D., 2009, Phys. Rev. A 79, 044501.
* Jentschura (2011a) Jentschura, U. D., 2011a, private communication.
* Jentschura (2011b) Jentschura, U. D., 2011b, Ann. Phys. (N.Y.) 326(2), 500.
* Jentschura (2011c) Jentschura, U. D., 2011c, Ann. Phys. (N.Y.) 326(2), 516.
* Jentschura _et al._ (2005a) Jentschura, U. D., A. Czarnecki, and K. Pachucki, 2005a, Phys. Rev. A 72, 062102.
* Jentschura _et al._ (2006) Jentschura, U. D., A. Czarnecki, K. Pachucki, and V. A. Yerokhin, 2006, Int. J. Mass Spectrom. 251(2-3), 102.
* Jentschura _et al._ (2002) Jentschura, U. D., J. Evers, C. H. Keitel, and K. Pachucki, 2002, New J. Phys. 4, 49.1.
* Jentschura _et al._ (2011a) Jentschura, U. D., A. Matveev, C. G. Parthey, J. Alnis, R. Pohl, T. Udem, N. Kolachevsky, and T. W. Hänsch, 2011a, Phys. Rev. A 83, 042505.
* Jentschura and Mohr (2004) Jentschura, U. D., and P. J. Mohr, 2004, Phys. Rev. A 69, 064103.
* Jentschura and Mohr (2005) Jentschura, U. D., and P. J. Mohr, 2005, Phys. Rev. A 72, 014103.
* Jentschura _et al._ (1999) Jentschura, U. D., P. J. Mohr, and G. Soff, 1999, Phys. Rev. Lett. 82(1), 53.
* Jentschura _et al._ (2001) Jentschura, U. D., P. J. Mohr, and G. Soff, 2001, Phys. Rev. A 63, 042512\.
* Jentschura and Nándori (2002) Jentschura, U. D., and I. Nándori, 2002, Phys. Rev. A 66, 022114.
* Jentschura _et al._ (2011b) Jentschura, U. D., M. Puchalski, and P. J. Mohr, 2011b, Phys. Rev. A 84, 064102.
* Jentschura and Yerokhin (2010) Jentschura, U. D., and V. A. Yerokhin, 2010, Phys. Rev. A 81, 012503.
* Jentschura _et al._ (2005b) Jentschura, U. J., S. Kotochigova, E. O. Le Bigot, and P. J. Mohr, 2005b, NIST Technical Note 1469, physics.nist.gov/hdel.
* Jiang _et al._ (2011) Jiang, Z., O. Francis, L. Vitushkin, V. Palinkas, A. Germak, M. Becker, G. D’Agostino, M. Amalvict, R. Bayer, M. Bilker-Koivula, S. Desogus, J. Faller, _et al._ , 2011, Metrologia 48, 246.
* K. Clothier _et al._ (1989) K. Clothier, W., G. J. Sloggett, H. Bairnsfather, M. F. Currey, and D. J. Benjamin, 1989, Metrologia 26(1), 9.
* Karagioz and Izmailov (1996) Karagioz, O. V., and V. P. Izmailov, 1996, Izmer. Tekh. 39(10), 3, [Meas. Tech. 39(10), 979 (1996)].
* Karshenboim (1995) Karshenboim, S. G., 1995, J. Phys. B 28(4), L77.
* Karshenboim (1996) Karshenboim, S. G., 1996, J. Phys. B 29(2), L29.
* Karshenboim (1997) Karshenboim, S. G., 1997, Z. Phys. D 39(2), 109.
* Karshenboim (2000) Karshenboim, S. G., 2000, Phys. Lett. A 266(4,5,6), 380\.
* Karshenboim and Ivanov (2002) Karshenboim, S. G., and V. G. Ivanov, 2002, Can. J. Phys. 80(11), 1305.
* Karshenboim _et al._ (2005) Karshenboim, S. G., V. G. Ivanov, Y. I. Neronov, B. P. Nikolaev, and Y. N. Tolparov, 2005, Can. J. Phys. 83(4), 405.
* Karshenboim _et al._ (2001a) Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 2001a, Can. J. Phys. 79(1), 81.
* Karshenboim _et al._ (2001b) Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 2001b, Zh. Eksp. Teor. Fiz. 120(3), 546, [JETP 93(3), 477 (2001)].
* Karshenboim and Milstein (2002) Karshenboim, S. G., and A. I. Milstein, 2002, Phys. Lett. B 549(3-4), 321.
* Karshenboim _et al._ (2008) Karshenboim, S. G., V. A. Shelyuto, and A. I. Vainshtein, 2008, Phys. Rev. D 78, 065036.
* Karshenboĭm (1993) Karshenboĭm, S. G., 1993, Zh. Eksp. Teor. Fiz. 103(4), 1105, [JETP 76(4), 541 (1993)].
* Keller (2008) Keller, M. W., 2008, Metrologia 45(1), 102.
* Keller _et al._ (2008) Keller, M. W., F. Piquemal, N. Feltin, B. Steck, and L. Devoille, 2008, Metrologia 45(3), 330.
* Keller _et al._ (2007) Keller, M. W., N. M. Zimmerman, and A. L. Eichenberger, 2007, Metrologia 44(6), 505.
* Kessler _et al._ (1999) Kessler, E. G., Jr., M. S. Dewey, R. D. Deslattes, A. Henins, H. G. Börner, M. Jentschel, C. Doll, and H. Lehmann, 1999, Phys. Lett. A 255(4-6), 221.
* Khriplovich and Sen’kov (1998) Khriplovich, I. B., and R. A. Sen’kov, 1998, Phys. Lett. A 249(5-6), 474.
* Khriplovich and Sen’kov (2000) Khriplovich, I. B., and R. A. Sen’kov, 2000, Phys. Lett. B 481(2,3,4), 447\.
* Kibble (1975) Kibble, B. P., 1975, in _Atomic Masses and Fundamental Constants 5_ , edited by J. H. Sanders and A. H. Wapstra (Plenum Press, New York), pp. 545–551.
* Kibble and Hunt (1979) Kibble, B. P., and G. J. Hunt, 1979, Metrologia 15(1), 5.
* Kibble _et al._ (1990) Kibble, B. P., I. A. Robinson, and J. H. Belliss, 1990, Metrologia 27(4), 173.
* Kinoshita (2010) Kinoshita, T., 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 3, pp. 69–117.
* Kinoshita and Nio (2006) Kinoshita, T., and M. Nio, 2006, Phys. Rev. D 73, 013003\.
* Kinoshita _et al._ (1990) Kinoshita, T., B. Nizic, and Y. Okamoto, 1990, Phys. Rev. D 41(2), 593.
* Kleinevoß (2002) Kleinevoß, U., 2002, _Bestimmung der Newtonschen Gravitationskonstanten $G$_, Ph.D. thesis, University of Wuppertal.
* Kleinvoß _et al._ (2002) Kleinvoß, U., H. Meyer, H. Piel, and S. Hartmann, 2002, private communication and to be published.
* Kleppner (1997) Kleppner, D., 1997, private communication.
* Kolachevsky _et al._ (2004) Kolachevsky, N., M. Fischer, S. G. Karshenboim, and T. W. Hänsch, 2004, Phys. Rev. Lett. 92, 033003.
* Kolachevsky _et al._ (2009) Kolachevsky, N., A. Matveev, J. Alnis, C. G. Parthey, S. G. Karshenboim, and T. W. Hänsch, 2009, Phys. Rev. Lett. 102, 213002.
* Korobov (2006) Korobov, V. I., 2006, private communication.
* Korobov (2008) Korobov, V. I., 2008, Phys. Rev. A 77, 042506.
* Korobov (2010) Korobov, V. I., 2010, private communication.
* Kotochigova (2006) Kotochigova, S., 2006, private communication.
* Kotochigova and Mohr (2006) Kotochigova, S., and P. J. Mohr, 2006, private communication.
* Kotochigova _et al._ (2002) Kotochigova, S., P. J. Mohr, and B. N. Taylor, 2002, Can. J. Phys. 80(11), 1373.
* Kramer _et al._ (2010) Kramer, T., C. Kreisbeck, V. Krueckl, E. J. Heller, R. E. Parrott, and C.-T. Liang, 2010, Phys. Rev. B 81, 081410.
* Krause (1997) Krause, B., 1997, Phys. Lett. B 390(1-4), 392.
* Krüger _et al._ (1999) Krüger, E., W. Nistler, and W. Weirauch, 1999, Metrologia 36(2), 147.
* Kuroda (1995) Kuroda, K., 1995, Phys. Rev. Lett. 75(15), 2796.
* Kuroda (1999) Kuroda, K., 1999, Meas. Sci. Technol. 10(6), 435.
* Łach _et al._ (2004) Łach, G., B. Jeziorski, and K. Szalewicz, 2004, Phys. Rev. Lett. 92, 233001.
* Laporta (2001) Laporta, S., 2001, Phys. Lett. B 523(1-2), 95.
* Laporta (2008) Laporta, S., 2008, Int. J. Mod. Phys. A 23(31), 5007.
* Laporta _et al._ (2004) Laporta, S., P. Mastrolia, and E. Remiddi, 2004, Nucl. Phys. B 688(1-2), 165.
* Laporta and Remiddi (1996) Laporta, S., and E. Remiddi, 1996, Phys. Lett. B 379(1-4), 283.
* Lee _et al._ (2005) Lee, R. N., A. I. Milstein, I. S. Terekhov, and S. G. Karshenboim, 2005, Phys. Rev. A 71, 052501.
* Lemarchand _et al._ (2011) Lemarchand, C., M. Triki, B. Darquié, C. J. Bordé, C. Chardonnet, and C. Daussy, 2011, New J. Phys. 13, 073028.
* Li _et al._ (2012) Li, S., B. Han, Z. Li, and J. Lan, 2012, Measurement 45(1), 1.
* Liu _et al._ (1995) Liu, R., H. Liu, T. Jin, Z. Lu, X. Du, S. Xue, J. Kong, B. Yu, X. Zhou, T. Liu, and W. Zhang, 1995, Acta Metrol. Sin. 16(3), 161.
* Liu _et al._ (1999) Liu, W., M. G. Boshier, S. Dhawan, O. van Dyck, P. Egan, X. Fei, M. G. Perdekamp, V. W. Hughes, M. Janousch, K. Jungmann, D. Kawall, F. G. Mariam, _et al._ , 1999, Phys. Rev. Lett. 82(4), 711.
* Louchet-Chauvet _et al._ (2011) Louchet-Chauvet, A., S. Merlet, Q. Bodart, A. Landragin, F. Pereira Dos Santos, H. Baumann, G. D’Agostino, and C. Origlia, 2011, IEEE Trans. Instrum. Meas. 60(7), 2527.
* Lundeen and Pipkin (1986) Lundeen, S. R., and F. M. Pipkin, 1986, Metrologia 22(1), 9.
* Luo (2010) Luo, J., 2010, private communication.
* Luo (2011) Luo, J., 2011, private communication.
* Luo _et al._ (2009) Luo, J., Q. Liu, L.-C. Tu, C.-G. Shao, L.-X. Liu, S.-Q. Yang, Q. Li, and Y.-T. Zhang, 2009, Phys. Rev. Lett. 102, 240801.
* Luther (1986) Luther, G., 1986, private communication.
* Luther (2010) Luther, G., 2010, private communication.
* Luther and Towler (1982) Luther, G. G., and W. R. Towler, 1982, Phys. Rev. Lett. 48(3), 121.
* Mana (2011) Mana, G., 2011, private communication.
* Manohar and Stewart (2000) Manohar, A. V., and I. W. Stewart, 2000, Phys. Rev. Lett. 85(11), 2248.
* Mariam (1981) Mariam, F. G., 1981, _High Precision Measurement of the Muonium Ground State Hyperfine Interval and the Muon Magnetic Moment_ , Ph.D. thesis, Yale University.
* Mariam _et al._ (1982) Mariam, F. G., W. Beer, P. R. Bolton, P. O. Egan, C. J. Gardner, V. W. Hughes, D. C. Lu, P. A. Souder, H. Orth, J. Vetter, U. Moser, and G. zu Putlitz, 1982, Phys. Rev. Lett. 49(14), 993.
* Martynenko (2006) Martynenko, A. P., 2006, Yad. Fiz. 69(8), 1344, [Phys. At. Nucl. 69(8), 1309 (2006)].
* Massa _et al._ (2011a) Massa, E., G. Mana, L. Ferroglio, E. G. Kessler, D. Schiel, and S. Zakel, 2011a, Metrologia 48(2), S44.
* Massa _et al._ (2009a) Massa, E., G. Mana, and U. Kuetgens, 2009a, Metrologia 46(3), 249.
* Massa _et al._ (2009b) Massa, E., G. Mana, U. Kuetgens, and L. Ferroglio, 2009b, New J. Phys. 11, 053013.
* Massa _et al._ (2011b) Massa, E., G. Mana, U. Kuetgens, and L. Ferroglio, 2011b, Metrologia 48(2), S37.
* Massa and Nicolaus (2011) Massa, E., and A. Nicolaus, 2011, Metrologia 48(2).
* Mastrolia and Remiddi (2001) Mastrolia, P., and E. Remiddi, 2001, in _The Hydrogen Atom: Precision Physics of Simple Atomic Systems_ , edited by S. G. Karshenboim, F. S. Pavone, G. F. Bassani, M. Inguscio, and T. W. Hänsch (Springer, Berlin), pp. 776–783.
* Matsumura _et al._ (1998) Matsumura, S., N. Kanda, T. Tomaru, H. Ishizuka, and K. Kuroda, 1998, Phys. Lett. A 244(1-3), 4.
* Melnikov and van Ritbergen (2000) Melnikov, K., and T. van Ritbergen, 2000, Phys. Rev. Lett. 84(8), 1673.
* Melnikov and Yelkhovsky (1999) Melnikov, K., and A. Yelkhovsky, 1999, Phys. Lett. B 458(1), 143.
* Merlet _et al._ (2010) Merlet, S., Q. Bodart, N. Malossi, A. Landragin, F. Pereira Dos Santos, O. Gitlein, and L. Timmen, 2010, Metrologia 47(4), L9.
* Mills _et al._ (2011) Mills, I. M., P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams, 2011, Phil. Trans. R. Soc. A 369(1953), 3907\.
* Milstein _et al._ (2002) Milstein, A. I., O. P. Sushkov, and I. S. Terekhov, 2002, Phys. Rev. Lett. 89, 283003.
* Milstein _et al._ (2003a) Milstein, A. I., O. P. Sushkov, and I. S. Terekhov, 2003a, Phys. Rev. A 67, 062103.
* Milstein _et al._ (2003b) Milstein, A. I., O. P. Sushkov, and I. S. Terekhov, 2003b, Phys. Rev. A 67, 062111.
* Milstein _et al._ (2004) Milstein, A. I., O. P. Sushkov, and I. S. Terekhov, 2004, Phys. Rev. A 69, 022114.
* Mohr (1975) Mohr, P. J., 1975, in _Beam-Foil Spectroscopy_ , edited by I. A. Sellin and D. J. Pegg (Plenum Press, New York), volume 1, pp. 89–96.
* Mohr (1982) Mohr, P. J., 1982, Phys. Rev. A 26(5), 2338.
* Mohr (1983) Mohr, P. J., 1983, At. Data. Nucl. Data Tables 29(3), 453.
* Mohr and Newell (2010) Mohr, P. J., and D. B. Newell, 2010, Am. J. Phys. 78(4), 338.
* Mohr and Taylor (2000) Mohr, P. J., and B. N. Taylor, 2000, Rev. Mod. Phys. 72(2), 351.
* Mohr and Taylor (2005) Mohr, P. J., and B. N. Taylor, 2005, Rev. Mod. Phys. 77(1), 1.
* Mohr _et al._ (2008) Mohr, P. J., B. N. Taylor, and D. B. Newell, 2008, Rev. Mod. Phys. 80(2), 633.
* Moldover (2009) Moldover, M. R., 2009, C. R. Physique 10(9), 815.
* Moldover _et al._ (1988) Moldover, M. R., J. P. M. Trusler, T. J. Edwards, J. B. Mehl, and R. S. Davis, 1988, J. Res. Natl. Bur. Stand. 93(2), 85.
* Mondéjar _et al._ (2010) Mondéjar, J., J. H. Piclum, and A. Czarnecki, 2010, Phys. Rev. A 81, 062511.
* Mount _et al._ (2010) Mount, B. J., M. Redshaw, and E. G. Myers, 2010, Phys. Rev. A 82, 042513.
* Nagy _et al._ (2006) Nagy, S., T. Fritioff, M. Björkhage, I. Bergström, and R. Schuch, 2006, Europhys. Lett. 74(3), 404.
* Nakamura _et al._ (2010) Nakamura, K., K. Hagiwara, K. Hikasa, H. Murayama, M. Tanabashi, T. Watari, C. Amsler, M. Antonelli, D. M. Asner, H. Baer, H. R. Band, R. M. Barnett, _et al._ , 2010, J. Phys. G 37, 075021.
* Neronov and Aleksandrov (2011) Neronov, Y. I., and V. S. Aleksandrov, 2011, Pis’ma. Zh. Eksp. Teor. Fiz. 94(6), 452, [JETP Lett. 94(6), 418 (2011)].
* Neronov and Karshenboim (2003) Neronov, Y. I., and S. G. Karshenboim, 2003, Phys. Lett. A 318(1-2), 126.
* Nevado and Pineda (2008) Nevado, D., and A. Pineda, 2008, Phys. Rev. C 77, 035202.
* Newton _et al._ (1979) Newton, G., D. A. Andrews, and P. J. Unsworth, 1979, Philos. Trans. R. Soc. London, Ser. A 290(1373), 373\.
* Odom _et al._ (2006) Odom, B., D. Hanneke, B. D’Urso, and G. Gabrielse, 2006, Phys. Rev. Lett. 97, 030801.
* Olver _et al._ (2010) Olver, F. W. J., D. W. Lozier, R. F. Boisvert, and C. W. Clark, 2010, _NIST Handbook of Mathematical Functions_ (National Institute of Standards and Technology and Cambridge University Press, Gaithersburg and Cambridge).
* Pachucki (1993a) Pachucki, K., 1993a, Phys. Rev. A 48(4), 2609.
* Pachucki (1993b) Pachucki, K., 1993b, Phys. Rev. A 48(1), 120.
* Pachucki (1994) Pachucki, K., 1994, Phys. Rev. Lett. 72(20), 3154.
* Pachucki (1995) Pachucki, K., 1995, Phys. Rev. A 52(2), 1079.
* Pachucki (2001) Pachucki, K., 2001, Phys. Rev. A 63, 042503.
* Pachucki (2006) Pachucki, K., 2006, Phys. Rev. Lett. 97, 013002.
* Pachucki (2008) Pachucki, K., 2008, Phys. Rev. A 78, 012504.
* Pachucki _et al._ (2005a) Pachucki, K., A. Czarnecki, U. D. Jentschura, and V. A. Yerokhin, 2005a, Phys. Rev. A 72, 022108.
* Pachucki and Grotch (1995) Pachucki, K., and H. Grotch, 1995, Phys. Rev. A 51(3), 1854.
* Pachucki and Jentschura (2003) Pachucki, K., and U. D. Jentschura, 2003, Phys. Rev. Lett. 91, 113005.
* Pachucki _et al._ (2004) Pachucki, K., U. D. Jentschura, and V. A. Yerokhin, 2004, Phys. Rev. Lett. 93, 150401, erratum: see Pachucki _et al._ (2005b).
* Pachucki _et al._ (2005b) Pachucki, K., U. D. Jentschura, and V. A. Yerokhin, 2005b, Phys. Rev. Lett. 94, 229902.
* Pachucki and Karshenboim (1995) Pachucki, K., and S. G. Karshenboim, 1995, J. Phys. B 28, L221.
* Pachucki and Karshenboim (1999) Pachucki, K., and S. G. Karshenboim, 1999, Phys. Rev. A 60(4), 2792.
* Pachucki and Sapirstein (2010) Pachucki, K., and J. Sapirstein, 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 7, pp. 219–272.
* Pachucki and Yerokhin (2009) Pachucki, K., and V. A. Yerokhin, 2009, Phys. Rev. A 79, 062516, erratum: see Pachucki and Yerokhin (2010b).
* Pachucki and Yerokhin (2010a) Pachucki, K., and V. A. Yerokhin, 2010a, Phys. Rev. Lett. 104, 070403.
* Pachucki and Yerokhin (2010b) Pachucki, K., and V. A. Yerokhin, 2010b, Phys. Rev. A 81, 039903.
* Pachucki and Yerokhin (2011a) Pachucki, K., and V. A. Yerokhin, 2011a, Can. J. Phys. 89(1), 95.
* Pachucki and Yerokhin (2011b) Pachucki, K., and V. A. Yerokhin, 2011b, J. Phys. Conf. Ser. 264, 012007.
* Park _et al._ (1999) Park, P. G., C. H. Choi, C. S. Kim, V. Y. Shifrin, and V. N. Khorev, 1999, J. Korean Phys. Soc. 34(4), 327.
* Parks and Faller (2010) Parks, H. V., and J. E. Faller, 2010, Phys. Rev. Lett. 105, 110801.
* Parthey _et al._ (2011) Parthey, C. G., A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, _et al._ , 2011, Phys. Rev. Lett. 107, 203001.
* Parthey _et al._ (2010) Parthey, C. G., A. Matveev, J. Alnis, R. Pohl, T. Udem, U. D. Jentschura, N. Kolachevsky, and T. W. Hänsch, 2010, Phys. Rev. Lett. 104, 233001.
* Penin (2009) Penin, A. A., 2009, Phys. Rev. B 79, 113303, erratum: see Penin (2010).
* Penin (2010) Penin, A. A., 2010, Phys. Rev. B 81, 089902.
* Peres (2010) Peres, N. M. R., 2010, Rev. Mod. Phys. 82(3), 2673.
* Petley and Donaldson (1984) Petley, B. W., and R. W. Donaldson, 1984, Metrologia 20(3), 81.
* Phillips _et al._ (1977) Phillips, W. D., W. E. Cooke, and D. Kleppner, 1977, Metrologia 13(4), 179.
* Phillips _et al._ (1984) Phillips, W. D., D. Kleppner, and F. G. Walther, 1984, private communication.
* Pitre (2011) Pitre, L., 2011, private communication.
* Pitre _et al._ (2009) Pitre, L., C. Guianvarc’h, F. Sparasci, A. Guillou, D. Truong, Y. Hermier, and M. E. Himbert, 2009, C. R. Physique 10(9), 835.
* Pitre _et al._ (2011) Pitre, L., F. Sparasci, D. Truong, A. Guillou, L. Risegari, and M. E. Himbert, 2011, Int. J. Thermophys. 32(9), 1825.
* de Podesta (2011) de Podesta, M., 2011, private communication.
* de Podesta _et al._ (2010) de Podesta, M., E. F. May, J. B. Mehl, L. Pitre, R. M. Gavioso, G. Benedetto, P. A. Giuliano Albo, D. Truong, and D. Flack, 2010, Metrologia 47(5), 588.
* Pohl _et al._ (2011) Pohl, R., F. D. Amaro, A. Antognini, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, T. W. Hänsch, _et al._ , 2011, J. Phys. Conf. Ser. 264, 012008.
* Pohl _et al._ (2010) Pohl, R., A. Antognini, F. Nez, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, _et al._ , 2010, Nature 466(7303), 213.
* Prades _et al._ (2010) Prades, J., E. de Rafael, and A. Vainshtein, 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 9, pp. 303–317.
* Puchalski _et al._ (2011) Puchalski, M., U. D. Jentschura, and P. J. Mohr, 2011, Phys. Rev. A 83, 042508.
* Quinn (1989) Quinn, T. J., 1989, Metrologia 26(1), 69.
* Quinn (2001) Quinn, T. J., 2001, Metrologia 38(1), 89.
* Quinn _et al._ (2001) Quinn, T. J., C. C. Speake, S. J. Richman, R. S. Davis, and A. Picard, 2001, Phys. Rev. Lett. 87, 111101.
* Rainville _et al._ (2005) Rainville, S., J. K. Thompson, E. G. Myers, J. M. Brown, M. S. Dewey, E. G. Kessler Jr, R. D. Deslattes, H. G. Börner, M. Jentschel, P. Mutti, and D. E. Pritchard, 2005, Nature 438(7071), 1096\.
* Ramsey (1990) Ramsey, N. F., 1990, in _Quantum Electrodynamics_ , edited by T. Kinoshita (World Scientific, Singapore), chapter 13, pp. 673–695.
* Redshaw _et al._ (2008) Redshaw, M., J. McDaniel, and E. G. Myers, 2008, Phys. Rev. Lett. 100, 093002.
* Roberts (2009) Roberts, B. L., 2009, private communication.
* Robinson (2012) Robinson, I. A., 2012, Metrologia 49(1), 113.
* Robinson and Kibble (2007) Robinson, I. A., and B. P. Kibble, 2007, Metrologia 44(6), 427.
* Ron _et al._ (2011) Ron, G., X. Zhan, J. Glister, B. Lee, K. Allada, W. Armstrong, J. Arrington, A. Beck, F. Benmokhtar, B. L. Berman, W. Boeglin, E. Brash, _et al._ , 2011, Phys. Rev. C 84, 055204.
* Rudziński _et al._ (2009) Rudziński, A., M. Puchalski, and K. Pachucki, 2009, J. Chem. Phys. 130, 244102.
* Sapirstein (2010) Sapirstein, J., 2010, J. Phys. B 43, 074015.
* Sapirstein _et al._ (1984) Sapirstein, J. R., E. A. Terray, and D. R. Yennie, 1984, Phys. Rev. D 29(10), 2290.
* Sapirstein and Yennie (1990) Sapirstein, J. R., and D. R. Yennie, 1990, in _Quantum Electrodynamics_ , edited by T. Kinoshita (World Scientific, Singapore), chapter 12, pp. 560–672.
* Schlamminger _et al._ (2006) Schlamminger, S., E. Holzschuh, W. Kündig, F. Nolting, R. E. Pixley, J. Schurr, and U. Straumann, 2006, Phys. Rev. D 74, 082001.
* Schmidt _et al._ (2007) Schmidt, J. W., R. M. Gavioso, E. F. May, and M. R. Moldover, 2007, Phys. Rev. Lett. 98, 254504.
* Schwob _et al._ (1999) Schwob, C., L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A. Clairon, 1999, Phys. Rev. Lett. 82(25), 4960, erratum: see Schwob _et al._ (2001).
* Schwob _et al._ (2001) Schwob, C., L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A. Clairon, 2001, Phys. Rev. Lett. 86(18), 4193.
* Shabaev _et al._ (1998) Shabaev, V. M., A. N. Artemyev, T. Beier, and G. Soff, 1998, J. Phys. B 31(8), L337.
* Shifrin _et al._ (1998a) Shifrin, V. Y., V. N. Khorev, P. G. Park, C. H. Choi, and C. S. Kim, 1998a, Izmer. Tekh. 1998(4), 68.
* Shifrin _et al._ (1998b) Shifrin, V. Y., P. G. Park, V. N. Khorev, C. H. Choi, and C. S. Kim, 1998b, IEEE Trans. Instrum. Meas. 47(3), 638\.
* Shifrin _et al._ (1999) Shifrin, V. Y., P. G. Park, V. N. Khorev, C. H. Choi, and S. Lee, 1999, IEEE Trans. Instrum. Meas. 48(2), 196.
* Shiner and Dixson (1995) Shiner, D. L., and R. Dixson, 1995, IEEE Trans. Instrum. Meas. 44(2), 518.
* Sick (2003) Sick, I., 2003, Phys. Lett. B 576(1-2), 62.
* Sick (2007) Sick, I., 2007, Can. J. Phys. 85(5), 409.
* Sick (2008) Sick, I., 2008, in _Precision Physics of Simple Atoms and Molecules_ , edited by S. G. Karshenboim (Springer, Lect. Notes Phys. 745, Berlin, Heidelberg), pp. 57–77.
* Sick (2011) Sick, I., 2011, Few-Body Syst. 50(1-4), 367.
* Sick (2012) Sick, I., 2012, to be published, Prog. Part. Nuc. Phys.
* Sienknecht and Funck (1985) Sienknecht, V., and T. Funck, 1985, IEEE Trans. Instrum. Meas. IM-34(2), 195.
* Sienknecht and Funck (1986) Sienknecht, V., and T. Funck, 1986, Metrologia 22(3), 209.
* Small _et al._ (1997) Small, G. W., B. W. Ricketts, P. C. Coogan, B. J. Pritchard, and M. M. R. Sovierzoski, 1997, Metrologia 34(3), 241.
* Smiciklas and Shiner (2010) Smiciklas, M., and D. Shiner, 2010, Phys. Rev. Lett. 105, 123001.
* Solders _et al._ (2008) Solders, A., I. Bergström, S. Nagy, M. Suhonen, and R. Schuch, 2008, Phys. Rev. A 78, 012514.
* Steiner _et al._ (2007) Steiner, R. L., E. R. Williams, R. Liu, and D. B. Newell, 2007, IEEE Trans. Instrum. Meas. 56(2), 592.
* Stock (2011) Stock, M., 2011, Phil. Trans. R. Soc. A 369(1953), 3936\.
* Stöckinger (2010) Stöckinger, D., 2010, in _Lepton Dipole Moments_ , edited by B. L. Roberts and W. J. Marciano (Advanced Series on Directions in High Energy Physics, Vol. 20, World Scientific, Singapore), chapter 12, pp. 393–438.
* Sturm _et al._ (2011) Sturm, S., A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. H. Keitel, and K. Blaum, 2011, Phys. Rev. Lett. 107, 023002.
* Suhonen _et al._ (2007) Suhonen, M., I. Bergström, T. Fritioff, S. Nagy, A. Solders, and R. Schuch, 2007, J. Instrum. 2, P06003.
* Sutton _et al._ (2010) Sutton, G., R. Underwood, L. Pitre, M. de Podesta, and S. Valkiers, 2010, Int. J. Thermophys. 31(7), 1310.
* Taylor (2011) Taylor, B. N., 2011, J. Res. Natl. Inst. Stand. Technol. 116(6), 797.
* Taylor and Mohr (2001) Taylor, B. N., and P. J. Mohr, 2001, IEEE Trans. Instrum. Meas. 50(2), 563.
* Tomaselli _et al._ (2002) Tomaselli, M., T. Kühl, W. Nörtershäuser, S. Borneis, A. Dax, D. Marx, H. Wang, and S. Fritzsche, 2002, Phys. Rev. A 65, 022502.
* Trapon _et al._ (2003) Trapon, G., O. Thévenot, J. C. Lacueille, and W. Poirier, 2003, Metrologia 40(4), 159.
* Trapon _et al._ (2001) Trapon, G., O. Thévenot, J.-C. Lacueille, W. Poirier, H. Fhima, and G. Genevès, 2001, IEEE Trans. Instrum. Meas. 50(2), 572.
* Tu _et al._ (2010) Tu, L.-C., Q. Li, Q.-L. Wang, C.-G. Shao, S.-Q. Yang, L.-X. Liu, Q. Liu, and J. Luo, 2010, Phys. Rev. D 82, 022001\.
* Uzan (2011) Uzan, J.-P., 2011, Living Rev. Relativity 14(2), 1.
* Valkiers _et al._ (2011) Valkiers, S., G. Mana, K. Fujii, and P. Becker, 2011, Metrologia 48(2), S26.
* Valkiers _et al._ (2010) Valkiers, S., D. Vendelbo, M. Berglund, and M. de Podesta, 2010, Int. J. Mass Spectrom. 291(1-2), 41.
* Van Dyck (2010) Van Dyck, R. S., Jr., 2010, private communication.
* Van Dyck _et al._ (2006) Van Dyck, R. S., Jr., D. B. Pinegar, S. V. Liew, and S. L. Zafonte, 2006, Int. J. Mass Spectrom. 251(2-3), 231.
* Van Dyck _et al._ (1987) Van Dyck, R. S., Jr., P. B. Schwinberg, and H. G. Dehmelt, 1987, Phys. Rev. Lett. 59(1), 26.
* Verdú (2006) Verdú, J., 2006, private communication.
* Wapstra _et al._ (2003) Wapstra, A. H., G. Audi, and C. Thibault, 2003, Nucl. Phys. A 729(1), 129.
* Weis and von Klitzing (2011) Weis, J., and K. von Klitzing, 2011, Phil. Trans. R. Soc. A 369(1953), 3954\.
* Weitz _et al._ (1995) Weitz, M., A. Huber, F. Schmidt-Kaler, D. Leibfried, W. Vassen, C. Zimmermann, K. Pachucki, T. W. Hänsch, L. Julien, and F. Biraben, 1995, Phys. Rev. A 52(4), 2664.
* Werth (2003) Werth, G., 2003, private communication.
* Wichmann and Kroll (1956) Wichmann, E. H., and N. M. Kroll, 1956, Phys. Rev. 101(2), 843.
* Wicht _et al._ (2002) Wicht, A., J. M. Hensley, E. Sarajlic, and S. Chu, 2002, Phys. Scr. T102, 82\.
* Williams _et al._ (1989) Williams, E. R., G. R. Jones, Jr., S. Ye, R. Liu, H. Sasaki, P. T. Olsen, W. D. Phillips, and H. P. Layer, 1989, IEEE Trans. Instrum. Meas. 38(2), 233.
* Williams _et al._ (1998) Williams, E. R., R. L. Steiner, D. B. Newell, and P. T. Olsen, 1998, Phys. Rev. Lett. 81(12), 2404.
* Winkler _et al._ (1972) Winkler, P. F., D. Kleppner, T. Myint, and F. G. Walther, 1972, Phys. Rev. A 5(1), 83.
* Wood and Solve (2009) Wood, B. M., and S. Solve, 2009, Metrologia 46(6), R13.
* Wundt and Jentschura (2008) Wundt, B. J., and U. D. Jentschura, 2008, Phys. Lett. B 659(3), 571.
* Yerokhin (2000) Yerokhin, V. A., 2000, Phys. Rev. A 62, 012508.
* Yerokhin (2010) Yerokhin, V. A., 2010, Eur. Phys. J. D 58(1), 57.
* Yerokhin (2011) Yerokhin, V. A., 2011, Phys. Rev. A 83, 012507.
* Yerokhin _et al._ (2005a) Yerokhin, V. A., A. N. Artemyev, V. M. Shabaev, and G. Plunien, 2005a, Phys. Rev. A 72, 052510.
* Yerokhin _et al._ (2002) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2002, Phys. Rev. Lett. 89, 143001.
* Yerokhin _et al._ (2003) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2003, Phys. Rev. Lett. 91, 073001.
* Yerokhin _et al._ (2005b) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2005b, Phys. Rev. A 71, 040101.
* Yerokhin _et al._ (2005c) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2005c, Zh. Eksp. Teor. Fiz. 128(2), 322, [JETP 101(2), 280 (2005)].
* Yerokhin _et al._ (2007) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2007, Can. J. Phys. 85(5), 521.
* Yerokhin _et al._ (2008) Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2008, Phys. Rev. A 77, 062510.
* Yerokhin and Jentschura (2008) Yerokhin, V. A., and U. D. Jentschura, 2008, Phys. Rev. Lett. 100, 163001.
* Yerokhin and Jentschura (2010) Yerokhin, V. A., and U. D. Jentschura, 2010, Phys. Rev. A 81, 012502.
* Yi _et al._ (2012) Yi, H., T. Li, X. Wang, F. Zhang, W. Huo, and W. Yi, 2012, Metrologia 49(1), 62.
* Zelevinsky _et al._ (2005) Zelevinsky, T., D. Farkas, and G. Gabrielse, 2005, Phys. Rev. Lett. 95, 203001.
* Zhan _et al._ (2011) Zhan, Z., K. Allada, D. S. Armstrong, J. Arrington, W. Bertozzi, W. Boeglin, J.-P. Chen, K. Chirapatpimol, S. Choi, E. Chudakov, E. Cisbani, P. Decowski, _et al._ , 2011, Phys. Lett. B 705(1-2), 59.
* Zhang (2011) Zhang, J. T., 2011, private communication.
* Zhang _et al._ (2011) Zhang, J. T., H. Lin, X. J. Feng, J. P. Sun, K. A. Gillis, M. R. Moldover, and Y. Y. Duan, 2011, Int. J. Thermophys. 32(7-8), 1297.
* Zhang _et al._ (1995) Zhang, Z., X. Wang, D. Wang, X. Li, Q. He, and Y. Ruan, 1995, Acta Metrol. Sin. 16(1), 1.
|
arxiv-papers
| 2012-03-24T15:49:42 |
2024-09-04T02:49:29.002792
|
{
"license": "Public Domain",
"authors": "Peter J. Mohr, Barry N. Taylor, David B. Newell",
"submitter": "Peter Mohr",
"url": "https://arxiv.org/abs/1203.5425"
}
|
1203.5529
|
###### Abstract
I review the phenomena associated with pairing in nuclear physics, most
prominently the ubiquitous presence of odd-even mass differences and the
properties of the excitation spectra, very different for even-even and odd-$A$
nuclei. There are also significant dynamical effects of pairing, visible in
the inertias associated with nuclear rotation and large-amplitude shape
deformation.
## Chapter 0 Nuclear pairing: basic phenomena revisited
### 1 Basic phenomena
In this section I will present some of the basic manifestations of pairing in
nuclei, using contemporary sources [1, 2] for the experimental data. In later
sections, I will describe in broad terms the present-day theoretical
understanding of nuclear pairing, emphasizing the many-body aspects rather
than the aspects related to the underlying Hamiltonian.
#### 1 Pairing gaps: odd-even binding energy differences
The basic hallmarks of pair condensates are the odd-even staggering in binding
energies, the gap in the excitation spectrum of even systems, and the
compressed quasiparticle spectrum in odd systems. To examine odd-even
staggering, it is convenient to define the even and odd neutron pairing gaps
with the convention
$\Delta^{(3)}_{o,Z}(N)={1\over
2}(E_{b}(Z,N+1)-2E_{b}(Z,N)+E_{b}(Z,N-1)),\,\,\,{\rm for}\,\,N\,\,{\rm odd,}$
(1) $\Delta^{(3)}_{e,Z}(N)=-{1\over
2}(E_{b}(Z,N+1)-2E_{b}(Z,N)+E_{b}(Z,N-1)),\,\,{\rm for}\,\,N\,\,\,{\rm even.}$
(2)
where $N$ and $Z$ are the neutron and proton numbers and $E_{b}$ is the
binding energy of the nucleus. The proton pairing gaps are defined in a
similar way. With the above definition, the gaps are positive for normal
pairing. The neutron pairing gaps are shown as a function of neutron number in
Fig. 1. The data for this plot was obtained from nuclear binding energies
given in the 2003 mass table [1]. The upper panel shows the gaps centered on
odd $N$. Typically, the odd-$N$ nuclei are less bound than the average of
their even-$N$ neighbors by about 1 MeV. However, one sees that there can
Figure 1: Upper panels: odd-N pairing gaps. Lower panels: even-N pairing gaps.
be about a factor of two scatter around the average value at a given $N$. Note
that there are two exceptional cases with negative $\Delta^{(3)}$ for odd
neutrons, at $N=23$ and $N=31$. I will come back them later. One can also see
a systematic trend in the gap values as a function of $N$, namely the gaps get
smaller in heavier nuclei. I will also come back to this behaviour in the
theory discussion. Another feature of the odd-$N$ gap systematics is the
occurance of dips at particular values of $N$. In fact the dips occur adjacent
to the well-known magic numbers $N=28,50,82$ and $126$. In addition there is a
dip adjacent to $N=14$, which corresponds to $n=2$ in the magic number
sequence $\frac{1}{3}(n+1)(n^{2}+2n+6)$.
The systematics of the even-$N$ gaps shown in the lower panel is similar with
respect to the following: average values, the fluctuations at each $N$, and
the smooth trend downward with increasing $N$. However, the magic number
anomolies are now very striking spikes that occur exactly at the magic
numbers. Also, the average values in lighter nuclei appear to be larger for
the even-$N$ gaps than for the odd-$N$. I will also come back to this feature
in the theory section.
The corresponding systematics of proton gaps is shown in Fig. 2. The same
qualitative features are present here as well, but the magic number effects
are less pronounced. I do not know of any explanation of this difference
between neutron and proton pairing.
Figure 2: Upper panels: odd-$Z$ pairing gaps. Lower panels: even-$Z$ pairing
gaps.
The table below gives some fits to the pairing gap systematics. Shown are the
fitted values of the gap parameterizations and the rms errors of the fits, in
units of MeV. The simplest model is a constant gap, $\Delta^{(3)}=C$, shown on
the line labeled $C$. One sees that a typical gap size is 1 MeV, and typical
fluctuations about that are smaller by a factor of 3. Beyond that, there are
differences between protons and neutrons and between the odd and the even
gaps. The even gaps are somewhat larger and have somewhat larger fluctuations,
which is to be expected in view of the shell effects exhibited in Fig. 1. The
odd proton gap is smaller than the odd neutron gaps which might be expected
from the repulsive Coulomb contribution to the pairing interaction. There is
also a mean-field contribution of the Coulomb that has opposite signs for even
and odd protons. Indeed the even proton gaps are actually larger than their
neutron counterparts.
For the next lines in the table, I come back to the broad trend in Fig. 1, a
systematic decrease in gaps with increasing mass number. It is conventional to
describe this with a fractional power dependence, $\Delta^{(3)}=c/A^{1/2}$.
This decreases the rms errors somewhat, but there is no theoretical basis for
the fractional power of $A$. In the last line I show the result of a two-
parameter fit to the functional form $\Delta^{(3)}=c_{1}/A+c_{2}$. This
functional form is more justified by theoretical considerations, as will be
discussed in the theory section below.
$\Delta^{(3)}$ | protons | protons | neutrons | neutrons
---|---|---|---|---
$o/e$ | odd | even | odd | even
data set | 418 | 407 | 443 | 442
$C$ | $0.96\pm 0.28$ | $1.64\pm 0.46$ | $1.04\pm 0.31$ | $1.32\pm 0.42$
$c/A^{1/2}$ | | | $12/A^{1/2}\pm 0.25$ | $12/A^{1/2}\pm 0.28$
$c_{1}/A+c_{2}$ | | | $24/A+0.82\pm 0.27$ | $41/A+0.94\pm 0.31$
#### 2 Basic spectral properties
The other strong signatures of pairing are in excitation spectra. In the
simple BCS theory, the lowest excited states in an even system requires
breaking two pairs giving an excitation energy
$E_{ex}\approx 2\Delta_{\rm BCS}.$ (3)
On the other hand, in the odd particle number system, the quasiparticle level
density diverges at the Fermi energy. This contrasting behavior is very
obvious in the nuclear spectrum. As an example, the isotope chain at proton
magic number $Z=50$ (the element Sn) has been a favorite for exhibiting and
studying pairing effects. Figure 3 shows the low-lying spectra of odd-$N$
members of the chain. One can see that there are several levels within one MeV
of the ground state. The spins and parities of the levels (not shown)
correspond very well with the single-particle orbitals near the neutron Fermi
level. However, the spectrum is very compressed with respect to orbital
energies calculated with a shell model potential well. In contrast, the even
members of the chain have no excited states at all within the excitation
energy range displayed in the Figure.
Figure 3: Energy levels of odd-$N$ Sn isotopes
Let us look now look at the global systematics for excitations in the even-
even nuclei. The estimate in Eq. (3) is too naive because there can be
collective excitations within the gap, as is well-known from early days of BCS
theory [3]. For example, there are longitudinal sound modes in an uncharged
superfluid fermionic liquid. These have a phonon spectrum allowing frequencies
within the quasiparticle gap. One might expect that such modes would be absent
in finite systems when the size of the system is small compared to the
coherence length of the pairing field. In fact the situation for nuclear
excitations is much more complicated. However, just for presenting the
systematics, we use the right-hand side of Eq. (3) to scale the excitation
energies, taking the ratio $E_{2}/2\Delta$, where $E_{2}$ is the excitation
energy and $\Delta$ is the smaller of $\Delta_{eZ}^{(3)}$ and
$\Delta_{eN}^{(3)}$.
The scaled excitation energies of the first excited states in even-even are
shown in Fig. 4. With only a few exceptions these states have angular momentum
and parity quantum numbers $J^{\pi}=2^{+}$ and can be considered to be
collective quadrupolar excitations of the ground state.
Figure 4: Energy gap in the excitation spectrum of even-even nuclei, scaled
to $2\Delta^{(3)}$. See text for details.
All of the ratios are smaller than one, with most in the range 0.1-0.5. The
very small excitations in the mass ranges $A=160-180$ and $220-250$ correpond
to nuclei with static quadrupole deformations.
The physics underlying these excitations is the softness of a typical nucleus
with respect to quadrupolar deformations. On a qualitative level, the
collectivity is similar to the phonon collectivity in the infinite Fermi gas.
A quantitative measure of the collectivity is the sum-rule fraction contained
in the excitation, using the energy-weighted sum rule for some density
operator. For the phonon case, the sum rule fraction approaches 100% when the
frequency of the collective mode is small compared to the gap [3]. The
collectivity in the nuclear quadrupole excitations is quite different. The sum
rule fraction carried by the lowest $2^{+}$ excitation is more or less
constant over the entire range of nuclear masses, but it only about 10% of the
total (for isoscalar quadrupole transitions. This is known as the Grodzins
systematics [4, 5]. The observed distribution of sum rule fractions is plotted
as a histogram in Fig. 5.
Figure 5: Sum rule fraction for the first excited $2^{+}$ state in even-even
nuclei. See text for details.
Turning to odd-$A$ spectrao, some systematics related to the level density are
shown in Fig. 6. The average excitation energy of the first excited state is
plotted for each odd mass number $A$, averaging over even values of $Z$. For
comparison, the solid line is the expected spacing in the Fermi gas formula
for the single-particle level density,
${dn_{s}\over dE}=V{mk_{F}\over 2\pi^{2}}\approx{A\over 100}\,\,\,\,{\rm
MeV}.$ (4)
The subscript $s$ on $N_{s}$ indicates that only one spin projection is
counted, and $k_{F}$ is the Fermi momentum. $V$ is the volume of the nucleus,
which is (roughly) proportional to the number of nucleons $A$. One can see
from the Figure that a typical spacing is a factor of 10 smaller than that
given by the Fermi gas formula. Clearly interaction effects are at work to
increase the level density near the ground state.
Figure 6: Average energy of the first excited state in odd-$A$ nuclei. The
dashed line is the Fermi gas estimate, Eq. 4
### 2 Theory
Mean-field theory has made enormous strides in nuclear physics; the self-
consistent mean field theory based on the Hartree-Fock-Bogoliubov
approximation and using semi-phenomenological energy functionals is now the
tool of choice for the global description of nuclear structure. It is not my
intention to review this subject since it is well covered elsewhere in this
volume.
Neverless, there are number of aspects of nuclear pairing that can be can
rather easily understood using only the more qualitative aspects of pairing
theory. Besides the pairing gaps and the effect on level densities, there are
important consequences for two-nucleon transfer reactions and on dynamic
properties such as radioactive decay modes. This section presents an overview
of some of these aspects.
#### 1 Mean-field considerations
BCS pairing is not the only source of odd-even staggering in binding energies.
As is well-known in the physics of finite electronic systems, the Kramers
degeneracy of single-particle orbitals gives rise to an odd-even effect. In a
fixed potential well, the pair-wise filling of the orbitals makes to a
contribution to $\Delta^{(3)}_{e}$ that varies with system size as the single
particle level spacing, $\Delta^{(3)}_{e}\sim A^{-1}$. In addition, the
diagonal matrix elements of the two-body interaction in the Hartree-Fock
orbitals also contribute to the odd-even staggering, both in
$\Delta^{(3)}_{o}$ and in $\Delta^{(3)}_{e}$ [6]. In the nuclear context, the
volume occupied by the orbitals is (approximately) proportional to the mass
number $A$, so this interaction contribution also varies as $A^{-1}$.
The last line of the Table shows a fit to the neutron pairing gaps including
an $A^{-1}$ term in the parameterization. It does almost as well as the
phenomenological $A^{1/2}$ form. Note also that the coefficient of $A^{-1}$ is
larger for the even gaps than the odd ones. This is just what is to be
expected from the contribution of the two-fold degenerate orbital energies.
Another mean field effect can be interpreted by Eq. (5,6) below, exhibiting
the dependence of the pairing gap on the single-particle level densities. In
general, level densities at the Fermi level are higher in spherical nuclei
than in deformed nuclei because of the spherical shell degeneracy. Thus, one
expects larger pairing gaps in spherical nuclei than in deformed. Even more
dramatic is the shell quenching seen in the odd-$N$ gaps in Fig. 1. The Fermi
level in the spherical nuclei showing quenched gaps turns out to be in the
$p_{1/2}$ or $s_{1/2}$ shell, which have low degeneracy. Thus, the occurance
of the shell quenching is only partly due to the adjacent magic number.
#### 2 Strength of the pairing interaction
The BCS theory gives the following formula for the gap parameter [7],
$\Delta_{\rm BCS}=(E_{max}-E_{min})\exp(-1/g)$ (5)
where
$g=-G{dn_{s}\over dE}.$ (6)
Here the prefactor of the exponential is the window of single-particle
energies for orbitals participating in the pairing and $G$ is the strength of
the pairing interaction. Eq. (6) defines the dimensionless quantity $g$ that
characterizes the strength of the pairing condensate.
In present-day theory, the qualitative formula Eq. (5) is superceded by
detailed calculations of the orbitals and the pairing interaction, based on
Hartree-Fock (HF) mean-field theory or Hartree-Fock-Bogoliubov (HFB) theory.
This permits the treatment of the interaction by a two-nucleon potential and
replacing of a generic level density by computed single-particle level
spectra. However, there are significant uncertainties about both these
aspects, and the pairing interaction is often parameterized in a simple way.
As an example, I show results of a global study of pairing systematics that
used the Skyrme energy functional for the mean field and a contact interaction
for the pairing [8]. The odd-$N$ pairing gaps were calculated for two
strengths of the pairing interaction, giving average gaps shown as the filled
circles in Fig. 7. In the HFB calculations, the energy window was taken to be
$E_{max}-E_{min}=100$ MeV. Using this value in Eq. 5, the calculated average
gap at $V_{0}/V_{0}^{sd}=1$ is reproduced for $g=0.20$. Noting that $g$
depending linearly on the pairing strength, Eq. 5 gives the dashed line as a
function of $V_{0}$. One sees that there is a very strong dependence of the
gap on the pairing strength which is reproduced by the simple theory of Eq. 5.
It is interesting to note also that Ref. [9][9] also estimated $g$ as
$g\approx 0.2$ using the meager data available at the time.
Figure 7: Filled circles: average pairing gaps for 443 odd-$N$ neutron gaps,
calculated for two strengths of a contact pairing interaction [8]. Dot-dashed
curve shows the dependence on strength according to eq. 5.
##### Origin of the pairing interaction
It is no surprise that conditions for pairing are satisfied in nuclei. The
nuclear interaction between identical nucleons is strongly attractive in the
spin-zero channel, almost to the degree to form a two-neutron bound state.
While this explains the origin at a qualitative level, the many-body aspects
of the nuclear interaction make it difficult to derive a quantitative theory
starting from basic interactions. The progress one has made so far is reviewed
in other chapters of this book, so I won’t go into detail here. But just for
perspective, I mention some of the major issues.
I first recall problems with the mean-field interaction to use at the Hartree-
Fock level. Most obviously, the effective interaction between nucleons in the
nuclear medium is strongly modified by the Pauli principle. The Pauli
principle suppresses correlations between nucleons and that in turn make the
effective interaction less attractive. Beyond that, it seems unavoidable to
introduce three-body interactions in a self-consistent mean-field theory.
These interactions have two origins. The first is the three-nucleon
interaction arising from sub-nucleon degrees of freedom. It has been
convincingly demonstrated that such interactions are needed to reproduce
binding energies of light nuclei and to calculated the bulk properties of
nuclear matter. Besides this more fundamental three-body interaction, there
may an induced interaction associated with the short-ranged correlations and
their suppression in the many-body environment. In the popular
parameterization of the effective interaction for use in mean-field theory,
the three-body interaction energy has the same order of magitude as the two-
body interaction energy. It would thus seem to be a great oversimplication to
ignore the three-body effects in the pairing interaction.
The last issue is the role of the induced interaction associated with low-
frequency excitations. We have seen that the nucleus is rather soft to surface
deformations. the virtual excitation of these modes would contribute to the
pairing in exactly the same way that phonon provide an attractive pairing
interaction for the electrons in a superconductor. The size of the induced
interaction is estimated in Ref. [10][10]; it may well have the same
importance as the two-particle interaction. Note that if low-frequency phonons
were dominant, the energy scale in Eq. (5) would be greatly reduced.
##### Spin-triplet pairing
The strong attraction between identical nucleons was the starting point for
the discussion of the pairing interaction in the previous section. In fact,
the attraction is even stronger between neutrons and protons in the spin $S=1$
channel. Here the interaction gives rise to the deuteron bound state.
Nevertheless, all the pairing phenomena seen above are a result of $S=0$
pairing between identical nucleons.
This connundrum is resolved in two ways. First of all, pairing is only favored
when all the particles can participate. The spin triplet interaction is only
strong in neutron-proton pairs, so it would be suppressed in nuclei with a
large imbalance between neutron and proton numbers. The other factor working
against spin-triplet pairing is the spin-orbit field of the nucleus. It breaks
the spin coupling of the pair wave function, but it is more effective in the
spin-triplet channel [11]. In any case, an increase in nuclear binding
energies is seen along the $N=Z$ line, called the “Wigner energy” [12].
#### 3 Dynamics
The dynamic properties of an extended fermionic system depend crucially on the
presence of a pairing condensate, changing it from a highly viscous fluid to a
superfluid. The effects in nuclei are not quite as dramatic as in extended
systems because the pairing coherence length in nuclei exceeds the size of the
nucleus. Nevertheless, the presence of a highly deformable surface in nuclei
requires that pairing be treated in a dynamical way.
##### Rotational inertia
Figure 8: Distribution of nuclei with respect to deformation indicator
$R_{42}$
The most clearly documented dynamic influence of pairing is its effect on the
moment of inertia of deformed nuclei. Without pairing, the rotational spectrum
of a deformed fermionic droplet is believed to follow the spectrum of a rigid
rotor,
$E_{J}={\hbar^{2}\over 2{\cal I}}J(J+1).$ (7)
Here $\hbar J$ is the angular momentum and the moment of inertia ${\cal I}$
would be close to the rigid value
${\cal I}\approx\frac{2}{3}Am\langle
r^{2}\rangle\approx\frac{2}{5}A^{5/3}mr_{0}^{2}.$ (8)
The author knows of no proof of this assertion, but it can derived from the
cranking approximation applied to a many-particle wave function in a (self-
consistent) deformed harmonic oscillator potential [13, pp. 77-78]. If the
pairing were strong enough to make the coherence length small compared to the
size of the system, the system would be a superfluid having irrotational flow
and a corresponding inertial dynamics. What is somewhat surprising is that the
weak pairing that is characteristic of nuclei still has a strong effect on the
inertia.
One can separate out the deformed nuclei from the others by making use of the
ratio excitation energies
$R_{42}={E_{4}\over E_{2}}.$ (9)
It is a good indicator of the character of the nucleus and has the value
$R_{42}=10/3$ for an axial rotor. A histogram of $R_{42}$ for all the nuclei
for which the energies are known is shown in Fig. 8. There is a sharp peak
around the rotor value. The $E_{2}$ excitation energies of the nuclei
corresponding to the peak are plotted in Fig. 9 as a function of $A$. Also
plotted (dashed line) is the predicted value assuming a rigid rotor, Eq. (8).
Figure 9: Excitation energy of the first $2^{+}$ state in deformed nuclei. The
line shows the prediction assuming a rigid rotor.
The experimental energies are systematically higher by a factor $\sim 2$, thus
requiring inertias about half the rigid values. Present-day self-consistent
mean-field theory is very successful in reproducing the experimental inertias,
calculating them is what is called the self-consistent cranking approximation.
As an example, the lower panel of Fig. 9 shows the calculated $2^{+}$ energies
using the HFB theory with an interaction that includes pairing[14]. The
average energies are very well reproduced, and the rms errors in the energies
are only $\pm 10\%$. While the theory works very well, it does not provide a
parametric understanding of the dependence of the inertia on the pairing
strength. Naively one might have expected that the effects would controlled by
the ratio of the size of the nuclei to the coherence length of the Cooper
pairs, which is a small number. We will also see in the next section another
dynamic property showing a large influence of pairing.
##### Large amplitude collective motion
Figure 10: Potential energy curve for the decay 223Rn $\rightarrow$ 209Pb +
14C. The outside potential is a combination of Coulomb and nuclear heavy-ion
potentials. The dots show the assumed Hartree-Fock states that connect the
ground state ${223}$Rn configuration to the final-state cluster configuration.
Also of interest, particularly in the theory of fission, is the effect of
pairing on large-amplitude shape changes. Qualitatively, it is clear that
pairing promotes fluidity. The degree to which this happens can be examined in
one of the important observables of nuclear fission induced by low-energy
excitation, such as occurs in neutron capture. The observable is the internal
energy of the fission fragments. With nonviscous fluid dynamics, the internal
energy would be largely deformation energy. With more viscous dynamics, there
would be additional thermal energy. So far, one has not been able to perform
realistic enough calculations to compare theory and experiment. But the
computational tools for the time-dependent HFB theory are now reaching the
point where such a test can be made. (See Chapter X in this book).
Spontaneous fission is a decay mode that requires the nucleus to tunnel under
a barrier as it is changing shape. This kind of under-the-barrier dynamics is
extremely sensitive to the character of the system, whether it is normal or
superfluid. If the system is normal, the relevant configurations under the
barrier are close to Hartree-Fock with relatively small interaction matrix
elements mixing different configurations. On the other hand, if there is
pairing condensate, the interaction between configurations can be enhanced by
a factor $2\Delta^{2}/G^{2}$ [10, p. 159, Eq. (7.8)], where here $G$ is a
typical interaction matrix element between neighboring mean-field
configurations. Numerically, the pairing enhancement factor can be an order of
magnitude or more. One should also keep in mind that in tunneling, the
lifetime depends exponentially on the inertial parameters of the dynamics. As
an example, the nucleus 234U is observed to decay by many different channels,
ranging from alpha decay to spontaneous fission, and including exotic modes
such as emission of a Neon isotope. The observed lifetimes of these decays
range over 12 order of magnitude. Theory including the enhancement factor is
able to reproduce the lifetimes to within one or two orders of magnitude[10,
p. 163, Table 7.1]. Without the enhancement factor, there would be no
possibility to explain them.
### Acknowledgment
I wish to thank A. Sogzogni for access to the NNDA data resources. I also
thank A. Steiner and S. Reddy for helpful comments on the manuscript. This
work was supported by the US Department of Energy under grant DE-
FG02-00ER41132.
## References
* 1. G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A729, 337 (2003).
* 2. Brookhaven data base, http://www.nndc.bnl.gov/.
* 3. P.W. Anderson, Phys. Rev. 112 1900 (1958).
* 4. L. Grodzins, Phys. Lett. 2, 88 (1962).
* 5. S. Raman, C. Nestor, and P. Tikkanen, At. D. Nucl. D. Tables 78 1 (2001).
* 6. T. Duguet, P. Bonche, P.H. Heenen, J. Meyer, Phys. Rev. C 65 014311 (2001).
* 7. “Theory of Superconductivity”, J.R. Schrieffer, (Benjamin, New York, 1964) p. 41, Eq. (34).
* 8. G.F. Bertsch, C.A. Bertulani, W. Nazarewicz, N. Schunck, and M.V. Stoitsov, Phys. Rev. C 79 034306 (2009).
* 9. A. Bohr, B. Mottelson, and D. Pines, Phys. Rev. 110 936 (1958).
* 10. “Nuclear superfluidity : pairing in finite systems”, D.M. Brink and R.A. Broglia., (Cambridge University Press, 2005).
* 11. A. Poves and G. Martinez-Pinedo, Phys. Lett. B 430, 203 (1998).
* 12. D. Lunney, J.M. Pearson and C. Thibault, Rev. Mod. Phys. 75 1021 (2003).
* 13. A. Bohr and B. Mottelson, Nuclear Structure, Vol. II, (Benjamin, 1974).
* 14. J.-P. Delaroche, et al., Phys. Rev. C 81 014303 (2010).
|
arxiv-papers
| 2012-03-25T18:40:52 |
2024-09-04T02:49:29.058302
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. F. Bertsch",
"submitter": "George F. Bertsch",
"url": "https://arxiv.org/abs/1203.5529"
}
|
1203.5552
|
# A metric theory of gravity with torsion in extra-dimension
Karthik H. Shankar Center for Memory and Brain, Boston University Anand
Balaraman Department of Physics, Georgia Southern University Kameshwar C.
Wali Department of Physics, Syracuse University
###### Abstract
We consider a theory of gravity with a hidden extra-dimension and metric-
dependent torsion. A set of physically motivated constraints are imposed on
the geometry so that the torsion stays confined to the extra-dimension and the
extra-dimension stays hidden at the level of four dimensional geodesic motion.
At the kinematic level, the theory maps on to General Relativity, but the
dynamical field equations that follow from the action principle deviate
markedly from the standard Einstein equations. We study static spherically
symmetric vacuum solutions and homogeneous-isotropic cosmological solutions
that emerge from the field equations. In both cases, we find solutions of
significant physical interest. Most notably, we find positive mass solutions
with naked singularity that match the well known Schwarzschild solution at
large distances but lack an event horizon. In the cosmological context, we
find oscillatory scenario in contrast to the inevitable singular big bang of
the standard cosmology.
## I Introduction
Einstein viewed the space-time as a pseudo-Riemannian differentiable manifold
in order to generalize the special-relativistic flat space-time to include
gravity. This was primarily motivated by the fact that the local flatness of
the manifold structure naturally implemented his principle of equivalence. The
generalization came along with the revolutionary idea that the trajectory of
any freely moving test body is simply a geodesic in the curved manifold, and
that gravity is not a Newtonian instantaneous action force, but an effect of
the curvature of the space-time manifold.
The basic constituents of the manifold structure are the _metric_ , that
defines the distance between any two points of the manifold, and the
_connection_ , that defines the covariant derivative and the curvature of the
manifold. Any theory of gravity should couple the dynamics of these quantities
to the dynamics of the matter moving in the space-time manifold. Among the
existing theories, the ensuing field equations of general relativity (GR) are
perhaps the simplest. Since in GR, _torsion_ , the antisymmetric combination
of connection coefficients, is identically zero, and since GR has withstood
numerous precise experimental tests Will ; Dicke , introduction of torsion has
seemed superfluous except in the presence of matter with intrinsic spin as in
Einstein-Cartan formulations Hehl ; Watanabe ; Shapiro ; Deser ; Poplawski .
However, for two major reasons, alternate theories of gravity that reduce to
GR in the weak field limit are seriously pursued. The first reason is that GR
leads to inevitable singularities - black holes (death of a massive star) and
big bang (birth of the universe). Though it is conventionally assumed that
quantization would eliminate these singularities, GR is not readily amenable
to quantization. The second reason is that the standard model of cosmology
based on GR requires most of the universe to be composed of unknown dark
energy in order to account for various cosmological observations Trodden . A
common strategy to construct modified theories of gravity is to make the
Lagrangian density a nontrivial function of the Ricci scalar SotiFara ; Olmo
and use the action principle to derive the modified field equations. Another
common strategy is to introduce extra-dimensions while constraining the
physical particles to a (3+1) dimensional hyper-surface as in the brane-world
theories Maartens ; DGP ; RandSund . In this paper we explore a different
approach by introducing metric-dependent torsion in Kaluza-Klein type theories
Wesson ; Kalinowski with one extra dimension.
In our approach, we consider a five-dimensional (5D) manifold foliated by a
family of 4D hyper-surfaces, whose geometries are virtually indistinguishable
from that of the 4D space-time of GR. The axis of foliation is special in the
sense that there could exist non-vanishing torsion components along that
dimension. We impose constraints on the connection so that any motion in the
fifth dimension does not affect observations based on the geodesic motions
along the 4D hyper-surfaces, thus keeping the fifth dimension essentially
hidden. The imposed constraints determine uniquely all the non-vanishing
torsion components in terms of the 5D metric fields, making this a purely
metric theory of gravity. Besides uniquely determining the torsion in the 5D
geometry, the imposed constraints lead to interesting equivalence between the
5D geometry with torsion and the torsion-free 4D geometry of GR. In
particular, the components of the connection and the Ricci tensor along the 4D
hypersurfaces turn out to exactly match what would arise from GR on a 4D
space-time. Consequently, any test of this theory based on geodesic motions
will yield the same results as GR.
Though by construction, the extra-dimension is hidden at the level of geodesic
motion, its effect is clearly reflected in the field equations. The field
equations are obtained by imposing the constraints on the action and varying
it with respect to the metric. This leads to global solutions qualitatively
distinct from those obtained from GR. Most notably, we find positive mass
naked singularity solutions that match the Schwarzschild solution at large
distances but lack an event horizon. In the cosmological context, we find
oscillatory solutions in contrast to the inevitable singular big bang in
standard cosmology.
We begin section II with a review of the general framework of the 5D geometry.
Section III deals with the specification of the constraints and the
determination of the torsion and connection in terms of the metric. Section IV
is devoted to the derivation of modified Einstein equations from the standard
action principle using Ricci scalar as the Lagrangian density. In section V,
we apply the modified Einstein equations to the homogeneous and isotropic
cosmology and identify numerical solutions pointing to accelerating and
oscillatory solutions to the universe. In section VI, we discuss static
spherically symmetric vacuum solutions and demonstrate the existence of
positive mass naked singularity solutions. The final section is devoted to a
summary and discussion of the results.
## II General Framework of 5D geometry
We denote the coordinates of the 5D manifold by the latin indices, $i,j,k,...$
that take values 0,1,2,3 and 5, and the coordinates along the 4D hypersurfaces
by the greek indices, $\mu,\nu,\lambda,...$ that take values 0,1,2 and 3. Fig.
1 is a schematic representation of the 5D geometry. With $x^{5}$ denoting the
axis of foliation, the metric of the foliated 5D geometry has the form:
$\mathbf{g}_{ij}=\left[\begin{array}[]{ccc|c}&&&\\\
&\mathrm{g}_{\mu\nu}+\epsilon\mathrm{A}_{\mu}\mathrm{A}_{\nu}\Phi^{2}&&\,\,\,\epsilon\mathrm{A}_{\mu}\Phi^{2}\\\
&&&\\\ \hline\cr&\epsilon\mathrm{A}_{\nu}\Phi^{2}&&\epsilon\Phi^{2}\\\
\end{array}\right]$ (1)
$\mathbf{g}^{ij}=\left[\begin{array}[]{ccc|c}&&&\\\
&\,\,\,\,\mathrm{g}^{\mu\nu}&&-\mathrm{A}^{\mu}\\\ &&&\\\
\hline\cr&-\mathrm{A}^{\nu}&&\mathrm{A}_{\lambda}\mathrm{A}^{\lambda}+\epsilon\Phi^{-2}\\\
\end{array}\right]$ (2)
$\displaystyle\mathbf{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathrm{g}_{\mu\nu}+\epsilon\mathrm{A}_{\mu}\mathrm{A}_{\nu}\Phi^{2},\,\mathbf{g}_{\mu
5}=\epsilon\mathrm{A}_{\mu}\Phi^{2},\,\mathbf{g}_{55}=\epsilon\Phi^{2},$
$\displaystyle\mathbf{g}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathrm{g}^{\mu\nu},\,\mathbf{g}^{\mu
5}=-\mathrm{A}^{\mu},\,\mathbf{g}^{55}=\mathrm{A}_{\lambda}\mathrm{A}^{\lambda}+\epsilon\Phi^{-2}.$
(3)
Here $\mathrm{A}_{\mu}$ is a 4D vector, whose indices are raised and lowered
with respect to the 4D metric $\mathrm{g}^{\mu\nu}$ and $\mathrm{g}_{\mu\nu}$.
The fifth dimension is space-like if $\epsilon=+1$ and it is time-like if
$\epsilon=-1$. Note that the 5D metric is denoted by the bold face
$\mathbf{g}$ and the 4D metric is light faced $\mathrm{g}$.
Figure 1: Schematic representation of the 5D geometry.
Let us denote the connection in the 5D geometry by $\tilde{\Gamma}_{\cdot\,}$
and its antisymmetric part, the torsion by $\mathrm{T}_{\cdot\,}$.
$\mathrm{T}^{i}_{\cdot\,jk}=\tilde{\Gamma}^{i}_{\cdot\,jk}-\tilde{\Gamma}^{i}_{\cdot\,kj}.$
(4)
Denoting the covariant derivative induced by the connection by
$\tilde{\nabla}$, the metricity condition is expressed as
$\tilde{\nabla}_{k}\mathbf{g}_{ij}=0$. With the metricity condition, the
connection $\tilde{\Gamma}_{\cdot\,}$ can be expressed as a sum of the Levi-
Civita connection $\hat{\Gamma}_{\cdot\,}$ and the contorsion
$\mathrm{K}_{\cdot\,}$,
$\tilde{\Gamma}^{i}_{\cdot\,jk}=\hat{\Gamma}^{i}_{\cdot\,jk}+\mathrm{K}^{i}_{\cdot\,jk},$
(5)
where the Levi-Civita connection is expressed purely in terms of the metric
$\hat{\Gamma}^{i}_{\cdot\,jk}=\Big{\\{}{}_{j}{}^{i}{}_{k}\Big{\\}}=\frac{1}{2}\mathbf{g}^{im}[{\partial_{j}\mathbf{g}_{km}+\partial_{k}\mathbf{g}_{jm}-\partial_{m}\mathbf{g}_{jk}}],$
(6)
and the contorsion in terms of the torsion Nakaharabook .
$\mathrm{K}^{i}_{\cdot\,jk}=\frac{1}{2}\left[\mathrm{T}^{i}_{\cdot\,jk}+\mathrm{T}^{\cdot\,i}_{j\,\cdot
k}+\mathrm{T}^{\cdot\,i}_{k\,\cdot j}\right].$ (7)
In the absence of torsion, the connection is simply the Levi-Civita part. In
order to compare the dynamics of this geometry to GR, we consider a reference
space-time in four dimensions with the metric $\mathrm{g}_{\mu\nu}$ and
torsion-free 4D Levi-Civita connection $\Gamma^{\lambda}_{\cdot\,\mu\nu}$.
$\Gamma^{\lambda}_{\cdot\,\mu\nu}=\frac{1}{2}\mathrm{g}^{\lambda\alpha}[{\partial_{\mu}\mathrm{g}_{\alpha\nu}+\partial_{\nu}\mathrm{g}_{\mu\alpha}-\partial_{\alpha}\mathrm{g}_{\mu\nu}}].$
(8)
This connection is different from the 4D components of the 5D Levi-Civita
connection that contains additional terms
$\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ (see eq. 69 in the appendix) that
depend on the extra-dimensional metric fields $\mathrm{A}_{\mu}$ and $\Phi$.
Hence the 4D components of the 5D Levi-Civita connection can be written as
$\hat{\Gamma}^{\lambda}_{\cdot\,\mu\nu}=\Gamma^{\lambda}_{\cdot\,\mu\nu}+\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$
(9)
In the presence of torsion, with the inclusion of contorsion, the 4D
components of the 5D connection takes the form
$\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}=\Gamma^{\lambda}_{\cdot\,\mu\nu}+\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}+\mathrm{K}^{\lambda}_{\cdot\,\mu\nu}.$
(10)
We note that the additional terms
$(\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}+\mathrm{K}^{\lambda}_{\cdot\,\mu\nu})$
do not generally vanish. However, in the next section we impose constraints on
the connection and find that these terms do vanish.
## III Constraints on the Connection
With minimal modifications to standard GR in mind, we first assume that the 4D
components of the connection $\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ are
symmetric, that is (i) $\mathrm{T}^{\lambda}_{\cdot\,\mu\nu}=0$. Next, we
require that geodesic motion and its observable effects in 4D are not affected
by any motion in the fifth dimension. This requirement essentially ensures
that the fifth dimension stays hidden at the level of 4D geodesics. For this
purpose, considering the 4D components of the geodesic equations in the 5D
geometry, namely,
$\displaystyle\overset{..}{x}^{\lambda}$
$\displaystyle+\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\left(\tilde{\Gamma}^{\lambda}_{\cdot\,\mu
5}+\tilde{\Gamma}^{\lambda}_{\cdot\,5\mu}\right)\dot{x}^{\mu}\dot{x}^{5}+\tilde{\Gamma}^{\lambda}_{\cdot\,55}\left(\dot{x}^{5}\right)^{2}=0,$
we are led to the second constraint (ii)
$\tilde{\Gamma}^{\lambda}_{\cdot\,i5}=\tilde{\Gamma}^{\lambda}_{\cdot\,5i}=0$.
An alternative formulation of these constraints in terms of vielbeins is
worked out in MPLA . These constraints are clearly not tensorial in nature
because the fifth dimension is singled out. It turns out however, that they
are sufficient to determine uniquely all the non-vanishing torsion components
in terms of the metric (see Appendix A for details).
$\displaystyle\mathrm{T}^{\mu}_{\cdot\,ij}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\mathrm{T}^{5}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle
2\partial_{[\mu}\mathrm{A}_{\nu]}+2\mathrm{J}_{[\mu}\mathrm{A}_{\nu]},$
$\displaystyle\mathrm{T}^{5}_{\cdot\,\mu 5}$ $\displaystyle=$
$\displaystyle\mathrm{J}_{\mu}-\partial_{5}\mathrm{A}_{\mu}-\mathrm{A}_{\mu}\mathrm{J}_{5},$
(11)
where $\mathrm{J}_{i}\equiv\Phi^{-1}\partial\Phi/\partial x^{i}$.
Using the above results for torsion and equations 5, 6 and 7 we find the
connection coefficients,
$\displaystyle\tilde{\Gamma}^{\lambda}_{\cdot\,55}$ $\displaystyle=$
$\displaystyle\tilde{\Gamma}^{\lambda}_{\cdot\,\nu
5}=\tilde{\Gamma}^{\lambda}_{\cdot\,5\nu}=0,$
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\mathrm{A}_{\nu}+\mathrm{J}_{\mu}\mathrm{A}_{\nu},$
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,5\mu}$ $\displaystyle=$
$\displaystyle\partial_{5}\mathrm{A}_{\mu}+\mathrm{J}_{5}\mathrm{A}_{\mu},$
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,\mu 5}$ $\displaystyle=$
$\displaystyle\mathrm{J}_{\mu},\,\,\,\tilde{\Gamma}^{5}_{\cdot\,55}=\mathrm{J}_{5},\,\,\,\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}=\Gamma^{\lambda}_{\cdot\,\mu\nu}.$
(12)
Here $\nabla_{\mu}$ is the covariant derivative operator in the torsion free
4D geometry with metric $\mathrm{g}_{\mu\nu}$. This connection has a very
special property: its 4D components are exactly the 4D Levi-Civita connection.
That is, $\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ and
$\mathrm{K}^{\lambda}_{\cdot\,\mu\nu}$ in eq. 10 exactly cancel each other.
In addition to determining the torsion and the connection in terms of the
metric fields, the constraints also imply that the 4D metric on all the
hypersurfaces are identical. As a consequence, the 4D components of the
connection also do not depend on $x^{5}$.
$\frac{\partial\mathrm{g}_{\mu\nu}}{\partial
x^{5}}=0\,\,\,\Longrightarrow\,\,\,\frac{\partial\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}}{\partial
x^{5}}=0.$ (13)
This should be contrasted with the Kaluza-Klein type theories where it is _a
priori_ assumed that $\mathrm{g}_{\mu\nu}$, $\mathrm{A}_{\mu}$ and $\Phi$ are
independent of $x^{5}$, known as the cylindrical condition. In our framework,
though $\mathrm{g}_{\mu\nu}$ is required to be independent of $x^{5}$,
$\mathrm{A}_{\mu}$ and $\Phi$ can in principle depend on $x^{5}$.
Substituting the connection (eq. 12) in the Ricci tensor defined by
$\tilde{R}_{ik}=\partial_{k}\tilde{\Gamma}^{j}_{\cdot\,ji}-\partial_{j}\tilde{\Gamma}^{j}_{\cdot\,ki}+\tilde{\Gamma}^{j}_{\cdot\,km}\tilde{\Gamma}^{m}_{\cdot\,ji}-\tilde{\Gamma}^{j}_{\cdot\,jm}\tilde{\Gamma}^{m}_{\cdot\,ki},$
(14)
we find
$\tilde{R}_{\mu\nu}=R_{\mu\nu},\,\,\tilde{R}_{\mu
5}=\tilde{R}_{5\mu}=\tilde{R}_{55}=0.$ (15)
Here $R_{\mu\nu}$ represents the Ricci tensor constructed from the torsion-
free 4D Levi-Civita connection. Hence the 4D components of the Ricci tensor
exactly match the Ricci tensor in GR with the metric $\mathrm{g}_{\mu\nu}$. It
also follows that the 5D Ricci scalar is exactly the same as the Ricci scalar
in the torsion-free 4D space-time, that is $\tilde{R}=R$.
An important point to emphasize is that at the level of geometry, this
framework is virtually indistinguishable from the torsion-free 4D space-time
of GR. Any observable geodesic motion or geodesic deviations between particles
would match what we expect based on GR. However, this is true only to the
extent the metric $\mathrm{g}_{\mu\nu}$ is identical to the solution of the
Einstein’s equations in GR. In section VI, we will see that this is indeed
true in the weak field limit for spherically symmetric vacuum solutions that
are relevant for experimental observations within the solar system.
## IV Action principle and Modified Einstein Equations
We start with the standard Einstein-Hilbert action with Ricci scalar as the
Lagrangian density,
$S=\int\tilde{R}\sqrt{-\mathbf{g}}\,d^{5}x.$ (16)
In varying the action, we note that the Ricci scalar and the connection
coefficients described in the previous section are functions of the metric
components alone.
$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\left[\tilde{R}\,\delta\sqrt{-\mathbf{g}}+\tilde{R}_{ik}\,\delta\mathbf{g}^{ik}\,\sqrt{-\mathbf{g}}\right]d^{5}x$
(17) $\displaystyle+$
$\displaystyle\int\delta\tilde{R}_{ik}\,\,\mathbf{g}^{ik}\sqrt{-\mathbf{g}}\,\,d^{5}x.$
The first term gives rise to the usual Einstein tensor,
$\tilde{\mathrm{G}}_{ik}=\tilde{R}_{ik}-(1/2)\mathbf{g}_{ik}\tilde{R}.$
In the absence of torsion, the second term becomes a boundary integral which
vanishes when the variation is fixed at the boundary and hence will not
contribute to the equations of motion. But in the presence of torsion, the
second term gives a nonzero contribution.
From eq. 14, we find the variations of the Ricci tensor to be
$\displaystyle\delta\tilde{R}_{ik}$ $\displaystyle=$
$\displaystyle\partial_{k}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}-\partial_{j}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}+\tilde{\Gamma}^{j}_{\cdot\,km}\delta\tilde{\Gamma}^{m}_{\cdot\,ji}+\tilde{\Gamma}^{m}_{\cdot\,ji}\delta\tilde{\Gamma}^{j}_{\cdot\,km}-\tilde{\Gamma}^{j}_{\cdot\,jm}\delta\tilde{\Gamma}^{m}_{\cdot\,ki}-\tilde{\Gamma}^{m}_{\cdot\,ki}\delta\tilde{\Gamma}^{j}_{\cdot\,jm}$
(18) $\displaystyle=$
$\displaystyle\left[\tilde{\nabla}_{k}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}-\tilde{\nabla}_{j}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}\right]+\mathrm{T}^{m}_{\cdot\,kj}\delta\tilde{\Gamma}^{j}_{\cdot\,mi}.$
Then, the second term in the r.h.s of eq. 17 takes the form
$\int\delta\tilde{R}_{ik}\,\mathbf{g}^{ik}\sqrt{-\mathbf{g}}\,d^{5}x=\int\left[\tilde{\nabla}_{k}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji})-\tilde{\nabla}_{j}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki})\right]\sqrt{-\mathbf{g}}\,d^{5}x+\int\mathbf{g}^{ik}\mathrm{T}^{m}_{\cdot\,kj}\delta\tilde{\Gamma}^{j}_{\cdot\,mi}\sqrt{-\mathbf{g}}\,d^{5}x.$
(19)
In deriving the above equation, we have used the metricity condition, namely
$\tilde{\nabla}_{j}\mathbf{g}^{ik}=0$. Substituting for the covariant
derivative in the first term of the r.h.s of eq. 19,
$\displaystyle\int\left[\tilde{\nabla}_{k}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji})\right.$
$\displaystyle-$
$\displaystyle\left.\tilde{\nabla}_{j}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki})\right]\sqrt{-\mathbf{g}}\,d^{5}x=\int\left[\partial_{k}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}\sqrt{-\mathbf{g}}\,)-\partial_{j}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}\sqrt{-\mathbf{g}}\,)\right]\,d^{5}x$
$\displaystyle+$
$\displaystyle\int\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}\left[\tilde{\Gamma}^{m}_{\cdot\,mk}-\frac{\partial_{k}\sqrt{-\mathbf{g}}}{\sqrt{-\mathbf{g}}}\right]\sqrt{-\mathbf{g}}\,d^{5}x-\int\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}\left[\tilde{\Gamma}^{m}_{\cdot\,mj}-\frac{\partial_{j}\sqrt{-\mathbf{g}}}{\sqrt{-\mathbf{g}}}\right]\sqrt{-\mathbf{g}}\,d^{5}x.$
The first term in the r.h.s of the above equation is a boundary term, an
integral of a total divergence. This will vanish when the variation is fixed
at the boundary, and hence can be ignored. The second and third terms in the
r.h.s can be simplified by noting
$\tilde{\Gamma}^{m}_{\cdot\,mk}=\hat{\Gamma}^{m}_{\cdot\,mk}+\mathrm{T}^{m}_{\cdot\,mk}$,
and
$\hat{\Gamma}^{m}_{\cdot\,mk}=(\partial_{k}\sqrt{-\mathbf{g}})/\sqrt{-\mathbf{g}}$,
leading to
$\int\left[\tilde{\nabla}_{k}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji})-\tilde{\nabla}_{j}(\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki})\right]\sqrt{-\mathbf{g}}\,d^{5}x=-\int\mathrm{T}^{m}_{\cdot\,km}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}\sqrt{-\mathbf{g}}\,d^{5}x+\int\mathrm{T}^{m}_{\cdot\,jm}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}\sqrt{-\mathbf{g}}\,d^{5}x,$
(21)
and eq. 19 becomes
$\int\delta\tilde{R}_{ik}\,\mathbf{g}^{ik}\sqrt{-\mathbf{g}}\,d^{5}x=\int\left[-\mathrm{T}^{m}_{\cdot\,km}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}+\mathrm{T}^{m}_{\cdot\,jm}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}+\mathrm{T}^{m}_{\cdot\,kj}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,mi}\right]\sqrt{-\mathbf{g}}\,d^{5}x.$
(22)
The second term in the r.h.s of eq.17 is thus a function of torsion and
variations in the connection given by eq. 22. Clearly, this term vanishes if
we assume that torsion is zero, and action principle would yield the standard
Einstein equations. Alternatively, if we treat the variation in the connection
to be composed of independent variations in metric and torsion, we obtain the
Einstein-Cartan equations Hehl , which ultimately leads to zero torsion when
the matter is not coupled to the connection. However, since torsion is not an
independent degree of freedom in our framework and is a function of the metric
components given by eq. 11, we first substitute its components in terms of the
metric and then carry out the variation with respect to the metric. To this
end, we note,
$\displaystyle\mathrm{T}^{m}_{\cdot\,km}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ji}$
$\displaystyle=$ $\displaystyle\mathrm{T}^{5}_{\cdot\,\nu
5}[\mathbf{g}^{\mu\nu}(\delta\tilde{\Gamma}^{\alpha}_{\cdot\,\alpha\mu}+\delta\tilde{\Gamma}^{5}_{\cdot\,5\mu})+\mathbf{g}^{5\nu}\delta\tilde{\Gamma}^{5}_{\cdot\,55}],$
$\displaystyle\mathrm{T}^{m}_{\cdot\,jm}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,ki}$
$\displaystyle=$ $\displaystyle\mathrm{T}^{5}_{\cdot\,\nu
5}\mathbf{g}^{\mu\alpha}\delta\tilde{\Gamma}^{\nu}_{\cdot\,\mu\alpha},$
$\displaystyle\mathrm{T}^{m}_{\cdot\,kj}\mathbf{g}^{ik}\delta\tilde{\Gamma}^{j}_{\cdot\,mi}$
$\displaystyle=$ $\displaystyle\mathrm{T}^{5}_{\cdot\,\nu
5}[\mathbf{g}^{\mu\nu}\delta\tilde{\Gamma}^{5}_{\cdot\,5\mu}+\mathbf{g}^{5\nu}\delta\tilde{\Gamma}^{5}_{\cdot\,55}].$
(23)
Taking these terms together, eq. 22 takes the form
$\displaystyle\int\delta\tilde{R}_{ik}\,\mathbf{g}^{ik}\sqrt{-\mathbf{g}}\,d^{5}x=$
$\displaystyle\int\mathrm{T}^{5}_{\cdot\,\nu
5}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\
g^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}\,d^{5}x,$
(24)
Note that the variations in the connection,
$\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}$ and
$\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}$, involve only the 4D components.
Since these are the 4D Levi-Civita components that only depend on the 4D
metric $\mathrm{g}_{\mu\nu}$, the above equation takes the form
$\int\delta\tilde{R}_{ik}\,\mathbf{g}^{ik}\sqrt{-\mathbf{g}}\,d^{5}x=\int\mathrm{H}_{\mu\nu}\delta\mathrm{g}^{\mu\nu}\,\sqrt{-\mathbf{g}}\,d^{5}x,$
(25)
where (see Appendix B for details),
$\displaystyle\mathrm{H}_{\mu\nu}=\nabla_{(\mu}\mathrm{B}_{\nu)}-(\nabla\cdot\mathrm{B})\mathrm{g}_{\mu\nu}+\mathrm{J}_{(\mu}\mathrm{B}_{\nu)}-(\mathrm{J}\cdot\mathrm{B})\mathrm{g}_{\mu\nu},$
$\displaystyle\qquad\mathrm{B}_{\mu}\equiv\mathrm{T}^{5}_{\cdot\,\mu
5}=\mathrm{J}_{\mu}-\partial_{5}\mathrm{A}_{\mu}-\mathrm{A}_{\mu}\mathrm{J}_{5}.$
(26)
Taking together the variations in both terms in eq. 17, we obtain the modified
Einstein tensor.
$\displaystyle\tilde{\mathrm{G}}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}(\mathrm{g}_{\mu\nu}+\mathrm{A}_{\mu}\mathrm{A}_{\nu}\epsilon\Phi^{2})R+\mathrm{H}_{\mu\nu}=\Sigma_{\mu\nu},$
$\displaystyle\tilde{\mathrm{G}}_{\mu 5}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathrm{A}_{\mu}\epsilon\Phi^{2}R=\Sigma_{\mu 5},$
$\displaystyle\tilde{\mathrm{G}}_{55}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon\Phi^{2}R=\Sigma_{55},$
where $\Sigma$ is the stress energy tensor that arises from the variations of
the assumed matter fields in the Lagrangian. Our focus being on gravity, we
will not further discuss the origin of $\Sigma$. Since the physical
interpretation of the stress- energy is more transparent with one covariant
and one contravariant indices, we express the above equations in an alternate
form, by noting
$\tilde{\mathrm{G}}_{i}^{\,\,j}=\mathbf{g}^{kj}\tilde{G}_{ik}$,
$\displaystyle
R_{\mu}^{\,\,\nu}-\frac{1}{2}R\delta^{\,\,\nu}_{\mu}+\mathrm{H}^{\,\,\nu}_{\mu}$
$\displaystyle=$ $\displaystyle\Sigma_{\mu}^{\,\,\nu},$ (27)
$\displaystyle-\mathrm{A}^{\alpha}R_{\mu\alpha}-\mathrm{A}^{\alpha}\mathrm{H}_{\mu\alpha}$
$\displaystyle=$ $\displaystyle\Sigma_{\mu}^{\,\,5},$ (28) $\displaystyle
0=\Sigma_{5}^{\,\,\mu},\qquad-\frac{1}{2}R$ $\displaystyle=$
$\displaystyle\Sigma_{5}^{\,\,5}.$ (29)
These are the modified Einstein equations in our framework. Since by
construction the fifth dimension is hidden with respect to the observable 4D
motion, the 5D components of the stress tensor $\Sigma_{\mu}^{\,\,5}$ and
$\Sigma_{5}^{\,\,5}$ are unobservable. It is not possible to solve equations
28 and 29 unless these components are theoretically known from the 5D matter
Lagrangian. In the present formulation, for simplicity, we shall ignore these
equations as though they simply serve to evaluate the components
$\Sigma_{\mu}^{\,\,5}$ and $\Sigma_{5}^{\,\,5}$, and treat only eq. 27 with
the observable 4D stress tensor to be relevant to physical solutions. In the
absence of specified matter fields in the Lagrangian, an alternate way to
interpret the modified Einstein equations is to regard
$-\mathrm{H}^{\,\,\nu}_{\mu}$ as _extra-dimensionally induced matter_.
When $\mathrm{H}^{\,\,\nu}_{\mu}=0$, eq. 27 reduces to the standard Einstein
equations for the 4D metric components $\mathrm{g}_{\mu\nu}$. In this case the
4D Bianchi identity necessarily implies the conservation of matter
$\nabla_{\nu}\Sigma^{\,\,\nu}_{\mu}=0$. But in general when
$\mathrm{H}^{\,\,\nu}_{\mu}$ is non-vanishing and dependent on the extra-
dimensional metric fields $\mathrm{A}_{\mu}$ and $\Phi$, eq. 27 by itself may
not be sufficient to solve for $\mathrm{g}_{\mu\nu}$ along with
$\mathrm{A}_{\mu}$ and $\Phi$, even after fixing the gauge. However, an
important physical simplification can be achieved by generalizing the
cylindrical condition to assume that $\mathrm{A}_{\mu}$ and $\Phi$ do not
depend on $x^{5}$. With this assumption, $\mathrm{B}_{\mu}=\mathrm{J}_{\mu}$
and $\mathrm{H}^{\,\,\nu}_{\mu}$ depends only on $\Phi$ and not on
$\mathrm{A}_{\mu}$, and eq. 27 is sufficient to solve for both
$\mathrm{g}_{\mu\nu}$ and $\Phi$. The vector $\mathrm{A}_{\mu}$ can in
principle be evaluated from eq. 28 by setting $\Sigma_{\mu}^{\,\,5}$ to zero,
but this would be inconsequential as $\mathrm{A}_{\mu}$ is decoupled from the
physically relevant equation that solves for the 4D metric
$\mathrm{g}_{\mu\nu}$. Hence, in the rest of the paper we will make the
assumption of cylindrical condition in order to explore solutions of physical
interest to the modified Einstein equations.
Finally, when $\mathrm{H}^{\,\,\nu}_{\mu}$ is non-vanishing, we note that
$\Sigma^{\,\,\nu}_{\mu}$ does not necessarily have to satisfy the 4D matter
conservation. However, with minimum modifications to GR and the empirical
conservation laws in mind, it is reasonable to assert the conservation of
$\Sigma^{\,\,\nu}_{\mu}$. Since the standard Einstein tensor satisfies the 4D
Bianchi identity independently of $\mathrm{H}^{\,\,\nu}_{\mu}$, the 4D matter
conservation implies,
$\nabla_{\nu}\Sigma^{\,\,\nu}_{\mu}=0\Longrightarrow\nabla_{\nu}\mathrm{H}^{\,\,\nu}_{\mu}=0.$
(30)
In the reminder of the paper, we study the solutions to the modified Einstein
equations (eq. 27) in two extremely symmetric situations, namely, the
homogeneous-isotropic geometry and the static spherically symmetric geometry.
## V Homogeneous-Isotropic Cosmology
The 4D metric of a homogeneous and isotropic universe has the form
$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right)$
(31)
The values of $k=0,+1,-1$ correspond respectively to flat, closed, and
hyperbolic spatial geometries. The standard Einstien tensor for this metric is
given by Waldbook
$\displaystyle\mathrm{G}^{t}_{t}$ $\displaystyle=$ $\displaystyle
3(\dot{a}/a)^{2}+3k/a^{2},$ $\displaystyle\mathrm{G}^{r}_{r}$ $\displaystyle=$
$\displaystyle 2(\overset{..}{a}/a)+(\dot{a}/a)^{2}+k/a^{2},$
$\displaystyle\mathrm{G}^{\theta}_{\theta}$ $\displaystyle=$
$\displaystyle\mathrm{G}^{\phi}_{\phi}=\mathrm{G}^{r}_{r},$ (32)
where over-dot denotes a derivative with respect to time. Since the geometry
is spatially homogeneous and isotropic, the metric fields including
$\mathrm{A}_{\mu}$ and $\Phi$ in the 5D geometry only depends on time. Hence
the only non-vanishing component of $\mathrm{J}_{\mu}$ is $\mathrm{J}_{t}$.
The induced matter terms given in eq. 26 are
$\displaystyle\mathrm{H}^{\,\,t}_{t}$ $\displaystyle=$ $\displaystyle
3\mathrm{J}_{t}(\dot{a}/a),$ $\displaystyle\mathrm{H}^{\,\,r}_{r}$
$\displaystyle=$ $\displaystyle
2\mathrm{J}_{t}(\dot{a}/a)+\dot{\mathrm{J}}_{t}+\mathrm{J}^{2}_{t},$
$\displaystyle\mathrm{H}^{\,\,\theta}_{\theta}$ $\displaystyle=$
$\displaystyle\mathrm{H}^{\,\,\phi}_{\phi}=\mathrm{H}^{\,\,r}_{r}.$ (33)
Figure 2: Solutions for $a(t)$ for different values of deceleration parameter
for $k=0$ and $k=+1$.
Before writing out the modified Einstein equations, we note that the
conservation equation (eq. 30) now reduces to
$\displaystyle\nabla_{\nu}\mathrm{H}^{\,\,\nu}_{\mu}=3\mathrm{J}_{t}\left[(\overset{..}{a}/a)-(\dot{a}/a)\mathrm{J}_{t}\right]=0$
$\displaystyle\Longrightarrow\qquad\mathrm{J}_{t}=0\qquad\mathrm{or}\qquad\mathrm{J}_{t}=\overset{..}{a}/\dot{a}$
(34)
These are the only two possibilities. From the definition of
$\mathrm{J}_{\mu}$, this implies that either $\Phi$ is a constant, which would
give rise to the usual Friedman-Robertson-Walker cosmology, or
$\Phi=\dot{a}(t)$. Focusing on the latter case, $\mathrm{H}^{\,\,\nu}_{\mu}$
simplifies to
$\mathrm{H}^{\,\,t}_{t}=3\overset{..}{a}/a,\qquad\mathrm{H}^{\,\,r}_{r}=2(\overset{..}{a}/a)+(\overset{...}{a}/\dot{a}).$
(35)
Taking the stress tensor to be that of a perfect fluid, the modified Einstein
equations (eq. 27) take the form
$\displaystyle 3(\dot{a}/a)^{2}+3k/a^{2}+3\overset{..}{a}/a$ $\displaystyle=$
$\displaystyle 8\pi\rho$ (36) $\displaystyle
4(\overset{..}{a}/a)+(\dot{a}/a)^{2}+(\overset{...}{a}/\dot{a})+k/a^{2}$
$\displaystyle=$ $\displaystyle-8\pi P$ (37)
where $\rho$ and $P$ are the density and pressure of the 4D matter. Combining
the above equations, we find
$\dot{\rho}+3(\rho+P)\dot{a}/a=0,$ (38)
which is just a restatement of the 4D matter conservation equation. For matter
dominated universe, $P=0$, and consequently eq. 38 yields
$\rho a^{3}=constant\qquad\mathrm{or}\qquad\rho=\rho_{o}/a^{3}.$ (39)
In effect, eqns. 36 and 39 are sufficient to solve for $a(t)$, which needs two
initial conditions along with the specification of $\rho_{o}$.
$a^{2}\overset{..}{a}+a(\dot{a})^{2}+ka=8\pi\rho_{o}/3,$ (40)
Without loss of generality we choose the current epoch to be at $t=0$, set the
current size of the universe $a(0)=1$, and the unit of time such that
$\dot{a}(0)=1$. In this unit of time, the current value of the Hubble’s
constant will be 1. Rather than specifying the value of $\rho_{o}$, we choose
to specify the current value of $\overset{..}{a}$. The effective equation for
$a(t)$ then takes the form
$a^{2}\overset{..}{a}+a(\dot{a})^{2}+ka=1+k-q_{o},$ (41)
where $q_{o}=-a(0)\overset{..}{a}(0)/\dot{a}^{2}(0)$, the current value of the
deceleration parameter, is the only free parameter to be specified. Figure 2
shows the behavior of $a(t)$ for various values of $q_{o}$ for spatially flat
and closed topologies. For the spatially flat topology, shown in the top
panels of the figure, we find that the universe does not originate from a
singular big bang for all $q_{o}<+0.5$. For the spatially closed topology
shown in the bottom panels, we find oscillatory solutions for all $q_{o}<+1$.
Oscillatory solutions in the spatially closed topology exhibit a scale factor
that oscillates between a maximum $a_{max}$ and a minimum $a_{min}$. The
acceleration reaches a positive value in a narrow interval around $a_{min}$,
and then becomes negative for the rest of the cycle until it gets back near
$a_{min}$. By taking the value of $q_{o}$ arbitrarily close to 1, we can make
$a_{min}$ arbitrarily close to zero. This can be seen from the bottom-left and
bottom-middle panels of fig. 2. Thus one could construct a universe that
collapses and bounces back to expand when it reaches an arbitrarily small
size, or equivalently arbitrarily high energy densities. It remains to be seen
if such solutions would fit the empirical red shift data.
## VI Static Spherically Symmetric Vacuum Solutions
The most general static spherically symmetric 4D metric has the form111 The
scalar functions $A(r)$ and $B(r)$ defined in this section should not be
confused with the vectors $\mathrm{A}_{\mu}$ and $\mathrm{B}_{\mu}$ defined in
eqns. 3 and 26 respectively.
$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}d\Omega^{2},$ (42)
and the standard Einstein tensor for this metric is Waldbook
$\displaystyle\mathrm{G}^{t}_{t}$ $\displaystyle=$
$\displaystyle\frac{rB^{\prime}+B^{2}-B}{r^{2}B^{2}},$
$\displaystyle\mathrm{G}^{r}_{r}$ $\displaystyle=$ $\displaystyle\frac{AB-
rA^{\prime}-A}{r^{2}AB},$ $\displaystyle\mathrm{G}^{\theta}_{\theta}$
$\displaystyle=$
$\displaystyle\frac{2A^{2}B^{\prime}-2ABA^{\prime}-2rABA^{\prime\prime}+rBA^{\prime
2}+rAA^{\prime}B^{\prime}}{4rA^{2}B^{2}},$
$\displaystyle\mathrm{G}^{\phi}_{\phi}$ $\displaystyle=$
$\displaystyle\mathrm{G}^{\theta}_{\theta},$ (43)
where a prime in the above equations denotes a derivative with respect to $r$.
The additional term $\mathrm{H}^{\,\,\nu}_{\mu}$ in the modified Einstein
equations (eq. 27) depends only on $\Phi$ when the cylindrical condition is
imposed on all metric components and is given by
$\mathrm{H}^{\,\,\nu}_{\mu}=\nabla_{\mu}\mathrm{J}^{\nu}-(\nabla\cdot\mathrm{J})\delta_{\mu}^{\nu}+\mathrm{J}_{\mu}\mathrm{J}^{\nu}-(\mathrm{J}\cdot\mathrm{J})\delta_{\mu}^{\nu}.$
(44)
Since $\mathrm{J}_{\mu}=\Phi^{-1}\partial_{\mu}\Phi$, the quantity
$\nabla_{\mu}\mathrm{J}^{\nu}$ is intrinsically symmetric in $\mu$ and $\nu$.
The static spherical symmetry of the geometry implies that $\mathrm{J}_{r}$ is
the only non-vanishing component, which we denote by $J(r)$. With this,
$\displaystyle\mathrm{H}^{\,\,t}_{t}$ $\displaystyle=$
$\displaystyle\frac{rJB^{\prime}-2B\left(rJ^{\prime}+2J+rJ^{2}\right)}{2rB^{2}},$
$\displaystyle\mathrm{H}^{\,\,r}_{r}$ $\displaystyle=$
$\displaystyle\frac{-J\left(rA^{\prime}+4A\right)}{2rAB},$
$\displaystyle\mathrm{H}^{\,\,\theta}_{\theta}$ $\displaystyle=$
$\displaystyle\frac{rJAB^{\prime}-2rABJ^{\prime}-rJBA^{\prime}-2ABJ\left(1+rJ\right)}{2rAB^{2}},$
$\displaystyle\mathrm{H}^{\,\,\phi}_{\phi}$ $\displaystyle=$
$\displaystyle\mathrm{H}^{\,\,\theta}_{\theta}.$ (45)
In order to obtain vacuum solutions, we set $\Sigma_{\mu}^{\nu}=0$ in eq. 27
and find the following three equations.
$\displaystyle J^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{J(1+B)}{r},$
$\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{2A(1-B+2rJ)}{r(2+Jr)},$ $\displaystyle B^{\prime}$
$\displaystyle=$
$\displaystyle\frac{2B\left(r^{2}J^{2}+(1+rJ)(1-B)\right)}{r(2+Jr)}.$ (46)
A close examination of the above equations reveals two basic properties of the
function $J(r)$. First, if $J(r)$ is a constant, it has to be identically
zero. Secondly, if $J(r)$ vanishes at some point, it has to vanish identically
everywhere. The simplest solution to the coupled equations (eq. 46) is when
$J(r)$ vanishes everywhere,
$J(r)=0,\,A(r)=\left(1-\frac{2M}{r}\right),\,B(r)=\left(1-\frac{2M}{r}\right)^{-1},$
(47)
which of course is the well-known Schwarzschild solution as expected.
### VI.1 General solution to $J(r)$
Let $F(r)\equiv 1/rJ(r)$ when $J(r)$ is non-vanishing. Substituting for $J(r)$
in terms of $F(r)$, the coupled equations (eq. 46) lead to the following
equation for $F(r)$,
$r\left[(2F^{2}+F)F^{\prime\prime}+F^{\prime 2}\right]=F^{\prime}(F+2),$ (48)
whose solution in turn determines the 4D metric functions $A(r)$ and $B(r)$.
The obvious solution of eq. 48 is $F(r)=$constant. This leads to
$J(r)=c/r\,,\,A(r)=(r)^{-\frac{2+4c}{2+c}}\,,\,B(r)=0,$ (49)
which is clearly unacceptable because $B(r)$ is identically zero.
Assuming that $F(r)$ is not a constant, we can obtain solutions to the second
order differential equation (eq. 48). In principle, the solution would have
two integration constants that would be determined by the boundary conditions,
one of which immediately follows from the form of the equation. It can be
easily seen that if $F(r)$ is a solution, then $F(\lambda r)$ is also a
solution for any scaling constant $\lambda$. We find a general solution in the
implicit form
$\lambda^{2}r^{2}=\gamma^{2}\left|F/\gamma+\beta+1\right|^{1+\beta}\left|F/\gamma+\beta-1\right|^{1-\beta},$
(50)
where $\beta$ and $\gamma$ are defined in terms of an independent arbitrary
constant $c$.
$\gamma=\sqrt{1+c+c^{2}},\,\,\,\beta=\frac{1+c}{\sqrt{1+c+c^{2}}}.\,\,$ (51)
Figure 3: $\beta$ is plotted as a function of $c$.
With $\lambda$ and $c$ as two arbitrary constants, eq. 50 represents the
general solution to the second order differential equation (eq. 48). From eq.
50, we find the derivatives of $F(r)$ to be
$F^{\prime}=\frac{1}{r}\frac{F^{2}+2(1+c)F+c}{F},\qquad
F^{\prime\prime}=-\frac{c}{r}\frac{F^{\prime}}{F^{2}},$ (52)
and substituting them in eq. 46, we obtain the metric functions $A(r)$ and
$B(r)$ in terms of $F(r)$.
$\displaystyle A^{\prime}(r)$ $\displaystyle=$ $\displaystyle 2cA(r)/rF(r),$
(53) $\displaystyle B(r)$ $\displaystyle=$ $\displaystyle
1+2(1+c)/F(r)+c/F^{2}(r).$ (54)
In order to obtain asymptotically flat solutions, we shall impose the boundary
conditions $A(r\rightarrow\infty)=1$ and $B(r\rightarrow\infty)=1$. To
understand the behavior of the functions $A(r)$ and $B(r)$ which define the
observable 4D geometry, we start with the properties of $F(r)$.
Figure 4: Numerical solution to $F(r)$ with $\lambda=-1$.
Unfortunately eq. 50 does not yield an explicit functional form for $F(r)$
except for simple cases when $c$ is either 0 or -1. Nevertheless the relevant
properties of $F(r)$ can be inferred from analyzing this implicit function.
First note that the quantity $(1+c+c^{2})$ is positive definite and $\beta$ is
finite and bounded for all values of $c$. Figure. 3 plots the behavior of
$\beta$ to show that it asymptotically reaches +1 and -1 at $c=+\infty$ and
$-\infty$ respectively. The following observations summarize the qualitative
properties of $F(r)$.
1) At $r=0$, $F$ can take one of two possible values. If $c<0$ ($\beta<1$),
then $F(0)$ can be either $\gamma(-\beta-1)$ or $\gamma(-\beta+1)$, while if
$c>0$ ($\beta>1$), then $F(0)$ can only be $\gamma(-\beta-1)$.
2) In the limit $r\rightarrow\infty$, $F$ necessarily has to diverge in order
to satisfy the boundary condition $B(r\rightarrow\infty)=1$. Eq. 50 then
implies for large $r$, $\lambda^{2}r^{2}=F^{2}$, implying $F$ could be either
positive or negative, such that
$F(r\rightarrow\infty)=\lambda r\Rightarrow
J(r\rightarrow\infty)=\lambda^{-1}/r^{2}.$ (55)
The behavior of $F(r)$ at the extremities is summarized in the following
table.
| $c<0\,\,\,(\beta<1)$ | $c>0\,\,\,(\beta>1)$
---|---|---
$r=0$ | $F=\begin{cases}\gamma(-\beta+1)\,>0\\\ \gamma(-\beta-1)\,<0\end{cases}$ | $F=\gamma(-\beta-1)\,<0$
$r\rightarrow\infty$ | $F=\lambda r$ | $F=\lambda r$
3) From eq. 52, it can be shown that $F^{\prime}=0$ when $F$ is either
$\gamma(-\beta+1)$ or $\gamma(-\beta-1)$, which can happen only at $r=0$.
Hence $F^{\prime}$ is either positive definite or negative definite, and so
$F(r)$ is a monotonic function.
4) Though $F$ is monotonic and finite for any finite $r$, it can reach zero at
$r_{o}$ given by
$\lambda^{2}r_{o}^{2}=|c|\left|\frac{\beta+1}{\beta-1}\right|^{\beta},$ (56)
and from eq. 52, $F^{\prime}$ diverges at $r_{o}$. Considering the physical
relevance of these solutions, we shall only focus on solutions that are non-
vanishing everywhere. Such solutions do indeed exist for a range of parameter
values. Rewriting eq. 50 at $r=r_{o}$,
$\left|\frac{F(r_{o})}{\gamma(\beta+1)}+1\right|^{1+\beta}\left|\frac{F(r_{o})}{\gamma(\beta-1)}+1\right|^{1-\beta}=1,$
(57)
we note that $F(r_{o})=0$ is not the only solution. Numerical plots in fig. 4
demonstrates the existence of non-vanishing $F(r)$ solutions.
5) From the table above, (i) for $c>0$, since $F(0)$ is negative, $F(r)$ has
to be negative definite which requires $\lambda$ to be negative. (ii) For
$c<0$, $\lambda$ can be either positive or negative, making $F(r)$ either
positive definite or negative definite respectively.
The functional form of $F(r)$ described by the above five properties along
with equations 53 and 54 will yield the functional form of the metric
functions $A(r)$ and $B(r)$.
### VI.2 Metric functions $A(r)$ and $B(r)$
With the boundary condition $A(\infty)=B(\infty)=1$, eqns. 53 and 54 yield
$A(r)=\exp{\left(-\int_{r}^{\infty}\frac{2c}{rF(r)}dr\right)},$ (58)
$B(r)=1+2(1+c)/F(r)+c/F^{2}(r).$ (59)
The following observations summarize the qualitative behavior of $A(r)$ and
$B(r)$.
1) From the asymptotic behavior of $F(r)\rightarrow\lambda r$ for large $r$,
we note that
$\displaystyle A(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2c\lambda^{-1}}{r}+\mathcal{O}(1/r^{2}),$ (60) $\displaystyle B(r)$
$\displaystyle=$ $\displaystyle
1+\frac{2(1+c)\lambda^{-1}}{r}+\mathcal{O}(1/r^{2}).$ (61)
Hence, when $|c|\gg 1$ and $r\rightarrow\infty$, the above solutions
approximate the Schwarzschild solution with mass $M\equiv[c\lambda^{-1}]$.
When both $c$ and $\lambda$ are either positive or negative, the gravity is
attractive, while when one is positive and the other is negative, the gravity
is repulsive.
2) Since $F(r)$ is either positive definite or negative definite, both $A(r)$
and $B(r)$ are finite and positive for all $r>0$. At $r=0$, since $F(0)$ is
either $\gamma(-\beta-1)$ or $\gamma(-\beta+1)$, eq. 59 implies $B(0)=0$.
3) As $r\rightarrow 0$, the integral in eq. 58 diverges as $[2c/F(0)]\ln(r)$.
When $[c/F(0)]$ is positive, then $A(0)=0$, and when $[c/F(0)]$ is negative
$A(0)=\infty$. The sign of $[c/F(0)]$ is the same as the sign of
$M=[c\lambda^{-1}]$. For $M>0$, $A(r)$ monotonically increases from $A(0)=0$
to $A(\infty)=1$; for $M<0$, $A(r)$ monotonically decreases from
$A(0)=+\infty$ to $A(\infty)=1$.
4) Irrespective of the sign of $M$, $B(0)=0$ and $B(\infty)=1$. However,
$B(r)$ is not necessarily monotonic. From eq. 59, we see that $B^{\prime}=0$
when either $F^{\prime}=0$ or when $F(r)=-c/(1+c)$. From the previous
subsection, $F^{\prime}\neq 0$ for all $r>0$, but $F(r)$ could attain the
value $-c/(1+c)$ for certain values of $c$ and $\lambda$. Since $F(r)$ is a
non-vanishing monotonic function taking all values from $F(0)$ to $\pm\infty$,
it is straightforward to check if it would attain the value $-c/(1+c)$. When
$\lambda<0$, $F(r)$ is negative definite, and $-c/(1+c)$ needs to be a
negative number lesser than $F(0)=\gamma(-\beta-1)$, which happens only when
$c<-1$. When $\lambda>0$, $F(r)$ is positive definite, and $-c/(1+c)$ needs to
be a positive number greater than $F(0)=\gamma(-\beta+1)$, which happens only
when $-1<c<0$.
The qualitative behavior of the functions $A(r)$ and $B(r)$ for the various
allowed ranges of $c$ and $\lambda$ are shown in figure 5. When
$M=[c\lambda^{-1}]$ is positive, $A(r)$ is a monotonically increasing function
leading to attractive gravity, and corresponds to Schwarzschild solution at
large $r$ when $|c|\gg 1$.
Figure 5: Schematic behavior of $A(r)$ and $B(r)$
### VI.3 Naked singularity at $r=0$
An important point to note from figure 5 is that these solutions do not have
an event horizon because both $A(r)$ and $B(r)$ are finite and positive for
all $r>0$.
Clearly, these solutions are smooth for all $r>0$. However, the point $r=0$ is
a physical singularity. Explicit calculations show that the Ricci scalar
$R_{\mu}^{\mu}$ vanishes everywhere, but the quantity $R_{\mu\nu}R^{\mu\nu}$
is nonvanishing. It turns out that
$R_{\mu\nu}R^{\mu\nu}=\frac{2(3F^{2}(r)+2cF(r)+c^{2})}{F^{4}(r)B^{2}(r)r^{4}}.$
(62)
At $r=0$, the numerator does not generally vanish, but the denominator
vanishes, making $R_{\mu\nu}R^{\mu\nu}$ diverge. Hence these solutions
correspond to a naked singularity at $r=0$ with no event horizon to censor it.
## VII Summary and Discussion
Metric and torsion are two independent constituents of metric compatible
Riemannian geometry. Because of the immense successes of torsion-free GR,
torsion has not played a significant role in theories of gravity. However,
when gravity is to be included with other interactions of elementary particles
with intrinsic spin, a more general theory including torsion becomes
imperative Hehl .
In the present work, torsion is incorporated in a novel way in higher
dimensional Kaluza-Klein type theories. Here torsion is not an independent
degree of freedom coupled to spin, rather it is determined in terms of metric
through a set of physically motivated constraints, which serve (i) to confine
torsion to the extra dimension, leaving the 4D space-time torsion free, and
(ii) to ensure that geodesic motions in 4D remain totally unaffected by the
presence of the extra-dimension. These constraints have previously been
imposed in terms of veilbeins MPLA ; Viet , but here it is realized that they
impose essentially the requirement that the fifth dimension is hidden at the
level of geodesic motion. It turns out that the non-vanishing torsion
components are functions of the 5D metric components with the 4D metric
$\mathrm{g}_{\mu\nu}$ obeying the so called cylindrical condition, namely, it
is independent of $x^{5}$. In the resulting geometry, all the 4D hypersurfaces
are equivalent, and the 4D components of the connection and the Ricci tensor
exactly match those of the standard 4D GR. Hence, at the level of geodesics,
this geometry is virtually indistinguishable from that of the standard GR.
To proceed further, we derive modified Einstein equations from the action
principle with Ricci scalar as the Lagrangian density. In this respect, an
alternate approach presents itself. In the Palatini formulation of GR, the
action is varied independently with respect to the connection, and in the
absence of torsion, the metricity follows from the equations of motion.
Recently Sotiriou ; DadhichPons , it has been shown that even without assuming
the absence of torsion, variations of the action independently with respect to
the metric and the connection lead to GR equations along with metricity,
provided the matter Lagrangian is not coupled to the connection. In our case,
with the constraints imposed on the connection, it is more convenient and
natural to impose metricity prior to action variation. Since the entire
connection is determined to be a function of the metric, we only need to vary
the action with respect to the metric to obtain the modified equations, making
the theory a purely metric theory of gravity. However, adopting a Palatini-
style approach, one could relax the assumption of metricity and vary the
action independently with respect to metric and connection along with the
imposed constraints, which might lead to a different set of modified Einstein
equations.
We apply the ensuing modified Einstein equations to study the cosmology of a
homogenous-isotropic universe. In matter dominated phase of the universe (zero
pressure), we obtain a second order differential equation for the scale factor
$a(t)$ in contrast to the first order differential equation in the usual FRW
cosmology. In FRW cosmology, the second derivative of $a(t)$ cannot be
independently prescribed as an initial condition, and decelerating expansion
is a necessary outcome in the absence of a cosmological constant. However in
our case, we have the choice of an initial condition for the second derivative
of $a(t)$, which can be tuned to fit the observed acceleration of the
universe. Figure 2 presents the behavior of $a(t)$ for various choices of the
current acceleration.
In an earlier version of the present work oldPaper , the field equations were
derived differently; by varying the action with respect to the metric prior to
expressing the Ricci tensor in terms of the metric. In the cosmology equations
generated from those field equations, acceleration was not an independently
prescribable initial condition. Chen and Jing ChenJing showed that those
equations yield accelerating universe solutions without resorting to dark
energy. They show that the model not only fits the supernovae data, but also
solves the cosmic age problem of old high redshift objects hiZshift . Whether
the cosmological solutions described in the current work would fit empirical
results just as well needs to be investigated.
In the case of spherically symmetric vacuum solutions to the modified Einstein
equations, we find some remarkably interesting results. As is well known, in
the unique Schwarzschild solution of GR, when the mass is positive, an event
horizon censors the central singularity. In contrast, we find positive mass
naked singularity solutions that lack an event horizon. Recently, similar
positive mass solutions without a horizon have been found KalyanaRama in a
simpler setting of torsion-free GR with multiple extra-dimensions. It would be
interesting to see how the particular vacuum solutions in the torsion induced
geometry in the present case match with those solutions in the torsion free
geometry.
The existence of positive mass solutions with naked singularity have immediate
consequences on gravitational collapse, opening up the possibility of an
arbitrarily large star collapsing to an arbitrarily small non-singular state.
Since trapped surfaces would not necessarily form in such collapses, finite
matter pressure could be sufficient to withstand a total collapse to
singularity. This suggests a detailed analysis of such solutions by treating
$-\mathrm{H}^{\,\,\nu}_{\mu}$ in eq. 27 as extra-dimensionally induced matter
in standard GR. It then raises the possibility of a sufficiently strong
gravitational collapse that stops short of collapsing to a singularity with a
finite induced stress-energy tensor that potentially violates the weak energy
condition in the region near the center. It remains to be seen if an
arbitrarily small static model star with finite stress-energy tensor can be
constructed with the external geometry matching the type of solutions
discussed in this paper.
In conclusion, inclusion of torsion in the context of extra-dimensions
presents a novel way of obtaining modified Einstein equations that have
significant physical consequences. For clarity and simplicity, we have
confined the treatment to five dimensions. However, the framework can be
generalized to arbitrary dimensions $D$, producing torsion-free $D-1$
dimensional metric theory. We could also consider many extra-dimensions and
generalize the constraints so as to hide all the extra-dimensions using
torsion, leading to a more general theory.
## Acknowledgments
The authors thank Venky Krishnan, Ramesh Anishetty, Johannes Noller and
Stanley Deser for helpful discussions.
## References
* (1) C. M. Will, Living Rev. Relativity 9, 3 (2006) [arXiv:gr-qc/0510072].
* (2) R.H. Dicke, The Theoretical Significance of Experimental Relativity, Gordon and Breach, New York, 1964.
* (3) F. W. Hehl, P. v.d. Hyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976).
* (4) T. Watanabe and M. J. Hayashi, arXiv:gr-qc/0409029 (2004).
* (5) I. L. Shapiro, Phys. Rept. 357, 113 (2002).
* (6) S. Deser and B. Zumino, Phys. Lett. B 62 335 (1976).
* (7) N. Poplawski, General Relativity and Gravitation 44 (4),1007 (2012).
* (8) A. Silvestri and M. Trodden, Rep. Prog. Phys. 72, 096901 (2009).
* (9) T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).
* (10) G. J. Olmo, Int. J. Mod. Phys.D20, 413-462 (2011).
* (11) R. Maartens and K. Koyama, Living Rev. Relativity 13, 5, (2010).
* (12) G. R. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B 485, 208 (2000).
* (13) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).
* (14) J. M. Overduin and P. S. Wesson, Phys. Rept. 283, 303 (1997).
* (15) M. W. Kalinowski, Int. J. Theo. Phys. 20, 563 (1981).
* (16) M. Nakahara, sections 7.8.1 and 7.8.4, Geometry, Topology and Physics, Second edition (IOP Publishing Ltd., London, 2003).
* (17) K. H. Shankar and K. C. Wali, Mod. Phys. Lett. A25, 2121(2010).
* (18) N. A. Viet and K. C. Wali, Phys. Rev. D 67, 124029 (2003).
* (19) R. M. Wald, section 5.2, General Relativity (University of Chicago Press, Chicago, 1984).
* (20) T. P. Sotiriou, Class. Quant. Grav. 26, 152001(2009).
* (21) N. Dadhich and J. M. Pons, arXiv:1010.0869v3[gr-qc] (2011).
* (22) K. H. Shankar and K. C. Wali, arXiv:gr-qc/0904.4038v1 (2009).
* (23) S. Chen and J. Jing, JCAP09, 001 (2009), arXiv:0905.0302v3.
* (24) A. C. S. Friaca, J. S. Alcaniz, and J. A. S. Lima, Mon. Not. Roy. Astron. Soc. 362, 1295 (2005).
* (25) S. Kalyana Rama, arXiv:1111.1897v3 [hep-th] (2011).
## Appendix A Computing the geometric quantities in the 5D manifold
In this appendix, expression for torsion, connection coefficients and the
Ricci tensor of the 5D geometry are computed in terms of the metric. It will
be shown that the physical constraints imposed on the connection will be
sufficient to determine uniquely all the components of torsion and hence the
other geometric quantities in terms of the metric.
### A.1 5D Levi-Civita Connections
The 5D Levi-Civita connection is given by
$\hat{\Gamma}^{j}_{\cdot\,ik}=\frac{1}{2}\mathbf{g}^{jm}\left(\partial_{k}\mathbf{g}_{im}+\partial_{i}\mathbf{g}_{km}-\partial_{m}\mathbf{g}_{ik}\right)$
(63)
Expressing the 5D metric in terms of the 4D metric and the extra-dimensional
metric fields given by eq. 3, we find the 5D Levi-Civita connection to be
$\displaystyle\hat{\Gamma}^{5}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\nabla_{\mu}\mathrm{A}_{\nu}+\nabla_{\nu}\mathrm{A}_{\mu}\right)+\left(\mathrm{A}_{\mu}\mathrm{J}_{\nu}+\mathrm{A}_{\nu}\mathrm{J}_{\mu}\right)+\frac{1}{2}\epsilon\Phi^{2}\mathrm{A}^{\lambda}\left(\mathrm{A}_{\mu}\mathrm{F}_{\lambda\nu}+\mathrm{A}_{\nu}\mathrm{F}_{\lambda\mu}\right)+\epsilon\Phi^{2}\mathrm{A}_{\mu}\mathrm{A}_{\nu}\mathrm{A}^{\lambda}\mathrm{J}_{\lambda}$
(64) $\displaystyle-$
$\displaystyle\frac{1}{2}\left(\mathrm{A}^{\lambda}\mathrm{A}_{\lambda}+\epsilon\Phi^{-2}\right)\partial_{5}\left(\epsilon\mathrm{A}_{\mu}\mathrm{A}_{\nu}\Phi^{2}\right)-\frac{1}{2}\left(\mathrm{A}^{\lambda}\mathrm{A}_{\lambda}+\epsilon\Phi^{-2}\right)\partial_{5}\mathrm{g}_{\mu\nu}$
$\displaystyle\hat{\Gamma}^{\mu}_{\cdot\,55}$ $\displaystyle=$
$\displaystyle-\epsilon\Phi^{2}\mathrm{J}^{\mu}+\epsilon\Phi^{2}\partial_{5}\mathrm{A}^{\mu}+\epsilon\Phi^{2}\mathrm{A}^{\mu}\mathrm{J}_{5}-\epsilon\Phi^{2}\mathrm{A}_{\sigma}\partial_{5}\mathrm{g}^{\mu\sigma}$
(65) $\displaystyle\hat{\Gamma}^{5}_{\cdot\,55}$ $\displaystyle=$
$\displaystyle\epsilon\Phi^{2}\mathrm{A}^{\lambda}\left(\mathrm{J}_{\lambda}-\partial_{5}\mathrm{A}_{\lambda}\right)-\epsilon\Phi^{2}\mathrm{A}^{\lambda}\mathrm{A}_{\lambda}\mathrm{J}_{5}+\mathrm{J}_{5}$
(66) $\displaystyle\hat{\Gamma}^{\mu}_{\cdot\,5\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon\Phi^{2}\mathrm{F}_{\nu}^{\cdot\mu}-\epsilon\Phi^{2}\mathrm{A}_{\nu}\mathrm{J}^{\mu}+\frac{1}{2}\mathrm{g}^{\mu\sigma}\partial_{5}\left(\epsilon\Phi^{2}\mathrm{A}_{\sigma}\mathrm{A}_{\nu}\right)+\frac{1}{2}\mathrm{g}^{\mu\sigma}\partial_{5}\mathrm{g}_{\nu\sigma}$
(67) $\displaystyle\hat{\Gamma}^{5}_{\cdot\,5\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon\Phi^{2}\mathrm{A}^{\lambda}\mathrm{F}_{\lambda\nu}+\epsilon\Phi^{2}\mathrm{A}_{\nu}\mathrm{A}^{\lambda}\mathrm{J}_{\lambda}+\mathrm{J}_{\nu}-\frac{1}{2}\mathrm{A}^{\lambda}\partial_{5}\left(\epsilon\mathrm{A}_{\lambda}\mathrm{A}_{\nu}\Phi^{2}\right)-\frac{1}{2}\mathrm{A}^{\lambda}\partial_{5}\mathrm{g}_{\nu\lambda}$
(68) $\displaystyle\hat{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\lambda}_{\cdot\,\mu\nu}+\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$
$\displaystyle\bar{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon\Phi^{2}\left(\mathrm{A}_{\nu}\mathrm{F}_{\mu}^{\cdot\lambda}+\mathrm{A}_{\mu}\mathrm{F}_{\nu}^{\cdot\lambda}\right)-\epsilon\Phi^{2}\mathrm{A}_{\mu}\mathrm{A}_{\nu}\mathrm{J}^{\lambda}+\frac{1}{2}\mathrm{A}^{\lambda}\partial_{5}\left(\epsilon\mathrm{A}_{\mu}\mathrm{A}_{\nu}\Phi^{2}\right)+\frac{1}{2}\mathrm{A}^{\lambda}\partial_{5}\mathrm{g}_{\mu\nu}$
(69)
where
$\mathrm{F}_{\mu\nu}\equiv\partial_{\mu}\mathrm{A}_{\nu}-\partial_{\nu}\mathrm{A}_{\mu}$,
$\mathrm{J}_{i}\equiv\epsilon\Phi^{-1}\partial_{i}\epsilon\Phi$ and
$\Gamma^{\lambda}_{\cdot\,\mu\nu}$ is the 4D Levi-Civita connection obtained
from the metric $\mathrm{g}_{\mu\nu}$. and $\nabla_{\mu}$ is the derivative
operator with the 4D Levi-Civita connection. The raising and lowering of
indices on $\mathrm{F}_{\mu\nu}$, $\mathrm{A}_{\mu}$ and $\mathrm{J}_{\mu}$
are performed with respect to the 4D metric.
### A.2 Torsion components
We start with the conditions that $\mathrm{T}^{\lambda}_{\cdot\,\mu\nu}=0$ and
$\tilde{\Gamma}^{\lambda}_{\cdot\,i5}=\tilde{\Gamma}^{\lambda}_{\cdot\,5i}=0$.
Together these conditions imply $\mathrm{T}^{\lambda}_{\cdot\,ij}=0$. The
remaining non-vanishing components of torsion that need to be determinned are
$\mathrm{T}^{5}_{\cdot\,ij}$, a total of 10 independent components. The
conditions $\tilde{\Gamma}^{\lambda}_{\cdot\,5i}=0$, consisting of 20
equations are sufficient to determine uniquely all the non-vanishing torsion
components.
$\tilde{\Gamma}^{\mu}_{\cdot\,i5}=\hat{\Gamma}^{\mu}_{\cdot\,i5}+\mathrm{K}^{\mu}_{\cdot\,i5}=0$
(70)
From the above equation, the non vanishing components of contorsion and
torsion can be determined in terms of components of the components of 5D Levi-
Civita connection $\hat{\Gamma}_{\cdot\,}$. First taking $i=5$,
$\mathrm{K}^{\mu}_{\cdot\,55}=\mathbf{g}^{\mu
j}\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,j5}=\mathbf{g}^{\mu\nu}\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,\nu
5}=-\hat{\Gamma}^{\mu}_{\cdot\,55}$ (71)
From the 5D metric (eq. 2), we have $\mathbf{g}^{\mu\nu}=\mathrm{g}^{\mu\nu}$.
Multiplying both sides by $\mathrm{g}_{\mu\sigma}$ and using the orthogonality
relations of the metric,
$\displaystyle\mathrm{g}_{\mu\sigma}\mathrm{g}^{\mu\nu}\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,\nu
5}$ $\displaystyle=$
$\displaystyle-\mathrm{g}_{\mu\sigma}\hat{\Gamma}^{\mu}_{\cdot\,55}$
$\displaystyle\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,\sigma 5}$
$\displaystyle=$
$\displaystyle-\mathrm{g}_{\mu\sigma}\hat{\Gamma}^{\mu}_{\cdot\,55}$
$\displaystyle\Rightarrow\mathrm{T}^{5}_{\cdot\,\sigma 5}$ $\displaystyle=$
$\displaystyle-\mathrm{g}_{\mu\sigma}\hat{\Gamma}^{\mu}_{\cdot\,55}\epsilon\Phi^{-2}$
(72)
Substituting for $\hat{\Gamma}^{\mu}_{\cdot\,55}$ from eq. 65, we obtain
$\mathrm{T}^{5}_{\cdot\,\sigma
5}=\mathrm{g}_{\mu\sigma}[\mathrm{J}^{\mu}-\partial_{5}\mathrm{A}^{\mu}-\mathrm{A}^{\mu}\mathrm{J}_{5}+\mathrm{A}_{\sigma}\partial_{5}\mathrm{g}^{\mu\sigma}]$
(73)
We have thus far used 4 equations and solved for 4 of the 10 independent
torsion components. Next, take $i=\nu$ in eq. 70. The contorsion components
$\mathrm{K}^{\mu}_{\cdot\,\nu 5}$ can be expressed in terms of the torsion
components as follows:
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,\nu 5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{g}^{\mu
j}\left(\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,j\nu}+\mathbf{g}_{\nu
5}\mathrm{T}^{5}_{\cdot\,j5}\right)$ (74) $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{g}^{\mu
5}\left(\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,5\nu}\right)+\frac{1}{2}\mathbf{g}^{\mu\sigma}\left(\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,\sigma\nu}+\mathbf{g}_{\nu
5}\mathrm{T}^{5}_{\cdot\,\sigma 5}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon\mathrm{A}^{\mu}\Phi^{2}\mathrm{T}^{5}_{\cdot\,\nu
5}+\frac{1}{2}\epsilon\mathrm{A}_{\nu}\Phi^{2}\mathrm{g}^{\mu\sigma}\mathrm{T}^{5}_{\cdot\,\sigma
5}+\frac{1}{2}\epsilon\Phi^{2}\mathrm{g}^{\mu\sigma}\mathrm{T}^{5}_{\cdot\,\sigma\nu}$
From $\mathrm{K}^{\mu}_{\cdot\,\nu 5}=-\hat{\Gamma}^{\mu}_{\cdot\,\nu 5}$, and
using eq. 72, we have
$-2\hat{\Gamma}^{\mu}_{\cdot\,\nu
5}+\mathrm{A}^{\mu}\mathrm{g}_{\alpha\nu}\hat{\Gamma}^{\alpha}_{\cdot\,55}+\mathrm{A}_{\nu}\hat{\Gamma}^{\mu}_{\cdot\,55}=\epsilon\Phi^{2}\mathrm{g}^{\mu\sigma}\mathrm{T}^{5}_{\cdot\,\sigma\nu}$
(75)
and hence,
$\mathrm{T}^{5}_{\cdot\,\sigma\nu}=\epsilon\Phi^{-2}\mathrm{g}_{\mu\sigma}\left[-2\hat{\Gamma}^{\mu}_{\cdot\,\nu
5}+\mathrm{A}^{\mu}\mathrm{g}_{\lambda\nu}\hat{\Gamma}^{\lambda}_{\cdot\,55}+\mathrm{A}_{\nu}\hat{\Gamma}^{\mu}_{\cdot\,55}\right]$
(76)
These are 16 equations. Though the torsion in the l.h.s is antisymmetric, the
r.h.s is a combination of symmetric and antisymmetric terms. Substituting for
the 5D Levi-Civita connection from section (A.1), the above equation has the
form,
$\displaystyle\mathrm{T}^{5}_{\cdot\,\sigma\nu}$ $\displaystyle=$
$\displaystyle\left[\partial_{\sigma}\mathrm{A}_{\nu}-\partial_{\nu}\mathrm{A}_{\sigma}\right]+\left[\mathrm{J}_{\sigma}\mathrm{A}_{\nu}-\mathrm{J}_{\nu}\mathrm{A}_{\sigma}\right]$
(77) $\displaystyle+$
$\displaystyle\left[\epsilon\Phi^{-2}\partial_{5}\mathrm{g}_{\sigma\nu}+\mathrm{A}_{\sigma}\mathrm{A}^{\lambda}\partial_{5}\mathrm{g}_{\lambda\nu}+\mathrm{A}_{\nu}\mathrm{A}^{\lambda}\partial_{5}\mathrm{g}_{\lambda\sigma}\right]$
The first two terms in the r.h.s above are antisymmetric in $\sigma$ and
$\nu$, while the third term is symmetric. The antisymmetry of torsion implies
that the symmetric terms in the r.h.s must be zero, and hence
$\partial_{5}\mathrm{g}_{\sigma\nu}=0.$ (78)
Consequently,
$\mathrm{T}^{5}_{\cdot\,\sigma\nu}=\left[\partial_{\sigma}\mathrm{A}_{\nu}-\partial_{\nu}\mathrm{A}_{\sigma}\right]+\left[\mathrm{J}_{\sigma}\mathrm{A}_{\nu}-\mathrm{J}_{\nu}\mathrm{A}_{\sigma}\right].$
(79)
Thus the 20 equations of imposed condition (eq. 70) have determined all 10
independent non-vanishing components of the torsion (equations 73 and 79), and
in addition imposed a constraint on the 10 independent components of the 4D
metric $\mathrm{g}_{\sigma\nu}$ making them independent of $x^{5}$.
### A.3 The contorsion, connection and the Ricci tensor
Some components of contorsion are directly prescribed by the imposed
condition, namely
$\mathrm{K}^{\mu}_{\cdot\,i5}=\mathrm{K}^{\mu}_{\cdot\,5i}=-\hat{\Gamma}^{\mu}_{\cdot\,i5}$.
The remaining components of the contorsion can be calculated from the torsion
components by using eq. 7. Since $\mathrm{T}^{5}_{\cdot\,ij}$ are the only non
vanishing components of torsion, it follows that
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,i5}$ $\displaystyle=$
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,5i}=-\hat{\Gamma}^{\mu}_{\cdot\,i5},$
$\displaystyle\mathrm{K}^{5}_{\cdot\,i5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{T}^{5}_{\cdot\,i5}+\frac{1}{2}\mathbf{g}^{5j}\left(\mathbf{g}_{i5}\mathrm{T}^{5}_{\cdot\,j5}+\mathbf{g}_{55}\mathrm{T}^{5}_{\cdot\,ji}\right)$
$\displaystyle\mathrm{K}^{5}_{\cdot\,5i}$ $\displaystyle=$
$\displaystyle\mathrm{K}^{5}_{\cdot\,i5}+\mathrm{T}^{5}_{\cdot\,5i}$
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,\nu\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathbf{g}^{\mu j}\left(\mathbf{g}_{\nu
5}\mathrm{T}^{5}_{\cdot\,j\lambda}+\mathbf{g}_{\lambda
5}\mathrm{T}^{5}_{\cdot\,j\nu}\right)$
$\displaystyle\mathrm{K}^{5}_{\cdot\,\nu\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{T}^{5}_{\cdot\,\nu\lambda}+\frac{1}{2}\mathbf{g}^{5j}\left(\mathbf{g}_{\nu
5}\mathrm{T}^{5}_{\cdot\,j\lambda}+\mathbf{g}_{\lambda
5}\mathrm{T}^{5}_{\cdot\,j\nu}\right)$ (80)
Substituting for the nonvanishing torsion components given by equations 73 and
79, along with the requirement that the 4D metric is independent of the fifth
dimension (eq. 78), the components of contorsion are found to be
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,i5}$ $\displaystyle=$
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,5i}=-\hat{\Gamma}^{\mu}_{\cdot\,i5},$
$\displaystyle\mathrm{K}^{5}_{\cdot\,i5}$ $\displaystyle=$
$\displaystyle-\hat{\Gamma}^{5}_{\cdot\,i5}+\mathrm{J}_{i},\qquad\mathrm{K}^{5}_{\cdot\,5i}=\mathrm{K}^{5}_{\cdot\,i5}+\mathrm{T}^{5}_{\cdot\,5i},$
$\displaystyle\mathrm{K}^{\mu}_{\cdot\,\nu\lambda}$ $\displaystyle=$
$\displaystyle\epsilon\Phi^{2}\mathrm{F}^{\mu}_{\cdot(\lambda}\mathrm{A}_{\nu)}+\mathrm{A}_{\lambda}\mathrm{A}_{\nu}\mathrm{J}^{\mu}\epsilon\Phi^{2}-\frac{1}{2}\mathrm{A}^{\mu}\partial_{5}(\epsilon\mathrm{A}_{\nu}\mathrm{A}_{\lambda}\Phi^{2}),$
$\displaystyle\mathrm{K}^{5}_{\cdot\,\nu\lambda}$ $\displaystyle=$
$\displaystyle-\epsilon\Phi^{2}\mathrm{A}^{\sigma}\mathrm{F}_{\sigma(\lambda}\mathrm{A}_{\nu)}-(\mathrm{A}_{\sigma}\mathrm{J}^{\sigma})\epsilon\mathrm{A}_{\nu}\mathrm{A}_{\lambda}\Phi^{2}+\frac{1}{2}(\mathrm{A}_{\sigma}\mathrm{A}^{\sigma}+\epsilon\Phi^{-2})\partial_{5}(\epsilon\mathrm{A}_{\nu}\mathrm{A}_{\lambda}\Phi^{2})-\mathrm{A}_{\nu}\mathrm{J}_{\lambda}+\frac{1}{2}\mathrm{F}_{\nu\lambda}.$
(81)
Now from eq. 5, we obtain all the connection coefficients,
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\mathrm{A}_{\nu}+\mathrm{J}_{\mu}\mathrm{A}_{\nu},$
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,5\mu}$ $\displaystyle=$
$\displaystyle\partial_{5}\mathrm{A}_{\mu}+\mathrm{J}_{5}\mathrm{A}_{\mu},$
$\displaystyle\tilde{\Gamma}^{5}_{\cdot\,\mu 5}$ $\displaystyle=$
$\displaystyle\mathrm{J}_{\mu},\,\,\tilde{\Gamma}^{5}_{\cdot\,55}=\mathrm{J}_{5}$
$\displaystyle\tilde{\Gamma}^{\lambda}_{\cdot\,\mu\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\lambda}_{\cdot\,\mu\nu},$
$\displaystyle\tilde{\Gamma}^{\mu}_{\cdot\,55}$ $\displaystyle=$
$\displaystyle\tilde{\Gamma}^{\mu}_{\cdot\,\nu
5}=\tilde{\Gamma}^{\mu}_{\cdot\,5\nu}=0.$ (82)
Taking $i=\mu$ and $k=\nu$ in the Ricci tensor defined by eq. 15, we have
$\begin{array}[]{cccc}\tilde{R}_{\mu\nu}=&+\partial_{\nu}\tilde{\Gamma}^{\sigma}_{\cdot\,\sigma\mu}-\partial_{\sigma}\tilde{\Gamma}^{\sigma}_{\cdot\,\nu\mu}&+\tilde{\Gamma}^{\sigma}_{\cdot\,\nu\lambda}\tilde{\Gamma}^{\lambda}_{\cdot\,\sigma\mu}&-\tilde{\Gamma}^{\sigma}_{\cdot\,\sigma\lambda}\tilde{\Gamma}^{\lambda}_{\cdot\,\nu\mu}\\\
&+\partial_{\nu}\tilde{\Gamma}^{5}_{\cdot\,5\mu}-\partial_{5}\tilde{\Gamma}^{5}_{\cdot\,\nu\mu}&+\tilde{\Gamma}^{5}_{\cdot\,\nu\lambda}\tilde{\Gamma}^{\lambda}_{\cdot\,5\mu}&-\tilde{\Gamma}^{5}_{\cdot\,5\lambda}\tilde{\Gamma}^{\lambda}_{\cdot\,\nu\mu}\\\
&&+\tilde{\Gamma}^{\sigma}_{\cdot\,\nu
5}\tilde{\Gamma}^{5}_{\cdot\,\sigma\mu}&-\tilde{\Gamma}^{\sigma}_{\cdot\,\sigma
5}\tilde{\Gamma}^{5}_{\cdot\,\nu\mu}\\\ &&+\tilde{\Gamma}^{5}_{\cdot\,\nu
5}\tilde{\Gamma}^{5}_{\cdot\,5\mu}&-\tilde{\Gamma}^{5}_{\cdot\,55}\tilde{\Gamma}^{5}_{\cdot\,\nu\mu}\end{array}$
(83)
Since $\tilde{\Gamma}_{\cdot\,}$ is the same as $\Gamma_{\cdot\,}$ when all
the indices are four dimensional, the first line is clearly the 4D Ricci
tensor. The terms in the subsequent lines can be re-expressed in terms of the
4D covariant derivative operator as follows
$\tilde{R}_{\mu\nu}=R_{\mu\nu}+\nabla_{\nu}\tilde{\Gamma}^{5}_{\cdot\,5\mu}-\partial_{5}\tilde{\Gamma}^{5}_{\cdot\,\nu\mu}-\tilde{\Gamma}^{5}_{\cdot\,55}\tilde{\Gamma}^{5}_{\cdot\,\nu\mu}+\tilde{\Gamma}^{5}_{\cdot\,\nu
5}\tilde{\Gamma}^{5}_{\cdot\,5\mu}$ (84)
Substituting for the connection $\tilde{\Gamma}_{\cdot\,}$ from eq. 83, we
find after some algebra this can be simplified as
$\displaystyle\tilde{R}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}+\nabla_{\nu}(\partial_{5}\mathrm{A}_{\mu}+\mathrm{J}_{5}\mathrm{A}_{\mu})-\partial_{5}(\nabla_{\nu}\mathrm{A}_{\mu}+\mathrm{J}_{\nu}\mathrm{A}_{\mu})-\mathrm{J}_{5}(\nabla_{\nu}\mathrm{A}_{\mu}+\mathrm{J}_{\nu}\mathrm{A}_{\mu})+\mathrm{J}_{\nu}(\partial_{5}\mathrm{A}_{\mu}+\mathrm{J}_{5}\mathrm{A}_{\mu})$
(85) $\displaystyle=$ $\displaystyle
R_{\mu\nu}+\mathrm{A}_{\lambda}\partial_{5}\Gamma^{\lambda}_{\cdot\,\mu\nu}$
$\displaystyle=$ $\displaystyle R_{\mu\nu}$
Similarly, the other components of the Ricci tensor are found to be
$\displaystyle\tilde{R}_{\mu 5}$ $\displaystyle=$
$\displaystyle\partial_{5}\tilde{\Gamma}^{\sigma}_{\cdot\,\sigma\mu}=0$ (86)
$\displaystyle\tilde{R}_{5\nu}$ $\displaystyle=$
$\displaystyle\partial_{\nu}\tilde{\Gamma}^{5}_{\cdot\,55}-\partial_{5}\tilde{\Gamma}^{5}_{\cdot\,\nu
5}=0$ (87) $\displaystyle\tilde{R}_{55}$ $\displaystyle=$ $\displaystyle 0$
(88)
Note that neither the connection nor the Ricci tensor depend on the signature
($\epsilon$) of the fifth dimension.
## Appendix B Computing the modified Einstein tensor
Here we provide some intermediate steps to go from equations 24 and 25 to eq.
26 and obtain a simplified expression for $\mathrm{H}_{\mu\nu}$. With
$\mathrm{B}_{\nu}\equiv\mathrm{T}^{5}_{\cdot\,\nu 5}$, and
$\mathrm{H}_{\mu\nu}$ defined by
$\int\mathrm{B}_{\nu}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\mathrm{g}^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}\,d^{5}x=\int\mathrm{H}_{\mu\nu}\delta\mathrm{g}^{\mu\nu}\,\sqrt{-\mathbf{g}}\,d^{5}x,$
(89)
we show that
$\mathrm{H}_{\mu\nu}=\nabla_{(\mu}\mathrm{B}_{\nu)}-(\nabla\cdot\mathrm{B})\mathrm{g}_{\mu\nu}+\mathrm{J}_{(\mu}\mathrm{B}_{\nu)}-(\mathrm{J}\cdot\mathrm{B})\mathrm{g}_{\mu\nu}.$
(90)
Proof:
Consider the first term in the integrand of the l.h.s of eq. 89.
$2\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}=\mathrm{g}^{\mu\alpha}\mathrm{g}^{\nu\lambda}(\partial_{\mu}\delta\mathrm{g}_{\lambda\alpha}+\partial_{\alpha}\delta\mathrm{g}_{\lambda\mu}-\partial_{\lambda}\delta\mathrm{g}_{\mu\alpha})+\mathrm{g}^{\mu\alpha}(\partial_{\mu}\mathrm{g}_{\lambda\alpha}+\partial_{\alpha}\mathrm{g}_{\lambda\mu}-\partial_{\lambda}\mathrm{g}_{\mu\alpha})\delta\mathrm{g}^{\nu\lambda}.$
(91)
The variations of the covariant metric in the above equation can be re-
expressed in terms of the variations of the contravariant metric using the
identity
$\delta\mathrm{g}_{\alpha\lambda}=-\mathrm{g}_{\alpha\mu}\mathrm{g}_{\lambda\nu}\delta\mathrm{g}^{\mu\nu}$.
We note that
$\displaystyle\mathrm{g}^{\mu\alpha}\mathrm{g}^{\nu\lambda}\partial_{\mu}\delta\mathrm{g}_{\lambda\alpha}$
$\displaystyle=$
$\displaystyle-\mathrm{g}^{\nu\lambda}\partial_{\mu}\mathrm{g}_{\lambda\sigma}(\delta\mathrm{g}^{\sigma\mu})-\mathrm{g}^{\mu\alpha}\partial_{\mu}\mathrm{g}_{\beta\alpha}(\delta\mathrm{g}^{\nu\beta})-\partial_{\mu}(\delta\mathrm{g}^{\nu\mu})$
$\displaystyle\mathrm{g}^{\mu\alpha}\mathrm{g}^{\nu\lambda}\partial_{\alpha}\delta\mathrm{g}_{\lambda\mu}$
$\displaystyle=$
$\displaystyle\mathrm{g}^{\mu\alpha}\mathrm{g}^{\nu\lambda}\partial_{\mu}\delta\mathrm{g}_{\lambda\alpha}$
$\displaystyle-\mathrm{g}^{\mu\alpha}\mathrm{g}^{\nu\lambda}\partial_{\lambda}\delta\mathrm{g}_{\mu\alpha}$
$\displaystyle=$
$\displaystyle\mathrm{g}^{\nu\lambda}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\sigma\mu})+\mathrm{g}^{\nu\lambda}\partial_{\lambda}\mathrm{g}_{\beta\sigma}(\delta\mathrm{g}^{\sigma\beta})+\mathrm{g}^{\nu\lambda}\mathrm{g}_{\beta\sigma}\partial_{\lambda}(\delta\mathrm{g}^{\sigma\beta})$
Taken together, we obtain
$2\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}=-2\mathrm{g}^{\nu\lambda}\partial_{\mu}\mathrm{g}_{\lambda\sigma}(\delta\mathrm{g}^{\sigma\mu})-2\partial_{\mu}(\delta\mathrm{g}^{\nu\mu})+2\mathrm{g}^{\nu\lambda}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\sigma\mu})+\mathrm{g}^{\nu\lambda}\mathrm{g}_{\beta\sigma}\partial_{\lambda}(\delta\mathrm{g}^{\sigma\beta})-\mathrm{g}^{\mu\alpha}\partial_{\lambda}\mathrm{g}_{\mu\alpha}(\delta\mathrm{g}^{\nu\lambda}).$
(92)
Next consider the second term in the integrand of the l.h.s of eq. 89. Since
$2\Gamma^{\alpha}_{\cdot\,\alpha\mu}=\mathrm{g}^{\alpha\lambda}\partial_{\mu}\mathrm{g}_{\alpha\lambda}$,
$\displaystyle 2\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}$ $\displaystyle=$
$\displaystyle(\delta\mathrm{g}^{\alpha\lambda})\partial_{\mu}\mathrm{g}_{\alpha\lambda}+\mathrm{g}^{\alpha\lambda}\partial_{\mu}(\delta\mathrm{g}_{\alpha\lambda})$
(93) $\displaystyle=$
$\displaystyle-(\delta\mathrm{g}^{\alpha\lambda})\partial_{\mu}\mathrm{g}_{\alpha\lambda}-\mathrm{g}_{\alpha\lambda}\partial_{\mu}(\delta\mathrm{g}^{\alpha\lambda})$
Using equations 92 and 93, we find
$\displaystyle\mathrm{B}_{\nu}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\mathrm{g}^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}$
$\displaystyle=$
$\displaystyle\mathrm{B}_{\nu}\left[-\partial_{\mu}(\delta\mathrm{g}^{\mu\nu})+\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\partial_{\mu}(\delta\mathrm{g}^{\alpha\beta})\right]\sqrt{-\mathbf{g}}$
(94) $\displaystyle+$
$\displaystyle\mathrm{B}_{\nu}\left[-\mathrm{g}^{\nu\lambda}\partial_{\mu}\mathrm{g}_{\lambda\sigma}(\delta\mathrm{g}^{\sigma\mu})+\frac{3}{2}\mathrm{g}^{\nu\lambda}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\sigma\mu})-\frac{1}{2}\mathrm{g}^{\mu\sigma}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\nu\lambda})\right]\sqrt{-\mathbf{g}}$
The first line in the r.h.s of the above equation contains terms with the
derivatives of the variation. We note that these terms are eventually going to
be integrated. By integrating them by parts and ignoring the boundary terms,
the above equation takes the form
$\displaystyle\mathrm{B}_{\nu}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\mathrm{g}^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}$
$\displaystyle=$
$\displaystyle\left[\partial_{\mu}(\mathrm{B}_{\nu}\sqrt{-\mathbf{g}})\delta\mathrm{g}^{\mu\nu}-\partial_{\mu}(\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\mathrm{B}_{\nu}\sqrt{-\mathbf{g}})\delta\mathrm{g}^{\alpha\beta}\right]$
(95) $\displaystyle+$
$\displaystyle\mathrm{B}_{\nu}\left[-\mathrm{g}^{\nu\lambda}\partial_{\mu}\mathrm{g}_{\lambda\sigma}(\delta\mathrm{g}^{\sigma\mu})+\frac{3}{2}\mathrm{g}^{\nu\lambda}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\sigma\mu})-\frac{1}{2}\mathrm{g}^{\mu\sigma}\partial_{\lambda}\mathrm{g}_{\mu\sigma}(\delta\mathrm{g}^{\nu\lambda})\right]\sqrt{-\mathbf{g}}$
To simplify the r.h.s of eq. 95 it is useful to note the following identities
$\partial_{\mu}(\mathrm{g}^{\mu\nu})=\mathrm{g}^{\sigma\nu}\mathrm{g}_{\sigma\lambda}\partial_{\mu}(\mathrm{g}^{\mu\lambda})=-\mathrm{g}^{\sigma\nu}\mathrm{g}^{\mu\lambda}\partial_{\mu}(\mathrm{g}_{\sigma\lambda}).$
(96)
$\partial_{\mu}(\sqrt{-\mathbf{g}})=\frac{\partial_{\mu}\mathbf{g}}{2\mathbf{g}}\sqrt{-\mathbf{g}}=\frac{\sqrt{-\mathbf{g}}}{2}\mathbf{g}^{ij}\partial_{\mu}(\mathbf{g}_{ij})$
(97) $\displaystyle\mathbf{g}^{ij}\partial_{\mu}(\mathbf{g}_{ij})$
$\displaystyle=$
$\displaystyle\mathrm{g}^{\sigma\lambda}\partial_{\mu}(\mathrm{g}_{\sigma\lambda})+\mathrm{g}^{\sigma\lambda}\partial_{\mu}(\epsilon\mathrm{A}_{\sigma}\mathrm{A}_{\lambda}\Phi^{2})+2\mathbf{g}^{\sigma
5}\partial_{\mu}(\epsilon\mathrm{A}_{\sigma}\Phi^{2})+\mathbf{g}^{55}\partial_{\mu}(\epsilon\Phi^{2})$
(98) $\displaystyle=$
$\displaystyle\mathrm{g}^{\sigma\lambda}\partial_{\mu}(\mathrm{g}_{\sigma\lambda})+2\mathrm{J}_{\mu}$
Using the above identities, the r.h.s of eq. 95 becomes
$\displaystyle\mathrm{B}_{\nu}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\mathrm{g}^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}$
$\displaystyle=$
$\displaystyle\left[\partial_{\mu}(\mathrm{B}_{\nu})\delta\mathrm{g}^{\mu\nu}-\partial_{\mu}(\mathrm{B}_{\nu})\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\delta\mathrm{g}^{\alpha\beta}\right.$
(99) $\displaystyle+$
$\displaystyle\mathrm{B}_{\nu}\mathrm{g}_{\alpha\beta}\mathrm{g}^{\sigma\nu}\mathrm{g}^{\mu\lambda}\partial_{\mu}(\mathrm{g}_{\sigma\lambda})\delta\mathrm{g}^{\alpha\beta}-\frac{1}{2}\mathrm{B}_{\nu}\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\mathrm{g}^{\sigma\lambda}\partial_{\mu}(\mathrm{g}_{\sigma\lambda})\delta\mathrm{g}^{\alpha\beta}$
$\displaystyle-$
$\displaystyle\mathrm{B}_{\nu}\mathrm{g}^{\nu\sigma}\partial_{\mu}(\mathrm{g}_{\sigma\alpha})\delta\mathrm{g}^{\alpha\mu}+\frac{1}{2}\mathrm{B}_{\nu}\mathrm{g}^{\nu\sigma}\partial_{\sigma}(\mathrm{g}_{\alpha\mu})\delta\mathrm{g}^{\alpha\mu}$
$\displaystyle+$
$\displaystyle\left.\mathrm{B}_{\nu}\mathrm{J}_{\mu}\delta\mathrm{g}^{\mu\nu}-\mathrm{B}_{\nu}\mathrm{J}_{\mu}\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\delta\mathrm{g}^{\alpha\beta}\right]\sqrt{-\mathbf{g}}.$
Rewriting the derivatives of the metric in terms of Levi-Civita connection, we
find
$\displaystyle\mathrm{B}_{\nu}\left[\mathrm{g}^{\mu\alpha}\delta\Gamma^{\nu}_{\cdot\,\mu\alpha}-\mathrm{g}^{\mu\nu}\delta\Gamma^{\alpha}_{\cdot\,\alpha\mu}\right]\sqrt{-\mathbf{g}}$
$\displaystyle=$
$\displaystyle\left[\partial_{\mu}(\mathrm{B}_{\nu})\delta\mathrm{g}^{\mu\nu}-\partial_{\mu}(\mathrm{B}_{\nu})\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\delta\mathrm{g}^{\alpha\beta}\right.$
$\displaystyle-$
$\displaystyle\mathrm{B}_{\nu}\Gamma^{\nu}_{\cdot\,\mu\sigma}\delta\mathrm{g}^{\sigma\mu}+\mathrm{B}_{\nu}\Gamma^{\nu}_{\cdot\,\mu\sigma}\mathrm{g}^{\sigma\mu}\mathrm{g}_{\alpha\beta}\delta\mathrm{g}^{\alpha\beta}$
$\displaystyle+$
$\displaystyle\left.\mathrm{B}_{\nu}\mathrm{J}_{\mu}\delta\mathrm{g}^{\mu\nu}-\mathrm{B}_{\nu}\mathrm{J}_{\mu}\mathrm{g}^{\mu\nu}\mathrm{g}_{\alpha\beta}\delta\mathrm{g}^{\alpha\beta}\right]\sqrt{-\mathbf{g}}.$
$=\left[\nabla_{\mu}\mathrm{B}_{\nu}-(\nabla\cdot\mathrm{B})\mathrm{g}_{\mu\nu}+\mathrm{J}_{\mu}\mathrm{B}_{\nu}-(\mathrm{J}\cdot\mathrm{B})\mathrm{g}_{\mu\nu}\right]\delta\mathrm{g}^{\mu\nu}\sqrt{-\mathbf{g}}$
(100)
Since the variation $\delta\mathrm{g}^{\mu\nu}$ is symmetric in the indices
$\mu$ and $\nu$, only the symmetric part of the r.h.s of the above equation
will contribute to the equations of motion. Hence $\mathrm{H}_{\mu\nu}$ will
be given by eq. 90.
|
arxiv-papers
| 2012-03-25T23:15:43 |
2024-09-04T02:49:29.064090
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Karthik H. Shankar, Anand Balaraman and Kameshwar C. Wali",
"submitter": "Karthik Shankar",
"url": "https://arxiv.org/abs/1203.5552"
}
|
1203.5667
|
# Simulating galactic outflows with thermal supernova feedback
Claudio Dalla Vecchia1,2 and Joop Schaye2
1Max Planck Institute for Extraterrestrial Physics, Gissenbachstraße 1, 85748
Garching, Germany
2Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the
Netherlands E-mail: caius@mpe.mpg.deE-mail: schaye@strw.leidenuniv.nl
###### Abstract
Cosmological simulations make use of sub-grid recipes for the implementation
of galactic winds driven by massive stars because direct injection of
supernova energy in thermal form leads to strong radiative losses, rendering
the feedback inefficient. We argue that the main cause of the catastrophic
cooling is a mismatch between the mass of the gas in which the energy is
injected and the mass of the parent stellar population. Because too much mass
is heated, the temperatures are too low and the cooling times too short. We
use analytic arguments to estimate, as a function of the gas density and the
numerical resolution, the minimum heating temperature that is required for the
injected thermal energy to be efficiently converted into kinetic energy. We
then propose and test a stochastic implementation of thermal feedback that
uses this minimum temperature increase as an input parameter and that can be
employed in both particle- and grid-based codes. We use smoothed particle
hydrodynamics simulations to test the method on models of isolated disc
galaxies in dark matter haloes with total mass $10^{10}$ and
$10^{12}h^{-1}~{}\mbox{M}_{\odot}$. The thermal feedback strongly suppresses
the star formation rate and can drive massive, large-scale outflows without
the need to turn off radiative cooling temporarily. In accord with
expectations derived from analytic arguments, for sufficiently high resolution
the results become insensitive to the imposed temperature jump and also agree
with high-resolution simulations employing kinetic feedback.
###### keywords:
methods: numerical — ISM: bubbles — ISM: jets and outflows — galaxies:
evolution — galaxies: formation — galaxies: ISM
## 1 Introduction
It is widely accepted that star formation (SF) feeds back energy into the
interstellar medium (ISM). The energy released by massive stars, both through
stellar winds and core-collapse supernova (SN) explosions, can efficiently
suppress SF by evaporating dense, star-forming clouds, by generating
supersonic turbulence and, eventually, by generating powerful, large-scale
outflows that eject gas from galaxies and enrich the intergalactic medium.
Modern cosmological simulations that follow the formation and evolution of
galaxies still lack both the resolution and the physics that is required to
model the multi-phase ISM and individual massive stars or SN explosions. The
same is true for simulations of individual galaxies, although the resolution
of such models is now sufficient to begin to crudely disentangle the relative
roles of the different mechanisms through which massive stars inject energy
and momentum (e.g. radiation pressure vs. SN explosions, Hopkins, Quataert, &
Murray, 2012). Thus, in naive implementations of stellar feedback, “star”
particles representing simple stellar populations (SSPs) distribute “SN
energy” over neighbouring resolution elements at each time step. This
procedure is well-known to be inefficient, in that most of the thermal energy
is radiated away before it can be converted to kinetic energy (e.g. Katz,
Weinberg, & Hernquist, 1996).
Three types of sub-grid recipes are commonly used to solve the over-cooling
problem: injecting the energy in kinetic form (e.g. Navarro & White, 1993;
Mihos & Hernquist, 1994; Kawata, 2001; Kay et al., 2002; Springel & Hernquist,
2003; Oppenheimer & Davé, 2006; Dalla Vecchia & Schaye, 2008; Dubois &
Teyssier, 2008; Hopkins, Quataert, & Murray, 2012), suppressing radiative
cooling by hand (e.g. Gerritsen, 1997; Mori et al., 1997; Thacker & Couchman,
2000; Kay et al., 2002; Sommer-Larsen, Götz, & Portinari, 2003; Brook et al.,
2004; Stinson et al., 2006; Piontek & Steinmetz, 2011), and decoupling the
different thermal phases by hand (e.g. Marri & White, 2003; Scannapieco et
al., 2006; Murante et al., 2010). Each solution has its own pros and cons and
all require the specification of sub-grid parameters. The different approaches
should converge when the resolution is increased, although it is not obvious
that this will in fact happen.
The inefficiency of thermal feedback is usually attributed to a lack of
resolution: the energy is deposited in gas that is too dense, because the hot,
low-density, bubbles that fill much of the volume of the multiphase ISM are
missing. However, as we pointed out in Dalla Vecchia & Schaye (2008, hereafter
DS08), a more fundamental problem is the fact that the SN energy is
distributed over too much mass, which implies that the temperature of the SN
heated gas is too low and hence the cooling time too short. Indeed, in reality
one SNII is produced for every $\sim 10^{2}~{}\mbox{M}_{\odot}$ of stars, and
the energy released in the explosion is initially carried by $\ll
10^{2}~{}\mbox{M}_{\odot}$ of ejecta. Hence, the ratio of the mass of the
ejecta and the mass of the stellar population that released the energy is
small $\ll 1$. In contrast, in simulations this ratio is $\gg 1$, unless the
mass of star particles is very large compared with that of the surrounding gas
resolution elements, which is typically not the case. Increasing the
resolution does not change the ratio between the mass of a star particle and
the mass of the neighbouring resolution elements (although it may in the case
of grid simulations) and hence is unlikely to solve the over-cooling problem
by itself.
The temperature jump of the gas receiving feedback energy can be increased by
storing the energy until it suffices to heat the gas by a desired amount.
Indeed, this strategy is for example used in some sub-grid recipes for
feedback from active galactic nuclei (AGN), which store the AGN energy in the
black hole until the neighbouring gas can be heated to a desired temperature
(Booth & Schaye, 2009). This approach is, however, not suitable for SN
feedback. Storing the energy in a star particle would not help because
standard implementations of thermal SN feedback are inefficient even if all
the SN energy of an SSP is released at once. Storing energy in gas particles
is undesirable because it would make the feedback non-local, which would for
example mean that heavy elements released by the star particle are less likely
to be carried by outflows.
An alternative way to increase the temperature jump is to reduce the ratio of
the heated mass to the mass of a star particle. This can be done by reducing
the number of neighbouring resolution elements that are heated, but that does
not help if even a single resolution element contains too much mass. Moreover,
this approach would result in a range of temperatures if not all resolution
elements have the same mass. To guarantee the efficiency of the feedback, one
would like to be able to specify the temperature jump of the gas that receives
energy. This can be accomplished by making the thermal feedback stochastic:
the probability that a neighbouring resolution element is heated will then
depend on the desired temperature jump and on the ratio of the mass of the
star particle to that of the neighbouring gas resolution element.
A stochastic approach to thermal feedback was tried, and found to be
effective, by Kay, Thomas, & Theuns (2003) in smoothed particle hydrodynamics
(SPH) simulations of groups of galaxies. For each simulation time-step and
each star particle, they integrated the energy released by SNe and distributed
it stochastically to its nearest gas neighbour. They only considered the
limiting case in which the temperature jump was sufficiently high that the
number of particles that could be heated per time step was less than one.
In this paper we generalise the method of Kay, Thomas, & Theuns (2003) to work
also for temperature jumps sufficiently low for multiple neighbours to be
heated. Using SPH simulations of isolated disc galaxies embedded in dark
haloes with total mass $10^{10}$ and $10^{12}h^{-1}~{}\mbox{M}_{\odot}$, we
show that our implementation of thermal feedback is able to strongly suppress
SF, to alter the morphology of the galaxy and to generate galactic winds.
Reassuringly, for our high-resolution simulations we reproduce the results of
DS08, where we simulated the same disc galaxies with kinetic feedback.
This paper is organised as follows. We compute the energy provided by core
collapse SNe (SNII) in Section 2, where we show that for a standard initial
mass function (IMF), a single star particle produces enough SN energy to heat
a gaseous mass equal to the mass of the star particle by a few keV, a
temperature that is sufficiently high for radiative cooling to be relatively
inefficient. We present our numerical implementation for thermal SN feedback
in SPH in Section 3, where we also compute some useful quantities as a
function of the free parameters of the method: the amount of SN energy
injected per unit stellar mass and the desired temperature jump. We also
explain how the method could be adapted for grid simulations. We dedicate
Section 4 to the derivation of the resolution constraints of our feedback
recipe, which follow from the requirement that the radiative cooling time
exceeds the sound crossing time across the heated resolution element so that
the thermal energy is efficiently converted into kinetic form. After
describing our simulations of isolated discs galaxies in Section 5, we present
our results in Section 6. Finally, we summarise and discuss our conclusions in
Section 7.
Videos and high-resolution images can be found at:
http://www.strw.leidenuniv.nl/DS12/
## 2 Energy provided by SNII
Each star particle is treated as a simple (or single) stellar population.
Thus, its stellar content is simply described by an IMF, $\Phi(M)$. The number
of stars per unit stellar mass ending their life as SNII, $n_{\rm SNII}$, is
then the integral of the IMF over the mass range $[M_{0},M_{1}]$,
$n_{\rm SNII}=\int_{M_{0}}^{M_{1}}\Phi(M)\,{\rm d}M,$ (1)
where $M_{0}$ and $M_{1}$ are the minimum and maximum initial mass of stars
that will explode as core collapse SNe. Throughout the paper we will use a
Chabrier IMF and the mass interval $[M_{0},M_{1}]=[6,100]~{}\mbox{M}_{\odot}$,
although we also report calculations for a Salpeter IMF and for the widely
used mass range of $[8,100]~{}\mbox{M}_{\odot}$ ($6$-$8~{}\mbox{M}_{\odot}$
stars explode as electron capture SNe in models with convective overshoot;
e.g. Chiosi, Bertelli, & Bressan 1992). For the Chabrier (Salpeter) IMF we
obtain, for the mass range $[M_{0},M_{1}]=[6,100]~{}\mbox{M}_{\odot}$, $n_{\rm
SNII}=1.736\times 10^{-2}~{}\mbox{M}_{\odot}^{-1}$ ($1.107\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}$). For the mass range
$[M_{0},M_{1}]=[8,100]~{}\mbox{M}_{\odot}$, we have $n_{\rm SNII}=1.180\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}$ ($0.742\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}$).
The total available energy per unit stellar mass provided by SNII,
$\epsilon_{\rm SNII}=n_{\rm SNII}E_{\rm SNII}$, is given by
$\epsilon_{\rm SNII}=8.73\times
10^{15}~{}\mbox{erg}~{}\mbox{g}^{-1}\left(\frac{n_{\rm SNII}}{1.736\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}}\right)E_{51},$ (2)
where $E_{\rm SNII}\equiv E_{51}\times 10^{51}~{}{\rm erg}$ is the available
energy from a single SNII event and we will assume $E_{51}=1$. The amount of
energy from SNII available in a SSP particle is then $m_{\ast}\epsilon_{\rm
SNII}$, where $m_{\ast}$ is the initial mass of the star particle.
If the energy is used to heat a gas mass $m_{\rm g,heat}$, then the
corresponding temperature increase is given by
$\displaystyle\Delta T$ $\displaystyle=$ $\displaystyle(\gamma-1)\frac{\mu
m_{\rm H}}{k_{\rm B}}\epsilon_{\rm SNII}\frac{m_{\ast}}{m_{\rm g,heat}}$ (3)
$\displaystyle=$ $\displaystyle 4.23\times 10^{7}~{}\mbox{K}\left(\frac{n_{\rm
SNII}}{1.736\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}}\right)\left(\frac{\mu}{0.6}\right)\times$
$\displaystyle E_{51}\frac{m_{\ast}}{m_{\rm g,heat}},$
where $\gamma=5/3$ is the ratio of specific heats for an ideal monatomic gas,
$k_{\rm B}$ is the Boltzmann constant, $m_{\rm H}$ is the mass of the proton,
and $\mu m_{\rm H}$ is the mean particle mass. We have assumed the gas to be
monatomic and neglected the energy used to increase the degree of ionisation
of the plasma.
The standard SPH approach is to distribute the SN energy over all neighbours
of a star particle.111The standard approach is to weigh the contribution to
each receiving gas particle by the SPH kernel. We ignore this weighting here
for simplicity and because differences between SPH neighbours are by
definition poorly resolved. The heated mass is then $m_{\rm g,heat}=N_{\rm
ngb}m_{\rm g}$, where $N_{\rm ngb}$ is the number of neighbouring particles
(typically 32–64, we use 48 in our simulations) and $m_{\rm g}$ is the mass of
a single gas particle. Assuming $m_{\ast}=m_{\rm g}$, we can see from equation
(3), that the average temperature increase for the heated gas particles is
$\sim 10^{6}~{}\mbox{K}$, which falls in the temperature regime for which the
cooling time is relatively short (e.g. Wiersma, Schaye, & Smith, 2009a). Note
that this procedure leads to even lower temperature increases if
$m_{\ast}<m_{\rm g}$, which happens if multiple star particles are produced by
each star-forming gas particle. Heating only a single gas particle would give
a temperature increase of $\sim 10^{7.5}~{}\mbox{K}$ and, as we will show, a
much longer cooling time. As we will describe in the next section, we will
make the temperature increase $\Delta T$ a free parameter and stochastically
heat neighbouring gas particles.
## 3 Thermal feedback implementation
For simplicity and for consistency with DS08, we assume that all the SN energy
produced within a star particle becomes available in a single time step. Once
a stellar particle reaches an age $t_{\rm SN}=3\times 10^{7}~{}{\rm yr}$,
corresponding to the maximum lifetime of stars that end their lives as core
collapse SNe, it stochastically injects thermal energy into its surroundings.
Another reason why we prefer to impose this small time delay, is that it
prevents the injection of energy before the release of heavy elements by (the
progenitors of) SNe, a process that happens continuously in our stellar
evolution prescription (Wiersma et al., 2009b).
We note that it is straightforward to modify our implementation of thermal
feedback to the case where SN energy is released stochastically over multiple
time steps. All one needs to do, is to replace $\epsilon_{\rm SNII}$ in the
equations below by the energy per unit mass that is released by the star
particle over the current time step. We tested this approach and did not find
any significant difference.
Before providing a detailed description of our implementation of thermal SN
feedback, we first describe the two free parameters used in our model.
### 3.1 Free parameters
Our recipe uses two free parameters. The first parameter, $f_{\rm th}$, is the
fraction of the available SN energy that is actually used in performing the
feedback.222We do not treat the energy per SN, $E_{51}$, as a free parameter
because it is relatively well constrained by observations and because changes
in $E_{51}$ are in any case degenerate with changes in $f_{\rm th}$. The value
of $f_{\rm th}$ is between zero and unity, and can be used to control the
efficiency of the feedback. Note that this parameter is not particular to our
method, it is common to all implementations of SNII feedback.
We will present simulations that use either $f_{\rm th}=0.4$ or $f_{\rm
th}=1$. The former value is used for comparison with the kinetic feedback of
DS08, where we used the same value. It was also used by Schaye et al. (2010),
who already showed some results of cosmological simulations that used the
prescription for thermal feedback presented here. For most of our runs we will
use $f_{\rm th}=1$ because it is an interesting limiting case and because high
values can be justified for thermal feedback since we are in fact simulating
radiative losses (we do not at any point turn off radiative cooling).
As discussed in the introduction, we wish to control the temperature increase
of the heated gas particles in order to avoid the regime of short cooling
times. We accomplish this by making the temperature increase $\Delta T$, or
rather the increase of the thermal energy per unit mass $\Delta\epsilon$, the
second free parameter. Although $\Delta\epsilon$ is the parameter we actually
use, we will often refer to the more intuitive corresponding value of $\Delta
T$, computed assuming the gas to be fully ionised ($\mu=0.6$). Our fiducial
temperature increase is $\Delta T=10^{7.5}~{}\mbox{K}$, but we will explore a
range of values.
Simulations employing the recipe described here should use values of $\Delta
T$ sufficiently high to avoid catastrophic cooling. As shown analytically in
section 4, the required minimum value will depend on the resolution. The
remaining freedom, particularly the value of $f_{\rm th}$, including possible
dependencies on local physical conditions, can be used to calibrate the method
by comparing the predictions for the quantities of interest to observations.
Naturally, the best-fit parameter values may depend on other numerical and
physical parameters.
### 3.2 Distributing the energy
In this section we derive the stochastic formulation for the energy injection.
The energy released by a single star particle is shared among a fraction of
the $N_{\rm ngb}$ neighbouring resolution elements. We will hereafter refer to
resolution elements as “particles” but note that the same method will also
work for grid simulations (replace “particle” by “cell”). We give each gas
particle the same probability $p$ of receiving energy from the star,
irrespective of its mass (and, for the case of SPH, irrespective of its kernel
weight). We draw a random number $0\leq r\leq 1$ for each star-gas particle
pair, and increase the internal energy of the gas particle by $\Delta\epsilon$
if $r\leq p$. We will now derive the value of the probability $p$.
The expectation value for the total amount of energy from SNII injected by a
single star particle in the surrounding medium is
$p\sum_{i=1}^{N_{\rm ngb}}E_{i}=p\Delta\epsilon\sum_{i=1}^{N_{\rm ngb}}m_{i},$
(4)
where $E_{i}$ and $m_{i}$ are, respectively, the total energy given to and the
mass of gas particle $i$. We require the mean injected energy to equal the
energy contributed by the star particle, $f_{\rm th}m_{\ast}\epsilon_{\rm
SNII}$, from which the probability $p$ follows:
$p=f_{\rm th}\frac{\epsilon_{\rm
SNII}}{\Delta\epsilon}\frac{m_{\ast}}{\sum_{i=1}^{N_{\rm ngb}}m_{i}}.$ (5)
Thus, the probability that a gas particle is heated is proportional to $f_{\rm
th}$, the fraction of total available SNII energy that the star particle
shares with the surrounding gas, and inversely proportional to
$\Delta\epsilon$, the amount of thermal energy per unit mass that is given to
each heated gas particle.
The stochastic treatment breaks down if the probability is larger than one,
because in that case the average amount of injected energy is lower than the
required value. Imposing $p\leq 1$ puts a constraint on the value of the
parameter $\Delta\epsilon$:
$\Delta\epsilon\geq f_{\rm th}\epsilon_{\rm
SNII}\frac{m_{\ast}}{\sum_{i=1}^{N_{\rm ngb}}m_{i}}\simeq\frac{f_{\rm
th}\epsilon_{\rm SNII}}{N_{\rm ngb}},$ (6)
where the last equality holds exactly if all particles (gas and stars) have
the same mass, as is usually approximately true for SPH simulations.
For a Chabrier IMF, assuming $N_{\rm ngb}=48$ and $f_{\rm th}=1$, we obtain
$\Delta\epsilon\geq 1.82\times 10^{14}~{}\mbox{erg}~{}\mbox{g}^{-1}$, which
corresponds to a temperature increase $\Delta T\geq 8.8\times
10^{5}~{}\mbox{K}$. For our fiducial choice for the temperature increase of
$10^{7.5}~{}\mbox{K}$ we are well above this lower limit. For the case of grid
simulations it is in principle possible that $\sum_{i=1}^{N_{\rm ngb}}m_{i}\ll
m_{\ast}$ and hence that $p>1$. In that case it is necessary to increase
$\Delta\epsilon$ above the chosen value in order to limit the probability to
unity.
The sum of the probability over all neighbours gives the expectation value for
the number of heated neighbours:
$\left<N_{\rm heat}\right>=f_{\rm th}\frac{\epsilon_{\rm
SNII}}{\Delta\epsilon}\frac{m_{\ast}N_{\rm ngb}}{\sum_{i=1}^{N_{\rm
ngb}}m_{i}}\simeq\frac{f_{\rm th}\epsilon_{\rm SNII}}{\Delta\epsilon},$ (7)
where the last equality holds exactly if all particles have the same mass. The
average number of heated neighbours is inversely proportional to
$\Delta\epsilon$ and proportional to $f_{\rm th}$. Expressed in terms of the
temperature increase $\Delta T$, the mean number of heated neighbours is
$\displaystyle\left<N_{\rm heat}\right>$ $\displaystyle=$ $\displaystyle
1.34~{}E_{51}\left(\frac{n_{\rm SNII}}{1.736\times
10^{-2}~{}\mbox{M}_{\odot}^{-1}}\right)\left(\frac{\mu}{0.6}\right)\times$ (8)
$\displaystyle f_{\rm th}\left(\frac{\Delta
T}{10^{7.5}~{}\mbox{K}}\right)^{-1}.$
Ideally, the energy should on average be shared with at least one gas
neighbour to make the feedback local to the star particle and to ensure that
the metals released by massive stars can be driven outwards. By injecting all
the SNII energy from a SSP at once, we can satisfy this constraint for
temperature increases that are sufficiently large to make radiative cooling
(initially) inefficient.
## 4 Ensuring effective feedback: resolution requirements
The thermal feedback can only be effective if the heated gas responds
hydrodynamically to the temperature increase before the thermal energy is
radiated away. This implies that the sound-crossing time scale across a heated
resolution element, $t_{\rm s}$, must be short compared with the radiative
cooling time scale in the heated gas, $t_{\rm c}$. If this condition is
satisfied, then the gas will start to expand adiabatically, doing work on its
surroundings and converting thermal energy into kinetic energy. The ratio
$t_{\rm s}/t_{\rm c}$ can be decreased either by increasing $t_{\rm c}$, which
can usually be accomplished by increasing the temperature, or by decreasing
$t_{\rm s}$, which means increasing the temperature and/or the spatial
resolution.
The sound crossing time across a resolution element of length $h$ is
$\displaystyle t_{\rm s}$ $\displaystyle=$ $\displaystyle\frac{h}{c_{\rm
s}}=\left(\frac{\mu m_{\rm H}}{\gamma k_{\rm
B}}\right)^{1/2}\frac{h}{T^{1/2}}$ (9) $\displaystyle=$ $\displaystyle
1.15\times 10^{5}~{}{\rm
yr}\left(\frac{\mu}{0.6}\right)^{1/2}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{-1/2}\left(\frac{h}{100~{}{\rm
pc}}\right),$
where $c_{\rm s}$ is the local sound speed. This time scale is related to the
classical definition of the Courant time step, $\Delta t=Ch/c_{\rm s}=Ct_{\rm
s}$, where $C$ ($<1$) is the Courant factor.
Figure 1: Contour plot of the logarithm of the ratio between the cooling time
and the sound-crossing time over the SPH kernel as a function of the density
and the temperature of the gas. The cooling time scale is computed using
tabulated cooling rates for primordial (solid contours) and solar (dotted
contours) element abundances and assuming collisional ionisation equilibrium.
We show the ratio for the particle mass used in our simulations of the
$10^{12}~{}h^{-1}~{}\mbox{M}_{\odot}$ halo. The vertical dashed line marks the
fiducial heating temperature of $10^{7.5}~{}\mbox{K}$ and the horizontal
dashed line the star formation density threshold in our simulations.
We define the radiative cooling time as
$t_{\rm c}=\frac{u}{\Lambda}=\frac{\rho\epsilon}{\Lambda},$ (10)
where $u$ and $\Lambda$ are the internal energy and radiative cooling rate per
unit volume, respectively, and $\rho$ is the gas density. The internal energy
per unit volume can be written as
$u=\frac{1}{\gamma-1}\frac{k_{\rm B}T}{\mu m_{\rm
H}}\rho=\frac{1}{\gamma-1}\frac{k_{\rm B}T}{\mu X_{\rm H}}n_{\rm H},$ (11)
where $n_{\rm H}$ and $X_{\rm H}$ are the hydrogen number density and mass
fraction, respectively. Before showing results for general cooling functions,
we will consider the case of pure Brehmsstrahlung, which dominates the cooling
rate for $T\ga 10^{7}~{}\mbox{K}$ and for which the cooling rate is given by
(Osterbrock, 1989):
$\displaystyle\Lambda$ $\displaystyle\simeq$ $\displaystyle 7.99\times
10^{-24}~{}\mbox{erg}~{}\mbox{cm}^{-3}~{}\mbox{s}^{-1}\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{2}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{1/2}\times$
(12) $\displaystyle g_{\rm f}\eta_{\rm e}(\eta_{\rm HII}+\eta_{\rm
HeII}+\eta_{\rm HeIII})$ $\displaystyle\simeq$ $\displaystyle 1.12\times
10^{-23}~{}\mbox{erg}~{}\mbox{cm}^{-3}~{}\mbox{s}^{-1}\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{2}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{1/2}$
$\displaystyle\frac{(1+X_{\rm H})(1+3X_{\rm H})}{8X_{\rm H}^{2}},$
where $\eta_{\rm i}=n_{\rm i}/n_{\rm H}$ is the number density of species $i$
relative to hydrogen and we assumed the plasma to be fully ionised in the last
step. We assumed for simplicity that the Gaunt factor $g_{\rm f}\simeq 1.4$
(consistent with the approximated equation given by Theuns et al. (1998) for
our fiducial temperature increase) and a primordial composition. Hence, we
obtain a radiative cooling time of
$\displaystyle t_{\rm c}$ $\displaystyle\simeq$ $\displaystyle 3.26\times
10^{7}~{}{\rm yr}\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{1/2}\times$
(13) $\displaystyle\left(\frac{\mu}{0.6}\right)^{1/2}\left(\frac{f(X_{\rm
H})}{0.13}\right),$
where
$f(X_{\rm H})=X_{\rm H}(1+X_{\rm H})^{-1}(1+3X_{\rm H})^{-1}$ (14)
and $f(X_{\rm H}=0.752)\simeq 0.13$.
Thus, we can write for the ratio between the two time scales
$\displaystyle\frac{t_{\rm c}}{t_{\rm s}}$ $\displaystyle=$ $\displaystyle
2.8\times 10^{2}\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)\left(\frac{h}{100~{}{\rm
pc}}\right)^{-1}\times$ (15)
$\displaystyle\left(\frac{\mu}{0.6}\right)^{-3/2}\left(\frac{f(X_{\rm
H})}{0.13}\right),$
For the case of SPH simulations, the spatial resolution is given by the gas
particle’s smoothing kernel which can be approximated as
$h\simeq\left(\frac{3}{4\pi}\frac{\sum_{i=1}^{N_{\rm
ngb}}m_{i}}{\rho}\right)^{1/3}\simeq\left(\frac{3}{4\pi}\frac{N_{\rm
ngb}\left<m\right>}{m_{\rm H}n_{\rm H}}X_{\rm H}\right)^{1/3},$ (16)
where $\left<m\right>$ is the average mass of the gas particles. Substituting
the above approximation into equation (15) we obtain the ratio of time scales
$\displaystyle\frac{t_{\rm c}}{t_{\rm s}}$ $\displaystyle\simeq$
$\displaystyle 98\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{-2/3}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)\left(\frac{\left<m\right>}{7\times
10^{4}~{}\mbox{M}_{\odot}}\right)^{-1/3}\times$ (17)
$\displaystyle\left(\frac{N_{\rm
ngb}}{48}\right)^{-1/3}\left(\frac{\mu}{0.6}\right)^{-3/2}\left(\frac{g(X_{\rm
H})}{0.14}\right),$
where $g(X_{\rm H})=X_{\rm H}^{-1/3}f(X_{\rm H})$ and we used the particle
mass appropriate for our simulations of the
$10^{12}~{}h^{-1}~{}\mbox{M}_{\odot}$ halo.
We expect cooling losses to be important for $t_{\rm c}\lesssim t_{\rm s}$.
The exact value of the ratio $f_{\rm t}\equiv t_{\rm c}/t_{\rm s}$ required to
ensure efficient feedback can only be determined using simulations, but we
expect it to be similar to 10 and will therefore use this as our fiducial
value. This agrees well with the recent, independent work of Creasey et al.
(2011), who derived a resolution criterion for shock capturing in SPH and
adaptive mesh refinement (AMR) simulations. Their criterion is based on
comparing the rates of shock heating and radiative cooling in a shock front,
and ensures that shock heating overwhelms cooling in order to avoid numerical
over-cooling. Following the suggestion of DS08, they applied their results to
the injection of thermal energy, and obtained a criterion that basically
translates into $t_{\rm c}/t_{\rm s}>8$.333Note that Creasey et al. (2011)
derived their criterion by estimating the velocity of a blast wave at the
blast radius of half the mean inter-particle distance.
From the relation between the sound crossing time and the Courant time step,
one can estimate the number of simulation time steps it would take the gas to
radiate its thermal energy if it cooled isochorically: $n_{\rm step}\sim
t_{\rm c}/\Delta t=f_{\rm t}t_{\rm s}/\Delta t=f_{\rm t}/C$ ($\sim 30$ for the
fiducial value of $f_{\rm t}$ and $C=0.3$). In most implementations of SPH the
sound speed in the Courant criterion is replaced by a “signal velocity” (e.g.
Monaghan, 1997), $v_{\rm sig}\ga 2c_{\rm s}$, which would more than double the
number of time steps.
Inverting equation (17), we find the following maximum density for which the
feedback is expected to be effective,
$\displaystyle n_{{\rm H},t_{\rm c}=f_{\rm t}t_{\rm s}}$ $\displaystyle=$
$\displaystyle
31~{}\mbox{cm}^{-3}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{3/2}\left(\frac{f_{t}}{10}\right)^{-3/2}\times$
(18) $\displaystyle\left(\frac{\left<m\right>}{7\times
10^{4}~{}\mbox{M}_{\odot}}\right)^{-1/2}\left(\frac{N_{\rm
ngb}}{48}\right)^{-1/2}\times$
$\displaystyle\left(\frac{\mu}{0.6}\right)^{-9/4}\left(\frac{g(X_{\rm
H})}{0.14}\right)^{3/2}.$
The critical density is thus proportional to $T^{3/2}\left<m\right>^{-1/2}$.
For the case of AMR simulations, the spatial resolution is given by the linear
size of the grid cell, which is commonly decreased until it is some factor,
$f_{\rm J}$, smaller than the local Jeans length: $h\leq L_{\rm J}/f_{\rm J}$.
Expressing the Jeans length as a function of temperature and density, $L_{\rm
J}\equiv c_{\rm s}\sqrt{\pi/(G\rho)}$, and substituting this into equation
(15), we obtain
$\displaystyle\frac{t_{\rm c}}{t_{\rm s}}$ $\displaystyle\geq$ $\displaystyle
50\left(\frac{n_{\rm
H}}{1~{}\mbox{cm}^{-3}}\right)^{-1/2}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)\left(\frac{T_{0}}{10^{4}~{}\mbox{K}}\right)^{-1/2}\left(\frac{f_{\rm
J}}{4}\right)\times$ (19)
$\displaystyle\left(\frac{\mu}{0.6}\right)^{-1}\left(\frac{g\prime(X_{\rm
H})}{0.15}\right),$
where $T_{0}$ is the initial gas temperature (from which the Jeans mass, hence
the spatial resolution, is derived), $g\prime(X_{\rm H})=X_{\rm
H}^{-1/2}f(X_{\rm H})$, and we assumed that the Jeans length is resolved with
four resolution elements. The equality holds for $h=L_{\rm J}/f_{\rm J}$. Note
that the same relation applies to SPH simulations with particle mass at least
$f_{\rm J}^{3}$ times smaller than the Jeans mass in gas with density $n_{\rm
H}$ and temperature $T_{0}$.
Inverting equation (19), we obtain
$\displaystyle n_{{\rm H},t_{\rm c}=f_{\rm t}t_{\rm s}}$ $\displaystyle\geq$
$\displaystyle
25~{}\mbox{cm}^{-3}\left(\frac{T}{10^{7.5}~{}\mbox{K}}\right)^{2}\left(\frac{T_{0}}{10^{4}~{}\mbox{K}}\right)^{-1}\left(\frac{f_{\rm
J}}{4}\right)^{2}\times$ (20)
$\displaystyle\left(\frac{\mu}{0.6}\right)^{-2}\left(\frac{g\prime(X_{\rm
H})}{0.15}\right)^{2},$
which is the analogue of equation (18) for AMR (and for SPH simulations with
particle mass at least $f_{\rm J}^{3}$ times smaller than the Jeans mass in
gas with density $n_{\rm H}$ and temperature $T_{0}$).
We show in Fig. 1 a contour plot of the time-scale ratio $t_{\rm c}/t_{\rm s}$
in the temperature-density plane for the fiducial gas particle mass $m_{\rm
g}=5.1\times 10^{4}h^{-1}~{}\mbox{M}_{\odot}$. We computed $t_{\rm c}$ using
the tabulated values of the collisional cooling rates of (Wiersma, Schaye, &
Smith, 2009a), which are also used in the SPH simulations described below. We
combine in the same plot the two cases of primordial (solid contours) and
solar (dotted contours) chemical compositions. The vertical dashed line marks
the fiducial heating temperature of $10^{7.5}~{}\mbox{K}$, while the
horizontal dashed line indicates the density threshold for SF in our
simulations. For $T\geq 10^{7.5}~{}\mbox{K}$ the difference between primordial
and solar metallicity is very small, but at lower temperatures metals reduce
the ratio $t_{\rm c}/t_{\rm s}$. Once the temperature has dropped to $10^{7}$
K for the case of solar abundances (or to values smaller than $10^{6}$ K for
primordial abundances), collisional excitation processes cause the cooling
rate to increase as the temperature drops (e.g. Wiersma, Schaye, & Smith,
2009a). For such temperatures the equations above, which assumed the cooling
rate to be dominated by Brehmsstrahlung, will underestimate the radiative
losses and will therefore overestimate the minimum density for which the
feedback is expected to be efficient.
Interestingly, for purely adiabatic expansion, i.e. $\rho\propto T^{3/2}$, the
ratio $t_{\rm c}/t_{\rm s}$ remains constant for the case of SPH (eq. [17]).
In that case the gas will follow a track parallel to the high temperature part
of the contours in Fig. 1 and $t_{\rm c}\propto\rho^{-2/3}$. Hence, if
radiative losses were unimportant initially, so that the hot bubble will start
to expand adiabatically, then radiative losses will remain unimportant as long
as the cooling rate is dominated by Brehmsstrahlung. For AMR, on the other
hand, the ratio $t_{\rm c}/t_{\rm s}\propto T^{1/4}$ during the adiabatic
phase, which implies that radiative losses may already become important while
Brehmsstrahlung dominates.
## 5 Simulations
Table 1: Simulation parameters: total mass, $M_{\rm halo}$; fraction of SN energy injected, $f_{\rm th}$; temperature jump, $\log_{10}\Delta T$; total number of particles, $N_{\rm tot}$; total number of gas particles in the disc, $N_{\rm disc}$; mass of baryonic particles, $m_{\rm b}$; mass of dark matter particles, $m_{\rm DM}$; gravitational softening of baryonic particles, $\epsilon_{\rm b}$; gravitational softening of dark matter particles, $\epsilon_{\rm DM}$; thermal feedback included, (Feedback). Values different from the fiducial ones are shown in bold. Simulation | $M_{\rm halo}$ | $f_{\rm th}$ | $\log_{10}\Delta T$ | $N_{\rm tot}$ | $N_{\rm disc}$ | $m_{\rm b}$ | $m_{\rm DM}$ | $\epsilon_{b}$ | $\epsilon_{\rm DM}$ | Feedback
---|---|---|---|---|---|---|---|---|---|---
| $(h^{-1}~{}\mbox{M}_{\odot})$ | | $(\mbox{K})$ | | | $(h^{-1}~{}\mbox{M}_{\odot})$ | $(h^{-1}~{}\mbox{M}_{\odot})$ | $(h^{-1}~{}\mbox{pc})$ | $(h^{-1}~{}\mbox{pc})$ |
G10-NOFB | $10^{10}$ | — | — | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | N
G10-040-70 | $10^{10}$ | $\mathbf{0.4}$ | $\mathbf{7.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-65 | $10^{10}$ | $1.0$ | $\mathbf{6.5}$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-70 | $10^{10}$ | $1.0$ | $\mathbf{7.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-75 | $10^{10}$ | $1.0$ | $7.5$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-80 | $10^{10}$ | $1.0$ | $\mathbf{8.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-85 | $10^{10}$ | $1.0$ | $\mathbf{8.5}$ | 5 000 494 | 235 294 | $5.1\times 10^{2}$ | $2.4\times 10^{3}$ | 10 | 17 | Y
G10-100-75-LR08 | $10^{10}$ | $1.0$ | $7.5$ | 625 061 | 29 411 | $\mathbf{4.1\times 10^{3}}$ | $\mathbf{1.9\times 10^{4}}$ | 20 | 34 | Y
G10-100-75-LR64 | $10^{10}$ | $1.0$ | $7.5$ | 78 132 | 3 676 | $\mathbf{3.3\times 10^{4}}$ | $\mathbf{1.5\times 10^{5}}$ | 40 | 68 | Y
G12-NOFB | $10^{12}$ | — | — | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | N
G12-040-70 | $10^{12}$ | $\mathbf{0.4}$ | $\mathbf{7.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-65 | $10^{12}$ | $1.0$ | $\mathbf{6.5}$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-70 | $10^{12}$ | $1.0$ | $\mathbf{7.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-75 | $10^{12}$ | $1.0$ | $7.5$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-80 | $10^{12}$ | $1.0$ | $\mathbf{8.0}$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-85 | $10^{12}$ | $1.0$ | $\mathbf{8.5}$ | 5 000 494 | 235 294 | $5.1\times 10^{4}$ | $2.4\times 10^{5}$ | 46 | 79 | Y
G12-100-75-LR08 | $10^{12}$ | $1.0$ | $7.5$ | 625 061 | 29 411 | $\mathbf{4.1\times 10^{5}}$ | $\mathbf{1.9\times 10^{6}}$ | 92 | 158 | Y
G12-100-75-LR64 | $10^{12}$ | $1.0$ | $7.5$ | 78 132 | 3 676 | $\mathbf{3.3\times 10^{6}}$ | $\mathbf{1.5\times 10^{7}}$ | 184 | 316 | Y
We ran simulations of isolated disc galaxies embedded in dark matter haloes
with total masses of $10^{10}$ and $10^{12}h^{-1}~{}\mbox{M}_{\odot}$, where
$h=0.73$. The initial conditions are as in Schaye & Dalla Vecchia (2008) and
DS08, thus the models do not include gaseous haloes and all the gas is
initially in the discs. We adopted a larger gravitational softening length
than in the previous works for the massive galaxy. We also modified the
original code as described below.
### 5.1 Code and initial conditions
We use a modified version of the TreePM/SPH code gadget (Springel, 2005) for
all the simulations presented in this paper.
We employ the SF recipe of Schaye & Dalla Vecchia (2008), which we briefly
describe here. Gas denser than the critical density for the onset of the
thermo-gravitational instability ($n_{\rm H}\sim 10^{-2}-10^{-1}~{}{\rm
cm}^{-3}$) is expected to be multiphase and star-forming (Schaye, 2004). We
model such gas by imposing a minimum temperature floor given by an effective
equation of state with pressure $P\propto\rho_{g}^{\gamma_{\rm eff}}$ for
densities exceeding $n_{\rm H}=0.1~{}{\rm cm}^{-3}$, normalised to
$P/k=10^{3}~{}{\rm cm}^{-3}~{}\mbox{K}$ at the threshold. We use $\gamma_{\rm
eff}=4/3$ for which both the Jeans mass and the ratio of the Jeans length and
the SPH kernel are independent of the density, thus preventing spurious
fragmentation due to a lack of numerical resolution.
We introduce a different definition of star-forming particle. In the previous
works a gas particle was flagged as star-forming if it crossed the density
threshold $n_{\rm H}^{\ast}=0.1~{}{\rm cm}^{-3}$ while its temperature was
below $T=10^{5}~{}\mbox{K}$. The particle then remained star-forming until its
density fell below the threshold density or the particle was promoted to a
wind particle.
In the present work we proceed as follows. Each particle is free to cool to
lower temperatures, but not below the temperature $T_{\rm EoS}$ imposed by the
effective equation of state. A gas particle is star-forming if
$\left\\{\begin{array}[]{l}n_{\rm H}\geq n_{\rm H}^{\ast}\\\
\log_{10}T<\log_{10}T_{\rm EoS}+\Delta\log_{10}T_{\rm
EoS}\end{array}\right.\,,$ (21)
where $\Delta\log_{10}T_{\rm EoS}$ is a free parameter which we set to
$\Delta\log_{10}T_{\rm EoS}=0.5~{}\mbox{dex}$.
$\Delta T=10^{6.5}~{}\mbox{K}$ | $\Delta T=10^{7.5}~{}\mbox{K}$ | $\Delta T=10^{8.5}~{}\mbox{K}$
---|---|---
Figure 2: Projections of the gas density (top row: edge-on; middle row: face-
on) and temperature (bottom-row: edge-on) for models G10-100-65 (left column),
the fiducial model G10-100-75 (middle column) and G10-100-85 (right column) at
time $t=250~{}\mbox{Myr}$. The white arrows in the top row show the velocity
field. Images are $17.5h^{-1}~{}\mbox{kpc}$ on a side. The colour coding is
logarithmic in density ($-4.3<\log_{10}n_{\rm H}/\mbox{cm}^{-3}<-0.3$) and
temperature ($3.7<\log_{10}T/\mbox{K}<5.8$). The colour scale is indicated by
the colour bars in each column.
The Kennicutt-Schmidt SF law is analytically converted and implemented as a
pressure law. As we demonstrated in Schaye & Dalla Vecchia (2008), our method
allows us to reproduce arbitrary input SF laws for any equation of state
without tuning any parameters. We use the observed Kennicutt (1998) law
$\dot{\Sigma}_{\ast}=1.5\times
10^{-4}~{}\mbox{M}_{\sun}\,\mbox{yr}^{-1}\,\mbox{kpc}^{-2}\left({\Sigma_{g}\over
1~{}\mbox{M}_{\sun}\,\mbox{pc}^{-2}}\right)^{1.4},$ (22)
where the different normalisation accounts for the fact that we are using a
Chabrier IMF.
$\Delta T=10^{6.5}~{}\mbox{K}$ | $\Delta T=10^{7.5}~{}\mbox{K}$ | $\Delta T=10^{8.5}~{}\mbox{K}$
---|---|---
Figure 3: Projections of the gas density (top row: edge-on; middle row: face-
on) and temperature (bottom-row: edge-on) for models G12-100-65 (left column),
the fiducial model G12-100-75 (middle column) and G12-100-85 (right column) at
time $t=250~{}\mbox{Myr}$. The white arrows in the top row show the projected
velocity field. Images are $45h^{-1}~{}\mbox{kpc}$ on a side. The colour
coding is logarithmic in density ($-4.3<\log_{10}n_{\rm
H}/\mbox{cm}^{-3}<1.3$) and temperature ($4.3<\log_{10}T/\mbox{K}<6.3$). The
colour scale is indicated by the colour bars in each column.
Radiative cooling and heating were included using tables for hydrogen and
helium from Wiersma, Schaye, & Smith (2009a). The cooling tables were
generated using the publicly available package cloudy (version 07.02; Ferland,
2000), assuming ionisation equilibrium in the presence of the Haardt & Madau
(2001) model for the $z=0$ UV background radiation from quasars and galaxies,
and the cosmic microwave background.
We implemented a version of the time-stepping algorithm described by Durier &
Dalla Vecchia (2012). Inactive particles that receive feedback energy are
immediately activated so that they can respond promptly to their new energetic
state. Their and their active neighbours’ signal velocities (see e.g.
Monaghan, 1997) are also updated in order to calculate the size of the next
time-step consistently. The new time-step is then propagated to the inactive
neighbours following the scheme of Saitoh & Makino (2009). We refer the reader
to Durier & Dalla Vecchia (2012) for a detailed discussion of the benefits
introduced by the scheme.
However, we do not expect the integration accuracy of our simulations to
improve significantly compared with e.g. DS08. As argued by Durier & Dalla
Vecchia (2012), imposing a limit to the maximum allowed time-step (e.g., by
significantly reducing the gravitational softening, as we have done here and
in DS08), may also maintain good energy conservation in the case of strong
energy perturbations. Indeed, we found only small deviations in the global
properties of the outflows when running the same fiducial models without the
time-step limiter.
The initial conditions are based on the model of Springel, Di Matteo, &
Hernquist (2005) and are described in DS08. The model consists of a dark
matter halo, a stellar bulge, and an exponential disc of stars and gas. The
circular velocities at the virial radii are $35.1$ and $163~{}{\rm km}\,{\rm
s}^{-1}$ for the $10^{10}$ and $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ haloes,
respectively. The virial radii are $35.1$ and $163~{}h^{-1}~{}\mbox{kpc}$. The
halo is rotating and has a dimensionless spin parameter $\lambda=0.33$. The
disc contains 4 percent of both the total mass and the total angular momentum.
The bulge contains 1.4 percent of the total mass and has a scale length one
tenth of that of the disc. The bulge has no net rotation. The initial gas
fraction of the disc is 30 percent, the remaining 70 percent of the disc mass
is made up of stars. The vertical distribution of the stellar disc has a
constant scale height of 10 percent of the radial disc scale length.
Figure 4: Two-dimensional mass-weighted probability density distribution of
the gas within $0.2r_{\rm vir}$ of the dwarf galaxy in temperature-density
space at time $t=250$ Myr. The colour coding indicates $f_{\rm g}=({\rm
d}M/M)/{\rm d}\log_{10}n_{\rm H}/{\rm d}\log_{10}T$. The three panels
correspond to the same three models as were shown in Fig. 2. The twelve
contours of $f_{\rm g}$ (black, dotted) are equally spaced in the range showed
by the colour bar. The red, dashed curves indicate radiative cooling time
contours and their labels indicate $\log_{10}(t_{\rm c}/{\rm yr})$. Below the
solid, green line photo-heating by the UV background dominates over radiative
cooling. The horizontal, dashed line indicates the heating temperature $\Delta
T$. The vertical, dotted line marks the threshold for star formation. The
imposed, effective equation of state is visible for densities larger than the
threshold density.
Except for our low-resolution runs, the total number of particles in each
simulation is 5,000,494, of which 235,294 are gas particles in the disc. The
baryonic particle mass for the $10^{10}h^{-1}~{}\mbox{M}_{\odot}$
($10^{12}h^{-1}~{}\mbox{M}_{\odot}$) halo is $m_{\rm b}=5.1\times
10^{2}h^{-1}~{}\mbox{M}_{\odot}$ ($m_{\rm b}=5.1\times
10^{4}h^{-1}~{}\mbox{M}_{\odot}$). The gravitational softening length was set
to $\epsilon_{b}=10h^{-1}~{}\mbox{pc}$ for the baryons and to $(m_{\rm
DM}/m_{\rm b})^{1/3}\epsilon_{b}\approx 17h^{-1}~{}\mbox{pc}$ for the dark
matter in the $10^{10}h^{-1}~{}\mbox{M}_{\odot}$ halo. The softening for the
massive galaxy is scaled up in proportion to $m_{\rm DM}^{1/3}$.
### 5.2 Simulation parameters
Table 1 lists the simulations we have performed. Each simulation was evolved
for $500~{}\mbox{Myr}$. The simulations are labelled with the prefix G10 and
G12 for the $10^{10}$ and the $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ haloes,
respectively, followed by the percentage of the SN energy that is injected and
by the logarithm of the temperature increase. For example, G10-040-75 refers
to the $10^{10}h^{-1}~{}\mbox{M}_{\odot}$ halo, with the SN feedback injecting
$40\%$ of the total available energy and increasing the gas particle
temperature by $\Delta T=10^{7.5}~{}\mbox{K}$. We have run several variations
of the fiducial models, G10-100-75 and G12-100-75. The list of all the models
follows:
* •
One run without SN feedback (G[10,12]-NOFB).
* •
One run injecting $40$ percent of the available SN energy, G[10,12]-040-75.
This is used for comparison with the kinetic feedback simulations of DS08,
which also used 40 percent of the energy.
* •
One set of runs varying the temperature increase $\log_{10}\Delta
T=[6.5,7.0,7.5,8.0,8.5]$. These runs are labelled
G[10,12]-100-[65,70,75,80,85], respectively.
* •
Two runs in which the number of particles was decreased by factors of 8 and
64, respectively (G[10,12]-100-75-LR08, G[10,12]-100-75-LR64).
To eliminate differences other than the feedback recipe from the comparison
with DS08, we repeated the simulations of models m12 and m12nowind of DS08
with the new version of the code. We changed the softening to the one used in
this work, and used the time-step limiter for the feedback run.
Figure 5: As Fig. 4, but for the models of the massive galaxy shown in Fig. 3.
Figure 6: The Kennicutt-Schmidt SF relation for a selection of G10 (left
panel) and G12 (right panel) models at $t=250~{}\mbox{Myr}$. Surface densities
are computed in cylindrical annuli containing a constant number of particles
and including all particles with vertical coordinate $|z|<2~{}\mbox{kpc}$. The
tilted line shows the observed Kennicutt SF law, equation (22), whereas the
vertical line shows the gas surface density below which SF is observed to
become inefficient (for more details see Schaye & Dalla Vecchia, 2008). All
models are in excellent agreement with the observations.
## 6 Simulation results
In this section we describe tests of our implementation of thermal feedback.
We first show the effect of varying the temperature increase. We proceed with
a comparison to the kinetic feedback model of DS08. We conclude the section by
reporting the results of resolutions tests.
### 6.1 Dependence on the temperature increase
We ran a set of simulations with $f_{\rm th}=1$ and varying $\log_{10}\Delta
T=[6.5,7.0,7.5,8.0,8.5]$ to study the dependence on the temperature increase.
We first describe the morphology of the galaxies and their outflows, and
proceed with discussing the SF histories and the outflow properties
quantitatively.
#### 6.1.1 Morphology
For each halo, we compare here the fiducial model, which has $\log_{10}\Delta
T=7.5$, to our most extreme models, which use $\log_{10}\Delta T=[6.5,8.5]$.
Fig. 2 shows projections of the density (top and middle rows) and temperature
(bottom row) for the dwarf galaxy at time $t=250~{}\mbox{Myr}$ for three
different values of the temperature increase (from left to right,
$\log_{10}\Delta T=[6.5,7.5,8.5]$). Recall that all models inject the same
amount of energy per unit stellar mass.
The morphology of the galaxy is irregular in all cases. As the heating
temperature is increased, the low-density bubbles in the ISM and in the
circumgalactic medium increase in size and open up vertical channels through
which the outflows can escape. Consequently, the outflow becomes more
collimated along the vertical axis as $\Delta T$ increases, enhancing its
bipolarity. The velocity field overplotted in the top row shows that the flow
is faster for larger $\Delta T$, while the bottom row shows that the
outflowing gas is also hotter.
Fig. 3 shows edge-on projections of the density (top row) and temperature
(bottom row) for the massive galaxy for the same three different values of the
temperature increase. The dependence on $\Delta T$ is more evident for this
more massive galaxy. For $\Delta T=10^{6.5}~{}\mbox{K}$ (left column), most of
the outflowing gas is confined to a region around the disc. The disc looks
puffed up as the expelled gas is deposited just outside it, and the gas breaks
up in little blobs. The velocities are small and the velocity field does not
show a clear preferential orientation. There is a galactic fountain, but no
large-scale galactic wind.
In contrast, the fiducial model ($\Delta T=10^{7.5}~{}\mbox{K}$; middle
column) shows a clear bipolar outflow, which is sustained until the end of the
run. The outflow is mostly driven from the inner part of the disk where a
large fraction of the SF is taking place, and it is collimated by the disc
which impedes motion within its plane. The temperature map shows several cold
blobs above and below the galactic disc. The blobs come from the disc and are
moving outward, thus the outflow is ejecting parcels of cold gas. Cold blobs
are seen falling back onto the disc at large radii.
The highest $\Delta T$ run ($\Delta T=10^{8.5}~{}\mbox{K}$; right column)
looks similar to the fiducial model, but the wind is hotter and moving faster
and the circumgalactic medium contains fewer cold blobs.
Videos illustrating the time evolution of the model galaxies are available at
this web address: http://www.strw.leidenuniv.nl/DS12/
#### 6.1.2 Gas phase distribution
The mass-weighted probability distribution function (PDF) of the gas at
$t=250$ Myr in the $(n_{\rm H},T)$ plane is shown in Figs. 4 and 5 for the
models shown in Figs. 2 and 3, respectively. To limit the effects of the
vacuum boundary conditions (mainly adiabatic cooling to extremely low
temperatures and densities), we include only the gas inside a spherical volume
of radius $0.2r_{\rm vir}$, but we normalise the PDFs to the total gas mass.
The vertical, dotted line marks our threshold density for SF, and at higher
densities the imposed, effective equation of state is clearly visible.
Contours of constant radiative cooling time (dashed, red lines) are over-
plotted.444Note that the $t_{\rm c}$ contours have been calculated assuming a
constant mean molecular weight, $\mu=0.6$. The contour labels indicate integer
values of $\log_{10}(t_{\rm c}/{\rm yr})$. The equilibrium between radiative
cooling and photo-heating by the UV background is shown by the solid, green
curve. Below this curve radiative heating dominates over radiative cooling.
Note that gas can reach temperatures lower than the equilibrium value if it
cools adiabatically. In the absence of feedback, none of the gas below the SF
threshold would have temperatures significantly above the equilibrium value.
Finally, the horizontal, dashed (green) lines indicate the feedback heating
temperatures $\Delta T$. Gas in which feedback energy has been injected
initially resides near these lines and, as long as radiative cooling is
unimportant, will expands adiabatically, exiting the star-forming region at
different temperatures. Indeed, the phase diagrams show different
distributions of shock-heated gas for different $\Delta T$, with the low-
density, high-temperature regions being more populated for increasing $\Delta
T$.
For the dwarf galaxy (Fig. 4) the temperature-density distributions are
similar for all values of $\Delta T$, although there are some differences in
the high temperature regime. We will show later that the SF histories and the
outflows are also very similar for all values of $\Delta T$ and that, given
the resolution, this is in accord with the results of section 4. There is very
little hot gas. The surface density of the disc, and hence the pressure, is
too low to confine the heated gas, which therefore immediately blows out of
the disc and cools adiabatically.
For the massive galaxy changing $\Delta T$ has a much more dramatic impact on
the distribution of shock-heated gas ($T>10^{5}~{}\mbox{K}$). For the model
with the lowest value of $\Delta T$ (left panel of Fig. 5), the peak of the
PDF lies within the density range $10^{-3}$ to $10^{-2}~{}\mbox{cm}^{-3}$.
This confirms the qualitative result shown in Fig. 3: most of the outflowing
gas accumulates in a region around the disc. If we increase $\Delta T$ (middle
and right panels), the peak in the PDF moves to lower densities ($\sim
10^{-4}~{}\mbox{cm}^{-3}$) thanks to the development of a large-scale outflow
that moves gas away from the disc. The total fraction of shock-heated gas
decreases with the $\Delta T$ because less gas resides within $0.2r_{\rm vir}$
if the wind velocity is higher.
#### 6.1.3 Star formation history
Before discussing the SF histories, we show the predicted Kennicutt-Schmidt SF
relations in Fig. 6. The left (right) panel shows the same three different
models as were shown in Fig. 2 (Fig. 3) for the dwarf (massive) galaxy. Gas
mass and SF surface densities were computed in annuli containing a constant
number of gas particles and including all particles with vertical coordinate
$|z|<2~{}\mbox{kpc}$. The observed Kennicutt-Schmidt law (eq. (22); tilted
line) and the steepening at the SF threshold density (vertical line) are well
matched. This success is not unexpected, as we already showed in Schaye &
Dalla Vecchia (2008) that the observed SF law can be implemented directly in
the form of a pressure law and that this enables the simulations to reproduce
the observations without the need to tune any parameters and irrespective of
whether strong feedback is present. While feedback determines the surface
density of the gas, it does not affect the efficiency of star formation at a
fixed surface density in our models, since pressure and surface density are
closely related in self-gravitating systems.
Figure 7: SFR as a function of time for the galaxies in the $10^{10}$ and the
$10^{12}h^{-1}~{}\mbox{M}_{\odot}$ haloes (left- and right panels,
respectively). We vary the temperature increase due to feedback events in
steps of $0.5~{}\mbox{dex}$ over the range $\log_{10}\Delta
T/\mbox{K}=[6.5,8.5]$. The dwarf galaxy’s SF history is insensitive to the
value of $\Delta T$, while for the massive galaxy the feedback is less
efficient for the lowest heating temperature.
The dependence of the SF histories on $\Delta T$ is shown in Fig. 7. For the
dwarf galaxy (left panel), the star formation rate (SFR) drops sharply within
the first $100~{}\mbox{Myr}$ due to the strong feedback produced by the
initial burst of SF, and remains nearly constant thereafter. The fact that the
sharp drop is due to feedback can be seen by comparing to the nearly
horizontal black, dotted curve, which shows the SF history for a run without
feedback. The factor by which the feedback reduces the SFR is insensitive to
$\Delta T$. This is in accord with the calculations presented in Section 4.
The dwarf galaxy has a low surface density and forms most of its stars at
densities close to the SF threshold of $n_{\rm H}=10^{-1}~{}{\rm cm}^{-3}$
(see Fig. 4). For such densities and primordial abundances we expect cooling
losses to be small even for heating temperatures as low at $10^{6.5}$ K.
Indeed, for our particle mass of $7\times 10^{2}~{}$M⊙ and a heating
temperature of $T=10^{6.5}$ K, equation (18) tells us that cooling losses
should be unimportant for densities $n_{\rm H}<10~{}{\rm cm}^{-3}$.
However, as Fig. 1 shows, the situation could be different for solar
abundances. While the cooling time is insensitive to the metallicity for $T\ga
10^{7.5}$ K, for $T=10^{6.5}$ K it is about an order of magnitude smaller for
solar metallicity than it is for primordial abundances. A factor ten increase
in the cooling rate would reduce the maximum density for which the feedback is
expected to be effective to $n_{\rm H}\approx 0.3~{}{\rm cm}^{-3}$ (eqs. [17]
and [18]). We would therefore expect a significant fraction of the feedback
energy to be radiated away before it can be converted into kinetic form, if we
were to run the dwarf galaxy simulation with $\Delta T=10^{6.5}$ K and solar
metallicity. Indeed, we have performed such a run (not shown) and find the SFR
to be much higher. After 400 Myr it is about 0.04 M${}_{\odot}\,{\rm yr}^{-1}$
which is closer to the run without feedback than to the run with the same
$\Delta T$ but primordial abundances.
For the massive galaxy the decline in the SFR is more gradual (right panel).
After a few hundred Myr, all runs predict roughly the same SFR except for the
one adopting $\Delta T=10^{6.5}~{}$K, which has a substantially higher SFR.
The morphological comparison (Fig. 3) shows that in this run the gas is unable
to escape to large radii. Instead, it accumulates around the disc, and
eventually falls back onto it. This is expected, because equation (18) shows
that radiative losses will become important at densities that are 10 times
lower than for the dwarf galaxy, because the particle mass is 100 times
higher. Moreover, the densities in the ISM are higher in the massive galaxy,
with many star particles forming in gas with densities $n_{\rm H}\sim
1-10~{}{\rm cm}^{-3}$ (Fig. 5). Comparing these numbers to the maximum density
for which cooling losses are small at this resolution, $n_{\rm H}\lesssim
31~{}{\rm cm}^{-3}~{}(T/10^{7.5})^{3/2}$ (eq. [18]), we expect small changes
for $\Delta T=10^{7}~{}$K and a substantial reduction of the feedback
efficiency for $\Delta T=10^{6.5}~{}$K.
Models G12-100-70 and G12-100-75 have similar SF histories, suggesting
convergence of the results. However, models G12-100-80 and G12-100-85 have
SFRs that are larger than that of the fiducial model by factors of $\simeq 25$
and 40 percent, respectively. The trend for the largest $\Delta T$’s likely
arises from poor sampling of the distribution of SN energy in the disc.
Indeed, the expectation value for the number of heated neighbours decreases
with $\Delta T$, and is (from eq. [7]) 0.42 and 0.13 for G12-100-80 and
G12-100-85, respectively. This shows the importance of locally linking the
feedback events with the formation of star particles by depositing the star
particle SN energy into at least one of its neighbours. The effect is less
severe for the dwarf galaxy because star formation is restricted to smaller
scales, these scales are resolved with many more particles, and it is easier
to eject gas in this case due to the lower ISM pressure and the shallower
gravitational potential well.
Finally, we note that the SFRs are lower than for the kinetic feedback runs of
DS08. This difference is, however, due to the different amount of energy
injected rather than to the manner in which this is done (i.e. thermal vs.
kinetic). We inject more energy here ($f_{\rm th}=1$ whereas DS08 used $f_{\rm
th}=0.4$), thus a stronger quenching is expected. In Section 6.2 we will show
that the two methods are in fact in good agreement with each other.
Figure 8: Mass outflow rate (left column) and average outflow velocity (right
column) measured through a spherical shell at radius $r=0.2r_{\rm vir}$ as a
function of time (top row) and at $t=500~{}\mbox{Myr}$ as a function of radius
(bottom row) for models G10-100-[65,70,75,80,85]. The dotted line in the top-
left panel indicates the SFR of model G10-100-75. All other curves are
labelled in the legends.
#### 6.1.4 Mass outflow rate and wind velocity
In this section we measure the wind velocity and mass loading as a function of
time and radius. We first briefly describe the method, which is identical to
the one employed in DS08.
Figure 9: Mass outflow rate (left column) and average outflow velocity (right
column) measured through a spherical shell at radius $r=0.2r_{\rm vir}$ as a
function of time (top row) and at $t=500~{}\mbox{Myr}$ as a function of radius
(bottom row) for models G12-100-[65,70,75,80,85]. The dotted line in the top-
left panel indicates the SFR of model G12-100-75. All other curves are
labelled in the legends.
We discretise the following integral equation of the net mass outflow rate
through a surface $S$:
$\dot{M}=\int_{S}\rho\mathbf{v}\cdot{\rm d}\mathbf{S},$ (23)
where $\rho$ is the gas density and $\mathbf{v}$ is the gas velocity at any
position on the surface. Given a spherical shell of radius $r$ and thickness
$\Delta r$ centred on the origin, the above equation becomes
$\dot{M}(r,\Delta r)=\frac{1}{\Delta r}\sum_{i=1}^{N_{\rm
shell}}m_{i}\mathbf{v}_{i}\cdot\frac{\mathbf{r}_{i}}{r_{i}},$ (24)
where $N_{\rm shell}$ is the total number of particles within the shell, and
$m_{i}$ and $\mathbf{r}_{i}$ are their mass and position, respectively. We
consider all particles and use $\Delta r=r_{\rm vir}/150$.
The average outflow velocity is also given in discrete form, and is the mass-
weighted, average radial velocity:
$\left<v\right>(r,\Delta r)=\frac{\sum_{i=1}^{N_{\rm
shell}}m_{i}(\mathbf{v}_{i}\cdot\frac{\mathbf{r}_{i}}{r_{i}})_{+}}{\sum_{i=1}^{N_{\rm
shell}}m_{i}},$ (25)
where $\mathbf{v}_{i}$ is the particle velocity. We consider only particles
moving away from the origin, as indicated by the subscript “$+$” in the above
equation.
To investigate the dependence of the outflow properties on the temperature
increase $\Delta T$, we show in Figs. 8 (dwarf galaxy) and 9 (massive galaxy)
the mass outflow rate (left column) and the average outflow velocity (right
column). The top row shows the evolution of the wind at radius $r=0.2r_{\rm
vir}$ and the bottom row shows the dependence on radius at time
$t=500~{}\mbox{Myr}$.
We first consider the dwarf galaxy (Fig. 8). Apart from the peak velocity,
which is reached after only a few tens of Myr, all models look very similar.
Apparently, the properties of the wind are nearly independent of the
temperature increase $\Delta T$. As discussed in the previous section, this is
expected for such a high resolution and for a primordial composition. However,
for solar abundances the equations derived in Section 4 predict that radiative
losses do become important for $\Delta T=10^{6.5}$ K. Indeed, we find that for
this heating temperature the mass outflow rate is reduced by more than an
order of magnitude if we assume the metallicity to be solar (not shown).
The wind is highly mass-loaded, with mass outflow rates at $0.2r_{\rm vir}$
that are a factor of $10-100$ higher than the SFR (the SFR of the fiducial
model is shown as the black, dotted curve in the top-left panel). The mass
flux builds up quickly in the first 100 Myr and declines gradually thereafter.
The top-right panel shows that the wind velocity peaks at different values at
the beginning of the simulations, but converges to similar values after $\sim
50~{}\mbox{Myr}$. The peak outflow velocity depends strongly on the
temperature increase, with the smallest (largest) value of $\sim 400~{}{\rm
km}\,{\rm s}^{-1}$ ($\sim 600~{}{\rm km}\,{\rm s}^{-1}$) corresponding to the
smallest (largest) $\Delta T$. The peak velocity is a measure of the average
velocity of gas particles that are able to move freely to large radii. Because
our simulations start without a gaseous halo, it is not clear how meaningful
the early evolution is, given that the high wind velocities would have
resulted in strong shocks if a gaseous halo had been present. However, because
the winds fill the haloes with gas, the results quickly become insensitive to
the artificial initial conditions. After about 100 Myr the wind velocities of
all models converge at about $100~{}{\rm km}\,{\rm s}^{-1}$ after which they
decline gently to several tens of kilometres per seconds.
The convergence in the SFR, mass outflow rate and outflow velocity for
different values of $\Delta T$ indicates that similar amounts of star-forming
gas has been extracted from the disc over time. The plots in the bottom row of
Fig. 8 confirm that. At each radius the outflow rates and velocities are
similar for all models, and the gas can efficiently escape into the halo, and
eventually beyond the virial radius.
As was already noted by DS08, the fact that the wind velocity becomes
proportional to the radius as we move away from the galaxy (bottom-right
panel) is due to travel time effects. Because the wind has only been blowing
for a finite amount of time $t$, only gas with a velocity greater than
$v=98~{}{\rm km}\,{\rm s}^{-1}~{}\left(\frac{r}{10~{}{\rm
kpc}}\right)\left(\frac{t}{10^{8}~{}{\rm yr}}\right)^{-1},$ (26)
would be present at radius $r$ if the wind velocity is constant with radius.
Although the wind may in reality accelerate or decelerate, our results suggest
that travel time effects may well be important too. Hence, wind velocities
that are observed to increase with distance do not necessarily imply that the
wind is accelerating.
Fig. 9 shows that the picture is different for the massive galaxy. The initial
wind velocity is sensitive to the heating temperature and determines the time
at which the outflow first passes through the shell at $r=0.2r_{\rm vir}$
(corresponding to the sharp rise in the top panels). For $\Delta T\geq
10^{7.5}$ K the mass outflow rates converge at about five times the SFR (the
SFR of model G12-100-75 is indicated by the dotted curve in the top-left
panel), so the winds are much less mass-loaded than for the dwarf galaxy. For
lower heating temperatures the mass flux is substantially lower. The outflow
rate of model G12-100-65 even becomes negative after about 320 Myr, indicating
net infall (dotted curve in the top-left panel). The wind cannot escape the
inner region of the halo, and is confined within a fraction of the virial
radius. This model therefore also predicts a much higher SFR (Fig. 7). The
fact that the mass flux drops strongly for $\Delta T<10^{7.5}$ K can be
understood by noting that the gas in the central regions and spiral arms has
densities $n_{\rm H}\sim 1-10~{}{\rm cm}^{-3}$ (Fig. 3) and that equation (18)
shows that cooling losses should make the feedback inefficient for densities
$n_{\rm H}<31~{}{\rm cm}^{-3}~{}(T/10^{7.5})^{3/2}$.
The wind velocities are a factor of a few higher than for the low-mass galaxy
and, at least for $\Delta T\geq 10^{7.5}$ K, depend strongly on the
temperature increase. Figure 3 showed that for these high heating
temperatures, the wind blows channels through which it can freely escape to
very large distances. The effect of drag by halo gas is therefore smaller than
for the dwarf galaxy and for the low-$\Delta T$ versions of the massive
galaxy, which all predict much puffier gas disks. Hence, in the high-$\Delta
T$ simulations of the massive galaxy the outflow speed is predominantly
regulated by gravity (as opposed to gas drag) and by the initial velocity.
While the potential is always the same, the initial wind velocity is set by
the heating temperature.
The radial dependence of the outflow is shown in the bottom row of Fig. 9.
Interestingly, the models that predict similar SF histories, also predict
similar outflow rates for $r<0.2r_{\rm vir}$. However, at larger radii the
outflow rates diverge, with higher heating temperatures giving larger mass
fluxes. Except for model G12-100-65, the velocities increase linearly with
radius beyond about $0.5~{}r_{\rm vir}$, suggesting that they are determined
by travel time constraints, as was the case for the dwarf galaxy.
Figure 10: Comparison of the SF histories in simulations employing thermal
(G10-040-70 and G12-040-70) and kinetic (m10 and m12) feedback. All models
inject 40 percent of the SNII energy. The kinetic feedback assumes an initial
wind velocity of $600~{}{\rm km}\,{\rm s}^{-1}$ and the thermal feedback a
temperature increase of $10^{7}$ K, which is close to the post-shock
temperature for a shock velocity of $600~{}{\rm km}\,{\rm s}^{-1}$. The left
and right panels show the SFR as a function of time for the $10^{10}$ and
$10^{12}h^{-1}~{}\mbox{M}_{\odot}$ haloes, respectively. The very small
difference between the no-feedback models (that are only noticeable for the
massive galaxy) are due to the different treatment of star-forming gas (see
section 5.1). The two feedback implementations result in very similar SF
histories.
Figure 11: Comparison of the mass outflow rate (left column) and average
outflow velocity (right column) measured through a spherical shell at radius
$r=0.2r_{\rm vir}$ as a function of time (top row) and at $t=500~{}\mbox{Myr}$
as a function of radius (bottom row) between model G10-040-70, which employs
thermal feedback, and model m10, which uses kinetic feedback. The dotted curve
in the top-left panel indicates the SFR of model G10-040-70. All other curves
are labelled in the legends. The agreement between the outflows predicted by
simulations that inject the SNII energy in thermal and kinetic form is
generally excellent.
Figure 12: As Fig. 11, but for the $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ halo.
The agreement between the thermal and kinetic implementations of SNII feedback
is very good for the wind velocities, but the mass outflow rates differ
substantially at late times and intermediate radii. This probably reflects the
fact that these wind parameter values yield results that are intermediate
between the galactic fountain and efficient, large-scale winds regimes, making
the outcome very sensitive to the exact values of the wind parameters.
### 6.2 Comparison with the kinetic feedback model
In this section we will compare the results of simulations using our new
thermal feedback prescription with runs employing the kinetic feedback of
DS08. As discussed in detail in DS08, the kinetic feedback recipe works as
follows. Once a star particle reaches an age of $3\times 10^{7}$ yr, its
neighbouring gas particles each have a probability of $\eta
m_{\ast}/\Sigma_{i=1}^{N_{\rm ngb}}m_{i}$ of receiving a randomly oriented
kick of velocity $v_{\rm w}$. For the case of equal mass particles, the mass
loading factor $\eta$ equals the average number of particles kicked per star
particle. We use DS08’s fiducial values of $v_{\rm w}=600~{}{\rm km}\,{\rm
s}^{-1}$ and $\eta=2$, which correspond to 40 percent of the available energy.
We compare the kinetic feedback runs to thermal feedback simulations that use
the same fraction of the available SN energy (i.e. $f_{\rm th}=0.4$ as opposed
to 1.0 for our fiducial model). We use a temperature increase of $\Delta
T=10^{7}$ K (as opposed to $10^{7.5}$ K for our fiducial model) as this is
close to the post-shock temperature for a shock velocity of $600~{}{\rm
km}\,{\rm s}^{-1}$.
To eliminate potential differences other than the feedback recipe, we re-ran
models m12 and m12nowind of DS08 with the new code and employing the same
softening lengths as used here.
#### 6.2.1 Star formation history
Fig. 10 compares the SF histories of the thermal feedback runs G10-040-70 and
G12-040-70 with the equivalent kinetic feedback runs m10 and m12 of DS08.
We first verify that the new criterion for identifying star-forming gas (see
Section 5.1) gives results that are consistent with the implementation used in
DS08. In order to eliminate differences due to the feedback implementation, we
compare runs without feedback. Comparison of the black and orange dotted
curves in Fig. 10, which show the SF histories predicted with the new and old
prescriptions, respectively, shows that differences due the slight change in
the recipe for SF are negligible.
Comparing the orange and black solid curves, which show the SF histories in
the runs with kinetic and thermal feedback, respectively, we see that the two
methods for injecting the energy from SNII are generally in good agreement.
#### 6.2.2 Mass outflow rate and wind velocity
Fig. 11 shows the mass outflow rate (left column) and the mean outflow
velocity (right column) for models G10-040-70 and m10. The top row shows the
evolution measured at $r=0.2r_{\rm vir}$, while the bottom row illustrates the
dependence on radius at time $t=500~{}\mbox{Myr}$. The agreement is generally
excellent. This implies that, at the resolution used for the dwarf galaxy, the
average quantities of the outflow are insensitive to the form in which the
energy is injected. Note that this is not a consequence of a fortunate choice
of parameters, as we showed in Section 6.1.4 that the outflow is insensitive
to the temperature increase $\Delta T$. The agreement between simulations
injecting kinetic and thermal energy is consistent with the results of Durier
& Dalla Vecchia (2012).
Fig. 12 shows the same plots for the $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ halo.
While the agreement for the velocities is again very good, there are in this
case some noticeable differences between the mass outflow rates. Compared with
the thermal feedback simulation, the kinetic model predicts a higher outflow
rate at radii $0.2\lesssim r/r_{\rm vir}\lesssim 0.4$ at late times. The
differences may imply that the resolution is too low to achieve convergence
between kinetic and thermal feedback prescription (recall that the mass
resolution is two orders of magnitude lower than for the dwarf galaxy).
Another reason, is however, that, unlike for the dwarf galaxy, for the massive
galaxy the outflow does depend on the wind parameters. Moreover, for this
massive halo $\Delta T=10^{7}$ K (or $v_{\rm w}=600~{}{\rm km}\,{\rm s}^{-1}$)
marks the transition between the regimes of galactic fountains (lower $\Delta
T$ or $v_{\rm w}$) and efficient large-scale winds (higher $\Delta T$ or
$v_{\rm w}$), as can be seen from Figs. 7 and 9. Hence, the results are very
sensitive to small differences in the input parameters and we could have
obtained better agreement by fine-tuning the value of $\Delta T$ (or $v_{\rm
w}$).
### 6.3 Resolution tests
Figure 13: Numerical convergence of the SF histories of the
$10^{10}h^{-1}~{}\mbox{M}_{\odot}$ and $10^{12}h^{-1}~{}\mbox{M}_{\odot}$
haloes (left- and right panels, respectively). The mass resolution is
decreased by factors of 8 (dashed curves) and 64 (dash-dotted curves). The
convergence is good, although there is a small, systematic increase of the SFR
with decreasing resolution for the massive galaxy.
Figure 14: Resolution dependence of the mass outflow rate (left column) and
mean outflow velocity (right column) measured through a spherical shell at
radius $r=0.2r_{\rm vir}$ as a function of time for the $10^{10}$ (top row)
and the $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ (bottom row) haloes. Note that the
particle mass in G10-100-75-LR064 is still lower than that in G12-100-75.
While the predictions for the low-mass galaxy are converged, decreasing the
resolution yields mostly higher outflow velocities for the high-mass galaxy,
but has little effect on the mass outflow rates.
We tested the numerical convergence of our implementation of thermal feedback
by decreasing the particle numbers by factors of 8 and 64. We will denote the
corresponding runs by appending ‘LR008’ resp. ‘LR064’ to the simulation names.
Hence, the particle masses are increased by factors of 8 and 64, while the
gravitational softening lengths and, for a fixed density, the SPH smoothing
kernels are increased by factors of 2 and 4, respectively. Before showing the
results, it is useful to consider what we may expect. To do so, we have to
check whether the simulations resolve the Jeans scales and whether we expect
radiative cooling losses to be significant.
As discussed in Schaye & Dalla Vecchia (2008), for star-forming gas in our
fiducial simulation of the massive halo the ratio of the SPH kernel mass to
the Jeans mass is smaller than $1/6$ and the ratio of the SPH kernel size to
the Jeans length is at most $1/(48)^{1/3}\approx 0.28$. For the low-mass halo
the mass and length ratios are lower by factors of 100 and $100^{1/3}\approx
4.64$, respectively. Note that the maximum possible values of these ratios are
independent of the density because star-forming particles cannot have
temperatures below a power-law effective equation of state with polytropic
index $\gamma_{\rm eff}=4/3$. Thus, while even our lowest-resolution
simulation of the dwarf galaxy resolves the Jeans scales, the same is only
true for our highest-resolution simulation of the massive galaxy.
In Section 4 we demonstrated that, for $\Delta T=10^{7.5}$ K and our fiducial
resolution for the massive galaxy, radiative losses should have little impact
on the efficiency of the feedback if the energy is injected in gas with
density $n_{\rm H}<31~{}{\rm cm}^{-3}$ and that this critical density is
inversely proportional to the squareroot of the particle mass (eq. [18]).
Hence, for the intermediate- and low-resolution models the densities above
which radiative losses may prevent efficient feedback are about 11 and
$4~{}{\rm cm}^{-3}$, respectively, and for the low-mass galaxy these densities
are 10 times higher. Comparing this to the actual densities of the star-
forming gas in the high-resolution simulations (Figs. 4 and 5), we see that
radiative losses may not be negligible for the lower-resolution simulations of
the high-mass galaxy, but that the feedback should remain efficient in the
low-resolution models of the dwarf galaxy.
Thus, both Jeans and radiative cooling arguments suggest that even the lowest-
resolution simulation of the dwarf galaxy should give converged results. On
the other hand, we do expect the lower-resolution versions of the massive
galaxy to show some difference. Furthermore, given that the lowest-resolution
models have fewer than 3700 gas particles in the disk, we expect the results
to become noisy due to the poor sampling.
Figs. 13 shows how the SF history depends on the numerical resolution for the
fiducial models G10-100-75 (left) and G12-100-75 (right). Similarly, Fig. 14
illustrates the resolution dependence of the predicted evolution of the mass
outflow rate (left column) and wind velocity (right column) for the dwarf (top
row) and massive (bottom row) galaxies. Clearly, our expectations are borne
out.
Whereas the dwarf galaxy runs are very well converged, for the massive galaxy
the predicted SFR increases with resolution, although the effect is small. The
first 250 Myr the mass outflow rate is somewhat higher in the lower resolution
simulations of the massive galaxy, but the situation reverses at later times.
Except for the first 10 Myr, the differences are, however, small. The wind
velocity is more sensitive to the resolution and is generally higher in the
lower-resolution models. The predictions for the outflow properties become
noisy for the lowest resolution simulations.
It is interesting that the convergence with numerical resolution is better
than found by DS08 for the kinetic feedback versions of the same simulations
(c.f. Figs. 10 and 11 of DS08). For example, with kinetic feedback the outflow
rate for the massive galaxy increased by a factor of about 3 as the mass
resolution was decreased by a factor of 8 and the SFR of the dwarf galaxy
increased substantially when the particle mass was decreased by a factor of
64. It should be kept in mind, however, that this could in part be due to the
fact that the fiducial wind velocity used by DS08 ($600~{}{\rm km}\,{\rm
s}^{-1}$) corresponds to post-shock temperature jumps that are a factor of a
few lower than our fiducial value of $\Delta T=10^{7.5}$ K.
## 7 Discussion
Models of galaxy formation and evolution require feedback from star formation
to reproduce the observed properties of galaxies. Cosmological simulations do
not have sufficient resolution to resolve individual SN explosions and must
therefore resort to sub-grid recipes. The simplest method, injecting the SN
energy released by a star particle (i.e. a simple stellar population; SSP)
during each time step into its surroundings, does not lead to efficient
feedback because the injected thermal energy is quickly radiated away.
Successful recipes generally either inject the energy in kinetic form, turn
off radiative cooling temporarily, or inject the energy in a hot sub-grid
phase that is decoupled from the colder phases by hand.
We demonstrated that the catastrophic radiative losses suffered by simple
thermal feedback recipes are due to a mismatch between the gas mass in which
the energy is injected and the mass of the SSP that produced the energy. In
the real Universe, one SNII is produced for every $\sim
10^{2}~{}\mbox{M}_{\odot}$ of stars, and the energy released in the explosion
is initially carried by $\ll 10^{2}~{}\mbox{M}_{\odot}$ of ejecta. Because the
ratio between the mass of the ejecta and the mass of the stellar population
that released the energy is small ($\ll 1$), the SN energy per unit mass of
ejecta is high. Hence, the ejecta move at very high velocity ($\gg
10^{3}~{}{\rm km}\,{\rm s}^{-1}$) and the post-shock temperatures are
sufficiently large for the radiative cooling time to be long. Indeed,
observations do provide evidence for hot gas associated with fast galactic
outflows (e.g. Heckman, Armus, & Miley 1987; Strickland et al. 2000; see also
the review by Veilleux, Cecil, & Bland-Hawthorn 2005).
In simulations, however, the ratio between the gas mass receiving the SN
energy and the stellar mass that produced it is much larger. Even if all the
SNII energy of an SSP is injected at once, the ratio between the mass of the
‘ejecta’ and that of the SSP will typically be large. For example, if, for the
case of SPH, the energy is shared by all $N_{\rm SPH}$ neighbours, then the
ratio will be $N_{\rm SPH}\gg 10$ and even larger if multiple star particles
are spawned per gas particle or if the feedback energy is released over
multiple time steps. Consequently, the temperature of the heated particles
will be relatively low and most of the thermal energy will be radiated away
before it is able to do $PdV$ work on its surroundings. It is important to
note that the ratio between the gas mass receiving the energy and the mass of
the SSP that produced it is independent of the resolution.
The realisation that the cause of the inefficient thermal feedback is a
mismatch between the simulated and observed ratios of heated mass to stellar
mass, rather than a straightforward lack of numerical resolution, immediately
indicates the solution: we need to decrease this ratio in the simulations. By
decreasing the mass ratio, we increase the temperature of the heated gas and
hence its cooling time. If the cooling time is long compared with the sound
crossing time scale across a resolution element, the heated gas will begin to
expand adiabatically and the injected thermal energy will be efficiently
converted into kinetic energy. We showed that in the temperature regime for
which Brehmsstrahlung dominates the radiative cooling rate ($\ga 10^{7}$ K for
solar abundances), the ratio of the cooling time and the sound crossing time
remains constant if the gas expands adiabatically, so that adiabatic cooling
does not invalidate the argument.
We showed analytically (see eq. [18]) that, for the case of SPH, the maximum
density for which radiative losses are small is $n_{\rm H}\sim 26~{}{\rm
cm}^{-3}(T/10^{7.5}~{}{\rm K})^{3/2}(m/10^{5}~{}\mbox{M}_{\odot})^{-1/2}$,
where $T$ is the temperature of the heated gas after receiving the feedback
energy and $m$ is the mass of a gas resolution element (and we assumed that
$\Delta T$ is sufficiently high for Brehmsstrahlung to dominate the radiative
cooling). The maximum density is nearly the same for AMR simulations with cell
sizes that are at least a factor of 4 smaller than the local Jeans length
(evaluated in gas with this density and a temperature of $10^{4}$ K, see eq.
[20]). Hence, by specifying the desired temperature jump $\Delta T$, we can
guarantee that the feedback is efficient up to some gas density that we can
estimate analytically.
We implemented this idea in the SPH code gadget in the form of stochastic
thermal feedback. A fixed $\Delta T$ then translates into a fixed probability
of receiving energy, which we evaluate for each SPH neighbour of a star
particle that has just crossed the critical age $t_{\rm SN}=3\times
10^{7}~{}{\rm yr}$, corresponding to the maximum lifetime of stars that end
their lives as SNII. For a Chabrier IMF and assuming equal mass particles, the
expectation value of the number of heated particles is $1.34f_{\rm th}(\Delta
T/10^{7.5}~{}{\rm K})^{-1}$, where $f_{\rm th}$ is the fraction of the SNII
energy that is injected (see eq. [8]).
Note that the parameter $\Delta T$ plays a similar role as the initial wind
velocity $v_{\rm w}$ for the case of kinetic feedback. For a fixed $v_{\rm
w}$, the mass loading factor used in kinetic feedback implementations sets
$f_{\rm th}$. The fraction of the injected SN energy can also be used as a
second free parameter for the case of thermal feedback. For a fixed
temperature jump, the “initial mass loading” is then the ratio of the heated
mass per unit stellar mass formed and this ratio is proportional to the
parameter $f_{\rm th}$.
The combination of stochastic feedback and a fixed increase in the energy per
unit mass of the gas receiving the feedback energy also works for kinetic
feedback: we can specify $v_{\rm w}$ and give each neighbouring resolution
element of a star particle a probability of being kicked in the wind that is
proportional to the mass loading factor or, equivalently, to $f_{\rm th}$.
This is in fact exactly the implementation of kinetic feedback that we used in
DS08.
The equivalence of the roles of $\Delta T$ and $v_{\rm w}$ suggests that the
efficiency of the kinetic feedback does not depend directly on the ratio of
the initial wind velocity and the escape velocity, contrary to what is often
assumed. Indeed, in DS08 we showed that if $v_{\rm w}$ is too low, the wind
stalls in the ISM, before it has even begun to climb out of the gravitational
potential. As we have shown, the efficiency of the wind depends on the
radiative losses and hence, for a fixed value of $v_{\rm w}$, on the numerical
resolution and the gas density. Because the typical gas densities increase
with the gas pressure and thus with the depth of the potential well, the
escape velocity does matter indirectly (also because drag forces increase with
the pressure). If the wind manages to blow out of the ISM, then the ultimate
efficiency of the feedback does depend on the ratio of the velocity of the
wind leaving the ISM and the escape velocity, because the ejected gas will
rain back onto the galaxy if it cannot escape the galaxy’s potential well.
We presented analytic derivations of the resolution criteria, both for SPH and
AMR simulations. We tested our recipe for thermal feedback on SPH simulations
of isolated disc galaxies in dark matter haloes of total mass
$10h^{-1}~{}\mbox{M}_{\odot}$ and $10^{12}h^{-1}~{}\mbox{M}_{\odot}$ using the
same set-up as we used to study kinetic SN feedback in DS08. We explored the
effect of the feedback on the gas distribution, the star formation history,
the mass outflow rate, and the wind velocity. The results were in accord with
our analytic predictions.
For sufficiently high $\Delta T$ and for sufficiently high resolution, the
thermal feedback strongly reduces the star formation rate and results in a
strong, large-scale, bi-polar outflow. Reassuringly, the results converge with
both $\Delta T$ and resolution and the converged results also agree well with
simulations employing kinetic feedback555The thermal feedback models only
agree with kinetic feedback simulations if the kicked wind particles are _not_
temporarily decoupled from the hydrodynamics, see DS08 for a critical
discussion of this common practice. (with sufficiently high $v_{\rm w}$).
However, if $\Delta T$ and/or the resolution are too low, then the results
become sensitive to both. For a fixed resolution, higher values of $\Delta T$
will then result in more efficient feedback. Hence, in this regime the ability
to choose $\Delta T$ or, for the case of kinetic feedback $v_{\rm w}$, implies
a considerable freedom. This freedom associated with the implementation of
feedback from star formation is currently the limiting factor for the
predictive power of cosmological simulations (e.g. Schaye et al., 2010;
Scannapieco et al., 2011).
Given that a higher $\Delta T$ yields smaller radiative losses, one may ask
why we do not use ultra-high values. Indeed, even in low-resolution
simulations the feedback could be made efficient locally by increasing $\Delta
T$ (or $v_{\rm w}$ for the case of kinetic feedback). There are, however,
several reasons why it is undesirable to increase this parameter to values
$>10^{8}$ K. First, even for $f_{\rm th}=1$ such high heating temperatures
imply that, on average, each star particle will heat less than one
neighbouring gas particle. Such a situation breaks the locality of the
feedback and may lead to sampling problems. That this can have grave
consequences is easy to see by considering the limiting case in which the
number of heated gas particles per star particle (i.e. the mean initial mass
loading) is $\ll 1$. Most heavy elements released by massive stars will then
no longer be injected in a wind. Many generations of star particles can form
in a given gas cloud before a single feedback event takes place. Conversely,
if, despite the low probability, a star particle forming in a region with a
low star formation density does heat a neighbour, the energy injected may be
sufficiently large to do catastrophic damage. Clearly, we should avoid the
regime in which the expectation value for the number of heated gas elements
per star particle formed is much less than one, at least for galaxies resolved
with relatively small numbers of particles.
At present, large-volume cosmological simulations typically have particle
masses $\ga 10^{6}~{}\mbox{M}_{\odot}$. At this resolution even heating one
gas particle per star particle (which corresponds to $\Delta T\sim 10^{7.5}$ K
for a Chabrier IMF and $f_{\rm th}=1$) results in strong radiative losses for
densities $n_{\rm H}>10~{}{\rm cm}^{-3}$, which are routinely reached in such
simulations. Hence, the predictions are still sensitive to the values of the
feedback prescription and are thus uncertain. This undesirable limitation can
be turned into an advantage if one takes an approach similar in spirit to
semi-analytic models: by varying $\Delta T$ (or $v_{\rm w}$) with halo mass or
with the local physical conditions, the feedback can be tuned to reproduce the
desired galaxy formation efficiency. However, the arguments given above
demonstrate that the results may change with increasing resolution666Turning
off the hydrodynamical forces in the high-density regime could make the
results insensitive to resolution for kinetic feedback (Springel & Hernquist,
2003), but the predictions will in that case disagree with converged, self-
consistent high-resolution simulations.. If the resolution and the value of
$\Delta T$ are sufficiently high for cooling losses to be small, then the
feedback can still be tuned by varying $f_{\rm th}$ with the local conditions.
We have demonstrated that, contrary to common wisdom, thermal feedback can be
efficient without turning off radiative cooling. For sensible parameter
choices overcooling can already be avoided for densities typical of the warm
ISM (i.e. $n_{\rm H}\sim 1~{}{\rm cm}^{-3}$) at the resolution achievable for
large-scale cosmological simulations ($m\sim 10^{7}~{}\mbox{M}_{\odot}$) and
for simulations of individual (low- to intermediate-mass) galaxies we can
already afford the resolution ($m\sim 10^{2}~{}\mbox{M}_{\odot}$) required for
the feedback to remain efficient up to densities typical of molecular clouds
($n_{\rm H}\sim 10^{3}~{}{\rm cm}^{-3}$). We have also shown that with
sufficient resolution, the results become insensitive to the problematic
parameter of the feedback implementation (i.e. $\Delta T$ for thermal feedback
and $v_{\rm w}$ for kinetic feedback) and the form in which the energy is
injected, thus removing some of the most important uncertainties in the
ingredients of hydrodynamical simulations of galaxy formation.
## Acknowledgements
We are very grateful to Volker Springel for allowing us to use gadget and his
initial conditions code for the simulations presented here. We thank Rob Crain
and Jarrett Johnson for a careful reading of the manuscript, and the anonymous
referee for a helpful report. The simulations presented here were run on the
Cosmology Machine at the Institute for Computational Cosmology in Durham as
part of the Virgo Consortium research programme and on the TMoX cluster at the
Rechenzentrum Garching of the Max Planck Society. This work was supported by
Marie Curie Reintegration Grant FP7-RG-256573 and by the Marie Curie Initial
Training Network CosmoComp (PITN-GA-2009-238356).
## References
* Booth & Schaye (2009) Booth C. M., Schaye J., 2009, MNRAS, 398, 53
* Brook et al. (2004) Brook C. B., Kawata D., Gibson B. K., Flynn C., 2004, MNRAS, 349, 52
* Chiosi, Bertelli, & Bressan (1992) Chiosi C., Bertelli G., Bressan A., 1992, ARA&A, 30, 235
* Creasey et al. (2011) Creasey P., Theuns T., Bower R. G., Lacey C. G., 2011, MNRAS, 415, 3706
* Dalla Vecchia & Schaye (2008) Dalla Vecchia C., Schaye J., 2008, MNRAS, 387, 1431 (DS08)
* Dubois & Teyssier (2008) Dubois Y., Teyssier R., 2008, A&A, 477, 79
* Durier & Dalla Vecchia (2012) Durier F., Dalla Vecchia C., 2012, MNRAS, 419, 465
* Ferland (2000) Ferland, G. J. 2000, Revista Mexicana de Astronomia y Astrofisica Conference Series, 9, 153
* Gerritsen (1997) Gerritsen J. P. E., 1997, PhDT
* Haardt & Madau (2001) Haardt, F., & Madau, P. 2001, in the proceedings of XXXVI Rencontres de Moriond, astro-ph/0106018
* Heckman, Armus, & Miley (1987) Heckman T. M., Armus L., Miley G. K., 1987, AJ, 93, 276
* Hopkins, Quataert, & Murray (2012) Hopkins P. F., Quataert E., Murray N., 2012, MNRAS, 421, 3522
* Kawata (2001) Kawata D., 2001, ApJ, 558, 598
* Katz, Weinberg, & Hernquist (1996) Katz N., Weinberg D. H., Hernquist L., 1996, ApJS, 105, 19
* Kay et al. (2002) Kay S. T., Pearce F. R., Frenk C. S., Jenkins A., 2002, MNRAS, 330, 113
* Kay, Thomas, & Theuns (2003) Kay S. T., Thomas P. A., Theuns T., 2003, MNRAS, 343, 608
* Kennicutt (1998) Kennicutt R. C., Jr., 1998, ApJ, 498, 541
* Marri & White (2003) Marri S., White S. D. M., 2003, MNRAS, 345, 561
* Mihos & Hernquist (1994) Mihos J. C., Hernquist L., 1994, ApJ, 437, 611
* Monaghan (1997) Monaghan J. J., 1997, JCoPh, 136, 298
* Mori et al. (1997) Mori M., Yoshii Y., Tsujimoto T., Nomoto K., 1997, ApJ, 478, L21
* Murante et al. (2010) Murante G., Monaco P., Giovalli M., Borgani S., Diaferio A., 2010, MNRAS, 405, 1491
* Navarro & White (1993) Navarro J. F., White S. D. M., 1993, MNRAS, 265, 271
* Oppenheimer & Davé (2006) Oppenheimer B. D., Davé R., 2006, MNRAS, 373, 1265
* Osterbrock (1989) Osterbrock D. E., 1989, agna.book,
* Piontek & Steinmetz (2011) Piontek F., Steinmetz M., 2011, MNRAS, 410, 2625
* Saitoh & Makino (2009) Saitoh T. R., Makino J., 2009, ApJ, 697, L99
* Scannapieco et al. (2006) Scannapieco C., Tissera P. B., White S. D. M., Springel V., 2006, MNRAS, 371, 1125
* Scannapieco et al. (2011) Scannapieco C., Wadepuhl M., Parry O. H., et al., 2011, arXiv:1112.0315
* Schaye (2004) Schaye J., 2004, ApJ, 609, 667
* Schaye & Dalla Vecchia (2008) Schaye J., Dalla Vecchia C., 2008, MNRAS, 383, 1210
* Schaye et al. (2010) Schaye J., Dalla Vecchia C., Booth C. M., et al., 2010, MNRAS, 402, 1536
* Sommer-Larsen, Götz, & Portinari (2003) Sommer-Larsen J., Götz M., Portinari L., 2003, ApJ, 596, 47
* Springel & Hernquist (2003) Springel V., Hernquist L., 2003, MNRAS, 339, 289 (SH03)
* Springel (2005) Springel V., 2005, MNRAS, 364, 1105
* Springel, Di Matteo, & Hernquist (2005) Springel V., Di Matteo T., Hernquist L., 2005, MNRAS, 361, 776
* Stinson et al. (2006) Stinson G., Seth A., Katz N., Wadsley J., Governato F., Quinn T., 2006, MNRAS, 373, 1074
* Strickland et al. (2000) Strickland D. K., Heckman T. M., Weaver K. A., Dahlem M., 2000, AJ, 120, 2965
* Thacker & Couchman (2000) Thacker R. J., Couchman H. M. P., 2000, ApJ, 545, 728
* Theuns et al. (1998) Theuns T., Leonard A., Efstathiou G., Pearce F. R., Thomas P. A., 1998, MNRAS, 301, 478
* Veilleux, Cecil, & Bland-Hawthorn (2005) Veilleux S., Cecil G., Bland-Hawthorn J., 2005, ARA&A, 43, 769
* Wiersma, Schaye, & Smith (2009a) Wiersma R. P. C., Schaye J., Smith B. D., 2009a, MNRAS, 393, 99
* Wiersma et al. (2009b) Wiersma R. P. C., Schaye J., Theuns T., Dalla Vecchia C., Tornatore L., 2009b, MNRAS, 399, 574
|
arxiv-papers
| 2012-03-26T13:54:36 |
2024-09-04T02:49:29.074232
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Claudio Dalla Vecchia and Joop Schaye",
"submitter": "Claudio Dalla Vecchia",
"url": "https://arxiv.org/abs/1203.5667"
}
|
1203.5740
|
# Inter-pixel crosstalk in Teledyne Imaging Sensors (TIS) H4RG-10 detectors
Rachel P. Dudik ,1 Margaret E. Jordan,7 Bryan N. Dorland,1 Daniel Veillette,1
Augustyn Waczynski,2 Benjamin F. Lane,3 Markus Loose,5 Emily Kan,2 James
Waterman,4 Chris Rollins,8 and Steve Pravdo6
###### Abstract
CMOS-hybrid arrays have become competitive optical detectors for use in
ground- and space-based astronomy. Inter-pixel capacitance is one source of
error that appears in most CMOS arrays. In this paper we use a single pixel
reset method to model inter-pixel capacitance (IPC). We combine this IPC model
with a model for charge diffusion to estimate the total crosstalk on H4RG-10
arrays. Finally, we compare our model results to 55Fe data obtained using an
astrometric camera built to test the H4RG-10 B0 generation detectors.
1United States Naval Observatory,
3450 Massachusetts Avenue, NW, Washington, D.C. 20392, USA 2Goddard Space
Flight Center, NASA
8800 Greenbelt Road, Greenbelt, MD 20771, USA 3Charles Stark Draper
Laboratory, Inc.,
555 Technology Sq., Cambridge, MA 02139, USA 4Optical Sciences Division, Naval
Research Laboratory,
4555 Overlook Ave., SW , Washington, D.C. 20375, USA 5Markury Scientific,
Inc.,
518 Oakhampton Street, Thousand Oaks, CA 91361, USA 6Jet Propulsion
Laboratory, NASA,
4800 Oak Grove Drive, Pasadena, CA 91109, USA 7Computational Physics, Inc.,
8001 Braddock Rd., Springfield, VA 22151, USA 8Research Support Instruments,
Inc.,
4325-B Forbes Boulevard. Lanham, MD 20706, USA
040.0040, 040.1240, 040.3060, 040.5160, 040.6040, 040.7480.
## 1 Introduction
Complementary Metal-Oxide Semiconductor (CMOS) sensors have become a
competitive astronomical ground- and space-based detector solution. CMOS
sensors have a flexible readout structure that allows a single pixel or group
of pixels on the array to be read out or reset at any time without disturbing
or reading out the rest of the array. This random-access, non-destructive read
capability is ideal for dynamic range-driven astronomical applications, since
it permits bright and faint objects to be observed simultaneously using a
single detector. In addition to the flexible readout, CMOS sensors are
naturally less sensitive to radiation than more traditional detectors like
Charge Coupled Devices (CCDs), since damage to one pixel in the array does not
adversely affect subsequent pixels in a row or column of the array. This
inherent radiation hardness is particularly appealing for space-based
applications.
While the readout capabilities of CMOS sensors are ideal for a variety of
observing strategies, the fill factor of each pixel is significantly reduced
because the readout circuitry is implanted directly on the photodetector
material. Janesick et al. [1] describe how the CMOS-hybrid focal planes have
been developed to address this issue. A CMOS-hybrid sensor is a CMOS device or
Readout Integrated Circuit (ROIC), mated with a layer of photodetector
material. The two layers are typically joined together using indium bump
bonds. The resultant hybrid SCA is back-illuminated, and combines the flexible
readout of the CMOS with a CCD-like fill factor of 100%. The United States
Naval Observatory (USNO) has used CMOS-hybrid detectors to take advantage of
this performance and flexibility, as part of its development of very large
format focal plane technologies, in support of the Joint Milli-Arcsecond
Pathfinder Survey (JMAPS) astrometry mission.
USNO has been testing large format, Teledyne Imaging Sensors (TIS) H4RG-10
Hybrid Visible Silicon Imager (HyViSI) Sensor Chip Assemblies (SCAs) since the
development of the first generation-A1 detector in 2006, as described by
Dorland et al. in 2007 [2]. In 2008 USNO supported development of a second
generation-A2 H4RG-10 with significantly lower dark current than the A1
predecessor, again described by Dorland et al. in 2009 [3] A third generation
detector, the H4RG-10 B0 detector, was fabricated in 2010 with JMAPS support
to address yield and pixel operability requirements. This third generation of
detector has very low dark current and noise properties comparable to many
CCDs. The quantum efficiency is in excess of 80% across multiple wavelengths
and the non-linearity is $<1\%$. Additionally, with JMAPS risk reduction
support, Hubbs et al. in 2011 [4], describes how TIS was able to increase
pixel operability to better than 99.9%. Here, operability specifically refers
to the percent of pixels that are fully connected without shorting.
While these flexible and low-noise CMOS hybrid arrays are excellent for most
astronomical applications, the H4RG-10 show higher levels of crosstalk, or
more specifically: the Inter-Pixel Capacitance (IPC) component of crosstalk
than CCDs. We also note that some CMOS pixel circuit designs do not exhibit
high levels of IPC. However, the chosen design of the H4RG-10 has advantages
that are difficult to achieve with these other designs, namely low read noise,
low dark current and low power consumption. Indeed, most low background
applications prefer the source-follower approach despite the IPC problem (eg.
JMAPS H4RG-10 B0), while higher background applications typically use one of
the other design options.
As discussed in detail below, crosstalk can be problematic for applications
like astrometry, because it has the effect of blurring the point spread
function (PSF) of the photon source, resulting in lower effective signal to
noise (S/N) in each pixel and high centroiding errors (See also [5, 6, 7]).
For this reason JMAPS has supported development of a fourth generation
detector, the H4RG-10 B1, designed in part to reduce IPC to values that are
negligible for astrometry.
In this paper we discuss the data analysis and modeling that have been used to
understand crosstalk for astronomical detectors. In Section 2 we describe the
models for charge diffusion and IPC that were combined to create a 55Fe model
based on the H4RG-10 B0 detector (third generation). In Section 3 we describe
the JPL camera used to collect the data. In Section 4 we discuss the analysis
methodology for single-pixel-reset data and compare the results with the
modeled data. In Section 5 we discuss the analysis methodology used for 55Fe
data obtained with the astrometric camera, and again compare the analysis and
modeling results. Finally, in Section 6 we summarize our findings. (In a
subsequent paper we show how crosstalk affects astrometry and photometry using
detailed detector and optics models for a realistic astrometric telescope (See
[5] for details.))
## 2 Crosstalk models
### 2.1 Introduction to charge diffusion and IPC
Crosstalk between pixels is caused by two independent phenomena, charge
diffusion, and IPC. Charge diffusion is the lateral movement (i.e. pixel-to-
pixel) of charge between the points of charge production and charge collection
in the bulk substrate of the detector. Charge diffusion occurs in all
photosensitive material, including the silicon substrate of all CCDs. IPC is
the capacitance that arises between adjacent detector pixels in the source-
follower CMOS design, and leads to coupling of signal between those pixels via
displacement currents flowing from the collection node.
For optical detectors with photosensitive silicon material, crosstalk can be
measured by exposing the detector to a radioactive source with a known charge
production rate in bulk silicon, and then measuring the voltage in pixels
surrounding a central hit. 55Fe is a soft X-ray source, quickly decaying
(half-life: 2.7 years) to Mn when a K shell electron is absorbed into the
nucleus. An electron from either the L or M shells drops to fill the hole
created in the K shell. This drop to a lower energy level causes the emission
of either a Kα (5.9 KeV) or Kβ (6.5 KeV) X-ray. The absorption of a Kα photon
in the bulk silicon of the detector produces, on average, 1620 e-, while Kβ
photon absorption produces an average of 1778 e-.
In 55Fe testing, the detector, under controlled environmental conditions, is
briefly exposed to the radioactive source and the resulting charge is
collected and read out for each exposure. When analyzed, photon hits
approximately normal to the detector surface and centered on a single pixel
are averaged and normalized to create a kernel representative of pixel
crosstalk. 55Fe testing and analysis is described in more detail in Section 5
of this document.
In addition to 55Fe testing, single-pixel-reset testing (SPR) allows for the
direct characterization of IPC alone, requiring no illumination source [7]. In
SPR, after setting all pixels in the SCA to a single voltage and making an
initial readout, a well-spaced grid of single pixels is reset to a second
voltage level. A subsequent readout of the pixels will reveal any IPC as
signal in pixels adjacent to the reset pixels. As with 55Fe testing, multiple
SPR test results are averaged (discarding any bad pixels, edge-affected
pixels, etc.) to create a representative detector IPC kernel. SPR testing and
data analysis methodology is described in more detail in Section 4.
The IPC kernel obtained through SPR testing, when combined with charge
diffusion modeling, can be used for 55Fe test verification and prediction.
This resultant 55Fe model is based on the convolution of a charge diffusion
kernel, resulting from an incident Kα photon, with an IPC kernel built on SPR-
measured IPC. This new, modeled kernel can be used to approximate the average
pixel crosstalk expected in 55Fe testing for a given detector. The subsections
below describe the modeling of IPC and charge diffusion.
### 2.2 IPC model
A simple model of the IPC expected within a detector array can be developed
based on the approach described by Moore, et al. in 2004 [8]. The detector
array of photodiodes is modeled as an array of capacitors, as shown in Figure
1 , each identical and with a capacitance that is unchanging with voltage.
xtalk00F1.ps
Figure 1: Inter-pixel capacitance (Cip). xtalk00F1.ps
The detector is constructed in a way that allows an electrical field to exist
between neighboring collection nodes, essentially creating small coupling
capacitors between the nodes. Charge entering a single nodal capacitor
$Q_{total}$ causes a voltage change in that node, and through the coupling
capacitors, causes voltage changes in neighboring nodes. $Q_{total}$ is the
sum of all the apparent charge seen both in the voltage of the original node
and the voltages of $n$ neighboring nodes.
$\sum_{n}V_{n}=\frac{Q_{total}}{C_{node}}$ (1)
In Moore s approach, the impulse response of each node $h(n)$ is a ratio of
the charge that appears electrically in a node $Q_{n}$, to the photocurrent
that entered the original node $Q_{total}$,
$h(n)=\frac{Q_{n}}{Q_{total}}.$ (2)
Using this approach, and assuming two-dimensional symmetry in nearest
neighboring pixels and diagonal neighboring pixels, a simple model of IPC can
be constructed. Charge appearing electrically in surrounding pixels is defined
in terms of a single variable $\alpha$, defined as the percent of total charge
seen in any of the four nearest neighbor pixels. Symmetry in nearest neighbors
and diagonal neighbors is assumed in this model. An example is shown in Figure
2 for a 3x3 pixel array.
xtalk00F2.eps
Figure 2: IPC model for 3x3 pixel array. xtalk00F2.eps
The sum of the charge in all pixels in the array = 1.0. The above model
kernel, with $\alpha$ = 0.075, is an IPC model for a general CMOS detector.
The model can be extended to larger kernel models to account for detectors
with IPC that is more broadly spread.
### 2.3 Charge diffusion
Lateral charge diffusion occurs while charge moves between the point of
generation and the point of collection, in the detector substrate. The process
begins with the absorption of a Kα photon, which, for 55Fe, produces a cloud
of 1620 charge pairs within the substrate, as illustrated in Figure 3.
xtalk00F3.ps
Figure 3: Charge diffusion decreases with increasing absorption depth along
$\bf{z}$. xtalk00F3.ps
The photon attenuation length $l_{a}$ for a Kα (5.9 KeV) photon is:
$l_{a}=\frac{\lambda}{4\pi\rm{Im}\left(n\right)},$ (3)
using the absorptive term Im($n$) of the complex refractive index of silicon,
$n$
$n=1-\delta-i\beta,$ (4)
where $\delta$ is the refractive index decrement and $\beta$ is the absorptive
index. For a photon with $E$=5.9 KeV, $\beta\sim 6.0\times 10^{-7}$ [12] and
$l_{a}\sim 28\mu$m. In the 55Fe model, charge diffusion is calculated for thin
slices of the bulk Si substrate. The probability of photon absorption is
calculated for each slice:
$P_{\Delta
z}=\frac{1}{l_{a}}\left(e^{\frac{-d_{1}}{l_{a}}}-e^{\frac{-d_{2}}{l_{a}}}\right)$
(5)
where $d_{1}$ and $d_{2}$ are positions along the $\bf{z}$ axis (a direction
normal to the pixel array plane ($xy$ plane), as defined for illustration in
Figure 3 above), and $d_{1}$ is further from the detector. (In this model, the
incident photon is always normal to the detector surface. This orientation was
selected to mimic the symmetry selection criteria for good hits used in actual
55Fe test data analysis, as described below in Section 4.4.)
We now consider charge diffusion for a detector with a fully depleted
substrate. Charge diffusion, as a function of depth and weighted for
absorption probability, is well represented by the two dimensional Gaussian
$f_{\rm{CD}}=\frac{P_{\Delta
z}}{\sqrt{2\pi\sigma\left(z\right)}}e^{\frac{-x^{2}-y^{2}}{2\sigma\left(z\right)^{2}}}$
(6)
with $\sigma\left(z\right)$ the Root Mean Square (RMS) standard deviation of
charge spreading in $x$ and $y$. This spreading is described in terms of the
diffusion constant $D_{\rm{P}}$ and the transit time $t_{\rm{tr}}$ for charges
(holes, for the detectors under consideration) to move from the point of
charge pair generation to the point of collection
$\sigma\left(z\right)=\sqrt{2D_{\rm{P}}t_{\rm{tr}}}.$ (7)
The transit time $t_{\rm{tr}}$ can be found using the hole drift velocity
$v_{\rm{drift}}$ as described by [9, 10]:
$v_{\rm{drift}}=\frac{dz}{dt}=\mu
E\left(z\right)=\mu\left(E_{\rm{max}}+\frac{qN_{\rm{d}}}{\epsilon_{\rm{Si}}}z\right),$
(8)
and integrating over the entire depletion depth,
$t_{\rm{tr}}=\frac{\epsilon_{\rm{Si}}}{qN_{\rm{d}}}\ln{\left(\frac{E_{\rm{max}}}{E\left(z\right)}\right)}$
(9)
where $\epsilon_{\rm{Si}}$ is the dielectric constant for silicon ($11.9\times
8.854\times 10^{-12}$ C/V), $q$ is the fundamental charge unit (C), and
$N_{\rm{d}}$ is the doping density for the bulk silicon substrate.
$E_{\rm{max}}$ includes the effects of both the substrate bias voltage $V$,
and the depletion voltage $V_{\rm{D}}$ [11],
$V_{\rm{D}}=\frac{qN_{\rm{d}}}{2\epsilon_{\rm{Si}}}z^{2}_{\rm{D}}$ (10)
$E_{\rm{max}}=-\left(\frac{V}{z_{\rm{D}}}+\frac{qN_{\rm{d}}}{2\epsilon_{\rm{Si}}}z_{\rm{D}}\right).$
(11)
When V is much larger than $V_{\rm{D}}$ [10, 11],
$\sigma\left(z\right)\approx\sqrt{\frac{2z^{2}_{\rm{D}}k_{\rm{B}}T}{qV}}$ (12)
where the Einstein relation, $D_{\rm{p}}/\mu=k_{\rm{B}}T/q$ has been applied
to equation 7, above, and where $T$ is the temperature of the detector (K) and
$k_{\rm{B}}$ is Boltzmann s constant (J/K). For absorption of the photon at a
specific $z$ within the substrate:
$\sigma\left(z\right)=\left[\frac{2k_{\rm{B}}T\epsilon_{\rm{Si}}}{q^{2}N_{\rm{d}}}\ln\left(\frac{E_{\rm{max}}}{\frac{VqN_{\rm{d}}}{z_{\rm{D}}2\epsilon_{\rm{Si}}}\left(2z-z_{\rm{D}}\right)}\right)\right]^{1/2}.$
(13)
In the model, the charge diffusion function $f_{\rm{CD}}$ is calculated for
each incremental slice through $z$, using the above equation for
$\sigma\left(z\right)$, with: $T=193$ K, $V=40$ V, and $z_{\rm{D}}=100\mu$m.
The charge cloud produced by absorption of a Kα photon is resolved at a
higher-than-pixel resolution, and diffusion to each neighboring pixel is
determined by the position of the cloud center projected somewhere on the
surface of the central pixel. Because of this, the charge diffusion is
calculated at a subpixel resolution, with an absorption probability weighting
for each slice. The high-resolution kernel is then summed and rebinned to a
lower resolution kernel, ready for convolution with the IPC kernel. Examples
of the charge diffusion model to both subpixel and pixel resolutions are shown
in Figure 4.
xtalk00F4.eps
Figure 4: Charge diffusion example images, with
$Q_{\rm{pixel}}/Q_{\rm{total}}$ for each pixel. xtalk00F4.eps
## 3 JMAPS test camera
The data described here were taken at the Jet Propulsion Laboratory (JPL)
using a test camera specifically designed for ground based astrometric testing
of H4RG-10 detectors for JMAPS. The camera functions with 32 outputs
controlled by a TIS non-cryogenic SIDECAR (system image, digitizing,
enhancing, controlling, and retrieving) ASIC (Application Specific Integrated
Circuit). The measurements were taken at 193K and the detector substrate
voltage was set to 10, 20, 30 or 40V depending on the measurement. The camera
voltage settings were the same for IPC SPR and 55Fe data acquisition. Table 1
lists the primary voltage parameters used for these measurements. An image of
this camera is shown in Figure 5.
Table 1: JMAPS Camera Voltage Parameters Parameter | Symbol | Voltage (V)
---|---|---
Bias Voltage | VSUB | 10-40
Digital Positive Power Supply | VDD | 3.12
Analog Positive Power Supply | VDDA | 3.12
Drain Node of Pixel SF | CELLDRAIN | 0.00
Drain Node of Output SF | DRAIN | 0.00
Source Node of Internal Current for SF | VBIASPOWER | 3.00
Detector Substrate Voltage | DSUB | 0.50
Detector Reset Voltage | VRESET | 0.32
Bias Voltage for Current Source of SF | VBIASGATE | 2.00
xtalk00F5.ps
Figure 5: JMAPS camera and H4RH-10 B0 in the dewar. xtalk00F5.ps
## 4 Single pixel reset (SPR) data
### 4.1 SPR data acquisition and analysis
The JMAPS astrometric test camera is equipped with microcode that can perform
the single pixel reset (SPR) technique described by Seshadri [12] and Finger
[13]. The single pixel reset function enables reset of single pixels in the
array to a different reset voltage than the rest of the pixels in the array.
The effective result is charge on the integration node for these reset pixels.
This method is useful for isolating IPC, since the charge is not generated in
the photodetector material, and is therefore not susceptible to the effects of
charge diffusion.
IPC maps are created using the following data acquisition method. First the
entire detector is reset to a given reset voltage and the pixels are read out
to generate an offset frame. Then, a grid of widely spaced single pixels is
reset to a new voltage. Following this reset, all pixels are again read out.
If IPC is present, the difference of these two images will show voltage in
pixels neighboring the reset single pixels. A cropped single pixel reset image
of a section of an H4RG-10 B0 detector taken with the JMAPS camera is shown in
Figure 6.
xtalk00F6.eps
Figure 6: Image of pixels that have been reset using the SPR method for
estimating IPC. xtalk00F6.eps
For this analysis the initial reset voltage was first set to 300mV for all
pixels. The SPR grid of pixels was then reset to 0, 100, 200, 300, 400, 500,
600mV. This resulted in 7 measurements of varying signal levels per substrate
voltage (10V, 20V, 30V, 40V). To analyze the data, the following procedure for
finding good hits was adopted:
1. 1.
First the image was offset (bias) corrected.
2. 2.
A small range of central pixel values were identified as hits . The range of
values for the central pixel was identified as a Gaussian distribution in the
histogram of the image, and was well above the noise floor of the image.
Defining this range is important to ensure that hot or bad pixels are excluded
from the IPC estimate. On average, each image contains approximately 64,000
good hits that are used for the analysis.
3. 3.
All pixels defined as hits based on the criteria above were background
corrected using a local background correction method. An annulus of 2 pixels
outside the central hit was extracted. The median of the pixels in this
annulus was subtracted from the values of the inner 9x9 pixels. An image
illustrating the annulus and kernel is shown in Figure 7.
4. 4.
The background-corrected kernels were then stacked and averaged.
5. 5.
The resultant measured IPC kernel was normalized, so that the charge in a
single pixel was represented as a percent of the total charge in the kernel.
For this detector, the noise floor was reached in both IPC and 55Fe crosstalk
data at the 9x9 pixel boundary.
xtalk00F7.eps
Figure 7: Illustration of background area used for correction. The shaded
region of 2-pixel width is the annulus used for correction. xtalk00F7.eps
### 4.2 IPC data and model
#### 4.2.1 Data
Figure 8 shows the fraction of signal in the central pixel as a function of
the SPR voltage. In this plot, the saturation limit is indicated by the drop-
off in fractional signal at 400 mV. However, in order to be certain that no
saturated pixels were included in the final averaged kernel, and to ensure
high S/N, the 200 mV SPR dataset was chosen for this analysis (as opposed to
0, 100 and 300 mV SPR data).
xtalk00F8.ps
Figure 8: Fractional voltage in the central pixel as a function of the reset
voltage level for SPR data. xtalk00F8.ps
Figure 9 below shows the 9x9 pixel kernel resulting from the data taken at 40V
and 200mV SPR. Despite the fact that the outer 9x9 annulus in the kernel below
appears to be statistically indistinguishable for the errors, we note that all
values in this annulus are systematically above zero indicating that the noise
floor has not been reached. For this reason, the 9x9 annulus is included in
the kernel. The values listed represent the percentage of the total signal for
an average kernel, based on approximately 64,000 events. Measurement errors
represent the standard deviation across all SPR events.
xtalk00F9.ps
Figure 9: IPC values for H4RG-10 B0 based on single pixel reset method.
xtalk00F9.ps
#### 4.2.2 Model
As can be seen in the IPC data kernel in Figure 9, IPC can spread
significantly beyond the 3x3 grid depicted in Section 2.2. In fact, for each
detector tested using SPR, the IPC distribution extended to at least a 9x9
grid. Additionally, each detector type (where type is determined by changing
design and processing) was found to have a unique IPC distribution signature.
Based on these findings, the IPC model is an averaged set of SPR data for each
detector type, and given in terms of $\alpha$ (the average fractional signal
in nearest neighboring pixels, per Section 2.2). The use of $\alpha$ is in
keeping with the standard modeling approach for IPC and provides a variable to
capture the standard deviation of the measured and summed SPR kernels in the
55Fe model (discussed in Section 4.5). Variation of $\alpha$ is also needed to
model the effects of varying levels of optical crosstalk on centroiding
precision.
In the example above (Figure 9), $\alpha$ = (4.51 + 4.61 + 4.61 + 4.51)/4.0 =
4.56. Using this $\alpha$, an IPC model (Figure 10) was developed to capture
the signal-spreading signature of the B0 generation. Each cell is a factor of
$\alpha$:
[width=16 cm] xtalk00F10.ps
Figure 10: IPC model based on SPR data. xtalk00F10.ps
where the central pixel, $C=(1/\alpha)\sum$ (of the grid of coefficients).
### 4.3 IPC asymmetry findings
Given the ROIC readout asymmetry in these detectors, we specifically set out
to determine if a corresponding asymmetry exists in the IPC. The JMAPS test
camera at JPL is designed to read out the full 4096x4096 pixel detector using
32 equally spaced outputs. This means that the detector is virtually divided
into 32, 128x4096 pixel columns . The columns are read out alternately from
the right and left, resulting in all odd columns being read out in one
direction and even columns in the other direction. Figure 11 shows this
left/right, odd/even dichotomy.
xtalk00F12.ps
Figure 11: Readout ”columns” determined by alternating readout direction.
xtalk00F12.ps
We conducted a column-by-column comparison of the IPC charge distribution for
B0 detectors. To perform this analysis the hits from all even and all odd
columns were separately averaged. The first and last columns of the array were
excluded from this analysis, since they suffer high dark current at the edges
that could contaminate the results. A clear horizontal dichotomy resulting
from the direction of the readout is visible in the odd vs even columns, as
shown in Figure 12.
We performed this analysis for all of the SPR data taken for 0-300mV Vreset
values and for all 4 substrate voltages. Table 2 summarizes our findings. In
the readout direction (horizontal direction) the offset induced by the readout
is approximately 5% depending on whether the column is odd or even. In
addition, a vertical bias is evident in every column (both even and odd) of
approximately 3.5%. The fifth and eighth columns of Table 2 show the residuals
resulting when hits from both odd and even columns are averaged. We find the
residuals to be low in the horizontal/read direction after averaging odd and
even columns, but a 3.5% residual is still present in the vertical direction.
These findings will be used in the selection criteria of 55Fe hits, as
described in Section 4.4.
xtalk00F13.eps
Figure 12: SPR-generated IPC kernels from even (left) and odd (right) columns.
xtalk00F13.eps
Table 2: Results of the odd/even column systematic offset test Vsub111substrate voltage (V) | Vreset222SPR reset voltage (V) | xe333$\rm{x_{right}/x_{left}}$ for all even columns | xo444$\rm{x_{right}/x_{left}}$ for all odd columns | xerror555% residual error from hits in both columns | ye666$\rm{y_{upper}/y_{lower}}$ for even columns | yo777$\rm{y_{upper}/y_{lower}}$ for odd columns | yerror888% residual error from hits in both columns
---|---|---|---|---|---|---|---
10.00 | 0.30 | 1.08 | 0.93 | -0.23% | 0.94 | 0.94 | 6.01%
10.00 | 0.20 | 1.07 | 0.94 | -0.29% | 0.97 | 0.97 | 3.27%
10.00 | 0.10 | 1.06 | 0.94 | -0.28% | 0.97 | 0.97 | 3.34%
10.00 | 0.00 | 1.05 | 0.96 | -0.39% | 0.96 | 0.96 | 3.52%
20.00 | 0.30 | 1.08 | 0.93 | -0.29% | 0.97 | 0.97 | 2.87%
20.00 | 0.20 | 1.07 | 0.94 | -0.28% | 0.97 | 0.97 | 3.16%
20.00 | 0.10 | 1.06 | 0.94 | -0.27% | 0.97 | 0.97 | 3.32%
20.00 | 0.00 | 1.05 | 0.96 | -0.42% | 0.97 | 0.97 | 3.39%
30.00 | 0.30 | 1.08 | 0.93 | -0.33% | 0.97 | 0.97 | 2.96%
30.00 | 0.20 | 1.07 | 0.94 | -0.26% | 0.97 | 0.97 | 3.27%
30.00 | 0.10 | 1.06 | 0.94 | -0.26% | 0.97 | 0.97 | 3.38%
30.00 | 0.00 | 1.05 | 0.95 | -0.34% | 0.97 | 0.97 | 3.43%
40.00 | 0.30 | 1.08 | 0.92 | -0.28% | 0.97 | 0.97 | 3.07%
40.00 | 0.20 | 1.07 | 0.93 | -0.21% | 0.97 | 0.97 | 3.32%
40.00 | 0.10 | 1.06 | 0.94 | -0.24% | 0.96 | 0.97 | 3.49%
40.00 | 0.00 | 1.05 | 0.95 | -0.30% | 0.96 | 0.96 | 3.53%
Averages | | | | -0.29% | | | 3.46%
### 4.4 55Fe data acquisition and analysis
55Fe measurements were taken at the same voltage settings as the SPR
measurements, using the JMAPS camera. For these measurements the source was
exposed for up to 12 seconds. During the first two seconds, a mechanical
shutter shielded the detector from the source. This was done to measure the
offset from the zero point, due to reset, without contamination from the 55Fe
hits. In the remaining 10 seconds the shield was removed and the detector was
non-destructively read out every 2 seconds, resulting in 5 frames of a
progressively increasing population of 55Fe hits.
The charge diffusion component of crosstalk is heavily dependent on the depth
at which the Kα or Kβ photon is absorbed. Two things must be taken into
consideration when selecting good hits. First, the primary goal is to select
hits that are as perpendicular to the detector surface plane as possible.
Secondly, the hits must be selected so that all absorption depths are sampled.
We define an accurate measurement of 55Fe as the average over all absorption
depths. To ensure this, we have used the following method to define normal
55Fe hits:
1. 1.
First the image was offset corrected. To avoid any noise associated with
settling, the frames used for this analysis were the fourth and fifth frames.
The fourth frame was taken as the offset frame and subtracted from the fifth
frame.
2. 2.
A small range of central pixel values were identified as hits. The range
chosen is in ADU and strongly depends on the gain of the system. Therefore,
for this initial rough estimate, the range was taken directly from a subsample
of hits on the image without regard to the symmetry of the hit. The sum in a
9x9 region around the central pixel represents the majority of the analog
digital units (ADUs) resulting from the hit. A Gaussian distribution centered
on the Kα peak results when these sums are plotted in a histogram. We defined
the median of a Gaussian fit to this distribution divided by the total
electrons expected for a Kα hit (1620e-) as the gain of the system. In our
case the gain was 1.19 +/- 0.04 ADU per electron. On average $\sim 23,000$
hits proceed to Step 3 as potential Kα hits.
3. 3.
Double hits or hits falling inside the parameter of the targeted hit region,
can skew the crosstalk results. To remove double hits, all of the potential Kα
hits from Step 2 were centered and stacked in the $\bf{z}$ direction, a
process depicted in Figure 13. Kernels with double hits were removed by
fitting a Gaussian distribution along the z-axis of the stack for each pixel
except the central 3x3 in the kernel stack. Hits in the stack with pixel
values greater than the mean +/- 3$\sigma$ over the stack were considered
double hits and the hit was removed from the analysis. The remaining hits in
the stack are considered single hits and were used in the following steps of
the analysis.
4. 4.
A local dark correction was then performed on the remaining single hits. The
pixels in an annulus of 2 pixels outside of the 9x9 central hit are defined as
background pixels (see Figure 7). The median of the pixels in this annulus was
subtracted from the values of the inner 9x9 pixels.
5. 5.
The symmetry criteria are perhaps the most challenging criteria to incorporate
into the 55Fe reduction method. Noise and detector effects prevent even a
perfectly symmetric hit from appearing symmetric in the data. This leads to
confusion of off-center hits with perfectly centered hits contaminated by
noise. To choose perfectly centered hits we use the following criteria:
1. (a)
The noise of the system can be measured by fitting a Gaussian distribution to
all pixels in the image except those affected by 55Fe hits. The sigma of this
distribution represents the 1-sigma noise of the measurement. Two times this
sigma is the maximum difference in counts that the left/right and upper/lower
pixels can have, due to the noise. For all of our measurements, this noise was
$\sim 15$ ADU (1$\sigma$).
2. (b)
As discussed in Section 4.3, the B0 detectors show both horizontal and
vertical systematic readout asymmetry. The asymmetry in the horizontal
direction is column-dependent while the asymmetry in the vertical direction is
not. We used the normalized kernels from our IPC analysis to estimate the
offset in ADU expected to result from the asymmetry induced by the readout. In
most cases this asymmetry in effect contributed $\sim 5-10$ADU to the total
noise.
3. (c)
With the noise and asymmetry tabulated we defined centered hits as those hits
having left/right and upper/lower differences that are less than 2-sigma plus
the asymmetry offset. In most cases this resulted in a total permissible
variation between left/right and upper/lower pixels of $\sim 35-40$ ADU.
6. 6.
Finally, in order to ensure only Kα hits are included in the final crosstalk
kernel (for the purposes of comparing the data to the model), the sum over all
9x9 pixels of each symmetric hit was taken. Using the gain from step 2, any
hit with a sum from the 9x9 that had values less than or equal to the Kβ
escape peak or greater than or equal to the Kβ peak [10] were excluded from
the stack. We note that the peaks of the Kβ lines are used, rather than the
edges of these peaks (based on Janesick [10]), in choosing our range. We do
this because the noise is high enough for our system that the Kβ peaks cannot
be distinguished from the Kα peak when plotted. To avoid preferentially
excluding Kα hits that overlap with the Kβ hits due to noise, we established
this definition. We recommend using only Kα hits for those systems with noise
that is low enough to distinguish the three peaks.
7. 7.
After applying these criteria, the remaining hits were averaged to generate a
9x9 kernel (in the case of the 40V data) in units of ADU. The crosstalk kernel
was defined as the charge in a single pixel divided by the total charge in the
9x9 kernel.
8. 8.
This test was repeated 9 times, resulting in 9 kernels for all 9 datasets. The
standard deviation across all nine measurements for each pixel represents the
error on the final kernel due to statistical differences between each data
set.
[width=15 cm] xtalk00F14.ps
Figure 13: Double hits are removed by fitting a Gaussian to stacked pixels.
Pixels with crosses were excluded from iteration. xtalk00F14.ps
This method describes the reduction strategy for the 40V 55Fe data. With lower
substrate voltage settings we find that a kernel larger than 9x9 is needed to
capture all of the escaped charge from the central pixel. We discuss this in
more detail in Section 4.6.
### 4.5 55Fe 40V data and model
#### 4.5.1 Data
Figure 14 shows the 55Fe kernel resulting from the acquisition and analysis
strategy described in Section 4.4 for the H4RG-10 B0 detector. Kernel pixel
values are the percent measured signal for an average of 9 tests. The total
number of good hits used for this kernel was 1094. The errors are also shown
for each pixel value and represent the standard deviation of the pixel values
over all 9 datasets. We define pixel crosstalk as the value of the average
four nearest neighbor pixels (shown in grey) divided by the central value
(shown in yellow). For the H4RG-10 B0 detectors we obtain a measured crosstalk
value of 9.83% for this 55Fe data set.
xtalk00F15.ps
Figure 14: Kernel and errors, with pixel values resulting from 55Fe tests of
H4RG-10 B0. xtalk00F15.ps
#### 4.5.2 Model
To generate the 55Fe model, the average IPC kernel ($f_{\rm{IPC},ij}$) , as
described above in Section 4.2, is used to represent the IPC spreading typical
of the detector. The uncertainty associated with the standard deviation
($\sigma_{\rm{cp}}$) of the averaged single-pixel-reset (SPR) results is
included in the 55Fe model through convolution of the charge diffusion kernel
with a set of IPC kernels derived from new alphas ($\alpha_{\rm{new}}$). These
$\alpha_{\rm{new}}$ values are calculated from a set of central pixel values
($C_{\rm{new}}$) selected from a Gaussian distribution built around the
averaged central pixel value ($C_{\rm{ave}}$) and the standard deviation of
that averaged value, $\sigma_{\rm{cp}}$:
$C_{\rm{new}}=\sigma_{\rm{cp}}\left[-2\ln{\left(1-rn_{1}\right)}\right]^{1/2}\cos{\left(2\pi
rn_{2}\right)}+C_{\rm{ave}}$ (14)
$\alpha_{\rm{new}}=\frac{\left(1-C_{\rm{new}}\right)}{\sum_{ij}f_{\rm{IPC},ij}}$
(15)
where, $rn_{1}$ and $rn_{2}$ are random numbers in the interval (0, 1)
(uniform deviates). The convolution is performed for a suitably large number
of $\alpha_{\rm{new}}$ values, with noise added to each convolved 55Fe kernel.
(Noise values are randomly selected from a Gaussian distribution, with a noise
standard deviation ($\sigma_{\rm{noise}}$),
$\sigma_{\rm{noise}}=\frac{\epsilon_{q}}{1620G}$ (16)
where $G$ is the measured gain, $\epsilon_{q}$ is the expected background
noise (in e-), and 1620 is the number of charge pairs generated by a Kα photon
absorbed in silicon.) The 55Fe kernels are summed, and the final set of pixels
is normalized so that the sum of charge across all pixels in the final kernel
= 1. The two-dimensional convolution of the charge diffusion and IPC kernels
is performed using the Python routine convolve2d.py. Figure 15 shows the
results of the 55Fe model.
xtalk00F16.ps
Figure 15: Predicted kernel, with pixel values resulting from the 55Fe model.
xtalk00F16.ps
A comparison of this 55Fe model kernel with the measured 55Fe data in Figure
14 indicates that the combination our SPR model and charge diffusion model
predicts the actual 55Fe kernel very well at 40V. The central values for the
model kernel are within the errors of the actual 55Fe data as both tables
above show. The SPR and charge diffusion model also predicts the values in the
surrounding pixels very well. We note that some of the pixels around the
central pixel do not quite match the model to within the errors. This is
likely because the signal in the surrounding pixels is much less than the
central pixel. When these pixels are normalized to the total in the kernel,
the effect of noise in the normalization is much higher for these pixels than
the central pixel. Thus, in addition to the statistical errors across all
hits, there is error in the normalization due to the noise of the system. A
system with lower noise than the camera presented here will likely find model
values even closer to the observed values. We explore the predictive power of
our model in more detail in the next section.
### 4.6 55Fe modeling across voltages
55Fe measurements were taken at substrate voltages ranging from 10 to 40V.
This section describes the crosstalk results for all voltage settings and
compares these results with the model predictions.
When extracting the kernel for the 10, 20, and 30 V measurements, it is
critical to take into consideration broader spreading induced by charge
diffusion. For 10, 20 and 30 V settings, we found that 15x15, 13x13, and 11x11
square pixel kernels respectively were needed to capture all of the charge
from the 55Fe hit. Because the gain of the system should not change with
substrate voltage, the gain from the 40V measurement was used for the lower
voltage settings, since here the Kα distribution is much tighter.
Table 3 shows the central and neighboring pixels resulting from the 55Fe
measurements and those predicted by the model.
Table 3: 55Fe model and data results for substrate voltages 10-40V Model and
Data Comparison for 10-40V 55Fe
---
| | Central Pixel
Vsub(V) | kernel size | model | data | std dev
10 | 15x15 | 32.24% | 29.16% | 2.61%
20 | 13x13 | 47.63% | 46.85% | 2.79%
30 | 11x11 | 55.57% | 54.65% | 1.65%
40 | 9x9 | 60.51% | 60.43% | 1.58%
| | X-right Pixel
10 | 15x15 | 9.52% | 8.88% | 0.57%
20 | 13x13 | 7.79% | 7.65% | 0.38%
30 | 11x11 | 6.48% | 6.63% | 0.27%
40 | 9x9 | 5.72% | 5.92% | 0.17%
| | X-left Pixel
10 | 15x15 | 9.50% | 8.88% | 0.25%
20 | 13x13 | 7.77% | 7.67% | 0.41%
30 | 11x11 | 6.48% | 6.57% | 0.36%
40 | 9x9 | 5.71% | 5.94% | 0.16%
Figure 16 contains plots of the central pixel value and nearest neighbor
value, both for the measurements and for the model. While the model comes
close to the results we have for the 10V data, it does not predict the data to
within the errors. At 10V the distributed charge inhabits a very large kernel:
15x15 pixels, for which the charge is very close to the noise floor of the
system. We therefore conclude that the 10V data presented here is not high
fidelity.
xtalk00F17.ps
Figure 16: Comparison of central and nearest neighbor (right and left) pixel
values: predicted by the model, and taken from the final 55Fe kernel.
xtalk00F17.ps
## 5 Summary
While the design of source-follower CMOS-hybrid arrays eliminates some noise
sources such as charge transfer inefficiency (CTI), read noise and dark
current, it also results in an increase in IPC, in contrast to other
detectors. IPC can be problematic for some astronomical applications like
astrometry and photometry, which relies on high S/N to obtain accurate and
scientifically useful results.
In this paper we have used the SPR method to model IPC in TIS H4RG-10 arrays.
We combined a charge diffusion model with this IPC model to understand the
global effect of crosstalk on these arrays. Our primary findings are as
follows:
* •
The IPC in the H4RG-10 B0 detectors extends out to 9x9 pixels based on our
single pixel reset method, requiring a extension to Moore s IPC model to
capture all of the charge spread.
* •
The single pixel reset method shows a read out bias that is column dependent.
This effect introduces an inherent systematic bias in the IPC pattern. This
bias must be included in the 55Fe reduction strategy in order to extract all
good hits from a 55Fe image.
* •
We have developed a model for combining IPC and charge diffusion effects as a
function of substrate voltage. Our model results closely match our empirical
results.
* •
Our results suggest that the SPR method alone combined with the model of
charge diffusion described here can provide enough information to simulate the
dominant properties of H4RG-10 B0 crosstalk to high accuracy without need for
a radioactive source. In addition the SPR data with this charge diffusion
model can be used to very accurately predict astrometric and photometric
performance [4].
* •
We find that the H4RG-10 B0 generation of detectors shows crosstalk of $\sim
10\%$, and a signal loss of $\sim 40\%$ in the central pixel at 40V. A risk
reduction effort is currently underway to reduce this crosstalk by half in a
new generation (B1) of H4RG-10 detectors.
The model based on SPR data presented here reproduces the 55Fe data to within
the statistical errors and noise properties of our system. In a companion
paper we use the models to explore the effects crosstalk at various levels
(5-15%) have on astrometric and photometric applications [4].
## 6 Acknowledgements
The research described in this paper was performed in part by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration. The authors are very
grateful to Chris Paine, Allan Runkle, Stuart Shaklan, and Larry Hovland, for
all of their help designing, building and operating the JMAPS H4RG-10
astrometric camera at JPL. We are also very grateful to Andre Wong, Bill
Tennant, and Yibin Bai at TIS for their informative thoughts and suggestions
that have greatly improved this paper.
## References
* [1] J. Janesick, J. T. Andrews, and T. Elliot, Fundamental performance differences between CMOS and CCD imagers: Part 2, SPIE Proceedings, 6267 (2006).
* [2] B. N. Dorland, et al., Laboratory and sky testing results for the TIS H4RG-10 4k x 4k 10-micron visible CMOS-hybrid detector, SPIE Proceedings Vol. 6690 (2007).
* [3] B. N. Dorland, et al., Initial laboratory and sky testing results for the second generation H4RG-10 4k x 4k, 10 micron visible CMOS-Hybrid detector, SPIE Proceedings Vol. 7439 (2009).
* [4] J. Hubbs, et al., High performance 4096x4096 pixel imaging sensors for the JMAPS Astrometry Mission, presented at the Military Science Symposium, Orlando FLorida USA 2011.
* [5] B. N.Dorland, US Naval Observatory, 3450 Massachusetts Avenue, Washington, D.C. 20392, USA, et al are preparing a manuscript to be called ” Effects of pixel crosstalk on astronomical measurements using a large format CMOS-hybrid detector.”
* [6] L. M. Simms, Hybrid CMOS SiPIN Detectors as Astronomical Imagers (ProQuest, UMI Dissertation Publishing, 2011).
* [7] L. Cheng, ”Interpixel Capacitive Coupling,” (thesis) RIT Digital Media Library Repository, https://ritdml.rit.edu/handle/1850/8414.
* [8] A. C. Moore, Z. Ninkov and W. J. Forrest, Interpixel capacitance in non-destructive focal plane arrays, Focal Plane Arrays for Space Telescopes, Proceedings of the SPIE, Vol. 5167, pp. 204-215 (2004).
* [9] S. M. Sze, Physics of Semiconductor Devices, Second Edition (Wiley-Interscience Publication, 1981).
* [10] S. E. Holland, et al., Development of Back-Illuminated, Fully-Depleted CCD Image Sensors for Use in Astronomy and Astrophysics, in Proceedings of IEEE Workshop on Charge-Coupled Devices and Advanced Image Sensors , (1997), pp. R26-1 R26-4.
* [11] J. R. Janesick, Scientific Charge-Coupled Devices , (SPIE Press, 2001).
* [12] http://henkle.lbl.gov/optical_constants/getdb2.html.
* [12] S. Seshadri, D. M. Cole, B. R. Hancock and R. M. Smith, Mapping electrical crosstalk in pixelated sensor arrays, in High Energy, Optical, and Infrared Detectors for Astronomy III, Proceedings of the SPIE, Vol. 7021 , D. A. Dorn and A. D. Holland, ed. (2008), pp. 1-10.
* [13] G. Finger, R. Dorn, M. Meyer, and L. Mehrgan, Interpixel capacitance in large format CMOS hybrid arrays, SPIE 6276, 62760F (2006)
|
arxiv-papers
| 2012-03-26T17:39:21 |
2024-09-04T02:49:29.085123
|
{
"license": "Public Domain",
"authors": "R. P. Dudik, M. Jordan, B. N. Dorland, D. Veillette, A. Waczynski, B.\n Lane, M. Loose, E. Kan, J. Waterman, S. Pravdo",
"submitter": "Rachel Dudik",
"url": "https://arxiv.org/abs/1203.5740"
}
|
1203.5770
|
###### Abstract
We establish the boundedness character of solutions of a system of rational
difference equations with a variable coefficient.
On the Boundedness of Solutions of a Rational System with a Variable
Coefficient
E. CAMOUZIS
American College of Greece, Deree College, 6 Gravias Street, Aghia Paraskevi,
15342 Athens, Greece
## 1 Introduction
Consider the system of difference equations
$x_{n+1}=\frac{x_{n}}{y_{n}}\;\;\text{and}\;\;y_{n+1}=x_{n}+\gamma_{n}y_{n},\;\;n=0,1,\ldots$
(1.1)
where $\displaystyle\\{\gamma_{n}\\}_{n=0}^{\infty}$ is an arbitrary sequence
of positive real numbers and the initial conditions $\displaystyle x_{0}$ and
$\displaystyle y_{0}$ are positive real numbers.
When $\displaystyle\gamma_{n}=\gamma>1$, the solution
$\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ converges to
$\displaystyle(0,\infty)$ and so it is unbounded. When
$\displaystyle\gamma=1$, the solution
$\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ satisfies the identity
$\displaystyle
x_{n}+y_{n}+\frac{x_{n}}{y_{n}}+\frac{1}{y_{n}}=x_{0}+y_{0}+\frac{x_{0}}{y_{0}}+\frac{1}{y_{0}}=A>2$
and it is easy to see that it converges to
$\displaystyle(0,\frac{A+\sqrt{A^{2}-4}}{2})$
and so is bounded. Finally, when $\displaystyle 0<\gamma<1$, it was
established in [2] that both components of every solution
$\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ are bounded from above by a
positive constant. The proof that was presented in [2] was based on the
properties of the double sequence of finite sums
$\displaystyle\phi(i,n)=\sum_{k=0}^{n}\gamma^{k}x_{k+i+1},\;\;i=0,1,\ldots,\;\;n=0,1,\ldots,$
for which, as it was shown in [2], it holds that
$\displaystyle\lim_{n\rightarrow\infty}\phi(i,n)=\frac{\gamma+x_{i}}{y_{i}},\;\;i=0,1,\ldots\;.$
In this paper we extend the ideas of the proof presented in [2] to establish
that when $\displaystyle\\{\gamma_{n}\\}_{n=0}^{\infty}$ is bounded from below
and from above by two positive constants $\displaystyle\gamma^{\prime}$ and
$\displaystyle\gamma$, and more precisely,
$\displaystyle 0<\gamma^{\prime}\leq\gamma_{n}\leq\gamma<1,$
both components of every solution of System (1.1) are bounded from above by a
positive constant. It was also shown in [2] that when
$\displaystyle\gamma_{n}=\gamma\in(0,1)$ and the initial conditions are
positive real numbers, the dynamics of System (1.1), in terms of boundedness,
are equivalent with the dynamics of the system
$x_{n+1}=\frac{x_{n}y_{n}}{x_{n}+\gamma}\;\;\text{and}\;\;y_{n+1}=\frac{y_{n}}{x_{n}+\gamma},\;\;n=0,1,\ldots\;.$
(1.2)
More precisely, as it was shown in [2], given a solution
$\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ of System (1.1) with
$\displaystyle\gamma_{n}=\gamma>0$, the sequence
$\displaystyle\\{x_{n},w_{n}\\}_{n=0}^{\infty}$, for which,
$\displaystyle w_{n}=\frac{\gamma+x_{n}}{y_{n}},\;\;n=0,1,\ldots,$
satisfies
$x_{n+1}=\frac{x_{n}w_{n}}{x_{n}+\gamma}\;\;\text{and}\;\;w_{n+1}=\frac{w_{n}}{x_{n}+\gamma},\;\;n=0,1,\ldots\;.$
(1.3)
This is also true for System (1.1) with the variable coefficient
$\displaystyle\gamma_{n}$. That is, given a solution
$\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ of System (1.1), the sequence
$\displaystyle\\{x_{n},w_{n}\\}_{n=0}^{\infty}$, where
$\displaystyle w_{n}=\frac{\gamma_{n-1}+x_{n}}{y_{n}},\;\;n=0,2,\ldots,$
with $\displaystyle\gamma_{-1}=\gamma_{0}$, satisfies the system
$x_{n+1}=\frac{x_{n}w_{n}}{x_{n}+\gamma_{n-1}}\;\;\text{and}\;\;w_{n+1}=\frac{w_{n}}{x_{n}+\gamma_{n-1}},\;\;n=0,1,\ldots\;.$
(1.4)
Furthermore,
$w_{n+1}=\frac{1}{y_{n}},\;\;\text{for all}\;\;n\geq 0.$ (1.5)
The following definitions and theorems for double sequences will be useful in
the sequel. Assume that $\displaystyle\\{\phi(k,n)\\}_{k,n=1}^{\infty}$, is a
double sequence of positive real numbers. Then we say that
$\displaystyle\phi(k,n)$ converges to $\displaystyle L\in[0,\infty)$, if for
every $\displaystyle\epsilon>0$, there exists $\displaystyle N(\epsilon)$ such
that
$\displaystyle|\phi(k,n)-L|<\epsilon,\;\;\text{for all}\;\;k,n\geq N.$
We write
$\displaystyle\lim_{k,n\rightarrow\infty}\phi(k,n)=L,$
and $\displaystyle L$ is called the double limit of the sequence. The two
limits
$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\phi(k,n)\;\;\text{and}\;\;\lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty}\phi(k,n)$
are called iterated limits.
Assume that $\displaystyle\\{\phi(k,n)\\}$ is a double sequence of positive
real numbers and
$\displaystyle(k_{1},n_{1})<(k_{2},n_{2})<\ldots<(k_{s},n_{s})<\ldots$
is a strictly increasing sequence of pairs of positive integers. Then
$\displaystyle\\{\phi(k_{s},n_{t})\\}$ is a double subsequence of
$\displaystyle\\{\phi(k,n)\\}$.
The following three theorems will be useful in the sequel. For the proof see
[3].
###### Theorem 1.1.
Assume that $\displaystyle\\{\phi(k,n)\\}_{k,n=1}^{\infty}$ is a double
sequence of positive real numbers which is bounded from above by a positive
constant. Also, assume that for each $\displaystyle k\geq 1$
$\displaystyle\lim_{n\rightarrow\infty}\phi(k,n)=w_{k}\;\;\text{exists}.$
Then for any subsequence $\displaystyle\\{\phi(k_{s},n_{t})\\}$ of
$\displaystyle\\{\phi(k,n)\\}$,
$\displaystyle\lim_{t\rightarrow\infty}\phi(k_{s},n_{t})=w_{k_{s}}\;\;\text{exists}\;\;\text{for
all}\;\;s.$
Furthermore, if
$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\phi(k,n)=L\;\;\text{exists},$
then for any subsequence $\displaystyle\\{\phi(k_{s},n_{t})\\}$ of
$\displaystyle\\{\phi(k,n)\\}$,
$\displaystyle\lim_{s\rightarrow\infty}\lim_{t\rightarrow\infty}\phi(k_{s},n_{t})=L.$
###### Theorem 1.2.
Assume that $\displaystyle\\{\phi(k,n)\\}_{k,n=1}^{\infty}$ is a double
sequence of positive real numbers, which is bounded from above by a positive
constant. Also, assume that $\displaystyle\\{\phi(k_{s},n_{t})\\}$ is a double
subsequence of $\displaystyle\\{\phi(k,n)\\}$ which strictly decreases (resp.
increases) to a nonnegative value $\displaystyle L$ and also
$\displaystyle\phi(k_{s},n_{t})<\phi(i,j),\;(\text{resp.}\phi(k_{s},n_{t})>\phi(i,j))\;\text{for
all}\;\;(i,j)<(k_{s},n_{t})$
and for all $\displaystyle(k_{s},n_{t})$. Then
$\displaystyle\lim_{s,t\rightarrow\infty}\phi(k_{s},n_{t})=\lim_{s\rightarrow\infty}\lim_{t\rightarrow\infty}\phi(k_{s},n_{t})=\lim_{t\rightarrow\infty}\lim_{s\rightarrow\infty}\phi(k_{s},n_{t})=L\in[0,\infty).$
###### Theorem 1.3.
Assume that $\displaystyle\\{\phi(k,n)\\}_{k,n=1}^{\infty}$ is a double
sequence of positive real numbers such that
$\displaystyle\lim_{n\rightarrow\infty}\phi(k,n)\;\;\text{exists
uniformly}\;\;\text{in}\;\;k$
and that
$\displaystyle\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}\phi(k,n)=L.$
Then the double limit of the sequence $\displaystyle\\{\phi(k,n)\\}$ exists
and
$\displaystyle\lim_{k,n\rightarrow\infty}\phi(k,n)=L.$
## 2 Boundedness
In this section we establish that both components of every solution of System
(1.1) are bounded from above by a positive constant.
###### Theorem 2.1.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of System
(1.1) with positive initial conditions $\displaystyle x_{0}$ and
$\displaystyle y_{0}$ and such that
$\displaystyle 0<\gamma^{\prime}\leq\gamma_{n}\leq\gamma<1,\;\;\text{for
all}\;\;n\geq 0$
and $\displaystyle\gamma^{\prime},\gamma\in(0,1)$. Then both components of the
solution $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ are bounded from
above by a positive constant.
The proof of the theorem will be presented at the end of this section.
Set $\displaystyle\gamma_{-1}=\gamma_{0}$. Consider the double sequence of
finite sums
$\displaystyle\phi(i,n)=x_{i+1}+\gamma_{i-1}x_{i+2}+\gamma_{i-1}\gamma_{i}x_{i+3}+\ldots+\gamma_{i-1}\cdots\gamma_{i+n-3}x_{i+n},$
with
$\displaystyle i=0,1,\ldots\;\;\text{and}\;\;n=1,2,\ldots,$
or equivalently,
$\phi(i,n)=\sum_{k=0}^{n-1}\mu(i,k)x_{i+k+1},\;\;i=0,1,\ldots,\;n=1,2,\ldots,$
(2.1)
where for each $\displaystyle i\geq 0$,
$\displaystyle\mu(i,k)=\prod_{j=i-1}^{k+i-3}\gamma_{j},\;\;k=2,3,\ldots$
and
$\displaystyle\mu(i,1)=1.$
The following lemmas will be useful in the sequel.
###### Lemma 2.2.
It holds that
$\displaystyle\lim_{i\rightarrow\infty}\lim_{k\rightarrow\infty}\mu(i,k)=\lim_{i,k\rightarrow\infty}\mu(i,k)=0.$
###### Proof.
In view of Theorem 1.3, it suffices to show that
$\displaystyle\lim_{k\rightarrow\infty}\mu(i,k)=0$
uniformly for each $\displaystyle i$. Indeed, for a given positive number
$\displaystyle\epsilon$ and $\displaystyle i$ arbitrary but fixed, we choose
$\displaystyle k>\frac{\ln\epsilon}{\ln\gamma}+1$, or equivalently
$\displaystyle\gamma^{k-1}<\epsilon$. Then
$\displaystyle\mu(i,k)<\gamma^{k-1}<\epsilon$
from which the result follows. ∎
###### Lemma 2.3.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of System
(1.1). Then for each $\displaystyle i\geq 0$,
$\displaystyle\lim_{n\rightarrow\infty}\frac{y_{i+n}}{\mu(i,n+2)}=\infty.$
###### Proof.
From the first equation of System (1.1) we see that
$\displaystyle\frac{y_{i+n+1}}{\mu(i,n+3)}=\frac{x_{i+n}}{\mu(i,n+3)}+\frac{y_{i+n}}{\mu(i,n+2)},\;\;n=0,1,\ldots$
and so the sequence
$\displaystyle\left\\{\frac{y_{i+n}}{\mu(i,n+2)}\right\\}_{n=0}^{\infty}$ is
strictly increasing. Now assume for the sake of contradiction that
$\displaystyle\lim_{n\rightarrow\infty}\frac{y_{i+n}}{\mu(i,n+2)}=L\in(0,\infty).$
Then, there exists a positive number $\displaystyle\epsilon$ arbitrarily small
and a positive integer $\displaystyle N$ sufficiently large such that
$\displaystyle y_{n+i}<(L+\epsilon)\mu(i,n+2),\;\;\text{for all}\;\;n\geq N.$
From Lemma 2.2, the sequence $\displaystyle\\{\mu(i,n+2)\\}_{n=0}^{\infty}$
converges to zero. Thus, the sequence
$\displaystyle\\{y_{i+n}\\}_{n=0}^{\infty}$ goes to zero as well. Furthermore,
$\displaystyle\mu(i,n+2)=\prod_{j=i-1}^{n+i-1}\gamma_{j}\leq\gamma^{n+1},\;\;\text{for
all}\;\;n\geq 0$
implies that
$\displaystyle y_{i+n}\leq(L+\epsilon)\gamma^{n+1},\;\;\text{for all}\;\;n\geq
N$
and so
$\displaystyle
x_{i+n+1}=\frac{x_{i+n}}{y_{i+n}}\geq\frac{1}{(L+\epsilon)\gamma^{n+1}}\cdot
x_{i+n},\;\;\text{for all}\;\;n\geq N$
from which it follows that
$\displaystyle\lim_{n\rightarrow\infty}x_{i+n+1}=\infty.$
However, from the second equation
$\displaystyle y_{i+n+1}>x_{i+n},\;\;\text{for all}\;\;n\geq 0$
which contradicts the fact that the sequence
$\displaystyle\\{y_{i+n}\\}_{n=0}^{\infty}$ converges to 0.
∎
###### Lemma 2.4.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of (1.1).
Then for all $\displaystyle i\geq 0$,
$\frac{x_{i}+\gamma_{i-1}}{y_{i}}=w_{i}=\phi(i,n)+\mu(i,n+1)w_{i+n},\;\;n=1,2,\ldots\;.$
(2.2)
###### Proof.
Let $\displaystyle i\geq 0$ be given. Clearly, in view of (1.4),
$\displaystyle w_{i}=x_{i+1}+\gamma_{i-1}w_{i+1}$
and so the result is true when $\displaystyle n=1$. Assume that $\displaystyle
k>1$ and that
$\displaystyle
w_{i}=x_{i+1}+\gamma_{i-1}x_{i+2}+\ldots+\gamma_{i-1}\cdots\gamma_{i+k-3}x_{i+k}+\gamma_{i-1}\cdots\gamma_{i+k-2}w_{i+k}$
$\displaystyle=\phi(i,k)+\mu(i,k+1)w_{k+i}.$
Then
$\displaystyle w_{i}=\phi(i,k)+\mu(i,k+1)(x_{i+k+1}+\gamma_{i+k-1}w_{i+k+1})$
$\displaystyle=\phi(i,k)+\mu(i,k+1)x_{i+k+1}+\mu(i,k+1)\gamma_{i+k-1}w_{i+k+1}$
$\displaystyle=\phi(i,k+1)+\mu(i,k+2)w_{i+k+1}.$
The proof is complete.
∎
###### Lemma 2.5.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of (1.1).
Then for all $\displaystyle i\geq 0$,
$\lim_{n\rightarrow\infty}\phi(i,n)=\sum_{k=0}^{\infty}\mu(i,k)x_{i+k+1}=\frac{x_{i}+\gamma_{i-1}}{y_{i}}=w_{i}.$
(2.3)
###### Proof.
The result follows from (2.2) together with the fact, in view of (1.5) and
Lemma 2.3, that
$\displaystyle\lim_{n\rightarrow\infty}\mu(i,n+1)w_{i+n}=\lim_{n\rightarrow\infty}\frac{\mu(i,n+1)}{y_{i+n-1}}=0.$
∎
###### Lemma 2.6.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of (1.1) and
assume that for an infinite sequence of positive integers
$\displaystyle\\{k_{i}\\}_{i=1}^{\infty}$, $\displaystyle\\{x_{k_{i}}\\}$ is a
bounded subsequence of $\displaystyle\\{x_{n}\\}$ and
$\displaystyle\lim_{i\rightarrow\infty}\frac{x_{k_{i}}+\gamma_{k_{i}-1}}{y_{k_{i}}}=\lim_{i\rightarrow\infty}w_{k_{i}}=M\in(0,\infty).$
Then the following statements are true:
1.
$\displaystyle\lim_{i\rightarrow\infty}\lim_{n\rightarrow\infty}\phi(k_{i},n)=M.$
2\. For any subsequence $\displaystyle\\{\phi(k_{i_{s}},n_{j})\\}$ of
$\displaystyle\\{\phi(k_{i},n)\\}$, it holds
$\displaystyle\lim_{s\rightarrow\infty}\lim_{j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=M.$
3.
$\displaystyle\limsup_{i,n\rightarrow\infty}\phi(k_{i},n)\leq M.$
4.
$\displaystyle\liminf_{i,n\rightarrow\infty}\phi(k_{i},n)>0.$
5.
$\displaystyle\liminf_{i\rightarrow\infty}x_{k_{i}+1}>0.$
###### Proof.
1\. The proof follows from Lemma 2.5 and the hypothesis.
2\. The proof is an immediate consequence of the result of Part 1 and Theorem
1.1, which is presented in the Introduction.
3\. The proof is an immediate consequence of the fact that, for each
$\displaystyle i\geq 1$,
$\displaystyle\phi(k_{i},n)<\sum_{k=0}^{\infty}\mu(k_{i},n)x_{k+k_{i}+1}=w_{k_{i}},\;\;\text{for
all}\;\;n\geq 1$
and the hypothesis that $\displaystyle w_{k_{i}}\rightarrow M$.
4\. The proof will be by contradiction. Assume for the sake of contradiction
that there exists a decreasing subsequence
$\displaystyle\\{\phi(k_{i_{s}},n_{j})\\}_{s,j=1}^{\infty}$ of
$\displaystyle\\{\phi(k_{i},n)\\}$, for which
$\displaystyle\lim_{s,j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=0$
and
$\displaystyle\phi(k_{i_{s}},n_{j})<\phi(p,q),\;\;\text{for
all}\;\;(p,q)<(k_{i_{s}},n_{j}).$
We claim that both $\displaystyle\\{k_{i_{s}}\\}$ and
$\displaystyle\\{n_{j}\\}$ must increase to infinity. Otherwise, for
$\displaystyle k_{i_{s}}$ finite and fixed,
$\displaystyle\lim_{j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=0.$
In view of the result of Part 2 and the hypothesis, we see that
$\displaystyle\lim_{j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=w_{k_{i_{s}}}>0$
which is a contradiction.
On the other hand assume that there exists a positive integer $\displaystyle
N$ such that
$\displaystyle\lim_{s\rightarrow\infty}\phi(k_{i_{s}},j)=0,\;\;j=1,2,\ldots,N\;\;\text{and}\;\;\liminf_{s\rightarrow\infty}\phi(k_{i_{s}},N+1)>0.$
In view of (2.1), as $\displaystyle s\rightarrow\infty$, it is easy to see
that
$\displaystyle x_{k_{i_{s}}+t}\rightarrow 0,\;\;\text{for
all}\;\;t=1,\ldots,N.$
By choosing a further subsequence of
$\displaystyle\\{k_{i_{s}}\\}_{s=1}^{\infty}$, which for economy in notation
we still denote it as $\displaystyle\\{k_{i_{s}}\\}$, it holds that for each
$\displaystyle j=-1,0,\ldots,N-2,\;$ the sequence
$\displaystyle\\{\gamma_{k_{i_{s}}+j}\\}_{s=1}^{\infty}$ converges to a
positive number. Set
$\displaystyle
m=\lim_{s\rightarrow\infty}\prod_{j=-1}^{N-2}\gamma_{k_{i_{s}}+j}\in(0,\infty).$
Clearly, and in view of (1.4),
$\displaystyle w_{k_{i_{s}}+N}\rightarrow\frac{M}{m}>0.$
Therefore,
$\displaystyle
x_{k_{i_{s}}+N+1}=\frac{x_{k_{i_{s}}+N}w_{k_{i_{s}}+N}}{\gamma_{k_{i_{s}}+N-1}+x_{k_{i_{s}}+N}}\rightarrow
0$
and so, in view of (2.1),
$\displaystyle\lim_{s\rightarrow\infty}\phi(k_{i_{s}},N+1)=0$
which is a contradiction. Therefore, the sequences
$\displaystyle\\{k_{i_{s}}\\}$ and $\displaystyle\\{n_{j}\\}$ are infinite
sequences of positive integers and both increase to infinity. By applying
Theorem 1.2, we get
$\displaystyle\lim_{s,j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=\lim_{s\rightarrow\infty}\lim_{j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=0.$
On the other hand, by applying the result of Part 2, we see that
$\displaystyle\lim_{s\rightarrow\infty}\lim_{j\rightarrow\infty}\phi(k_{i_{s}},n_{j})=M\in(0,\infty)$
which is a contradiction.
5\. From Part 4, clearly, there exists a positive number $\displaystyle I$
such that
$\displaystyle\phi(k_{i},n)>I,\;\;\text{for all}\;\;i,n\geq N.$
In particular,
$\phi(k_{i},N)=x_{k_{i}+1}+\gamma_{k_{i}-1}x_{k_{i}+2}+\ldots+\gamma_{k_{i}-1}\cdots\gamma_{k_{i}-3+N}x_{k_{i}+N}>I>0,$
(2.4)
for all $\displaystyle i\geq N$. Now assume for the sake of contradiction and
without loss of generality that
$\displaystyle x_{k_{i}+1}\rightarrow 0.$
Note that
$\displaystyle w_{k_{i}}=x_{k_{i}+1}+\gamma_{k_{i}-1}w_{k_{i}+1}\Rightarrow
w_{k_{i}+1}=\frac{w_{k_{i}}}{\gamma_{k_{i}-1}}-\frac{x_{k_{i}+1}}{\gamma_{k_{i}-1}}$
and so there exists a further subsequence of
$\displaystyle\\{k_{i}\\}_{i=1}^{\infty}$, which for economy in notation we
still denote as $\displaystyle\\{k_{i}\\}$, such that
$\displaystyle\gamma_{k_{i}-1}\rightarrow
m>0\;\;\text{and}\;\;w_{k_{i}+1}\rightarrow\frac{M}{m},$
and so
$\displaystyle
x_{k_{i}+2}=\frac{x_{k_{i}+1}w_{k_{i}+1}}{\gamma_{k_{i}}+x_{k_{i}+1}}\rightarrow
0.$
By induction, we see that
$\displaystyle\lim_{i\rightarrow\infty}x_{k_{i}+j}=0,\;\;\text{for
all}\;\;j=1,2,\ldots,N.$
By taking limits in (2.4), as $\displaystyle i\rightarrow\infty$, we get a
contradiction.
∎
We now present the proof of Theorem 2.1
###### Proof.
Let $\displaystyle\\{x_{n},y_{n}\\}_{n=0}^{\infty}$ be a solution of System
(1.1). First we establish that the component
$\displaystyle\\{y_{n}\\}_{n=0}^{\infty}$ of the solution is bounded from
below by a positive constant. Assume for the sake of contradiction that there
exists an infinite sequence of indices
$\displaystyle\\{n_{i}\\}_{i=1}^{\infty}$ such that
$\displaystyle y_{n_{i}+1}=x_{n_{i}}+\gamma_{n_{i}}y_{n_{i}}\rightarrow 0.$
Clearly,
$\displaystyle x_{n_{i}-t}\rightarrow
0\;\;\text{and}\;\;y_{n_{i}-t}\rightarrow 0,\;\;\text{for
all}\;\;t=0,1,\ldots\;.$
In addition, there exists a sequence of indices
$\displaystyle\\{k_{i}\\}_{i=1}^{\infty}$ such that
$\displaystyle k_{i}\leq n_{i},\;\;\text{for all}\;\;i,$
for which
$(y_{k_{i}-1}\geq
1\;\text{and}\;y_{k_{i}}<1)\;\text{and}\;(y_{t}<1,\;\;\text{for
all}\;\;t\in\\{k_{i}+1,\ldots,n_{i}\\}),$ (2.5)
because otherwise,
$\displaystyle x_{n_{i}}=\frac{x_{0}}{\prod_{j=0}^{n_{i}-1}y_{j}}>x_{0},$
which is a contradiction. From
$\displaystyle
y_{k_{i}}=x_{k_{i}-1}+\gamma_{k_{i}-1}y_{k_{i}-1}\;\;\text{and}\;\;y_{k_{i}-1}\geq
1,\;\;\text{for all}\;\;i,$
it follows that
$\displaystyle y_{k_{i}}\geq\gamma_{k_{i}-1}\geq\gamma^{\prime},\;\;\text{for
all}\;\;i,$
and so
$\displaystyle y_{k_{i}}\in[\gamma^{\prime},1),\;\;\text{for all}\;\;i.$
For $\displaystyle i$ sufficiently large, when $\displaystyle
r\in\\{k_{i}+1,\ldots,n_{i}\\}$,
$\displaystyle x_{r}=\frac{x_{r-1}}{y_{r-1}}>x_{r-1}$
and more precisely,
$\displaystyle x_{n_{i}}>x_{n_{i}-1}>\ldots>x_{k_{i}+1}>x_{k_{i}}.$
Therefore,
$\displaystyle x_{k_{i}}<x_{n_{i}},$
from which it follows that $\displaystyle x_{k_{i}}\rightarrow 0$. By
utilizing the fact that
$\displaystyle y_{k_{i}}\in[\gamma^{\prime},1),\;\;\text{for all}\;\;i,$
we may select a further subsequence of $\displaystyle\\{k_{i}\\}$, still
denoted as $\displaystyle\\{k_{i}\\}$ such that
$\displaystyle y_{k_{i}}\rightarrow
L\in[\gamma^{\prime},1]\;\;\text{and}\;\;\gamma_{k_{i}-1}\rightarrow
l_{-1}\in[\gamma^{\prime},\gamma].$
Therefore,
$\displaystyle x_{k_{i}+1}=\frac{x_{k_{i}}}{y_{k_{i}}}\rightarrow
0\;\;\text{and}\;\;w_{k_{i}}=\frac{\gamma_{k_{i}-1}+x_{k_{i}}}{y_{k_{i}}}\rightarrow\frac{l_{-1}}{L}=M\in\left[\gamma^{\prime},\frac{\gamma}{\gamma^{\prime}}\right].$
By applying Lemma 2.6, we get
$\displaystyle\liminf_{i\rightarrow\infty}x_{k_{i}+1}>0$
which is a contradiction. Hence, the component
$\displaystyle\\{y_{n}\\}_{n=0}^{\infty}$ of the solution is bounded from
below by a positive constant $\displaystyle m$. In view of
$\displaystyle
x_{n+1}=\frac{x_{n}}{y_{n}}=\frac{1}{y_{n-1}}\cdot\frac{x_{n}}{x_{n}+\gamma_{n-1}},\;\;\text{for
all}\;\;n\geq 1,$
we see that
$\displaystyle x_{n+1}<\frac{1}{m},\;\;\text{for all}\;\;n\geq 1,$
and so the component $\displaystyle\\{x_{n}\\}_{n=0}^{\infty}$ is bounded from
above. From the second equation of the system, clearly
$\displaystyle y_{n+1}<\frac{1}{m}+\gamma y_{n},\;\;\text{for all}\;\;n\geq
2,$
and so
$\displaystyle\limsup_{n\rightarrow\infty}y_{n}\leq\frac{1}{m(1-\gamma)}.$
The proof of the Theorem is complete.
∎
## References
* [1]
* [2] E. Camouzis, On the Boundedness of Solutions of a Rational System, _International Journal of Difference Equations and Applications_ , (to appear).
* [3] Habil, E. D., Double sequences and double series, submitted to the Islamaic University Journal, (2005).
|
arxiv-papers
| 2012-03-26T19:41:21 |
2024-09-04T02:49:29.091688
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Elias Camouzis",
"submitter": "Elias Camouzis",
"url": "https://arxiv.org/abs/1203.5770"
}
|
1203.5779
|
# Magneto-transport in mesoscopic rings and cylinders: Effects of electron-
electron interaction and spin-orbit coupling
Santanu K. Maiti santanu@post.tau.ac.il School of Chemistry, Tel Aviv
University, Ramat-Aviv, Tel Aviv-69978, Israel
###### Abstract
We undertake an in-depth analysis of the magneto-transport properties in
mesoscopic single-channel rings and multi-channel cylinders within a tight-
binding formalism. The main focus of this review is to illustrate how the long
standing anomalies between the calculated and measured current amplitudes
carried by a small conducting ring upon the application of a magnetic flux
$\phi$ can be removed. We discuss two different cases. First, we examine the
combined effect of second-neighbor hopping integral and Hubbard correlation on
the enhancement of persistent current in presence of disorder. A significant
change in current amplitude is observed compared to the traditional nearest-
neighbor hopping model and the current amplitude becomes quite comparable to
experimental realizations. In the other case we verify that in presence of
spin-orbit interaction a considerable enhancement of persistent current
amplitude takes place, and the current amplitude in a disordered ring becomes
almost comparable to that of an ordered one. In addition to these, we also
present the detailed band structures and some other related issues to get a
complete picture of the phenomena at the microscopic level.
###### pacs:
73.23.-b, 73.23.Ra, 73.21.Hb
## I Introduction
The study of magneto-transport properties in low-dimensional systems is always
an interesting problem in mesoscopic physics. This is relatively a new branch
of condensed matter physics which deals with systems whose dimensions are
intermediate between the microscopic and macroscopic length scales imrybook ;
datta1 ; datta2 ; metalidis ; weinmann . In contrast to the macroscopic
objects where we generally use the laws of classical mechanics, the meso-scale
systems are treated quantum mechanically since in this region fluctuations
play the very crucial role. Over the last many years low-dimensional model
quantum systems have been the objects of intense research, both in theory and
in experiments, mainly due to the fact that these simple looking systems are
prospective candidates for nano devices in electronic as well as spintronic
engineering aharon08 ; ahar2 ; ore1 ; ore2 ; yey ; torio ; buks96 ; fuhrer07 ;
kob ; popp03 ; foldi08 ; berc04 ; berc2 ; vidal1 ; vidal2 ; san10 ; san11 ;
san12 .
Several striking spectral properties are also exhibited by such systems owing
to the quantum interference which is specially observed in quantum geometries
with closed loop structures. The existence of dissipationless current in a
mesoscopic metallic ring threaded by an Aharonov-Bohm (AB) flux $\phi$ is a
direct consequence of quantum phase coherence. In this new quantum regime, two
important aspects appear at low temperatures. The first one is that the phase
coherence length $L_{\phi}$ i.e., the length scale over which an electron
maintains its phase memory, increases significantly with the lowering of
temperature and becomes comparable to the system size $L$. The other one is
that the energy levels of such small finite size systems are discrete. These
two are the most essential criteria for the appearance of persistent charge
current in a small metallic ring/cylinder due to the application of an
external magnetic flux $\phi$. In the pioneering work of Büttiker, Imry and
Landauer butt , the appearance of persistent current in metallic rings has
been explored. Later, many excellent experiments levy ; chand ; mailly ; jari
; deb ; reu have been carried out in several ring and cylindrical geometries
to reveal the actual mechanisms of persistent current. Though much efforts
have been paid to study persistent current both theoretically cheu1 ; cheu2 ;
peeters1 ; peeters2 ; peeters3 ; mont ; mont1 ; alts ; von ; schm ; ambe ;
abra ; bouz ; giam ; yu ; belu ; ore ; xiao1 ; xiao2 ; san1 ; san2 ; san3 ;
san8 as well as experimentally levy ; chand ; mailly ; jari ; deb ; reu , yet
several anomalies still exist between the theory and experiment, and the full
knowledge about it in this scale is not well established even today.
The results of the single loop experiments are significantly different from
those for the ensemble of isolated loops. Persistent currents with expected
$\phi_{0}$ periodicity have been observed in isolated single Au rings chand
and in a GaAs-AlGaAs ring mailly . Levy et al. levy found oscillations with
period $\phi_{0}/2$ rather than $\phi_{0}$ in an ensemble of $10^{7}$
independent Cu rings. Similar $\phi_{0}/2$ oscillations were also reported for
an ensemble of disconnected $10^{5}$ Ag rings deb as well as for an array of
$10^{5}$ isolated GaAs-AlGaAs rings reu . In a recent experiment, Jariwala et
al. jari obtained both $\phi_{0}$ and $\phi_{0}/2$ periodic persistent
currents in an array of thirty diffusive mesoscopic Au rings. Except for the
case of the nearly ballistic GaAs-AlGaAs ring mailly , all the measured
currents are in general one or two orders of magnitude larger than those
expected from the theory.
Free electron theory predicts that at absolute zero temperature ($T=0\,$K), an
ordered one-dimensional ($1$D) metallic ring threaded by magnetic flux $\phi$
supports persistent current with maximum amplitude $I_{0}=ev_{F}/L$, where
$v_{F}$ is the Fermi velocity and $L$ is the circumference of the ring. Metals
are intrinsically disordered which tends to decrease the persistent current,
and the calculations show that the disorder-averaged current $\langle
I\rangle$ crucially depends on the choice of the ensemble cheu2 ; mont ; mont1
. The magnitude of the current $\langle I^{2}\rangle^{1/2}$ is however
insensitive to the averaging issues, and is of the order of $I_{0}l/L$, $l$
being the elastic mean free path of the electrons. This expression remains
valid even if one takes into account the finite width of the ring by adding
contributions from the transverse channels, since disorder leads to a
compensation between the channels cheu2 ; mont . However, the measurements on
an ensemble of $10^{7}$ Cu rings levy reported a diamagnetic persistent
current of average amplitude $3\times 10^{-3}$ $ev_{F}/L$ with half a flux-
quantum periodicity. Such $\phi_{0}/2$ oscillations with diamagnetic response
were also found in other persistent current experiments consisting of ensemble
of isolated rings deb ; reu .
Measurements on single isolated mesoscopic rings on the other hand detected
$\phi_{0}$-periodic persistent currents with amplitudes of the order of
$I_{0}\sim ev_{F}/L$, (closed to the value for an ordered ring). Theory and
experiment mailly seem to agree only when disorder is weak. In another recent
nice experiment Bluhm et al. blu have measured the magnetic response of $33$
individual cold mesoscopic gold rings, one ring at a time, using a scanning
SQUID technique. They have measured $h/e$ component and predicted that the
measured current amplitude agrees quite well with theory cheu1 in a single
ballistic ring mailly and an ensemble of $16$ nearly ballistic rings raba .
However, the amplitudes of the currents in single-isolated-diffusive gold
rings chand were two orders of magnitude larger than the theoretical
estimates. This discrepancy initiated intense theoretical activity, and it is
generally believed that the electron-electron correlation plays an important
role in the disordered diffusive rings abra ; bouz ; giam . An explanation
based on the perturbative calculation in presence of interaction and disorder
has been proposed and it seems to give a quantitative estimate closer to the
experimental results, but still it is less than the measured currents by an
order of magnitude, and the interaction parameter used in the theory is not
well understood physically.
To remove the controversies regarding the persistent current amplitude between
theoretical and experimental verifications we can proceed in two different
ways. In literature almost all the theoretical results have been done based on
a tight-binding (TB) framework within the nearest-neighbor hopping (NNH)
approximation. It has been shown that in the NNH model electronic correlation
provides a small enhancement of current amplitude in disordered materials
i.e., a weak delocalizing effect is observed in presence of electron-electron
(e-e) interaction. As a first attempt, we modify the traditional NNH model by
incorporating the effects of higher order hopping integrals, at least second-
neighbor hopping (SNH), in addition to the NNH integral. It is also quite
physical since electrons have some finite probabilities to hop from one site
to other sites apart from nearest-neighbor with reduced strengths. We will
show that the inclusion of higher order hopping integrals gives significant
enhancement of current amplitude and it reaches quite closer to the current
amplitude of ordered systems. This is one approach. In the other way we
examine that in presence of spin-orbit (SO) interaction a considerable
enhancement of persistent current amplitude takes place, and the current
amplitude in the disordered ring is almost comparable to that of an ordered
ring. The spin-orbit fields in a solid are called the Rashba spin-orbit
interaction (RSOI) or the Dresselhaus spin-orbit interaction (DSOI) depending
on whether the electric field originates from a structural inversion asymmetry
or the bulk inversion asymmetry respectively meier . Quantum rings formed at
the interface of two semiconducting materials are ideal candidates where the
interplay of the two kinds of SOI might be observed. A quantum ring in a
heterojunction is realized when a two dimensional gas of electrons is trapped
in a quantum well due to the band offset at the interface of two different
semiconducting materials. This band offset creates an electric field which may
be described by a potential gradient normal to the interface premper . The
potential at the interface is thus asymmetric, leading to the presence of a
RSOI. On the other hand, at such interfaces, the bulk inversion symmetry is
naturally broken.
The other important controversy comes for the determination of the sign of
low-field currents and still it is an unresolved issue between theoretical and
experimental results. In an experiment on persistent current Levy et al. levy
have shown diamagnetic nature for the measured currents at low-field limit.
While, in other experiment Chandrasekhar et al. chand have obtained
paramagnetic response near zero field limit. Jariwala et al. jari have
predicted diamagnetic persistent current in their experiment and similar
diamagnetic response in the vicinity of zero field limit were also supported
in an experiment done by Deblock deb et al. on Ag rings. Yu and Fowler yu
have shown both diamagnetic and paramagnetic responses in mesoscopic Hubbard
rings. Though in a theoretical work Cheung et al. cheu2 have predicted that
the direction of current is random depending on the total number of electrons
in the system and the specific realization of the random potentials. Hence,
prediction of the sign of low-field currents is still an open challenge and
further studies on persistent current in mesoscopic systems are needed to
remove the existing controversies.
In the present review we address several important issues of magneto-transport
in single-channel mesoscopic rings and multi-channel mesoscopic cylinders
which are quite challenging from the standpoint of theoretical as well as
experimental research. A brief outline of the presentation is as follows.
First, we address magnetic response in mesoscopic Hubbard rings threaded by AB
flux $\phi$. We try to propose an idea to remove the unexpected discrepancy
between the calculated and measured current amplitudes by incorporating the
effect of second-neighbor hopping (SNH) in addition to the traditional
nearest-neighbor hopping (NNH) integral in the tight-binding Hamiltonian.
Using a generalized Hartree-Fock (HF) approximation kato ; kam ; san4 ; san5 ;
san6 , we numerically compute persistent current ($I$), Drude weight ($D$) and
low-field magnetic susceptibility ($\chi$) as functions of AB flux $\phi$,
total number of electrons $N_{e}$ and system size $N$. With this (HF) approach
one can study magnetic response in a much larger system since here a many-body
Hamiltonian is decoupled into two effective one-body Hamiltonians. One is
associated with up spin electrons and other is related to down spin electrons.
But the point is that, the results calculated using generalized HF mean-field
theory may deviate from exact results with the reduction of dimensionality. So
we should take care about the mean-field calculation, specially, in $1$D
systems. To trust our predictions, here we also we make a comparative study
between the results obtained from mean-field theory and exactly diagonalizing
the full many-body Hamiltonian. The later approach where a complete many-body
Hamiltonian is diagonalized to get energy eigenvalues is not suitable to study
magnetic response in larger systems since the size of the matrices increases
very sharply with the total number of up and down spin electrons.
Next, we explore the behavior of persistent current in an interacting
mesoscopic ring with finite width threaded by an Aharonov-Bohm flux $\phi$.
For this cylindrical system we also see that the inclusion of higher order
hopping integrals leads to a possibility of getting enhanced persistent
current and the current is quite comparable to the ordered one. Our results
can be utilized to study magnetic response in any interacting mesoscopic
system.
Finally, in the last part, we focus our attention on the behavior of
persistent current in a one-dimensional mesoscopic ring threaded by a magnetic
flux in presence of the Rashba and Dresselhaus SO interactions. Here, the
effect of electron-electron interaction is neglected. We show that the
presence of the SO interaction leads to a significant enhancement of
persistent current san7 . In addition to these, we also describe very briefly
the energy band structures and the oscillations of persistent current as the
RSOI is varied to make the present communication a self contained study.
Throughout the review we perform all the essential features of magneto-
transport at absolute zero temperature and set $c=e=h=1$ for numerical
calculations.
## II A Hubbard ring in absence of SO interactions
In this section we describe the magneto-transport properties in a single-
channel $1$D mesoscopic ring in the presence of on-site Coulomb interaction.
The effect of SO interaction is not taken into account.
### II.1 Model and theoretical formulation
We start by referring to Fig. 1, where a normal metal ring is threaded by a
magnetic flux $\phi$. To describe the system we use a tight-binding framework.
For a $N$-site ring, penetrated by a magnetic flux $\phi$ (measured in unit of
the elementary flux quantum $\phi_{0}=ch/e$), the tight-binding Hamiltonian in
Wannier basis looks in the form,
$\displaystyle H_{R}$ $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\epsilon_{i\sigma}c_{i\sigma}^{\dagger}c_{i\sigma}+\sum_{ij,\sigma}t\left[e^{i\theta}c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.\right]+$
(1)
$\displaystyle\sum_{ik,\sigma}t_{1}\left[e^{i\theta_{1}}c_{i\sigma}^{\dagger}c_{k\sigma}+h.c.\right]+\sum_{i}Uc_{i\uparrow}^{\dagger}c_{i\uparrow}c_{i\downarrow}^{\dagger}c_{i\downarrow}$
where, $\epsilon_{i\sigma}$ is the on-site energy of an electron at the site
$i$ of spin $\sigma$ ($\uparrow,\downarrow$). The variable $t$ corresponds to
the nearest-neighbor ($j=i\pm 1$) hopping strength, while $t_{1}$ gives the
second-neighbor ($k=i\pm 2$) hopping integral. $\theta=2\pi\phi/N$ and
$\theta_{1}=4\pi\phi/N$ are the phase factors associated with the hopping of
an electron from one site to its neighboring
Figure 1: (Color online). Schematic view of a $1$D mesoscopic ring penetrated
by a magnetic flux $\phi$. The filled black circles correspond to the
positions of the atomic sites.
site and next-neighboring site, respectively. $c_{i\sigma}^{\dagger}$ and
$c_{i\sigma}$ are the creation and annihilation operators, respectively, of an
electron at the site $i$ with spin $\sigma$. $U$ is the strength of on-site
Hubbard interaction.
Decoupling of the interacting Hamiltonian: In order to determine the energy
eigenvalues of the interacting model quantum system described by the tight-
binding Hamiltonian given in Eq. 1, first we decouple the interacting
Hamiltonian using generalized Hartree-Fock approach, the so-called mean field
approximation. In this approach, the full Hamiltonian is completely decoupled
into two parts. One is associated with the up-spin electrons, while the other
is related to the down-spin electrons with their modified site energies. For
up and down spin Hamiltonians, the modified site energies are expressed in the
form, $\epsilon_{i\uparrow}^{\prime}=\epsilon_{i\uparrow}+U\langle
n_{i\downarrow}\rangle$ and
$\epsilon_{i\downarrow}^{\prime}=\epsilon_{i\downarrow}+U\langle
n_{i\uparrow}\rangle$, where $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is
the number operator. With these site energies, the full Hamiltonian (Eq. 1)
can be written in the decoupled form as,
$\displaystyle H_{R}$ $\displaystyle=$
$\displaystyle\sum_{i}\epsilon_{i\uparrow}^{\prime}n_{i\uparrow}+\sum_{ij}t\left[e^{i\theta}c_{i\uparrow}^{\dagger}c_{j\uparrow}+e^{-i\theta}c_{j\uparrow}^{\dagger}c_{i\uparrow}\right]$
(2) $\displaystyle+$
$\displaystyle\sum_{ik}t_{1}\left[e^{i\theta_{1}}c_{i\uparrow}^{\dagger}c_{k\uparrow}+e^{-i\theta_{1}}c_{k\uparrow}^{\dagger}c_{i\uparrow}\right]$
$\displaystyle+$
$\displaystyle\sum_{i}\epsilon_{i\downarrow}^{\prime}n_{i\downarrow}+\sum_{ij}t\left[e^{i\theta}c_{i\downarrow}^{\dagger}c_{j\downarrow}+e^{-i\theta}c_{j\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle+$
$\displaystyle\sum_{ik}t_{1}\left[e^{i\theta_{1}}c_{i\downarrow}^{\dagger}c_{k\downarrow}+e^{-i\theta_{1}}c_{k\downarrow}^{\dagger}c_{i\downarrow}\right]$
$\displaystyle-$ $\displaystyle\sum_{i}U\langle n_{i\uparrow}\rangle\langle
n_{i\downarrow}\rangle$ $\displaystyle=$ $\displaystyle
H_{\uparrow}+H_{\downarrow}-\sum_{i}U\langle n_{i\uparrow}\rangle\langle
n_{i\downarrow}\rangle$
where, $H_{\uparrow}$ and $H_{\downarrow}$ correspond to the effective tight-
binding Hamiltonians for the up and down spin electrons, respectively. The
last term is a constant term which provides an energy shift in the total
energy.
Self consistent procedure: With these decoupled Hamiltonians ($H_{\uparrow}$
and $H_{\downarrow}$) of up and down spin electrons, we start our self
consistent procedure considering initial guess values of $\langle
n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$. For these initial
set of values of $\langle n_{i\uparrow}\rangle$ and $\langle
n_{i\downarrow}\rangle$, we numerically diagonalize the up and down spin
Hamiltonians. Then we calculate a new set of values of $\langle
n_{i\uparrow}\rangle$ and $\langle n_{i\downarrow}\rangle$. These steps are
repeated until a self consistent solution is achieved.
Calculation of ground state energy: Using the self consistent solution, the
ground state energy $E_{0}$ for a particular filling at absolute zero
temperature ($T=0\,$K) can be determined by taking the sum of individual
states up to Fermi energy ($E_{F}$) for both up and down spins. Thus, we can
write the final form of ground state energy as,
$E_{0}=\sum_{n}E_{n\uparrow}+\sum_{n}E_{n\downarrow}-\sum_{i}U\langle
n_{i\uparrow}\rangle\langle n_{i\downarrow}\rangle$ (3)
where, the index $n$ runs for the states upto the Fermi level. $E_{n\uparrow}$
($E_{n\downarrow}$) is the single particle energy eigenvalue for $n$-th
eigenstate obtained by diagonalizing the Hamiltonian $H_{\uparrow}$
($H_{\downarrow}$).
Calculation of persistent current: At absolute zero temperature, total
persistent current of the system is obtained from the expression,
$I(\phi)=-c\frac{\partial E_{0}(\phi)}{\partial\phi}$ (4)
where, $E_{0}(\phi)$ is the ground state energy.
Calculation of Drude weight: The Drude weight for the ring can be calculated
through the relation,
$D=\left.\frac{N}{4\pi^{2}}\left(\frac{\partial{{}^{2}E_{0}(\phi)}}{\partial{\phi}^{2}}\right)\right|_{\phi\rightarrow
0}$ (5)
where, $N$ gives total number of atomic sites in the ring. Kohn kohn has
shown that for an insulating system $D$ decays exponentially to zero, while it
becomes finite for a conducting system.
Determination of low-field magnetic susceptibility: The general expression of
magnetic susceptibility $\chi$ at any flux $\phi$ is written in the form,
$\chi(\phi)=\frac{N^{3}}{16\pi^{2}}\left(\frac{\partial
I(\phi)}{\partial\phi}\right).$ (6)
Evaluating the sign of $\chi(\phi)$ we can able to predict whether the current
is paramagnetic or diamagnetic in nature. Here we will determine $\chi(\phi)$
only in the limit $\phi\rightarrow 0$, since we are interested to know the
magnetic response in the low-field limit.
### II.2 Numerical results and discussion
In this sub-section throughout our numerical work we set the nearest-neighbor
hopping strength $t=-1$ and second-neighbor hopping strength $t_{1}=-0.7$.
Energy scale is measured in unit of $t$.
#### II.2.1 Perfect Hubbard Rings Described with NNH Integral
For perfect rings we choose $\epsilon_{i\uparrow}=\epsilon_{i\downarrow}=0$
for all $i$ and since here we consider the rings with only NNH integral, the
second-neighbor hopping strength $t_{1}$ is fixed at zero.
Energy-flux characteristics: To explain the relevant features of magnetic
response we begin with the energy-flux characteristics. As illustrative
examples, in Fig. 2 we plot the ground state energy levels as a function of
magnetic flux $\phi$ for some typical mesoscopic rings in the half-filled
case, where (a) and (b) correspond to $N=5$ and $6$,
Figure 2: (Color online). Ground state energy levels as a function of flux
$\phi$ for some typical mesoscopic rings in half-filled case. The red, green
and blue curves correspond to $U=0$, $1$ and $2$, respectively. (a) $N=5$ and
(b) $N=6$.
respectively. The red curves represent the energy levels for the non-
interacting ($U=0$) rings, while the green and blue lines correspond to the
energy levels for the interacting rings where the electronic correlation
strength $U$ is fixed to $1$ and $2$, respectively. From the spectra it is
observed that the ground state energy level shifts towards the positive energy
and it becomes much flatter with the increase of the correlation strength $U$.
Both for the two different ring sizes ($N=5$ and $6$) the ground state energy
levels vary periodically with AB flux $\phi$, but a significant difference is
observed in their periodicities depending on the oddness and evenness of the
ring size $N$. For $N=6$ (even), the energy levels show conventional
$\phi_{0}$ ($=1$, in our chosen unit system $c=e=h=1$) flux-quantum
periodicity. On the other hand, the period becomes half i.e., $\phi_{0}/2$ for
$N=5$ (odd). This $\phi_{0}/2$ periodicity disappears as long as the filling
is considered away from the half-filling. At the same time, it also vanishes
if impurities are introduced in the system, even if the ring is half-filled
with odd $N$. Therefore, $\phi_{0}/2$ periodicity is a special feature for odd
half-filled perfect rings irrespective of the Hubbard strength $U$, while for
all other cases traditional $\phi_{0}$ periodicity is obtained.
To judge the accuracy of the mean-field calculations in our ring geometry, in
Fig. 3 we show the variation of lowest energy levels where the eigenenergies
are determined through exact diagonalization of the full many-body Hamiltonian
for the identical rings
Figure 3: (Color online). Ground state energy levels as a function of flux
$\phi$ for some typical mesoscopic rings in half-filled case, where
eigenenergies are determined through exact diagonalization of the full many-
body Hamiltonian. The red, green and blue curves correspond to $U=0$, $1$ and
$2$, respectively. (a) $N=5$ and (b) $N=6$.
as given in Fig. 2, considering the same parameter values. Comparing the
results presented in Figs. 2 and 3, we see that the mean-field results agree
very well with the exact diagonalization method. Thus we can safely use mean-
field approach to study magnetic response in our geometry.
Current-flux characteristics: Following the above energy-flux characteristics
now we describe the behavior of persistent current in mesoscopic Hubbard
rings. As representative examples, in Fig. 4 we display the variation of
persistent currents as a function of flux $\phi$ for some typical single-
channel mesoscopic rings in the half-filled case, where (a) and (b) correspond
to $N=15$ and $20$, respectively. The red, green and blue curves in Fig. 4(a)
correspond to the currents for $U=0$, $1.5$ and $2$, respectively, while these
curves in Fig. 4(b) represent the currents for $U=0$, $1$ and $1.5$,
respectively. In the absence of any e-e interaction ($U=0$), persistent
current shows saw-tooth like nature as a function of flux $\phi$ with sharp
transitions at $n\phi_{0}/2$ (red line of Fig. 4(a)) or $n\phi_{0}$ (red line
of Fig. 4(b)), where $n$ being an integer, depending on whether $N$ is odd or
even. The saw-tooth like behavior disappears as long as the electronic
correlation is introduced into the system. This is clearly observed from the
green and blue curves of Fig. 4. Additionally, in the presence of $U$, the
current amplitude gets suppressed compared to the current amplitude in the
non-interacting case, and it decreases gradually with increasing $U$. This
provides the lowering
Figure 4: (Color online). Persistent current as a function of flux $\phi$ for
single-channel mesoscopic rings in half-filled case. (a) $N=15$. The red,
green and blue curves correspond to $U=0$, $1.5$ and $2$, respectively. (b)
$N=20$. The red, green and blue curves correspond to $U=0$, $1$ and $1.5$,
respectively.
of electron mobility with the rise of $U$ and the reason behind this can be
much better understood from our forthcoming discussion. Both for two different
rings with sizes $N=15$ (odd) and $20$ (even), persistent currents vary
periodically with AB flux $\phi$ showing different periodicities, following
the energy-flux characteristics. For $N=15$, current shows $\phi_{0}/2$ flux-
quantum periodicity, while for the other case ($N=20$), current exhibits
$\phi_{0}$ flux-quantum periodicity.
Variation of electronic mobility-Drude weight: To reveal the conducting
properties of Hubbard rings, we study the variation of Drude weight $D$ for
these systems. Drude weight can be calculated by using Eq. 5. Finite value of
$D$ predicts the metallic phase, while for the insulating phase it drops
exponentially to zero kohn .
As illustrative examples, in Fig. 5 we show the variation of Drude weight $D$
as a function of electronic correlation strength $U$ for some typical single-
channel Hubbard rings. In Fig. 5(a) the results are shown for three different
half-filled rings, where the red, green and blue lines correspond to the rings
with $N=10$, $30$ and $50$, respectively. From the curves it is evident that
for smaller values of $U$, the half-filled rings show finite value of $D$
which reveals that they are in the metallic phase. On the other hand, $D$
drops sharply to zero when $U$ becomes high. Thus the rings become insulating
when $U$ is quite large. The results for the non-half filled case are shown in
Fig. 5(b), where we fix the ring size $N=20$ and vary the electron filling.
The red, green and blue curves represent $N_{e}=10$, $14$ and $18$,
respectively, where $N_{e}$ gives the total number of electrons in the ring.
For these three choices of $N_{e}$, the ring is always less than half-filled
(since $N_{e}<N$) and the ring
Figure 5: (Color online). Drude weight as a function of Hubbard interaction
strength $U$ for single-channel mesoscopic rings. (a) Half-filled case. The
red, green and blue curves correspond to $N=10$, $30$ and $50$, respectively.
(b) Non-half-filled case with $N=20$. The red, green and blue curves
correspond to $N_{e}=10$, $14$ and $18$, respectively.
is in the conducting phase irrespective of the correlation strength $U$. Now
we try to justify the dependence of the Hubbard strength $U$ on the electronic
mobility for these different fillings. To understand the effect of $U$ on
electron mobility here we measure a quantity called ‘average spin density’
(ASD) which is defined by the factor
$\sum_{i}|(n_{i\uparrow}-n_{i\downarrow})|/N$. The integer $i$ is the site
index and it runs from $1$ to $N$. By calculating ASD we can estimate the
occupation probability of electrons in the ring and it supports us to explain
whether the ring lies in the metallic phase or in the insulating one. For the
rings those are below half-filled, ASD is always less than unity irrespective
of the value of $U$ as shown by the curves in Fig. 6(b). It reveals that for
these systems, ground state always supports an empty site and electron can
move along the ring avoiding double occupancy of two different spin electrons
at any site $i$ in the presence of e-e correlation which provides the metallic
phase ($D>0$). For a fixed ring size and a particular strength of $U$, the ASD
increases as the filling is increased towards half-filling which is noticed by
comparing the three different curves in Fig. 6(b). On the other hand, in the
half-filled rings, ASD is less than unity for small value of $U$, while it
reaches to unity when $U$ is large. This behavior is clearly shown by the
curves given in Fig. 6(a), where the red, green and blue lines correspond to
ASDs for the half-filled rings with $N=10$, $30$ and $50$, respectively. Thus,
for low $U$ there is some finite probability of getting two opposite spin
electrons in a same site which allows electrons to move along the ring and the
metallic phase is obtained. But for large $U$, ASD reaches to
Figure 6: (Color online). Average spin density (ASD) as a function of Hubbard
interaction strength $U$ for single-channel mesoscopic rings. (a) Half-filled
case. The red, green and blue curves correspond to $N=10$, $30$ and $50$,
respectively. (b) Non-half-filled case with $N=20$. The red, green and blue
curves correspond to $N_{e}=10$, $14$ and $18$, respectively.
unity which means that each site is singly occupied either by an up or down
spin electron with probability $1$. In this case ground state does not support
any empty site and due to strong repulsive e-e correlation one electron
sitting in a site does not allow to come other electron with opposite spin
from the neighboring site which provides the insulating phase ($D=0$). The
situation is somewhat analogous to Mott localization in one-dimensional
infinite lattices. In perfect Hubbard rings the conducting nature has been
studied exactly quite a long ago using the ansatz of Bethe by Shastry and
Sutherland shastry . They have calculated charge stiffness constant ($D_{c}$)
and have predicted that $D_{c}$ goes to zero as the system approaches towards
half-filling for any non-zero value of $U$. Our numerical results clearly
justify their findings.
Low-field magnetic susceptibility: Now, we discuss the variation of low-field
magnetic susceptibility which can be calculated from Eq. 6 by setting
$\phi\rightarrow 0$. With the help of this parameter we can justify whether
the current is paramagnetic ($+$ve slope) or diamagnetic ($-$ve slope) in
nature. For our illustrative purposes, in Fig. 7 we show the variation of low-
field magnetic susceptibility with system size $N$ for some typical single-
channel mesoscopic rings in the half-filled case. Figure 7(a) correspond to
the variation of low-field magnetic susceptibility for the non-interacting
($U=0$) rings,
Figure 7: (Color online). Low-field magnetic susceptibility as a function of
system size $N$ for single-channel mesoscopic rings in half-filled case. (a)
$U=0$. $N$ is an odd ($2n+1$) or an even ($2n+2$) number, where $n$ is an
integer. (b) $U=1$. $N$ is an even number obeying the relation $N=4n+2$. (c)
$U=1$. $N$ is an even number satisfying the relation $N=4n$. (d) $U=1$. $N$ is
an odd number following the relation $N=2n+1$.
where the ring size can by anything i.e., either odd, following the relation
$N=2n+1$ ($n$ is an integer), or even, obeying the expression $N=2n+2$. It is
observed that both for odd and even $N$, low-field current exhibits
diamagnetic nature. The behavior of the low-field currents changes
significantly when the e-e interaction is taken into account. Depending on the
ring size $N$, the sign becomes $+$ve and $-$ve as shown by the curves given
in Figs. 7(b)-(d). For the interacting rings where the relation $N=4n+2$ is
satisfied, the low-field current becomes diamagnetic (Fig. 7(b)). The sign
becomes paramagnetic when $N=4n$ (Fig. 7(c)) and $N=2n+1$ (Fig. 7(d)). Thus,
in brief, we say that for non-interacting half-filled rings low-field current
exhibits diamagnetic response irrespective of $N$ i.e., whether $N$ is odd or
even. For the interacting half-filled rings with odd $N$, low-field current
provides only the paramagnetic behavior, while for even $N$, depending on the
particular value of $N$, the response becomes either diamagnetic or
paramagnetic. These natures of low-field currents change for the cases of
other electron fillings. Hence, it can be emphasized that the behavior of the
low-field currents is highly sensitive on the Hubbard correlation, electron
filling, evenness and oddness of $N$, etc. The behavior of zero-field magnetic
susceptibility in Hubbard rings has been studied extensively quite a long back
using the Bethe ansatz by Shiba shiba . In this work, he has studied magnetic
susceptibility per electron as functions of electron filling and Hubbard
correlation strength and provided several interesting results. From his
findings we can clearly justify our presented results.
#### II.2.2 Disordered Hubbard Rings Described with NNH and SNH Integrals
Now, we explore the combined effect of electron-electron correlation and
second-neighbor hopping (SNH) integral on persistent current in disordered
mesoscopic rings.
To get a disordered ring, we choose site energies ($\epsilon_{i\uparrow}$ and
$\epsilon_{i\downarrow}$) randomly from a “Box” distribution function of width
$W$. As the site energies are chosen randomly it is needed to consider the
average over a large number of disordered configurations (from the stand point
of statistical average). Here, we determine the currents by taking the average
over $50$ random disordered configuration in each case to achieve much
accurate results.
As illustrative examples, in Fig. 8 we display the variation of persistent
currents for some single-channel mesoscopic rings considering $1/3$ electron
filling. In (a) the results are given for the rings characterized by the NNH
integral model. The red curve represents the current for the ordered ($W=0$)
non-interacting ($U=0$) ring. It shows saw-tooth like nature with AB flux
$\phi$ providing $\phi_{0}$ flux-quantum periodicity. The situation becomes
completely different when impurities are introduced in the ring as clearly
seen by the other two colored curves. The green curve represents the current
for the case only when impurities are considered but the effect of Hubbard
interaction is not taken into account. It varies continuously with $\phi$ and
gets much reduced amplitude, even an order of magnitude, compared to the
perfect case. This is due to the localization of the energy eigenstates in the
presence of impurity, which is the so-called Anderson localization. Hence, a
large difference exists between the current amplitudes of an ordered and
disordered non-interacting rings and it was the main controversial issue among
the theoretical and experimental predictions. Experimental results suggest
that the measured current amplitude is quite comparable to the theoretically
estimated current amplitude in a perfect system. To remove this controversy,
as a first attempt, we include the effect of Hubbard interaction in the
disordered ring described by the NNH model. The result is shown by the blue
curve where $U$ is fixed at $0.5$. It is observed that the current amplitude
gets increased compared to the non-interacting disordered ring, though the
increment is too small. Not only that the enhancement can take place only for
small values of $U$, while for large enough $U$ the current amplitude rather
decreases. This phenomenon can be explained as follows. For the non-
interacting disordered ring the probability of getting two opposite spin
electrons becomes higher at the atomic sites where the site energies are lower
than the other sites since the electrons get pinned at the lower site energies
to minimize the ground state energy, and this pinning of electrons becomes
increased with the rise of impurity strength $W$. As a result the mobility of
electrons and hence the current amplitude gets reduced with the increase of
impurity strength $W$. Now, if we introduce electronic correlation in the
system then it tries to depin two opposite spin electrons those are situated
together due to the Coulomb repulsion. Therefore, the electronic mobility is
enhanced which provides quite larger current amplitude. But,
Figure 8: (Color online). Persistent current as a function of flux $\phi$ for
single-channel mesoscopic rings with $N=15$ considering $1/3$ electron
filling. (a) Rings with only NNH integral. The red line corresponds to the
ordered non-interacting ring, while the green and blue lines correspond to the
disordered ($W=2$) rings with $U=0$ and $0.5$, respectively. (b) Rings with
NNH and SNH integrals. The red line represents the ordered non-interacting
ring, whereas the green and blue line correspond to the disordered ($W=2$)
rings with $U=0$ and $1.5$, respectively.
for large enough interaction strength, mobility of electrons gradually
decreases due to the strong repulsive interaction. Accordingly, the current
amplitude gradually decreases with $U$. So, in short, we can say that within
the nearest-neighbor hopping (NNH) model electron-electron interaction does
not provide any significant contribution to enhance the current amplitude, and
hence the controversy regarding the current amplitude still persists.
To overcome this controversy, finally we make an attempt by incorporating the
effect of second-neighbor hopping (SNH) integral in addition to the nearest-
neighbor hopping (NNH) integral. With this modification a significant change
in current amplitude takes place which is clearly observed from Fig. 8(b). The
red curve refers to the current for the perfect ($W=0$) non-interacting
($U=0$) ring and it achieves much higher amplitude compared to the NNH model
(see red curve of Fig. 8(a)). This additional contribution comes from the SNH
integral since it allows electrons to hop further. The main focus of this sub-
section is to interpret the combined effect of SNH integral and Hubbard
correlation on the enhancement of persistent current in disordered ring. To do
this first we narrate the effect of SNH integral in disordered non-interacting
ring. The nature of the current for this particular case is shown by the green
curve of Fig. 8(b). It shows that the current amplitude gets reduced compared
to the perfect case (red line), which is expected, but the reduction of the
current amplitude is very small than the NNH integral model (see green curve
of Fig. 8(a)). This is due the fact that the SNH integral tries to delocalize
the electronic states, and therefore, the mobility of the electrons is
enriched. The situation becomes more interesting when we include the effect of
Hubbard interaction. The behavior of the current in the presence of
interaction is plotted by the blue curve of Fig. 8(b) where we fix $U=1.5$.
Very interestingly we see that the current amplitude is enhanced moderately
and quite comparable to that of the perfect ring. Therefore, it can be
predicted that the presence of SNH integral and Hubbard interaction can
provide a persistent current which may be comparable to the measured current
amplitudes. In the above analysis we consider the effect of only SNH integral
in addition to the NNH model, and, illustrate how such a higher order hopping
integral leads to an important role on the enhancement of current amplitude in
presence of Hubbard correlation for disordered rings. Instead of considering
only the SNH integral we can also take the contributions from all possible
higher order hopping integrals with reduced hopping strengths. Since the
strengths of other higher order hopping integrals are too small, the
contributions from these factors are reasonably small and they will not
provide any significant change in the current amplitude.
## III A Hubbard cylinder in absence of SO interactions
In this section we extend our discussion for an interacting mesoscopic ring
with finite width threaded by an AB flux $\phi$. Here also we ignore the
effect of SO interaction on magneto-transport properties like the previous
section.
### III.1 Model and the Hamiltonian
Let us start by referring to Fig. 9, where a small metallic cylinder is
threaded by a magnetic flux $\phi$. The filled black circles correspond to the
positions of the atomic sites in the cylinder. To predict the size of a
cylinder we use two parameters $N$ and $M$, where the $1$st one ($N$)
represents total number of atomic sites in each circular ring and the other
one ($M$) gives total number of identical circular rings. For the description
of our model quantum system we use a tight-binding framework and in order to
incorporate the effect of higher order hopping integrals to the Hamiltonian
here we consider second-neighbor hopping (SNH) (shown by the red dashed line
in Fig. 9) in
Figure 9: (Color online). Schematic view of a $1$D mesoscopic cylinder
penetrated by a magnetic flux $\phi$. The red dashed line corresponds to the
second-neighbor hopping integral and the filled black circles represent the
positions of the atomic sites. A persistent current $I$ is established in the
cylinder.
addition to the nearest-neighbor hopping (NNH) of electrons. Considering both
NNH and SNH integrals the TB Hamiltonian for the cylindrical system in Wannier
basis looks in the form,
$\displaystyle H_{\mbox{c}}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\sigma}\epsilon_{i,j,\sigma}c_{i,j,\sigma}^{\dagger}c_{i,j,\sigma}+\sum_{i,j,\sigma}t_{l}\left[e^{i\theta_{l}}c_{i,j,\sigma}^{\dagger}c_{i,j+1,\sigma}\right.$
(7) $\displaystyle+$
$\displaystyle\left.h.c.\right]+\sum_{i,j,\sigma}t_{d}\left[e^{i\theta_{d}}c_{i,j,\sigma}^{\dagger}c_{i+1,j+1,\sigma}+h.c.\right]$
$\displaystyle+$
$\displaystyle\sum_{ij}Uc_{i,j,\uparrow}^{\dagger}c_{i,j,\uparrow}c_{i,j,\downarrow}^{\dagger}c_{i,j,\downarrow}$
where, ($i,j$) represent the co-ordinate of a lattice site. The index $i$ runs
from $1$ to $M$, while the integer $j$ goes from $1$ to $N$.
$\epsilon_{i,j,\sigma}$ is the on-site energy of an electron at the site
($i,j$) of spin $\sigma$ ($\uparrow,\downarrow$). $t_{l}$ and $t_{d}$ are the
NNH and SNH integrals, respectively. Due to the presence of magnetic flux
$\phi$ (measured in unit of the elementary flux quantum $\phi_{0}=ch/e$), a
phase factor $\theta_{l}=2\pi\phi/N$ appears in the Hamiltonian when an
electron hops longitudinally from one site to its neighboring site, and
accordingly, a negative sign comes when the electron hops in the reverse
direction. $\theta_{d}$ is the associated phase factor for the diagonal motion
of an electron between two neighboring concentric rings. No phase factor
appears when an electron moves along the vertical direction which is set by
proper choice of the gauge for the vector potential $\vec{A}$ associated with
the magnetic field $\vec{B}$, and this choice makes the phase factors
($\theta_{l}$, $\theta_{d}$) identical to each other for the longitudinal and
diagonal motions. Since the magnetic field corresponding to the AB flux $\phi$
does not penetrate anywhere of the surface of the cylinder, we ignore Zeeman
term in the above tight-binding Hamiltonian (Eq. 7).
$c_{i,j,\sigma}^{\dagger}$ and $c_{i,j,\sigma}$ are the creation and
annihilation operators, respectively, of an electron at the site ($i,j$) with
spin $\sigma$. $U$ is the on-site Hubbard interaction term.
### III.2 Theoretical formulation
To calculate energy eigenvalues, persistent current and related issues here we
follow exactly the same prescription which we illustrate in the earlier
section (Sec. II) i.e., Hartree-Fock mean-field approach.
### III.3 Numerical results and discussion
Throughout the numerical analysis, in this sub-section, we set the nearest-
neighbor hopping strength $t_{l}=-1$ and fix $M=2$ i.e., cylinders with two
identical rings. Energy scale is measured in the unit of $t_{l}$. We describe
the results in three different parts. In the first part, we consider perfect
cylinders with only nearest-neighbor hopping integral. In the second part,
disordered cylinders described with only NNH integral are considered. Finally,
in the third part we discuss the effect of second-neighbor hopping (SNH)
integral on the enhancement of persistent current in disordered cylinders.
#### III.3.1 Perfect cylinders with NNH integral
For perfect cylinders we choose
$\epsilon_{i,j,\uparrow}=\epsilon_{i,j,\downarrow}=0$ for all ($i,j$). Since
here we consider the cylinders described with NNH integral only, the second-
neighbor hopping strength $t_{d}$ is fixed to zero.
Energy-flux characteristics: As illustrative examples, in Fig. 10 we show the
variation of ground state energy levels as a function of magnetic flux $\phi$
for some typical mesoscopic cylinders where $N$ is fixed at $5$ (odd $N$). In
(a) the results are given for the quarterly-filled ($N_{e}=5$) cylinders,
while in (b) the curves correspond to the results for the half-filled
($N_{e}=10$) cylinders. The red, green and blue lines represent the ground
state energy levels for $U=0$, $0.5$ and $1$, respectively. It is observed
that the ground state energy shows oscillatory behavior as a function of
$\phi$ and the energy increases as the electronic correlation strength $U$
gets increased. Most significantly we see that the ground state energy levels
provide two different types of periodicities depending on the electron
filling. At quarter-filling, ground state energy level gives $\phi_{0}$ ($=1$,
since $c=e=h=1$ in our chosen unit system) flux-quantum periodicity. On the
other hand, at half-filling it shows $\phi_{0}/2$ flux-quantum periodicity.
The situation becomes quite different when the total number of atomic sites
$N$ in individual rings is even. For our illustrative purposes in Fig. 11 we
plot the lowest energy levels as a function of $\phi$ for some typical
mesoscopic cylinders considering $N=8$ (even $N$). The curves of different
colors correspond to the identical meaning as in Fig. 10. From the spectra
given in Figs. 11(a) (quarter-filled case) and (b) (half-filled case) it is
clearly observed that the ground state energy levels vary
Figure 10: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders with $N=5$ and $M=2$. The red, green and
blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.
Figure 11: (Color online). Ground state energy levels as a function of flux
$\phi$ for some perfect cylinders considering $N=8$ and $M=2$. The red, green
and blue curves correspond to $U=0$, $0.5$ and $1$, respectively. (a) Quarter-
filled case and (b) Half-filled case.
periodically with AB flux $\phi$ exhibiting only $\phi_{0}$ flux-quantum
periodicity. Thus it can be emphasized that the appearance of half flux-
quantum periodicity strongly depends on the electron filling as well as on the
oddness and evenness of the total number of atomic sites $N$ in individual
rings. Only for the half-filled cylinders with odd $N$, the lowest energy
level gets $\phi_{0}/2$ periodicity with flux $\phi$. Now it is important to
note that this half flux-quantum periodicity does not depend on the width
($M$) of the cylinder and also it is independent of the Hubbard correlation
strength $U$. Hence, depending on the system size and filling of electrons
variable periodicities are observed in the variation of lowest energy level.
It may provide an important signature in studying magnetic response in nano-
scale loop geometries.
Current-flux characteristics: In Fig. 12 we display the current-flux
characteristics for some impurity free mesoscopic cylinders considering $M=2$.
In (a) the
Figure 12: (Color online). Persistent current as a function of flux $\phi$ for
some ordered mesoscopic cylinders considering $M=2$. (a) Half-filled case with
$N=15$. The red, green and blue curves correspond to $U=0$, $1.5$ and $2$,
respectively. (b) Quarter-filled case with $N=20$. The red, green and blue
curves correspond to $U=0$, $2$ and $3$, respectively.
results are given for the half-filled case where we set $N=15$. The red line
corresponds to the current for the non-interacting ($U=0$) case, while the
green and blue lines represent the currents when $U=1.5$ and $2$,
respectively. From the curves we notice that the current amplitude gradually
decreases with the increase of electronic correlation strength $U$. The reason
is that at half-filling each site is occupied by at least one electron of up
spin or down spin, and the placing of a second electron of opposite spin needs
more energy due to the repulsive effect of $U$. Thus conduction becomes
difficult as it requires more energy when an electron hops from its own site
and situates at the neighboring site. Now both for the non-interacting and
interacting cases, current shows half flux-quantum periodicity as a function a
$\phi$ obeying the energy-flux characteristics since here we choose odd $N$
($N=15$). The behavior of the persistent currents for even $N$ is shown in (b)
where we set $N=20$. The currents are drawn for the quarter-filled case i.e.,
$N_{e}=20$, where the red, green and blue curves correspond to $U=0$, $2$ and
$3$, respectively. The reduction of current amplitude with the increase of
Hubbard interaction strength is also observed for this quarter-filled case,
similar to the case of half-filled as described earlier. But the point is that
at quarter-filling, the reduction of current amplitude is much smaller
compared to the half-filled situation. This is quite obvious in the sense that
at less than half-filling ‘empty’ lattice sites are available where electrons
can hop easily without any cost of extra energy and the conduction becomes
much easier than the half-filled situation. In this quarter-filled case,
persistent currents provide only $\phi_{0}$ flux-quantum periodicity following
the $E$-$\phi$ diagram. From these current-flux characteristics it can be
concluded that for ‘ordered’ cylinders current amplitude always decreases with
the enhancement in Hubbard correlation strength $U$.
#### III.3.2 Disordered cylinders with NNH integral
In order to describe the effect of impurities on electron transport now we
focus our attention on the results of some typical disordered cylinders
described with NNH integral. Here we consider the diagonal disordered
Figure 13: (Color online). Ground state energy level as a function of flux
$\phi$ for half-filled disordered mesoscopic cylinders ($M=2$) considering
$U=1$ and $W=2$. (a) $N=5$ and (b) $N=8$.
cylinders i.e., impurities are introduced only at the site energies without
disturbing the hopping integrals. The site energies in each concentric ring
are chosen from a correlated distribution function which looks in the form,
$\epsilon_{j,\uparrow}=\epsilon_{j,\downarrow}=W\cos\left(j\lambda\pi\right)$
(8)
where, $W$ is the impurity strength. $\lambda$ is an irrational number and we
choose $\lambda=(1+\sqrt{5})/2$, for the sake of our illustration. Setting
$\lambda=0$, we get back the pure system with uniform site energy $W$. Now,
instead of considering site energies from a correlated distribution function,
as mentioned above in Eq. 8, we can also take them randomly from a “Box”
distribution function of width $W$. But in the later case we have to take the
average over a large number of disordered configurations (from the stand point
of statistical average) and since it is really a difficult task in the aspect
of numerical computation we select the other option. Not only that in the
averaging process several mesoscopic phenomena may disappear. Therefore, the
averaging process is an important issue in low-dimensional systems.
In presence of disorder, energy levels get modified significantly. For our
illustrative purposes in Fig. 13 we plot ground state energy levels as a
function of magnetic flux $\phi$ for some disordered mesoscopic cylinders when
they are half-filled. The Hubbard interaction strength $U$ is set at $1$ and
the impurity strength $W$ is fixed to $2$. In (a) the ground state energy
level is shown for a cylinder with $N=5$ (odd), while in (b) it is presented
for a cylinder taking $N=8$ (even). Quite interestingly we see that for the
cylinder with odd $N$, the half flux-quantum periodicity of the lowest energy
level disappears in the presence of impurity and it provides conventional
$\phi_{0}$ periodicity. Hence, for cylinders with odd $N$, $\phi_{0}/2$ flux-
quantum periodicity will be observed only when they are free from any
impurity. For the disordered cylinder with even $N$ ($N=8$), the lowest energy
level as usual provides $\phi_{0}$ periodicity similar to the impurity free
cylinders containing even $N$. Apart from this periodic nature, impurities
play another significant role in the determination of the slope of the energy
levels. The slope of the lowest energy level decreases significantly compared
to the perfect case, and therefore, a prominent change in current amplitude
also takes place.
To justify the above facts, in Fig. 14 we present the variations of persistent
currents with AB flux $\phi$ for a half-filled mesoscopic cylinder, described
in the framework of NNH model, considering $N=15$ and $M=2$. The red curve
represents the current for the ordered ($W=0$) non-interacting ($U=0$)
cylinder. It shows saw-tooth like nature with flux $\phi$ providing
$\phi_{0}/2$ flux-quantum periodicity. The situation becomes completely
different when impurities are introduced in the cylinder as seen by the other
two curves. The green curve represents the current for the case only when
impurities are considered but the effect of electronic correlation is not
taken into account. It shows a continuous like nature with $\phi_{0}$ flux-
quantum periodicity. The most important observation is that the current
amplitude gets reduced enormously, even an order of magnitude, compared to the
perfect cylinder. This is due to the localization of the energy eigenstates in
the presence of impurity, which is the so-called Anderson localization. Hence,
a large difference exists in the current amplitudes of an ordered and
disordered non-interacting cylinders and it was the main controversial issue
among the theoretical and experimental predictions. Experimental verifications
suggest that the measured current amplitude is quite comparable to the
theoretical current amplitude obtained in a perfect system. To remove this
controversy, as a first attempt, we include the effect of e-e correlation in
the disordered cylinder described by the NNH model. The result is shown by the
blue curve where $U$ is fixed at $1.5$. It is observed that the current
amplitude gets increased compared to the non-interacting disordered cylinder,
though the increment is too small. Not only that the enhancement can take
place only for small values of $U$, while for large enough $U$ the current
amplitude rather decreases. This phenomenon can be implemented as follows.
Figure 14: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder considering $N=15$ and $M=2$. The red line
corresponds to the ordered case when $U=0$, whereas the green and blue lines
correspond to the disordered case ($W=2$) when $U=0$ and $1.5$, respectively.
For the non-interacting disordered cylinder the probability of getting two
opposite spin electrons becomes higher at the atomic sites where the site
energies are lower than the other sites since the electrons get pinned at the
lower site energies to minimize the ground state energy, and this pinning of
electrons becomes increased with the rise of impurity strength $W$. As a
result the mobility of electrons and hence the current amplitude gets reduced
with the increase of impurity strength $W$. Now, if we introduce electronic
correlation in the system then it tries to depin two opposite spin electrons
those are situated together due to the Coulomb repulsion. Therefore, the
electronic mobility is enhanced which provides larger current amplitude. But,
for large enough interaction strength, no electron can able to hop from one
site to other at the half-filling since then each site is occupied either by
an up or down spin electron which does not allow other electron of opposite
spin due to the repulsive term $U$. Accordingly, the current amplitude
gradually decreases with $U$. On the other hand, at less than half-filling
though there is some finite probability to hop an electron from one site to
the other available ‘empty’ site but still it is very small. So, in brief, we
can say that within the nearest-neighbor hopping (NNH) approximation electron-
electron interaction does not provide any significant contribution to enhance
the current amplitude, and hence the controversy regarding the current
amplitude still persists.
#### III.3.3 Disordered cylinders with NNH and SNH integrals
To overcome the existing situation regarding the current amplitude, in this
sub-section, finally we make an attempt by incorporating the effect of second-
neighbor hopping (SNH) integral in addition to the nearest-neighbor hopping
(NNH) integral.
A significant change in current amplitude takes place when we include the
contribution of second-neighbor hopping (SNH) integral in addition to the NNH
integral. As representative examples, in Fig. 15 we plot the current-flux
characteristics for a half-filled mesoscopic cylinder considering $N=15$ and
$M=2$. The black,
Figure 15: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.6$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 14) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.
magenta and gold lines correspond to the results in the presence of SNH
integral, while the other three colored curves (red, green and blue) represent
the currents in the absence of SNH integral. Here we choose $t_{d}=-0.6$. The
black curve refers to the persistent current for the perfect ($W=0$) non-
interacting ($U=0$) cylinder and it achieves much higher amplitude compared to
the NNH model (red curve). This additional contribution comes from the SNH
integral since it allows electrons to hop further. In addition it is also
noticed that the current varies periodically with $\phi$ providing $\phi_{0}$
flux-quantum periodicity, instead of $\phi_{0}/2$ as in the case of NNH
integral model (red curve). Thus, it can be emphasized that $\phi_{0}/2$
periodicity will be observed only when the cylinder is (a) free from impurity,
(b) half-filled, (c) made with odd $N$, and (d) described by the nearest-
neighbor hopping model. The main focus of this sub-section is to interpret the
combined effect of SNH integral and electron-electron correlation on the
enhancement of persistent current amplitude in disordered cylinder. To do this
first we narrate the effect of SNH integral in disordered non-interacting
cylinder. The nature of the current for this particular case is shown by the
magenta curve of Fig. 15. It shows that the current amplitude gets reduced
compared to the perfect case (black line), which is expected, but the
reduction of the current amplitude is quite small than the NNH integral model.
This is due the fact that the SNH integral tries to delocalize the electronic
states, and therefore, the mobility of the electrons is enriched. The
situation becomes more interesting when we include the effect of Hubbard
interaction. The behavior of the current in the presence of interaction
Figure 16: (Color online). Persistent current as a function of flux $\phi$ for
a half-filled mesoscopic cylinder taking $N=15$ and $M=2$ in the presence of
NNH and SNH integrals. The black line corresponds to the ordered case when
$U=0$, whereas the magenta and gold lines correspond to the disordered case
($W=2$) when $U=0$ and $1.5$, respectively. Here SNH integral is fixed at
$-0.8$. The currents shown by the red, green and blue lines for the ring
described with NNH model (identical to Fig. 14) are re-plotted to judge the
effect of SNH integral over NNH model much clearly.
is plotted by the gold curve of Fig. 15 where we fix $U=1.5$. Very
interestingly we see that the current amplitude is enhanced significantly and
quite comparable to that of the perfect cylinder.
For better clarity of the results discussed above, in Fig. 16 we also present
the similar feature of persistent current for other hopping strength of SNH
integral. Here we set $t_{d}=-0.8$. From these curves we see that the current
amplitude gets enhanced more as we increase the SNH strength.
Thus, it can be emphasized that the presence of higher order hopping integrals
and electron-electron correlation may provide a persistent current which can
be comparable to the measured current amplitudes. Throughout the above
analysis we set the width of the cylinders at a fixed value ($M=2$), for the
sake of our illustration. But, all these results are also valid for cylinders
of larger widths.
## IV A mesoscopic ring with Rashba and Dresselhaus SO interactions
Finally, in this section we address the magneto-transport properties in a
mesoscopic ring, threaded by an AB flux $\phi$, in the presence of Rashba and
Dresselhaus SO interactions. We establish that the presence of SO interaction,
in general, leads to an enhanced amplitude of the persistent current. This is
another one approach through which we can justify the appearance of larger
current amplitude in a disordered ring. For this discussion we neglect the
effect of electron-electron interaction.
### IV.1 Model, TB Hamiltonian and the theoretical formulation
The schematic view of a mesoscopic ring subjected to an AB flux $\phi$
(measured in unit of the elementary flux quantum $\phi_{0}=ch/e$) is shown in
Fig. 17.
Figure 17: (Color online). A mesoscopic ring threaded by an AB flux $\phi$.
Within a TB framework the Hamiltonian for such an $N$-site ring is sheng ;
splett ; moca (and references therein),
$H=H_{0}+H_{so}.$ (9)
Here,
$H_{0}=\sum_{n}\mbox{\boldmath$c_{n}^{\dagger}\epsilon_{0}c_{n}$}+\sum_{n}\left(\mbox{\boldmath$c_{n}^{\dagger}t$}\,e^{i\theta}\mbox{\boldmath$c_{n+1}$}+h.c.\right)$
(10)
and,
$H_{so}=-\sum_{n}\left[\mbox{\boldmath$c_{n}^{{\dagger}}t_{so}$}e^{i\theta}\mbox{\boldmath$c_{n+1}^{{\dagger}}$}+h.c.\right]$
(11)
where,
$t_{so}$ $\displaystyle=$ $\displaystyle
it_{Rso}\left(\mbox{\boldmath$\sigma_{x}$}\cos\varphi_{n,n+1}+\mbox{\boldmath$\sigma_{y}$}\sin\varphi_{n,n+1}\right)$
(12) $\displaystyle-$ $\displaystyle
it_{Dso}\left(\mbox{\boldmath$\sigma_{y}$}\cos\varphi_{n,n+1}+\mbox{\boldmath$\sigma_{x}$}\sin\varphi_{n,n+1}\right).$
$n=1$, $2$, $\dots$, $N$ is the site index along the azimuthal direction
$\varphi$ of the ring. The other factors in Eqs. 10 and 12 are as follows.
$c_{n}$=$\left(\begin{array}[]{c}c_{n\uparrow}\\\
c_{n\downarrow}\end{array}\right);$
$\epsilon_{0}$=$\left(\begin{array}[]{cc}\epsilon_{0}&0\\\
0&\epsilon_{0}\end{array}\right);$ $t$=$t\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$.
Here $\epsilon_{0}$ is the site energy of each atomic site of the ring. $t$ is
the nearest-neighbor hopping integral and $\theta=2\pi\phi/N$ is the phase
factor due to the AB flux $\phi$ threaded by the ring. $t_{Rso}$ and $t_{Dso}$
are the isotropic nearest-neighbor transfer integrals which measure the
strengths of Rashba and Dresselhaus SO couplings, respectively, and
$\varphi_{n,n+1}=\left(\varphi_{n}+\varphi_{n+1}\right)/2$, where
$\varphi_{n}=2\pi(n-1)/N$. $\sigma_{x}$ and $\sigma_{y}$ are the Pauli spin
matrices. $c_{n\sigma}^{\dagger}$ ($c_{n\sigma}$) is the creation
(annihilation) operator of an electron at the site $n$ with spin $\sigma$
($\uparrow,\downarrow$). Throughout the numerical analysis, in this sub-
section, we choose $t=1$ and measure the SO coupling strength in unit of $t$.
At absolute zero temperature ($T=0\,$K), the persistent current in the ring
described with fixed number of electrons $N_{e}$ is determined by,
$I\left(\phi\right)=-c\frac{\partial E_{0}(\phi)}{\partial\phi}$ (13)
where, $E_{0}(\phi)$ is the ground state energy. We compute this quantity by
exactly diagonalizing the TB Hamiltonian (Eq. 9) to understand unambiguously
the role of the RSOI interaction alone on persistent current.
### IV.2 Numerical results and discussion
Energy-flux characteristics: Before presenting the results for $I(\phi)$, to
make the present communication a self contained study, we first take a look at
the energy
Figure 18: (Color online). $E$-$\phi$ curves of a $12$-site ring, where the
$1$st and $2$nd columns correspond to the results for the ordered ($W=0$) and
disordered ($W=1$) cases, respectively. The red, green and blue lines
correspond to $t_{Rso}=t_{Dso}=0$; $t_{Rso}=1$, $t_{Dso}=0$ and $t_{Rso}=1$,
$t_{Dso}=0.5$, respectively.
spectrum of both an ordered and a disordered ring with and without the SO
interactions, as the flux through the ring is varied. In Fig. 18 the flux
dependent spectra are shown for a $12$-site ordered ring and a randomly
disordered one (with diagonal disorder) in the left and the right columns
respectively. Clearly, disorder destroys the band crossings observed in the
ordered case. The presence of the RSOI and the DSOI also lifts the degeneracy
and opens up gaps towards the edges of the spectrum.
Enhancement of persistent current:
$\bullet$ An ordered ring: In Fig. 19 we examine the effect of the RSOI on the
persistent current of an ordered ring with $80$ sites. The DSOI is set equal
to zero. We have examined both the non-half-filled and half-filled band cases,
but present results for the latter only to save space. With increasing
strength
Figure 19: (Color online). Current-flux characteristics of a $80$-site ordered
($W=0$) half-filled ring for different values of $t_{Rso}$ when $t_{Dso}$ is
set at $0$.
of the RSOI the persistent current exhibits a trend of an increase in its
amplitude. Local phase reversals take place together with the appearance of
kinks in the current-flux diagrams which are however, not unexpected even
without the RSOI, and are results of the band crossings observed in the
spectra of such rings. The amplitude of the persistent current at a specific
value of the magnetic flux is of course not predictable in any simple manner,
and is found to be highly
Figure 20: (Color online). Persistent current at a particular AB flux
($\phi=0.25$) as a function Rashba SO interaction strength for an ordered
($W=0$) half-filled ring with $N=60$ when $t_{Dso}$ is set to zero.
sensitive to the number of electrons $N_{e}$ (i.e., the filling factor).
Issues related to the dependence of the persistent current on the filling
factor have been elaborately discussed by Splettstoesser et al. splett .
The persistent current in an ordered ring also exhibits interesting
oscillations in its amplitude as the RSOI is varied keeping the magnetic flux
fixed at a particular value. The oscillations persist irrespective of the
band-filling factor $N_{e}$, with or without the presence of the DSOI. In Fig.
20 the oscillating nature of persistent current is presented for a $60$-site
ordered ring in the half-filled band case when $\phi$ is set at $\phi_{0}/4$.
The current exhibits oscillations with growing amplitude as the strength of
the RSOI is increased.
$\bullet$ A disordered ring: We now present the results for a disordered ring
of $80$ sites in Fig. 21. Disorder is introduced via a random distribution
(width $W=2$) of the values of the on-site potentials (diagonal disorder), and
results averaged over sixty disorder configurations have been presented. The
DSOI remains
Figure 21: (Color online). Current-flux characteristics of a $80$-site
disordered ($W=2$) half-filled ring for different values of $t_{Rso}$ when
$t_{Dso}$ is fixed at $0$.
zero. Without any spin-orbit interaction, disorder completely suppresses the
persistent current (an effect of the localization of the electronic states in
the ring), as it is observed in Fig. 21 (black curve). With the introduction
of the RSOI, the current starts increasing, and for $t_{Rso}=3$ (blue curve),
increases significantly,
Figure 22: (Color online). Persistent current at a particular AB flux
($\phi=0.25$) as a function Rashba SO interaction strength for a disordered
($W=2$) half-filled ring with $N=60$ when $t_{Dso}$ is set to zero.
attaining a magnitude comparable to that in a perfectly ordered ring. It is to
be noted that the strength of the RSOI is strongly dependent on gate voltage.
An enhancement of the persistent current in the presence of disorder can be
achieved even with much lower values of the RSOI parameter compared to what
have already been presented in the figures. To achieve this one needs to
increase the size of the mesoscopic ring. We have checked this with a
$100$-site ring where even with $t_{Rso}=0.5$ the current increases by an
order of magnitude compared to the case when $t_{Rso}=0$. However, we present
the results using a somewhat larger values of $t_{Rso}$ for a better viewing
of the results. Similar observations are made by setting $t_{Rso}=0$ and
varying $t_{Dso}$.
Disorder introduces quantum interference which leads to localization of the
electronic states. RSOI, on the other hand, introduces spin flip scattering in
the system, which can destroy quantum interference effect, leading to a
possible delocalization of the electronic states. This leads to an enhancement
of the persistent current in the presence of disorder. The competition between
the strength of disorder and the RSOI is also apparent in Fig. 22. For small
values of the RSOI, the disorder dominates. As the strength of the RSOI is
increased, the spin flip scattering starts dominating over the quantum
interference effect, and finally the oscillations become quite similar to that
in a ballistic ring. As the SO interaction is a natural interaction for a
quantum ring grafted at a heterojunction, we are thus tempted to propose that
the spin-orbit interaction is responsible for an enhanced persistent current
in such mesoscopic disordered rings.
Before we end this section, we would like to mention that the presence of DSOI
alone leads to exactly similar results as expected, since the Rashba and the
Dresselhaus Hamiltonians are related by a unitary transformation. This does
not change the physics. We also examine the behavior of persistent current in
presence of both the interactions. The amplitude of the current does not
increase significantly compared to the case where only one interaction is
present. However, the precise magnitude of the current is sensitive to the
strength of the magnetic flux threading the ring. The observation remains
valid even when the strengths of the RSOI and DSOI are the same.
## V Summary and Conclusions
In this review we have demonstrated the magneto-transport properties in
single-channel rings and multi-channel cylinders based on the tight-binding
framework. Several anomalies between the theoretical and experimental results
have been pointed out and we have tried to remove some of these discrepancies.
The main controversy is associated with the proper determination of persistent
current amplitude. We have addressed two different possibilities of getting
enhanced current amplitudes and proved that the currents are quite comparable
to the experimentally predicted results. In one approach we have concluded
after an exhaustive numerical calculation that the presence of higher order
hopping integrals and electron-electron interaction can provide a persistent
current which may be comparable to the actual measured values. We have
justified these results both for the single-channel and multi-channel cases.
In other approach we have established that in presence of spin-orbit
interaction a considerable enhancement of current amplitude takes place, even
for the non-interacting nearest-neighbor hoping model, and the magnitude of
the current in a disordered ring becomes almost comparable to that of an
ordered one. In addition to these, we have also analyzed the detailed band
structures, low-field magnetic susceptibilities, variation of electronic
mobility and some other issues to get the full picture of the phenomena at the
meso-scale and nano-scale levels.
Here we have considered several important approximations by ignoring the
effects of temperature, electron-phonon interaction, etc. Due to these
factors, any scattering process that appears in the systems would have
influence on electronic phases. At the end, we would like to say that we need
further study in such systems by incorporating all these effects.
Future directions and opportunities: Although the studies involving the
mesoscopic rings and cylinders have already generated a wealth of literature
there is still need to look deeper into the problems both from the point of
view of fundamental physics and to resolve a few issues that have not yet been
answered in an uncontroversial manner. For example, it may be interesting to
study the combined effect of electron-electron interaction and spin-orbit
interaction on magneto-transport properties. Specially, the effect of SO
interaction on electron transport in mesoscopic and nano-scale semiconductor
structures should be carefully examined. The principal reason is its potential
application in spintronics, where the possibility of manipulating and
controlling the spin of the electron rather than its charge, plays the all
important role zutic ; ding ; bellucci ; san9 .
## References
* (1) Y. Imry. Introduction to Mesoscopic Physics. Oxford University Press, New York (1997).
* (2) S. Datta, Electronic transport in mesoscopic systems, Cambridge University Press, Cambridge (1995).
* (3) S. Datta, Quantum Transport: Atom to Transistor, Cambridge University Press, Cambridge (2005).
* (4) G. Metalidis, Electronic Transport in Mesoscopic Systems. PhD thesis, Martin-Luther-Universität Halle-Wittenberg (2007).
* (5) D. Weinmann, The Physics of Mesoscopic Systems, (2005), based on the Lectures held during the Seventh Petra School of Physics during 17-22 September 2000 at the University of Jordon.
* (6) A. Aharony, O. Entin-Wohlman, Y. Tokura, and S. Katsumoto, Phys. Rev. B 78, 125328 (2008).
* (7) A. Aharony, O. Entin-Wohlman, Y. Tokura, and S. Katsumoto, Proceedings, FQMT08.
* (8) A. Rodriguez, F. Dominguez-Adame, I. Gomez, and P. A. Orellana, Phys. Lett. A 320, 242 (2003).
* (9) I. Gomez and M. L. Ladrón de Guevara, Phys. Rev. B 67, 085321 (2003).
* (10) A. Levy Yeyati and M. Büttiker, Phys. Rev. B 52, R14360 (1995).
* (11) M. E. Torio, K. Hallberg, S. Flach, A. E. Miroshnichenko, and M. Titov, Eur. Phys. J. B 37, 399 (2004).
* (12) E. Buks, R. Schuster, M. Heiblum, D. Mahalu, V. Umansky, and H. Shtrikman, Phys. Rev. Lett. 77, 4664 (1996).
* (13) A. Fuhrer, P. Brushein, T. Ihn, M. Sigrist, K. Ensslin, W. Wegscheider, and M. Bichler, Phys. Rev. B 73, 205326 (2006).
* (14) K. Kobayashi, H. Aikawa, A. Sano, S. Katsumoto, and Y. Iye, Phys. Rev. B 70, 035319 (2004).
* (15) M. Popp, D. Frustaglia, and K. Richter, Nanotechnology 14, 347 (2003).
* (16) P. Földi, O. Kalman, M. G. Benedict, and F. M. Peeters, Nano Lett. 8, 2556 (2008),
* (17) D. Bercioux, M. Governale, V. Cataudella, and V. M. Ramaglia, Phys. Rev. Lett. 93, 056802 (2004).
* (18) D. Bercioux, M. Governale, V. Cataudella, and V. M. Ramaglia, Phys. Rev. B 72, 075305 (2005).
* (19) J. Vidal, B. Doucot, R. Mosseri, and P. Butaud, Phys. Rev. Lett. 85, 3906 (2000).
* (20) B. Doucot and J. Vidal, Phys. Rev. Lett. 88, 227005 (2002).
* (21) S. K. Maiti, J. Phys. Soc. Jpn. 78, 114602 (2009).
* (22) S. K. Maiti, Phys Lett. A 373, 4470 (2009).
* (23) S. K. Maiti, Solid State Commun. 149, 1623 (2009).
* (24) M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett. 96A, 365 (1983).
* (25) L. P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074 (1990).
* (26) V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher, and A. Kleinsasser, Phys. Rev. Lett. 67, 3578 (1991).
* (27) D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993).
* (28) E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, and R. A. Webb, Phys. Rev. Lett. 86, 1594 (2001).
* (29) R. Deblock, R. Bel, B. Reulet, H. Bouchiat, and D. Mailly, Phys. Rev. Lett. 89, 206803 (2002).
* (30) B. Reulet, M. Ramin, H. Bouchiat, and D. Mailly, Phys. Rev. Lett. 75, 124 (1995).
* (31) H. F. Cheung, Y. Gefen, E. K. Riedel, and W. H. Shih, Phys. Rev. B 37, 6050 (1988).
* (32) H. F. Cheung and E. K. Riedel, Phys. Rev. Lett. 62, 587 (1989).
* (33) L. K. Castelano, G.-Q. Hai, B. Partoens, and F. M. Peeters, Phys. Rev. B 78, 195315 (2008).
* (34) M. Zarenia, M. J. Pereira, F. M. Peeters, and G. de Farias, Phys. Rev. B 81, 045431 (2010).
* (35) D. Y. Vodolazov, F. M. Peeters, T. T. Hongisto, and K. Yu. Arutyunov, Europhys. Lett. 75, 315 (2006).
* (36) G. Montambaux, H. Bouchiat, D. Sigeti, and R. Friesner, Phys. Rev. B 42, 7647 (1990).
* (37) H. Bouchiat and G. Montambaux, J. Phys. (Paris) 50, 2695 (1989).
* (38) B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. 66, 88 (1991).
* (39) F. von Oppen and E. K. Riedel, Phys. Rev. Lett. 66, 84 (1991).
* (40) A. Schmid, Phys. Rev. Lett. 66, 80 (1991).
* (41) V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 (1990).
* (42) M. Abraham and R. Berkovits, Phys. Rev. Lett. 70, 1509 (1993).
* (43) G. Bouzerar, D. Poilblanc, and G. Montambaux, Phys. Rev. B 49, 8258 (1994).
* (44) T. Giamarchi and B. S. Shastry, Phys. Rev. B 51, 10915 (1995).
* (45) N. Yu and M. Fowler, Phys. Rev. B 45, 11795 (1992).
* (46) S. Bellucci and P. Onorato, Physica E 41, 1393 (2009).
* (47) P. A. Orellana and M. Pacheco, Phys. Rev. B 71, 235330 (2005).
* (48) H.-M. Li and J.-L. Xiao, Physica B 396, 91 (2007).
* (49) H.-M. Li, Y.-L. Huang, and J.-L. Xiao, Int. J. Mod. Phys. B 22, 2255 (2008).
* (50) S. K. Maiti, Physica E 31, 117 (2006).
* (51) S. K. Maiti, Solid State Phenomena 155, 87 (2009).
* (52) S. K. Maiti, S. Saha, and S. N. Karmakar, Eur. Phys. J. B 79, 209 (2011).
* (53) S. K. Maiti, J. Appl. Phys. 110, 064306 (2011).
* (54) H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, and K. A. Moler, Phys. Rev. Lett. 102, 136802 (2009).
* (55) W. Rabaud, L. Saminadayar, D. Mailly, K. Hasselbach, A. Benoit, and B. Etienne, Phys. Rev. Lett. 86, 3124 (2001).
* (56) L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Schön, and K. Ensslin, Nature Physics 3, 650 (2007).
* (57) J. Premper, M. Trautmann, J. Henk, and P. Bruno, Phys. Rev. B 76, 073310 (2007).
* (58) H. Kato and D. Yoshioka, Phys. Rev. B 50, 4943 (1994).
* (59) A. Kambili, C. J. Lambert, and J. H. Jefferson, Phys. Rev. B 60, 7684 (1999).
* (60) S. K. Maiti and A. Chakrabarti, Phys. Rev. B 82, 184201 (2010).
* (61) S. K. Maiti, Solid State Commun. 150, 2212 (2010).
* (62) S. K. Maiti, Phys. Status Solidi B 248, 1933 (2011).
* (63) S. K. Maiti, M. Dey, S. Sil, A. Chakrabarti, and S. N. Karmakar, Europhys. Lett. 95, 57008 (2011).
* (64) W. Kohn, Phys. Rev. 133, A171 (1964).
* (65) B. S. Shastry and B. Sutherland, Phys. Rev. Lett. 65, 243 (1990).
* (66) H. Shiba, Phys. Rev. B 6, 930 (1972).
* (67) J. S. Sheng and K. Chang, Phys. Rev. B 74, 235315 (2006).
* (68) J. Splettstoesser, M. Governale, and U. Zülicke, Phys. Rev. B 68, 165341 (2003).
* (69) C. P. Moca and D. C. Marinescu, J. Phys.: Condens. Matter 18, 127 (2006).
* (70) I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
* (71) G.-H. Ding and B. Dong, Phys. Rev. B 76, 125301 (2007).
* (72) S. Bellucci and P. Onorato, Phys. Rev. B 78, 235312 (2008).
* (73) M. Dey, S. K. Maiti, and S. N. Karmakar, J. Appl. Phys. 109, 024304 (2011).
|
arxiv-papers
| 2012-03-26T10:01:38 |
2024-09-04T02:49:29.097506
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Santanu K. Maiti",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1203.5779"
}
|
1203.5910
|
††thanks: zhang_ping@iapcm.ac.cn††thanks: sslee@semi.ac.cn
# Theory of multiple magnetic scattering for quasiparticles on a gapless
topological insulator surface
Zhen-Guo Fu SKLSM, Institute of Semiconductors, CAS, P. O. Box 912, Beijing
100083, China LCP, Institute of Applied Physics and Computational
Mathematics, P.O. Box 8009, Beijing 100088, China Ping Zhang LCP, Institute
of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing
100088, China Zhigang Wang LCP, Institute of Applied Physics and
Computational Mathematics, P.O. Box 8009, Beijing 100088, China Fawei Zheng
LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box
8009, Beijing 100088, China Shu-Shen Li SKLSM, Institute of Semiconductors,
CAS, P. O. Box 912, Beijing 100083, China
###### Abstract
We develop a general low-energy multiple-scattering partial-wave theory for
gapless topological insulator (TI) surfaces in the presence of magnetic
impurities. As applications, we discuss the differential cross section (CS)
$d\Lambda/d\varphi$, the total CS $\Lambda_{tot}$, the Hall component of
resistivity $\Omega$, and inverse momentum relaxation time $\Gamma_{M}$ for
single- and two-centered magnetic scattering. We show that differing from the
nonmagnetic impurity scattering, $s\mathtt{-}$wave approximation is not
advisable and convergent in the present case. The symmetry of CS is reduced
and the backscattering occurs and becomes stronger with increasing the
effective magnetic moment $M$ of single magnetic impurity. We show a non-zero
perpendicular resistivity component $\Omega$, which may be useful for tuning
the Hall voltage of the sample. Consistent with the analysis of
$d\Lambda/d\varphi$, by comparing $\Gamma_{M}$ with $\Lambda_{tot}$, we can
determine different weights of backscattering and forward scattering. Similar
to CS, $\Omega$ and $\Gamma_{M}$ also exhibit oscillating behavior for
multiple magnetic scattering centers due to interference effect.
###### pacs:
72.10.-d, 72.10.Fk, 73.20.-r, 73.50.Bk
## I Introduction
A topic of fundamental importance in condensed matter physics is how the
presence of defects or impurities induce strong modifications on the local
electronic properties of crystalline solids. These modifications, with the
stunning development of scanning tunneling microscopy (STM), have been
extensively investigated on metal surfaces where they are well known as
Friedel oscillations and manifest as standing waves in the local electronic
density spanning regions up to $\sim$10 nm from the defects on the metal
surfaces. One kind of particularly suitable prototype that have been used in a
large amount of STM measurements to study the effects of impurity and the
formation of adsorbate superstructure are the (111) surfaces of noble metals,
on which the surface-state electrons form a two-dimensional (2D) nearly free-
electron gas. These Shockley-type surface states are dispersed as
$\epsilon=\hbar^{2}k^{2}/2m_{eff}$ (measured relative to the bottom of the
surface-state band) and localized in narrow band gaps in the center of the
first Brillouin zone of the (111)-projected bulk band structure. Thus they
have extremely small Fermi wave vectors
$k_{f}=\sqrt{2m_{eff}\epsilon_{F}}/\hbar$ and consequently the Friedel
oscillations of the surface state have a significantly larger wavelength than
those of the bulk states.
Recently, topological insulator (TI) has attracted tremendous experimental and
theoretical studies Hasan ; Qi2011 . Unlike (111) surfaces of noble metals, a
peculiar characteristic of TI is the presence of strong spin-orbit coupling
(SOC), which results in a variety of unique properties. One intriguing fact is
that the ideal TI surface is described at low energies by a 2D massless-Dirac
wave equation with an additional locking between momentum and spin of surface
electron. Because of the Dirac spectrum and SOC induced fermionic chirality,
the impurity scattering effect in TIs is naturally expected to display novel
behavior that should be absent from the conventional semiconductor or metal-
surface 2D electron gases. Many experimental and theoretical efforts towards
this issue have been payed. The anomalous Friedel oscillations in the vicinity
of a single localized impurity Liu2009 ; Balatsky2010 ; Balatsky2011 ;
Balatsky2012 , as well as the identification of the nature and the precise
location of impurities on TI surface using STM Xue2009 ; Roushan2009 ; Xue2011
; Alpichshev2011 ; Biswas2011 , have been discussed. However, when impurities
are located close to each other, multiple scattering effects should be
important, such as the issue of the long-range interactions between the
adsorbates mediated by the Dirac electrons of TI surface ZGFu2011 ; ZGFu20112
. In particular, since the quasiparticle’s spin is strongly coupled to its
momentum, quantum interference between different spin states during multiple
scattering process could then display prominent phenomena such as electric
conductance weak (anti-)localization He and Aharonov-Bohm effect Fu2011 in
STM signals.
In the presence of the time-reversal symmetry (TRS), the backscattering
induced by nonmagnetic impurities is forbidden on the gapless TI surface
because of a $\pi$ Berry phase associated with the $2\pi$ adiabatic rotation
of Dirac electron spin along the Fermi energy surface. However, considering
the magnetic impurities on the gapless TI surface, one would like to observe
the backscattering since the TRS is broken. Many efforts have been devoted to
exploring this issue. For example, very recently, quasiparticle interference
induced by a magnetic Co adatom on gapless Bi2Se3 surface has been found in
STM experiments Ye2011 . Furthermore, a magnetic field can be generated when
the TI sample is deposited on a lithographically patterned ferromagnetic
layer, which could also induce backscattering of massless Dirac electrons in
TI Zazunov2010 .
Because of its importance both from basic point of interest and to TI-based
chemical catalysis and electronics, in the present paper we address this issue
by presenting a first attempt at the theoretical evaluation of the multiple
scattering problem of the massless Dirac electrons on the TI surface in the
presence of localized and identical magnetic impurities. Specially, we present
the analytical expressions for multiple partial-wave scattering of massless
Dirac electrons with magnetic impurities, based on which the asymptotic
multiple scattering amplitude for random arraying magnetic impurities are
obtained under the particular large distance approximations (the identical
impurities are treated as a large scattering center). The differential and
total cross sections (CSs), the inverse momentum relaxation time, and the Hall
component of resistivity are discussed. We find that differing from the
nonmagnetic scattering, for the magnetic impurity scattering, the CS is not
convergent under $s\mathtt{-}$wave approximation. Therefore, higher partial
waves should be introduced. For the single magnetic impurity scattering, we
show the fact that the backscattering becomes much stronger when increasing
the effective magnetic moment $M$. By comparing the inverse momentum
relaxation time with total CS, we can determine whether there exist more
backscattering than forward scattering or not. Similar to CS, the inverse
momentum relaxation time and Hall factor display oscillating behavior for
multiple magnetic scattering centers due to interference.
## II Model and theory
The eigenstates of the effective low-energy Hamiltonian of TI surface near the
Dirac point Zhang2009
$H_{0}\mathtt{=}\hbar
v_{f}\left(\boldsymbol{\sigma}\mathtt{\times}\boldsymbol{k}\right)\mathtt{\cdot}\hat{z}$
(1)
are given by the spinors
$\Psi_{0,\pm}\left(\boldsymbol{r}\right)\mathtt{=}\frac{e^{i\boldsymbol{k}\cdot\boldsymbol{r}}}{\sqrt{2}}\left(\begin{array}[c]{cc}1,&\mp
ie^{i\theta_{\boldsymbol{k}}}\end{array}\right)^{\text{T}}$, where the
upper/lower sign corresponds to the electron/hole part of the spectrum. Here,
$v_{f}\sim 5\times 10^{5}$ m/s is the Fermi velocity, $\boldsymbol{\sigma}$
are Pauli matrices, and $\boldsymbol{k}$ is the in-plane wavevector. The
impurity potential can be expressed as
$V_{i}\mathtt{=}\frac{1}{2}J_{i}\boldsymbol{S}_{i}\mathtt{\cdot}\boldsymbol{\sigma}\Theta\left(a-r\right),$
(2)
where $\boldsymbol{S}_{i}\mathtt{=}S\boldsymbol{n}_{i}$ is the classical spin
(with its orientation vector $\boldsymbol{n}_{i}$) of the $i$th magnetic
impurity, $J_{i}$ is the exchange coupling strengths, $a$ is the radius of the
scatterer, and $\Theta\left(r\right)$ is the Heaviside function. If the
measurements are performed at a temperature higher than the Kondo temperature
Cui , the coupling between impurity spins will not exceed the critical value
$J_{cr}$ before a Kondo effect occurs. In this work we assume that the
exchange coupling $J_{i}<J_{cr}$, so that the Ruderman-Kittel-Kasuya-Yosida
interactions between impurity spins and Kondo screening of the impurity spin
by the band electrons are neglected, and the impurity spin acts as a classical
local magnetic moment under mean-field approximation Liu2009 ; Balatsky2010 .
In order to obtain the analytical expressions of wavefunctions, we just
consider the component of the classical spin along the normal line of TI
surface, i.e., $V_{i}\mathtt{=}M\sigma_{z}\Theta\left(a-r\right)$. The fact
that a magnetic Co impurity with only perpendicular spin component on the TI
surface does not open a gap has been experimentally observed Ye2011 . To
develop a scattering theory from localized, cylindrically-symmetric
scatterers, it is convenient to resolve the problem in cylindrical
coordinates. By considering the continuity of the wavefunction at the boundary
of the magnetic scattering potential, one can immediately obtain the
analytical expression of the scattered wave, written as
$\Psi_{\text{sc}}\left(\boldsymbol{r},\boldsymbol{k},\pm\right)\mathtt{=}s_{0}G_{+0}T_{0}^{-}\Phi_{\pm}^{in}\mathtt{+}\sum_{l=1}^{\infty}\left[s_{l}G_{+l}T_{l}^{-}\mathtt{+}s_{-l}G_{-l}T_{l}^{+}\right]\Phi_{\pm}^{in}.$
(3)
Here, $\Phi_{\pm}^{in}$ denotes the incident plane-wave centered about a
single scatterer located at $\boldsymbol{r}_{n}$. The cylindrically-symmetric
Green’s functions take the form
$\displaystyle G_{+l}$
$\displaystyle=\frac{i^{l}e^{il\theta_{n}}}{2}\left(\begin{array}[c]{cc}H_{l}^{(1)}\left(k\rho_{n}\right)&\pm
H_{l}^{(1)}\left(k\rho_{n}\right)\\\ \pm
H_{l+1}^{(1)}\left(k\rho_{n}\right)e^{i\theta_{n}}&H_{l+1}^{(1)}\left(k\rho_{n}\right)e^{i\theta_{n}}\end{array}\right),$
(6) $\displaystyle G_{-l}$
$\displaystyle=\frac{i^{l}e^{-il\theta_{n}}}{2}\left(\begin{array}[c]{cc}H_{l}^{(1)}\left(k\rho_{n}\right)&\mp
H_{l}^{(1)}\left(k\rho_{n}\right)\\\ \mp
H_{l-1}^{(1)}\left(k\rho_{n}\right)e^{i\theta_{n}}&H_{l-1}^{(1)}\left(k\rho_{n}\right)e^{i\theta_{n}}\end{array}\right),$
(9)
for $\left|\epsilon\right|>M$, where upper and lower signs in the right side
of these expressions denote the $\epsilon>0$ and $\epsilon<0$ parts of the
spectrum, $k=\frac{\epsilon}{\hbar v_{f}}$,
$\boldsymbol{\rho}_{n}\mathtt{=}\boldsymbol{r}\mathtt{-}\boldsymbol{r}_{n}$
and
$e^{i\theta_{n}}\mathtt{=}\frac{\boldsymbol{\rho}_{n}\cdot\left(\hat{x}+i\hat{y}\right)}{\rho_{n}}$.
For the energy regin of $\left|\epsilon\right|<M$, the Hankel functions
$H_{l}^{(1)}\left(k\rho_{n}\right)$ in $G_{\pm l}$ should be replaced by the
modified Bessel functions of first kind $I_{l}\left(k\rho_{n}\right)$. The
scattering amplitude is expressed as
$s_{l}=\frac{A_{+}J_{l}\left(k^{\prime}a\right)J_{l+1}\left(ka\right)-A_{-}J_{l}\left(ka\right)J_{l+1}\left(k^{\prime}a\right)}{A_{-}H_{l}^{(1)}\left(ka\right)J_{l+1}\left(k^{\prime}a\right)-A_{+}H_{l+1}^{(1)}\left(ka\right)J_{l}\left(k^{\prime}a\right)}$
(10)
for $\left|\epsilon\right|>M,$ where $A_{\pm}=\sqrt{\left|\epsilon\pm
M\right|}$, $k^{\prime}=\frac{\sqrt{\left|\epsilon^{2}-M^{2}\right|}}{\hbar
v_{f}}$, and $J_{l}$ is the Bessel function of order $l$. Whereas $s_{l}$
should also be replaced by
$\widetilde{s}_{l}=\frac{A_{+}I_{l}\left(k^{\prime}a\right)J_{l+1}\left(ka\right)+A_{-}J_{l}\left(ka\right)I_{l+1}\left(k^{\prime}a\right)}{-A_{-}H_{l}^{(1)}\left(ka\right)I_{l+1}\left(k^{\prime}a\right)-A_{+}H_{l+1}^{(1)}\left(ka\right)I_{l}\left(k^{\prime}a\right)}\text{
}$ (11)
for the case of for $\left|\epsilon\right|<M.$ Note that $s_{l}$
($\widetilde{s}_{l}$) satisfies the unitarity condition
$\operatorname{Re}[s_{l}]=-\left|s_{l}\right|^{2}$
($\operatorname{Re}[\widetilde{s}_{l}]=-\left|\widetilde{s}_{l}\right|^{2}$),
and
$\lim_{\epsilon\rightarrow M^{+}}s_{l}=\lim_{\epsilon\rightarrow
M^{-}}\widetilde{s}_{l}=-J_{l+1}\left(ka\right)/H_{l+1}\left(ka\right)$ (12)
for all $l$. The $l^{th}$-partial-wave $t\mathtt{-}$matrix is
$T_{l}^{\pm}\mathtt{=}$diag$(\begin{array}[c]{cc}\hat{P}_{l}^{\pm},&\mp
i\hat{P}_{l\mp 1}^{\pm}\end{array})$ with $\hat{P}_{l}^{\pm}=\frac{e^{\pm
il\theta}}{i^{l}k^{l}}(\partial_{r}\pm\frac{i}{r}\partial_{\theta})^{l}$. A
detailed derivation is given in Appendix A.
It is easy to extend scattering theory of massless Dirac fermions to the
realistic and reasonable case of multiple magnetic impurities, where the
quantum interference effect in the propagation process of Dirac fermions on TI
surface can be observed. This has not been discussed in previous studies, such
as Ref. Zazunov2010 . Taking into account all of partial waves, for $N$
magnetic scatterers located at positions
$\boldsymbol{r}_{1},\boldsymbol{r}_{2},\cdots\boldsymbol{r}_{N}$, the
scattered wavefunction is given by
$\Psi_{\text{sc}}\left(\boldsymbol{r}\right)=\mathbb{G}\left(\boldsymbol{r}\right)\mathbf{S}D^{-1}\vec{\phi}.$
(13)
Here, $\mathbb{G}\left(\boldsymbol{r}\right)$ (a
$2\mathtt{\times}2N\left(2l_{\max}\mathtt{+}1\right)$ matrix) contains the
propagation information from detector to impurities $G_{\pm
l}\left(\boldsymbol{r},\boldsymbol{r}_{i}\right)$. $S$ is a diagonal matrix
with nonzero element $s_{\pm l}^{\left(n\right)}$. The $\mathbf{G}$ matrix, is
constructed by $T_{l}^{-}\left[G_{\pm l^{\prime}}\left(n,m\right)\right]$,
describing the propagation between impurities. $\overrightarrow{\phi}$ can be
written as a $2N\left(2l_{\max}\mathtt{+}1\right)\mathtt{\times}1$ vector
which imposes informations of incident waves (see details in Appendix B).
At this stage, we should point out that the above equations for multiple
magnetic scattering of massless Dirac quasiparticles are similar to those for
multiple nonmagnetic scattering of massive Dirac quasiparticles, but totally
different from those for multiple nonmagnetic scattering of massless Dirac
quasiparticles, since the expressions therein can be simplified as a more
compact form due to $s_{-\left(l+1\right)}^{\prime}=s_{l}^{\prime}$.
The above theory enables to solve multiple magnetic scattering problems in
gapless TI surfaces with higher partial waves, which could be important as
distances between scatterers decrease or the scattering potential is
strengthened. One simple application is to calculate the magnetic scattering
CSs. To calculate the CSs, we have to take the approximations
$H_{l}^{(1)}\left(z\right)\rightarrow\sqrt{\frac{2}{\pi
z}}e^{i\left(z-\frac{l\pi}{2}-\frac{\pi}{4}\right)}$, and
$e^{il\theta_{n}}=\left[\frac{\boldsymbol{\rho}_{n}\cdot\left(\hat{x}+i\hat{y}\right)}{\rho_{n}}\right]^{l}\approx\left[\frac{\boldsymbol{r}\cdot\left(\hat{x}+i\hat{y}\right)}{r}\right]^{l}=e^{il\varphi}$
for large distance. As a result,
$\Psi_{\text{sc}}\left(\boldsymbol{r}\right)\rightarrow
f\left(\boldsymbol{k},\varphi\right)\frac{e^{ikr}}{\sqrt{2r}}\left(\begin{array}[c]{c}1\\\
\mp ie^{i\varphi}\end{array}\right),$ (14)
where $f\left(\boldsymbol{k},\varphi\right)$ is the scattering amplitude, from
which we have the differential and total CSs as follows:
$\displaystyle\frac{d\Lambda}{d\varphi}$
$\displaystyle=\left|f\left(\boldsymbol{k},\varphi\right)\right|^{2},$ (15)
$\displaystyle\Lambda_{tot}$
$\displaystyle=\int_{0}^{2\pi}d\varphi\left|f\left(\boldsymbol{k},\varphi\right)\right|^{2}=\sqrt{\frac{8\pi}{k}}\operatorname{Im}\left[e^{-i\pi/4}f\left(\boldsymbol{k},\varphi=0\right)\right].$
(16)
Here, we have used the two-dimensional optical theorem.
Besides, we could also obtain the transverse component of resistivity (or say
the analog of Hall component in the case with magnetic field)
$\Omega=\int_{0}^{2\pi}d\varphi\left|f\left(\boldsymbol{k},\varphi\right)\right|^{2}\sin\varphi,$
(17)
and the inverse electron momentum relaxation time (the quantity proportional
to the dissipative component of resistivity)
$\Gamma_{M}=\int_{0}^{2\pi}d\varphi\left|f\left(\boldsymbol{k},\varphi\right)\right|^{2}\left(1-\cos\varphi\right).$
(18)
## III Results and discussions
In the following calculations, without losing the general properties, we shall
just consider the incident wave
$\Phi_{+}^{in}\mathtt{=}\frac{e^{ikx}}{\sqrt{2}}(\begin{array}[c]{cc}1\mathtt{,}&\mathtt{-}i\end{array})^{\text{T}}$
propagating along the positive $\hat{\boldsymbol{x}}$ direction. In
particular, for a single magnetic impurity scattering, we can obtain the
scattering amplitude including all of the partial waves, which is written as
$f\left(\boldsymbol{k},\varphi\right)=\left\\{f_{0}\left(\varphi\right)+\sum_{l=1}^{\infty}\left[f_{l}\left(\varphi\right)+f_{-l}\left(\varphi\right)\right]\right\\}e^{i\left(\boldsymbol{k}-k\hat{r}\right)\cdot\boldsymbol{r}^{\prime}}$
(19)
with $f_{0}\left(\varphi\right)=\sqrt{\frac{2}{i\pi k}}s_{0}$ and $f_{\pm
l}\left(\varphi\right)=\sqrt{\frac{2}{i\pi k}}s_{\pm l}e^{\pm il\varphi}$. The
differential and total CSs are given by
$\displaystyle\frac{d\Lambda}{d\varphi}$ $\displaystyle=\frac{2}{\pi
k}\left|\left\\{s_{0}+\sum_{l=1}^{\infty}\left[s_{l}e^{il\varphi}+s_{-l}e^{-il\varphi}\right]\right\\}\right|^{2},$
(20) $\displaystyle\Lambda_{tot}$
$\displaystyle=-\frac{4}{k}\left[\operatorname{Re}\left(s_{0}\right)+\sum_{l=1}^{\infty}\operatorname{Re}\left(s_{l}+s_{-l}\right)\right].$
(21)
This total CS equation is obtained from the optical theorem. It is clear that
the $s-$wave is independent on the direction of scattered wave, therefore, we
have to introduce higher partial waves, such as $p\mathtt{-}$ and
$d\mathtt{-}$waves, and so on. If one just considers the $s-$wave in
calculations, differential CS may lead to an unreasonable result of
$\left.d\Lambda/d\varphi\right|_{\varphi=\pi}=0$ (backscattering is forbidden)
for some particular effective magnetic moment $M$. This is different from the
nonmagnetic impurity scattering on TI surface ZGFu2011 as well as on
conventional 2DEG with weak Rashba SOC Walls2006 , where the $s\mathtt{-}$wave
approximation should be a reasonable choice.
Figure 1: (Color online) The normalized differential CSs $d\Lambda/d\phi$ for
(a) a single and (b) two magnetic scatterers located at
$\boldsymbol{r}_{1,2}=(\pm 3,0)$ on TI surface with effective magnetic moment
$M=60$ meV. The radius of scatterer $a\mathtt{=}1$ nm, and $l_{\max}=2$ are
chosen.
The results of normalized differential CS as a function of energy $\epsilon$
for the massless Dirac electron scattered by a single magnetic impurity
absorbed on TI surface with effective magnetic moment $M\mathtt{=}60$ meV are
shown in Fig. 1(a). In the calculations we take $l_{\max}=2$, which works out
convergent results. Different from the nonmagnetic impurity scattering case,
the backscattering is obvious in the differential CS, i.e.,
$\left.d\Lambda/d\varphi\right|_{\varphi=\pi}\neq 0$, in present case since
the time-reversal symmetry is broken by magnetic impurity scattering.
Furthermore, we find that for the weak effective magnetic moment $M$ (such as
the values chosen in this work $M\leq 100$ meV), the backscattering is greater
than forward scattering. However, if the effective magnetic moment is large
enough (for example, when $M\sim 500$ meV and $\epsilon=450$ meV) we find the
backscattering is weaker than the forward scattering (not shown here).
Two-impurity scattering provides a good test-bed to highlight the coexistence
of various scattering phenomena, including transmission, reflection,
interference, and resonance. The corresponding CS offers a measure of
interaction events between the two impurity centers, and interference effects
are useful in revealing actual electron density currents on TI surfaces. For
instance, if two impurities are close to each other, the electronic
wavefunctions will be scattered from both impurities, resulting in quantum
interference. From the above theory, we can obtain the simple expression of
scattering amplitude just containing the $s\mathtt{-}$wave ($l_{\max}=0$),
which is given by
$\displaystyle f_{0}^{N}\left(\boldsymbol{k},\varphi\right)$
$\displaystyle=\sum_{n,m=1}^{N}\frac{s_{0}e^{i\left(\boldsymbol{k}\cdot\boldsymbol{r}_{m}-k\hat{r}\cdot\boldsymbol{r}_{n}\right)}}{\sqrt{2i\pi
k}}$ (22)
$\displaystyle\times\left\\{\left[D^{-1}\right]_{\left(2n-1\right),\left(2m-1\right)}+\left[D^{-1}\right]_{2n,2m}\right.$
$\displaystyle\left.\pm\left[\left[D^{-1}\right]_{2n,\left(2m-1\right)}+\left[D^{-1}\right]_{\left(2n-1\right),2m}\right]\right\\}.$
Figure 2: (Color online) The magnetic moment $M$ dependence of differential
CS $d\Lambda/d\phi$ along the negative direction of $x$ axis $\phi=\pi$ for
one magnetic scatterer (red line), two scatterers (green line) located at
$\boldsymbol{r}_{1,2}=(\pm 3,0)$, and three scatterers (blue) located at
$\boldsymbol{r}_{1}=(-3,0)$, $\boldsymbol{r}_{2}=(3,-1)$,
$\boldsymbol{r}_{3}=(2,4)$. The Fermi energy is chosen as $\epsilon=70$ meV.
However, if higher partial waves ($l_{\max}\geq 1$) are taken into account,
the scattering amplitude expression for $N$ magnetic impurities becomes
tedious and complex since $D-$matix becomes a lager one.
Typical numerical results of differential CSs for two magnetic impurities
located at $\boldsymbol{r}_{1,2}=\left(\pm 3,0\right)$ are presented in Fig.
1(b). We also note that the interference effect is related not only to the
effective magnetic moment $M$ but also to the configuration of impurities. For
the present considered impurity locations $\boldsymbol{r}_{1,2}$, on one hand,
we find from Fig. 1(b) that the backscattering is more prominent than the
forward scattering. On the other hand, comparing with the nonmagnetic double-
impurity scattering on gapless TI surface, the symmetry of differential CSs
for two identical magnetic scatterers is reduced.
On one hand, independent on the impurity locations, with increasing the
effective magnetic moment $M$, we find that the relative strength of
backscattering becomes more and more remarkable since the differential CS
along the negative $\hat{\boldsymbol{x}}$ direction
$\left.d\Lambda/d\varphi\right|_{\varphi=\pi}$ increases with $M$, see Fig. 2
with the Fermi energy $\epsilon=70$ meV. On the other hand, comparing with the
scattering from a single (red curve) magnetic impurity, two (green curve) or
three (blue curve) impurities will weaken or strengthen the backscattering due
to interference effect, which is dependent on the impurity configurations
relative to the direction of incident wave. We must point out that in the
calculations, we should use $\tilde{s}_{l}$ for the energy region of
$\left|\epsilon\right|<M$, while $s_{l}$ for the energy region of
$\left|\epsilon\right|>M$, which are denoted in Fig. 2.
Figure 3: (Color online) The total CS $\Lambda_{tot}$ for a single (a-b) and
two (c-d) magnetic impurities located at $\boldsymbol{r}_{1,2}=(\pm 3,0)$. The
effective magnetic moment is chosen as $M=60$ meV in (a)and (c), while $M=100$
meV in (b) and (d), respectively.
Now let us turn to discuss the total CSs, which are exhibited in Fig. 3. As
mentioned above, the $s\mathtt{-}$wave approximation cannot give out
convergent result, see the black curves in Fig. 3, whereas, when we introduce
higher partial waves (such as $l_{\max}=2$ chosen in our calculations), the
total CSs becomes convergent ultimately. Differing from the nonmagnetic
impurity scattering, although the higher partial waves can induce remarkable
corrections, we have not found additional resonant peaks in total CSs due to
higher partial waves. Moreover, it is obvious that interference between double
impurities brings about oscillations in total CSs (see Figs. 3(c) and 3(d)),
which cannot be observed in the case of nonmagnetic impurity scattering
ZGFu2011 . From numerical calculations, we find on one hand that, the optical
theorem is correct and should characterize the general multiple-scattering
processes, since the results obtained by the numerical integration of the
first equality in Eq. (16) are in good agreement with that obtained from the
second equality; On the other hand, the curves of total CSs are smooth at the
energy of $\epsilon=M$, which indicates that the limiting function of $s_{l}$
and $\tilde{s}_{l}$, i.e., Eq. (12) is reasonable. Besides, for the much
strong effective magnetic impurity scattering, with increasing the Fermi
energy we find that the total CS for single- (double-) impurity is convergent
to 4 nm (8 nm). This is also different from the nonmagnetic impurity
scattering, where the total CSs converge to zero with increasing the energy of
Dirac electrons ZGFu2011 .
Figure 4: (Color online) Hall component of resistivity $\Omega$ for a single
(a) and two (b) magnetic impurities on TI surface; Inverse momentum relation
time $\Gamma_{M}$ (thick solid curves) and total CS $\Lambda_{tot}$ (thin
dashed curved) for a single (c) and two (d) magnetic impurities as functions
of energy $\epsilon$.
In spite of the differential and total CSs, we also calculated the transverse
component of resistivity $\Omega$, which is analogous to Hall component in the
case with external magnetic field. The typical results of $\Omega$ as a
function of $\epsilon$ for single- and double-impurity with different $M$ are
listed in Figs. 4(a) and 4(b), respectively. We find that the Hall component
of the resistivity $\Omega$ always keeps its sign as negative (i.e.,
$\Omega<0$), which is independent on the impurity locations. In the numerical
calculations, we take $l_{\max}\geq 2$ which results in a convergent result,
however, taking into account higher partial waves, it is difficult to be
obtained analytically from Eqs. (17) and (19) since the expression for
integral result is tedious and complex. Therefore, the low-energy Dirac
electrons are deflected to one side of TI sample due to magnetic impurity
scattering. This fact may be helpful for tuning the Hall voltage of sample.
Interestingly, we note that this type of Hall component also occurs in the
nonmagnetic impurity scattering on gapless TI surface. We now switch gears and
consider the case of double magnetic impurities on TI surface again. Different
from the single case, $\Omega$ exhibits oscillating behavior due to the
interference during the multiple impurities scattering processes, as shown in
Fig. 4(b).
Before ending this paper, we would like to discuss another important quantity,
the inverse electron momentum relaxation time $\Gamma_{M}$, which is
proportional to the dissipative component of resistivity. The numerical
results are plotted in Figs. 4(c) and 4(d) for single- and double-impurity
cases, respectively. The behavior of $\Gamma_{M}$ (thick solid curves) is
qualitatively similar to the one for the total CSs $\Lambda_{tot}$ (thin
dashed curves), and the interference effect is also clear in the $\Gamma_{M}$
induced by double-impurity, see Fig. 4(d). The inverse electron momentum
relaxation time $\Gamma_{M}$ is a fairly sensitive quantity that determines
whether the charge carrier is attracted to an impurity or is repelled from it.
It is also useful for determining whether the backscattering is greater than
the forward scattering by comparing it with the total CSs (note that both
$\Lambda_{tot}$ and $\Gamma_{M}$ have the dimension of an length in two
dimensional scattering). If $\Lambda_{tot}\mathtt{<}\Gamma_{M}$
($\Lambda_{tot}\mathtt{>}\Gamma_{M}$), there is more (less) backscattering
than forward scattering. Taking $M=60$ meV as an example, we find
$\Lambda_{tot}\mathtt{<}$ $\Gamma_{M}$, as revealed by the red curves in Fig.
4(c), which indicates that the backscattering is greater than the forward
scattering in the low-energy region. This fact is consistent with the behavior
of differential CSs shown in Fig. 1(a). Particularly, by observing the green
curves in Fig. 4(d) of energy region of $80\mathtt{\sim}120$ meV (the shadow
region), one can find $\Lambda_{tot}\mathtt{>}\Gamma_{M}$. This suggests that
due to the interference the backscattering is weaker than forward scattering
in this energy region for the present double magnetic impurity locations,
which is also consistent with the analysis of differential CSs. Consequently,
we believe that our results shown here are reasonable, and we hope our
findings could be detected in the future experiments.
## IV Conclusions
In summary, we have proposed a general low-energy multiple-scattering partial-
wave theory for quasiparticles on the gapless topological insulator (TI)
surfaces in the presence of magnetic impurities. Based on this theory, one can
solve the scattering problems of $N$ magnetic impurities. As an application,
we have calculated the CSs, the inverse momentum relaxation time, and the
transverse resistivity component for a single and two circular magnetic
scattering. We have found that the usual $s\mathtt{-}$wave approximation is
not sufficient, while higher partial waves must be introduced to obtain
convergent results. On the gapless TI surfaces, differing from the single
nonmagnetic impurity case, the backscattering occurs and becomes much stronger
with increasing the effective magnetic moment $M$. Interference effects are
obvious in CSs from quasiparticle scattering off two magnetic scattering
centers, and oscillating behaviors are introduced in $\Lambda_{tot}$
associated with higher-order partial-waves. A non-zero perpendicular
resistivity component has also been shown. Similar to the total CS, the
inverse momentum relaxation time and the transverse resistivity component
exihibit oscillations for multiple magnetic scattering centers due to
interference. Furthermore, our theory could be extended to spin-polarized
case. It could also be applied to simulate the electron flow (charge current
and spin current) through a quantum point contact on a TI surface by
monitoring the changes in conductance through the quantum point contact as a
moveable STM tip is scanned above the surface of TI.
###### Acknowledgements.
This work was supported by NSFC under Grants No. 90921003, No. 60776063, and
No. 60821061, and by the National Basic Research Program of China (973
Program) under Grants No. 2009CB929103 and No. G2009CB929300.
## APPENDIX A: DERIVATION DETAILS OF SCATTERED WAVEFUNCTION
Starting from the model considered in the main text, one can find the spinor
spherical wavefunctions inside the magnetic scattering potential ($r<a$) as
follows:
$\displaystyle\zeta_{l}^{(1,2)}\left(\boldsymbol{r},k^{\prime},\pm\right)$
$\displaystyle=\frac{we^{il\theta}}{\sqrt{2\left|\epsilon\right|k^{\prime}}}\left(\begin{array}[c]{c}\sqrt{\left|\epsilon+M\right|}H_{l}^{(1,2)}\left(k^{\prime}r\right)\\\
\pm\sqrt{\left|\epsilon-M\right|}H_{l+1}^{(1,2)}\left(k^{\prime}r\right)e^{i\theta}\end{array}\right),\text{
\ \ }(\left|\epsilon\right|>M),$ (A1)
$\displaystyle\widetilde{\zeta}_{l}\left(\boldsymbol{r},k^{\prime},\pm\right)$
$\displaystyle=\frac{we^{il\theta}}{\sqrt{2\left|\epsilon\right|k^{\prime}}}\left(\begin{array}[c]{c}\sqrt{\left|\epsilon+M\right|}I_{l}\left(k^{\prime}r\right)\\\
\mp\sqrt{\left|\epsilon-M\right|}I_{l+1}\left(k^{\prime}r\right)e^{i\theta}\end{array}\right),\text{
\ \ }(\left|\epsilon\right|<M),$ (A2)
where $k^{\prime}=\frac{\sqrt{\left|\epsilon^{2}-M^{2}\right|}}{\hbar v_{f}}$
and $w=e^{i\phi}$ is an overall phase meaning of the nonrelativistic
wavefunction in the rest frame of $\epsilon=M$. Here,
$H_{l}^{(1,2)}\left(z\right)$ and $I_{l}\left(z\right)$ are $l^{th}$-order
Hankel functions and modified Bessel functions of first kind, respectively. We
would like to point out that since the wavefunctions should be zero at $r=0$,
we have neglected the modified Bessel functions of second kind
$K_{l}\left(z\right)$ for $\left|\epsilon\right|<M$, which are emanative in
the limit of $z\rightarrow 0$. Whereas, the wavefunctions outside the magnetic
potential ($r>a$) can be expressed as
$\chi_{l,\pm}^{(1,2)}\left(\boldsymbol{r}\right)=\frac{1}{\sqrt{2k}}\left(\begin{array}[c]{c}H_{l}^{(1,2)}\left(kr\right)e^{il\theta}\\\
\pm
H_{l+1}^{(1,2)}\left(kr\right)e^{i\left(l+1\right)\theta}\end{array}\right),$
(A3)
with $k=\frac{\epsilon}{\hbar v_{f}}$. $\chi^{(1)}$ ($\chi^{(2)}$) denotes the
outgoing (incoming) cylindrical wave about $\boldsymbol{r}$=$0$. The incident
plane-wave centered about a single scatterer located at $\boldsymbol{r}_{n}$
is given by
$\Phi_{\pm}^{in}\left(\boldsymbol{r}\right)\mathtt{=}\sum_{l=-\infty}^{\infty}\frac{\sqrt{k}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}e^{il\left(\theta_{n}-\theta_{\boldsymbol{k}}\right)}}{2}i^{l}\left(\chi_{l,\pm}^{(1)}\left(\boldsymbol{\rho}_{n}\right)\mathtt{+}\chi_{l,\pm}^{(2)}\left(\boldsymbol{\rho}_{n}\right)\right),$
(A4)
where
$\boldsymbol{\rho}_{n}\mathtt{=}\boldsymbol{r}\mathtt{-}\boldsymbol{r}_{n}$
and
$e^{i\theta_{n}}\mathtt{=}\frac{\boldsymbol{\rho}_{n}\cdot\left(\hat{x}+i\hat{y}\right)}{\rho_{n}}$.
Remember that $\theta_{\boldsymbol{k}}$ is the angle defining the direction of
the wave vector, and $\theta_{n}$ is related to the direction of
$\boldsymbol{r}\mathtt{-}\boldsymbol{r}_{n}$ in this equation. Then, the fully
scattered wave function in the region $\rho_{n}>a$ can be written as
explicitly
$\displaystyle\Psi_{\text{I}}\left(\boldsymbol{r},\boldsymbol{k},\pm\right)$
$\displaystyle=\Phi_{\pm}^{in}\left(\boldsymbol{r}\right)+\Psi_{\text{sc}}\left(\boldsymbol{r},\boldsymbol{k},\pm\right)$
$\displaystyle=\sqrt{k}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}\sum_{l=-\infty}^{\infty}i^{l}\left[\frac{1}{2}e^{2i\delta_{l}}\chi_{l}^{\left(1\right)}\left(\boldsymbol{\rho}_{n},\boldsymbol{k},\pm\right)+\frac{1}{2}\chi_{l}^{\left(2\right)}\left(\boldsymbol{\rho}_{n},\boldsymbol{k},\pm\right)\right]e^{il\left(\theta_{n}-\theta_{\boldsymbol{k}}\right)},$
(A5)
where $\delta_{l}$ are phase shifts of the outgoing cylindrical partial waves,
$\chi_{l}^{\left(1\right)}\left(\boldsymbol{\rho}_{n},\boldsymbol{k},\pm\right)$.
In the region of $\rho_{n}\leq a$, the fully scattered wave function is given
by
$\Psi_{\text{II}}\left(\boldsymbol{r},\boldsymbol{k^{\prime}},\pm\right)=\left\\{\begin{array}[c]{c}\sqrt{k^{\prime}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}\sum_{l=-\infty}^{\infty}i^{l}d_{l}\left[\frac{1}{2}\zeta_{l}^{\left(1\right)}\left(\boldsymbol{\rho}_{n},\boldsymbol{k^{\prime}},\pm\right)-\frac{1}{2}\zeta_{l}^{\left(2\right)}\left(\boldsymbol{\rho}_{n},\boldsymbol{k^{\prime}},\pm\right)\right]e^{il\left(\theta_{n}-\theta_{\boldsymbol{k}}\right)},\text{
\ \ }\left(\left|\epsilon\right|>M\right)\\\
\sqrt{k^{\prime}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}\sum_{l=-\infty}^{\infty}i^{l}\widetilde{d}_{l}\widetilde{\zeta}_{l}\left(\boldsymbol{r},k^{\prime},\pm\right)e^{il\left(\theta_{n}-\theta_{\boldsymbol{k}}\right)},\text{
\ \ }\left(\left|\epsilon\right|<M\right)\end{array}\right..$ (A6)
By the continuity of the wavefunction at $\rho_{n}=a$,
$\Psi_{\text{I}}\left(\boldsymbol{a},\boldsymbol{k},\pm\right)=\Psi_{\text{II}}\left(\boldsymbol{a},\boldsymbol{k}^{\prime},\pm\right)$,
we can obtain the scattered wavefunction Eq. (3), and the scattering amplitude
$s_{l}$ and $\widetilde{s}_{l}$ shown in Eqs. (10) and (11) in main text. It
is clear that the scattered wavefunctions shown in Eq. (3) are different from
those for nonmagnetic impurity scattering on gapless TI surface, where
$\Psi_{\text{sc}}\left(\boldsymbol{r}\right)\mathtt{=}\sum_{l=0}^{\infty}\frac{4i\hbar
v_{f}s_{l}^{\prime}}{k}G_{l}\left(\boldsymbol{r},\boldsymbol{r}_{n},\epsilon\right)T_{l}^{{}^{\prime}}\left[\Phi_{\pm}^{in}\right]$
(A7)
with
$G_{l}\mathtt{\propto}\left(\begin{array}[c]{cc}H_{l}^{(1)}e^{il\theta_{n}}&\mp
H_{l+1}^{(1)}e^{-i\left(l+1\right)\theta_{n}}\\\ \pm
H_{l+1}^{(1)}e^{i\left(l+1\right)\theta_{n}}&H_{l}^{(1)}e^{-il\theta_{n}}\end{array}\right),$
(A8)
$T_{l}^{{}^{\prime}}\mathtt{=}\mathtt{diag}(\begin{array}[c]{cc}\hat{P}_{l}^{-},&\hat{P}_{l}^{+}\end{array})$,
and
$s_{l}^{\prime}=\frac{J_{l}\left(\kappa^{\prime}a\right)J_{l+1}\left(ka\right)-J_{l}\left(ka\right)J_{l+1}\left(\kappa^{\prime}a\right)}{H_{l}^{(1)}\left(ka\right)J_{l+1}\left(\kappa^{\prime}a\right)-H_{l+1}^{(1)}\left(ka\right)J_{l}\left(\kappa^{\prime}a\right)}.$
(A9)
Here $\kappa^{\prime}\mathtt{=}\frac{\epsilon-V_{0}}{\hbar v_{f}}$, and
$V_{0}$ is the scalar potential.
### APPENDIX B: DETAILS OF THE EXPEDITION FOR MULTIPLE SCATTERED WAVE
In order to understand the extending operation, we would like to start from
the $s\mathtt{-}$wave scattering for two identical magnetic impurities. The
total wavefunction can be written as
$\Psi\left(\boldsymbol{r}\right)=\Phi\left(\boldsymbol{r}\right)+\sum_{n=1}^{2}s_{0}^{\left(n\right)}G_{+0}\left(\boldsymbol{r},\boldsymbol{r}_{n},\epsilon\right)T_{0}^{-}\left[\Psi_{n}\left(\boldsymbol{r}_{n},\boldsymbol{k},\pm\right)\right],$
(B1)
where
$\Psi_{n}\left(\boldsymbol{r}\right)=\Phi\left(\boldsymbol{r}\right)+\sum_{m\neq
n}^{2}s_{0}^{\left(m\right)}G_{+0}\left(\boldsymbol{r},\boldsymbol{r}_{m},\epsilon\right)T_{0}^{-}\left[\Psi_{m}\left(\boldsymbol{r}_{m},\boldsymbol{k},\pm\right)\right].$
(B2)
Equation (B1) indicates that if the value of $\Psi\left(\boldsymbol{r}\right)$
and its derivatives due to the $\hat{P}_{0,1}^{\pm}$ dependence of
$T_{0}^{\pm}$ at each scatterer is known, the entire wavefunction
$\Psi\left(\boldsymbol{r}\right)$ is completely determined. We can calculate
the derivatives of $\Psi_{1}$ at $\boldsymbol{r}_{1}$ and $\Psi_{2}$ at
$\boldsymbol{r}_{2}$ and combine the result into a matrix equation, which are
given by
$\left(\begin{array}[c]{c}T_{0}^{-}\left[\Psi_{1}\left(\boldsymbol{r}_{1}\right)\right]\\\
T_{0}^{-}\left[\Psi_{2}\left(\boldsymbol{r}_{2}\right)\right]\end{array}\right)=D^{-1}\left(\begin{array}[c]{c}T_{0}^{-}\left[\Phi\left(\boldsymbol{r}_{1}\right)\right]\\\
T_{0}^{-}\left[\Phi\left(\boldsymbol{r}_{2}\right)\right]\end{array}\right),$
(B3)
where
$D=\mathbf{1}_{4\times 4}-\mathbf{G}_{4\times 4}\mathbf{S}_{4\times 4},$ (B4)
with
$\displaystyle\mathbf{G}$
$\displaystyle\mathbf{=}\left(\begin{array}[c]{cc}\mathbf{0}&T_{0}^{-}\left[G_{+0}\left(\boldsymbol{r}_{1},\boldsymbol{r}_{2},\epsilon\right)\right]\\\
T_{0}^{-}\left[G_{+0}\left(\boldsymbol{r}_{2},\boldsymbol{r}_{1},\epsilon\right)\right]&\mathbf{0}\end{array}\right),$
(B5) $\displaystyle\mathbf{S}$
$\displaystyle\mathbf{=}\left(\begin{array}[c]{cc}s_{0}^{\left(1\right)}&0\\\
0&s_{0}^{\left(2\right)}\end{array}\right)\otimes\mathbf{1}_{2\times 2}.$ (B6)
Finally, the total wavefunction is written as
$\Psi\left(\boldsymbol{r}\right)\mathtt{=}\Phi_{\pm}^{in}\mathtt{+}\mathbb{G}\left(\boldsymbol{r}\right)\mathbf{S}D^{-1}\left(\begin{array}[c]{c}T_{0}^{-}\left[\Phi\left(\boldsymbol{r}_{1}\right)\right]\\\
T_{0}^{-}\left[\Phi\left(\boldsymbol{r}_{2}\right)\right]\end{array}\right)$
(B7)
with
$\mathbb{G}\left(\boldsymbol{r}\right)=\left(\begin{array}[c]{cc}G_{+0}\left(\boldsymbol{r},\boldsymbol{r}_{1}\right),&G_{+0}\left(\boldsymbol{r},\boldsymbol{r}_{2}\right)\end{array}\right).$
(B8)
Taking into account $l_{\max}\geq 1$ partial waves, for $N$ magnetic
scatterers located at positions
$\boldsymbol{r}_{1},\boldsymbol{r}_{2},\cdots\boldsymbol{r}_{N}$, the
scattered wavefunction is given by Eq. (13) in main text. For numerical
calculations, we have to align the matrix elements reasonably, thereby, we
define symbols $\mu_{0}=2\left(n-1\right)\left(2l_{\max}+1\right)+1$,
$\lambda_{0}=\mu_{0}+1$,
$\alpha_{0}=2\left(n-1\right)\left(2l_{\max}+1\right)+4l-1$,
$\gamma_{0}=\alpha_{0}+1$,
$\tau_{0}=2\left(n-1\right)\left(2l_{\max}+1\right)+4l+1$,
$\eta_{0}=\tau_{0}+1$, $\nu_{0}=2\left(m-1\right)\left(2l_{\max}+1\right)+1$,
$\beta_{0}=\nu_{0}+1$,
$\alpha=2\left(m-1\right)\left(2l_{\max}+1\right)+4l^{\prime}-1$,
$\gamma=\alpha+1$,
$\nu=2\left(m-1\right)\left(2l_{\max}+1\right)+4l^{\prime}+1$, and
$\beta=\nu+1$. Following this way $\mathbb{G}\left(\boldsymbol{r}\right)$ (a
$2\mathtt{\times}2N\left(2l_{\max}\mathtt{+}1\right)$ matrix) is aligned as
$\mathbb{G}\left(\boldsymbol{r}\right)=\left(\begin{array}[c]{ccc}\widetilde{G}\left(\boldsymbol{r},\boldsymbol{r}_{1}\right),&\widetilde{G}\left(\boldsymbol{r},\boldsymbol{r}_{2}\right),&\cdots,\end{array}\begin{array}[c]{c}\widetilde{G}\left(\boldsymbol{r},\boldsymbol{r}_{N}\right)\end{array}\right)$
(B9)
with
$\widetilde{G}\left(\boldsymbol{r},\boldsymbol{r}_{i}\right)\mathtt{=}[\begin{array}[c]{ccc}G_{+0},&G_{+1},&G_{-1},\end{array}\cdots,\begin{array}[c]{cc}G_{+l_{\max}},&G_{-l_{\max}}\end{array}]$.
Explicitly, for $l=0$, we align
$\left(\begin{array}[c]{cc}\mathbb{G}\left(\boldsymbol{r}\right)_{1,\mu_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{1,\lambda_{0}}\\\
\mathbb{G}\left(\boldsymbol{r}\right)_{2,\mu_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{2,\lambda_{0}}\end{array}\right)=G_{+0}\left(\boldsymbol{r},\boldsymbol{r}_{n}\right),$
(B10)
and for $l\geq 1$,
$\displaystyle\left(\begin{array}[c]{cc}\mathbb{G}\left(\boldsymbol{r}\right)_{1,\alpha_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{1,\gamma_{0}}\\\
\mathbb{G}\left(\boldsymbol{r}\right)_{2,\alpha_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{2,\gamma_{0}}\end{array}\right)$
$\displaystyle=G_{+l}\left(\boldsymbol{r},\boldsymbol{r}_{n}\right),$ (B11)
$\displaystyle\left(\begin{array}[c]{cc}\mathbb{G}\left(\boldsymbol{r}\right)_{1,\tau_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{1,\eta_{0}}\\\
\mathbb{G}\left(\boldsymbol{r}\right)_{2,\tau_{0}}&\mathbb{G}\left(\boldsymbol{r}\right)_{2,\eta_{0}}\end{array}\right)$
$\displaystyle=G_{-l}\left(\boldsymbol{r},\boldsymbol{r}_{n}\right).$ (B12)
The $S$ matrix is diagonal,
$\mathbf{S}=\text{diag}\left(s_{\pm
l}^{\left(n\right)}\right)_{2N\left(2l_{\max}+1\right)\times
2N\left(2l_{\max}+1\right)},$ (B13)
with
$\mathbf{S}_{\mu_{0},\mu_{0}}=\mathbf{S}_{\lambda_{0},\lambda_{0}}=s_{+0}^{\left(n\right)}$,
$\mathbf{S}_{\alpha_{0},\alpha_{0}}=\mathbf{S}_{\gamma_{0},\gamma_{0}}=s_{+l}^{\left(n\right)}$,
$\mathbf{S}_{\tau_{0},\tau_{0}}=\mathbf{S}_{\eta_{0},\eta_{0}}=s_{-l}^{\left(n\right)}$.
Then we construct the $\mathbf{G}$ matrix, which is written as
$\mathbf{G}=\left(\begin{array}[c]{cccc}0&G\left(1,2\right)&\cdots&G\left(1,N\right)\\\
G\left(2,1\right)&0&\cdots&G\left(2,N\right)\\\ \vdots&\vdots&\ddots&\vdots\\\
G\left(N,1\right)&G\left(N,2\right)&\cdots&0\end{array}\right),$ (B14)
where $G\left(n,m\right)$ is a $2\left(2l_{\max}+1\right)\times
2\left(2l_{\max}+1\right)$ matrix, which are constructed by
$T_{l}^{-}\left[G_{\pm l^{\prime}}\left(n,m\right)\right]$. Explicitly, one
would align the $\mathbf{G}$ matrix by the following way
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\mu_{0},\nu_{0}\right)&\mathbf{G}\left(\mu_{0},\beta_{0}\right)\\\
\mathbf{G}\left(\lambda_{0},\nu_{0}\right)&\mathbf{G}\left(\lambda_{0},\beta_{0}\right)\end{array}\right)$
$\displaystyle=T_{0}^{-}\left[G_{+0}\left(n,m\right)\right],$ (B15)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\mu_{0},\alpha\right)&\mathbf{G}\left(\mu_{0},\gamma\right)\\\
\mathbf{G}\left(\lambda_{0},\alpha\right)&\mathbf{G}\left(\lambda_{0},\gamma\right)\end{array}\right)$
$\displaystyle=T_{0}^{-}\left[G_{+l^{\prime}}\left(n,m\right)\right],$ (B16)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\mu_{0},\nu\right)&\mathbf{G}\left(\mu_{0},\beta\right)\\\
\mathbf{G}\left(\lambda_{0},\nu\right)&\mathbf{G}\left(\lambda_{0},\beta\right)\end{array}\right)$
$\displaystyle=T_{0}^{-}\left[G_{-l^{\prime}}\left(n,m\right)\right],$ (B17)
for $l=0$ and $n\neq m$, and
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\alpha_{0},\nu_{0}\right)&\mathbf{G}\left(\alpha_{0},\beta_{0}\right)\\\
\mathbf{G}\left(\gamma_{0},\nu_{0}\right)&\mathbf{G}\left(\gamma_{0},\beta_{0}\right)\end{array}\right)$
$\displaystyle=T_{l}^{-}\left[G_{+0}\left(n,m\right)\right],$ (B18)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\alpha_{0},\alpha\right)&\mathbf{G}\left(\alpha_{0},\gamma\right)\\\
\mathbf{G}\left(\gamma_{0},\alpha\right)&\mathbf{G}\left(\gamma_{0},\gamma\right)\end{array}\right)$
$\displaystyle=T_{l}^{-}\left[G_{+l^{\prime}}\left(n,m\right)\right],$ (B19)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\alpha_{0},\nu\right)&\mathbf{G}\left(\alpha_{0},\beta\right)\\\
\mathbf{G}\left(\gamma_{0},\nu\right)&\mathbf{G}\left(\gamma_{0},\beta\right)\end{array}\right)$
$\displaystyle=T_{l}^{-}\left[G_{-l^{\prime}}\left(n,m\right)\right],$ (B20)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\tau_{0},\nu_{0}\right)&\mathbf{G}\left(\tau_{0},\beta_{0}\right)\\\
\mathbf{G}\left(\eta_{0},\nu_{0}\right)&\mathbf{G}\left(\eta_{0},\beta_{0}\right)\end{array}\right)$
$\displaystyle=T_{l}^{+}\left[G_{+0}\left(n,m\right)\right],$ (B21)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\tau_{0},\alpha\right)&\mathbf{G}\left(\tau_{0},\gamma\right)\\\
\mathbf{G}\left(\eta_{0},\alpha\right)&\mathbf{G}\left(\eta_{0},\gamma\right)\end{array}\right)$
$\displaystyle=T_{l}^{+}\left[G_{+l^{\prime}}\left(n,m\right)\right],$ (B22)
$\displaystyle\left(\begin{array}[c]{cc}\mathbf{G}\left(\tau_{0},\nu\right)&\mathbf{G}\left(\tau_{0},\beta\right)\\\
\mathbf{G}\left(\eta_{0},\nu\right)&\mathbf{G}\left(\eta_{0},\beta\right)\end{array}\right)$
$\displaystyle=T_{l}^{+}\left[G_{-l^{\prime}}^{k}\left(n,m\right)\right],$
(B23)
for $l\neq 0$ and $n\neq m$, while $\mathbf{G}\left(n,n\right)=0$ for $n=m.$
$\overrightarrow{\phi}$ can be written as a
$2N\left(2l_{\max}\mathtt{+}1\right)\mathtt{\times}1$ vector,
$\overrightarrow{\phi}=\left(\begin{array}[c]{cccc}\phi_{1},&\phi_{2},&\cdots,&\phi_{N}\end{array}\right)^{\text{T}},$
(B24)
where
$\phi_{i}=\left[\begin{array}[c]{cccc}T_{0}^{-}\left[\Phi\left(\boldsymbol{r}_{i}\right)\right],&T_{1}^{-}\left[\Phi\left(\boldsymbol{r}_{i}\right)\right],&T_{1}^{+}\left[\Phi\left(\boldsymbol{r}_{i}\right)\right],&\mathtt{\cdots},\end{array}\right.$
$\left.\begin{array}[c]{cc}T_{l_{\max}}^{-}\left[\Phi\left(\boldsymbol{r}_{i}\right)\right],&T_{l_{\max}}^{+}\left[\Phi\left(\boldsymbol{r}_{i}\right)\right]\end{array}\right]^{\text{T}}$.
Explicitly, for $l^{\prime}=0,$
$\overrightarrow{\phi}_{\nu_{0}}=P_{0}^{-}\left[e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right],\overrightarrow{\phi}_{\beta_{0}}=iP_{1}^{-}\left[\mp
ie^{i\theta_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right].$
(B25)
For $l^{\prime}>0,$
$\displaystyle\overrightarrow{\phi}_{\alpha}$
$\displaystyle=P_{l^{\prime}}^{-}\left[e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right],\overrightarrow{\phi}_{\gamma}=iP_{l^{\prime}+1}^{-}\left[\mp
ie^{i\theta_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right],$
(B26) $\displaystyle\overrightarrow{\phi}_{\nu}$
$\displaystyle=P_{l^{\prime}}^{+}\left[e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right],\overrightarrow{\phi}_{\beta}=iP_{l^{\prime}+1}^{+}\left[\pm
ie^{i\theta_{k}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}_{n}}/\sqrt{2}\right].$
(B27)
## References
* (1) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
* (2) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
* (3) Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. 102, 156603 (2009).
* (4) R. R. Biswas and A. V. Balatsky, Phys. Rev. B 81, 233405(R) (2010).
* (5) R. R. Biswas and A. V. Balatsky, Phys. Rev. B 83, 075439 (2011).
* (6) A. M. Black-Schaffer, and A. V. Balatsky, Phys. Rev. B 85, 121103(R) (2012).
* (7) T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, Phys. Rev. Leet. 103, 266803 (2009).
* (8) P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, A. Yazdani, Nature 460, 1106 (2009).
* (9) J. Wang, W. Li, P. Cheng, C. Song, T. Zhang, P. Deng, X. Chen, X. Ma, K. He, J.-F. Jia, Q.-K. Xue, and B.-F. Zhu, Phys. Rev. B 84, 2135447 (2011).
* (10) Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, Phys. Rev. Lett. 104, 016401 (2011).
* (11) Z. Alpichshev, R. R. Biswas, A. V. Balatsky, J. G. Analytis, J.-H. Chu, I. R. Fisher, A. Kapitulnik, Phys. Rev. Lett. 108, 206402 (2012).
* (12) Z.-G. Fu, P. Zhang, Z. Wang, and S.-S. Li, Appl. Phys. Lett. 99, 232109 (2011).
* (13) Z.-G. Fu, P. Zhang, Z. Wang, and S.-S. Li, Phys. Rev. B, 84, 235438 (2011).
* (14) H.-T. He, G. Wang, T. Zhang, I.-K. Sou, G. K. L Wong, J.-N. Wang, H.-Z. Lu, S.-Q. Shen, and F.-C. Zhang, Phys. Rev. Lett. 106, 166805 (2011).
* (15) Z.-G. Fu, P. Zhang, and S.-S. Li, Appl. Phys. Lett. 99, 243110 (2011).
* (16) M. Ye, S. V. Eremeev, K. Kuroda, E. E. Krasovskii, E. V. Chulkov, Y. Takeda, Y. Saitoh, K. Okamoto, S. Y. Zhu, K. Miyamoto, M. Arita, M. Nakatake, T. Okuda, Y. Ueda, K. Shimada, H. Namatame, M. Taniguchi, A. Kimura, phys. Rev. B 85, 205317 (2012).
* (17) A. Zazunov, A. Kundu, A. Hütten, and R. Egger, Phys. Rev. B 82, 155431 (2010).
* (18) H.-J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S-.C. Zhang, Nature Phys. 5, 438 (2009).
* (19) J. J. Cha, J. R. Williams, D. Kong, S. Meister, H. Peng, A. J. Bestwick, P. Gallagher, D. Goldhaber-Gordon, and Y. Cui, Nano Lett. 10, 1076, (2010).
* (20) J. D. Walls, J. Huang, R. M. Westervelt, and E. J. Heller, Phys. Rev. B 73, 035325 (2006).
|
arxiv-papers
| 2012-03-27T09:39:31 |
2024-09-04T02:49:29.107512
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhen-Guo Fu, Ping Zhang, Zhigang Wang, Fawei Zheng, and Shu-Shen Li",
"submitter": "Zhen-Guo Fu",
"url": "https://arxiv.org/abs/1203.5910"
}
|
1203.5955
|
# Empirical Likelihood for Right Censored Lifetime Data
Shuyuan He1∗, Wei Liang2∗, Junshan Shen3∗ and Grace Yang 4 111Research of
authors 1, 2 and 3 were Supported by National Natural Science Foundation of
China(11171230, 10801003). Research of author 4 was supported by the US
National Science Foundation, while working at the Foundation.
Capital Normal University1, Xiamen University2,
Peking University3, University of Maryland4
Abstract
This paper considers the empirical likelihood (EL) construction of confidence
intervals for a linear functional ${\theta}$ based on right censored lifetime
data. Many of the results in literature show that $-2\log$(empirical
likelihood ratio) has a limiting scaled-$\chi^{2}_{1}$ distribution, where the
scale parameter is a function of the unknown asymptotic variance. The scale
parameter has to be estimated for the construction. Additional estimation
would reduce the coverage accuracy for ${\theta}$. This diminishes a main
advantage of the EL method for censored data. By utilizing certain influence
functions in an estimating equation, it is shown that under very general
conditions, $-2\log$(EL ratio) converges weakly to a standard $\chi^{2}_{1}$
distribution and thereby eliminates the need for estimating the scale
parameter. Moreover, a special way of employing influence functions eases the
otherwise very demanding computations of the EL method. Our approach yields
smaller asymptotic variance of the influence function than those comparable
ones considered by Wang & Jing (2001) and Qin & Zhao (2007). Thus it is not
surprising that confidence intervals using influence functions give a better
coverage accuracy as demonstrated by simulations.
Key words and phrases. Empirical likelihood, Right censored lifetimes,
Influence function, Parameter estimation, Confidence intervals
AMS 1991 subject classifications. Primary 62H99; Secondary 62H05.
## 1 Introduction
Let $Y$ be a non-negative random variable with distribution function
$F(y)=P[Y\leq y]$. Let $C$ be a nonnegative random variable with distribution
function $G(x)=P[C\leq x]$ and $C$ is independent of $Y$. Instead of $Y$, we
observe $Z=\min(Y,C)$ and the indicator $\delta=I[Y\leq C]$ of the event
$[Y\leq C].$ Both $F$ and $G$ are assumed continuous but unknown. In this
paper we use the empirical likelihood method with right-censored data to study
the problem of constructing confidence intervals for ${\theta}$, a functional
of the distribution $F$ defined by $\text{E}g(Y,{\theta})=0$. For example, if
$g(x,{\theta})=\xi(x)-{\theta}$, for some function $\xi$ we get
${\theta}=\text{E}\,\xi(Y)=\int_{0}^{\infty}\xi(x)\,\text{d}F(x).$ (1. 1)
The problem of estimating ${\theta}$ in (1.1) with a sample of $n$ i.i.d.
observations of $(Z,\delta)$ has been studied by many authors. If
$\xi(Y)=I[Y\leq y]$ for a fixed $y$, then ${\theta}=F(y)$. A well-known
nonparametric maximum likelihood and asymptotically optimal estimator of
$F(y)$ is the Kaplan-Meier (KM) estimator $F_{n}$ as defined in (2.3). A
natural estimator of ${\theta}$ is
${\theta}_{n}=\int_{0}^{\infty}\xi(s)\,\text{d}F_{n}(s).$ (1. 2)
For an arbitrary function $\xi$, several authors, e.g. Yang (1994) have shown
that under the condition of finite second moment,
$\int_{0}^{\infty}\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)<\infty,$
(1. 3)
the asymptotic distribution of $\sqrt{n}({\theta}_{n}-{\theta})$, as $n$ goes
to infinity, is normal $N(0,\,\sigma^{2})$, where
$\sigma^{2}=\int_{0}^{\infty}\frac{(\overline{F}(s)\xi(s)-\psi(s))^{2}}{\overline{F}^{2}(s)\overline{G}(s)}\,\text{d}F(s),\
\psi(s)=\int_{s}^{\infty}\xi(x)\,\text{d}F(x),s\geq 0,$ (1. 4)
and $\overline{G}=1-G,\overline{F}=1-F$.
Confidence intervals for ${\theta}$ can be constructed using the asymptotic
normal distribution $N(0,\,\sigma^{2})$. Alternatively, the EL method can be
used as to be investigated in this paper. Employing either method, one needs
to deal with a rather complicated form of the asymptotic variance
$\sigma^{2}$. Among other things, it is computationally demanding.
To use the normal distribution $N(0,\,\sigma^{2})$, it is necessary to
estimate the unknown variance $\sigma^{2}$. Stute (1996) proposed a jackknife
estimator to replace $\sigma^{2}$ in the calculation. Although any consistent
estimator $\sigma_{n}^{2}$ of $\sigma^{2}$ can be used, the convergence rate
of $\sigma_{n}^{2}$ is generally unknown. Substitution by the estimate
$\sigma_{n}^{2}$ tends to reduce the coverage accuracy for ${\theta}$ as
compare to the case of known $\sigma^{2}$.
The usefulness of the EL method for constructing confidence interval/regions
has been well established in a wide variety of situations, see e.g., DiCiccio
et al. (1991) and Chen (1994), and an extensive literature review in Owen
(2001) to that day. Let $R({\theta})$ denote the EL ratio function of a one-
dimensional parameter ${\theta}$ for $n$ i.i.d “complete” observations. Owen
(1988) proved that under certain regularity conditions, $-2\log R({\theta})$
converges to a chi-squared distribution with one degree of freedom. The EL
method gives confidence intervals for ${\theta}$ as $\\{{\theta}:\,-2\log
R({\theta})\leq c_{1-\alpha}\\}$, where $c_{1-\alpha}$ is the $(1-\alpha)$th
quantile of the $\chi_{1}^{2}$ distribution. Here the construction of
confidence intervals does not require estimation of asymptotic variance. In
view of a complicated variance formula in (1.4), this would have provided a
welcome method for censored data. However, as far as we know, for censored
data, the asymptotic standard chi-squared distribution holds only in some
special cases see, e.g. Owen (Chapter 6, 2001). More recent literature shows
that most of the asymptotic distributions involve weights which are functions
of unknown variances or covariance matrices. This is the case, for example, in
the following papers. Li and Wang (2003) studied right-censored regression
models, Ren (2008) used weighted EL under a variety of censoring models, Wang
& Jing (2001) and Qin & Zhao (2007) estimated functionals ${\theta}$, and
Hjort& McKeague & van Keilegom, in their extension of the scope of the EL
method (2009), obtained an asymptotic distribution (Theorem 2.1) which is a
sum of weighted chi-squared distributions with unknown weights. Therefore,
using these results to construct confidence intervals for ${\theta}$ still
require an additional estimation of the unknown $\sigma^{2}$. This diminishes
a main advantage of the EL method for censored data.
The EL ratio $R({\theta})$ is obtained by utilizing auxiliary information on
${\theta}$ through a set of estimating equations. In this paper, we show that
by using certain influence functions with a special construction of estimating
equations in the EL ratio, the asymptotic distribution of -2$\log R({\theta})$
of the functional ${\theta}$ in (1.1) is a standard $\chi_{1}^{2}$ without
involving any unknown scale parameter. Our approach transfers the problem of
estimating $\sigma^{2}$ to the influence functions. As a result, it also
significantly simplifies the often intensive computations of the EL method for
censored data.
Our work is motivated by the work of Wang & Jing (2001) and Qin & Zhao (2007).
Wang & Jing (2001) obtained an EL ratio by first finding an estimating
equation for a certain complete sample and then modifying the estimating
equation for the right censored sample. The resulting estimating equation is a
sum instead of a product (inherent of the product limit estimator). With this
approach, Wang & Jing (2001) use the estimating function
$M_{1}(Z,\delta,{\theta})$ for ${\theta}$ (see (3.7) ) and Qin & Zhao (2007)
use $M_{2}(Z,\delta,{\theta})$ (see (3.8)) for estimating the mean residual
life $\text{E}(Y-t_{0}|Y\geq t_{0})$ at age $t_{0}.$ However, both of these
papers obtain an asymptotic scaled $\chi^{2}_{1}$ distribution for $-2\log
R({\theta}).$
Instead of $M_{1}$ and $M_{2}$, we use influence functions. We compute the
influence functions $W(Z,\delta,\theta)$ of
$\mu_{n}=\int_{0}^{\infty}g(x,{\theta})\,\text{d}F_{n}(x),$
as defined by (3.6), where $F_{n}$ is the Kaplan-Meier estimator. The
influence functions $W^{\prime}s$ are to be utilized to construct an
estimation function for the EL method. Numerous examples of the function $g$
are given in Section 2.
The paper is organized as follows. Preliminaries assumptions and examples of
${\theta}$ and $g$ are given in Section 2. The influence function
$W(Z,\delta,{\theta})$ are given in Section 3. It is shown in Theorem 3.1 that
asymptotically $\sqrt{n}(\mu_{n}-\mu)$ is a partial sum of $n$ independent
influence functions $W(Z_{j},\delta_{j},{\theta})$ (an IID representation), or
is asymptotically linear. Here $\mu$ denotes $E\,g(Y,{\theta})$ and the the
condition $\text{E}\,g(X,{\theta})=0$ is not imposed in Section 3. These
results are general for any $\xi(x)$ having finite second moment (1.3) and no
restrictions are placed on the upper boundaries of $X$ and $C$.
An IID representation of the Kaplan-Meier estimator has been obtained by many
authors e.g., Lo & Singh (1986), Stute & Wang (1993) and Chen & Lo (1997)
using different approaches, under different conditions and in different forms.
See Yang (1997) and references therein. Here, we use an influence function
$W(Z,\delta,{\theta})$ of ${\theta}_{n}$ in (1.2) obtained in He &
Huang(2003). We show that the variance of the influence function is smaller
than that of $M_{1}$ and $M_{2}$ (see Remark 3.1 in Section 3), which results
in an improvement of the asymptotic coverage accuracies of ${\theta}$. In
Section 4, estimation of the influence functions is carried out in Theorem
4.1. The weak convergence of $-2\log R({\theta})$ to the standard
$\chi_{1}^{2}$ distribution without any scale parameter is proved in Theorem
4.2 which justifies the EL construction of confidence intervals for censored
data. In Section 5, simulation comparison of the new method with that of the
scaled $\chi_{1}^{2}$ distribution is presented. The amount of improvement
depends on the form of ${\theta}$. For survival function or mean, the coverage
ratios computing from the traditional normal approximation and the EL method
are about the same. The EL method performs better for more complicated
${\theta}$. Most of the proofs are relegated to the Appendix.
## 2 Preliminaries, Assumptions and Examples
For any right continuous monotone function $h({x})$, let $h({x-})$ or
$h_{-}(x)$ denote the left continuous version of $h({x})$ and the curly
brackets $h\\{{x}\\}$ denote the difference $h({x})-h({x-})$. Then
$h\\{x\\}=\text{d}h(x)$. For any cumulative distribution function $F$, let
$\overline{F}=1-F$. • Assume that
$F(x)=P(Y\leq x),\quad G(x)=P(C\leq x)\ \ \text{and}\ \ H(x)=P(Z\leq x)$ (2.
1)
are continuous cumulative distributions of $Y$, $C$ and $Z=\min(Y,C)$,
respectively. Let $[0,b_{H}]$ be the range of $H$, where
$b_{H}=\sup\\{x:H(x)<1\\}.$
$b_{F}$ and $b_{G}$ are similarly defined for $F$ and $G$. Then
$b_{H}=\min(b_{F},b_{G})$.
Given a sample of $n$ i.i.d. random vectors $(Z_{i},\delta_{i})$,
$i=1,2,\cdots,n$, of $(Z,\delta)$, their empirical distribution functions are
given by:
$\displaystyle H_{n}^{1}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x,\delta_{j}=1],$ $\displaystyle
H_{n}^{0}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x,\delta_{j}=0],$ (2. 2) $\displaystyle
H_{n}(x)=H_{n}^{0}(x)+H_{n}^{1}(x)=\frac{1}{n}\displaystyle\sum^{n}_{j=1}I[Z_{j}\leq
x].$
Asymptotic optimal nonparametric estimators of $F(x)$ and $G(x)$ are the well-
known Kaplan-Meier estimators given by
$F_{n}(x)=1-\prod_{s\leq
x}\left[1-\frac{H^{1}_{n}\\{s\\}}{{\overline{H}}_{n}(s-)}\right]\hbox{ and }\
\ G_{n}(x)=1-\prod_{s\leq
x}\left[1-\frac{H^{0}_{n}\\{s\\}}{{\overline{H}}_{n}(s-)}\right],$ (2. 3)
respectively, where an empty product is set equal to one. It can be checked
that for all $x$,
${\overline{H}}_{n}(x)={\overline{F}}_{n}(x){\overline{G}}_{n}(x).$ (2. 4)
Applying (2.3) and (2.4) we get
$\displaystyle\text{d}F_{n}(x)=F_{n}(x)-F_{n}(x-)=\overline{F}_{n}(x-)\frac{\text{d}H_{n}^{1}(x)}{{\overline{F}}_{n}(x-){\overline{G}}_{n}(x-)}.$
(2. 5)
It follows that
$\text{d}H_{n}^{1}(x)={\overline{G}}_{n}(x-)\,\text{d}F_{n}(x),\quad\text{d}H_{n}^{0}(x)={\overline{F}}_{n}(x-)\,\text{d}G_{n}(x).$
(2. 6)
Put
$\displaystyle H^{0}(x)=P(Z\leq x,\delta=0),\quad H^{1}(x)=P(Z\leq
x,\delta=1),\mbox{and}$ (2. 7) $\displaystyle\overline{H}(x)=P(Z>x).$
Then
$\displaystyle
H^{0}(x)=\text{E}\,H_{n}^{0}(x)=\int_{0}^{x}\overline{F}(s)\,\text{d}G(s),$
$\displaystyle
H^{1}(x)=\text{E}\,H_{n}^{1}(x)=\int_{0}^{x}\overline{G}(s)\,\text{d}F(s),$
(2. 8)
$\displaystyle\overline{H}(x)=\text{E}\,\overline{H}_{n}(x)=\overline{F}(x)\overline{G}(x).$
Here and after, the integral sign $\int^{b}_{a}$ stands for $\int_{(a,b]}$ and
$\int$ stands for $\int_{0}^{\infty}$.
Examples of ${\theta}$ and $g(x,{\theta})$
1\. $g(x,{\theta})=I[x>y]-{\theta}$ with $y$ fixed. Then
$\text{E}\,g(Y,{\theta})=\int(I[x>y]-{\theta})\,\text{d}F(x)=0.$
Solving this equation yields ${\theta}=\overline{F}(y)$, the survival function
of $Y$.
2\. $g(x,{\theta})=x^{k}-{\theta}$. Then ${\theta}=\text{E}\,Y^{k}$, the $k$th
moments of $Y$.
3\. $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$ with $t_{0}$ fixed. Then
${\theta}=\text{E}(Y-t_{0}|Y\geq t_{0})=\frac{\text{E}(Y-t_{0})I[Y\geq
t_{0}]}{P(Y\geq t_{0})},$
the mean residual life of $Y$.
4\. $g(x,{\theta})=x(I[x>y]-{\theta})$. Then
${\theta}=\frac{1}{\text{E}\,Y}\int_{y}^{\infty}s\,\text{d}F(s),$
the length biased survival function of $Y$. See Vardi (1982), for example.
5\. $g(x,{\theta})=x^{2}-{\theta}x$. Then
${\theta}=\frac{1}{\text{E}\,Y}\int_{0}^{\infty}x^{2}\,\text{d}F(x),$
the mean of the length-biased lifetime.
6\. $g(x,{\theta})=x^{2}-2{\theta}x$. Then
${\theta}=\frac{1}{2\text{E}\,Y}\int_{0}^{\infty}x^{2}\,\text{d}F(x),$
the mean of the length biased residual lifetime.
7\. $g(x,{\theta})=I[x\leq{\theta}]-p$ with $p\in(0,1)$. Then
${\theta}=F^{-1}(p)$, the $p$th quantile of $Y$.
Examples (4)-(6) often appear in renewal processes and their applications.
## 3 Influence function of $\mu_{n}$
Throughout Section 3, ${\theta}$ is a fixed value. Then it is convenient to
suppress ${\theta}$ in the exposition, by setting $\xi(x)=g(x,{\theta})$,
$\mu=\text{E}\,g(X,{\theta})$, and
$\displaystyle\mu_{n}=\int\xi(x)\,\text{d}F_{n}(x)=\int\frac{\xi(x)}{\overline{G}_{n}(x-)}\,\text{d}H_{n}^{1}(x).$
(3. 1)
Likewise, set $W=W(Z,\delta)=W(Z,\delta,{\theta}).$
In Theorem 3.1, we prove that the estimator $\mu_{n}$ for $\mu$ is asymptotic
linear. That is, there is a function $W=W(Z,\delta)$, such that
$\text{E}\,W=0$, $\text{Var}(W)<\infty$ and
$\sqrt{n}(\mu_{n}-\mu)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W(Z_{i},\delta_{i})+o_{p}(1).$
The function $W(Z,\delta)$ is defined with respect to the true distributions
$(F,G)$. Following literature, we call $W(Z_{i},\delta_{i})$ the $i$-th
influence function of $\mu_{n}$. See, for e.g. van der Vaart (1998) or Tsiatis
(2006).
Theorem 3.1 will be proved by first establishing a similar result for the
truncated $W(Z_{i},\delta_{i})$ as defined in (3.3). Let
$\xi_{b}(x)=\xi(x)I[x\leq b]$ be the restriction of $\xi(x)$ on $(-\infty,b]$,
where $b<b_{H}$ is an arbitrarily chosen constant. By similar truncation, put
$\mu_{b}=\int\xi_{b}(x)\,\text{d}F(x),\ \ \psi_{b}(s)=\int_{x\geq
s}\xi_{b}(x)\,\text{d}F(x),\ \
{\overline{\delta}}_{i}=I[Y_{i}>C_{i}]=1-\delta_{i}.$ (3. 2)
We consider the i.i.d. random variables,
$W_{i}(b)=\frac{\xi_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\mu_{b}+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi_{b}(Z_{i})-\int\psi_{b}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}\,\text{d}H^{0}(s),$ (3. 3)
for $i=1,\cdots,n.$
Under finite variance condition (1.3), it can be calculated that,
$\displaystyle\text{E}\,\frac{\xi_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}$
$\displaystyle=$ $\displaystyle\mu_{b},\ \ \text{E}\,W_{i}(b)=0,$
$\displaystyle\text{Var}(W_{i}(b))$ $\displaystyle=$
$\displaystyle\int\frac{\xi_{b}^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu_{b}^{2}-\int\frac{\psi_{b}^{2}(s)}{\overline{F}(s)\overline{G}^{2}(s)}\,\text{d}G(s).$
(3. 4)
As $b$ approaches the upper bound $b_{H}$, we have
$\mu_{b}\to\mu,\quad\psi(s)=\lim_{b\to b_{H}}\psi_{b}(s)=\int_{x\geq
s}\xi(x)\,\text{d}F(x),\quad\mbox{and}$ (3. 5)
$W_{i}=\lim_{b\to
b_{H}}W_{i}(b)=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\mu+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i})-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}\,\text{d}H^{0}(s),$ (3. 6)
for $i=1,\cdots,n$. The $W_{i}^{\prime}s$, for $i=1,\cdots,n,$ are i.i.d.
random variables.
Wang & Jing (2001) use the estimating function based on
$M_{1}(Z,\delta,{\theta})=\frac{\xi(Z)\delta}{\overline{G}(Z)}-{\theta}$ (3.
7)
to estimate ${\theta}$ in (1.1). Qin & Zhao (2007) used the estimating
function based on
$M_{2}(Z,\delta,{\theta})=\frac{g(Z,{\theta})\delta}{\overline{G}(Z)},$ (3. 8)
where $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$, to estimate the mean
residual life ${\theta}=\text{E}(Y-t_{0}|Y\geq t_{0})$ at a specified age
$t_{0}.$ This case is covered in our formulation, see example (3) in Section
2\. Comparing with $M_{1}$ and $M_{2}$, our $W_{i}^{\prime}s$ contain two
additional terms. Note that $W_{i}^{\prime}s$ are not observable random
variables and whose estimation will be addressed in Section 4.
Under finite variance condition (1.3), applying the dominated convergence
theorem and the Lebesgue-Stieltjes integration by parts, we obtain
$\displaystyle\text{E}\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}$
$\displaystyle=$ $\displaystyle\mu,\ \ \text{E}\,W_{i}=0,$ (3. 9)
$\displaystyle\text{Var}(W_{i})$ $\displaystyle=$
$\displaystyle\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2}-\int\frac{\psi^{2}(s)}{\overline{F}(s)\overline{G}^{2}(s)}\,\text{d}G(s)$
(3. 10) $\displaystyle=$
$\displaystyle\int\frac{(\overline{F}(s)\xi(s)-\psi(s))^{2}}{\overline{F}^{2}(s)\overline{G}(s)}\,\text{d}F(s).$
(3. 11)
Remark 3.1 Formulas (3.6) are (3.10) are obtained in He $\&$ Huang (2003), and
(3.11) is given in Yang (1994). Under condition (1.3), it can be shown that
(3.10) and (3.11) are equal. The variance of $W_{i}$ is smaller than that of
$M_{1}$ and $M_{2}$ defined in (3.7) and (3.8). The variance of the latter two
equals
$\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2}$
with the corresponding choices of $\xi(z)=g(z,{\theta})$, in $M_{1}$ and
$M_{2}$.
We proceed to prove Theorem 3.1. For the restricted $W_{i}(b)^{\prime}s$, the
following lemma is taken from (3.11) of He & Huang (2003).
Lemma 3.1 Let $F$ and $G$ be continuous. For each ${\theta}$ fixed, set
$\xi(x)=g(x,{\theta})$. Assume that $\int\xi^{2}(x)\,\text{d}F(x)<\infty$ and
$b<b_{H}$. Let $\xi_{b}(x)$ be the restriction of $\xi$ on $(0,b]$ for
$b<b_{H}$. Then, as $n\to\infty$,
$\sqrt{n}\int\xi_{b}(x)\,\text{d}(F_{n}(x)-F(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}(b)+o_{p}(1).$
The following result will be used repeatedly and for easy reference, it is
stated in Lemma 3.2. Its proof is given the appendix.
Lemma 3.2 For $b<b_{H}$, let $\\{h_{n}(b)\\}$ be a random sequence such that
$h_{n}(b)\to h(b)$ in distribution as $n\to\infty$, and $h(b)=o_{p}(1)$ as
$b\to b_{H}$. As $n\to\infty$, if $V_{n}=O_{p}(1)$ and the random sequence
$\\{S_{n}\\}$ can be written as $S_{n}=o_{p}(1)+V_{n}h_{n}(b)$ for any
$b<b_{H}$, then $S_{n}=o_{p}(1)$.
Remark 3.2 In what follows, $h_{n}(b)$ is used as a generic notation to denote
any random sequence $\\{h_{n}(b)\\}$ that satisfies the assumptions of Lemma
3.2. This simplifies many of the statements later. For example, under
condition (1.3) and $b<b_{H}$, put
$\displaystyle
h_{n}(b)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s).$
(3. 12)
Then, by the SLLN and (2.8)
$\displaystyle\lim_{n\to\infty}h_{n}(b)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H^{1}(s)$
$\displaystyle=$
$\displaystyle\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\overline{G}(s)\,\text{d}F(s)=h(b)\to
0,\ \mbox{as $b\to b_{H}.$}$
###### Theorem 3.1.
Let $W_{i}$ be given by (3.6). Suppose $F$ and $G$ are continuous, and for
each fixed ${\theta}$ set $\xi(x)=g(x,{\theta})$. Then under condition (1.3)
as $n\to\infty$,
$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$
###### Proof.
For $b<b_{H}$, put $\overline{\xi}_{b}=\xi(x)-\xi_{b}(x)=\xi(x)I[x>b]$.
Decompose the following difference as
$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}\equiv
J_{1}(b)+J_{2}(b)-J_{3}(b),$ (3. 13)
where
$\displaystyle
J_{1}(b)=\sqrt{n}\int\xi_{b}(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}(b),$
$\displaystyle
J_{2}(b)=\sqrt{n}\int\overline{\xi}_{b}(x)\,\text{d}(F_{n}(x)-F(x)),$
$\displaystyle J_{3}(b)=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(W_{i}-W_{i}(b)).$
Let $\overline{\psi}_{b}(x)=\int_{s\geq
x}\overline{\xi}_{b}(s)\,\text{d}F(s)$. It follows that
$\overline{\psi}_{b}^{2}(x)\leq\overline{F}(x)^{2}\int_{s\geq
x}\overline{\xi}_{b}^{2}(s)\,\text{d}F(s).$ (3. 14)
By Lemma 3.1, $J_{1}(b)=o_{p}(1)$. We shall show that $J_{2}(b)=h_{n}(b)$ and
$J_{3}(b)=h_{n}(b)$ as $n\to\infty$. Applying Corollary 1 of Yang (1994),
$J_{2}(b)$ converges weakly to $N(0,\,\overline{\sigma}_{b}^{2})$, where
$\overline{\sigma}_{b}^{2}$ is similarly defined as in (3.12) with $\xi$ and
$\psi$ replacing by their restrictions $\overline{\xi}_{b}$ and
$\overline{\psi}_{b}$ respectively.
Now
$\displaystyle\overline{\sigma}_{b}^{2}\leq\int\frac{\overline{\xi}_{b}^{2}(x)}{\overline{G}(x)}\,\text{d}F(x)\to
0,\ \text{as}\ b\to b_{H}.$ (3. 15)
Therefore $J_{2}(b)$ converges to $h(b)=Z_{0}\overline{\sigma}_{b}$ in
distribution, and $h(b)=o_{p}(1)$ as $b\to b_{H}$, where $Z_{0}$ is a
$N(0,\,1)$ random variable. It follows that $J_{2}(b)=h_{n}(b)$.
To prove $J_{3}(b)=h_{n}(b)$, note that the difference $W_{i}-W_{i}(b)$, as
given in (3.6) and (3.3) equals to
$\frac{\overline{\xi}_{b}(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}-\int\overline{\xi}_{b}(x)\,\text{d}F(x)+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\overline{\psi}_{b}(Z_{i})-\int\overline{\psi}_{b}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s).$ (3. 16)
Therefore $W_{i}-W_{i}(b)$ are i.i.d. random variables with mean zero and
variance $\overline{\sigma}_{b}^{2}$. Hence, $J_{3}(b)=h_{n}(b)$ follows for
the same reason as that of $J_{2}(b)$.
We conclude that the following holds for (3.13),
$\sqrt{n}\int\xi(x)\,\text{d}(F_{n}(x)-F(x))-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}=o_{p}(1)+h_{n}(b)+h_{n}(b).$
Theorem 3.1 follows from Lemma 3.2. ∎
Remark 3.3 If $b_{F}<b_{G}$, then (1.3) is equivalent to $\xi$ having finite
second moment. If $\xi$ is bounded and away from zero, then (1.3) is
equivalent to $\int_{0}^{\infty}\,\text{d}F(s)/{\overline{G}(s)}.$
## 4 Empirical Likelihood Ratios and Confidence Intervals for ${\theta}$
To develop an EL inference procedure, we consider a specific $g(x,{\theta})$.
For each fixed ${\theta}$, as before, set $\xi(x)=g(x,{\theta})$. We shall
utilize the i.i.d. random variables
$\displaystyle
W_{i}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})}+\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i})-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s).$ (4. 1)
to obtain an estimating equation for the EL ratio. Recall that
$\mu=\text{E}\,W_{i}=\int_{0}^{\infty}\xi(x)\,\text{d}F(x)$ and
$\text{Var}(W_{i})$ are given by (3.10). Note that setting
$\xi(x)=g(x,{\theta})$ above has nothing to do with defining ${\theta}$ from
the equation $\text{E}\,g(X,{\theta})=0$ as given in (1.5). If, however, the
true parameter ${\theta}_{0}$ is the solution of the equation
$\text{E}\,g(Y,{\theta})=\int g(x,{\theta})\,\text{d}F(x)=0,$ (4. 2)
then $\xi(x)=g(x,{\theta}_{0})$ is such that
$\mu=\text{E}\,W_{i}=\int_{0}^{\infty}\xi(x)\,\text{d}F(x)=0.$
Regarding $W_{i}$ for $i=1,\cdots,n$ as a “complete” random sample, one could
formulate an EL likelihood ratio $R({\theta}_{0})$ with multinomial
probability $p_{i}$ assigned to $W_{i}$ and the constraint
$\sum_{i=1}^{n}W_{i}p_{i}=0$. However, $W_{i}^{\prime}s$ are not observable
because of the unknown distributions $G$, $F$ $H$ and $H^{0}$. We shall
replace them by the KM estimates, $F_{n}$, $G_{n}$ given by (2.3) and an
estimate of $\psi$,
$\psi_{n}(x)=\int_{s\geq x}\xi(s)\,\text{d}F_{n}(s).$ (4. 3)
Replacing $\overline{G},\overline{H},H^{0}$ in (4.1) by their corresponding
empirical distributions in (2.2) gives an approximation of $W_{i}$ in (4.1) by
$W_{ni}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}\,\text{d}H_{n}^{0}(s).$ (4. 4)
The price to pay for the estimation is that ${W_{ni}}^{\prime}s$ are not
stochastically independent which complicates the ensuing analysis.
The following theorem indicates the possibility of using $W_{ni}$ to construct
empirical likelihood ratio and to obtain asymptotically a standard $\chi^{2}$
distribution.
###### Theorem 4.1.
Let $W_{ni}$ be given by (4.4) and $\text{E}\,\xi(Y)=0$. Then under condition
(1.3), as $n\to\infty$, we have
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$
(4. 5)
###### Proof.
By (2.6), we have
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}W_{ni}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Big{(}\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)\Big{)}$ $\displaystyle=$
$\displaystyle\int\frac{\xi(s)}{\overline{G}_{n}(s-)}\,\text{d}H^{1}_{n}(s)+\int\frac{\psi_{n}(s)}{{\overline{H}_{n}}(s-)}\,\text{d}H_{n}^{0}(s)-\int\psi_{n}(s)\frac{\overline{H}_{n}(s-)}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)$
$\displaystyle=$
$\displaystyle\int\frac{\xi(s)}{\overline{G}_{n}(s-)}\,\text{d}H^{1}_{n}(s)=\int\xi(s)\,\text{d}F_{n}(s).$
Applying $\int_{0}^{\infty}\xi(s)\,\text{d}F(s)=\text{E}\xi(Y)=0$ and Theorem
3.1, we arrive at
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\sqrt{n}\int\xi(s)\,\text{d}(F_{n}(s)-F(s))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1).$
(4. 6)
∎
Following Owen (2001), define the EL ratio of ${\theta}$ by a multinomial
likelihood subject to constraints as
$R({\theta})=\sup_{\\{p_{i}\\}}\left\\{\ \prod_{i=1}^{n}np_{i}\Big{|}\
\sum_{i=1}^{n}p_{i}=1,\,\sum_{i=1}^{n}p_{i}W_{ni}=0,\,p_{i}\geq
0,\,i=1,2,\cdots,n\right\\}.$ (4. 7)
To determine $R({\theta})$, we solve, as usual, for the Lagrange multipliers
$\mu$ and $\lambda$ in
$A=\sum_{i=1}^{n}\log(np_{i})-n\lambda(\sum_{i=1}^{n}p_{i}W_{ni})-\mu(1-\sum_{i=1}^{n}p_{i}).$
Then $\mu=-n$ and $p_{i}=\dfrac{1}{n}(1+\lambda W_{ni})^{-1},\
i=1,2,\cdots,n,$ where $\lambda$ is the solution of
$\dfrac{1}{n}\sum\limits_{i=1}^{n}\dfrac{W_{ni}}{1+\lambda W_{ni}}=0.$
The uniqueness of $\lambda$ will be addressed in the proof of Theorem 4.2. The
EL ratio of ${\theta}$ can be written as
$R({\theta})=\prod_{i=1}^{n}(np_{i})=\prod_{i=1}^{n}(1+\lambda W_{ni})^{-1}.$
(4. 8)
###### Theorem 4.2.
Suppose that ${\theta}_{0}$ is the unique solution of (4.2) and finite second
moment (1.3) holds. Set $\xi(x)=g(x,{\theta}_{0})$. Then
$l({\theta}_{0})=-2\log R({\theta}_{0})$ converges in distribution to a
$\chi_{1}^{2}$ random variable with one degree of freedom, as $n\to\infty$.
Applying Theorem 4.2, confidence intervals for ${\theta}$ can be constructed
as
$I_{1}=\\{{\theta}:l({\theta})\leq c_{1-\alpha}\\},$ (4. 9)
where $c_{1-\alpha}$ is the $(1-\alpha)$th quantile of the $\chi_{1}^{2}$
distribution. $I_{1}$ has asymptotic coverage probability of $1-\alpha$, as
$n\to\infty.$
To prove Theorem 4.2, we shall make use of the following Taylor’s expansion of
$-2\log R({\theta}_{0})$,
$\displaystyle-2\log R({\theta}_{0})$ $\displaystyle=$ $\displaystyle
2\sum_{i=1}^{n}\ln(1+\lambda_{n}W_{ni})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}2\Big{(}\lambda_{n}W_{ni}-\frac{1}{2}\lambda_{n}^{2}W_{ni}^{2}+\eta_{i}\Big{)}\qquad(\
\text{here}\ |\eta_{i}|\leq|X_{i}|^{3})$ $\displaystyle=$ $\displaystyle
2\lambda_{n}n\overline{W}_{n}-\lambda_{n}^{2}nS_{n}^{2}+2\sum_{i=1}^{n}\eta_{i}$
$\displaystyle=$ $\displaystyle
2n\frac{\overline{W}_{n}^{2}}{\sigma^{2}}-\frac{1}{\sigma^{4}}n\overline{W}_{n}^{2}(\sigma^{2}+o_{p}(1))+o_{p}(1)$
$\displaystyle=$
$\displaystyle\frac{n\overline{W}_{n}^{2}}{\sigma^{2}}+o_{p}(1)\to\chi_{1}^{2},\qquad\text{in
distribution},$
where
$\displaystyle\overline{W}_{n}=\sum_{i=1}^{n}W_{ni},\quad\mbox{and \;
$S_{n}^{2}=\sum_{i=1}^{n}W_{ni}^{2}$.}$ (4. 10)
Asymptotic analysis of $\overline{W}_{n}=\sum_{i=1}^{n}W_{ni}$ and
$S_{n}^{2}=\sum_{i=1}^{n}W_{ni}^{2}$ are needed for establishing the last two
equalities in the expansion. It will be proven in Lemma 4.3 and Theorem 4.2
that both of these averages are related to the asymptotic variance (1.4) or
(3.10). The following lemmas are needed for proving Theorem 4.2.
Lemma 4.1 Let $f_{n}(x)$ and $f(x)$ be monotone functions defined on the range
of $Z$. If $f(x)$ is continuous and for $x\in[0,b_{H}]$, $f_{n}(x)\to f(x)$,
as $n\to\infty$, then $f_{n}(x)$ converges to $f(x)$ uniformly on $[0,b_{H}]$.
The proof is omitted.
Lemma 4.2 Let $V_{ni}=W_{ni}{\overline{H}}_{n}(Z_{i}-){\overline{H}}(Z_{i})$
and $V_{i}=W_{i}{\overline{H}}^{2}(Z_{i})$. Under the conditions of Theorem
4.2 and ${\theta}={\theta}_{0}$, as $n\to\infty$,
(1) $\dfrac{1}{n}\sum\limits_{i=1}^{n}(W_{ni}-W_{i})^{2}=o_{p}(1)$,
(2) $\dfrac{1}{n}\sum\limits_{i=1}^{n}(V_{ni}-V_{i})^{2}\to 0,\ \text{a.s.}$.
The proof is relegated to the Appendix. Lemma 4.2 is needed for showing that
with probability 1, for large $n$ the set $\\{W_{ni}\\}$ contains a positive
and a negative value. To facilitate the proof, $V_{ni}^{\prime}s$, a
modification of $W_{ni}^{\prime}s$, are introduced to deal with the problem at
the boundary $b_{H}.$ It follows that for large $n$ there exists a unique
$\lambda_{n}$ for $R({\theta})$ in (4.8).
Lemma 4.3 Let $W_{ni}$ and $W_{i}$ be given by (4.4) and (4.1), respectively.
Under the conditions of Theorem 4.2 and ${\theta}={\theta}_{0}$, as
$n\to\infty$,
$\max\limits_{1\leq i\leq n}|W_{ni}|=o_{p}(\sqrt{n}),\ \ \
\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\sigma^{2}+o_{p}(1),$
and
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}\to N(0,\,\sigma^{2}),\ \ \text{in
dist.},$
where $\sigma^{2}=\text{Var}(W_{i})$ is given by (3.10).
The proof is relegated to the Appendix. We now prove Theorem 4.2.
###### Proof of Theorem 4.2.
The Lagrange multiplier $\lambda$ in (4.6) appears in the equation
$\displaystyle h(\lambda)=\frac{1}{n}\sum_{i=1}^{n}\frac{W_{ni}}{1+\lambda
W_{ni}}=0.$ (4. 11)
We shall show that for large $n$, $h(\lambda)=0$ has a unique solution
$\lambda_{n}$ such that $\lambda_{n}W_{ni}>-1$ for all $i$. Put
$U_{i}=\begin{cases}-W_{ni}^{-1},&W_{ni}\neq 0,\\\
\infty,&W_{ni}=0.\end{cases}$
Let $U_{(1)}\leq U_{(2)}\leq\cdots\leq U_{(n)}$ be the ordered statistics of
$U_{1},U_{2},\cdots,U_{n}$. Then
$h(\lambda)=\frac{1}{n}\sum_{i=1}^{n}\frac{W_{ni}}{1+\lambda
W_{ni}}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\lambda-U_{(i)}}$
is monotone and differentiable in $\lambda$ on each nonempty interval
$(U_{(i)},U_{(i+1)})$. We claim that for large $n$, there exists an $i$ such
that $U_{(i)}<0<U_{(i+1)}$. To see this, we note that for every
$\varepsilon>0$, by Lemma 4.2(2)
$\frac{1}{n}\sum_{i=1}^{n}I[\,|V_{ni}-V_{i}|\geq\varepsilon\,]\leq\frac{1}{n}\sum_{i=1}^{n}(V_{ni}-V_{i})^{2}/\varepsilon^{2}=o(1),\
\text{a.s.}.$
Using the fact that $I[V_{i}\geq\varepsilon]\leq
I[V_{ni}\geq\varepsilon/2]+I[{|V_{ni}-V_{i}|\geq\varepsilon/2}]$, we get
$\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\geq\varepsilon/2]\geq\frac{1}{n}\sum_{i=1}^{n}I[V_{i}\geq\varepsilon]+o(1),\
\text{a.s.}.$
Using the fact that $P(V_{1}>0)=P(W_{1}>0)>0$, it is seen that for some
$\varepsilon>0$,
$\displaystyle\liminf_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\geq\varepsilon/2]\geq
P(V_{1}\geq\varepsilon)>0,\ \text{a.s.}.$
Similarly, we have
$\liminf_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}I[V_{ni}\leq-\varepsilon/2]\geq
P(V_{1}\leq-\varepsilon)>0,\ \text{a.s.}.$
Since $W_{ni}$ and $V_{ni}$ have the same sign, hence the claim is true. It
follows that there is a unique
$\lambda_{n}\in(U_{(i)},U_{(i+1)})=(-1/\max\\{W_{ni}\\},-1/\min\\{W_{ni}\\})$
such that $h(\lambda_{n})=0$ and $\lambda_{n}\max\\{W_{ni}\\}>-1$ and
$\lambda_{n}\min\\{W_{ni}\\}>-1$.
The rest of the proof is similar to that of Owen (2001). In fact, setting
$X_{i}=\lambda_{n}W_{ni},\ \
\overline{W}_{n}=\frac{1}{n}\sum_{i=1}^{n}W_{ni},\ \
S_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}W_{ni}^{2},$
we have $S_{n}^{2}=\sigma^{2}+o_{p}(1)$,
$\frac{1}{n}\sum_{i=1}^{n}\frac{X_{i}^{2}}{1+X_{i}}=\frac{1}{n}\sum_{i=1}^{n}\frac{(X_{i}+1-1)X_{i}}{1+X_{i}}=\overline{X}_{n}-\lambda_{n}h(\lambda_{n})=\lambda_{n}\overline{W}_{n}$
and
$\lambda_{n}^{2}S_{n}^{2}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}\leq\frac{1}{n}\sum_{i=1}^{n}\frac{X_{i}^{2}}{1+X_{i}}\big{(}1+\max_{1\leq
j\leq n}|X_{j}|\big{)}=\lambda_{n}\overline{W}_{n}+\lambda_{n}^{2}o_{p}(1).$
(4. 12)
It follows that
$\lambda_{n}=\frac{\overline{W}_{n}}{\sigma^{2}+o_{p}(1)}=O_{p}(n^{-1/2})$
and
$\overline{W}_{n}=\lambda_{n}\sigma^{2}+o_{p}(n^{-1/2}).$ (4. 13)
Applying Lemma 4.3, we have
$\sum_{i=1}^{n}|X_{i}|^{3}\leq\lambda_{n}^{3}\sum_{i=1}^{n}|W_{ni}|^{2}\max_{1\leq
j\leq n}|W_{ni}|=O_{p}(n^{-3/2})O_{p}(n)o_{p}(n^{1/2})=o_{p}(1).$
Therefore the Taylor expansion (above (4.10)) is valid from which the theorem
follows.
∎
Remark 4.1 We are able to obtain the standard asymptotic $\chi^{2}_{1}$
distribution for $-2\log R({\theta}_{0})$ is because the asymptotic variance
of
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1)$
(which is $\sigma^{2}=\text{Var}(W_{i})$) equals the limit of (see Lemma 4.3)
$\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\frac{1}{n}\sum_{i=1}^{n}W^{2}_{i}+o_{p}(1).$
If $-2\log(EL\,ratio)$ is based on the estimating function
$M_{2}=M_{2}(Z,\delta,{\theta})$ in (3.8) ( or in (3.7)), then
${V}_{ni}=\frac{g(Z_{i},{\theta})\delta_{i}}{1-G_{n}(Z_{i})}$
will be used to construct $-2\log(EL\,ratio)$. Now, the asymptotic variance of
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}V_{ni}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}W_{i}+o_{p}(1)$
is $\sigma^{2}$ (see(3.10)), but the limit of
$\frac{1}{n}\sum_{i=1}^{n}V^{2}_{ni}$ or
$\frac{1}{n}\sum_{i=1}^{n}\Big{(}V_{ni}-\frac{1}{n}\sum_{j=1}^{n}V_{nj}\Big{)}^{2}$
is (see Remark 3.1)
$\sigma_{1}^{2}=\int\frac{\xi^{2}(s)}{\overline{G}(s)}\,\text{d}F(s)-\mu^{2},\
\ \mu=0.$
Therefore, a scaled parameter $r=\sigma_{1}^{2}/\sigma^{2}$ must be introduced
in order to obtain the asymptotic distribution for $-2\log(EL\,ratio)$ as in
Wang $\&$ Jing (2001).
## 5 Simulation
Simulations are carried out to study and compare finite sample performance of
confidence intervals $I_{1}$ in (4.9) derived from Theorem 4.2 and $I_{2}$
from the scaled $\chi_{1}^{2}$ distribution given by Wang & Jing (2001) and
Qin & Zhao (2007).
To calculate $I_{1}$, $W_{ni}$ in (4.4) is used, where
$W_{ni}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}+\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\frac{1}{n}\sum_{j=1}^{n}\psi_{n}(Z_{j})\frac{I[Z_{i}\geq
Z_{j}]{\overline{\delta}}_{j}}{\overline{H}_{n}^{2}(Z_{j}-)},$
and $\psi_{n}(x)$ is given by (4.3).
Confidence intervals $I_{2}$ are calculated as follows. Let $F_{n},$ and
$G_{n}$ be the Kaplan-Meier estimators defined by (2.3). Suppose
$\hat{\theta}$ is the unique solution of $\int
g(s,\,\theta)\,\mathrm{d}F_{n}(s)=0$. Set
$\xi_{i}=g(Z_{i},\,\theta),\qquad\hat{\xi}_{i}=g(Z_{i},\,\hat{\theta}),$
$V_{ni}=\frac{\xi_{i}\,\delta_{i}}{1-G_{n}(Z_{i})},\qquad\hat{V}_{ni}=\frac{\hat{\xi}_{i}\,\delta_{i}}{1-G_{n}(Z_{i})},$
(5. 1)
$\sigma_{1}^{2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{V}_{ni}-\overline{V}_{n})^{2},\qquad\overline{V}_{n}=\frac{1}{n}\sum_{i=1}^{n}\hat{V}_{ni},$
$\hat{r}=\frac{\sigma_{1}^{2}}{n\,\widehat{\mathrm{Var}}^{*}(jack)},$
where $n\,\widehat{\mathrm{Var}}^{*}(jack)$ is the modified jackknife
estimator of the asymptotic variance of $\hat{\xi}$ given in Stute(1996).
Then, the EL-based confidence interval for ${\theta}$ is
$I_{2}=\left\\{\theta:\,\,2\,\hat{r}\sum_{i=1}^{n}\log(1+\lambda V_{ni})\leq
c_{1-\alpha}\right\\},$ (5. 2)
where $\lambda$ is the solution of $\sum_{i=1}^{n}V_{ni}/(1+\lambda
V_{ni})=0$.
Simulations were performed in two scenarios. In scenario I, the parameter of
interest is ${\theta}_{0}=\text{E}\,Y$ and in scenario II, the mean residual
lifetime. Scenario I: The parameter of interest is ${\theta}_{0}=\text{E}\,Y$
and $\xi(x)=g(x,{\theta})=x-{\theta}$ is used for calculating $I_{1}$. Two
cases were simulated:
(i) The lifetime $Y$ is uniformly distributed on $(0,1)$ and the censoring
time $C$ is uniformly distributed on (0, c). We selected $c=2.5$ and $c=1.3$
which corresponds respectively to 20% and 30% censoring proportions.
(ii) $Y$ has a Weibull(1, 10) distribution and $C$ has an Exp($\lambda$)
distribution. Then for $\lambda=4.3$ and $\lambda=2.7$, the corresponding
censoring proportions are 20% and 30%. The simulated observations are $n$
i.i.d. copies of $Z=\min(Y,\,C),\delta=I[Y\leq C]$. Based on the simulated
observations, confidence intervals $I_{1}$ derived from Theorem 4.2 and
$I_{2}$ from (5.2) were calculated. The process was repeated for $N=2\times
10^{4}$ times and the coverage proportions and the average width of the
confidence intervals were calculated using the $N$ data sets. The results are
summarized in Table 1 and Table 2.
Table 1: The coverage proportions for the true ${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion | | | | |
---|---|---|---|---|---
Nominal Value | Sample Size | Uniform(0, 1) | | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ | | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.876 | 0.881 | | 0.871 | 0.871
| 40 | 0.895 | 0.897 | | 0.889 | 0.890
| 60 | 0.897 | 0.897 | | 0.893 | 0.893
| 80 | 0.897 | 0.898 | | 0.896 | 0.896
0.95 | 20 | 0.928 | 0.935 | | 0.922 | 0.924
| 40 | 0.946 | 0.949 | | 0.939 | 0.941
| 60 | 0.947 | 0.948 | | 0.945 | 0.946
| 80 | 0.947 | 0.947 | | 0.947 | 0.948
30% censoring proportion | | | | |
Nominal Value | Sample Size | Uniform(0, 1) | | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ | | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.841 | 0.861 | | 0.867 | 0.869
| 40 | 0.885 | 0.890 | | 0.890 | 0.891
| 60 | 0.888 | 0.892 | | 0.890 | 0.891
| 80 | 0.897 | 0.900 | | 0.893 | 0.894
0.95 | 20 | 0.897 | 0.916 | | 0.916 | 0.924
| 40 | 0.934 | 0.941 | | 0.939 | 0.943
| 60 | 0.941 | 0.946 | | 0.944 | 0.946
| 80 | 0.945 | 0.947 | | 0.945 | 0.947
Table 2: The average width of confidence intervals for ${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion | width | | width
---|---|---|---
Nominal Value | Sample Size | Uniform(0, 1) | | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ | | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.217 | 0.218 | | 0.092 | 0.091
| 40 | 0.157 | 0.157 | | 0.066 | 0.065
| 60 | 0.129 | 0.129 | | 0.054 | 0.053
| 80 | 0.112 | 0.112 | | 0.046 | 0.046
0.95 | 20 | 0.258 | 0.259 | | 0.110 | 0.109
| 40 | 0.187 | 0.187 | | 0.079 | 0.078
| 60 | 0.154 | 0.154 | | 0.064 | 0.064
| 80 | 0.133 | 0.133 | | 0.056 | 0.055
30% censoring proportion | width | | width
Nominal Value | Sample Size | Uniform(0, 1) | | Weibull(1, 10)
$1-\alpha$ | $n$ | $I_{2}$ | $I_{1}$ | | $I_{2}$ | $I_{1}$
0.90 | 20 | 0.220 | 0.227 | | 0.097 | 0.096
| 40 | 0.162 | 0.164 | | 0.069 | 0.069
| 60 | 0.134 | 0.134 | | 0.057 | 0.057
| 80 | 0.116 | 0.116 | | 0.049 | 0.049
0.95 | 20 | 0.260 | 0.270 | | 0.116 | 0.116
| 40 | 0.192 | 0.196 | | 0.083 | 0.083
| 60 | 0.159 | 0.160 | | 0.068 | 0.068
| 80 | 0.138 | 0.139 | | 0.059 | 0.059
The following are noted.
(1) As the sample size $n$ increases, all of the coverage proportions converge
to the nominal level $1-\alpha$.
(2) For Uniform(0, 1) distribution, $I_{1}$ has better coverage proportions.
In 8/16 of the cases, the average width of $I_{2}$ is slightly shorter than
that of $I_{1}$. In 8/16 of the cases, $I_{2}$ and $I_{1}$ have the same
average width.
(3) For Weibull(1, 10) distribution, $I_{1}$ has better coverage proportion
and width.
In the $j$th simulation, $\\{W_{ni}\\}$ and $\\{V_{ni}\\}$ were calculated
according to (4.4) and (5.1) respectively. Then the sample means of
$\\{W_{ni}\\}$ and $\\{V_{ni}\\}$ are the same (see the proof of Theorem 4.1).
But the sample variance $s_{W}^{2}(j)$ of $\\{W_{ni}\\}$ and the sample
variance $s_{V}^{2}(j)$ of $\\{V_{ni}\\}$ are different. Let
$s_{W}^{2}=\frac{1}{N}\sum_{j=1}^{N}s_{W}^{2}(j),\ \ \
s_{V}^{2}=\frac{1}{N}\sum_{j=1}^{N}s_{V}^{2}(j).$
Table 3 shows that the sample variance of $\\{W_{ni}\\}$ is smaller than that
of $\\{V_{ni}\\}$. This is proved in Remark 3.1 for the population variances.
Table 3: The sample variances of $\\{W_{ni}\\}$ and $\\{V_{ni}\\}$,
${\theta}_{0}=\text{E}\,Y$ 20% censoring proportion
---
| Uniform(0, 1) | | Weibull(1, 10)
Sample Size $n$ | $s_{W}^{2}$ | $s_{V}^{2}$ | | $s_{W}^{2}$ | $s_{V}^{2}$
20 | 0.0935 | 0.1121 | | 0.0157 | 0.0163
40 | 0.0938 | 0.1115 | | 0.0157 | 0.0162
60 | 0.0937 | 0.1107 | | 0.0157 | 0.0161
80 | 0.0934 | 0.1100 | | 0.0158 | 0.0162
30% censoring proportion
| Uniform(0,1) | | Weibull(1, 10)
Sample Size $n$ | $s_{W}^{2}$ | $s_{V}^{2}$ | | $s_{W}^{2}$ | $s_{V}^{2}$
20 | 0.1005 | 0.1386 | | 0.0175 | 0.0185
40 | 0.1013 | 0.1401 | | 0.0176 | 0.0184
60 | 0.1016 | 0.1402 | | 0.0176 | 0.0183
80 | 0.1012 | 0.1393 | | 0.0176 | 0.0183
Scenario II: Let $g(x,{\theta})=(x-t_{0}-{\theta})I[x\geq t_{0}]$. Then by
solving the equation $\text{E}\,g(Y,{\theta})=0$, we obtain the mean residual
life of $Y$,
${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})=\frac{\text{E}(Y-t_{0})I[Y\geq
t_{0}]}{P(Y\geq t_{0})},$ (5. 3)
as studied in Qin & Zhao (2007). Let $Y$ have a Weibull (1, 10) distribution
and $C$ have an Exp($\lambda$) distribution. By setting $\lambda=4.3$ and
$\lambda=2.7$, we achieved 20% and 30% censoring proportions respectively. As
in Scenario I, each simulation was repeated $N=2\times 10^{4}$ times. The
coverage proportion of the $N$ data sets and their average width were
calculated. The results are summarized in Table 4 and Table 5, respectively.
Table 4: The coverage proportion and average width of confidence intervals for ${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})$ under the assumptions of $Y\sim$ Weibull(1, 10), 20% censoring proportion, and $1-\alpha=0.90$. Sample Size $n$ | Method | Coverage Ratio | | Average Width
---|---|---|---|---
| | $P(Y\geq t_{0})$ | | $P(Y\geq t_{0})$
| | $0.90$ | $0.70$ | $0.50$ | $0.30$ | | $0.90$ | $0.70$ | $0.50$ | $0.30$
20 | $I_{2}$ | 0.878 | 0.851 | 0.795 | 0.659 | | 0.074 | 0.062 | 0.054 | 0.044
| $I_{1}$ | 0.881 | 0.863 | 0.820 | 0.701 | | 0.074 | 0.062 | 0.056 | 0.048
40 | $I_{2}$ | 0.889 | 0.878 | 0.859 | 0.800 | | 0.053 | 0.046 | 0.042 | 0.039
| $I_{1}$ | 0.891 | 0.884 | 0.874 | 0.833 | | 0.053 | 0.046 | 0.043 | 0.041
60 | $I_{2}$ | 0.897 | 0.892 | 0.877 | 0.839 | | 0.044 | 0.037 | 0.035 | 0.034
| $I_{1}$ | 0.898 | 0.897 | 0.888 | 0.863 | | 0.044 | 0.038 | 0.035 | 0.035
80 | $I_{2}$ | 0.895 | 0.888 | 0.884 | 0.853 | | 0.038 | 0.033 | 0.031 | 0.030
| $I_{1}$ | 0.896 | 0.892 | 0.892 | 0.871 | | 0.038 | 0.033 | 0.031 | 0.031
Table 5: The coverage ratio and average width of confidence intervals for ${\theta}_{0}=\text{E}(Y-t_{0}|Y\geq t_{0})$, under the assumptions $Y\sim$ Weibull (1, 10), 30% censoring proportion and $1-\alpha=0.90$. Sample Size $n$ | Method | Coverage Ratio | | Average Width
---|---|---|---|---
| | $P(Y\geq t_{0})$ | | $P(Y\geq t_{0})$
| | $0.90$ | $0.70$ | $0.50$ | $0.30$ | | $0.90$ | $0.70$ | $0.50$ | $0.30$
20 | $I_{2}$ | 0.864 | 0.833 | 0.760 | 0.605 | | 0.079 | 0.065 | 0.055 | 0.043
| $I_{1}$ | 0.872 | 0.851 | 0.793 | 0.659 | | 0.079 | 0.065 | 0.058 | 0.048
40 | $I_{2}$ | 0.887 | 0.872 | 0.846 | 0.777 | | 0.057 | 0.048 | 0.045 | 0.041
| $I_{1}$ | 0.891 | 0.882 | 0.867 | 0.818 | | 0.057 | 0.049 | 0.046 | 0.043
60 | $I_{2}$ | 0.892 | 0.888 | 0.870 | 0.822 | | 0.046 | 0.040 | 0.037 | 0.036
| $I_{1}$ | 0.895 | 0.895 | 0.884 | 0.851 | | 0.046 | 0.040 | 0.038 | 0.037
80 | $I_{2}$ | 0.892 | 0.888 | 0.878 | 0.845 | | 0.040 | 0.035 | 0.033 | 0.032
| $I_{1}$ | 0.895 | 0.895 | 0.887 | 0.869 | | 0.040 | 0.035 | 0.033 | 0.033
The following are noted from Tables 4 and 5.
(1) As the sample size $n$ increases, all of the coverage proportions increase
and are close to the nominal levels.
(2) The coverage proportions of $I_{1}$ are much better than that of $I_{2}$.
(3) In 15/32 of the cases, the average width of $I_{2}$ is slightly shorter
than that of $I_{1}$. In 17/32 of the cases, $I_{2}$ and $I_{1}$ have the same
average width.
## 6 Appendix
###### Proof of Lemma 3.2.
Put $\eta_{n}=o_{p}(1)$. By assumptions, for any $\varepsilon>0$ and
$\delta>0$, there exist $M>0$, $b<b_{H}$ and $n_{0}>1$ such that for $n\geq
n_{0}$, $P(|V_{n}|\geq M)\leq\delta$, $P(|h_{n}(b)|\geq\varepsilon/M)\leq
P(|h(b)|\geq\varepsilon/M)+\delta/2\leq\delta$ and
$P(|\eta_{n}|\geq\varepsilon)<\delta$ . It follows that for $n\geq n_{0}$,
$\displaystyle P(|S_{n}|\geq 2\varepsilon)$ $\displaystyle\leq$ $\displaystyle
P(|\eta_{n}|\geq\varepsilon)+P(|V_{n}h_{n}(b)|\geq\varepsilon)$
$\displaystyle\leq$
$\displaystyle\delta+P(|V_{n}h_{n}(b)|\geq\varepsilon,|h_{n}(b)|\leq\varepsilon/M)+\delta$
$\displaystyle\leq$ $\displaystyle P(|V_{n}|\geq M)+2\delta$
$\displaystyle\leq$ $\displaystyle 3\delta.$
∎
###### Proof of Lemma 4.2.
The differences $W_{ni}-W_{i}$ in eq. (4.4) and (4.1) can be expressed in
terms of
$\displaystyle\gamma_{i}=\frac{\xi(Z_{i})\delta_{i}}{\overline{G}_{n}(Z_{i}-)}-\frac{\xi(Z_{i})\delta_{i}}{\overline{G}(Z_{i})},$
$\displaystyle\eta_{i}=\frac{{\overline{\delta}}_{i}}{{\overline{H}_{n}}(Z_{i}-)}\psi_{n}(Z_{i})-\frac{{\overline{\delta}}_{i}}{{\overline{H}}(Z_{i})}\psi(Z_{i}),$
$\displaystyle\nu_{i}=\int\psi_{n}(s)\frac{I[Z_{i}\geq
s]}{\overline{H}_{n}^{2}(s-)}dH_{n}^{0}(s)-\int\psi(s)\frac{I[Z_{i}\geq
s]}{\overline{H}^{2}(s)}dH^{0}(s),$
as
$W_{ni}-W_{i}=\gamma_{i}+\eta_{i}-\nu_{i}.$
Applying an elementary inequality $(a+b+c)^{2}\leq 3(a^{2}+b^{2}+c^{2})$, we
obtain
$(W_{ni}-W_{i})^{2}=(\gamma_{i}+\eta_{i}-\nu_{i})^{2}\leq
3(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}).$ (6. 1)
The lemma will be proven by showing that the sample means of
$\gamma^{2}_{i},\eta_{i}^{2}$ and $\nu_{i}^{2}$ tend to zero in probability.
The proofs will be presented in (A), (B) and (C) below.
(A) The sample mean of $\gamma_{i}^{2}$ is $o_{p}(1)$.
Proof: Let $G_{n}(x)$ be the K-M estimator defined in (2.3) and $b<b_{H}$.
Then as $n\to\infty$,
$U_{n}=\sup_{s\leq b}\frac{|G_{n}(s-)-G(s)|}{\overline{G}_{n}(s)}=o_{p}(1),\ \
V_{n}=\sup_{s\leq\max\\{Z_{i}\\}}\frac{|G_{n}(s-)-G(s)|}{\overline{G}_{n}(s-)}=O_{p}(1).$
(6. 2)
See Zhou (1992). To apply this result, we shall in the following proof split
the integrals into two intervals $[0,b]$ and $(b,b_{H}]$ accordingly.
For any $b<b_{H}$, using (2.2), we have
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\gamma_{i}^{2}$ $\displaystyle=$
$\displaystyle\int\Big{(}\frac{\xi(s)}{\overline{G}_{n}(s-)}-\frac{\xi(s)}{\overline{G}(s)}\Big{)}^{2}\,\text{d}H_{n}^{1}(s)$
(6. 3) $\displaystyle\leq$ $\displaystyle
U_{n}^{2}\int_{0}^{b}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)+V_{n}^{2}\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s)$
$\displaystyle=$ $\displaystyle o_{p}(1)O_{p}(1)+O_{p}(1)h_{n}(b)=o_{p}(1),\
\text{as}\ n\to\infty,$
where $U_{n}^{2}$ and $V_{n}^{2}$ are given by (6.2) and
$\displaystyle
h_{n}(b)=\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}^{2}(s)}\,\text{d}H_{n}^{1}(s).$
(6. 4)
Recall that $h_{n}(b)$ is explained in (3.12). It was shown that $h_{n}(b)$
satisfies the conditions in Lemma 3.2. The proof follows by invoking Lemma
3.2.
(B) The sample mean of $\eta_{i}^{2}$ is $o_{p}(1)$.
Proof: For $b<b_{H}$, define
$T_{n}(b,t]=\int_{b}^{t}\frac{\psi_{n}^{2}(s)}{\overline{H}_{n}^{2}(s-)}\,\text{d}H^{0}_{n}(s),\
\
S_{n}(b,t]=\int_{b}^{t}\frac{\psi^{2}(s)}{\overline{H}^{2}(s)}\,\text{d}H^{0}_{n}(s).$
Observe that
$\psi_{n}^{2}(s)=\Big{(}\int_{u\geq
s}\xi(u)\,\text{d}F_{n}(u)\Big{)}^{2}\leq\overline{F}_{n}(s-)\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u),$
and $F_{n}$ and $G_{n}$ have no common jumps. It follows that
$\displaystyle T_{n}(b,b_{H}]$ $\displaystyle\leq$
$\displaystyle\int_{b}^{b_{H}}\Big{(}\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u)\Big{)}\overline{F}_{n}^{2}(s-)\frac{\,\text{d}G_{n}(s)}{\overline{H}_{n}^{2}(s-)}$
(6. 5) $\displaystyle\leq$ $\displaystyle\int_{b}^{b_{H}}\Big{(}\int_{u\geq
s}\xi^{2}(u)\,\text{d}F_{n}(u)\Big{)}\,\text{d}\Big{(}\frac{1}{\overline{G}_{n}(s)}\Big{)}$
$\displaystyle\leq$ $\displaystyle\lim_{s\to
b_{H}}\frac{1}{\overline{G}_{n}(s)}\int_{s}^{b_{H}}\xi^{2}(u)\,\text{d}F_{n}(u)+\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}_{n}(s-)}\,\text{d}F_{n}(s)$
$\displaystyle\leq$ $\displaystyle
2\int_{b}^{b_{H}}\frac{\xi^{2}(s)}{\overline{G}_{n}(s-)}\,\text{d}F_{n}(s)$
$\displaystyle\leq$ $\displaystyle
2\int_{b}^{b_{H}}\Big{(}\frac{1}{\overline{G}_{n}(s-)}-\frac{1}{\overline{G}(s)}+\frac{1}{\overline{G}(s)}\Big{)}^{2}\xi^{2}(s)\,\text{d}H_{n}^{1}(s)$
$\displaystyle=$ $\displaystyle o_{p}(1)+O_{p}(1)h_{n}(b).$
The first inequality follows from (2.6) the second and the third from
integration by parts, the fifth from (2.5) and the last equality from
$(a+b)^{2}\leq 2(a^{2}+b^{2})$, (6.3) and (6.4).
By the same token, we conclude that
$S_{n}(b,b_{H}]=o_{p}(1)+O_{p}(1)h_{n}(b)$.
Write
$\xi=\xi^{+}-\xi^{-},\ \psi_{n}(x)=\int_{s\geq
x}\xi^{+}(s)\,\text{d}F_{n}(s)-\int_{s\geq x}\xi^{-}(s)\,\text{d}F_{n}(s),$
where $\xi^{+}$ and $\xi^{-}$ are the positive and negative part of $\xi$.
Define monotone functions:
$\psi_{n}^{\pm}(x)=\int_{s\geq x}\xi^{\pm}(s)\,\text{d}F_{n}(s),\
\psi^{\pm}(x)=\int_{s\geq x}\xi^{\pm}(s)\,\text{d}F(s).$
$\psi_{n}^{\pm}(x)$ converges to $\psi^{\pm}(x)$ almost surely for
$x\in[0,b_{H}]$ as shown by Stute & Wang (1993). Furthermore, by Lemma 4.1,
the convergence is uniform on $[0,b_{H}]$.
From these we conclude the uniform convergence of $\psi_{n}$ to $\psi$,
$\displaystyle\sup_{0\leq x\leq b_{H}}|\psi_{n}(x)-\psi(x)|=o(1),\
\text{a.s.}.$ (6. 6)
Therefore, for $b<b_{H}$,
$\sup_{s\leq
b}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\to
0,\ \text{a.s.}.$
Applying $(a+b)^{2}\leq 2a^{2}+2b^{2}$ and Lemma 3.2, we have
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\eta_{i}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{b}+\int_{b}^{b_{H}}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\,\text{d}H^{0}_{n}(s)$
(6. 7) $\displaystyle\leq$
$\displaystyle\int_{0}^{b}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}\,\text{d}H^{0}_{n}(s)+2T_{n}(b,b_{H}]+2S_{n}(b,b_{H}]$
$\displaystyle=$ $\displaystyle
o_{p}(1)+O_{p}(1)h_{n}(b)+O_{p}(1)h_{n}(b)=o_{p}(1).$
(C) The sample mean of $\nu_{i}^{2}$ is $o_{p}(1)$.
Proof: Write, for $0\leq a<t$,
$B_{n}(a,t]=\int_{a}^{t}\frac{\psi_{n}(s)\,\text{d}H_{n}^{0}(s)}{\overline{H}_{n}^{2}(s-)},\
\ D(a,t]=\int_{a}^{t}\frac{\psi(s)\,\text{d}H^{0}(s)}{\overline{H}^{2}(s)}.$
Then, for $b<b_{H}$, we have
$\displaystyle\Delta_{n}^{2}$ $\displaystyle\equiv$
$\displaystyle\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)=\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}(-\overline{H}_{n}(t))$
$\displaystyle\leq$
$\displaystyle\overline{H}_{n}(b)B_{n}^{2}(b,b]+2\int_{b}^{b_{H}}\overline{H}_{n}(t-)B_{n}(b,t]\frac{\psi_{n}(t)}{\overline{H}_{n}^{2}(t-)}\,\text{d}H_{n}^{0}(t)$
$\displaystyle\leq$ $\displaystyle
0+2\Big{(}\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)\Big{)}^{1/2}\Big{(}\int_{b}^{b_{H}}\frac{\psi_{n}^{2}(t)}{\overline{H}_{n}^{2}(t-)}\,\text{d}H_{n}^{0}(t)\Big{)}^{1/2}$
$\displaystyle=$ $\displaystyle\Delta_{n}2[T_{n}(b,b_{H})]^{1/2}.$
The second term in the first inequality is obtained using the Lebesgue-
Stieltjes integration by parts.
Applying (6.5), we get
$\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)=\Delta_{n}^{2}\leq
4T_{n}(b,b_{H}]=o_{p}(1)+O_{p}(1)h_{n}(b).$
Similarly, for $S(b,b_{H}]=\text{E}S_{n}(b,b_{H}]$, we have
$\int_{b}^{b_{H}}D^{2}(b,t]\,\text{d}H_{n}(t)\leq
4S(b,b_{H}]+o_{p}(1)=o_{p}(1)+O_{p}(1)h_{n}(b).$
Applying the uniform convergence of $\psi_{n}$ to $\psi$ (see(6.6)), we
conclude that for any $b<b_{H}$, with probability 1, $B_{n}(0,t]\to D(0,t]$
uniformly on $[0,b]$. It follows that $B_{n}(0,b]\to D(0,b]$, and
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\nu_{i}^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{b}+\int_{b}^{b_{H}}\big{(}B_{n}(0,t]-D(0,t]\big{)}^{2}\,\text{d}H_{n}(t)$
$\displaystyle=$ $\displaystyle
o_{p}(1)+\int_{b}^{b_{H}}\big{(}B_{n}(0,b]-D(0,b]+B_{n}(b,t]-D(b,t]\big{)}^{2}\,\text{d}H_{n}(t)$
$\displaystyle\leq$ $\displaystyle
o_{p}(1)+4\int_{b}^{b_{H}}B_{n}^{2}(b,t]\,\text{d}H_{n}(t)+4\int_{b}^{b_{H}}D^{2}(b,t)\,\text{d}H_{n}(t)$
$\displaystyle=$ $\displaystyle
o_{p}(1)+O_{p}(1)h_{n}(b)+O_{p}(1)h_{n}(b)=o_{p}(1).$
Now, we prove result (2) of the lemma. Introduce
$A_{ni}=W_{i}{\overline{H}}_{n}(Z_{i}-){\overline{H}}(Z_{i})$. From (6.1) we
get
$(V_{ni}-A_{ni})^{2}\leq
3(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}){\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i}).$
Similar to (6.3) and (6.7), we have
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\gamma_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\Big{(}\frac{\xi(s)}{\overline{G}_{n}(s-)}-\frac{\xi(s)}{\overline{G}(s)}\Big{)}^{2}{\overline{H}}^{2}_{n}(s-){\overline{H}}^{2}(s)\,\text{d}H_{n}^{1}(s)=o(1),\
\text{a.s.},$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\eta_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\Big{(}\frac{\psi_{n}(s)}{\overline{H}_{n}(s-)}-\frac{\psi(s)}{\overline{H}(s)}\Big{)}^{2}{\overline{H}}^{2}_{n}(s-){\overline{H}}^{2}(s)\,\text{d}H^{0}_{n}(s)=o(1),\
\text{a.s.}.$
Since for any $b<b_{H}$, with probability 1,
$|B_{n}(0,t]-D(0,t]|{\overline{H}_{n}(t-)}{\overline{H}}(t)\to 0$
uniformly on $[0,b]$, and
$\sup_{t\leq b_{H}}|B_{n}(0,t]-D(0,t]|{\overline{H}_{n}(t-)}{\overline{H}}(t)$
is bounded by some constant, it follows that
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\nu_{i}^{2}{\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})$
$\displaystyle=$
$\displaystyle\int_{0}^{b_{H}}\big{[}(B_{n}(0,t]-D(0,t]){\overline{H}}_{n}(t-){\overline{H}}(t)\big{]}^{2}\,\text{d}H_{n}(t)=o(1),\
\text{a.s.}.$
Now we get
$\frac{1}{n}\sum_{i=1}^{n}(V_{ni}-A_{ni})^{2}\leq
3\frac{1}{n}\sum\limits_{i=1}^{n}(\gamma_{i}^{2}+\eta_{i}^{2}+\nu_{i}^{2}){\overline{H}}^{2}_{n}(Z_{i}-){\overline{H}}^{2}(Z_{i})=o(1),\
\text{a.s.}.$
At last, we have
$\displaystyle\frac{1}{n}\sum\limits_{i=1}^{n}(V_{ni}-V_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2}{n}\sum\limits_{i=1}^{n}(V_{ni}-A_{ni})^{2}+\frac{2}{n}\sum\limits_{i=1}^{n}(A_{ni}-V_{i})^{2}$
$\displaystyle=$ $\displaystyle
o(1)+\frac{2}{n}\sum\limits_{i=1}^{n}(H(Z_{i})-H_{n}(Z_{i}-))^{2}W_{i}^{2}\overline{H}^{2}(Z_{i})$
$\displaystyle=$ $\displaystyle o(1),\ \text{a.s.}.$
∎
###### Proof of Lemma 4.3.
Since $W_{i}$ are i.i.d. random variables with zero mean and finite variance
$\sigma^{2}$, hence $\max\limits_{1\leq i\leq n}|W_{i}|=o_{p}(\sqrt{n})$. It
follows from Lemma 4.2 that
$\displaystyle\max\limits_{1\leq i\leq n}|W_{ni}|$ $\displaystyle\leq$
$\displaystyle\Big{(}\max\limits_{1\leq i\leq
n}|W_{ni}-W_{i}|^{2}\Big{)}^{1/2}+\max\limits_{1\leq i\leq n}|W_{i}|$ (6. 8)
$\displaystyle\leq$
$\displaystyle\sqrt{n}\Big{(}\frac{1}{n}\sum_{i=1}^{n}(W_{ni}-W_{i})^{2}\Big{)}^{1/2}+o_{p}(\sqrt{n})$
$\displaystyle=$ $\displaystyle o_{p}(\sqrt{n}).$
Note that $W_{ni}^{2}$ is bounded by
$W^{2}_{i}+(W_{i}-W_{ni})^{2}-2|W_{i}(W_{i}-W_{ni})|\leq W_{ni}^{2}\leq
W_{i}^{2}+(W_{ni}-W_{i})^{2}+2|W_{i}(W_{i}-W_{ni})|.$
By Lemma 4.2, we get
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}W^{2}_{ni}=\sigma^{2}+o_{p}(1).$
The last result follows from Theorem 3.1, $\xi(x)=g(x,{\theta}_{0})$ and
(4.6).
∎
## References
* [1] Chen, K. and Lo, S-H. (1997). On the rate of uniform convergence of the product-limit estimator: strong and weak laws. Ann. Statist. 25. 1050–1087.
* [2] Chen, S. X. (1994). Empirical likelihood confidence intervals for linear regression coefficients. J. Multivariate Anal. 49. 24–40.
* [3] DiCiccio, T. J., Hall, P. and Romano, J.P. (1991). Empirical likelihood is Bartlett-correctable. Ann. Statist. 19. 1053–1061.
* [4] He, S. Y. and Huang, X. (2003). Central limit theorem of linear regression model under right censorship. Science in China. 46. 600–610.
* [5] Hjort, N. L., McKeague, I. W., and van Keilegom, I. (2009): Extending the scope of empirical likelihood. Ann. Statist. 37, 1079-1111
* [6] Li, G. and Wang, Qi-Hua (2003) Empirical likelihood methods for linear regression analysis of right censored data, Statist. Sinica 13 51-68.
* [7] Lo, S-H. and Singh, P. (1986). The product-limit estimator and the bootstrap: some asymptotic representations. Probab. Theory Related Fields. 71. 455–465.
* [8] Owen, A. (1988). Empirical likelihood ratio confidence intervals for single functional. Biometrika. 75. 237–249.
* [9] Owen, A. (2001). Empirical likelihood. Chapman and Hall, London.
* [10] Qin, G. S. and Zhao, Y. C. (2007). Empirical likelihood inference for the mean residual life under random censorship. Statist. Probab. Lett. 77. 549–557.
* [11] Ren, J. J. (2008). Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data. Ann. Statist. 36. 147–166.
* [12] Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
* [13] Stute, W. and Wang, J. L. (1993). The strong law under random censorship. Ann. Statist. 21. 1591–1607.
* [14] Stute, W. (1996). The Jackknife estimate of variance of a Kaplan-Meier integral. Ann. Statist. 24. 2679–2704.
* [15] Tsiatis A. A. (2006). Semiparametric Theory and Missing Data. New York: Springer.
* [16] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.
* [17] Vardi Y. (1982). Nonparametric estimation in the presence of length bias. Ann. Statist. 10. 616–620.
* [18] Wang, Q. H. and Jing, B. Y. (2001). Empirical likelihood for a class of functions of survival distribution with censored data. Ann. Inst. Statist. Math. 53. 517–527.
* [19] Yang, G. (1997). The Kaplan-Meier estimator. Encyclopedia of Statistical Sciences, update Vol. 1, Wiley, 334–343.
* [20] Yang, S. (1994). A central limit theorem for functionals of the Kaplan-Meier estimator. Statist. Probab. Lett. 21. 337–345.
* [21] Zhou, M. (1992). Asymptotic normality of the ’synthetic data’ regression estimator for censored survival data. Ann. Statist. 20. 1002–1021.
|
arxiv-papers
| 2012-03-27T12:33:10 |
2024-09-04T02:49:29.114628
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuyuan He, Wei Liang, Junshan Shen, Grace Yang",
"submitter": "Shuyuan He",
"url": "https://arxiv.org/abs/1203.5955"
}
|
1203.6122
|
# Diffusion of Real-Time Information in Social-Physical Networks
Dajun Qian1, Osman Yağan2, Lei Yang1 and Junshan Zhang1 1 School of ECEE,
Arizona State University, Tempe, AZ, USA
2 Cybersecurity Laboratory (CyLab), Carnegie Mellon University, Pittsburgh,
PA, USA
###### Abstract
We study the diffusion behavior of real-time information. Typically, real-time
information is valuable only for a limited time duration, and hence needs to
be delivered before its “deadline.” Therefore, real-time information is much
easier to spread among a group of people with frequent interactions than
between isolated individuals. With this insight, we consider a social network
which consists of many cliques and information can spread quickly within a
clique. Furthermore, information can also be shared through online social
networks, such as Facebook, twitter, Youtube, etc.
We characterize the diffusion of real-time information by studying the phase
transition behaviors. Capitalizing on the theory of inhomogeneous random
networks, we show that the social network has a critical threshold above which
information epidemics are very likely to happen. We also theoretically
quantify the fractional size of individuals that finally receive the message.
The numerical results indicate that real-time information could be much easier
to propagate in a social network when large size cliques exist.
## I Introduction
### I-A Motivation and Background
In today’s modern society, people are becoming increasingly tied together over
social networks. Thanks to _online social networks_ , such as Facebook and
Twitter, people can share messages quickly with their friends [1]. Meanwhile,
a _physical information network_ [2, 3, 4] based on traditional face-to-face
interactions still remains an important medium for message spreading. Very
recent work [5] has shown that different social networks are usually _coupled_
together, and the conjoining could greatly facilitate information diffusion.
As a result, today’s hot spot news or fashion behaviors are more likely to
generate pronounced influence over the population than ever before.
The main thrust of this study is dedicated to understanding the diffusion
behavior of real-time information. Typically, the real-time information is
valuable only for a limited time duration [6], and hence needs to be delivered
before its “deadline.” For example, once a limited-time coupon is released
from Groupon or Dealsea.com, people can share this news either by talking to
friends or posting it on Facebook. However, people would not have much
interest on this deal after it is not longer available.
Clearly, due to the timeliness requirement, the potential influenced scale of
real-time information in a social network depends on the speed of message
propagation. The faster the message passes from one to another, the more
people can learn this news before it expires. With this insight, in order to
characterize its diffusion behavior, a key step is to quantify how fast the
message can spread along different social connections.
In this study, we assume that information could spread amongst people through
both face-to-face contacts and online communications. In an online social
network, the message can spread quickly over long distance, and hence it is
reasonable to treat online connections as the same type of links regardless of
real-world distances.
On the other hand, face-to-face communications are largely constrained by
distance between individuals. Recent works in [4, 7] have explored the
structure of physical information network by tracking in-person interactions
over the population. Their findings indicate that such interactions would give
rise to a social graph made of a large number of small isolated _cliques_.
Each clique stands for a group of people who are close to each other. The
message can spread quickly within a clique via frequent interactions, but
takes longer time to spread across cliques separated by longer distances.
Clearly, constrained by its deadline, the real-time information could be less
likely to propagate across cliques via face-to-face contacts. Needless to say,
in order to characterize the diffusion behavior of real-time information, we
need to consider the impact of such clique structure, which has not been
studied in previous works on general information diffusions [5, 8, 9].
### I-B Summary of Main Contributions
We explore the diffusion of real-time information in an overlaying social-
physical network where the information could spread amongst people through
both face-to-face contacts (physical information network) and online
communications (online social network). Based on empirical observations in [4,
7], we assume that the physical information network consists of many isolated
cliques where each clique represents a group of people with frequent face-to-
face interactions, e.g., family in a house or colleagues in an office.
Clearly, the face-to-face contacts are less likely to happen across cliques.
We characterize the information diffusion process by studying the _phase
transition behaviors_ (see Section II-C for details). Specifically, we show
that the social network has a _critical threshold_ above which information
epidemics are very likely to happen, i.e., the information can reach a non-
trivial fraction of individuals. We also quantify the fraction of individuals
that finally receive the message by computing the size of _giant component_
(see Section II-C for details). The numerical results in Section V indicate
that real-time information could be much easier to propagate in a social
network when large size cliques exist. As illustrated in Figure 6, when the
average clique size increases from $1$ to $2$, the fraction of individuals
that receive the message can grow sharply from $14\%$ to $80\%$.
Note that our work here has significant differences from the previous studies
on information propagation. In [8, 9], it is assumed that message could
propagate at the same speed along different social relationships. Clearly,
such assumption would be inappropriate for the diffusion of real-time
information that depends on propagation speeds. Very recent work in [5]
considered online connections and face-to-face connections for general
information diffusion, but did not study the impact of clique structure on
information diffusion. To the best of our knowledge, this paper is the first
work on the diffusion of real-time information with consideration on the
clique structure in social networks. We believe that our work will offer
initial steps towards understanding the diffusion behaviors of real-time
information.
## II System model
We consider an overlying social-physical network $\mathbb{H}$ that consists of
a physical information network $\mathbb{W}$ and an online social network
$\mathbb{F}$. The nodes in the graph $\mathbb{W}$ represent the human beings
in the real world. Based on empirical studies in [4, 7], we assume that the
graph $\mathbb{W}$ consists of many isolated cliques where each clique
represents a group of closely located people, e.g., family in a house or
colleagues in an office. Meanwhile, each node in $\mathbb{W}$ can
independently participate the online social network $\mathbb{F}$ with
probability $\alpha$, and the nodes in $\mathbb{F}$ stand for their online
memberships. Throughout this paper, we also refer to the nodes in $\mathbb{W}$
and $\mathbb{F}$ as “individuals” and “online users,” respectively.
Furthermore, the links connecting the nodes in $\mathbb{W}$ stand for
traditional face-to-face connections, while the links in $\mathbb{F}$
represent online connections.
Figure 1: System model $\mathbb{H}$
### II-A Topology Structure in System Model
In what follows, we specify the topology structure in the system model
$\mathbb{H}$ in Fig. 1.
Cliques in the physical information network. The physical information network
has $N$ nodes and the nodes set is denoted by $\mathcal{N}=\\{1,2,...,N\\}$.
These nodes are gathered into many cliques with different sizes. We expect the
clique size follows the distribution $\\{\mu_{n}^{w},n=1,2,...,D\\}$,
$n=1,2,...D$, where $D$ is the largest possible size. Therefore, an arbitrary
clique could contain $n$ nodes with probability $\mu_{n}^{w}$. We generate
these cliques as follows: at step $t=1$, we randomly choose $n$ nodes from the
collection $\mathcal{N}$ and create a clique with the selected $n$ nodes,
where $n$ is a random number following the distribution
$\\{\mu_{n}^{w},n=1,2,...,D\\}$. We also denote the collection of the
remaining nodes in $\mathcal{N}$ by $\mathcal{N}_{1}$. At each step $t$, we
repeat the above procedure to create a new clique from the collection
$\mathcal{N}_{t-1}$111Note that the last generated clique may not follow the
expected size distribution, since there would be only too few nodes left to
choose. However, such impact on clique size distribution will be negligible if
the number of cliques is large enough., and assume that we can finally
generate $N_{c}$ cliques in $\mathbb{W}$ 222Throughout this paper, we use
“clique in $\mathbb{W}$” and “clique in $\mathbb{H}$” interchangeably, in the
sense that the network $\mathbb{W}$ is also a part of system model
$\mathbb{H}$.. Generally speaking, the existence of large size cliques
indicates that many individuals are close to each other. In other words, the
clique size distribution $\\{\mu_{n}^{w}\\}$ offers an abstract
characterization of personal distances in $\mathbb{W}$ from a macroscopic
perspective.
Type-$0$ (intra-clique) links in $\mathbb{W}$. Since the nodes within the same
cliques could interact to each other frequently, we assume these nodes are
fully connected by _type- $0$ links_.
Type-$1$ (inter-clique) links in $\mathbb{W}$. We assume that a face-to-face
interaction is still possible between cliques, e.g., a person may talk to a
remote friend by walking across a long distance. Suppose each node can
randomly connect to $k^{w}$ nodes from other cliques through _type- $1$ links_
where $k^{w}$ is a random variable drawn independently from the distribution
$\\{{p_{k}^{w},k=0,1,...}\\}$.
Online users and type-$2$ (online) links. The nodes in the online network
$\mathbb{F}$ represent the online users. As in [5], we assume each online user
randomly connects to $k^{f}$ online neighbors in $\mathbb{F}$, where $k^{f}$
is a random variable whose distribution is drawn independently from
$\\{p_{k}^{f},k=0,1,...\\}$. We denote such online connection as _type- $2$
link_. Furthermore, we draw a virtual _type- $3$ link_ from an online user in
$\mathbb{F}$ to the actual person it corresponds to in the physical
information network $\mathbb{W}$; this indicates that the two nodes actually
correspond to the same individual.
Online users associated with a clique. To avoid confusions, we say _“size- $n$
clique with $m$ online members”_ when referring to the case that a clique
contains $n$ individuals and only $m$ of them participate in the online social
network $\mathbb{F}$. With this insight, we can also differentiate among the
collection of size-$n$ cliques according to their affiliated online users.
Specifically, for the collection of size-$n$ cliques with $m$ online members,
$m\leq n\leq D$, we assume their fraction size in the whole collection of
cliques is $\mu_{nm}$. It is easy to see that
${\mu_{nm}}=\mu_{n}^{w}\left({\begin{array}[]{*{20}{c}}n\\\
m\end{array}}\right){\alpha^{m}}{\left({1-\alpha}\right)^{n-m}}~{}~{}\textrm{and}~{}~{}{\mu_{n}^{w}}=\sum\limits_{m=1}^{n}{\mu_{nm}}.$
(1)
Furthermore, for the collection of cliques with $m$ online users, their
fraction size can be given by
${\mu_{m}^{f}}=\sum\limits_{n=m}^{D}{\mu_{nm}}.$ (2)
### II-B Information Transmissibility
The message can propagate at different speeds along different types of social
connections in $\mathbb{H}$. Due to timeliness requirement, the real-time
information is easier to pass over a link that offers fast propagation speed.
Therefore, we assign each link with a _transmissibility_ as in [5, 9], i.e.,
the probability that the message can successfully pass through.
From practical scenarios, we set the transmissibility along type-$0$ link as
$T_{c}=1$ since the message spreads quickly within a clique. We also define
the transmissibilities along type-$1$ and type-$2$ links as $T_{w}$ and
$T_{f}$, respectively. Throughout this paper, we say a link is _occupied_ if
the message can successfully pass through that link. Hence, in $\mathbb{H}$
each type-$1$ link is occupied independently with probability $T_{w}$, whereas
each type-$2$ link is occupied independently with probability $T_{f}$.
### II-C Information Cascade
We give a brief description of the information diffusion process in the
following. Suppose that the message starts to spread from an arbitrary node
$i$ in a clique of $\mathbb{W}$. Then, the other nodes in this clique will
quickly receive that message through type-$0$ links. The message can also
propagate to other cliques through occupied type-$1$ and type-$2$ links. This
in turn may trigger further message propagation and may eventually lead to an
information epidemic; i.e., a non-zero fraction of individuals may receive the
information in the limit $N\to\infty$.
Clearly, an arbitrary individual can spread the information to nodes that are
reachable from itself via the occupied edges of $\mathbb{H}$. Hence, the size
of an information outbreak (i.e., the number of individuals that are informed)
is closely related to the size of the connected components of $\mathbb{{H}}$,
which contains only the _occupied_ type-$1$ and type-$2$ links [5, 9, 8] of
$\mathbb{H}$. Thus, the information diffusion process considered here is
equivalent to a heterogeneous bond-percolation process over the network
$\mathbb{H}$; the corresponding bond percolation is heterogeneous since the
occupation probabilities are different for type-$1$ and type-$2$ links. In
this paper, we will exploit this relation and find the condition and the size
of information epidemics by studying the phase transition properties of
$\mathbb{{H}}$. A key observation is that the system $\mathbb{{H}}$ exhibits a
_phase transition_ behavior at a _critical threshold_. Specifically, a _giant
connected component_ $G_{H}$ that covers a non-trivial fraction of nodes in
$\mathbb{{H}}$ is likely to appear above the critical threshold meaning that
_information epidemics_ are possible. Below that critical threshold, all
components are small indicating that the fraction of influenced individuals
tends to zero in the large system size limit.
It is easy to see that the influenced individuals and cliques correspond to
the nodes and cliques in $\mathbb{W}$ that are contained inside $G_{H}$.
Hence, we introduce two parameters to evaluate the scale of information
diffusion:
* •
$S_{c}$: The fractional size of influenced cliques in $\mathbb{W}$. Namely,
$S_{c}$ corresponds to the ratio of the number of cliques in $G_{H}$ to the
total number of cliques in $\mathbb{W}$.
* •
$S_{n}$: The fractional size of influenced individuals in $\mathbb{W}$.
Namely, $S_{n}$ corresponds to the ratio of the number of nodes that belong to
the cliques in $G_{H}$ to the total number of nodes in $\mathbb{W}$.
With this insight, we can explore the information diffusion process by
characterizing the phase transition behavior of the giant component $G_{H}$.
## III Equivalent graph: a clique level approach
In this study, we are particularly interested in the following two questions:
Figure 2: Equivalent graph $\mathbb{E}$. Nodes {a,b,c,d} in this graph
corresponds to the cliques {a,b,c,d} of the system $\mathbb{H}$ in Fig. 1. We
assign type-$1$ and type-$2$ links in $\mathbb{E}$ according to the same type
of links connecting cliques in Fig. 1.
* •
What is the critical threshold of the system $\mathbb{H}$? In other words,
under what condition, the information reaches a non-trival fraction of the
network rather than dying out quickly?
* •
What is the expected size of an information epidemic? In other words, to what
fraction of nodes and cliques does the information reach? Or, equivalently,
what are the sizes $S_{c}$ and $S_{n}$?
These two questions can be answered by quantifying the phase transition
behaviors of the random graph $\mathbb{H}$. Due to the clique structure in our
system model, the techniques employed in existing works [5, 8, 9] cannot be
directly applied here. To tackle this challenge, we develop an equivalent
random graph $\mathbb{E}$ that exhibits the same phase transition behavior as
the original model $\mathbb{H}$. Then, we characterize the phase transition
behaviors in the graph $\mathbb{E}$ by capitalizing on the recent results in
_inhomogeneous random graph_ [10, 11].
We first construct an equivalent graph $\mathbb{E}$ based on the topology
structure of $\mathbb{H}$. Since the nodes within the same clique can
immediately share the message, we treat each clique including affiliated
online users as a single virtual node in $\mathbb{E}$. Furthermore, we assign
type-$1$ and type-$2$ links between two virtual nodes according to the
original connections in $\mathbb{H}$. To get a more concrete sense, we depict
the equivalent graph in Fig. 2 that corresponds to the original model in Fig.
1. It is easy to see that the (type-$1$ and type-$2$) link degree of a virtual
node equals the total number of (type-$1$ and type-$2$) links that are
incident on the nodes within the corresponding clique. The equivalent graph
$\mathbb{E}$ is expected to exhibit the same phase transition behavior as the
original model $\mathbb{H}$ since both graphs have the similar connectivity
structure. In particular, the fractional size of the giant component $G_{E}$
in the equivalent graph $\mathbb{E}$ (the ratio of the number of nodes in
$G_{E}$ to the number of nodes in $\mathbb{E}$) is equal to the aforementioned
quantity $S_{c}$. Thus, with a slight abuse of notation, we use $S_{c}$ to
denote the fractional size of $G_{E}$.
The degree of an arbitrary node in $\mathbb{E}$ can be represented by a two-
dimensional vector $\boldsymbol{d}=[d^{w}\>\>d^{f}]$ where $d^{w}$ and $d^{f}$
correspond to the numbers of type-$1$ and type-$2$ links incident on that
node, respectively. For a node in $\mathbb{E}$ that corresponds to a size-$n$
clique in $\mathbb{W}$, we use $K_{n}^{w}$ to denote its type-$1$ link degree,
where $K_{n}^{w}$ is a random variable following the distribution
$\\{P_{nk}^{w},k=0,1,2,...\\}$. Similarly, for a node in $\mathbb{E}$ that
corresponds to a clique with $m$ online users, we use $K_{m}^{f}$ to denote
its type-$2$ link degree where $K_{m}^{f}$ follows the distribution
$\\{P_{mk}^{f},k=0,1,2,...\\}$. It is clear to see that an arbitrary node in
$\mathbb{E}$ has link degree $[i\>\>j]$ with probability
$p(i,j)=\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}P_{ni}^{w}P_{mj}^{f}}}~{}~{}~{}i,j\in
N.$ (3)
Let ${\rm{E}}[{d_{w}}]$ and ${\rm{E}}[{d_{f}}]$ be the mean numbers of
type-$1$ and type-$2$ links for a node in $\mathbb{E}$, i.e.,
${\rm{E}}[{d_{w}}]=\sum\nolimits_{i=0}^{\infty}{\sum\nolimits_{j=0}^{\infty}{p(i,j)i}}$
and
${\rm{E}}[{d_{f}}]=\sum\nolimits_{i=0}^{\infty}{\sum\nolimits_{j=0}^{\infty}{p(i,j)j}}$.
We also define
${\rm{E}}[{d_{w}}{d_{f}}]=\sum\nolimits_{i=0}^{\infty}{\sum\nolimits_{j=0}^{\infty}{p(i,j)ij}}$.
Furthermore, let ${\rm{E}}[{({d_{w}})^{2}}]$ and ${\rm{E}}[{({d_{f}})^{2}}]$
denote the second moments of the number of type-$1$ and type-$2$ links for a
node in $\mathbb{E}$, respectively; i.e.,
${\rm{E}}[{({d_{w}})^{2}}]=\sum\nolimits_{i=0}^{\infty}{\sum\nolimits_{j=0}^{\infty}{p(i,j){i^{2}}}}$
and
${\rm{E}}[{({d_{f}})^{2}}]=\sum\nolimits_{i=0}^{\infty}{\sum\nolimits_{j=0}^{\infty}{p(i,j){j^{2}}}}$.
## IV Analytical solutions
In this section, we analyze information diffusion process by characterizing
the phase transition behaviors in the equivalent random graph $\mathbb{E}$. We
present our analytical results in the following two steps. We first quantify
the conditions for the emergence of a giant component as well as the
fractional sizes $S_{c}$ and $S_{n}$ for the special case $T_{w}=1$ and
$T_{f}=1$. We next show that these results can be easily extended to a more
general case with $0\leq T_{w}\leq 1$ and $0\leq T_{f}\leq 1$.
### IV-A Special Case: $T_{w}=T_{f}=1$
We characterize the phase transition behavior of the giant component in
$\mathbb{E}$ by capitalizing the theory of inhomogeneous random graphs [10,
11, 12]. Specifically, we define
${a_{11}}={\rm{E[(}}{d_{w}}{{\rm{)}}^{2}}{\rm{]/E[}}{d_{w}}{\rm{]}}-1$,
${a_{12}}={\rm{E[}}{d_{w}}{{\rm{d}}_{f}}{\rm{]/E[}}{d_{w}}{\rm{]}}$,
${a_{21}}={\rm{E[}}{d_{w}}{{\rm{d}}_{f}}{\rm{]/E[}}{d_{f}}{\rm{]}}$ and
${a_{22}}={\rm{E[(}}{d_{f}}{{\rm{)}}^{2}}{\rm{]/E[}}{d_{f}}{\rm{]}}-1$. Along
the same line in [5, 10, 12], we have the following result.
###### Lemma 4.1
Let
$\sigma=\frac{1}{2}\left({{a_{11}}+{a_{22}}+\sqrt{{{({a_{11}}-{a_{22}})}^{2}}+4{a_{12}}{a_{21}}}}\right)$
(4)
if $\sigma>1$, with high probability (whp) there exists a giant component in
$\mathbb{E}$, i.e., a non-trival fraction of nodes in $\mathbb{E}$ are
connected; otherwise, a giant component does no exist in $\mathbb{E}$ whp.
The proof of Lemma 4.1 is relegated to Appendix A. As we discussed in Section
II-C, the existence of a giant component in $\mathbb{E}$ indicates that the
information can reach a non-trival fraction of cliques in $\mathbb{H}$ rather
than dying out quickly.
Next, let $h_{1}$ and $h_{2}$ in $(0,1]$ be given by the smallest solution to
the following recursive equations:
${h_{1}}=\frac{1}{{{{\rm{E}}[{d_{w}}]}}}\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}},$
(5)
${h_{2}}=\frac{1}{{{\rm{E}}[{d_{f}}]}}\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]}}.$
(6)
We have the following results on the size and probability of an information
epidemic.
###### Lemma 4.2
The fractional size of the giant component in $\mathbb{E}$ (equivalently, the
fractional size of influenced cliques in $\mathbb{W}$) is given by
${S_{c}}=\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}\left({1-{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}\right)}}.$
(7)
The fractional size of influenced nodes in $\mathbb{W}$ is given by
${S_{n}}=\frac{1}{C}\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{n{\mu_{nm}}\left({1-{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}\right)}},$
(8)
with the normalization term $C=\sum\limits_{n=1}^{D}{n{\mu_{n}}}$.
The proof of Lemma 4.2 is relegated to Appendix A. For any given set of
parameters, Lemma 4.2 reveals the fraction of individuals and cliques that are
likely to receive an information that is started from an arbitrary individual.
Namely, an information started from an arbitrary individual gives rise to an
information epidemic with probability $S_{n}$ (attributed to the possibility
that the arbitrary node belongs to the giant component $G_{H}$), and reaches a
fraction $S_{n}$ of nodes in the network. Similar conclusions can be drawn in
terms of $S_{c}$ for the fraction of cliques that receive the information.
Note that the condition (4) in Lemma 4.1 depends on the first/second moments
of $d_{w}$ and $d_{f}$, which boils down to the linear combinations of the
first/second moments of $k^{w}$ and $k^{f}$ in the following manner:
${\rm{E}}[{d_{w}}]=\sum\limits_{n=1}^{D}{\mu_{n}^{w}n{\rm{E}}[k^{w}]}~{}~{}~{}~{}{\rm{E}}[{d_{f}}]=\sum\limits_{m=1}^{D}{\mu_{m}^{f}m{\rm{E}}[k^{f}]},$
(9)
${\rm{E}}[{d_{w}}{d_{f}}]=\sum\limits_{n=1}^{D}{\sum\limits_{m=1}^{n}{{\mu_{nm}}nm{\rm{E}}[k^{w}]{\rm{E}}[k^{f}]}},$
(10)
${\rm{E}}[{({d_{w}})^{2}}]=\sum\limits_{n=1}^{D}{\mu_{n}^{w}\left({n{\rm{E}}[{{(k^{w})}^{2}}]+({n^{2}}-n){{\left({{\rm{E}}[k^{w}]}\right)}^{2}}}\right)},$
(11)
${\rm{E}}[{({d_{f}})^{2}}]=\sum\limits_{m=1}^{D}{\mu_{m}^{f}\left({m{\rm{E}}[{{(k^{f})}^{2}}]+({m^{2}}-m){{\left({{\rm{E}}[k^{f}]}\right)}^{2}}}\right)}.$
(12)
According to Lemma 4.2, $S_{c}$ and $S_{n}$ are determined by
${\rm{E}}[h_{1}^{K_{n}^{w}}]$, ${\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]$,
${\rm{E}}[h_{2}^{K_{m}^{f}}]$ and ${\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]$ in
(5)-(8), i.e., the integrals with respect to the distributions of $K_{n}^{w}$
and $K_{m}^{f}$ for different $n$ and $m$. The calculations can be simplified
by utilizing the following transformations:
${\rm{E}}[h_{1}^{K_{n}^{w}}]=({{\rm{E}}[h_{1}^{{k^{w}}}]})^{n}~{}~{}~{}{\rm{E}}[h_{2}^{K_{m}^{f}}]=({{\rm{E}}[h_{2}^{{k^{f}}}]})^{m},$
(13)
${\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]=n{\left({{\rm{E}}[h_{1}^{{k^{w}}}]}\right)^{n-1}}{\rm{E}}[{k^{w}}h_{1}^{{k^{w}}-1}],$
(14)
${\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]=m{\left({{\rm{E}}[h_{2}^{{k^{f}}}]}\right)^{m-1}}{\rm{E}}[{k^{f}}h_{2}^{{k^{f}}-1}].$
(15)
With the help of (13)-(15), we only need to calculate the integrals with
respect to the distributions of $k^{w}$ and $k^{f}$. In this way, we can find
$h_{1}$ and $h_{2}$ by numerically solving the recursive equations (5)-(6) and
compute $S_{c}$ and $S_{n}$ from (7)-(8), respectively. The detailed
derivations of (9)-(15) are omitted (see details in Appendix B).
### IV-B General Case: $0\leq T_{w}\leq 1$ and $0\leq T_{f}\leq 1$
We next generalize Lemma 4.1 and Lemma 4.2 to the case $0\leq T_{w}\leq 1$ and
$0\leq T_{f}\leq 1$. To this end, we maintain the _occupied_ links in the
equivalent graph $\mathbb{E}$ by deleting each type-$1$ and type-$2$ edge with
probability $1-T_{w}$ and $1-T_{f}$, respectively. Let ${\tilde{k}^{w}}$ and
${\tilde{k}^{f}}$ be the occupied link degrees (instead of $k^{w}$ and
$k^{f}$) with the distributions $\\{\tilde{p}_{k}^{w},k=0,1,...\\}$ and
$\\{\tilde{p}_{k}^{f},k=0,1,...\\}$. According to [9], the generating
functions corresponding to ${\tilde{k}^{w}}$ and ${\tilde{k}^{f}}$ can be
given by
$\tilde{g}(x)=g\left({1+{T_{w}}(x-1)}\right)~{}~{}\tilde{q}(x)=q\left({1+{T_{f}}(x-1)}\right).$
(16)
From (9)-(15), we observe that the critical threshold and the giant component
size are determined by the distributions of $k^{w}$ and $k^{f}$. Therefore,
Lemma 4.1 and Lemma 4.2 still hold if we replace the terms associated with
$k^{w}$ and $k^{f}$ in (9)-(15) by those associated with ${\tilde{k}^{w}}$ and
${\tilde{k}^{f}}$, respectively. To this end, by using the generating
functions (16), we find
$\displaystyle{\rm{E}}[{{\tilde{k}}^{w}}]$ $\displaystyle=$
$\displaystyle{T_{w}}{\rm{E}}[{k^{w}}],$
$\displaystyle{\rm{E}}[{({{\tilde{k}}^{w}})^{2}}]$ $\displaystyle=$
$\displaystyle
T_{w}^{2}\left({{\rm{E}}[{{({k^{w}})}^{2}}]-{\rm{E}}[{k^{w}}]}\right)+{T_{w}}{\rm{E}}[{k^{w}}].$
In the same manner, we can compute ${\rm{E}}[{{\tilde{k}}^{f}}]$ and
${\rm{E}}[{({{\tilde{k}}^{f}})^{2}}]$. The critical threshold (in the general
case) can now be computed by replacing ${\rm{E}}[{{k}^{w}}]$,
${\rm{E}}[{{k}^{f}}]$, ${\rm{E}}[{({{k}^{w}})^{2}}]$,
${\rm{E}}[{({{k}^{f}})^{2}}]$ with ${\rm{E}}[{{\tilde{k}}^{w}}]$,
${\rm{E}}[{{\tilde{k}}^{f}}]$, ${\rm{E}}[{({{\tilde{k}}^{w}})^{2}}]$,
${\rm{E}}[{({{\tilde{k}}^{f}})^{2}}]$, respectively, in (9)-(12).
In order to compute the giant component size, we only need to replace the
corresponding terms in (13)-(15) with ${\rm{E}}[h_{1}^{\tilde{k}^{w}}]$,
${\rm{E}}[h_{2}^{\tilde{k}^{f}}]$,
${\rm{E}}[{{\tilde{k}}^{w}}h_{1}^{{{\tilde{k}}^{w}}-1}]$ and
${\rm{E}}[{{\tilde{k}}^{f}}h_{2}^{{{\tilde{k}}^{f}}-1}]$. By using (16), we
have
${\rm{E}}[h_{1}^{\tilde{k}^{w}}]=\tilde{g}({h_{1}})={\rm{E}}[{(1+{T_{w}}({h_{1}}-1))^{{k^{w}}}}],$
${\rm{E}}[{{\tilde{k}}^{w}}h_{1}^{{{\tilde{k}}^{w}}-1}]={\left[{\tilde{g}({h_{1}})}\right]^{\prime}}={T_{w}}{\rm{E}}[{k_{w}}{(1+{T_{w}}({h_{1}}-1))^{{k^{w}}-1}}].$
Similar relations can be obtained for ${\rm{E}}[h_{1}^{\tilde{k}^{f}}]$ and
${\rm{E}}[{{\tilde{k}}^{f}}h_{1}^{{{\tilde{k}}^{f}}-1}]$. The size of the
giant component (in the general case) can now be computed by reporting the
updated (13)-(15) into (5)-(8).
## V Numerical results and simulations
In this section, we numerically study the diffusion of real-time information
by utilizing the analytical results derived in Section IV. In particular, we
are interested in how the clique structure can impact the scale of information
epidemic. To get a more concrete sense, we compare four system scenarios, each
with different clique size distribution as illustrated in Table I.
For the sake of fair comparison, the total number of nodes in $\mathbb{W}$ is
fixed at $12000$ in each scenario. From Table I, we can see that the average
clique size increases from scenario $1$ to scenario $4$, indicating that
individuals are getting closer to each other. We assume that the type-$1$ link
degree for each node in $\mathbb{W}$ follows a poisson distribution, i.e.,
$p_{k}^{w}=\frac{{{\lambda^{k}}}}{{k!}}\cdot{e^{-\lambda}}$, $k=0,1,2,...$,
where $\lambda$ is the average type-$1$ link degree. Meanwhile, the type-$2$
link degree for each online user in $\mathbb{F}$ follows a power-law
distribution with exponential cutoff, i.e., $p_{0}^{f}=0$ and
$p_{k}^{f}=\frac{1}{C}{k^{-\gamma}}{e^{-\frac{k}{\Gamma}}},\;\;k=1,2,\ldots,$
(17)
with the normalization factor
$C=\sum\limits_{k=1}^{\infty}{{k^{-\gamma}}{e^{-\frac{k}{\Gamma}}}}$.
TABLE I: The clique size distribution in four scenarios scenario | size-$1$ | size-$2$ | size-$3$ | average clique size
---|---|---|---|---
$1$ | $100\%$ | $0$ | $0$ | $1$
$2$ | $66.7\%$ | $33.3\%$ | $0$ | $1.333$
$3$ | $33.3\%$ | $66.7\%$ | $0$ | $1.666$
$4$ | $33.3\%$ | $33.3\%$ | $33.3\%$ | $2$
Figure 3: The minimum $T_{w}$ required for the existence of a giant component
in $\mathbb{E}$ versus $T_{f}$ in four scenarios. We let $\lambda=1.5$,
$\alpha=0.1$, $\gamma=3$ and $\Gamma=10$. Each curve corresponds to the
boundary of the phase transition in one scenario. A giant component is very
likely to emerge above the boundary. Figure 4: The empirical probability
$p_{inf}$ for the existence of giant component is plotted. The simulation
results are obtained with $N=12000$ by averaging $200$ experiments. We let
$T_{f}=0.4$ and other parameters follow the same setup as in Figure 3. From
scenario $1$ to scenario $4$, the values of $p_{inf}$ exhibit a sharp increase
at $T_{w}=0.64$, $T_{w}=0.4$, $T_{w}=0.35$ and $T_{w}=0.26$, respectively.
Such sharp increase of $p_{inf}$ corresponds to the phase transition. These
values are in good agreement with the minimum required $T_{w}$ from the
corresponding curves in Figure 3. We assume that a giant component exists if
more than $5\%$ of the cliques are connected.
We first compare these scenarios in term of the required conditions for the
existence of a giant component; in other words, in terms of the minimum
conditions for an information epidemic to take place. We let $\lambda=1.5$,
$\alpha=0.1$, $\gamma=3$ and $\Gamma=10$. By computing the system’s critical
threshold, we depict in Figure 3 the minimum required value of $T_{w}$ to have
a giant component in $\mathbb{E}$ versus $T_{f}$. Each scenario corresponds to
a curve in the figure that stands for the boundary of a phase transition;
above the boundary a giant component is very likely to emerge. By the
definition of transmissibility, a lower required $T_{w}$ indicates that the
system is more likely to give rise to a giant component (and thus, to an
information epidemic). From scenario $1$ to scenario $4$, this figure clearly
shows that larger clique sizes lead to smaller values for the minimum $T_{w}$
required for an information epidemic, meaning that information epidemics are
more likely to take place for larger clique sizes. The analytical results of
Figure 3 are also verified by simulations. For a fixed $T_{f}$, the
probability of the existence of giant component $p_{inf}$ is expected to have
a sharp increase as $T_{w}$ approaches to the corresponding minimum required
value in Figure 3. Indeed, we observe in Figure 4 that when $T_{f}=0.4$, for
each scenario, such sharp transition occur at $T_{w}$ close to the
corresponding minimum required value obtained in Figure 3. Such sharp increase
of $p_{inf}$ corresponds to the phase transition, i.e., the giant component
could exist with high probability above the critical threshold. Therefore, the
minimum required $T_{w}$ values obtained via simulations are in good agreement
with our analysis.
We next compare these scenarios in terms of the fractional sizes of influenced
cliques and influenced individuals. For each scenario, we plot the fractional
size of the giant component in $\mathbb{E}$ versus $T_{f}$ in Figure 5, which
indicates the fraction of cliques that will receive the information. We set
$T_{w}=0.3$, $\lambda=2$, $\alpha=0.3$, $\gamma=3$ and $\Gamma=10$. In this
Figure, the curves stand for analytical results obtained by (7), while the
marked points stand for the simulation results obtained by averaging $200$
experiments for each set of parameter. It is easy to check that the analytical
results are in good agreement with the simulations. Obviously, the fractional
size of influenced cliques in scenario $4$ (with average clique size $2$) is
much larger than that in scenario $1$ (with average clique size $1$), which
indicates that large cliques in the social network could greatly facilitate
the message propagation.
We finally compare the fractional size of influenced individuals in Figure 6.
In this figure, the curves stand for the fractional size of influenced nodes
obtained via (8), whereas the marked points stand for the simulation results.
Similar to Figure 5, the information is much easier to propagate in a social
network when larger size cliques exist. For instance, when $T_{f}=1$, the
fractional size of individuals that receive the message grows sharply from
$14\%$ (scenario $1$ with average clique size $1$) to $80\%$ (scenario $4$
with average clique size $2$). In conclusion, the above results agree with a
natural conjecture that the messages are more influential (i.e., more likely
to reach a large portion of the population) when people are close to each
other.
Figure 5: Fraction size of influenced cliques in $\mathbb{W}$ versus $T_{f}$.
We set $T_{w}=0.3$, $\lambda=2$, $\alpha=0.3$, $\gamma=3$ and $\Gamma=10$. The
curves stand for analytical results obtained by (7), while the marked points
stand for the simulation results with $N=12000$ by averaging $200$ experiments
for each set of parameter. The analytical results are in good agreement with
the simulations. Figure 6: Fraction size of influenced nodes in $\mathbb{W}$
versus $T_{f}$. All the parameters follow the same setup as in Figure 5. The
curves stand for analytical results obtained by (8) and the marked points
stand for the simulation results with $N=12000$. The analytical results are in
good agreement with the simulations. For comparison, we also plot the fraction
size of influenced cliques in scenario $1$ where the each clique has only one
node.
## VI conclusion
In this study, we explore the diffusion of real-time information in social
networks. We develop an overlaying social-physical network that consists of an
online social network and a physical information network with clique
structure. We theoretically quantify the condition and the size of information
epidemics. To the best of our knowledge, this paper is the first work on the
diffusion of real-time information with consideration on the clique structure
in social networks. We believe that our findings will offer initial steps
towards understanding the diffusion behaviors of real-time information.
## VII Appendix
### VII-A Proofs of Lemma 4.1 and Lemma 4.2
In [11, 12] Söderberg studied the phase transition behaviors of inhomogeneous
random graphs where nodes are connected by different types of edges. Such
graphs are also called “colored degree-driven random graphs” in the sense that
different types of edges correspond to different colors. In a graph with
$r$-types of edges, the edge degree of an arbitrary node can be represented by
a $r$-dimension vector $\boldsymbol{d}=[d^{1}\>\>\cdots\>\>d^{r}]$, where
$d^{j}$ stands for the number of type-$j$ edges incident on that node. In our
study, the equivalent graph $\mathbb{E}$ has two types of edges and the degree
distribution of an arbitrary node is denoted by
$p(i,j)={\rm{P[}}{d_{w}}=i,{d_{f}}=j{\rm{]}}$. Also, the generating function
of degree distribution $\\{p(i,j)\\}$ can be defined by
$H({x_{1}},{x_{2}})=\sum\nolimits_{i}{\sum\nolimits_{j}{p(i,j)x_{1}^{i}x_{2}^{j}}}$.
Clearly, the multivariable combinatorial moments can be achieved by partial
differentiation at $x_{1}=1$ and $x_{2}=1$, i.e.,
$\displaystyle{\rm{E[}}{d_{w}}{\rm{]}}$ $\displaystyle=$
$\displaystyle{\partial_{1}}H({x_{1}},{x_{2}}){|_{{x_{1}}={x_{2}}=1}},$
$\displaystyle{\rm{E[}}{d_{f}}{\rm{]}}$ $\displaystyle=$
$\displaystyle{\partial_{2}}H({x_{1}},{x_{2}}){|_{{x_{1}}={x_{2}}=1}},$
$\displaystyle{\rm{E[}}{d_{w}}{d_{f}}{\rm{]}}$ $\displaystyle=$
$\displaystyle{\partial_{1}}{\partial_{2}}H({x_{1}},{x_{2}}){|_{{x_{1}}={x_{2}}=1}},$
$\displaystyle{\rm{E[(}}{d_{w}}{{\rm{)}}^{2}}{\rm{]}}$ $\displaystyle=$
$\displaystyle\partial_{1}^{2}H({x_{1}},{x_{2}}){|_{{x_{1}}={x_{2}}=1}},$
$\displaystyle{\rm{E[(}}{d_{f}}{{\rm{)}}^{2}}{\rm{]}}$ $\displaystyle=$
$\displaystyle\partial_{2}^{2}H({x_{1}},{x_{2}}){|_{{x_{1}}={x_{2}}=1}}.$
Let $\\{{a_{k}}\\}$ denote size distribution of the largest connected
component that can be reached from an arbitrary node in $\mathbb{E}$, whose
generating function is defined by $g(z)=\sum\nolimits_{k}{{a_{k}}{z^{k}}}$.
Furthermore, we define a two-dimension vector
$\mathbf{h}(z)=[{h_{1}}(z),{h_{2}}(z)]$, where ${h_{i}}(z)$ stands for the
generating function of size distribution of the component connected by
type-$i$ edges. According to the existing results in [5, 11, 12], we have that
$g(z)=z\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j){h_{1}}{{(z)}^{i}}{h_{2}}{{(z)}^{j}}=zH(\mathbf{h}(z))}},$
(18)
where $\mathbf{h}(z)$ satisfies the following recursive equations
$\displaystyle{h_{1}}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{{{\rm{E[}}{d_{w}}{\rm{]}}}}{\partial_{1}}H(h(z)),$ (19)
$\displaystyle{h_{2}}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{{{\rm{E[}}{d_{f}}{\rm{]}}}}{\partial_{2}}H(h(z)).$ (20)
The emergence of giant component in $\mathbb{E}$ can be checked by examining
the stability of the recursive equations (19)-(20) at the point
${h_{1}}={h_{1}}(1)=1$ and ${h_{2}}={h_{2}}(1)=1$. Along the same line as in
[11, 12], we define a $2\times 2$ Jacobian matrix $\mathbf{J}$, i.e.,
$\mathbf{J}=\left[{\begin{array}[]{*{20}{c}}{{a_{11}}}&{{a_{12}}}\\\
{{a_{21}}}&{{a_{22}}}\\\ \end{array}}\right],$
where
$\displaystyle{a_{11}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{\rm{E[}}{d_{w}}{\rm{]}}}}\partial_{1}^{2}H(h(z)){|_{{h_{1}}={h_{2}}=1}}={{{\rm{E[(}}{d_{w}}{{\rm{)}}^{2}}{\rm{-}}{d_{w}}{\rm{]}}}\mathord{\left/{\vphantom{{{\rm{E[(}}{d_{w}}{{\rm{)}}^{2}}{\rm{-}}{d_{w}}{\rm{]}}}{{\rm{E[}}{d_{w}}{\rm{]}}}}}\right.\kern-1.2pt}{{\rm{E[}}{d_{w}}{\rm{]}}}},$
$\displaystyle{a_{12}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{\rm{E[}}{d_{w}}{\rm{]}}}}\partial_{1}\partial_{2}H(h(z)){|_{{h_{1}}={h_{2}}=1}}={{{\rm{E[}}{d_{w}}{d_{f}}{\rm{]}}}\mathord{\left/{\vphantom{{{\rm{E[}}{d_{w}}{d_{f}}{\rm{]}}}{{\rm{E[}}{d_{w}}{\rm{]}}}}}\right.\kern-1.2pt}{{\rm{E[}}{d_{w}}{\rm{]}}}},$
$\displaystyle{a_{21}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{\rm{E[}}{d_{f}}{\rm{]}}}}\partial_{1}\partial_{2}H(h(z)){|_{{h_{1}}={h_{2}}=1}}={{{\rm{E[}}{d_{w}}{d_{f}}{\rm{]}}}\mathord{\left/{\vphantom{{{\rm{E[}}{d_{w}}{d_{f}}{\rm{]}}}{{\rm{E[}}{d_{f}}{\rm{]}}}}}\right.\kern-1.2pt}{{\rm{E[}}{d_{f}}{\rm{]}}}},$
$\displaystyle{a_{22}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{\rm{E[}}{d_{f}}{\rm{]}}}}\partial_{2}^{2}H(h(z)){|_{{h_{1}}={h_{2}}=1}}={{{\rm{E[(}}{d_{f}}{{\rm{)}}^{2}}{\rm{-}}{d_{f}}{\rm{]}}}\mathord{\left/{\vphantom{{{\rm{E[(}}{d_{f}}{{\rm{)}}^{2}}{\rm{-}}{d_{f}}{\rm{]}}}{{\rm{E[}}{d_{f}}{\rm{]}}}}}\right.\kern-1.2pt}{{\rm{E[}}{d_{f}}{\rm{]}}}}.$
The spectral radius of $\mathbf{J}$ is given by
$\sigma=\frac{1}{2}\left({{a_{11}}+{a_{22}}+\sqrt{{{({a_{11}}-{a_{22}})}^{2}}+4{a_{12}}{a_{21}}}}\right).$
By [5, 10, 12], if $\sigma>1$, with high probability there exist a giant
component in the graph $\mathbb{E}$;otherwise, a giant component is very less
likely to exist in $\mathbb{E}$. Therefore, the condition (4) in Lemma 4.1 is
achieved. Furthermore, the fraction size $S_{c}$ equals $1-g(1)$ [5]. By (18),
we have that
$\displaystyle{S_{c}}$ $\displaystyle=$ $\displaystyle
1-g(1)=\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)\left({1-h_{1}^{i}h_{2}^{j}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{P_{ni}^{w}P_{mj}^{f}\left({1-h_{1}^{i}h_{2}^{j}}\right)}}}}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}\left({1-\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{P_{ni}^{w}P_{mj}^{f}h_{1}^{i}h_{2}^{j}}}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}\left({1-{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}\right)}}.$
In view of (19) and (20), we have that
$\displaystyle{h_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{{\rm{E}}[{d_{w}}]}}}\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)ih_{1}^{i-1}h_{2}^{j}}}$
$\displaystyle=$
$\displaystyle\frac{1}{{{{\rm{E}}[{d_{w}}]}}}\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}},$
$\displaystyle{h_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{{{{\rm{E}}[{d_{f}}]}}}\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)jh_{1}^{i}h_{2}^{j-1}}}$
$\displaystyle=$
$\displaystyle\frac{1}{{{\rm{E}}[{d_{f}}]}}\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]}}.$
Furthermore, (VII-A) can be rewritten in the following form:
$S_{c}=\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}\sum_{n=1}^{D}\sum_{m=0}^{n}\mu_{nm}P_{ni}^{w}P_{mj}^{f}{(1-h_{1}^{i}h_{2}^{j})}}.$
Clearly, the term in parentheses gives the probability that a node with
colored degree $[{d_{w}}=i,{d_{f}}=j]$ belongs to the giant component. In
other words, the term in parentheses is the expected number of cliques added
to the giant cluster by a degree $[{d_{w}}=i,{d_{f}}=j]$ clique. Hence,
summing over all such $i,j$’s we get an expression for the expected size of
the giant cluster (in terms of number of cliques).
In order to compute the expected giant component size in terms of the number
of nodes, namely to compute $S_{n}$, we can modify the above expression such
that the term $n(1-h_{1}^{i}h_{2}^{j})$ gives the expected number of nodes to
be included in the giant cluster by a degree $[{d_{w}}=i,{d_{f}}=j]$ clique.
In other words, with probability $(1-h_{1}^{i}h_{2}^{j})$ the clique under
consideration will belong to the giant component $G_{H}$ and will bring $n$
nodes to the actual giant size $S_{n}$. This yields
$\displaystyle{\bar{S}_{n}}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}}}P_{ni}^{w}P_{mj}^{f}n\left({1-h_{1}^{i}h_{2}^{j}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{n{\mu_{nm}}}}\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{P_{ni}^{w}P_{mj}^{f}\left({1-h_{1}^{i}h_{2}^{j}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{n{\mu_{nm}}\left({1-\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{P_{ni}^{w}P_{mj}^{f}h_{1}^{i}h_{2}^{j}}}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{n{\mu_{nm}}\left({1-{\rm{E}}[h_{1}^{K_{n}^{w}}]{\rm{E}}[h_{2}^{K_{m}^{f}}]}\right)}}.$
We next have
${S_{n}}=\frac{1}{C}{\bar{S}_{n}},~{}~{}~{}C=\sum\limits_{n=1}^{D}{n{\mu_{n}}},$
where the normalized term $C$ makes $S_{n}=1$ at $h_{1}=h_{2}=0$. Therefore,
the conclusions (7) and (8) in Lemma 4.2 have been obtained.
### VII-B Detailed Derivations for Equations (9)-(15)
As defined in Section III, $K_{n}^{w}$ is the sum of $n$ independent copies of
$k^{w}$ and $K_{m}^{f}$ is the sum of $m$ independent copies of $k^{f}$. It
follows that
${\rm{E}}[K_{n}^{w}]=n{\rm{E}}[k^{w}]~{}~{}~{}~{}{\rm{E}}[K_{m}^{f}]=m{\rm{E}}[k^{f}],$
$\displaystyle{\rm{E}}[{(K_{n}^{w})^{2}}]$ $\displaystyle=$
$\displaystyle{\mathop{\rm
var}}[K_{n}^{w}]+{\left({{\rm{E}}[K_{n}^{w}]}\right)^{2}}$ $\displaystyle=$
$\displaystyle
n{\rm{E}}[{(k^{w})^{2}}]+({n^{2}}-n){\left({{\rm{E}}[k^{w}]}\right)^{2}},$
$\displaystyle{\rm{E}}[{(K_{m}^{f})^{2}}]$ $\displaystyle=$
$\displaystyle{\mathop{\rm
var}}[K_{m}^{f}]+{\left({{\rm{E}}[K_{m}^{f}]}\right)^{2}}$ $\displaystyle=$
$\displaystyle
m{\rm{E}}[{(k^{f})^{2}}]+({m^{2}}-m){\left({{\rm{E}}[k^{f}]}\right)^{2}}.$
In view of this, we can rewrite the first/second moments of $d_{w}$ and
$d_{f}$ as follows:
$\displaystyle{\rm{E}}[{d_{w}}]$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)i}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}\left({\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{iP_{ni}^{w}P_{mj}^{f}}}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[}K_{n}^{w}}]=\sum\limits_{n=1}^{D}{\mu_{n}^{w}n{\rm{E}}[k^{w}]},$
$\displaystyle{\rm{E}}[{d_{f}}]$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)j}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}\left({\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{jP_{ni}^{w}P_{mj}^{f}}}}\right)}}$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=0}^{n}{{\mu_{nm}}{\rm{E}}[}K_{m}^{f}}]=\sum\limits_{m=1}^{D}{\mu_{m}^{f}m{\rm{E}}[k^{f}]},$
$\displaystyle{\rm{E}}[{d_{w}}{d_{f}}]$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j)i}}j$
$\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{D}{\sum\limits_{m=1}^{n}{{\mu_{nm}}nm{\rm{E}}[k^{w}]{\rm{E}}[k^{f}]}},$
$\displaystyle{\rm{E}}[{({d_{w}})^{2}}]=\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j){i^{2}}=\sum\limits_{n=1}^{D}{\mu_{n}^{w}E[{{(K_{n}^{w})}^{2}}]}}}$
$\displaystyle=\sum\limits_{n=1}^{D}{\mu_{n}^{w}\left({n{\rm{E}}[{{(k^{w})}^{2}}]+({n^{2}}-n){{\left({{\rm{E}}[k^{w}]}\right)}^{2}}}\right)},$
$\displaystyle{\rm{E}}[{({d_{f}})^{2}}]=\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{\infty}{p(i,j){j^{2}}=\sum\limits_{m=1}^{D}{\mu_{m}^{f}E[{{(K_{m}^{f})}^{2}}]}}}$
$\displaystyle=\sum\limits_{m=1}^{D}{\mu_{m}^{f}\left({m{\rm{E}}[{{(k^{f})}^{2}}]+({m^{2}}-m){{\left({{\rm{E}}[k^{f}]}\right)}^{2}}}\right)}.$
We next characterize the generating functions of $K_{n}^{w}$ and $K_{m}^{f}$.
Specifically, the generating functions of the type-$1$ and type-$2$ link
degree distribution for a single node in $\mathbb{H}$ can be defined by
$g(x)=\sum\nolimits_{k=1}^{\infty}{p_{k}^{w}{x^{k}}}$ and
$q(x)=\sum\nolimits_{k=1}^{\infty}{p_{k}^{f}{x^{k}}}$. Since $K_{n}^{w}$ and
$K_{m}^{f}$ are sums of i.i.d. random variables, their generating functions of
turn out to be
${G_{n}}(x)=\sum\limits_{k=0}^{\infty}{P_{nk}^{w}{x^{k}}}={\left[{g(x)}\right]^{n}}~{}~{}~{}~{}~{}1\leq
n\leq D,$ (22)
${Q_{m}}(x)=\sum\limits_{k=0}^{\infty}{P_{mk}^{f}{x^{k}}}=\left\\{{\begin{array}[]{*{20}{c}}{{{\left[{q(x)}\right]}^{m}}}~{}~{}~{}1\leq
m\leq D,\\\ 0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m=0\\\ \end{array}}\right.$ (23)
With (22) and (23), ${\rm{E}}[h_{1}^{K_{n}^{w}}]$,
${\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]$, ${\rm{E}}[h_{2}^{K_{m}^{f}}]$ and
${\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]$ can boil down to the expected value
with respect to the distribution of $k^{w}$ and $k^{f}$ as follows.
$\begin{array}[]{l}{\rm{E}}[h_{1}^{K_{n}^{w}}]={G_{n}}({h_{1}})={\left({g({h_{1}})}\right)^{n}}=({{\rm{E}}[h_{1}^{{k^{w}}}]})^{n}\\\
{\rm{E}}[h_{2}^{K_{m}^{f}}]=Q_{m}({h_{2}})={\left({q({h_{2}})}\right)^{m}}=({{\rm{E}}[h_{2}^{{k^{f}}}]})^{m}\end{array}$
$\displaystyle{\rm{E}}[K_{n}^{w}h_{1}^{K_{n}^{w}-1}]$ $\displaystyle=$
$\displaystyle{G_{n}}({h_{1}})^{\prime}=n{\left({g({h_{1}})}\right)^{n-1}}{\left({g({h_{1}})}\right)^{\prime}}$
$\displaystyle=$ $\displaystyle
n{\left({{\rm{E}}[h_{1}^{{k^{w}}}]}\right)^{n-1}}{\rm{E}}[{k^{w}}h_{1}^{{k^{w}}-1}]$
$\displaystyle{\rm{E}}[K_{m}^{f}h_{2}^{K_{m}^{f}-1}]$ $\displaystyle=$
$\displaystyle{Q_{m}}({h_{2}})^{\prime}=m{\left({q({h_{2}})}\right)^{m-1}}{\left({q({h_{2}})}\right)^{\prime}}$
$\displaystyle=$ $\displaystyle
m{\left({{\rm{E}}[h_{2}^{{k^{f}}}]}\right)^{m-1}}{\rm{E}}[{k^{f}}h_{2}^{{k^{f}}-1}].$
## References
* [1] A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, and B. Bhattacharjee. Measurement and analysis of online social networks. In Proceedings of the 7th ACM SIGCOMM Conference on Internet measurement, 2007.
* [2] J. Leskovec, L.A. Adamic, and B.A. Huberman. The dynamics of viral marketing. ACM Transactions on the Web (TWEB), 1(1):5, 2007.
* [3] L. Isella, J. Stehlé, A. Barrat, C. Cattuto, J.F. Pinton, and W. Van den Broeck. What’s in a crowd? Analysis of face-to-face behavioral networks. Journal of Theoretical Biology, 2010.
* [4] K. Zhao, J. Stehle, G. Bianconi, and A. Barrat. Social network dynamics of face-to-face interactions. Phys. Rev. E, 83(5):056109, 2011.
* [5] O. Yagan, D. Qian, J. Zhang, and D. Cochran. Conjoining speeds up information diffusion in overlaying social-physical networks. Technical report, Available online at arXiv:1112.4002v1[cs.SI],.
* [6] J. Yang and J. Leskovec. Modeling information diffusion in implicit networks. In Proceedings of the 10th IEEE International Conference on Data Mining, 2010.
* [7] J. Stehlé, A. Barrat, and G. Bianconi. Dynamical and bursty interactions in social networks. Phys. Rev. E, 81(3):035101, 2010.
* [8] M. E. J. Newman, S. H. Strogatz, and D. J. Watts. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E, 64(2), 2001.
* [9] M. E. J. Newman. Spread of epidemic disease on networks. Phys. Rev. E, 66(1), 2002.
* [10] B. Söderberg. General formalism for inhomogeneous random graphs. Phys. Rev. E, 66(066121), 2002.
* [11] B. Söderberg. Random graphs with hidden color. Phys. Rev. E, 68(015102(R)), 2003.
* [12] B. Söderberg. Random graph models with hidden color. Acta Phys. Pol. B, 34:5085–5102, 2003.
|
arxiv-papers
| 2012-03-28T00:49:36 |
2024-09-04T02:49:29.124894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dajun Qian, Osman Ya\\u{g}an, Lei Yang, Junshan Zhang",
"submitter": "Dajun Qian",
"url": "https://arxiv.org/abs/1203.6122"
}
|
1203.6184
|
# Leptonic and Digamma decay Properties of S-wave quarkonia states
Manan Shah mnshah09@gmail.com Arpit Parmar arpitspu@yahoo.co.in P C Vinodkumar
pothodivinod@yahoo.com Department of Physics, Sardar Patel University,Vallabh
Vidyanagar, INDIA.
###### Abstract
Based on Martin like potential, the S-wave masses of quarkonia have been
reviewed. Resultant wave functions at zero inter quark separation are employed
to compute the hyperfine splitting of the nS states and the leptonic and
digamma decay widths of $n{{}^{3}S_{1}}$ and $n{{}^{1}S_{0}}$ states of
quarkonia respectively. Analysis on the level differences of S-wave excited
states of quantum mechanical bound systems show a systematic behaviour as
n-increases. In view of such systematic behaviour expected for quarkonia, we
observe that $Y(4263)$ and $X(4630)$ $1^{--}$ states are closer to the 4S and
6S states while $\psi(4415)$ and $Z(4430)$ are closer to the 5S state of
$c\bar{c}$ systems. Similarly we find $\Upsilon(10865)$ is not fit to be the
5S state of $b\bar{b}$ system. while $Y_{b}(10880)$ observed by Belle or
(10996) observed by Babar fit to be the 6S state of bottonia. Our predicted
leptonic width, 0.242 keV of $\Upsilon(10579,4S)$ is in good agreement with
the experimental value of 0.272 $\pm$ 0.029 keV. We predict the leptonic
widths of the pure 5S and 6S states of upsilon states as 0.191 keV and 0.157
keV respectively. In the case of charmonia, we predict the leptonic widths of
the 4S, 5S and 6S states as 0.654 keV, 0.489 keV and 0.387 keV respectively.
## Introduction
The recent experimental observations particularly in quarkonia sector have
generated renewed interest in the study of hadron spectroscopy CLEO1 ; CLEO ;
PDG2010 ; babar . The discovery of the $\eta_{b}\ (1S)$ state babar ; CLEO1
(BaBar and CLEO collaboration) and $\eta_{c}\ (2S)$ state and many high
precision experimental observations of various hadronic states PDG2010 have
necessitated reconsideration of the parameters involved in the previous
studies Meinel ; Gray . The spectroscopic parameters like the interquark
potential parameters that provide the masses of the bound states and the
corresponding wave functions obtained from the phenomenology are detrimental
in the predictions of their decay widths. Most of the existing theoretical
values for the decay rates are based on potential model calculations that
employ different types of interquark potentials D.Ebert ; Horace ; Guo ;
N.Brambilla .
Till recently, all that was known above the $D\bar{D}$ threshold was the four
vector states $\psi(3770)$, $\psi(4040)$, $\psi(4160)$, $\psi(4415)$. The new
renaissance in hadron spectroscopy has come from the recent discovery of the
large numbers of new states X, Y, Z N.Brambilla-1 ; Chen1 ; Chen ; Kai ; Q.He
; T.E. . The challenges paused by these new states include the right
identification with the proper $J^{PC}$ values and their decay modes.
Eventhough the spectroscopy of quarkonium states are well recorded
experimentally, the S-wave masses of charmonium states beyond 3S and the
bottonium states beyond 4S are still not very well resolved. There seemed to
be mixing of other resonances nereby. For example The $1^{--}$ states such as
$\psi(3770)$, $Y(4008)$, $Y(4260)$, $Y(4360)$, $X(4630)$, $Y(4660)$,
$\Upsilon(10865)$, $\Upsilon(11020)$, $Y_{b}(10880)$ etc, may be the quarkonia
states either with or without mixing with the nearby resonance states. For
instance, $\Upsilon(11020)$ state has recently been analysed to be a mixed
bottonium $\Upsilon(6S)$ and $\Upsilon(5D)$ states with mixing angle of
$\theta=40^{o}\pm 5^{o}$ Badalian .
###
In this context, we reconsider the $\Upsilon(nS)$ states of bottonium and
$\psi(nS)$ states of charmonium to study their properties. The spectroscopic
parameters deduced using a phenomenological approach will be employed to
compute the decay properties such as the leptonic and di-gamma decay widths
with no additional parameters.
## Methodology
It has been shown that a purely phenomenological approach to the
nonrelativistic potential-model study of $\Upsilon$ spectra and $\psi$ spectra
can lead to a static non-Coulombic Power-law potential of the form a ; b
$V(r)=\lambda r^{\nu}+V_{0}$ (1)
where $\nu$ is close to 0.1 and $\lambda>0$.
Following general quantum mechanical rules as discussed in quigg , the binding
energy of a system with reduced mass $\mu$ in a power law potential, $\lambda
r^{\nu}$ is given by
$\displaystyle E_{nl}=\lambda^{2/(2+\nu)}$
$\displaystyle\left(2\mu\right)^{-\nu/(2+\nu)}$ (2)
$\displaystyle\left[A(\nu)\left(n+\frac{l}{2}-\frac{1}{4}\right)\right]^{2\nu/(2+\nu)}$
and the corresponding square of the probability amplitude of the S-waves at
the zero separation of the quark-antiquark system is given by
$\displaystyle|\psi_{n}(0)|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\left(\frac{2\mu\lambda}{\hbar^{2}}\right)^{3/(2+\nu)}\frac{\nu}{(2+\nu)}$
(3)
$\displaystyle[A(\nu)]^{3\nu/(2+\nu)}\left(n-\frac{1}{4}\right)^{2(\nu-1)/(2+\nu)}$
where
$A(\nu)=\left[2\nu\sqrt{\pi}\
\Gamma\left(\frac{3}{2}+\frac{1}{\nu}\right)\right]/\Gamma(1/\nu),\ \ \ \ \
\nu>0.$ (4)
The nonrelativistic Schrodinger bound-state mass (spin average mass) of the
$Q\bar{Q}$ ($Q\in b,c$) system follows as
$M_{SA}=2m_{Q}+V_{0}+E_{nl}$ (5)
For the hyperfine split we have considered the standard one gluon exchange
interaction Rai . Accordingly, the hyperfine mass split for the S-wave is
given by
$\Delta M=A_{hyp}|\psi_{n}(0)|^{2}/m_{Q}^{2}.$ (6)
The b and c quark mass parameters $m_{b}$ and $m_{c}$ are taken as 4.67 GeV
and 1.27 GeV respectively as given in PDG PDG2010 . The vector $\Upsilon(nS)$,
$\psi(nS)$ and the pseudoscalar $\eta_{b}(nS)$, $\eta_{c}(nS)$ masses are
obtained by adding $\Delta M/4$ and $-3\Delta M/4$ respectively to the
corresponding spin average mass of the nS state given by eq.(5). A fit to this
mass formula using the experimental masses of $\Upsilon(1S,2S)$ and the newly
discovered $\eta_{b}(1S)$ states provides us the potential parameters
$\lambda,V_{0}$ and the hyperfine parameter ($A_{hyp}$) in the case of
bottonium system. Similarly, a fit to this mass formula using the experimental
masses of $\psi(1S,2S)$ and $\eta_{c}(1S)$ states provides us the potential
parameters $\lambda,V_{0}$ and the hyperfine parameter ($A_{hyp}$) of the
charmonium systems. The predicted $\Upsilon(nS)$ , $\eta_{b}(nS)$ and
$\psi(nS)$ , $\eta_{c}(nS)$ states for n $\geq$ 2 are presented in Table 1 and
2 respectively.
To compare and identify our predicted states with the respective experimental
masses, the PDG PDG2010 average values as well as some of the $1^{--}$ states
of X,Y,Z N.Brambilla-1 are considered for the energy level difference,
$\Delta M=\\{\Gamma(n+1)S-\Gamma(nS)\\}$. These values are plotted against the
(n+1)S-nS for n$=$ 1 to 5 in the case of bottonia and charmonia in FIG.(1) and
(2) respectively. It is expected that the excited states must follow a
specific trend line representing its characteristic spectral property. So we
compare our predicted states with those which are closer to the systematic
expected behaviour shown by the solid line. The states which are widely off
from the expected behaviour are then identified as either mixed (disturbed)
states or exotic states.
Table 1: Results for $b\bar{b}$ spectrum
$nS$ | $M_{V}$ | $M_{V}$ | $M_{V}$ | $M_{P}$ | $M_{P}$ | $M_{P}$
---|---|---|---|---|---|---
| $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$
| [our] | 1 | Exp. | [our] | 1 | Exp.PDG2010
$1S$ | 9460.43 | 9460.38 | $\Upsilon$(9460) PDG2010 | 9392.43 | 9392.91 | $\eta_{b}$(9391)
$2S$ | 10021.60 | 10023.3 | $\Upsilon$(10023) PDG2010 | 9989.15 | 9987.42 | -
$3S$ | 10345.72 | 10364.2 | $\Upsilon$(10355) PDG2010 | 10323.46 | 10333.9 | -
$4S$ | 10574.91 | 10636.4 | $\Upsilon$(10579) PDG2010 | 10557.86 | 10609.4 | -
$5S$ | 10754.74 | - | $\Upsilon$(10860) PDG2010 | 10740.81 | - | -
$6S$ | 10903.15 | - | $\Upsilon$(11020) PDG2010 | 10891.33 | - | -
| | OR | $Y_{b}$(10888) Chen1 ; Chen | | |
| | OR | $\Upsilon$(10996) B.Aubert | | |
PDG2010 $\rightarrow$ PDG (2010); Chen1 ; Chen $\rightarrow$ Belle (2010);
1 $\rightarrow$ Radford & Repko (2011); B.Aubert $\rightarrow$ Babar (2009).
Table 2: Results for $c\bar{c}$ spectrum
$nS$ | $M_{V}$ | $M_{V}$ | $M_{V}$ | $M_{P}$ | $M_{P}$ | $M_{P}$
---|---|---|---|---|---|---
| $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$ | $(MeV)$
| [our] | 2007 | Exp. | [our] | 2007 | Exp.PDG2010
$1S$ | 3097.14 | 3096.92 | $J/\psi$(3097) PDG2010 | 2980.47 | 2981.7 | $\eta_{c}$(2980)
$2S$ | 3687.91 | 3686.1 | $\psi$(3687) PDG2010 | 3631.97 | 3619.2 | $\eta_{c}$(3637)
$3S$ | 4030.75 | 4102.0 | $\psi$(4040) PDG2010 | 3992.39 | 4052.5 | -
$4S$ | 4273.51 | 4446.8 | $Y$(4260) Yuan ; Aubert1 | 4244.13 | - | -
$5S$ | 4464.14 | - | $\psi$(4415) PDG2010 | 4440.15 | - | -
| | OR | $Z$(4430) Mizuk ; Choi | | |
$6S$ | 4621.55 | - | $X$(4630) BES08 | 4601.18 | - | -
PDG2010 $\rightarrow$ PDG (2010); 2007 $\rightarrow$ Radford & Repko (2007);
Yuan $\rightarrow$ Belle (2007); Aubert1 $\rightarrow$ Babar (2005);
BES08 $\rightarrow$ BES (2008); Mizuk $\rightarrow$ Belle (2009);Choi
$\rightarrow$ Belle (2008).
Table 3: The leptonic widths of the $\Upsilon(nS)$ and the di-gamma widths of $\eta_{b}(nS)$ states $(nS)$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{\gamma\gamma}$ | $\Gamma^{\gamma\gamma}$ | $\Gamma^{\gamma\gamma}$
---|---|---|---|---|---|---
| (keV) | (keV) | (keV) | (keV) | (keV) | (keV)
| [our] | 1 | Exp. | [our] | Schuler | Mohammad
$1S$ | 1.207 | 1.33 | 1.34$\pm$0.018 PDG2010 | 0.497 | 0.460 | 0.580
$2S$ | 0.513 | 0.62 | 0.612$\pm$0.011 PDG2010 | 0.209 | 0.20 | -
$3S$ | 0.330 | 0.48 | 0.443$\pm$0.008 PDG2010 | 0.134 | - | -
$4S$ | 0.242 | 0.40 | 0.272$\pm$0.029 PDG2010 | 0.098 | - | -
$5S$ | 0.191 | - | 0.310$\pm$0.07 PDG2010 | 0.077 | - | -
$6S$ | 0.157 | - | 0.130$\pm$0.030 PDG2010 | 0.064 | - | -
## Leptonic and di-gamma decay widths of Bottonium and charmonium states
Apart from the masses of the lowlying states, the hyperfine splits due to
chromomagnetic interaction and the right behaviour of the wave function that
provides as the correct predictions of the decay rates are important features
of any successful model. Accordingly,the radial wave functions of the
identified nS states of quarkonia ($c\bar{c},b\bar{b}$) obtained from eqn.(3)
are employed to predict the leptonic and digamma widths of the vector $1^{--}$
and $0^{-+}$ states respectively. The leptonic decay widths with the radiative
correction of $\Upsilon(nS)\rightarrow{l^{+}l^{-}}$ and
$\psi(nS)\rightarrow{l^{+}l^{-}}$ are computed as Ajay2008 ; Ajay2005 ;
Kwong1988
$\Gamma^{l^{+}l^{-}}=\frac{16\pi\alpha_{e}^{2}e_{Q}^{2}}{M_{V}^{2}}|\psi^{2}_{nl}(0)|\left[1-\frac{16}{3\pi}\alpha_{s}\right]$
(7)
And the di-gamma (two photon) decay widths of
$\eta_{b}(nS)\rightarrow{\gamma\gamma}$ and
$\eta_{c}(nS)\rightarrow{\gamma\gamma}$ with radiative correction are obtained
as
$\Gamma^{\gamma\gamma}=\frac{48\pi\alpha_{e}^{2}e_{Q}^{4}}{M_{P}^{2}}\
|\psi^{2}_{n}(0)|\left[1-\frac{\alpha_{s}}{\pi}\left(\frac{20-\pi^{2}}{3}\right)\right]$
(8)
$\alpha_{e}$ is the electromagnetic coupling constant and $\alpha_{s}$ is the
strong coupling constant. The predicted results in the case of bottonia and
charmonia are tabulated in Table 3 and 4 respectively.
Table 4: The leptonic widths of the $\psi(nS)$ and the di-gamma widths of $\eta_{c}(nS)$ states $nS$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{l^{+}l^{-}}$ | $\Gamma^{\gamma\gamma}$ | $\Gamma^{\gamma\gamma}$ | $\Gamma^{\gamma\gamma}$
---|---|---|---|---|---|---
| (keV) | (keV) | (keV) | (keV) | (keV) | (keV)
| [our] | 2007 | Exp. | [our] | Schuler | Mohammad
$1S$ | 4.944 | 1.89 | 5.55 $\pm$0.14 PDG2010 | 10.373 | 7.8 | 11.8
$2S$ | 1.671 | 1.04 | 2.35 $\pm$0.04 PDG2010 | 3.349 | 3.5 | -
$3S$ | 0.959 | 0.77 | 0.86 $\pm$0.07 PDG2010 | 1.900 | - | -
| | | 0.83 $\pm$0.07 BES08 | | | -
$4S$ | 0.654 | 0.65 | - | 1.288 | - | -
$5S$ | 0.489 | - | 0.58 $\pm$ 0.07 PDG2010 | 0.961 | - | -
| | | 0.35 $\pm$0.12 BES08 | | | -
$6S$ | 0.387 | - | - | 0.759 | - | -
Figure 1: Behavior of energy level shift of the (n+1)S$-$nS states Figure 2:
Behavior of energy level shift of the (n+1)S$-$nS states
## Results and discussion
We have been able to predict the charmonium and bottonium S-wave masses states
which are in good agreement with the reported PDG values as compared to the
predicted values of 1 . We have also predicted the $\eta_{b}(2S-6S)$ states
within the mass range 9.989 GeV to 10.891 GeV and $\eta_{c}(2S-6S)$ states
within the mass range 3631.97 GeV to 4601.18 GeV. We hope to find future
experimental support in favor of our predictions. With no additional
parameters we have been able to predict the leptonic decay widths of
$\Upsilon(1S-6S)$ as well as $\psi(1S-6S)$ states which are in good agreement
with the known experimental values PDG2010 . The predicted di-gamma decay
widths of $\eta_{b}(1S-6S)$ and $\eta_{c}(1S-6S)$ states would be helpful to
identify the $0^{-+}$ resonances in future experiments.
###
We do not compare our 5S and 6S states with those of $\Upsilon$(10860) and
$\Upsilon$(11020) states listed in PDG. They find it difficult to assign it as
the 5S and 6S states respectively as there seemed to be mixing of two Briet
Wigner resonances PDG2010 . The energy level shifts of the (n+1)S$-$nS states
clearly indicate that the state $\Upsilon$ (10860) is far from the expected
trend line as shown in FIG. 1. As expected it is seen from table 3 and 4, that
the leptonic decay widths falls off as nS goes from 1S to 6S. Such a behaviour
is seen even in the reported experimental values except in the case of
$\Upsilon(10860)$. Thus we identify $\Upsilon(10860)$ as either a mixed state
or exotic, while $Y_{b}(10888)$ observed very recently by Belle Chen and
$\Upsilon(10996)$ observed by Babar B.Aubert are closer to the predicted 6S
states of bottonia.
In the charmonium sector, many vector $(1^{--})$ states have been reported by
Belle and Babar N.Brambilla-1 ; Chen1 ; Chen ; Kai ; Q.He ; T.E. . It is
observed that the $1^{--}$ state, $\psi(3770)$ does not fit into the
$\psi(3S)$ state, while $\psi(4040)$ belong to the vector 3S state of
charmonia. Similarly, the newly discovered vector state $Y(4263)$ BES08 and
$\psi(4415)$ PDG2010 are closer to the 4S and 5S states of charmonia. Though
our predicted 6S state at 4621 MeV is close to the newly discovered $X(4630)$
$1^{--}$ state, the energy level difference as seen from Fig.(2) suggests it
to be unfit as the 6S state. The behaviour of the energy level difference also
suggests $\psi(4415)$ may be a mixed state. We expect $\psi(5S)$ state to be
about 200 MeV above $Y(4260)$ state. Such a state with mass of
$(4443^{+24}_{-18})$ represented by $Z(4430)$ has been reported by Belle,
though its $J^{PC}$ is not yet known Mizuk ; Choi . Then $X(4630)$ state
becomes the right candidate for the 6S state of charmonia. Thus future
experimental confirmation of vector charmonia state around 4464 MeV can
resolve the issue of the charmonium 5S and 6S vector states. And we predict
the leptonic decay widths of $\psi(5S,6S)$ states around 0.49 keV and 0.39 keV
as well as that for the $\Upsilon(5S,6S)$ states around 0.19 keV and 0.16 keV
respectively.
## Acknowledgments
The work is done under UGC Major research project NO. F.40-457/2011(SR).
## References
* (1) G.Bonvicini,et al.,CLEO collaboration,Phys. Rev.D 81 (2010) 031104.
* (2) K.M.Ecklund, et al.,CLEO collaboration,Phys. Rev.D 78 (2008) 091501.
* (3) B.Auger,et al.,BaBar collaboration,Phys. Rev. Lett. 103(2009)161801.
* (4) K Nakamura _et al._ (Particle Data Group), J.Phys. G, 37,075021 (2010).
* (5) S.Meinel,RBC Collaboration,UKQCD Collaboration,Phys. Rev. D 79(2009)094501.
* (6) A.Gray,et al.,HPQCD Collaboration,UKQCD Collaboration,Phys. Rev. D 72(2005)094507.
* (7) D.Ebert,R.N.Faustov,V.O.Galkin,Int.Mod. Phys. Lett. A 20(2005) 1887.
* (8) Horace W. Crater, Cheuk-Yin Wong, Peter Van Alstine, Phys. Rev. D 74 (2006) 054028.
* (9) Guo-Li Wang, Phys. Lett. B 653 (2007) 206.
* (10) N.Brambilla,E.Mereghetti, A. Vairo, Phys. Rev. D 79 (2009) 074002.
* (11) N.Brambilla et al.,Eur.Phys. J. C 71 (2011) 1534.
* (12) K.F. Chen et al. (Belle Collaboration), Phys. Rev. D 82(2010)091106(R). arXiv:0808.2445[hep-ex]
* (13) K.F. Chen et al. (Belle Collaboration), Phys. Rev. Lett. 100(2008)112001. arXiv:0710.2577 [hep-ex]
* (14) Kai Yi (for the CDF Collaboration), arXiv:1010.3470 [hep-ex]
* (15) Q. He et al. (CLEO Collaboration), Phys. Rev. D 74(2006)091104. arXiv:hep-ex/0611021
* (16) T.E. Coan et al. (CLEO Collaboration), Phys. Rev. Lett. 96(2006)162003 . arXiv:hep-ex/0602034
* (17) A.M. Badalian, B.L.G. Bakker and I.V. Danilkin. Physics of Atomic Nuclei,73(2010)138.
* (18) A.Martin, phys. Lett. 93B, 338 (1980).
* (19) N.Barik and S. N. Jena, Phys. Lett. 97B, 261 (1980).
* (20) C. Quigg and J.L. Rosner, Phys Rep. 56, No. 4 (1979) 167 235.
* (21) Ajay Kumar Rai,Bavin Patel and P C Vinodkumar, Phy.Rev. C78,055202(2008).
* (22) Rai A K, Pandya J N and Vinodkumar P C, Eur. Phys. J. A4, 77 (2008).
* (23) Rai A K, Pandya J N and Vinodkumar P C, J. Phys. G: Nucl. Part. Phys. 31, 1453 (2005).
* (24) Kwong W, Mackenzie P B, Rosenfeld R and Rosener J L, Phys. Rev. D37, 3210(1988).
* (25) Stanley F. Radford, Wayne W. Repko,Nucl. phys A 865(2011) 69-75.
* (26) B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 102(2009)012001. arXiv:0809.4120 [hep-ex]
* (27) Stanley F. Radford, Wayne W. Repko,Phys. Rev. D 75(2007) 074031.
* (28) G.A.Schuler,F.A.Berends,R.vanGulik,Nucl. Phy. B 523 (1998)423-438.
* (29) Mohammad R. Ahmady and Roberto R. Mendel, Phys. Rev. D 51 (1995)141.
* (30) C.Z. Yuan et al. (Belle Collaboration), Phys. Rev. Lett. 99(2007)182004. arXiv:0707.2541 [hep-ex]
* (31) B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 95(2005)142001. arXiv:hep-ex/0506081
* (32) M. Ablikim et al. (BES Collaboration), Phys. Lett. B 660 (2008) 315. arXiv:0705.4500 [hep-ex]
* (33) R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 80 (2009) 031104. arXiv:0905.2869 [hep-ex]
* (34) S.K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100 (2008)142001. arXiv:0708.1790 [hep-ex]
|
arxiv-papers
| 2012-03-28T07:36:39 |
2024-09-04T02:49:29.135420
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Manan Shah, Arpit Parmar, and P. C. Vinodkumar",
"submitter": "P C Vinodkumar",
"url": "https://arxiv.org/abs/1203.6184"
}
|
1203.6189
|
# Physical Parameters of Some Close Binaries: ET Boo, V1123 Tau, V1191 Cyg,
V1073 Cyg and V357 Peg
F. Ekmekçi A. Elmaslı M. Yılmaz T. Kılıçoğlu T. Tanrıverdi Ö. Baştürk H.
V. Şenavcı Ş. Çalışkan B. Albayrak S. O. Selam Ankara University, Faculty
of Science, Department of Astronomy and Space Sciences, TR-06100, Tandoğan,
Ankara, Turkey Niğde University, Faculty of Arts and Sciences, Department of
Physics, 51240, Niğde, Turkey Ankara University Observatory, 06837,
Ahlatlıbel, Ankara, Turkey
###### Abstract
With the aim of providing new and up-to-date absolute parameters of some close
binary systems, new BVR CCD photometry was carried out at the Ankara
University Observatory (AUG) for five eclipsing binaries, ET Boo, V1123 Tau,
V1191 Cyg, V1073 Cyg and V357 Peg between April, 2007 and October, 2008. In
this paper, we present the orbital solutions for these systems obtained by
simultaneous light and radial velocity curve analyses. Extensive orbital
solution and absolute parameters for ET Boo system were given for the first
time through this study. According to the analyses, ET Boo is a detached
binary while the parameters of four remaining systems are consistent with the
nature of contact binaries. The evolutionary status of the components of these
systems are also discussed by referring to their absolute parameters found in
this study.
###### keywords:
binaries: eclipsing; stars: fundamental parameters; stars: individual (ET Boo,
V1123 Tau, V1191 Cyg, V1073 Cyg, V357 Peg)
††journal: New Astronomy
## 1 Introduction
Eclipsing binary stars are essential key objects in the field of stellar
astrophysics to reach precise absolute stellar dimensions (i.e., masses, radii
etc.) and laboratories to test different theoretical aspects on stellar
structure and evolution.
Obtaining absolute parameters of close binary systems is very important in
examining/supporting the general consideration of the evolutionary history of
close binary stars (see e.g., Kraicheva, 1987, 1988; Zahn, 1989; Zahn and
Bouchet, 1989; Franstman, 1992). In this context, we present the results of
Wilson-Devinney (WD) light curve analysis of ET Boo, V1123 Tau, V1191 Cyg,
V1073 Cyg and V357 Peg based on new BVR CCD observations carried out at the
Ankara University Observatory (AUG) and spectroscopic observations by
Rucinski’s research team published in a series of fifteen papers on ”Radial
Velocity Studies of Close Binary Stars”. The current information on these five
systems could be summarized as follows:
ET Boo (SAO 45318, BD +47∘ 2190, HIP 73346, 2MASS $J14592031+4649036$) is a
$\beta$ Lyr type eclipsing binary, which is also a member of a quadruple
system (Pribulla et al., 2006), and found to be a close visual binary system
in 1978 by Couteau (1981). The spectral type of the system was given as F8 by
Bartkevičius and Gudas (2002). Up to now, the photometric observations of the
system have appeared to be rather unattended with the exception of the study
by Oja (1986) which only gives colour indices of ${\it B-V}=0.60$ and ${\it
U-B}=-0.01$ for ET Boo.
V1123 Tau (BD +17∘ 579, HIP 16706, ADS2624A) was discovered during the
HIPPARCOS mission (ESA, 1997) as a $\beta$ Lyr type eclipsing binary, and it
was classified as W UMa type eclipsing binary by Kazarovets et al. (2009). The
first ground-based photometry was made by Özdarcan et al. (2006). They only
gave the light and colour curves of V1123 Tau, obtained in the years 2003 and
2005, together with the new light elements. Recent radial velocity studies for
a number of close binary stars, including V1123 Tau, were carried out by
Rucinski et al. (2008). They pointed out that V1123 Tau is accompanied by a
fainter companion (ADS2624B, $\rho=4.3^{"}$, $\theta=136^{\circ}$, and $\Delta
V=1.77$), and the light contribution of this visual companion appears to have
affected the light amplitudes of the system. Gutiérrez (2009) presented the
results of optical spectroscopy of the visual binary ADS2624 and confirmed
that the spectral types of components A (V1123 Tau) and B are G0V and K0V,
respectively. The system has been classified as a W-subtype of W UMa systems
by Rucinski et al. (2008). Deb and Singh (2011) performed a light curve
analysis for V band observations of ASAS-3 project using Wilson-Devinney code.
Separately, Zahng et al. (2011) presented the absolute parameters of V1123 Tau
system based on their Wilson-Devinney analysis for the light curves obtained
with the high precision, multi-band CCD observations of this binary system.
V1191 Cyg (GSC 03159-01512, 2MASS J20165081+4157413, TYC 3159-1512-1) was
found to be a variable star by Mayer (1965). Pribulla et al. (2005a) gave new
CCD observations in BVRI filters, and carried out a light curve analysis using
Pribulla’s ROCHE code. They performed a grid search in the mass ratio (q)
parameter during the analysis since no spectroscopic mass ratio was known for
the system. They defined the system as W-subtype and gave the results for
geometric elements as ${\it i}=80.4(4)^{\circ}$, $q=0.094$, $f=0.46(2)$. Based
on the systematic deviation of the fit in R passband around primary minimum
they pointed out a probability of the presence of a third companion. Recently,
Ulaş et al. (2012) presented the parameters for the hot and cooler companions
together with the photometric and spectroscopic variations of V1191 Cyg binary
system. This system was also studied by Zhu et al. (2011) using their CCD
photometric light curves in BV(RI)c bands obtained in 2009. They derived the
absolute parametrs of V1191 Cyg based on their spectroscopic and photometric
solutions.
V1073 Cyg (BD +33∘ 4252, HD 204038, HIP 105739) is a W UMa type eclipsing
binary system. The variability of V1073 Cyg was first recognized by Strohmeier
(1960). Strohmeier et al. (1962) published photographic minima of the system
together with the first light elements and a photographic light curve.
Fitzgerald (1964) obtained first radial velocity curve and found the mass
ratio (q) to be 0.34 and spectral classification A3 Vm for the primary
component. Kondo (1966) solved the light curves of the system in Y (5410
$\AA$) and B (4250 $\AA$) bands using Russell-Merrill technique and proposed a
contact model for the system. Kruseman (1967) obtained light curves in blue
and yellow bands for which they haven’t given the effective wavelengths. They
also well fitted the data given by Kondo (1966) based on their results. Abt
and Bidelman (1969) classified the system as F0n III-IV or F0n V,
contradicting A3m V classification of Fitzgerald (1964) that used only
hydrogen lines. Leung and Schneider (1978) analyzed the light curves obtained
by Kondo (1966) with the Wilson-Devinney computer code. They have proposed
that V1073 Cyg was an overcontact system with a fill-out factor of 7%.
Niarchos (1978) reanalyzed the light curves of both Kondo (1966) and
Bendinelli et al. (1967) making use of Kopal’s method. Aslan and Herczeg
(1984) analyzed the orbital period behavior of the system and detected a
sudden period decrease of 0.4 seconds in 1976. Sezer (see Sezer, 1993, 1994),
found 8% overcontact and a mass ratio of 0.436 assuming $T{{}_{1}}=8570K$.
Wolf and Diethelm (1992) gave an O-C diagram and found that the period was
constantly decreasing. Ahn et al. (1992) analyzed three sets of previously
published light curves with the program LIGHT2, assuming convective behavior,
and $T_{1}=6700K$ (consistent with the previously derived spectral type of
F2), and found 8% overcontact with both components near the terminal-age main
sequence (TAMS). Morris and Naftilan (2000) observed V1073 Cyg for 4
consecutive nights in July 1998 and analyzed the light curves they obtained,
adopting the mass ratio published by Fitzgerald (1964) and using parameters
found by Sezer (1996) as initial parameters. Morris and Naftilan (2000)
indicated that most of the photometric analyses until the time of their study
relied on BD +33∘ 4248 as comparison star which they tested its variability
and found no evidence. Morris and Naftilan (2000) made an orbital period
analysis as well, and found that mass transfer was a more plausible
explanation for the orbital period change than mass loss, because of the
system’s high escape velocity. Another period analysis of the system was
published by Yang and Liu (2000) who also reported that the orbital period was
not stable but decreasing and that the light curves of the system show
unstable behavior too. They detected a positive O’Connell effect in Kondo
(1966)’s light curves while a negative O’Connell effect was observed by Sezer
(1993). Pribulla and Rucinski (2006) and Rucinski et al. (2007) could not
detect an additional component using adaptive optics observations. The system
is also defined as an A-subtype of W UMa systems (Pribulla et al., 2006).
V357 Peg (BD +24∘ 4828, HD 222994, HIP 117185, SAO 91468) was discovered and
classified as W UMa type binary system during the HIPPARCOS mission (ESA,
1997). The first photometric light curves of the system were obtained by
Yaşarsoy et al. (2000) but no analyses were performed. Selam (2004) analysed
the HIPPARCOS light curve of V357 Peg with Rucinski’s simplified light curve
synthesis method (Rucinski, 1993) and derived the mass ratio and inclination
of the system as 0.30 and 75∘, respectively. The first radial velocity curve
was given by Rucinski et al. (2008). They performed the first spectroscopic
observation and concluded that V357 Peg is an A-subtype contact binary system
with a mass ratio of 0.401 and spectral type of F2 V by using the spectra
taken between 1997 and 2005. Recently, Deb and Singh (2011) presented the
results for the V357 Peg binary system by using V band observations of ASAS-3
project in their Wilson-Devinney light curve analysis.
## 2 Observations
CCD observations of five eclipsing binaries (ET Boo, V1123 Tau, V1191 Cyg,
V1073 Cyg and V357 Peg) were carried out by using an Apogee ALTA U47+CCD
camera (1024x1024 pixels) with BVR filters mounted on a 40 cm Schmidt-
Cassegrain telescope of the Ankara University Observatory (AUG). The log of
observations is given in Table 1. The reduction of the CCD frames has been
performed with standard packages of IRAF111IRAF is distributed by the National
Optical Astronomy Observatory, which are operated by the Association of
Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation., and the individual differential BVR
observations were computed in the sense of variable minus comparison star. The
corresponding orbital phases for each variable were computed with the light
elements listed in Table 2. The light elements of V1123 Tau, listed in Table
2, were calculated by using the eight minima given by Yılmaz et al. (2009) and
three minima from the observations of this study. The light elements of V1191
Cyg were calculated by using the 11 published minima by various authors
(Pribulla et al., 2005b; Nelson, 2006; Hübscher et al., 2006; Nelson, 2007;
Parimucha et al., 2007; Hübscher, 2007). The light elements of V1073 Cyg were
calculated by using 19 minima obtained in various studies (Müyesseroğlu et
al., 1996; Nelson, 1998, 2003; Derman and Kalcı, 2003; Brát et al., 2007,
2008; Yılmaz et al., 2009) and those published in WEB sites
(http://astro.sci.muni.cz/variables/ocgate/; www.antonpaschke.com). For V357
Peg light elements were calculated by using 15 published minima by Keskin et
al. (2000), Aliş et al. (2002), Tanrıverdi et al. (2003), Albayrak et al.
(2005), Dróżdż and Ogloza (2005), Dvorak (2005), Parimucha et al. (2007) and
Hübscher and Walter (2007), and 5 minima obtained from the observations of
this study. Information for the comparison and check stars, used during the
observations, are given in Table 3.
Table 1: The log of CCD observations.
System | Obs. dates | # of data | Nightly mean errors
---|---|---|---
| | $\Delta B$ | $\Delta V$ | $\Delta R$ | $\sigma_{B}$(mag) | $\sigma_{V}$(mag) | $\sigma_{R}$(mag)
ET Boo | 16 Apr; 4,5,11 May | 562 | 682 | 619 | 0.004 | 0.004 | 0.005
| 2007 | | | | | |
V1123 Tau | 9,12,16,30 Sept; | 644 | 645 | 643 | 0.004 | 0.003 | 0.004
| 1 Oct 2008 | | | | | |
V1191 Cyg | 5,6,8,10,12,14,17 | 931 | 932 | 953 | 0.005 | 0.003 | 0.001
| Jul 2008 | | | | | |
V1073 Cyg | 9,13,18,19 Aug;2,14 | 1444 | 1326 | 1324 | 0.003 | 0.007 | 0.008
| Sept 2008 | | | | | |
V357 Peg | 10,16,23 Aug; | | | | | |
| 4,13,16,18 Sept; | 1583 | 1730 | 1867 | 0.001 | 0.001 | 0.001
| 8,19 Oct 2008 | | | | | |
Table 2: The light elements used in this study for each system. | Epoch | |
---|---|---|---
System | (HJD+2400000) | Period (days) | Ref.
ET Boo | 52701.5928 | 0.6450398 | Pribulla et al. (2006)
V1123 Tau | 54719.5749(5) | 0.3999441(98) | This study
V1191 Cyg | 54672.3431(19) | 0.3133885(5) | This study
V1073 Cyg | 54698.5000(51) | 0.7858500(70) | This study
V357 Peg | 54702.4652(5) | 0.5784511(1) | This study
Table 3: The comparison and check stars used during the observation for each
variables.
| Variable | Comparison | Check
---|---|---|---
| ET Boo | GSC 03474-00142 | -
$\alpha_{2000}$: | 14h59m20s.32 | 14h59m10s.00 | -
$\delta_{2000}$: | +46∘49${}^{{}^{\prime}}$03".61 | +46∘41${}^{{}^{\prime}}$29".20 | -
Spec. Typ.: | F8 | - | -
V(mag): | 9.09 | 10.9 | -
| V1123 Tau | GSC 1238-1039 | GSC 1238-1028
$\alpha_{2000}$: | 03h34m58s.55 | 03h34m28s.87 | 03h34m27s.48
$\delta_{2000}$: | +17∘42${}^{{}^{\prime}}$38".04 | +17∘42${}^{{}^{\prime}}$12".18 | +17∘35${}^{{}^{\prime}}$46".50
Spec. Typ.: | G0 | A5 | -
V(mag): | 9.97 | 10.9 | 11.44
| V1191 Cyg | HD 228669 | HD 228695
$\alpha_{2000}$: | 20h16m50s.81 | 20h16m01s.92 | 20h16m18s.55
$\delta_{2000}$: | +41∘57${}^{{}^{\prime}}$41".36 | +41∘58${}^{{}^{\prime}}$00".18 | +41∘58${}^{{}^{\prime}}$53".11
Spec. Typ.: | - | F8 | A0
V(mag): | 10.8 | 10.59 | 9.99
| V1073 Cyg | GSC 2711-2412 | GSC 2711-2014
$\alpha_{2000}$: | 21h25m00s.36 | 21h24m36s.99 | 21h24m53s.80
$\delta_{2000}$: | +33∘41${}^{{}^{\prime}}$14".94 | +33∘48${}^{{}^{\prime}}$21".15 | +33∘50${}^{{}^{\prime}}$12".59
Spec. Typ.: | F1V | A0 | -
V(mag): | 8.38 | 8.87 | 10.81
| V357 Peg | GSC 2254-1156 | GSC 2254-2715
$\alpha_{2000}$: | 23h45m35s.06 | 23h45m09s.02 | 23h45m29s.94
$\delta_{2000}$: | +25∘28${}^{{}^{\prime}}$18".94 | +25∘21${}^{{}^{\prime}}$12".05 | +25∘21${}^{{}^{\prime}}$46".20
Spec. Typ.: | F5 | - | -
V(mag): | 9.06 | 10.54 | 11.34
## 3 Analyses of the light curves
During the light curve analyses, the PHOEBE graphical user interface developed
by Prsa and Zwitter (2005) to visualize the well known Wilson-Devinney (WD)
light curve analysis code (Wilson and Devinney, 1971) was used. The adjustable
parameters are the inclination $i$, the non-dimensional potentials
($\Omega_{h,c}$), the surface temperature of the components ($T_{ph}$) and the
relative monochromatic luminosities ($L$) in each passband. The limb-darkening
coefficients of logarithmic law (the values of x and y in Table 4) were taken
from van Hamme (1993)’s tables. In the solutions, the rotation axis of the
components were taken to be perpendicular to the orbital plane, and
synchronized rotation was assumed for the component stars. The initial
estimations of the third light contribution for ET Boo and V1123 Tau were made
by using the corresponding $M_{v}$ taken from Table II of Straižyz and
Kuriliene (1981) on the adopted calibration of MK spectral types in absolute
magnitudes $M_{v}$. The differential correction program was initiated for the
simultaneous solution of the light and radial velocity curves and then a
visual inspection of the agreement between the synthetic and observational
light curves was made. The radial velocity (RV) data of all systems were taken
from the series of papers on ”Radial Velocity Studies of Close Binary Stars”
by Rucinski and his collaborators. RV data of ET Boo and V1073 Cyg were taken
from Pribulla et al. (2006), and RV data of V1123 Tau, V1191 Cyg, and of V357
Peg were taken from Rucinski et al. (2008). The goodness of fits
$\Sigma(O-C)^{2}$ to the light curves were checked for every run. After
reaching a satisfactory agreement, we fixed the parameters mentioned above and
adjusted for the various spot parameters if necessary. The results are
summarized in Table 4. The temperatures are given in units of Kelvin degrees
and the longitudes ($\lambda$), latitudes ($\beta$) and the radii (r) of spots
are in arc degrees. The ”latitude” of a star spot center, measured from 0
degrees at the ”north” (+ z) pole to 180 degrees at the ”south” pole of the
star. And, the longitude of a star spot center, measured counter-clockwise (as
viewed from above the + z axis) from the line of star centers from 0 to 360
degrees (Wilson, 1993). From Table 4, it can be seen that the achievement of
the results of spot parameters of the system were obtained with no spots for
ET Boo, one dark spot located on the cooler component of V1123 Tau, two dark
spots located on the cooler component of V1191 Cyg, one dark spot located on
the cooler component of V1073 Cyg, and one bright spot located on the hotter
component of V357 Peg. Rucinski et al. (2008) had determined the large
photospheric dark spot to be probably on the seconadry component of V357 Peg
around 0.75 orbital phase by using their observations made between August 25
and September 6, 2005. They also pointed out that the observatios made in 1997
did not show any indication of the photospheric spot. Therefore, at first the
WD analysis of the light curves of V357 Peg was attempted to be run with the
dark spot on the secondary component of the system, but no suitable result was
achieved. Then, it was seen that the analysis with bright spot on the hotter
component gave a satisfactory result for V357 Peg.
Table 4: Results of simultaneous WD code analysis of BVR CCD observations of
five close binary systems.
System | ET Boo | V1123 Tau | V1191 Cyg | V1073 Cyg | V357 Peg
---|---|---|---|---|---
q($M_{c}$/$M_{h})$ | 0.884 | 3.584 | 9.346 | 0.303 | 0.401
e (eccentricity) | 0 | 0 | 0 | 0 | 0
$V_{\gamma}$(kms-1) | \- 23.35 | 25.32 | \- 16.82 | \- 6.85 | \- 11.026
i(∘) | 76.3$\pm$0.1 | 74.01$\pm$1.05 | 83.2$\pm$2.2 | 69.85$\pm$0.72 | 73.34$\pm$1.57
Component | hot/cool | hot/cool | hot/cool | hot/cool | hot/cool
$\Omega$ | 4.657$\pm$0.006 | 7.194$\pm$0.005 | 14.144$\pm$0.05 | 2.44$\pm$0.08 | 2.60$\pm$1.44
| 3.817$\pm$0.004 | 7.194$\pm$0.005 | 14.144$\pm$0.05 | 2.44$\pm$0.08 | 2.60$\pm$1.44
$T_{ph}(K)$ | 6125 | 5920 | 6300 | 6700 | 7000
| 5758$\pm$40 | 5821$\pm$31 | 6215$\pm$55 | 6520$\pm$74 | 6687$\pm$971
Albedo | 0.5/0.5 | 0.5/0.5 | 0.5/0.5 | 0.5/0.5 | 0.65/0.65
g (gravity dark.) | 0.32/0.32 | 0.32/0.32 | 0.32/0.32 | 0.32/0.32 | 0.35/0.35
x(B) | 0.822/0.837 | 0.830/0.834 | 0.349/0.371 | 0.798/0.804 | 0.188/0.257
x(V) | 0.736/0.760 | 0.749/0.756 | 0.139/0.154 | 0.702/0.710 | 0.062/0.095
x(R) | 0.643/0.668 | 0.657/0.664 | 0.027/0.042 | 0.608/0.616 | -0.043/-0.016
y(B) | 0.197/0.141 | 0.166/0.151 | 0.536/0.516 | 0.255/0.240 | 0.694/0.624
y(V) | 0.261/0.233 | 0.248/0.239 | 0.674/0.664 | 0.283/0.278 | 0.724/0.701
y(R) | 0.271/0.253 | 0.263/0.257 | 0.695/0.687 | 0.287/0.284 | 0.734/0.719
$r_{pole}$ | 0.263/0.315 | 0.268/0.473 | 0.201/0.536 | 0.463/0.270 | 0.447/0.298
$r_{point}$ | 0.277/0.365 | —/— | —/— | —/— | —/—
$r_{side}$ | 0.267/0.326 | 0.281/0.512 | 0.210/0.599 | 0.499/0.282 | 0.481/0.313
$r_{back}$ | 0.273/0.346 | 0.324/0.541 | 0.249/0.619 | 0.527/0.321 | 0.513/0.358
$\lambda_{spot\\#1}$(∘) | —/— | —/237.56 | —/276.17 | —/333.14 | 310/—
$\lambda_{spot\\#2}$(∘) | —/— | —/— | —/166.35 | —/— | 60/—
$\beta_{spot\\#1}$(∘) | —/— | —/91.57 | —/102.29 | —/15.09 | 50/—
$\beta_{spot\\#2}$(∘) | —/— | —/— | —/90.15 | —/— | 60/—
rspot#1(∘) | —/— | —/11.69 | —/8.16 | —/50.66 | 28/—
rspot#2(∘) | —/— | —/— | —/9.89 | —/— | —/—
f(T)spot#1 | —/— | —/0.784 | —/0.786 | —/0.879 | 1.085/—
f(T)spot#2 | —/— | —/— | —/0.838 | —/— | —/—
f (fillout) | — | 0.165 | 0.295 | 0.174 | 0.312
$L_{1}$/($L_{Total}$)(in B) | 0.366 | 0.220 | 0.133 | 0.933 | 0.741
$L_{1}$/($L_{Total}$)(in V) | 0.345 | 0.211 | 0.130 | 0.928 | 0.730
$L_{1}$/($L_{Total}$)(in R) | 0.329 | 0.210 | 0.128 | 0.921 | 0.723
$L_{2}$/($L_{Total}$)(in B) | 0.389 | 0.609 | 0.868 | 0.077 | 0.258
$L_{2}$/($L_{Total}$)(in V) | 0.391 | 0.602 | 0.870 | 0.072 | 0.269
$L_{2}$/($L_{Total}$)(in R) | 0.392 | 0.610 | 0.872 | 0.079 | 0.277
$L_{3}$/($L_{Total}$)(in B) | 0.245$\pm$0.002 | 0.171$\pm$0.002 | — | — | —
$L_{3}$/($L_{Total}$)(in V) | 0.264$\pm$0.002 | 0.186$\pm$0.002 | — | — | —
$L_{3}$/($L_{Total}$)(in R) | 0.278$\pm$0.002 | 0.179$\pm$0.002 | — | — | —
$\Sigma(O-C)^{2}$ (in B) | 0.019 | 0.053 | 0.077 | 0.069 | 0.259
$\Sigma(O-C)^{2}$ (in V) | 0.045 | 0.033 | 0.043 | 0.141 | 0.106
$\Sigma(O-C)^{2}$ (in R) | 0.036 | 0.055 | 0.047 | 0.132 | 0.120
The absolute parameters of five binary stars, obtained by means of the WD
analyses, are given in Table 5. The optimum fit to each passband observed
light curves (Obs) to the synthetic ones (Theo) are shown in Fig. 1 for ET
Boo, in Fig. 2 for V1123 Tau and V1191 Cyg, and in Fig. 3 for V1073 Cyg and
V357 Peg. The final (O-C) residuals between the observed (Obs) and optimum
synthetic light curves are also given in these three Figures.
Table 5: Absolute parameters, obtained by means of the WD analyses, of the
five systems given in Table 4.
System | ET Boo | V1123 Tau | V1191 Cyg | V1073 Cyg | V357 Peg
---|---|---|---|---|---
a ($R_{\odot}$) | 4.06$\pm$0.01 | 2.68$\pm$0.02 | 2.18$\pm$0.01 | 4.70$\pm$0.02 | 3.09$\pm$0.03
$M_{h}(M_{\odot})$ | 1.15$\pm$0.02 | 0.35$\pm$0.01 | 0.14$\pm$0.01 | 1.73$\pm$0.10 | 0.85$\pm$0.03
$M_{c}(M_{\odot})$ | 1.02$\pm$0.03 | 1.27$\pm$0.03 | 1.28$\pm$0.02 | 0.53$\pm$0.06 | 0.34$\pm$0.02
$R_{h}(R_{\odot})$ | 1.096$\pm$0.003 | 0.78$\pm$0.03 | 0.48$\pm$0.05 | 2.33$\pm$0.11 | 1.48$\pm$0.13
$R_{c}(R_{\odot})$ | 1.369$\pm$0.004 | 1.36$\pm$0.03 | 1.27$\pm$0.06 | 1.36$\pm$0.12 | 0.99$\pm$0.14
$L_{h}(L_{\odot})$ | 1.51$\pm$0.01 | 0.66$\pm$0.04 | 0.32$\pm$0.07 | 9.77$\pm$0.95 | 4.73$\pm$0.85
$L_{c}(L_{\odot})$ | 1.85$\pm$0.02 | 1.91$\pm$0.11 | 2.16$\pm$0.27 | 3.01$\pm$0.67 | 1.77$\pm$1.54
$Logg_{h}(cgs)$ | 4.42 | 4.20 | 4.22 | 3.94 | 4.02
$Logg_{c}(cgs)$ | 4.17 | 4.27 | 4.34 | 3.89 | 3.97
$M_{bol,hot}$ | 4.30 | 5.19 | 5.98 | 2.28 | 3.06
$M_{bol,cool}$ | 4.08 | 4.05 | 3.91 | 3.55 | 4.13
Figure 1: The BVR light curves of ET Boo with the theoretical light curve
solutions (solid lines). The final O-C residuals from the fit are also shown
at the bottom of the figure. Figure 2: The BVR light curves of V1123 Tau (left
panel) and V1191 Cyg (right panel) with the theoretical light curve solutions
(solid lines). The final O-C residuals from the theoretical fits are also
shown at the bottom of each panel. Figure 3: The BVR light curves of V1073 Cyg
(left panel) and V357 Peg (right panel) with the theoretical light curve
solutions (solid lines). The final O-C residuals from the theoretical fits are
also shown at the bottom of each panel.
## 4 Results and conclusions
New CCD BVR observations of five eclipsing binaries (ET Boo, V1123 Tau, V1073
Cyg, V1191 Cyg and V357 Peg) were obtained and the light curve analyses
performed to acquire the absolute parameters of the components of these five
systems. According to the results of simultaneous WD analysis given in Table 4
and the absolute parameters given in Table 5, following evaluations can be
inferred:
ET Boo: Up to now, the absolute parameters of ET Boo seem to be given for the
first time in this study. The results given in Tables 4 and 5 are obtained
with the detached binary mode in WD code and no spots located on the
components of ET Boo binary system.
V1123 Tau: The results given in Tables 4 and 5 are obtained with the MODE 3
for overcontact binaries and one dark spot located on the cooler component of
V1123 Tau. The absolute parameters of V1123 Tau, with their error values,
obtained in this study (see Table 5) are consistent with those given by Zahng
et al. (2011), but they are a bit lower than those given by Deb and Singh
(2011) (see Table 6) due mainly to the different values of orbital inclination
and third light contribution. Deb and Singh (2011) obtained the orbital
inclination and the third light level as i(∘)= 68.10$\pm$0.24 and
$L_{3}=0.013\pm 0.009$ in V while our results gave the value of this
parameters as $74.01\pm 1.05$ and $0.186\pm 0.002$, respectively. Also, Deb
and Singh (2011) have not attempted to obtain any spot parameters in their
analysis.
Table 6: Absolute parameters, of V1123 Tau, V1191 Cyg and V357 Peg systems as
found in the literature.
System | V1123 Tau | V1191 Cyg | V357 Peg
---|---|---|---
a ($R_{\odot}$) | - | 2.779$\pm$0.011 | 2.20$\pm$0.08 | 2.194$\pm$0.012 | 3.920$\pm$0.016
$M_{h}(M_{\odot})$ | 0.40$\pm$0.02 | 0.392$\pm$0.015 | 0.13$\pm$0.01 | 0.139$\pm$0.08 | 0.690$\pm$0.013
$M_{c}(M_{\odot})$ | 1.36$\pm$0.05 | 1.404$\pm$0.070 | 1.29$\pm$0.08 | 1.306$\pm$0.022 | 1.720$\pm$0.015
$R_{h}(R_{\odot})$ | 0.80$\pm$0.01 | 0.789$\pm$0.004 | 0.52$\pm$0.15 | 0.518$\pm$0.003 | 1.250$\pm$0.024
$R_{c}(R_{\odot})$ | 1.37$\pm$0.02 | 1.389$\pm$0.006 | 1.31$\pm$0.18 | 1.307$\pm$0.007 | 2.120$\pm$0.015
$L_{h}(L_{\odot})$ | 0.67$\pm$0.04 | 0.695$\pm$0.129 | 0.46$\pm$0.25 | 0.463$\pm$0.006 | 2.406$\pm$0.333
$L_{c}(L_{\odot})$ | 2.01$\pm$0.07 | 2.130$\pm$0.385 | 2.71$\pm$0.80 | 2.731$\pm$0.029 | 9.673$\pm$1.195
References | (1) | (2) | (3) | (4) | (2)
(1)Zahng et al. (2011), (2)Deb and Singh (2011), (3)Ulaş et al. (2012), (4)Zhu
et al. (2011)
V1191 Cyg: The results given in Tables 4 and 5 are obtained with the MODE 3
for overcontact binaries and one dark spot located on the cooler component of
V1191 Cyg. The absolute parameters of V1191 Cyg, with their error values,
obtained in this study (see Table 5) are consistent with those given by Ulaş
et al. (2012) and Zhu et al. (2011)(see Table 6).
V1073 Cyg: The results given in Tables 4 and 5 are obtained with the MODE 3
for overcontact binaries and one dark spot located on the cooler component of
V1073 Cyg. The results given by Ahn et al. (1992) on the masses, radii,
temperatures, filling factor for V1073 Cyg binary system were somewhat
different from the results of this study. They found that the masses, radii,
temperatures, and filling factor as $M_{h}=0.51\pm 0.01M_{\odot}$,
$M_{c}=1.60\pm 0.02M_{\odot}$, $R_{h}=1.33\pm 0.02R_{\odot}$, $R_{c}=2.24\pm
0.02R_{\odot}$, $L_{h}=0.46\pm 0.03L_{\odot}$, $L_{c}=0.95\pm 0.03L_{\odot}$,
and f=0.92, respectively. Pribulla et al. (2003) gave f = 0.04 and
$T_{c}=6650K$ for V1073 Cyg in their ”Catalogue of the field contact binary
stars”. Jafari et al. (2006) also found $M_{h}=0.55M_{\odot}$,
$M_{c}=1.64M_{\odot}$, $R_{h}=1.40R_{\odot}$, $R_{c}=2.28R_{\odot}$,
$T_{h}=6700K$, $T_{c}=6494K$, f=0.19 with an unspotted model. In this study,
the masses, radii, luminosities and the temperatures of the hotter and cooler
components of V1073 Cyg are obtained as listed in Table 5.
Dumitrescu and Dinescu (1976) gave a light curve of the system obtained with
no filter. But they did not present any kind of solution to this light curve.
Ahn et al. (1992) obtained new Reticon spectral observations of the system and
computed the mass ratio as 0.32$\pm$0.01. Because they determined a late
spectral type for the system as noted by Hilditch and Hill (1975) from the
uvby colors and Hill et al. (1975)’s spectral classification (F2 IV-F1 V),
they assumed convective envelopes for both of the components. Sezer (1993)
obtained photoelectric light curves in B and V bands, and used the WD computer
program to find 3% overcontact and a mass ratio of 0.436, assuming radiative
behavior and $T_{h}=8570K$. Sezer (1996) revised the analyses of his previous
light curves, this time assuming convective behavior, and $T_{h}=6700K$,
consistent with the spectroscopic values of Ahn et al. (1992) instead of
taking A3Vm as the spectral type, and found 19% - 22% overcontact, a mass
ratio of 0.306 and i = 69∘.4. The light curves of the system were reported
also to be variable by Yang and Liu (2000). While a positive O’Connell effect
had been observed in 1963-1964 (Kondo, 1966), a negative O’Connell effect was
observed in 1988-1991 (Sezer, 1993). Yang and Liu (2000) could not find a
definite solution for the unstable behavior of the light curves of V1073 Cyg
and they grouped it with AU Ser and FG Hya as they are A-subtype W UMa stars
showing instability in both their orbital periods and light curves.
V357 Peg: The results given in Tables 4 and 5 are obtained with the MODE 3 of
WD code for overcontact binaries and one bright spot located on the hotter
component of V357 Peg binary system. The absolute parameters of V357 Peg, with
their error values, obtained in this study (see Table 5) were somewhat
different from those given by Deb and Singh (2011)(see Table 6). The main
difference between the WD light curve analysis by Deb and Singh (2011) and the
analaysis of this study is the bright spot we located on the hotter component
of V357 Peg system(see Table 4). Therefore, this hot spot may indicate the
effect of a mass transfer between the components of the system and this effect
may be the cause of somewhat different values of absolute parameters that we
have in our analysis.
MK classifications and evolutionary states: An evaluation of the log g and
$logT_{ph}$ values of the components of five eclipsing binaries(ET Boo, V1123
Tau, V1073 Cyg, V1191 Cyg and V357 Peg) in the $log~{}g-logT_{e}$ diagrams
given by Straižyz and Kuriliene (1981) and by Maeder and Meynet (1988) reveals
the following results:
The spectral types of both components of ET Boo could be F7-8 IV or MK
classification for ET Boo could be as F8 V+F8 V which is not far from one
another(aside from luminosity classes IV). In a sense, this estimation was a
verification of the F8 spectral type of the system given by Bartkevičius and
Gudas (2002). No theoretical study on the evolutionary status on detached
$\beta$ Lyr type close binary systems has been published yet. Therefore, we
could not evaluate the absolute parameters of the components of ET Boo to see
and examine the evolutionary characteristics of the component stars of ET Boo.
However, on the occasion of the similarity of ET Boo and some short period RS
CVn type binaries (e.g. CG Cyg, WY Cnc, RT And, and ER Vul), in point of their
mass ratio, and absolute parameters (see Dryomova et al., 2005; Budding et
al., 1996; Lastennet and Valls-Gabaud, 2002; Kjurkchieva et al., 2003, 2004),
a plausible inference can be made for the relationship of the evolutionary
status of ET Boo system. Lastennet and Valls-Gabaud (2002) showed that RT And
and CG Cyg have a secondary component far too cool to be matched by the same
isochrone as the primary. They also pointed out to the same difficulty with
the models by fitting simultaneously their effective temperatures, masses and
radii (see Pols et al., 1997). In addition to effective temperature revision,
they gave a possible explanation of the disagreement that may come from mass
transfer and starspot activity. Therefore, it can be inferred that ET Boo
system is more likely to undergo an evolutionary progress to be a short period
RS CVn-type system than to be a W UMa-type contact binary. In order to confirm
this point of view, it will be better to take some high resolution spectra of
ET Boo to evaluate this point together with the activity phenomena.
The spectral types of the components of V1123 Tau could be G0 V+G1-2 V or MK
classification for the cooler component of V1123 Tau could be as G1-2 V.
However, Rucinski et al. (2008) estimated that G0 V is the spectral type of
V1123 Tau.
The spectral types of the components of V1191 Cyg could be F6-7 V-IV or MK
classification for the cooler component of V1191 Tau could be $\sim$ F6-7V.
This prediction is in agreement with the estimation by Rucinski et al. (2008)
on the spectral type of V1191 Cyg as F6 V.
The spectral types of the components of V1073 Cyg could be F5 IV + F5 IV or MK
classification for the hotter component of V1073 Tau could be $\sim$ F5 V -
IV. But the results given by Ahn et al. (1992) on MK spectral type for V1073
Cyg binary system were somewhat different from the results of this study. They
found the spectral type as F1-F2. Hill et al. (1975) gave also the spectral
classification as F2 IV-F1 V.
The spectral types of the components of 357 Peg could be F2 IV+F3 IV. However,
Rucinski et al. (2008) estimated that F2V is the spectral type of V357 Peg.
Hilditch et al. (1988) showed that some primary components of the contact
W-type W UMa systems are located below the ZAMS in the mass-luminosity diagram
due to luminosity transfer to the secondary components, and the secondary
components of the W-type systems all have larger radii than expected for their
ZAMS masses. On the other hand, the A-type W UMa systems which are more
evolved than W-type counterparts, are located near or beyond the TAMS. And the
secondary components of these A-type systems have substantially larger radii
than expected for their ZAMS masses. By comparing and evaluating the masses,
radii, luminosities and the temperatures of W-type systems contact binaries
(V1123 Tau, V1191 Cyg), it can be seen that the characteristics of hotter and
cooler components of V1123 Tau and V1191 Cyg are in agreement with the results
of Hilditch et al. (1988) on the secondary and primary components of W-type
systems, respectively. Again, with the values of masses, radii, luminosities
and temperatures of the hotter and cooler components of A-type system contact
binaries (V1073 Cyg, V357 Peg) it can also seen that the characteristics of
hotter and cooler components of V1073 Cyg and V357 Peg are in agreement with
the results of Hilditch et al. (1988) on the primary and secondary components
of A-type systems, as well.
Consequently, the masses and the radii of the components together with their
bolometric absolute magnitudes, given in Table 5, are important indicators for
the evolutionary status of the components in the context of close binary
evolution.
## 5 Acknowledgments
This research has made use of the Simbad database, operated at CDS,
Strasbourg, France, and of NASA’s Astrophysics Data System Bibliographic
Services. ÖB would like to thank The Scientific and Technological Research
Council of Turkey (TÜBİTAK) for their support through BIDEB-2211 scholarship.
## References
* Abt and Bidelman (1969) Abt, H.A., Bidelman, W.P., 1969. ApJ 158, 1091.
* Ahn et al. (1992) Ahn, Y.S., Hill, G., Khalesseh, B., 1992. A&A 265, 597.
* Albayrak et al. (2005) Albayrak, B., Yüce, K., Selam, S.O., Tanrıverdi, T., Okan, A., Çınar, D., Topal, S., Özgür, E., Şener, H.T., Ergün, İ., Civelek, E., 2005. IBVS 5649.
* Aliş et al. (2002) Aliş, S., Keskin, M.M., Atay, M.E., 2002. IBVS 5282.
* Aslan and Herczeg (1984) Aslan, Z., Herczeg, T., 1984. IBVS 2478.
* Bartkevičius and Gudas (2002) Bartkevičius, A., Gudas, A., 2002. Baltic Astronomy 11, 153.
* Bendinelli et al. (1967) Bendinelli, O., Catalano, S., Cristaldi, S., Delli Ponti, C., 1967. Publ. Oss. Astron. Univ. Bologna 9, 11.
* Brát et al. (2007) Brát, L., Zejda, M., Svoboda, P., 2007. Open European Journal on Variable Stars 74, 1.
* Brát et al. (2008) Brát, L., Šmelcer, L., Kučáková, H., Ehrenberger, R., Kocián, R., Lomoz, F., Urbančok, L., Svoboda, P., Trnka, J., Marek, P., Dřevěný, R., Uhlář, R., Poddaný, S., Zasche, P., Skarka, M., 2008. Open European Journal on Variable Stars 94, 1.
* Budding et al. (1996) Budding, E., Butter, C.J., Doyle, J.G., Etzel, P.B., Oláh, K., Zeilik, M., Brown, D., 1996. Ap&SS 404, 611.
* Couteau (1981) Couteau, P., 1981. A&AS 43, 79.
* Deb and Singh (2011) Deb, S., Singh, H.P., 2011. MNRAS 412, 1787.
* Derman and Kalcı (2003) Derman, E., Kalcı, R., 2003. IBVS 5439.
* Dróżdż and Ogloza (2005) Dróżdż, M., Ogloza, W., 2005. IBVS 5623.
* Dryomova et al. (2005) Dryomova, G., Perevozkina, E., Svechnikov, M., 2005. A&A 437, 375.
* Dumitrescu and Dinescu (1976) Dumitrescu, A., Dinescu, R., 1976. IBVS 1116.
* Dvorak (2005) Dvorak, S.W., 2005. IBVS 5603.
* ESA (1997) ESA, 1997. The Hipparcos and Tycho Catalogues, ESA-SP-1200.
* Fitzgerald (1964) Fitzgerald, M.P., 1964. Publ. of the David Dunalp Obs. 2, 417.
* Franstman (1992) Frantsman, Y.L., 1992. Sov. Astr. 36, 155.
* Gutiérrez (2009) Gutiérrez, C.M., 2009. New Astronomy 14, 149.
* Hilditch and Hill (1975) Hilditch, R.W., Hill, G., 1975. Mem. R. Astr. Soc. 79, 101.
* Hilditch et al. (1988) Hilditch, R.W., King, D.T., McFarlane, T.M., 1988. MNRAS 231, 341.
* Hill et al. (1975) Hill, G., Hilditch, R.W., Younger, F., Fisher, W.A., 1975. Mem. R. Astr. Soc. 79, 131.
* Hübscher et al. (2006) Hübscher, J., Paschke, A., Walter, F., 2006. IBVS 5731.
* Hübscher and Walter (2007) Hübscher, J., Walter, F., 2007. IBVS 5761.
* Hübscher (2007) Hübscher, J., 2007. IBVS 5802.
* Jafari et al. (2006) Jafari, M., Khalesseh, B., Pazhouhesh, R., 2006. ApSS 306, 29.
* Kazarovets et al. (2009) Kazarovets, E.V., Samus, N. N., Durlevich, O.V., Frolov, M.S., Antipin, S. V.; Kireeva, N. N., Pastukhova, E.N., 2009. IBVS 4659.
* Keskin et al. (2000) Keskin, V., Yaşarsoy, B., Sipahi, E., 2000. IBVS 4855.
* Kjurkchieva et al. (2003) Kjurkchieva, D.P., Marchev, D.V., Zola, S., 2003. A&A 404, 611.
* Kjurkchieva et al. (2004) Kjurkchieva, D., Marchev, D., Ogloza, W., 2004. A&A 415, 231.
* Kondo (1966) Kondo, Y., 1966. AJ 71, 54.
* Kraicheva (1987) Kraicheva, Z.T., 1987. Nauchnye Informatsii 63, 105.
* Kraicheva (1988) Kraicheva, Z.T., 1988. Nauchnye Informatsii 65, 41.
* Kruseman (1967) Kruseman, P., 1967. Bull. of the Astron. Inst. of Netherlands Supp. Ser. 2, 377.
* Lastennet and Valls-Gabaud (2002) Lastennet, E., Valls-Gabaud, D., 2002. A&A 396, 551.
* Leung and Schneider (1978) Leung, K.C., Schneider, D.P., 1978. ApJ 222, 917.
* Maeder and Meynet (1988) Maeder, A., Meynet, G., 1988. A&AS 76, 411.
* Mayer (1965) Mayer, P., 1965. Bull. Astron. Obs. Czechosl., 16, 255.
* Morris and Naftilan (2000) Morris, S.L., Naftilan, S.A., 2000. PASP 112, 852.
* Müyesseroğlu et al. (1996) Müyesseroğlu, Z., Gürol, B., Selam, S.O., 1996. IBVS 4380.
* Nelson (1998) Nelson, R.H., 1998. IBVS 4621.
* Nelson (2003) Nelson, R.H., 2003. IBVS 5371.
* Nelson (2006) Nelson, R.H., 2006. IBVS 5672.
* Nelson (2007) Nelson, R.H., 2007. IBVS 5760.
* Niarchos (1978) Niarchos, P.G., 1978. Ap&SS 58, 301.
* Oja (1986) Oja, T., 1986. A&AS 65, 405.
* Özdarcan et al. (2006) Özdarcan, O., Topçu, A.T., Evren, S., Taş, G., 2006. IBVS 5688.
* Parimucha et al. (2007) Parimucha, S., Vanko, M., Pribulla, T., Hambalek, L., Dubovsky, P., Baludansky, D., Petrik, K., Chrastina, M., Urbancok, L., 2007. IBVS 5777.
* Pols et al. (1997) Pols, O.R., Tout, C.A., Schröder, K., Eggleton, P.P., Manners, J., 1997, MNRAS 289, 869.
* Pribulla et al. (2003) Pribulla, T., Kreiner, J.M., Tremko, J., 2003. Contib. Astron. Obs. Skalnaté Pleso 33, 38.
* Pribulla et al. (2005a) Pribulla, T., Vaňko, M., Chochol, D., Parimucha, S., Baluďanský, D., 2005a. Ap&SS 296, 281.
* Pribulla et al. (2005b) Pribulla, T., Baludansky, D., Chochol, D., Chrastina, M., Parimucha, S., Petrik, K., Szasz, G., Vanko, M., Zboril, M., 2005b. IBVS 5668.
* Pribulla et al. (2006) Pribulla, T., Rucinski, S.M., Lu, W., Mochnacki, S.W., Conidis, G., Blake, R.M., DeBond, H., Thomson, J.R., Pych, W., Ogloza, W., Siwak, M., 2006. AJ 132, 769.
* Pribulla and Rucinski (2006) Pribulla, T., Rucinski, S.M., 2006. AJ 131, 2986.
* Prsa and Zwitter (2005) Prsa, A., & Zwitter, R., 2005. ApJ, 628, 426.
* Rucinski (1993) Rucinski, S.M., 1993. PASP 105, 1433.
* Rucinski et al. (2007) Rucinski, S.M., Pribulla, T., van Kerkwijk, M.H., 2007. AJ 134, 2353.
* Rucinski et al. (2008) Rucinski, S.M., Pribulla, T., Mochnacki, S.W., Liokumovich, E., Lu, W., DeBond, H., De Ridder, A., Karmo, T., Rock, M., Thomson, J.R., Ogloza, W., Kaminski, K., Ligeza, P., 2008. AJ 136, 586.
* Selam (2004) Selam, S.O., 2004. A&A 416, 1097.
* Sezer (1993) Sezer, C., 1993. Ap&SS 208, 15.
* Sezer (1994) Sezer, C., 1994. Ap&SS 215, 153.
* Sezer (1996) Sezer, C., 1996. Ap&SS 245, 89.
* Straižyz and Kuriliene (1981) Straižyz, V., Kuriliene, G., 1981. Ap&SS 80, 353.
* Strohmeier (1960) Strohmeier, W., 1960. Veröffentlichungen der Remeis-Sternwarte zu Bamberg, Nr. 27. Veränderlichen-Colloquium Bamberg 1959, p.1.
* Strohmeier et al. (1962) Strohmeier, W., 1962. IBVS 9.
* Tanrıverdi et al. (2003) Tanrıverdi, T., Kutdemir, E., Elmaslı, A., Şenavcı, H.V., Albayrak, B., Selam, S.O., Aydın, C., Aksu, O., Bulca, I., Çınar, D., Kara, A., Demirhan, M., Yılmaz, M., Çetintaş, C., Gözler, A.P., Karakaş, T., Sezgin, A.S., Turhanoğlu, B., 2003. IBVS 5407.
* Ulaş et al. (2012) Ulaş, B., Kalomeni, B., Keskin, V., Köse, O., Yakut, K., 2012. New Astronomy 17, 46.
* van Hamme (1993) van Hamme, W., 1993. AJ 106, 2096.
* Wilson and Devinney (1971) Wilson, R.E., Devinney, E.J., 1971. ApJ 166, 605.
* Wilson (1993) Wilson, R.E., 1993. In: New Fronties in Binary Star Research, K.C. Leung, I.-S. Nha (Eds.), ASP Conference Series Vol.38, p.91.
* Wolf and Diethelm (1992) Wolf, M., Diethelm, R., 1992. Acta Astronomica 42, 363.
* Yang and Liu (2000) Yang, Y., Liu, Q., 2000. Ap&SS 274, 799.
* Yaşarsoy et al. (2000) Yaşarsoy, B., Sipahi, E., Keskin, V., 2000. IBVS 4866.
* Yılmaz et al. (2009) Yılmaz, M., Baştürk, Ö., Alan, N., Şenavcı, H.V., Tanrıverdi, T., Kılıçoğlu, T., Çalışkan, S., Çelik, L., Aydın, G., Çakan, D., Bilgiç, D., Uluş, N.D., Elmaslı, A., Selam, S.O., Albayrak, B., Ekmekçi, F., 2009. IBVS 5887.
* Zahn (1989) Zahn, J.-P., 1989. A&A 220, 112.
* Zahn and Bouchet (1989) Zahn, J.-P., Bouchet, L., 1989. A&A 223, 112.
* Zahng et al. (2011) Zahn, Xiao-Bin, Ren, An-Bin, Luo, Chang-Qing, Luo, Yang-Ping, 2011. RAA 11, 583.
* Zhu et al. (2011) Zhu, L.Y., Qian, S.B., Soonthornthum, B., He, J.J., Liu, L., 2011. AJ 142, 124.
|
arxiv-papers
| 2012-03-28T08:20:29 |
2024-09-04T02:49:29.140151
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Ekmek\\c{c}i, A. Elmasl{\\i}, M. Y{\\i}lmaz, T.\n K{\\i}l{\\i}\\c{c}o\\u{g}lu, T. Tanr{\\i}verdi, \\\"O. Ba\\c{s}t\\\"urk, H. V.\n \\c{S}enavc{\\i}, \\c{S}. \\c{C}al{\\i}\\c{s}kan, B. Albayrak, S. O. Selam",
"submitter": "Fehmi Ekmek\\c{c}i",
"url": "https://arxiv.org/abs/1203.6189"
}
|
1203.6319
|
11institutetext: (1) Dip. di Fisica and INFN, Università “Tor Vergata”, Via
della Ricerca Scientifica 1, I-00133 Roma, Italy.
(2)The Niels Bohr Institut, Blegdamsvej 17, DK-2100 Copenhagen, Denmark.
(3) Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA
(4) Department of Physics, Department of Mathematics and Computer Science, and
J.M. Burgerscentrum, Eindhoven University of Technology, 5600 MB Eindhoven,
The Netherlands;
and International Collaboration for Turbulence Research.
(5) Dept. de Fisica i Eng. Nuclear, Universitat Politecnica de Catalunya Edif.
GAIA, Rambla Sant Nebridi s/n, 08222 Terrassa, Barcelona, Spain
# Population dynamics in compressible flows
Roberto Benzi(1) Mogens H. Jensen(2) David R. Nelson(3) Prasad Perlekar (4)
Simone Pigolotti (5) Federico Toschi (4)
###### Abstract
Organisms often grow, migrate and compete in liquid environments, as well as
on solid surfaces. However, relatively little is known about what happens when
competing species are mixed and compressed by fluid turbulence. In these
lectures we review our recent work on population dynamics and population
genetics in compressible velocity fields of one and two dimensions. We discuss
why compressible turbulence is relevant for population dynamics in the ocean
and we consider cases both where the velocity field is turbulent and when it
is static. Furthermore, we investigate populations in terms of a continuos
density field and when the populations are treated via discrete particles. In
the last case we focus on the competition and fixation of one species compared
to another
## 1 Introduction
Challenging problems arise when spatial migrations of species are combined
with population genetics. Stochastic number fluctuations are inevitable at a
frontier, where the population size is small and the discrete nature of the
organisms becomes essential. Depending on the parameter values, these
fluctuations can produce important changes with respect to the deterministic
predictions 3 ; 4 . When two or more species undergo a Darwinian competition
in a spatial environment, one must deal with additional issues such as genetic
drift (stochastic fluctuations in the local fraction of one species compared
to another) and Fisher genetic waves, fisher which allow more fit species to
replace less fit ones. On solid surfaces, the complexities of spatial
population genetics are elegantly accounted for by the stepping stone model,
originally introduced by Kimura and Weiss 6 , 8 .
However, much of population genetics, from the distant past up to the present,
played out in liquid environments, such as lakes, rivers and oceans. For
example, there are fossil evidence for oceanic photosynthetic cyanobacteria
(likely pre- cursors of chloroplasts in plants and a major source of oxygen in
the atmosphere) that date back a billion years or more 11 . In addition, it
has recently become possible to perform satellite observations of chlorophyll
concentrations to identify fluid dynamical niches of phytoplankton types off
the eastern coast of the southern tip of South America 12 , where the domains
of the species are largely determined by the tangen- tial velocity field
obtained from satellite altimetry. In cases such as these, spatial growth and
evolutionary competition take place in the presence of advecting flows, some
of them at high Reynolds numbers 13 .
Phytoplankton needs light and nutrients to grow and many phytoplankton species
are able to adjust their density and swim to stay near the surface. Nutrients
are brought to the surface from deeper ocean layers, usually below 500 meters.
Therefore, oceanic circulation plays an important role in shaping spatial
growth and evolution of plankton species. To appreciate the complexity of the
problem, it is worthwhile to shortly review our present knowledge on the basic
mechanisms one should consider. In a three dimensional turbulent flow at high
Reynolds number, the velocity field is fluctuating over a range of scales
$[L,\eta]$ where $L$ is the scale of energy pumping in the system and
$\eta\equiv(\nu^{3}/\epsilon)^{1/4}$ is the Kolmogorov dissipation scale. The
velocity field is also fluctuating in time. According to Kolmogorov theory,
one can define the dissipation time scale as
$\tau_{\eta}\equiv\sqrt{\nu/\epsilon}$. In the upper oceanic mixed layer ,
forcing is provided by heat and momentum exchange with atmosphere and the
observed values dissipation of $\epsilon$ ranges from $10^{-7}cm^{2}/sec^{3}$
up to $50cm^{2}/sec^{3}$, which implies $\eta\in[0.01,2]cm$ and
$\tau_{\eta}\in[0.01,300]sec$. The phytoplankton size lies in the range
$[10,200]\mu m$ with a density difference respect to sea water density in the
range $[0.01,0.1]$. Advection of individuals in the ocean should be studied by
considering all forces acting on them. In particular, because of density
mismatch and finite size, individuals are not advected as simple Lagrangian
tracers Toschi 14 , i.e. the velocity field experienced by each individual
is not the Lagrangian velocity field, but an effective velocity field which
may be not incompressible. A suitable measure of compressibility can be
defined as $\kappa=\langle(div\
\vec{v})^{2}\rangle/\langle(\vec{\nabla}\vec{v})^{2}\rangle$, where
$\langle..\rangle$ stands for space and time average. Using the above
mentioned values of phytoplankton size $a$, density mismatch $\delta\rho/\rho$
and turbulent energy dissipation $\epsilon$, one obtains
$\sqrt{\kappa}=\frac{\delta\rho}{\rho}\frac{a^{2}}{\nu\tau_{\eta}}\in[10^{-9},.4]$
Another very important feature to be considered is the ability of individuals
to swim in a preferential direction towards the largest concentration of
nutrients (chemotaxis). The swimming velocity $V_{c}$ is presently estimated
in the range $[10,500]\mu m/sec$. Because of turbulent, individuals are
subject to external forces which try to change the direction. It is observed
that with a characteristic time $B\sim 5sec$, individuals try to recover the
preferential direction. This mechanism, named gyrotaxis Gyrotaxis1 and
Gyrotaxis2 , introduces an effective compressible flow with compressibility
$\sqrt{\kappa}=\frac{V_{c}B}{\eta}\in[2.510^{-3},1]$
It is important to remark that turbulent flows with an effective
compressibility can dramatically change population dynamics: concentration of
individuals increases in low pressure regions (sinks) and decreases in high
pressure regions (source) and the population is spatially characterized by
small scale patchiness. The above discussion shows that intense turbulent
activity in the oceanic upper layer may introduce non trivial effect, due to
compressibility, in the phytoplankton growth and evolution at rather small
scale.
The same considerations might be relevant for large scale motions. Very large
scale oceanic circulation ($100-300km$) are characterized by relatively small
Rossby number $Ro$, defined as $Ro=u_{H}/(fL)$, where $u_{H}$ is the
characteristic horizontal velocity, order $0.1m/sec$, $f=10^{-4}sec^{-1}$ is
the Coriolis frequency and $L$ is the characteristic large scale circulation.
For $Ro<<1$, the velocity field is close to the geostrophic balance, meaning
that the Coriolis force balances the pressure gradient. Under such
circumstances, the vertical velocity $w$ is rather small and it can be
estimated to be $0.1mm/sec$ or equivalently few meters/day. The horizontal
velocity can be decomposed in the geostrophic component $\vec{v}_{g}$ and the
non geostrophic part $\vec{v}_{a}$ where $div_{H}\vec{v}_{g}=0$ and
$div_{H}\vec{v}_{a}+\partial_{z}w=0$ with
$div_{H}\equiv\partial_{x}+\partial_{y}$. According to quasi-geostrophic
dynamics, near the surface there exists an effective compressible flow acting
on time scale order $div\ \vec{v}_{a}\sim 10^{-6}sec^{-1}$ much longer than
the longest population growth rate $\mu\sim 2\ 10^{-5}sec^{-1}$. Therefore, at
very large scale, population dynamics evolves under the advection of an
incompressible flow. The above picture changes dramatically if we consider
flows at $Ro$ close to $1$. Recent numerical simulations as well as direct
observations klein ,Mizobota , 15 have shown that surface density tends to
develop sharp horizontal gradients (fronts) especially near by the edge of
oceanic eddies. Formation of intense fronts, produced by the enhanced
filamentation of surface density SQG1 SQG2 , increases the vertical advection
and destroy geostrophic balance. As a results two important phenomena seem to
take place in the ocean at relatively large scale (order $10km$) 3docean
fronts1 : regions of relative large and positive vertical velocity (upwelling)
tends to increase nutrients for phytoplankton providing an increase of total
biological mass while regions of negative vertical velocity increases
concentration of the phytoplankton population. Frontogenesis, as it is usually
named the formation of sharp density gradients, can develop vertical velocity
up to few millimeter/sec. Consequently, the horizontal velocity near
frontogenetic regions is characterized by an effective compressibility with
$div_{H}\vec{v}\sim 10^{-4}$ 3docean , i.e. smaller than the population
growing rate. The above picture suggests the formation of plankton patchiness
on scale ranging from $100m$ to $10-30km$. As a tentative conclusion to our
short review of phytoplankton in the ocean, albeit the complexity of the
problem, it seems important to understand the role of turbulent compressible
flows in population dynamics and population genetics trying to understand, at
least in the simplest cases, if a new and non trivial phenomenology can be
discovered and its relevance to biological evolution.
In these lectures we review our recent work 17 ,18 ,18b on population
dynamics and population genetics in compressible velocity fields of one and
two dimensions, motivated by the above discussion. We consider cases both
where the velocity field is turbulent and when it is static. Furthermore, we
investigate populations in terms of a continuos density field and when the
populations are treated via discrete particles. In the last case we focus on
the competition and fixation of one species compared to another.
## 2 One dimensional case
In this section we shall discuss some qualitative and quantitative ideas
underlying the effect of compressible turbulence on population dynamics. We
restrict ourself to the one dimensional case where most concepts can be
discussed using rather simple analytical tools.
Upon specializing to one dimension, the Fisher equation reads 2
$\partial_{t}c+\partial_{x}(uc)=D\partial_{x}^{2}c+\mu c-bc^{2}$ (1)
Equation (1) is relevant for the case of compressible flows, where
$\partial_{x}u\neq 0$, and for the case when the field $c(x,t)$ describes the
population of inertial particles or biological species. By suitable rescaling
of $c(x.t)$, we can always set $b=\mu$. In the following, unless stated
otherwise, we shall assume $b=\mu$ whenever $\mu\neq 0$ and $b=0$ for $\mu=0$.
The Fisher equation for $u=0$ has travelling front solutions which can be
computed analytically:
$c(x,t)=\frac{1}{[1+Cexp(-5\mu t/6\pm x\sqrt{\mu/D}/6)]^{2}}$ (2)
From (2) we can see that the non linear wave propagates with velocity
$v_{F}\sim(D\mu)^{1/2}$ fisher , 2 . In Fig. (1) we show a numerical solution
of Eq. (1) with $D=0.005$, $\mu=1$ and $u=0$ obtained by numerical integration
on a space domain of size $L=1$ with periodic boundary conditions. The figure
shows the space-time behaviour of $c(x,t)$ for $c(x,t)=0.1,0.3.0,5.0,7$ and
$0.9$. With initial condition $c(x,t=0)$ nonzero on only a few grid points
centered at $x=L/2$, $c(x,t)$ spreads with a velocity $v_{F}\sim 0.07$ and,
after a time $L/v_{F}\sim 4$ reaches the boundary. Note that the
characteristic size of the Fisher’wave interface thichness is order
$\sqrt{D/\mu}$.
Figure 1: Contour plot of the numerical simulation of eq. 1 with $\mu=1$,
$D=0.005$ and with periodic boundary conditions. The initial conditions are
$c(x,t)=0$ everywhere expect for few grid points near $L/2=0.5$ where $c=1$.
The horizontal axis represents time while the vertical axis is space.
Let us consider first the case $\mu=b=0$. In this limit, Eq. (1) is just the
Fokker-Planck equation describing the probability distribution $P(x,t)\equiv
c(x,t)$ to find a particle in the range $(x,x+dx)$ at time $t$, whose dynamics
is given by the stochastic differential equation:
$\frac{dx}{dt}=u(x,t)+\sqrt{2D}\eta(t)$ (3)
where $\eta(t)$ is a white noise with
$\langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-t^{\prime})$. Let us assume
for the moment that $u(x,t)=u(x)$ is time independent and, moreover, let us
take $u(x,t)=-\Gamma(x-x_{0})$. Then, the stationary solution of (1) is given
by
$P(x,t)=A^{-1}exp[-\Gamma(x-x_{0})^{2}/2D]$ (4)
where $A$ is a normalization constant. $P(x,t)=P(x)$ is strongly peaked near
the points $x_{0}$ and (4) tells us that $P$ spreads around $x_{0}$ with a
characteristic length of order $\xi\equiv\sqrt{D/\Gamma_{0}}$. Hereafter, we
shall refer to $\xi$ as ”quasi-localization length”.
The same argument can be used to study the effect for a more generic turbulent
like one dimensional field $u(x)$, still time independent. We can identify
$\Gamma$ as a typical gradient of the turbulent velocity field $u$. In a
turbulent flow, the velocity field is correlated over spatial scale of order
$v_{*}/\Gamma$ where $v_{*}^{2}/2$ is the average kinetic energy of the flow.
For $P$ to be localized near a generic sink at the point $x_{0}$, despite
spatial variation in the turbulent field, we must require that the
localization length $\xi$ should be smaller than the turbulent correlation
scale $v_{*}/\Gamma$, i.e.
$\sqrt{\frac{D}{\Gamma}}<\frac{v_{*}}{\Gamma}\rightarrow\frac{v_{*}^{2}}{D\Gamma}>2$
(5)
Condition (5) can be easily understood by considering the simple case of a
periodic velocity field $u$, i.e. $u=v_{*}sin(xv_{*}/\Gamma)$. In this case,
condition (5) states that $D$ should be small enough for the probability $P$
not to spread over all the minima of $u$. For small $D$ or equivalently for
large $v_{*}^{2}/\Gamma$, the solution will be localized near the minima of
$u$, at least for the case of a frozen turbulent velocity field $u(x)$.
The above analysis can be extended for velocity field $u(x,t)$ that depend on
both space and time. The crucial observation is that, close to the sinks
$x_{i}$ of $u(x,t)$, we should have $u(x_{i},t)\sim 0$. Thus, although $u$ is
a time dependent function, sharp peaks in $P(x,t)$ move quite slowly, simply
because $u(x,t)\sim 0$ near the maximum of $P(x,t)$. One can consider a
Lagrangian path $x(t)$ such that $x(0)=x_{0}$, where $x_{0}$ is one particular
point where $u(x_{0},0)=0$ and $\partial_{x}u(x,0)|_{x=x_{0}}<0$. From direct
numerical simulation of Lagrangian particles in fully developed turbulence, we
know that the acceleration of Lagrangian particles is a strongly intermittent
quantitiy, i.e. it is small most of the time with large (intermittent) bursts.
Thus, we expect that the localized solution of $P$ follows $x(t)$ for quite
long times except for intermittent bursts in the turbulent flow. During such
bursts, the position where $u=0$ changes abruptly, i.e. almost discontinuosly
from one point, say $x(t)$, to another point $x(t+\delta t)$. During the short
time interval $\delta t$, $P$ will drift and spread, eventually reforming to
become localized again near $x(t+\delta t)$. The above discussion suggests
that the probability $P(x,t)$ will be localized most of the time in the
Lagrangian frame, except for short time intervals $\delta t$ during an
intermittent burst.
From (5) we conclude that for large value of $D$ $P(x,t)$ is spread out, while
for small enough $D$, $P$ should be a localized or sharply peaked function of
$x$ most of the time. An abrupt transition, or at least a sharp crossover,
from extended to sharply peaked functions $P$, should be observed for
decreasing $D$.
It is relatively simple to extend the above analysis for a non zero growth
rate $\mu>0$, see also vulpiani2 for a time independent flow. The requirement
(5) is now only a necessary condition to observe localization in $c$. For
$\mu>0$ we must also require that the characteristic gradient on scale $\xi$
must be larger than $\mu$, i.e. the effect of the small scale turbulent
fluctuations should act on a time scale smaller than $1/\mu$. We estimate the
gradient on scale $\xi$ as $\delta v(\xi)/\xi$, where $\delta v(\xi)$ is the
characteristic velocity difference on scale $\xi$. We invoke the Kolmogorov
theory, and set $\delta v(\xi)=v_{*}(\xi/L)^{1/3}$ to obtain:
$\mu<\frac{\delta
v(\xi)}{\xi}=\frac{v_{*}\xi^{-2/3}}{L^{1/3}}=v_{*}(\frac{\Gamma}{LD})^{1/3}$
(6)
In (6), we interpret $\Gamma$ as the characteristic velocity gradient of the
turbulent flow. Note also that $\delta v(\xi)/\xi\leq\Gamma$ on the average,
which leads to the inequality:
$\mu<\Gamma$ (7)
From (5) and (7) we also find
$\frac{v_{*}^{2}}{D\mu}>2$ (8)
a second necessary condition.
One may wonder whether a non zero growth rate $\mu$ can change our previous
conclusions about the temporal behavior, and in particular about its effect on
the dynamics of the Lagrangian points where $u(x,t)=0$. Consider the solution
of (1) at time $t$, allow for a spatial domain of size $L$, and introduce the
average position
$x_{m}\equiv\int_{0}^{L}dxx\frac{c(x,t)}{Z(t)}$ (9)
where $Z(t)=\int_{0}^{L}dxc(x,t)$. Upon assuming for simplicity a single
localized solution, we can think of $x_{m}$ just as the position where most of
the bacterial concentration $c(x,t)$ is localized. We can compute the time
derivative $v_{m}(t)=dx_{m}/dt$. After a short computation, we obtain:
$v_{m}(t)=Z\int_{0}^{L}dx(x_{m}-x)P(x,t)^{2}+\int_{0}^{L}u(x,t)P(x,t)dx$ (10)
where $P(x,t)\equiv c(x,t)/Z(t)$. Note that $v_{m}$ is independent of $\mu$.
Moreover, when $c$ is localized near $x_{m}$, both terms on the r.h.s. of (10)
are close to zero. Thus, $v_{m}$ can be significantly different from zero only
if $c$ is no longer localized and the first integral on the r.h.s becomes
relevant. We can now understand the effect of the non linear term in (1): when
$c(x,t)$ is localized, the non linear term does not affect the value of
$v_{m}$ simply because $v_{m}$ is close to $0$. On the other hand, when
$c(x,t)$ is extended the non linear term drives the system to the state $c=1$
which is an exact solution in the absence of turbulent convection $u(x,t)=0$.
We now discuss whether our previous analysis can be compared against numerical
simulations of (1) in the one dimensional case. To completely specify equation
(1) we must define the dynamics of the ”turbulent” velocity field $u(x,t)$.
Although we consider a one dimensional case, we want to study the statistical
properties of $c(x,t)$ subjected to turbulent fluctuations which are close to
those generated by the three dimensional Navier-Stokes equations. Hence, the
statistical properties of $u(x,t)$ should be characterized by intermittency
both in space and in time. Although intermittency is not a crucial point in
our investigations, we want to use a one dimensional velocity field with some
generic features in terms of space and time dynamics. For this reason, we
build the turbulent field $u(x,t)$ by appealing to a simplified shell model of
fluid turbulence biferale . The wavenumber space is divided into shells of
scale $k_{n}=2^{n-1}k_{0}$, $n=1,2,...$. For each shell with characteristic
wavenumber $k_{n}$, we describe turbulence by using the complex Fourier-like
variable $u_{n}(t)$, satisfing the following equation of motion:
$\displaystyle(\frac{d}{dt}$ $\displaystyle+$ $\displaystyle\nu
k_{n}^{2})u_{n}=i(k_{n+1}u_{n+1}^{*}u_{n+2}-\delta k_{n}u_{n-1}^{*}u_{n+1}$
(11) $\displaystyle+$ $\displaystyle(1-\delta)k_{n-1}u_{n-1}u_{n-2})+f_{n}\ .$
The model contains one free parameter, $\delta$, and it conserves two
quadratic invariants (when the force and the dissipation terms are absent) for
all values of $\delta$. The first is the total energy $\sum_{n}|u_{n}|^{2}$
and the second is $\sum_{n}(-1)^{n}k_{n}^{\alpha}|u_{n}|^{2}$, where
$\alpha=\log_{2}(1-\delta)$. In this note we fix $\delta=-0.4$. For this value
of $\delta$ the model reproduces intermittency features of the real three
dimensional Navier Stokes equation with surprising good accuracy biferale .
Using $u_{n}$, we can build the real one dimensional velocity field $u(x,t)$
as follows:
$u(x,t)=F\sum_{n}[u_{n}e^{ik_{n}x}+u^{*}_{n}e^{-ik_{n}x}],$ (12)
where $F$ is a free parameter to tune the strength of velocity fluctuations
(given by $u_{n}$) relative to other parameters in the model (see next
section). In all numerical simulations we use a forcing function
$f_{n}=(\epsilon(1+i)/u^{*}_{1})\delta_{n,1}$, i.e. energy is supplied only to
the largest scale corresponding to $n=1$. With this choice, the input power in
the shell model is simply given by
$1/2\sum_{n}[u^{*}_{n}f_{n}+u_{n}f^{*}_{n}]=\epsilon$ , i.e. it is constant in
time. To solve Eqs. (1) and (11) we use a finite difference scheme with
periodic boundary conditions. Theses model equations can be studied in detail
without major computational efforts. The free parameters of the model are the
diffusion constant $D$, the size of the periodic 1d spatial domain $L$, the
growth rate $\mu$, the viscosity $\nu$ (which fixes the Reynolds number $Re$),
the “strength’ of the turbulence $F$ and finally the power input in the shell
model, namely $\epsilon$. Note that according to the Kolmogorov theory frisch
, $\epsilon\sim u_{rms}^{3}/L$ where $u_{rms}^{2}$ is the mean square
velocity. Since $u_{rms}\sim F$, we obtain that $F$ and $\epsilon$ are related
as $\epsilon\sim F^{3}$. By rescaling of space, we can always put $L=1$. We
fix $\epsilon=0.04$ and $\nu=10^{-6}$, corresponding to an equivalent
$Re=u_{rms}L/\nu\sim 3\times 10^{5}$. Most of our numerical results are
independent of $Re$ when $Re$ is large enough. In the limit
$Re\rightarrow\infty$, the statistical properties of eq. (1) depend on the
remaining free parameters, $D$, $\mu$ and $F$. For future reference, we
compare the characteristic time scales for this simple model of homogeneous
isotropic turbulence with the local doubling times of microorganisms described
by Eqs. (1). Upon assuming the usual Kolmogorov scaling picture, we expect
fluid mixing time scales $t$ in the range
$\nu^{1/2}/\epsilon^{1/2}<t<L^{2/3}/\epsilon^{1/3}$, or $0.01<t<3.0$, for
typical parameter values of the shell model given above. On the other hand,
the characteristic doubling time $t_{2}$ of, say, bacteria, in our model is
$t_{2}\sim 1/\mu$. Our simulations typically take $\mu=1$ so that $0.2\leq
t_{2}\leq 1.0$, implying cell division times somewhere in the middle of the
Kolmogorov range. Microorganisms that grow rapidly compared to a range of
turbulent mixing times out to the Kolmogorov outer scale, as is the case here,
are crucial to the interesting effects we find when $\mu>0$. Bacteria or
yeast, often mechanically shaken at frequencies of order 1Hz in a test tube in
standard laboratory protocols, have cell division times of 20-90 minutes, and
do not satisfy this criterion. However, conditions that approximately match
our simulations can be found for, say, bacterioplankton in the upper layer of
the ocean, where large eddy turnover times do exceed microorganism doubling
times review , robinson .
Figure 2: Same parameters and initial condition as in Fig. (1) for equation
(1) with a ”strong turbulent” flow $u$ advecting $c(x,t)$. Figure 3: The
behavior in time of the total ”mass” $Z(t)\equiv\int dxc(x,t)$: circles show
the function $Z$ for the case of Fig. (1), i.e. a Fisher wave with no
turbulence; triangles show $Z$ for the case of Fig. (2) when a strong
turbulent flows is advecting $c(x,t)$
In agreement with our previous theoretical analysis, in figure (9) we show the
numerical solutions of Eq. (1) for a relatively ”strong” turbulent flow. A
striking result is displayed: we see no trace of a propagating front: instead,
a well-localized pattern of $c(x,t)$ forms and stays more or less in a
stationary position. For us, Fig. (2) shows a counter intuitive result. One
naive expectation might be that turbulence enhances mixing. The mixing effect
due to turbulence is usually parametrized in the literature frisch by
assuming an effective (eddy) diffusion coefficient $D_{eff}\gg D$. As a
consequence, one naive guess for Eq. (1) is that the spreading of an initial
population is qualitatively similar to the travelling Fisher wave with a more
diffuse interface of width $\sqrt{D_{eff}/\mu}$ As we have seen, this naive
prediction is wrong for strong enough turbulence: the solution of equation (1)
shows remarkable localized features which are preserved on time scales longer
than the characteristic growth time $1/\mu$ or even the Fisher wave
propagation time $L/v_{F}$. An important consequence of the localization
effect is that the global ”mass” (of growing microorganisms, say) ,
$Z\equiv\int dxc(x,t)$, behaves differently with and without turbulence. In
Fig. (3), we show $Z(t)$: the curve with circles refers to the conditions
shown in Fig. (1)), while the curve with triangles to Fig. (2).
The behavior of $Z$ for the Fisher equation without turbulence is a familiar
S-shaped curve that reaches the maximum $Z=1$ on a time scale $L/v_{F}$. On
the other hand, the effect of turbulence (because of localization) on the
Fisher equation dynamics reduces significantly $Z$ almost by one order of
magnitude
With biological applications in mind, it is important to determine conditions
such that the spatial distribution of microbial organisms and the carrying
capacity of the medium are significantly altered by convective turbulence.
Within the framework of the Fisher equation, localization effect has been
studied for a constant convection velocity and quenched time-independent
spatial dependence in the growth rate $\mu$ nelson1 , nelson2 , nelson3 ,
neicu . In our case, localization, when it happens, is a time-dependent
feature and depends on the statistical properties of the compressible
turbulent flows. It is worth noting that the localized ”boom and bust”
population cycles studied here may significantly effect ”gene surfing”
genesurf at the edge of a growing population, i.e. by changing the
probability of gene mutation and fixation in the population.
One prediction of eq.s (5) and (7) is that the limit $\mu\rightarrow 0$ should
be singular. More precisely, the quantity $\langle Z(t)\rangle$ must be equal
to $1$ for $\mu=0$, while our predictions based on (5) and (7) imply that
$\langle Z(t)\rangle<1$ for $\mu\neq 0$ because of ”quasi localization” of the
solutions. In the insert of figure (4) we show the time averaged $\langle
Z(t)\rangle$, computed for different values of $\mu$ for $F=0.5$. For large
$\mu$, $<Z>\rightarrow 1$, as predicted by our phenomenological approach,
while in the limit $\mu\rightarrow 0$ the values of $<Z>$ converges to $0.1$.
To predict the limit $\mu\rightarrow 0$ we can assume that $c_{\mu}(x,t)$ for
small enough $\mu$ can be obtained by the knowledge of the solution
$c_{0}(x,t)$ at $\mu=0$ by the relation
$c_{\mu}(x,t)=^{s}Z_{\mu}c_{0}(x,t)$ (13)
where in the above equation the symbol $=$ means ”in the statistical sense”
and $Z_{\mu}=\langle c_{\mu}\rangle_{x}$ (the subscript $x$ indicates average
on space). Since the solution $c_{\mu}(x,t)$ satisfies the constrain $\langle
c_{\mu}\rangle_{x}-\langle c_{\mu}^{2}\rangle_{x}=0$ for any $\mu$, we obtain:
$Z_{\mu}-Z_{\mu}^{2}\langle c_{0}^{2}\rangle_{x}=0\ \ \rightarrow\ \
Z_{\mu}=\frac{1}{\langle c_{0}^{2}\rangle_{x}}$ (14)
Once again we remark that eq. (14) should be interpreted in a statistical
sense, i.e. the time average of $Z_{\mu}$ should be equal for small $\mu$ to
the time average of $\langle c_{0}^{2}\rangle_{x}^{-1}$. In the insert of
figure (4) the blue dotted line corresponds to the time average of $\langle
c_{0}^{2}\rangle_{x}^{-1}$: equation (14) is clearly confirmed by our
numerical findings. As we shall see in the next section, the same argument can
be applied for two dimensional compressible flow.
Figure 4: Behavior of the carrying capacity $\langle Z\rangle_{\mu}$ as a
function of $\mu\tau$ from $128^{2}$ ( dots) and $512^{2}$ (squares) numerical
simulations with $Sc=1$. Note that for $\mu\tau\rightarrow 0.001$, the
carrying capacity approaches the limit $1/\langle P^{2}\rangle$ (dotted line)
predicted by Eq. (21). In the inset we show similar results for one
dimensional compressible turbulent flows.
## 3 Fisher equation in two dimensional compressible flows
As discussed in the previous section, an advecting compressible turbulent flow
leads to highly non-trivial dynamics for the Fisher equation. Although
previous results were obtained only in one dimension using a synthetic
advecting flow from a shell mdel of turbulence, two striking effects were
observed: the concentration field $c({\bm{x}},t)$ is strongly localized near
transient but long-lived sinks of the turbulent flows for small enough growth
rate $\mu$; in the same limit, the space-time average concentration (denoted
in the following as carrying capacity) becomes much smaller than its maximum
value $1$. Here, we present numerical results aimed at understanding the
behavior of the Fisher’equation for two dimensional compressible turbulent
flows and extending our previous results to more realistic two dimensional
turbulent flows. Our model consists in assuming that the microorganism
concentration field $c({\bm{x}},t)$, whose dynamics is described by the
equation
$\partial_{t}c+{\bf\nabla}\cdot({\bf u}c)=D\nabla^{2}c+\mu c(1-c)$ (15)
We assume that the population is constrained on a planar surface of constant
height in a three dimensional fully developed turbulent flow with periodic
boundary conditions. Such a system could be a rough approximation to
microorganisms that actively control their bouyancy to mantain a fixed depth
below the surface of a turbulent fluid. As a consequence of this choice, the
flow field in the two dimensional slice becomes compressible boffetta . We
consider here a turbulent advecting field ${\bf u}({\bf x},t)$ described by
the Navier-Stokes equations, and nondimensionalize time by the Kolmogorov
time-scale $\tau\equiv(\nu/\epsilon)^{1/2}$ and space by the Kolmogorov
length-scale $\eta\equiv(\nu^{3}/\epsilon)^{1/4}$. The non-dimensional numbers
charecterizing the evolution of the scalar field $C({\bf x},t)$ are then the
Schmidt number $Sc=\nu/D$ and the non-dimensional time $\mu\tau$. A
particularly interesting regime arises when the doubling time $\mu^{-1}$ is
somewhere in the middle of the range of eddy turnover times that characterize
the turbulence. Although the underlying turbulent energy cascade is somewhat
different mck09 , this situation arises for oceanic plankton, who double in
$\sim 12$ hours, in a medium with eddy turnover times varying from minutes to
months mar03 .
We conducted a three dimensional direct numerical simulation (DNS) of
homogeneous, isotropic turbulence at two different resolutions ($128^{3}$ and
$512^{3}$ collocation points) in a cubic box of length $L=2\pi$. The Taylor
microscale Reynolds number frisch for the full 3D simulation was
$Re_{\lambda}=75$ and $180$, respectively, the viscosities were $\nu=0.01$ and
$\nu=2.05\cdot 10^{-3}$, the total energy dissipation rate was around
$\epsilon\simeq 1$ in both cases. For the analysis of the Fisher equation we
focused only on the time evolution of a particular 2D slab taken out of the
full three dimensional velocity field and evolved a concentration field
$c({\bm{x}},t)$ constrained to lie on this plane only. A typical plot of the
$2d$ concentration field, along with the corresponding velocity divergence
field (taken at time $t=86$, $Re_{\lambda}=180$) in the plane is shown in Fig.
5 ($Sc=5.12$): the concentration $c(x,y,t)$ is highly peaked in small areas,
resembling one dimensional filaments. When the microorganisms grow faster than
the turnover times of a significant fraction of the turbulent eddies, $c({\bf
x},t)$ grows in a quasi-static compressible velocity field, and accumulates
near sinks and along slowly contracting eigendirections, leading to filaments.
The geometry of the concentration field suggests that $c(\bf x,t)$ is
different from zero on a set of fractal dimension $d_{F}$ much smaller than
$2$. A box counting analysis of the fractal dimension of $c(\bf x,t)$ supports
this view and provides evidence that $d_{F}=1.\pm 0.15$.
Note that for $\mu=0$, Eq. (15) reduces to the Fokker-Planck equation
describing the probability distribution $P(x,y,t)$ to find a Lagrangian
particle subject to a force field ${\bf u}({\bf x},t)$ at $x,y$ at time $t$:
${\frac{\partial P}{\partial t}}+{\bm{\nabla}}\cdot({\bm{u}}P)=D\nabla^{2}P$
(16)
The statistical properties of $P$ have been studied in several works (e.g.
bec03 and massimo ) and it is known that for compressible turbulence $P({\bf
x},t)$ exhibits a nontrivial multifractal scaling. Upon multiplying eqn. (16)
by $P$ and integrating in space we obtain:
$\frac{1}{2}\partial_{t}\left\langle
P^{2}\right\rangle_{s}+\frac{1}{2}\left\langle P^{2}(\nabla\cdot{\bf
u})\right\rangle_{s}=-D\left\langle(\nabla P)^{2}\right\rangle_{s}$ where
$\left\langle\dots\right\rangle_{s}$ denotes a spatial integration. In the
statistically stationary regime, the above equation reduces to:
$\frac{1}{2}\langle P^{2}(\nabla\cdot{\bf u})\rangle=-D\langle(\nabla
P)^{2}\rangle,$ (17)
where now $\langle\dots\rangle$ stands for space and time average. Eq. (17)
shows that for $\nabla\cdot{\bf u}=0$ the only possible solution is
$P={\mbox{c}onst}$. However, compressibility leads to nontrivial dynamics such
that $P^{2}$ and $\nabla\cdot{\bf u}$ are anticorrelated. We measure the
degree of compressibility by the factor $\kappa\equiv{\langle(\nabla\cdot{\bf
u})^{2}\rangle}/{\langle(\nabla{\bf u})^{2}\rangle}$, and estimate the l.h.s.
of Eq. (17) by assuming $\langle P^{2}(\nabla\cdot{\bf
u})\rangle=-A_{1}\langle P^{2}\rangle\langle(\nabla\cdot{\bf
u})^{2}\rangle^{1/2}$, where we used the so called one point closure for
turbulent flows frisch and $A_{1}$ is expected to be order unity. We estimate
the r.h.s of Eq. (17) by assuming:
$\langle(\nabla P)^{2}\rangle=A_{2}\frac{\langle P^{2}\rangle}{\xi^{2}}$ (18)
where we define $\xi$ the “quasi-localization” length of $P$, which is
expected to be of the same order of the width of the narrow filaments in Fig.
5. Finally we set $\langle(\nabla{\bf u})^{2}\rangle=\epsilon/\nu$ where
$\epsilon$ is the mean rate of energy dissipation and $\nu$ is the viscosity.
On putting everything together we find a localization length given by:
$\xi^{2}=\frac{2A_{2}D\sqrt{\nu}}{A_{1}\sqrt{\kappa\epsilon}}.$ (19)
One important quantity -from the biological point of view- is the carrying
capacity,
$Z(t)=\frac{1}{L^{2}}\int dxdyc({\bm{x}};t),$ (20)
and in particular its time average in the statistical steady state with growth
rate $\mu$, $\langle Z(t)\rangle_{\mu}$. We are interested to understand how
$\langle Z\rangle_{\mu}$ behaves as a function of $\mu$, in the two important
limits $\mu\rightarrow\infty$ and $\mu\rightarrow 0$. In the limit
$\mu\rightarrow\infty$, we expect the carrying capacity attains its maximum
value $\langle Z\rangle_{\mu\rightarrow\infty}=1$, because when the
characteristic time $1/\mu$ becomes much smaller than the Kolmogorov
dissipation time $\tau_{\eta}\equiv(\nu/\varepsilon)^{1/2}$, the effect of the
velocity field is a relatively small perturbation on the rapid growth of the
microorganisms. Indeed, consider a perturbation expansion of $c({\bf x},t)$ in
terms of $\delta=1/\mu$. On defining
$c\equiv\displaystyle\sum_{i}\delta^{i}c_{i}({\bf x},t)$, substituting in Eq.
(15), assuming steady state, and collecting the terms up to ${\cal
O}(\delta^{2})$ we find $c\approx
1-\epsilon({\bm{\nabla}}\cdot{\bm{u}})+\epsilon^{2}\\{{\bm{\nabla}}\cdot\left[{\bm{u}}({\bm{\nabla}}\cdot{\bm{u}})\right]-D\nabla^{2}(\nabla\cdot{\bm{u}})-({\bm{\nabla}}\cdot{\bm{u}})^{2}\\}+{\cal
O}(\delta^{3})$. The above analysis shows that in the limit $\mu\to\infty$ the
concentration field tends to become uniform with the leading correction coming
from the local compressibility. After substituting the expansion of $c$ in Eq.
(20) one gets $Z\approx
1-(\delta^{2}/L)\int({\bm{\nabla}}\cdot{\bm{u}})^{2}{\rm{d\bm{x}}}+{\cal
O}(\delta^{3})$. Note that the leading correction to the carrying capacity is
of order $\delta^{2}$, is consistent with the physical picture presented
above.
Figure 5: Plot of the concentration $c(x,y,t)$. The white color indicate
regions with low concentration while regions of high concentration are denoted
by black.
By defining $\Gamma\equiv\langle({\bm{\nabla}}\cdot{\bm{u}})^{2}\rangle^{1/2}$
as the r.m.s value of the velocity divergence, we expect a crossover in the
behavior of $\langle Z\rangle_{\mu}$ for $\mu<\Gamma$. In the limit
$\mu\rightarrow 0$, following our discussion in the previous section, we
expect that:
$\displaystyle\lim_{\mu\rightarrow 0}\langle Z\rangle_{\mu}=\frac{1}{\langle
P^{2}\rangle}.$ (21)
We have tested both Eq.(21) and the limit $\mu\rightarrow\infty$ against our
numerical simulations. In Fig. (4) we show the behavior of $\langle
Z\rangle_{\mu}$ for the numerical simulations discussed in this section. The
horizontal line represents the value $1/\langle P^{2}\rangle$ obtained by
solving Eq. (16) for the same velocity field and $\mu=0$. For our numerical
simulations we estimate $\Gamma=4.0$ and we observe, for $\mu>\Gamma$ the
carrying capacity $\langle Z\rangle_{\mu}$ becomes close to its maximum value
$1$. The limit $\mu\rightarrow 0$ requires some care. Let us define
$\tau_{b}\equiv 1/\mu$ to be the time scale for the bacteria to grow. The
effect of turbulence is relevant for $\tau_{b}$ longer than the Kolmogorov
dissipation time scale $\tau_{\eta}$. We also expect that $\tau_{b}$ must be
smaller than the large scale correlation time
$\tau_{L}\sim(L^{2}/\epsilon)^{1/3}$, which depends on the forcing mechanism
driving the turbulent flows and the large scale $L$. Thus, the limit
$\mu\rightarrow 0$ can be investigated either for $L\rightarrow\infty$ or by
forcing the system with a constant energy input which slows down the large
scale, as it is the case in our numerical simulations
The limit $\mu\rightarrow 0$ can be investigated more accurately as follows:
according to known results on Lagrangian particles in compressible turbulent
flows, we know that $P$ should have a multifractal structure in the inviscid
limit $\nu\rightarrow 0$ 14 . If our assumption leading to Eq. (21) is
correct, $c({\bf x},t)$ must show multifractal behavior in the same limit with
multifractal exponents similar to those of $P$. For analytical results, see
Ref. bec03 . Numerical evidence for the multifractal behavior of Lagrangian
tracers in compressible flows can be found in Ref. boffetta .
We perform a multifractal analysis of the concentration field $c(x,y,t)$ with
$\mu>0$ by considering the average quantity
$\tilde{c}_{\mu}(r)\equiv\frac{1}{r^{2}}\int_{B(r)}c(x,y,t)dxdy$ where $B(r)$
is a square box of size $r$. Then the quantities
$\langle\tilde{c}_{\mu}(r)^{p}\rangle$ are expected to be scaling functions of
r, i.e. $\langle\tilde{c}_{\mu}(r)^{p}\rangle\sim r^{a(p)}$, where $a(p)$ is a
non linear function of $p$ with $a(2)=-0.47$, see 18 for details.
Our multifractal analysis allow us to investigate the possible relation
between the localization length $\xi$ defined in Eq. (18) and the carrying
capacity $\langle Z\rangle_{\mu}$. The localization length $\xi$ can be
considered as the smallest scale below which one should observe fluctuations
of $c(x,t)$. Thus we can expect that $\langle
P^{2}(x,t)\rangle\sim\xi^{a(2)}$. Using (21) we obtain $\langle
Z\rangle\sim\xi^{-a(2)}$. In the inset of Figure (6) we show $\langle
Z\rangle$ as a function of $\xi$ (obtained by using (18) for $\mu=0.01$ and
different values of the diffusivitiy $D$. According to Eq. (19), reducing the
diffusivity $D$ will shrink the localization length $\xi$ and hence $\langle
Z\rangle_{\mu}$. From Figure (6) a clear power law behavior is observed with a
scaling exponent $0.46$ very close to the predicted behavior $-a(2)=0.47$.
Finally, we discuss bacterial populations subject to both turbulence and
uniform drift because of, e.g., sedimentation under the action of gravity
field. In this case, we can decompose the velocity field into zero mean
turbulence fluctuations plus a constant “wind” velocity $u_{0}$. In presence
of a mean drift velocity Eq. 15 becomes:
${\frac{\partial c}{\partial
t}}+{\bm{\nabla}}\cdot[({\bm{u}}+u_{0}\hat{e_{x}})c]=D\nabla^{2}c+\mu c(1-c)$
(22)
where $\hat{e_{x}}$ is the unit vector along the $x$-direction. Note that the
mean drift breaks the Galilean invariance as the concentration $c$ is advected
by the wind, while turbulent fluctuations ${\bf u}$ remain fixed. In Fig. 6 we
show the variation of carrying capacity versus $u_{0}$ for two different
values of $\mu$ and fixed diffusivity $D=0.015$. We find that for $u_{0}\leq
u_{rms}$ ($u_{rms}$ is the root-mean-square turbulent velocity) the carrying
capacity $Z$ saturates to a value equal to the value of $Z$ in absence of
$u_{0}$ i.e., quasilocalization by compressible turbulence dominate the
dynamics. For $u_{0}>u_{rms}$ the drift velocity delocalizes the bacterial
density thereby causing $Z\to 1$, in agreement with the results discussed in
1d .
Figure 6: Main figure: plot of $\langle Z\rangle$ as function of a super
imposed external velocity $u_{0}$ for $\mu=0.01$ (bullets) and $\mu=0.1$
(triangles). Inset: log-log plot of $\langle Z\rangle$ as a function of the
localization length $\xi$ defined in Eq. (18) for $u_{0}=0$. The slope is
consistent with the prediction $<Z>\sim\xi^{-a(2)}$ discussed in the text. The
numerical simulations are done for $\mu=0.01$ and different values of $D$ from
$D=0.05$ to $D=0.001$.
## 4 Discrete population dynamics
The population dynamics of a single species expanding into new territory was
first studied in the pioneering works of Fisher, Kolmogorov, Petrovsky and
Piscounov (FKPP) fisher ; 2 ; 3 . Later, Kimura and Weiss studied individual-
based counterparts of the FKPP equation 6 , revealing the important role of
number fluctuations. In particular, stochasticity is inevitable at a frontier,
where the population size is small and the discrete nature of the individuals
becomes essential. Depending on the parameter values, fluctuations can produce
radical changes with respect to the deterministic predictions 3 ; 4 . If
$f(x,t)$ is the population fraction of, say, a mutant species and $1-f(x,t)$
that of the wild type, the stochastic FKPP equation reads in one dimension 8 :
$\partial_{t}f(x,t)=D\partial^{2}_{x}f+sf(1-f)+\sqrt{D_{g}f(1-f)}\xi(x,t)$
(23)
where $D$ is the spatial diffusion constant, $D_{g}$ is the genetic diffusion
constant (inversely proportional to the local population size), $s$ is the
genetic advantage of the mutant and $\xi=\xi(x,t)$ is a Gaussian noise, delta-
correlated in time and space that must be interpreted using Ito calculus 8 .
In the neutral case ($s=0$), number fluctuations induce a striking effect in
the population dynamics, namely segregation of the two species. One can show
that the dynamics of competing species in 1D can be characterized by the
dynamics of boundaries between the $f=0$ and $f=1$ states of Eq. 23, which
perform a random walk. This effect is theoretically predicted by Eq.(23) and
confirmed experimentally in the linear inoculation experiments on neutral
variants of fluorescently labelled bacteria illustrated in Fig. (8a) hall .
We study the influence of advection on the dynamics of two distinct
populations consisting of discrete ’particles’. Due to competition and
stochasticity, interactions between two populations usually drive one of the
two populations to extinction. The average time of this event (the fixation
time) is a quantity of great biological interest since it determines the
amount of genetic and ecological diversity that the system can sustain.
Studying competition in a hydrodynamics context, where both a compressible
velocity field and stochasticity due to finite population sizes are present,
calls for a nontrivial generalization of Eq. (23). One complication is that,
because of compressibility, the sum of the concentrations of the two species
is no longer invariant during the dynamics.
Figure 7: The possible six birth and death processes in the particle model
consisting of two species, A (red) and B (green). Figure 8: (a) Experimental
range expansion of the two neutra E. coli strains used in Ref. hall , but run
about one day longer (D. Nelson, unpublished). The black bar at the bottom is
due a small crack in the agar substrate. (b) Space-time plot of the off-
lattice particle model with no advecting velocity field. A realization
characterized by a pattern similar to the experimental one has been selected
for illustrative purposes. (c) Particle model with a compressible turbulent
velocity field. Simulations are run until fixation (disappearance of one of
the two species); note the reduced carrying capacity and the much faster
fixation time in (c). Parameters: $N=10^{3}$, $D=10^{-4}$, $\mu=1$. Parameters
of the shell model are as in 17 .
We have overcome these problems through a off-lattice particle model designed
to explore how compressible velocity fields affect biological competition. Let
us consider two different organisms, $A$ and $B$, which advect and diffuse in
space while undergoing duplication (i.e. cell division) and density-dependent
annihilation (death), see Fig. 7. Specifically, we implement the following
stochastic reactions: each particle of species $i=A,B$ duplicates with rate
$\mu_{i}$ and annihilates with a rate $\bar{\mu}_{i}\widehat{n}_{i}$, where
$\widehat{n}_{i}$ is the number of neighboring particles (of both types) in an
interaction range $\delta$. Let $N$ be the total number of organisms that can
be accomodated in the unit interval with total density $c_{A}+c_{B}=1$. To
reduce the number of parameters, we fix $\delta=1/N$ as the average particle
spacing in the absence of flow. Further, we set
$\bar{\mu}_{A}=\bar{\mu}_{B}=\mu_{B}=\mu$, but take $\mu_{A}=\mu(1+s)$ to
allow for a selective advantage (faster reproduction rate) of species $A$. We
will start by analyzing in depth the neutral case $s=0$ and consider the
effect of $s>0$ in the end of the Letter. In one dimension and with these
choices of parameters, our macroscopic coupled equations for the densities
$c_{A}(x,t)$ and $c_{B}(x,t)$ of individuals of type $A$ and $B$ in an
advecting field $v(x,t)$ read
$\displaystyle\partial_{t}c_{A}\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\\!-\partial_{x}(vc_{A})\\!+\\!D\partial^{2}_{x}c_{A}\\!+\\!\mu
c_{A}(1\\!+\\!s\\!-\\!c_{A}\\!-\\!c_{B})\\!+\\!\sigma_{A}\xi$
$\displaystyle\partial_{t}c_{B}\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\\!-\partial_{x}(vc_{B})\\!+\\!D\partial^{2}_{x}c_{B}\\!+\\!\mu
c_{B}(1\\!-\\!c_{A}\\!-\\!c_{B})\\!+\\!\sigma_{B}\xi^{\prime}\ $ (24)
with $\sigma_{A}=\sqrt{\mu c_{A}(1+s+c_{A}+c_{B})/N}$ and
$\sigma_{B}=\sqrt{\mu c_{B}(1+c_{A}+c_{B})/N}$. $\xi(x,t)$ and
$\xi^{\prime}(x,t)$ are independent delta-correlated noise sources with an
Ito-calculus interpretation as in Eq. (23).
Figure 9: Average fixation time $\tau_{f}$ for neutral competitions in
compressible turbulence and sine wave advection, as a function of (left) the
reduced carrying capacity $\langle Z\rangle_{F}$ and (right) forcing intensity
$F$ (small $\langle Z\rangle_{F}$ in the left panel corresponds to large
forcing in the right panel). (left) Red circles and blue triangles are
particle simulations. Other symbols denote simulations of the continuum
equations with different resolutions on the unit interval. The black dashed
line is the mean field prediction, $\tau_{f}=N\langle Z\rangle_{F}/2$. In
(right), only particle simulations are shown and dashed lines are the
theoretical prediction $\tau_{f}=\tau_{0}+c/F$ based on boundary domains, with
fitted parameters $\tau_{0}=9.5$, $c=3.5$ in the case of the shell model and
$\tau_{0}=16$, $c=1.4$ in the case of the sine wave.
Simulations of the particle model corresponding to (4) with $v=s=0$ result in
a dynamics similar to the one observed in experiments, as shown in Fig.(8b).
In this simple limit, our model can be considered as a grand canonical
generalization of Eq (23), where the total density of individuals
$c_{A}+c_{B}$ is now allowed to fluctuate around an average value $1$. We fix
the following parameters: $N=10^{3}$, $D=10^{-4}$, $\mu=1$ and $L=1$ where $L$
is a one dimensional domain endowed with periodic boundary conditions. With
these parameters, the fixation time $\tau_{f}$ would be $\sim 10^{4}$ for the
one dimensional FKKP equation, and $\sim 10^{3}$ for the well-mixed case.
Introducing a compressible velocity field $v(x,t)$, via the shell model 11 as
shown in Fig.(8c), leads to radically different dynamics. Individuals tend to
concentrate at long-lived sinks in the velocity field. Further, extinction is
enhanced and the total number of individuals $n(t)$ present at time $t$ is on
average smaller than $N$.
In order to study how a velocity field changes $\tau_{f}$, we first analyze
two different velocity fields: The first is a velocity field $v(x,t)$
generated by a shell model (Eqs. 11) of compressible turbulence 17 ,
reproducing the power spectrum of high Reynolds number turbulence with forcing
intensity $F$. The second is a static sine wave, $v(x)=F\sin(2\pi x)$,
representing a simpler case in which only one Fourier mode is present, and
thus a single sink, in the advecting field. In both cases, periodic boundary
conditions on the unit interval are implemented.
Fig.(9) shows the average fixation time $\tau_{f}$ for $s=0$ in the first two
cases, while varying the intensity $F$ of advection. In the left panel, we
plot the fixation times as a function of the time-averaged reduced carrying
capacity $\langle Z\rangle_{F}$, where $Z(t)=n(t)/N$ is the carrying capacity
reduction, i.e. the ratio between the actual number of particles and the
average number of particles $N$ observed in absence of the velocity field.
Plotting vs. $\langle Z\rangle_{F}$ allows comparisons with the mean field
prediction, $\tau_{f}=2N\langle Z\rangle_{F}/\mu$, valid for well mixed
systems (black dashed line) 8 . For the shell model, we include simulations of
the macroscopic equations (4) with different resolutions ($256$ and $512$
lattice sites on the unit interval), obtaining always similar results for
$\tau_{f}$ vs. $\langle Z\rangle_{F}$.
In all cases, the presence of a spatially varying velocity field leads to a
dramatic reduction of $\tau_{f}$, compared to mean field theory. The fixation
time drops abruptly as soon as $\langle Z\rangle<1$, even for very small $F$.
Acknowledgment
We acknowledge computational support from CASPUR (Roma, Italy uner HPC Grant
2009 N. 310), from CINECA (Bologna, Italy) and SARA (Amsterdam, The
Netherlands). Support for D.R.N. was provided in part by the National Science
Foundation through Grant DMR-0654191 and by the Harvard Materials Research
Science and Engineering Center through NSF Grant DMR-0820484. Data from this
study are publicly available in unprocessed raw format from the iCFDdatabase
(http://cfd.cineca.it). M.H.J. was supported by Danish National Research
Foundation through ”Center for Models of Life”.
## References
* (1) W. van Saarloos, Phys. Rep. 386, 29-222 (2003).
* (2) O. Hallatschek and K. Korolev, Phys. Rev. Lett. 103, 108103 (2009), and references therein.
* (3) R. Fisher, Ann. Eugenics 7, 335 (1937); A. Kolmogorov, I. Petrovsky and N. Psicounoff, Moscow, Univ. Bull. Math, 1, 1, (1937).
* (4) M. Kimura and G. H. Weiss, Genetics 49, 561-576 (1964); J. F. Crow and M. Kimura An Introduction to Population Genetics, Blackburn Press, Caldwell, NJ (2009).
* (5) For a recent review, see K. Korolev et al. Rev. Mod. Phys. 820, 1691-1718 (2010).
* (6) B. A. Whitton and M. Potts The Ecology of Cyanobacteria: Their Diversity in Time and Space eds. Kluwer, Dordrecht, Netherland
* (7) F. D’Ovidio et al., Proc. Natl. Acad. Sci. 107, 18366-18370 (2010). ds (2000).
* (8) W. J. McKiver and Z. Neufeld, Phys. Rev. E 79, 061902 1-8 (2009).
* (9) F. Peters, C. Marraese, Marine Ecology Progress Series, 205, 291, (2000)
* (10) F. Toschi, E. Bodenschatz, Annual Rev. Fluid. Mech., 41, 375, (2008)
* (11) J. Bec, Phys. Fluids 15, L81-L84 (2003).
* (12) W. M. Durham, E. Ciment, R. Stocker, Phys. Rev. Lett., 106, 238102, (2011)
* (13) C. Torney, Z. Neufeld, Phys. Rev. Lett., 99, 078101,(2007)
* (14) P. Klein and G. Lapeyre, Annual Review of Marine Science, 1, 357, (2009)
* (15) K. Mizobota, Saitoh SI, Shiomoto A., Miyamura T, Shiga N, et. al , Prog. Oceanogr., 55 , 65, (2002)
* (16) A. P. Martin, Progr. Ocean. 57, 125-174 (2003).
* (17) I. M. Held, R. T. Pierrehumber, S.T. Garner, K.L. Swanson, J. Fluid. Mech., 282, 1, (1995)
* (18) X. Capet, P. Klein, B.L. Hua, G. Lapeyre, J. C. McWilliams, J. Fluid. Mech., 604, 165, (2008)
* (19) P. Klein, B. L. Hua, G. Lapeyre, X. Capet, S. Le Gentil, H. Sasaki, J. Phys. Oceanogr. 38, 1748, (2008)
* (20) L. Thomas, A. Tandon, A. Mahadevan, J. Geophys. Res., 177, 17, (2008)
* (21) R. Benzi, D. R. Nelson Physica D 238 2003-2015 (2009).
* (22) P. Perlekar, R. Benzi, D.R. Nelson, Phys. Rev. Lett. 105, 144501 (2010).
* (23) S. Pigolotti, R. Benzi, M.H. Jensen, D. R. Nelson, Phys. Rev. Lett. submitted.
* (24) A. Kolmogorov, N. Petrovsky, and N. Piscounov, Moscow Univ. Math. Bull. 1, 1-25 (1937).
* (25) S. Berti, D. Vergni, A. Vulpiani Europhys. Lett. 83, 54003 (2008)
* (26) L. Biferale, Annu., 2003, Rev. Fluid Mech. 35, 441.
* (27) U. Frisch, Turbulence the legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1996).
* (28) T. Tel et. al., Chemical and Biological Activity in Open Flows: A Dynamical Systems Approach, Phys. Reports, 413, 91, 2005
* (29) A. R. Robinson, Proc. R. Soc. Lond. A453, 2295 (1997); A455, 1813 (1999)
* (30) D. R. Nelson, and N. M. Shnerb. 1998. Non-hermitian localization and population biology. Phys. Rev. E. 58:1383.
* (31) K.A. Dahmen, D. R. Nelson, and N. M. Shnerb. 2000. Life and death near a windy oasis. J. Math. Biol. 41:1-23.
* (32) N. M. Shnerb, 2001, Extinction of a bacterial colony under forced convection in pie geometry. Phys. Rev. E 63:011906, and references therein.
* (33) T. Neicu, A. Pradhan, D. A. Larochelle, and A. Kudrolli. 2000. Extinction transition in bacterial colonies under forced convection. Phys. Rev. E. 62:1059 - 1062.
* (34) O. Hallatschek and D. R. Nelson, Theor. Popul. Biology, 73,1, 158, 2007.
* (35) J. R. Cressman et al., Europhys. Lett. 66, 219-225 (2004); G. Boffetta et al., Phys. Rev. Lett. 93, 134501 (2004).
* (36) J. ichi Wakita et al., J. Phys. Soc. Jpn. 63, 1205 (1994).
* (37) R. Benzi and D. Nelson, Physica D 238, 2003 (2009).
* (38) G. Boffetta, J. Davoudi, B. Eckhardt, and J. Schumacher, Phys. Rev. Lett. 93, 134501 (2004).
* (39) J. Bec, Phys. Fluids 15, L81 (2003); J. Bec, J. Fluid Mech., 528, 255 (2005).
* (40) G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Mod. Phys. 73, 914 (2001).
* (41) O. Hallatschek and D. Nelson, Proc. Natl. Acad. Sci 104, 19926-19930 (2007).
* (42) W. McKiver and Z. Neufeld, Phys. Rev. E 79, 061902 (2009).
* (43) A. Martin, Prog. in Oceanography 57, 125 (2003).
* (44) A. Kurganov and E. Tadmor, J. Comp. Phys. 160, 241 (2000).
|
arxiv-papers
| 2012-03-28T17:08:13 |
2024-09-04T02:49:29.147541
|
{
"license": "Public Domain",
"authors": "Roberto Benzi, Mogens H. Jensen, David R. Nelson, Prasad Perlekar,\n Simone Pigolotti, Federico Toschi",
"submitter": "Benzi Roberto",
"url": "https://arxiv.org/abs/1203.6319"
}
|
1203.6380
|
# The Diophantine Equation
$\mathbf{\arctan\left(\dfrac{1}{x}\right)+\arctan\left(\dfrac{\ell}{y}\right)=\arctan\left(\dfrac{1}{k}\right)}$
Konstantine Zelator
Mathematics, Statistics, and Computer Science
212 Ben Franklin Hall
Bloomsburg University of Pennsylvania
400 East 2nd Street
Bloomsburg, PA 17815
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
kzelator@bloomu.edu
e-mails: konstantine zelator@yahoo.com
## 1 Introduction
The subject matter of this work is the two-variable diophantine equation
$\arctan\left(\dfrac{1}{x}\right)+\arctan\left(\dfrac{\ell}{y}\right)=\arctan\left(\dfrac{1}{k}\right)$
for given positive integers $k$ and $\ell$, such that gcd $(\ell,k^{2}+1)=1$
(i.e., $\ell$ and $k^{2}+1$ are relatively prime). The main objective is to
determine all positive integer pairs $(x,y)$ which satisfy
$\left\\{\begin{array}[]{l}\arctan\left(\dfrac{1}{x}\right)+\arctan\left(\dfrac{\ell}{y}\right)=\arctan\left(\dfrac{1}{k}\right)\\\
x,y\in\mathbb{Z}^{+},\ {\rm gcd}(\ell,k^{2}+1)=1\ {\rm and}\\\ {\rm with\
gcd}\ (\ell,y)=1\ \left({\rm i.e.,}\ \ell\ {\rm and}\ y\ {\rm are}\right.\\\
\left.{\rm relatively\ prime}\right)\end{array}\right\\}$ (1)
This is done in Theorem 1, Section 4. As we will see, there are exacgtly $N$
distinct solutions to (1) where $N$ is the number of positive divisors of the
integer $k^{2}+1$. The $N$ pairs $(x,y)$, which are solutions to (1), are
expressed parametrically in terms of the positive divisors of $k^{2}+1$. Also,
note that when $\ell=1$, equation (1) is symmetric with respect to the two
variables $x$ and $y$. If $(a,b)$ is a solution, then so is $(b,a)$. The
motivating force behind this work is a recent article published in the journal
Mathematics and Computer Education (see [1]). The article, authored by Hasan
Unal, is entitled “Proof without words: an arctangent equality”. It consists
of four illustrations, a purely geometric proof of the equality,
$\arctan\left(\dfrac{1}{3}\right)+\arctan\left(\dfrac{1}{7}\right)=\arctan\left(\dfrac{1}{2}\right).$
From the point of view of (1), the last equality says that the pair $(3,7)$ is
a solution of (1), in the case $\ell=1$ and $k=2$.
According to Theorem 1, $(3,7)$ and $(7,3)$ are the only solutions to (1) for
$\ell=1$ and $k=2$.
This, then, is the other objective of this article. To generate more
arctangent type of equalities. This is done in Section 5, where a listing of
such equalities is offered; an immediate consequence of Theorem 1.
In Section 2, we list two trigonometric preliminaries: the well known identity
for the tangent of the sum of two angles and a couple of basic facts regarding
arctangent function.
In Section 3, we state two well known results from number theory: Euclid’s
lemma; and the formula that gives the number of positive divisors of a
positive integer. We use these in the proof of Theorem 1.
## 2 Trigonometric preliminaries
1. (a)
If $\theta_{1}$ and $\theta_{2}$ are two angles measured in radians, such that
neither $\theta_{1}$ nor $\theta_{2}$, nor their sum $\theta_{1}+\theta_{2}$
is of the form $k\pi+\dfrac{\pi}{2},\ k$ and integer. Then,
$\tan(\theta_{1}+\theta_{2})=\dfrac{\tan\theta_{1}+\tan\theta_{2}}{1-\tan\theta_{1}\theta_{2}}$
2. (b)
Let $f$ be the arctangent function, $f(x)=\arctan x$. Then,
1. (i)
$\arctan 1=\dfrac{\pi}{4}$
2. (ii)
$\left\\{\begin{array}[]{lr}0<\theta<\dfrac{\pi}{r}&\\\ {\rm and}&\\\
&\theta=\arctan
c\end{array}\right\\}\Leftrightarrow\left\\{\begin{array}[]{l}0<\theta<\dfrac{\pi}{4}\\\
\\\ 0<c=\tan\theta<1\end{array}\right\\}$
3. (iii)
$\left\\{\begin{array}[]{ll}0<\theta<\dfrac{\pi}{2}\\\ {\rm and}&\\\
&\theta=\arctan
c\end{array}\right\\}\Leftrightarrow\left\\{\begin{array}[]{l}0<\theta<\dfrac{\pi}{2}\\\
\\\ 0<c=\tan\theta\end{array}\right\\}$
## 3 Number theory preliminaries
The following result is commonly known as Euclid’s lemma, and is of great
significance in number theory.
Result 1 (Euclid’s lemma): Let $a,b,c$ be positive integers such that $a$ is a
divisor of the product $bc$; and with $a$ also being relatively prime to $b$.
Then, $a$ is a divisor of $c$.
The next result provides a formula that gives the exact number of positive
divisors of a positive integer.
Result 2 (number of divisors formula) Let $n\geq 2$ be a positive integer, and
let $p_{1},\ldots,p_{t}$ in increasing order, be the distinct prime bases that
appear in the prime factorization of $n$, so that $n=p^{e_{1}}_{1},\ldots
p^{e_{t}}_{t}$, with the exponents $e_{1},\ldots,e_{t}$ being positive
integers. Also, let $N$ be the number of positive divisors of $n$. Then,
1. (i)
$N=\overset{t}{\underset{i=1}{\Pi}}(e_{i}+1)\ldots(e_{1}+1)\ldots(e_{t}+1)$.
2. (ii)
In particular, when $e_{1}=\ldots=e_{t}=1$ (i.e., when $n$ is squarefree)
$N=2^{t}$
Both of these two results can be easily found in number theory books and
texts. For example, see reference [2].
## 4 Theorem 1 and its proof
###### Theorem 1.
Let $k$ and $\ell$ be fixed or given positive integers such that
gcd$(\ell,k^{2}+1)=1$. Consider the diophantine equation (1).
1. (a)
There are exactly $N$ distinct positive integer pairs $(x,y)$ which are
solutions to equation (1) where $N$ is the number of positive integer divisors
of the integer $k^{2}+1$. Specifically, if $(x,y)$ is a positive integer
solution of (1), then
$x=k+\ell\cdot\left(\dfrac{k^{2}+1}{d}\right)\ {\rm and}\ y=k\ell+d$ where $d$
is a positive integer divisor of $k^{2}+1$.
2. (b)
If $k^{2}+1=p$, a prime number, then equation (1) has exactly two distinct
positive integer solutions. These are
$(x,y)=(k+\ell(k^{2}+1),\ k\ell+1),\ \ (k+\ell,\ k\ell+k^{2}+1).$
3. (c)
If $k^{2}+1=p_{1}p_{2}$, a product of two distinct primes $p_{1}$ and $p_{2}$,
equation (1) has exactly four distinct positive integer solutions. These are,
$\begin{array}[]{ll}(x,y)=(k+\ell(k^{2}+1),k\ell+1),&(k+\ell,\
k\ell+k^{2}+1),\\\ \\\ (k+\ell p_{2}.\ k\ell+p_{1}),\ {\rm and}&{}\ (k+\ell
p_{1},\ k\ell+p_{2})\end{array}$
###### Proof.
First note that parts (b) and (c) are immediate consequences of part (a) and
Result 2. We omit the details. We prove part (a)
1. (a)
Let $d$ be a positive integer divisor of $k^{2}+1$. We will show that the
positive integer pair,
$(x_{d},y_{d})=\left(k+\ell\cdot\left(\dfrac{k^{2}+1}{d}\right),\
k\ell+d\right)$ is a solution to (1). First note that $y_{d}=k\ell+d$, is
relatively prime to $\ell$. Indeed, if $y_{d}$ and $\ell$ had a prime factor
$q$ in common then $q$ would divide $y_{d}-k\ell=d$; and thus (since $d$ is a
divisor of $k^{2}+1$) $y_{d}-k\ell=d$, then $q$ would divide $k^{2}+1$
contrary to the hypothesis that gcd$(\ell,k^{2}+1)=1$. Thus,
gcd$(\ell,y_{d})=1$.
It is clear that since $k,\ell$ and $d$ are positive integers, we have
$x_{d}>1,y_{d}>1$ and $k\geq 1$. So,
$\left(0<\dfrac{1}{x_{d}}<1,\ 0<\dfrac{\ell}{y_{d}}<1,\ 0<\dfrac{1}{k}\leq
1\right).$ (2)
Let
$\theta_{1}=\arctan\left(\dfrac{1}{x_{d}}\right),\
\theta_{2}=\arctan\left(\dfrac{1}{y_{d}}\right),\
\theta=\arctan\left(\dfrac{1}{k}\right).$ (3)
Then, by (2), (3) and part (b) of the trigonometric preliminaries, we have
$\left\\{\begin{array}[]{l}0<\theta_{1}<\dfrac{\pi}{4},\
0<\theta_{2}<\dfrac{\pi}{4},\ 0<\theta\leq\dfrac{\pi}{4}\\\ \\\ {\rm and}\
0<\theta_{1}+\theta_{2}<\dfrac{\pi}{2},\ \tan\theta_{1}=\dfrac{1}{x_{d}},\
\tan\theta_{2}=\dfrac{\ell}{y_{d}},\
\tan\theta=\dfrac{1}{k}\end{array}\right\\}$ (4)
From (4) and part (a) of trigonometric preliminaries, it follows that
$\begin{array}[]{rcl}\tan(\theta_{1}+\theta_{2})&=&\dfrac{\frac{1}{x_{d}}+\frac{\ell}{y_{d}}}{1-\frac{1}{x_{d}}\cdot\frac{\ell}{y_{d}}};\\\
\\\ \tan(\theta_{1}+\theta_{2})&=&\dfrac{y_{d}+\ell
x_{d}}{x_{d}y_{d}-\ell};\\\ \\\
\tan(\theta_{1}+\theta_{2})&=&\dfrac{d\cdot(y_{d}+\ell
x_{d})}{dx_{d}y_{d}-d\ell}.\end{array}$ (5)
By (5) and the expressions for $x_{d}$ and $y_{d}$ (see beginning of the
proof) we get
$\begin{array}[]{rcl}\tan(\theta_{1}+\theta_{2})&=&\dfrac{d^{2}+k\ell d+k\ell
d+\ell^{2}\cdot(k^{2}+1)}{[dk+\ell(k^{2}+1)](k\ell+d)-d\ell};\\\ \\\
\tan(\theta_{1}+\theta_{2})&=&\dfrac{d^{2}+2k\ell
d+\ell^{2}\cdot(k^{2}+1)}{d\ell k^{2}+k\ell^{2}(k^{2}+1)+kd^{2}+\ell
dk^{2}+d\ell-d\ell};\\\ \\\ \tan(\theta_{1}+\theta_{2})&=&\dfrac{d^{2}+2k\ell
d+\ell^{2}\cdot(k^{2}+1)}{k\cdot[2dk\ell+d^{2}+\ell^{2}(k^{2}+1)]}=\dfrac{1}{k}=\tan\theta;\\\
\\\ \tan(\theta_{1}+\theta_{2})&=&\tan\theta\end{array}$ (6)
By (6) and part (b) of the trigonometric preliminaries, it follows that
$\theta_{1}+\theta_{2}=\theta$, which combined with (3), clearly establishes
that the pair $(x_{d},y_{d})$ is a solution to (1).
Now, the converse. Suppose that $(x,y)$ is a positive integer solution to (1).
Then
$\left(0<\dfrac{1}{x}\leq 1,\ \ 0<\dfrac{\ell}{y}\leq\ell,\ \
0<\dfrac{1}{k}\leq 1\right)$ (7)
Using (7), the trigonometric preliminaries, parts (a) and (b) and by taking
tangent of both sides of (1), we obtain,
$\dfrac{\frac{1}{x}+\frac{\ell}{y}}{1-\frac{1}{x}\frac{\ell}{y}}=\dfrac{1}{k}$
or equivalently
(Note that since $0<\dfrac{1}{k}\leq 1$. The equal sides of (1) can be utmost
equal to $\dfrac{\pi}{4}$ )
$\begin{array}[]{rcl}xy-\ell&=&k(y+x\ell);\\\ \\\
y\cdot(x-k)&=&\ell\cdot(1+kx)\end{array}$ (8)
Equation (8) shows that $y$ is a divisor of the product $\ell(1+kx)$. But, by
(1), we know that gcd$(\ell,y)=1$. Thus, by Result 1 (Euclid’s lemma), it
follows that $y$ must divide $1+kx$; and so,
$\left\\{\begin{array}[]{l}1+kx=y\cdot v\\\ \\\ v\ {\rm a\ positive\
integer}\end{array}\right\\}$ (9)
By (9) and (8) we have that,
$x=\ell\cdot v+k$ (10)
From (9) and (10) we further get
$1+k(\ell v+k)=yv;$
or equivalently
$k^{2}+1=(y-\ell k)\cdot v$ (11)
Since $v$ is a positive integer, equation (11) shows that $(y-\ell k)$ is a
positive integer divisor of $k^{2}+1$. Let $y-\ell k=d$, $d$ a positive
divisor of $k^{2}+1$. Then $y=\ell k+d$ and by (11) and (10) we also get
$x=k+\ell\cdot\left(\dfrac{k^{2}+1}{d}\right),$
which proves that the solution $(x,y)$ has the required form.
Finally, we see by inspection that the $N$ (number of positive divisors of
$k^{2}+1$) positive integer solutions to (1) are distinct since, obviously,
all the $Ny$-coordinates are distinct. The proof is complete.
∎
## 5 A listing of nine equalities
Let $k$ and $\ell$ be positive integers such that gcd$(\ell,k^{2}+1)=1$.
Applying Theorem 1 with $d=1$ and $d=k^{2}+1$ produces two inequalities.
1. 1.
$\arctan\left(\dfrac{1}{k+\ell(k^{2}+1)}\right)+\arctan\left(\dfrac{\ell}{k\ell+1}\right)=\arctan\left(\dfrac{1}{k}\right)$
2. 2.
$\arctan\left(\dfrac{1}{k+\ell}\right)+\arctan\left(\dfrac{\ell}{k\ell+k^{2}+1}\right)=\arctan\left(\dfrac{1}{k}\right)$
Next, applying Theorem 1 with $k=\ell=1$, produces the equality:
1. 3.
$\arctan\left(\dfrac{1}{3}\right)+\arctan\left(\dfrac{1}{2}\right)=\dfrac{\pi}{4}$
For $\ell=1$ and $k=2$:
1. 4.
$\arctan\left(\dfrac{1}{3}\right)+\arctan\left(\dfrac{1}{7}\right)=\arctan\left(\dfrac{1}{2}\right)$.
For $\ell=1$ and $k=3$
1. 5.
$\arctan\left(\dfrac{1}{11}\right)+\arctan\left(\dfrac{1}{4}\right)=\arctan\left(\dfrac{1}{3}\right)$
2. 6.
$\arctan\left(\dfrac{1}{8}\right)+\arctan\dfrac{1}{5}=\arctan\left(\dfrac{1}{3}\right)$
For $\ell=2$ and $k=4$:
1. 7.
$\arctan\left(\dfrac{1}{38}\right)+\arctan\left(\dfrac{2}{9}\right)=\arctan\left(\dfrac{1}{4}\right)$
2. 8.
$\arctan\left(\dfrac{1}{6}\right)+\arctan\left(\dfrac{2}{25}\right)=\arctan\left(\dfrac{1}{4}\right)$
For $\ell=1$ and $k=6$:
1. 9.
$\arctan\left(\dfrac{1}{43}\right)+\arctan\left(\dfrac{1}{7}\right)=\arctan\left(\dfrac{1}{6}\right)$
## References
* [1] Unal, Hasan, Proof without words: an arctangent equality, Mathematics and Computer Education, Fall 2011, Vol. 45, No. 3, p 197.
* [2] Rosen, Kenneth H., Elementary Number Theory and Its Applications, 5th edition, Pearson, Addison Wesley, 2005. For Result 1 (Lemma 3.4 in the above book), see page 109. For Result 2 (Theorem 7.9 in the above book), see page 252.
|
arxiv-papers
| 2012-03-28T21:13:22 |
2024-09-04T02:49:29.154271
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1203.6380"
}
|
1203.6399
|
# EXPLICIT FORMULAS INVOLVING $q$-EULER NUMBERS AND POLYNOMIALS
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr and Jong Jin Seo
Department of Applied Mathematics, Pukyong National University, Busan 608-737,
Republic of Korea seo2011@pknu.ac.kr (Corresponding author)
###### Abstract.
In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive
some interesting relations for q-Euler numbers and polynomials by using their
generating function and derivative operator. Also, we show between the q-Euler
numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic
integer ring.
###### Key words and phrases:
Euler numbers and polynomials, $q$-Euler numbers and polynomials with weight
$0$, $q$-Bernoulli numbers with weight $0$, $p$-adic $q$-integral
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. PRELIMINARIES
Imagine that $p$ be a fixed odd prime number. Throughout this paper we use the
following notations, by $\mathbb{Z}_{p}$ denotes the ring of $p$-adic rational
integers, $\mathbb{Q}$ denotes the field of rational numbers, $\mathbb{Q}_{p}$
denotes the field of $p$-adic rational numbers, and $\mathbb{C}_{p}$ denotes
the completion of algebraic closure of $\mathbb{Q}_{p}$. Let $\mathbb{N}$ be
the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$.
The $p$-adic absolute value is defined by
$\left|p\right|_{p}=\frac{1}{p}.$
In this paper we assume $\left|q-1\right|_{p}<1$ as an indeterminate.
$\left[x\right]_{q}$ is a $q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q},$
we note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ (see[1-12]).
We say that $f$ is a uniformly differentiable function at a point
$a\in\mathbb{Z}_{p}$, if the difference quotient
$F_{f}\left(x,y\right)=\frac{f\left(x\right)-f\left(y\right)}{x-y}$
has a limit $f{\acute{}}\left(a\right)$ as
$\left(x,y\right)\rightarrow\left(a,a\right)$ and denote this by $f\in
UD\left(\mathbb{Z}_{p}\right)$.
Let $UD\left(\mathbb{Z}_{p}\right)$ be the set of uniformly differentiable
function on $\mathbb{Z}_{p}$. For $f\in UD\left(\mathbb{Z}_{p}\right)$, let us
start with the expressions
$\frac{1}{\left[p^{N}\right]_{q}}\sum_{0\leq\xi<p^{N}}f\left(\xi\right)q^{\xi}=\sum_{0\leq\xi<p^{N}}f\left(\xi\right)\mu_{q}\left(\xi+p^{N}\mathbb{Z}_{p}\right),$
represents $p$-adic $q$-analogue of Riemann sums for $f$. The integral of $f$
on $\mathbb{Z}_{p}$ will be defined as the limit
$\left(N\rightarrow\infty\right)$ of these sums, when it exists. The $p$-adic
$q$-integral of function $f\in UD\left(\mathbb{Z}_{p}\right)$ is defined by T.
Kim
(1.1)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{q}\left(\xi\right)=\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{q}}\sum_{\xi=0}^{p^{N}-1}f\left(\xi\right)q^{\xi}\text{
}$
The bosonic integral is considered as a bosonic limit $q\rightarrow 1,$
$I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right)$. Similarly, the
fermionic $p$-adic integral on $\mathbb{Z}_{p}$ is introduced by T. Kim as
follows:
(1.2)
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-q}\left(\xi\right)$
(for more details, see [9-12]).
In [6], the $q$-Euler polynomials with wegiht $0$ are introduced as
(1.3)
$\widetilde{E}_{n,q}\left(x\right)=\int_{\mathbb{Z}_{p}}\left(x+y\right)^{n}d\mu_{-q}\left(y\right)$
From (1.3), we have
$\widetilde{E}_{n,q}\left(x\right)=\sum_{l=0}^{n}\binom{n}{l}x^{l}\widetilde{E}_{n-l,q}$
where $\widetilde{E}_{n,q}(0)=\widetilde{E}_{n,q}$ are called $q$-Euler
numbers with weight $0$. Then, $q$-Euler numbers are defined as
$q\left(\widetilde{E}_{q}+1\right)^{n}+\widetilde{E}_{n,q}=\left\\{\QATOP{\left[2\right]_{q},\text{
if }n=0}{0,\text{ \ \ \ \ if\ }n\neq 0,}\right.$
with the usual convention about replacing $\left(\widetilde{E}_{q}\right)^{n}$
by $\widetilde{E}_{n,q}$ is used.
Similarly, the $q$-Bernoulli polynomials and numbers with weight $0$ are
defined, respectively
$\displaystyle\widetilde{B}_{n,q}\left(x\right)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q}}\sum_{y=0}^{p^{n}-1}\left(x+y\right)^{n}q^{y}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\left(x+y\right)^{n}d\mu_{q}\left(y\right)$
and
$\widetilde{B}_{n,q}=\int_{\mathbb{Z}_{p}}y^{n}d\mu_{q}\left(y\right)$
(for more informations, see [4]).
We, by using Kim’s et al. method in [2], will investigate some interesting
identities on the $q$-Euler numbers and polynomials from their generating
function and derivative operator. Consequently, we derive some properties on
$q$-Euler numbers and polynomials and $q$-Bernoulli numbers and polynomials by
using $q$-Volkenborn integral and fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$.
## 2\. ON KIM’S $q$-EULER NUMBERS AND POLYNOMIALS
Let us consider Kim’s $q$-Euler polynomials as follows:
(2.1)
$F_{x}^{q}=F_{x}^{q}\left(t\right)=\frac{\left[2\right]_{q}}{qe^{t}+1}e^{xt}=\sum_{n=0}^{\infty}\widetilde{E}_{n,q}\left(x\right)\frac{t^{n}}{n!}.$
Here $x$ is a fixed parameter. Thus, by expression of (2.1), we can readily
see the following
(2.2) $qe^{t}F_{x}^{q}+F_{x}^{q}=\left[2\right]_{q}e^{xt}.$
Last from equality, taking derivative operator $D$ as $D=\frac{d}{dt}$ on the
both sides of (2.2). Then, we easily see that
(2.3)
$qe^{t}\left(D+I\right)^{k}F_{x}^{q}+D^{k}F_{x}^{q}=\left[2\right]_{q}x^{k}e^{xt}$
where $k\in\mathbb{N}^{\ast}$ and $I$ is identity operator. By multiplying
$e^{-t}$ on both sides of (2.3), we get
(2.4)
$q\left(D+I\right)^{k}F_{x}^{q}+e^{-t}D^{k}F_{x}^{q}=\left[2\right]_{q}x^{k}e^{\left(x-1\right)t}$
Let us take derivative operator $D^{m}\left(m\in\mathbb{N}\right)$ on both
sides of (2.4). Then we get
(2.5)
$qe^{t}D^{m}\left(D+I\right)^{k}F_{x}^{q}+D^{k}\left(D-I\right)^{m}F_{x}^{q}=\left[2\right]_{q}x^{k}\left(x-1\right)^{m}e^{xt}$
Let $G\left[0\right]$ (not $G\left(0\right)$) be the constant term in a
Laurent series of $G\left(t\right)$. Then, from (2.5), we get
(2.6)
$\sum_{j=0}^{k}\binom{k}{j}\left(qe^{t}D^{k+m-j}F_{x}^{q}\left(t\right)\right)\left[0\right]+\sum_{j=0}^{m}\binom{m}{j}\left(-1\right)^{j}\left(D^{k+m-j}F_{x}^{q}\left(t\right)\right)\left[0\right]=\left[2\right]_{q}x^{k}\left(x-1\right)^{m}$
By (2.1), we see
(2.7)
$\left(D^{N}F_{x}^{q}\left(t\right)\right)\left[0\right]=\widetilde{E}_{N,q}\left(x\right)\text{
and\
}\left(e^{t}D^{N}F_{x}^{q}\left(t\right)\right)\left[0\right]=\widetilde{E}_{N,q}\left(x\right)$
By expressions of (2.6) and (2.7), we see that
(2.8)
$\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\widetilde{E}_{k+m-j,q}\left(x\right)=\left[2\right]_{q}x^{k}\left(x-1\right)^{m}.$
From (2.1), we note that
(2.9)
$\frac{d}{dx}\left(\widetilde{E}_{n,q}\left(x\right)\right)=n\sum_{l=0}^{n-1}\binom{n-1}{l}\widetilde{E}_{l,q}x^{n-1-l}=n\widetilde{E}_{n-1,q}\left(x\right)$
By (2.9), we easily see,
(2.10)
$\int_{0}^{1}\widetilde{E}_{n,q}\left(x\right)dx=\frac{\widetilde{E}_{n+1,q}\left(1\right)-\widetilde{E}_{n+1,q}}{n+1}=-\frac{\left[2\right]_{q^{-1}}}{n+1}\widetilde{E}_{n+1,q}$
Now, let us consider definition of integral from $0$ to $1$ in (2.8), then we
have
$\displaystyle-\left[2\right]_{q^{-1}}\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\frac{\widetilde{E}_{k+m-j+1,q}}{k+m-j+1}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\left(-1\right)^{m}B\left(k+1,m+1\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\left(-1\right)^{m}\frac{\Gamma\left(k+1\right)\Gamma\left(m+1\right)}{\Gamma\left(k+m+2\right)}$
where $B\left(m,n\right)$ is beta function which is defined by
$\displaystyle B\left(m,n\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}x^{m-1}\left(1-x\right)^{n-1}dx$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(m\right)\Gamma\left(n\right)}{\Gamma\left(m+n\right)},\text{
}m>0\text{ and\ }n>0.$
As a result, we obtain the following theorem
###### Theorem 1.
For $n\in\mathbb{N},$ we have
$\displaystyle\sum_{j=1}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\frac{\widetilde{E}_{k+m-j+1,q}}{k+m-j+1}$
$\displaystyle=$ $\displaystyle
q\frac{\left(-1\right)^{m+1}}{\left(k+m+1\right)\binom{k+m}{k}}-\left[2\right]_{q}\frac{\widetilde{E}_{k+m+1,q}}{k+m+1}.$
Substituting $m=k+1$ into Theorem 1, we readily get
$\displaystyle\sum_{j=1}^{k+1}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{k+1}{j}\right]\frac{\widetilde{E}_{2k+2-j,q}}{2k+2-j}$
$\displaystyle=$ $\displaystyle
q\frac{\left(-1\right)^{k}}{\left(2k+2\right)\binom{2k+1}{k}}-\left[2\right]_{q}\frac{\widetilde{E}_{2k+2,q}}{2k+2}.$
By (2.1), it follows that
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left(k+m-j\right)\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\widetilde{E}_{k+m-j-1,q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}x^{k-1}\left(x-1\right)^{m-1}\left(\left(k+m\right)x-k\right).$
Let $m=k$ in (2.1), we see that
$\sum_{j=0}^{k}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{k}{j}\right]\widetilde{E}_{2k-j,q}\left(x\right)=\left[2\right]_{q}x^{k}\left(x-1\right)^{k}$
Last from equality, we discover the following
(2.12)
$\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\widetilde{E}_{2k-2j,q}\left(x\right)+\left(q-1\right)\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\widetilde{E}_{2k-2j-1,q}\left(x\right)=\left[2\right]_{q}x^{k}\left(x-1\right)^{k}.$
Here $\left[.\right]$ is Gauss’ symbol. Then, taking integral from $0\ $to $1$
both sides of last equality, we get
$\displaystyle-\left[2\right]_{q^{-1}}\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\frac{\widetilde{E}_{2k-2j+1,q}}{2k-2j+1}+\left[2\right]_{q^{-1}}\left(1-q\right)\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\frac{\widetilde{E}_{2k-2j,q}}{2k-2j}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\left(-1\right)^{k}B\left(k+1,k+1\right)$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}\left(-1\right)^{k}}{\left(2k+1\right)\binom{2k}{k}}.$
Consequently, we derive the following theorem
###### Theorem 2.
The following identity
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\frac{\widetilde{E}_{2k-2j+1,q}}{2k-2j+1}+\left(q-1\right)\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\frac{\widetilde{E}_{2k-2j,q}}{2k-2j}$
$\displaystyle=$
$\displaystyle\frac{q\left(-1\right)^{k+1}}{\left(2k+1\right)\binom{2k}{k}}.$
is true.
In view of (2.1) and (2.12), we discover the following applications:
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k+1}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{k+1}{j}\right]\widetilde{E}_{2k+1-j,q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\widetilde{E}_{2k+1,q}\left(x\right)+\sum_{j=1}^{\left[\frac{k+1}{2}\right]}\left[q\binom{k}{2j}+\binom{k}{2j}+\binom{k}{2j-1}\right]\widetilde{E}_{2k+1-2j,q}\left(x\right)$
$\displaystyle+\sum_{j=0}^{\left[\frac{k+1}{2}\right]}\left[q\binom{k}{2j+1}-\binom{k}{2j+1}-\binom{k}{2j}\right]\widetilde{E}_{2k-2j,q}\left(x\right)$
$\displaystyle=$
$\displaystyle-\left[\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\widetilde{E}_{2k-2j,q}\left(x\right)+\frac{q-1}{1+q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\widetilde{E}_{2k-2j+1}\left(x\right)\right]$
$\displaystyle+\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\widetilde{E}_{2k+1-2j,q}\left(x\right)+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\widetilde{E}_{2k+1-2j,q}\left(x\right)$
$\displaystyle+\left(q-1\right)\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\widetilde{E}_{2k-2j,q}\left(x\right)+\frac{q-1}{1+q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\widetilde{E}_{2k-2j+1}\left(x\right)$
By expressions (2.12) and (2), we have the following Theorem
###### Theorem 3.
For $k\in\mathbb{N}$, we have
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\widetilde{E}_{2k+1-2j,q}\left(x\right)+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\widetilde{E}_{2k+1-2j,q}\left(x\right)$
$\displaystyle+\left(q-1\right)\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left[\widetilde{E}_{2k-2j,q}\left(x\right)+\frac{1}{1+q}\widetilde{E}_{2k-2j+1}\left(x\right)\right]$
$\displaystyle=$ $\displaystyle
x^{k}\left(x-1\right)^{k}\left(\left[2\right]_{q}x-q\right)$
## 3\. $p$-adic integral on $\mathbb{Z}_{p}$ associated with Kim’s $q$-Euler
polynomials
In this section, we consider Kim’s $q$-Euler polynomials by means of $p$-adic
$q$-integral on $\mathbb{Z}_{p}$. Now we start with the following assertion.
Let $m,k\in\mathbb{N}$, Then by (2.8),
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q}\int_{\mathbb{Z}_{p}}x^{k}\left(x-1\right)^{m}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\int_{\mathbb{Z}_{p}}x^{l+k}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\widetilde{E}_{l+k,q}$
On the other hand, right hand side of (2.8),
$\displaystyle I_{2}$
$\displaystyle=\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\widetilde{E}_{l,q}$
Equating $I_{1}$ and $I_{2}$, we get the following theorem
###### Theorem 4.
For $m,k\in\mathbb{N}$, we have
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\widetilde{E}_{l,q}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\widetilde{E}_{l+k,q}.$
Let us take fermionic $p$-adic $q$-inetgral on $\mathbb{Z}_{p}$ left hand side
of (3), we get
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}x^{k}\left(x-1\right)^{k}\left(\left[2\right]_{q}x-q\right)d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\int_{\mathbb{Z}_{p}}x^{k+l+1}d\mu_{-q}\left(x\right)-q\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\int_{\mathbb{Z}_{p}}x^{k+l}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\widetilde{E}_{k+l+1,q}-q\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\widetilde{E}_{k+l,q}$
In other word, we consider right hand side of (3) as follows:
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{-q}\left(x\right)$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{-q}\left(x\right)$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left[\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{-q}\left(x\right)\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{-q}\left(x\right)\end{array}\right]$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{E}_{l,q}$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{E}_{l,q}$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left[\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\widetilde{E}_{l,q}\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\widetilde{E}_{l,q}\end{array}\right]$
Equating $I_{3}$ and $I_{4}$, we get the following theorem
###### Theorem 5.
For $k\in\mathbb{N}$, we have
$\displaystyle\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\left[\left[2\right]_{q}\widetilde{E}_{k+l+1,q}-q\widetilde{E}_{k+l,q}\right]$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{E}_{l,q}$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{E}_{l,q}$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left\\{\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\widetilde{E}_{l,q}\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\widetilde{E}_{l,q}\end{array}\right\\}$
Now, we consider (2.8) and (2.1) by means of $q$-Volkenborn integral. Then, by
(2.8), we see
$\displaystyle\left[2\right]_{q}\int_{\mathbb{Z}_{p}}x^{k}\left(x-1\right)^{m}d\mu_{q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\int_{\mathbb{Z}_{p}}x^{l+k}d\mu_{q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\widetilde{B}_{l+k,q}$
On the other hand,
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{q}\left(x\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\widetilde{B}_{l,q}$
Therefore, we get the following theorem
###### Theorem 6.
For $m,k\in\mathbb{N}$, we have
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{m}\binom{m}{l}\left(-1\right)^{m-l}\widetilde{B}_{l+k,q}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\max\left\\{k,m\right\\}}\left[q\binom{k}{j}+\left(-1\right)^{j}\binom{m}{j}\right]\sum_{l=0}^{k+m-j}\binom{k+m-j}{l}\widetilde{E}_{k+m-j-l,q}\widetilde{B}_{l,q}$
By using fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ left hand side of
(3), we get
$\displaystyle I_{5}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q}\int_{\mathbb{Z}_{p}}x^{k}\left(x-1\right)^{k}\left(\left[2\right]x-q\right)d\mu_{q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\int_{\mathbb{Z}_{p}}x^{k+l+1}d\mu_{q}\left(x\right)-q\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\int_{\mathbb{Z}_{p}}x^{k+l}d\mu_{q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\widetilde{B}_{k+l+1,q}-q\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\widetilde{B}_{k+l,q}$
Also, we consider right hand side of (3) as follows:
$\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{q}\left(x\right)$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{q}\left(x\right)$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left[\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{q}\left(x\right)\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\int_{\mathbb{Z}_{p}}x^{l}d\mu_{q}\left(x\right)\end{array}\right]$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{B}_{l,q}$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{B}_{l,q}$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left[\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\widetilde{B}_{l,q}\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\widetilde{B}_{l,q}\end{array}\right]$
Equating $I_{5}$ and $I_{6}$, we get the following Corollary
###### Corollary 1.
For $k\in\mathbb{N}$, we get
$\displaystyle\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k-l}\left[\left[2\right]_{q}\widetilde{B}_{k+l+1,q}-q\widetilde{B}_{k+l,q}\right]$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{B}_{l,q}$
$\displaystyle+\sum_{j=1}^{\left[\frac{k}{2}\right]}\binom{k}{2j-1}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k+1-2j-l,q}\widetilde{B}_{l,q}$
$\displaystyle+\sum_{j=0}^{\left[\frac{k}{2}\right]}\binom{k}{2j+1}\left\\{\begin{array}[]{c}\left(q-1\right)\sum_{j=0}^{2k-2j}\binom{2k-2j}{l}\widetilde{E}_{2k-2j-l,q}\widetilde{B}_{l,q}\\\
+\frac{q-1}{1+q}\sum_{l=0}^{2k-2j+1}\binom{2k-2j+1}{l}\widetilde{E}_{2k-2j-l+1}\widetilde{B}_{l,q}\end{array}\right\\}$
## References
* [1] Araci, S., Erdal, D., and Seo, J-J., A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [2] T. Kim, B. Lee, S. H. Lee, S-H. Rim, Identities for the Bernoulli and Euler numbers and polynomials, Accepted in Ars Combinatoria.
* [3] Kim, D., Kim, T., Lee, S-H., Dolgy, D-V., and Rim, S-H., Some new identities on the Bernoulli numbers and polynomials, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 856132, 11 pages.
* [4] Kim, T., Choi, J., and Kim, Y-H., Some identities on the $q$-Bernoulli numbers and polynomials with weight $0,$ Abstract And Applied Analysis, Volume 2011, Article ID 361484, 8 pages.
* [5] Kim, T., On a$\ q$-analogue of the $p$-adic log gamma functions related integrals, J. Number Theory, 76 (1999) no. 2, 320-329.
* [6] Kim, T., and Choi, J., On the $q$-Euler numbers and polynomials with weight $0$, Abstract and Applied Analysis, Volume 2012, ID 795304, 7 pages, doi:10.1155/2012/795304.
* [7] Kim, T., On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
* [8] Kim, T., On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21(2011), no.2, p. 207-215, http://arxiv.org/abs/1011.5305.
* [9] Kim, T., $q$-Volkenborn integration, Russ. J. Math. phys. 9$\left(2002\right),$ 288-299.
* [10] Kim, T., $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27.
* [11] Kim, T., New approach to $q$-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225.
* [12] Kim, T., Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4,484–491.
|
arxiv-papers
| 2012-03-29T00:01:06 |
2024-09-04T02:49:29.158645
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Jong Jin Seo",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1203.6399"
}
|
1203.6488
|
March 29, 2012; Revised August 4, 2012
# Explosive Nucleosynthesis in Magnetohydrodynamical Jets from Collapsars. II
Heavy-Element Nucleosynthesis of s, p, r-Processes
Masaomi Ono ${}^{1,2,\,}$111E-mail: ono@yukawa.kyoto-u.ac.jp Masa-aki
Hashimoto 2 Shin-ichiro Fujimoto 3 Kei Kotake4 and Shoichi Yamada5 1Yukawa
Institute for Theoretical Physics1Yukawa Institute for Theoretical Physics
Kyoto University Kyoto University Kyoto 606-8502 Kyoto 606-8502 Japan
2Department of Physics Japan
2Department of Physics Kyushu University Kyushu University Fukuoka 812-8581
Fukuoka 812-8581 Japan
3Kumamoto National College of Technology Japan
3Kumamoto National College of Technology Kumamoto 861-1102 Kumamoto 861-1102
Japan
4Division of Theoretical Astronomy/Center for Computational Astrophysics Japan
4Division of Theoretical Astronomy/Center for Computational Astrophysics
National Astronomical Observatory of Japan National Astronomical Observatory
of Japan Mitaka 181-8588 Mitaka 181-8588 Japan
5Advanced Research Institute for Science and Engineering Japan
5Advanced Research Institute for Science and Engineering Waseda University
Waseda University Tokyo 113-0033 Tokyo 113-0033 Japan Japan
###### Abstract
We investigate the nucleosynthesis in a massive star of 70 $M_{\odot}$ with
solar metallicity in the main sequence stage. The helium core mass after
hydrogen burning corresponds to 32 $M_{\odot}$. Nucleosynthesis calculations
have been performed during the stellar evolution and the jetlike supernova
explosion of a collapsar model. We focus on the production of elements heavier
than iron group nuclei. Nucleosynthesis calculations have been accomplished
consistently from hydrostatic to dynamic stages by using large nuclear
reaction networks, where the weak s-, p-, and r-processes are taken into
account. We confirm that s-elements of $60<A<90$ are highly overproduced
relative to the solar abundances in the hydrostatic nucleosynthesis. During
oxygen burning, p-elements of $A>90$ are produced via photodisintegrations of
seed s-elements. However, the produced p-elements are disintegrated in later
stages except for 180Ta. In the explosive nucleosynthesis, elements of
$90<A<160$ are significantly overproduced relative to the solar values owing
to the r-process, which is very different from the results of spherical
explosion models. Only heavy p-elements ($N>50$) are overproduced via the
p-process because of the low peak temperatures in the oxygen- and neon-rich
layers. Compared with the previous study of r-process nucleosynthesis
calculations in the collapsar model of 40 $M_{\odot}$ by Fujimoto et al., [S.
Fujimoto, M. Hashimoto, K. Kotake and S. Yamada, Astrophys. J. 656 (2007),
382; S. Fujimoto, N. Nishimura and M. Hashimoto, Astrophys. J. 680 (2008),
1350], our jet model cannot contribute to the third peak of the solar
r-elements and intermediate p-elements, which have been much produced because
of the distribution of the lowest part of electron fraction in the ejecta.
Averaging the overproduction factors over the progenitor masses with the use
of Salpeter’s IMF, we suggest that the 70 $M_{\odot}$ star could contribute to
the solar weak s-elements of $60<A<90$ and neutron-rich elements of
$90<A<160$. We confirm the primary synthesis of light p-elements in the
ejected matter of high peak temperature. The ejected matter has [Sr/Eu]
$\sim-0.4$, which is different from that of a typical r-process-enriched star
CS22892-052 ([Sr/Eu] $\sim-1$). We find that Sr-Y-Zr isotopes are primarily
synthesized in the explosive nucleosynthesis in a similar process of the
primary production of light p-elements, which has been considered as one of
the sites of a lighter element primary process (LEPP).
## 1 Introduction
The origin of elements, particularly those heavier than iron, is still under
debate.[3] Since charged particle reactions are difficult to produce those
elements inside stars because of coulomb barriers, other nucleosynthesis
processes, that is, two neutron capture processes, are required. One is the r
(rapid)-process and the other is the s (slow)-process.[4] In the r (s)
-process, neutron captures are faster (slower) than beta decays. Since the
r-process requires high neutron exposure relative to seeds, the r-process
favors low electron fraction ($Y_{e}$) and/or relatively high-entropy
environments.[5]
One of the promising sites of the r-process has been thought to be the
neutrino-driven wind. [5, 6, 7, 8] However, recent one-dimensional
hydrodynamical simulations of the neutrino-driven wind with Boltzmann neutrino
transport have revealed [9] that the electron fraction of the wind becomes
high ($Y_{e}\gtrsim$ 0.5) and the entropy becomes low for the r-process.
Therefore, other astrophysical sites such as neutron star mergers [10, 11, 12]
or black hole winds [13] have been proposed. However, the properties of ejecta
such as densities, temperatures, and electron fractions are highly uncertain
except for those in Ref. ref:goriely_2011, which makes even the qualitative
analysis of the r-process difficult.
In general, the s-process occurs at the end of core helium burning in massive
stars and/or in AGB stars. Elements heavier than iron of $A<90$ are produced
in massive stars, which is called the weak component of the s-process (weak
s-process).[14] On the other hand, elements of $90<A<208$ are produced in AGB
stars called the main component (main s-process).[14] The s-process is very
sensitive to cross sections of neutron captures and $\beta$-decay rates,
especially at the branching points such as 79Se and 85Kr.[14] Therefore, the
nucleosynthesis in massive stars with the use of recent experimental cross
sections is worth investigating. From the view point of astrophysics, part of
the elements synthesized by the weak s-process could be ejected through the
subsequent supernova explosion, where the produced s-elements should be the
seeds of the p-process.[15]
Abundances of metal-poor stars provide a good opportunity for understanding
the nucleosynthesis because the abundances reflect the outcome only from a
small number of supernova explosions. Observations of metal-poor stars have
strongly suggested [16] that r-elements of the extremely metal-poor stars that
have [Eu/Fe]222We adopt the usual notation [A/B] = log($N_{\rm A}$/$N_{\rm
A}$) $-$ log($N_{\rm A}$/$N_{\rm A})_{\odot}$ for elements A and B. $\gtrsim
1$ (hereafter referred to as r-process-rich stars) have a “universal”
abundance pattern, which reproduces the pattern of the solar system r-process
abundances for $Z>56$. However, the abundances of $Z<56$ are not the case
(e.g., Ref. ref:sneden_2003), and the observed [Sr, Y, Zr/Ba, Eu] ratios have
dispersion in low-metallicity stars.[18] In particular, on the basis of the
models of the chemical evolution of galaxies, it has been suggested [18] that
the abundances of Sr-Y-Zr ($Z=38,\,39$, and 40, respectively) estimated from
the contributions of the s\- and r-processes are about 10 to 20% less than the
solar system abundances. As a consequence, a primary component from massive
stars is needed to explain 8% of the solar abundance of Sr and 18% of those of
Y and Zr, which should require a so-called lighter element primary process
(LEPP). [18]
On the other hand, the mechanism of core-collapse supernova explosions is
still a topic of debate. Pushed by recent observations revealing the
aspherical natures of supernovae, [19, 20] multidimensional studies of core-
collapse supernova explosions have been elaborately performed as described in
reviews. [21, 22, 23, 24, 25] Recent two/three-dimensional neutrino-radiation
hydrodynamic simulations have shown successful supernova explosions, although
in some of the models, the explosion energies are relatively small (1049–1050
erg). [26, 27, 28, 29] QCD phase transition with a mixed phase of quarks and
hadrons has also been reported as another possible supernova explosion
mechanism even though the explosion is assumed to be spherical.[30]
Magnetohydrodynamical (MHD) simulations with some approximate neutrino
transport schemes have shown [31, 32, 33, 34, 35] jetlike explosions under
some specific combinations of initial parameters for a strong magnetic field
and differentially rapid rotation. While neutron stars are expected to be left
after supernova explosions, it has been suggested that a star of more than 25
$M_{\odot}$ may collapse to a black hole (BH) [36]; an accretion disk is
formed around the BH if the star has enough angular momentum before the
collapse. This system could produce a relativistic jet of gamma-ray bursts
(GRBs) [37] due to MHD effects and/or neutrino heating around the rotational
axis,[38, 39] whose system is called a collapsar model.[40] MHD simulations in
the context of the collapsar model have shown the formation of jets [41, 42,
43, 44, 45, 46, 47, 48] due to winding-up effects of the magnetic field or the
Blandford-Znajek process.[49]
Nucleosynthesis calculations of the r-process with the use of a collapsar
model of 40 $M_{\odot}$ have been performed extensively by Fujimoto et al.,[1,
2] where it is shown that the r-process would operate inside the jets.
Explosive nucleosynthesis in GRB jets has also been investigated.[50, 51]
Recent nucleosynthesis calculations in a three-dimensional MHD supernova model
have suggested that such supernovae could be the sources of the r-process
elements in the early Galaxy.[52] However, in those calculations, the produced
nuclei are limited to primary synthesized ones inside the jets and comparisons
with the solar system abundances have been focused on elements heavier than
iron group nuclei. Nucleosynthesis calculations in spherical supernova
explosions and detailed hydrostatic ones of the progenitors have proved that
elements of $20<A<90$ are co-overproduced relative to the solar system
abundances. [53] However, elements of $A>90$ are not overproduced except for
some p-elements. Recently, explosive nucleosynthesis calculations for a 15
$M_{\odot}$ presupernova model [54] with the solar metallicity based on two-
dimensional hydrodynamical simulations have been performed,[55] in which
neutrino-driven explosions are triggered by adjusting the core neutrino
luminosity parametrically. They have concluded that the overproductions
relative to the solar abundances are similar to the results of spherical
explosion models.[53] In our previous paper (Paper I),[56] we performed
explosive nucleosynthesis calculations inside the jetlike explosions for the
collapsar of a 70 $M_{\odot}$ star with the solar metallicity. These
calculations include hydrostatic nucleosynthesis using a nuclear reaction
network, which has 464 nuclei (up to 94Kr). In the present paper, we revisit
the nucleosynthesis inside the jetlike explosion of the collapsar model and
hydrostatic one taking into account all of the weak s-, r-, and p-processes.
This makes it possible for us to estimate the consistent abundances of the
ejecta. In particular, we study whether the collapsar model could be the
source of elements heavier than iron.
In §2, we present the hydrostatic nucleosynthesis during the evolution from
the helium burning stage to the onset of the core collapse. In §3, we briefly
summarize the MHD explosion model for the explosive nucleosynthesis and show
the results. Section 4 is devoted to a summary of the overall results. We give
some discussions with respect to uncertainties and remarks about the
connection between metal-poor stars and the possibility for LEPP.
## 2 Hydrostatic nucleosynthesis
In Paper I, we investigated the nucleosynthesis in a massive star of 32
$M_{\odot}$ helium core corresponding to a main sequence star of 70
$M_{\odot}$.[57] We have used the evolutional tracks with a nuclear reaction
network, which includes 464 nuclei (up to 94Kr).[56] In massive stars, the
(weak) s-process should occur at the end of helium and carbon burning stages.
However, in the previous paper, we have included only nuclei of $A<94$; we
could not discuss the weak s-process. Therefore, in the present paper, we
perform a more detailed nucleosynthesis calculation with a larger nuclear
reaction network.
### 2.1 Stellar model, initial compositions, and physical inputs
A star of $M_{\rm ms}\sim 70~{}M_{\odot}$ with the solar metallicity could
correspond to the upper limit of accreting BH models (collapsars), because
more massive stars suffer from the strong mass loss.[36] As a result, the size
of the helium core will be affected considerably by the mass loss. However,
the rate of mass loss is still very uncertain [58]; we calculate the evolution
of a massive helium core, $M_{\alpha}=32~{}M_{\odot}$, without the mass loss
as an extreme case, which is worth studying to see the final fate for the
series of helium core evolution.
We calculate the nucleosynthesis along each evolutional track of the Lagrange
mass from the stage of gravitational contraction of the core to the initiation
of iron core collapse. The calculation has been carried out by using changes
in the density ($\rho$), temperature ($T$) and convective regions. This is the
so-called postprocess nucleosynthesis calculation. In convective regions,
elements are mixed and compositions become almost uniform. Therefore, the
region is calculated as one zone with averaged mass fractions and nuclear
reaction rates as in Ref. ref:prantzos_1987.
Toward s-process calculation, we construct a new reaction network including
1714 nuclei up to 241U, in which the reaction rates are based on a new REACLIB
compilation, namely, JINA REACLIB database [60]. The included elements are
given in Table 1. The experimental ($n$, $\gamma$) reaction rates in JINA
REACLIB are based on KADoNiS projects. [61] In stellar environments,
$\beta$-decay rates could be different from the values in laboratories.
Takahashi and Yokoi (hereafter referred to as TY87) [62] calculated
theoretically the $\beta$-decay and electron capture rates for elements
heavier than 59Ni and tabulated the rates taking into account thermally
enhanced ionized and excited states in stellar interiors. The table ranges
over 5$\times$107 $\leq T\leq$ 5$\times$108 K and 1026 $\leq n_{e}\leq$
3$\times$1027 cm-3, where $n_{e}$ is the number density of electrons. We adopt
the temperature and density dependence of the rates if available.
Unfortunately, the range of the table is limited only for $T$ and $\rho$ of
the helium burning stage. If the temperature and density are outside of the
table, we use the values of the edges of the table. Reaction rates concerning
180Ta are specially treated as noted in Appendix A, because 180Ta has a long-
lived isomeric state. As suggested by Prantzos et al.[59], the timescales of
neutron-induced reactions ($\approx$ $10^{-4}$ s) are much shorter than those
of the variation of abundances of the other elements ($\approx
10^{9}$–$10^{11}$ s) as well as convective timescales, and thereby, neutron
abundance is determined locally under thermal equilibrium conditions. This
indicates that neutrons are not mixed uniformly in convective regions.
Therefore, to calculate abundances in a convective region as one zone
including the effects of different neutron abundances over the region, we
adopt doubly averaged reaction rates for “neutron-induced” reactions of ($n$,
$\gamma$), ($n$, $p$), and ($n$, $\alpha$), according to the same method
described in Ref. ref:prantzos_1987.
Let $X(i)$ denote the mass fraction of the element $i$. The initial mass
fractions are assumed to be $X(^{4}$He$)=0.981$ and $X(^{14}$N$)=0.0137$,
where all the original CNO isotopes are assumed to be converted to 14N during
the core hydrogen burning. Mass fractions of the heavier elements are taken to
be proportional to the solar system abundances [63] (e.g., $X(^{56}\rm
Fe)=1.17\times 10^{-3}$).
The evolutionary changes in composition with respect to the stellar structure
such as 12C, 16O, and 20Ne are slightly different from the results with the
use of the reaction network of 464 nuclei [56] originating from the
differences in the initial abundances and adopted reaction rates. The produced
mass fractions of elements of $A>94$, which have not been included in the 464
network, amount to about 10-6. Therefore, differences of 10-6 in mass
fractions may be introduced. Although there are some differences in the main
composition as described in Paper I between the original stellar evolution
model [57] and postprocess hydrostatic nucleosynthesis calculations with the
reaction network of 464 or 1714 nuclei, the differences are not so large for
our purpose.
Table 1: 1714 nuclides contained in the nuclear reaction network for the hydrostatic nucleosynthesis. Nuclide | $A$ | Nuclide | $A$ | Nuclide | $A$ | Nuclide | $A$
---|---|---|---|---|---|---|---
H | 1 – 3 | Cr | 46 – 62 | Ag | 100 – 123 | Yb | 162 – 184
He | 3 – 6 | Mn | 48 – 65 | Cd | 103 – 126 | Lu | 165 – 187
Li | 6 – 9 | Fe | 50 – 68 | In | 105 – 129 | Hf | 168 – 189
Be | 7 – 12 | Co | 52 – 70 | Sn | 108 – 132 | Ta | 171 – 191
B | 8 – 14 | Ni | 54 – 73 | Sb | 111 – 135 | W | 174 – 194
C | 9 – 18 | Cu | 57 – 75 | Te | 113 – 137 | Re | 177 – 197
N | 11 – 21 | Zn | 59 – 78 | I | 117 – 140 | Os | 180 – 200
O | 13 – 23 | Ga | 62 – 80 | Xe | 120 – 142 | Ir | 183 – 203
F | 14 – 26 | Ge | 64 – 83 | Cs | 122 – 145 | Pt | 186 – 206
Ne | 17 – 28 | As | 67 – 86 | Ba | 125 – 148 | Au | 188 – 209
Na | 19 – 30 | Se | 69 – 89 | La | 128 – 150 | Hg | 191 – 212
Mg | 21 – 33 | Br | 72 – 91 | Ce | 131 – 153 | Tl | 194 – 215
Al | 23 – 35 | Kr | 74 – 93 | Pr | 133 – 156 | Pb | 198 – 217
Si | 25 – 38 | Rb | 76 – 96 | Nd | 136 – 158 | Bi | 202 – 220
P | 27 – 40 | Sr | 79 – 98 | Pm | 138 – 160 | Po | 205 – 222
S | 29 – 42 | Y | 81 – 101 | Sm | 141 – 163 | At | 209 – 224
Cl | 31 – 45 | Zr | 83 – 103 | Eu | 143 – 165 | Rn | 212 – 227
Ar | 33 – 48 | Nb | 86 – 106 | Gd | 146 – 168 | Fr | 215 – 229
K | 35 – 50 | Mo | 89 – 109 | Tb | 148 – 171 | Ra | 217 – 232
Ca | 37 – 53 | Tc | 91 – 112 | Dy | 151 – 174 | Ac | 222 – 234
Sc | 39 – 55 | Ru | 93 – 115 | Ho | 154 – 176 | Th | 225 – 237
Ti | 41 – 57 | Rh | 96 – 117 | Er | 157 – 179 | Pa | 227 – 239
V | 43 – 59 | Pd | 98 – 120 | Tm | 160 – 181 | U | 231 – 241
### 2.2 S\- and p-processes
Let us focus on the production of elements heavier than iron and the weak s\-
and p-processes in the hydrostatic nucleosynthesis. Figure 1 shows
overproduction factors $X(i)/X(i)_{\,\odot}$ at the beginning of the core
collapse, where $X(i)$ is the value averaged over the star including the
hydrogen-rich envelope of 38 $M_{\odot}$, and $X(i)_{\odot}$ is that of the
solar system abundances. It is noted that in the estimation of the
overproduction factors, we adopt the value of 4.55$\times$109 yr ago for 40K,
which is a long-lived radioactive nucleus (the half-life is 1.25$\times$109
yr) and the present abundance is about one order of magnitude less than that
when the solar system was born ($\sim$ 4.5 yr ago). There is inconsistency
between the initial abundance of 40K and the value in the estimation of the
overproduction factor. However, the overproduction of 40K is not determined by
the initial abundance of 40K but by the abundance of 39K as described below.
Overall, elements of $60<A<90$ are highly overproduced relative to the solar
ones; the overproduction level ranges over 102 – 103, which is similar to the
weak s-process scenario proposed by many previous studies of the s-process in
massive stars. [59, 64, 65]
Figure 2 shows the time evolution of mass fractions of selected elements
averaged over the whole star. 86Kr, which is one of the representative
s-elements, is overproduced at the helium burning stage. We confirm that the
overproduced elements of $60<A<90$ are mainly produced during the helium core
burning, and the neutrons are mainly supplied by the 22Ne($\alpha$, $n$)25Mg
reaction (Fig. 2) as pointed out in previous studies (see Ref.
ref:kappeler_2011 for a recent review). 22Ne is produced by the sequence of
14N($\alpha$, $\gamma$)18F($\beta^{-}\nu$)18O($\alpha$, $\gamma$)22Ne
reactions. 40K is produced by the 39K($n$, $\gamma$)40K reaction in the helium
burning stage. The solar system abundances of 39K and 40K are 3516 and 0.44
(the present value) (normalized as the abundance of silicon to be 106),[63]
respectively. Therefore, even if the small amount of 39K is converted to 40K,
we regard it to be much overproduced relative to the solar value. 180Ta is
also overproduced in the helium burning stage by the sequence of
179Hf($\beta^{-}$)179Ta($n$, $\gamma$)180Ta reactions (see Fig. 2). It is
noted that the reaction channel of 179Hf($\beta^{-}$)179Ta is closed if we do
not use the $\beta$-decay rates of TY87 because 179Hf is stable in the
laboratory. However, overproduced 180Ta decays from a thermally populated
ground state at the end of the helium burning, and the overproduction level
returns almost to the initial value (Fig. 2). In the helium burning stage,
96Zr is overproduced more than 10 times (Fig. 1) relative to the solar value
by neutron captures, which are faster than $\beta^{-}$-decays of 95Zr
($\tau_{1/2}\sim$ 64 d). However, Sr and Y are not so overproduced. Although
the overproduction levels of the elements of $10<A<90$ do not increase
significantly after the helium burning, some elements of $A>90$ are produced
after that.
In the carbon and neon burning stages, neutrons are mainly supplied by
reactions of 12C(12C, $n$)23Mg, 13C($\alpha$, $n$)16O, 17O($\gamma$, $n$)16O,
and 25Mg($\alpha$, $n$)28Si. The 22Ne($\alpha$, $n$)25Mg reaction is also
activated in the helium burning shell. Since 179Hf is produced by neutron
captures, 180Ta is overproduced again relative to the solar value via the
sequence of 179Hf($\beta^{-}$)179Ta($n$, $\gamma$)180Ta.
During the oxygen burning stage, the temperature becomes high ($T_{\rm c}\sim$
2$\times$109 K, where $T_{\rm c}$ is the temperature at the center) and
photodisintegrations of s-elements activate the p-process in the oxygen- and
neon-rich layers. The possibility of the p-process in hydrostatic evolution of
massive stars was proposed in previous studies. [66, 67, 68, 53] We find that
seed s-elements that have larger mass numbers tend to be disintegrated into
p-elements by ($\gamma$, $n$) reactions, and light p-elements of 74Se, 78Kr,
and 84Sr are produced at the oxygen- and neon-rich layers by ($\gamma$, $n$)
and ($\gamma$, $p$) reactions (Fig. 3). It is noted that 180Ta is produced
more and more by the 181Ta($\gamma$, $n$)180Ta reaction. Since the solar
abundances of 180Ta are much smaller than those of 181Ta, conversion of small
amounts of 181Ta leads to the overproduction of 180Ta relative to the solar
value.
In the later stages, produced p-elements of $A>90$ are disintegrated by
subsequent ($\gamma$, $n$) reactions and $\beta^{+}$-decays except for 180Ta.
After all, p-elements whose overproduction factors are greater than 10 are
only 74Se, 78Kr, and 180Ta at the beginning of the collapse (see Fig. 3). It
is noted that ($n$, $\gamma$) and ($\gamma$, $n$) reactions are in thermal
equilibrium at this stage.
The quantitative assessment of the adopted beta decay rates (TY87) should
require more experimental data.[14] Moreover, the table of the $\beta$-decay
rates of TY87 covers only the temperature and density of the helium burning
stage as mentioned in §2.1. Therefore, we have also calculated the
nucleosynthesis using laboratory $\beta$-decay rates instead of those of TY87.
Although overall overproduction levels change only by some factors as seen in
Fig. 4, those of more neutron-rich isotopes of $60<A<85$ tend to be decreased.
We find that the production of elements for $85<A<100$ is very sensitive to
the differences in $\beta$-decay rates, and Sr, Y, Zr, and Mo tend to be
overproduced. Although the channel of 179Hf($\beta^{-}$)179Ta($n$,
$\gamma$)180Ta is closed without TY87, 180Ta is produced by the
181Ta($\gamma$, $n$)180Ta reaction after the carbon burning and remains
through ($n$, $\gamma$) $\rightleftharpoons$ ($\gamma$, $n$) equilibrium in
the later stages.
Let us summarize the hydrostatic nucleosynthesis. 1) We confirm the weak
s-process scenario: s-elements of $60<A<90$ are highly overproduced relative
to the solar abundances. 2) High overproductions of 180Ta could be attributed
to the higher density and temperature evolutional tracks of the 70 $M_{\odot}$
star. 3) The p-process in the oxygen- and neon-rich layers occurs after the
carbon burning stage. However, the produced p-elements do not remain due to
subsequent ($\gamma$, $n$) reactions and $\beta^{+}$-decays. We suggest that
for smaller massive stars, the overproductions of 180Ta are significantly
decreased because of the lower density and temperature evolutional paths. On
the other hand, the p-process could become important because the produced
p-elements of larger mass numbers may survive in low-density and low-
temperature environments.
Figure 1: Overproduction factors $X(i)/X(i)_{\,\odot}$ against the mass number
$A$ averaged over the star including the hydrogen envelope at the beginning of
the core collapse. Distinguished symbols connected by lines indicate the
isotopes. Figure 2: Changes in mass fractions of selected elements averaged
over the helium core against remaining time before the core collapse ($t_{\rm
coll.}$ = 4.22$\times$105 yr). Upward (downward) arrows denote the time of the
start (end) of the burning stages. Figure 3: Same as Fig. 1 but for 35
p-elements. Figure 4: Same as Fig. 1 but for the results with use of
laboratory $\beta$-decay rates instead of those of TY87.
## 3 Explosive nucleosynthesis in a magnetohydrodynamical jet
In Paper I, we investigated the explosive nucleosynthesis with the nuclear
reaction network including 464 nuclei (up to 94Kr).[56] In the present paper,
we recalculate the explosive nucleosynthesis with a much larger reaction
network and focus on the production of heavier elements, that is, p\- and
r-elements. In this section, we present the review of the explosion model, the
method of explosive nucleosynthesis, and the results combined with that of the
hydrostatic nucleosynthesis.
### 3.1 Supernova explosion model
In Paper I, we have constructed supernova explosion models using a collapsar
model.[56] Here, we briefly summarize the explosion model for the
nucleosynthesis calculation. We have performed two-dimensional MHD simulations
of the collapsar model using a nonrelativistic MHD code, ZEUS-2D,[69, 70]
which is modified [71] for handling supernova simulations with a realistic
nuclear equation of state (EOS) based on the relativistic mean field
theory.[72]
For a low-density region of $\rho<10^{5}$ g cm-3, another EOS is connected
[73] smoothly at the density boundary, which consists of the nonrelativistic
ions, partially degenerate relativistic electrons, and radiation.
We have taken into account neutrino cooling by electron-positron ($e^{\pm}$)
pair captures on nucleons, $e^{\pm}$ pair annihilation, and nucleon-nucleon
bremsstrahlung. We neglect to include both detailed neutrino transport and
heating processes, because the maximum density remains less than 1010 g cm-3
in our calculations. We discuss the effects of neutrino absorptions on the
nucleosynthesis in §3.2. BH was mimicked as an inner free absorption boundary
and gravitational point source with pseudo Newtonian potential.[74] We adopted
the spherical coordinate ($r$, $\theta$, $\phi$), and the computation domain
was taken from the inner boundaries $r_{\rm in}=$ 50 – 200 km to 3$\times
10^{4}$ km, which covers inner oxygen-rich layers.
The initial presupernova model is the 32 $M_{\odot}$ helium core corresponding
to an $M_{\rm ms}$ = 70 $M_{\odot}$ star,[57] which is the same stellar
evolution model obtained in the previous section. The initial configuration of
angular velocity and magnetic field was implemented by analytical form with
parameters as in the previous study.[43, 31, 44, 2] The initial angular
velocity is written as follows:
$\Omega\,(r)=\Omega_{0}\frac{r^{2}_{0}}{r^{2}+r^{2}_{0}},$ (1)
where $r$ is the radius from the center, and $\Omega_{0}$ and $r_{0}$ are
model parameters. The initial toroidal magnetic field is given in proportional
to the angular velocity distribution as
$B_{\phi}\,(r)=B_{0}\frac{r^{2}_{0}}{r^{2}+r^{2}_{0}},$ (2)
where $B_{0}$ is a model parameter. We adopt the R51 model as in Paper I for
the explosive nucleosynthesis, which has the largest amount of the mass end
energy ejection rates among the investigated models. The parameters of the
initial angular velocity and magnetic field are $\Omega_{0}$ = 5 s-1, $r_{0}$
= 1500 km, $B_{0}=5.7\times 10^{12}$ G, and $B_{Z}=5\times 10^{11}$ G, where
$B_{Z}$ is the initial uniform poloidal magnetic field along the rotational
axis. The specified parameters of the rotation and magnetic field correspond
to the model of the rapid rotation and the strong magnetic field. Recent
stellar evolution models indicate [75] that if the magnetic field is taken
into account, the resultant specific angular momentum of the central region
becomes smaller than that required for the typical collapsar model ($j\sim$
1017 cm2 s-1 [37]). In our simulation, the jet is triggered by the central
magnetic pressure, which grows due to the compression and winding-up effects
of the magnetic field, and the amplified magnetic field reaches around $\sim$
1015 G. If the magnetorotational instability (MRI) [76] successfully operates
in the core-collapse phase, the magnetic field could be quickly amplified to
the same level from an initial magnetic field weaker than that ascribed in the
present paper. However, resolving MRI in a global simulation is very hard and
not feasible in the present calculations. Therefore, we assume that some
mechanisms such as MRI amplify the magnetic field rapidly from a weak initial
magnetic field and we mimic the situation by simply imposing a strong initial
magnetic field. Note that the reached magnetic field strength $\sim$ 1015 G is
comparable to that at saturation due to MRI.[77]
The resulting total ejection mass and explosion energy at the end of the
simulation ($t_{f}=1.504$ s) are 0.124 $M_{\odot}$ and 3.02$\times 10^{50}$
erg, respectively. The specific angular momentum $j$ after the formation of a
disklike structure is about 5 $\times$ 1016 cm2 s-1 at the radius of 500 km.
The disk extends to about 1000 km from the center at the accretion phase.
After the strong jet formation, an expanding bow shock is generated at the
outer region of the disk and the matter residing in the region begins to
expand outward along the equatorial axis (see the description of Fig. 6 in
§3.2). However, even after the formation of the jet and the expanding bow
shock, the accretion continues at the inner edge of the disk and the accretion
rate maintains the value of 0.1 – 1 $M_{\odot}$ s-1 with a few factors
declined in 1 s. Therefore, our model can be regarded as a collapsar model. It
is noted that the jet obtained by the simulation is mildly relativistic
($\lesssim$ 0.1 $c$) and baryon rich ($\gtrsim$ 0.1 $M_{\odot}$). In contrast,
ultrarelativistic GRB jets should be baryon poor ($\sim$ 10-5 $M_{\odot}$
[78]). Moreover, the event rate of mildly relativistic jets could be larger
than those of normal GRB jets.[79] Therefore, both the ejected mass and event
rate suggest that the contribution to the chemical evolution of galaxies could
be large compared with that of ultrarelativistic ones.
### 3.2 Computational method of nucleosynthesis inside the jet
For calculating the nucleosynthesis inside the MHD jet, 20,000 tracer
particles are distributed over the computational domain between 1000 km and
$3\times 10^{4}$ km from the center, which covers initially the region from
around the iron core surface to the inner oxygen-rich layer. All the matter
initially located at radii smaller than 1000 km is absorbed into the inner
boundary. The Lagrange evolution of density and temperature of each tracer
particle can be obtained from the method described in Refs. ref:nagataki_1997
and ref:fujimoto_2007, by which we calculate the nucleosynthesis and follow
the change in composition. Figure 6 shows the distribution of the tracer
particles at the end of the simulation ($t_{f}$ = 1.504 s). The particles
initially located at the inner iron core, Si-rich layers, and oxygen-rich
layers are indicated in red, green, and blue, respectively. Some fractions of
the particles initially located at the inner iron core are ejected by the jet.
It should be noted that in Fig. 6, we can see a “blank” region in which there
are no particles in the equatorial regions. The blank region corresponds to
the expanding region after the jet formation as mentioned in §3.1. The density
of the equatorial blank region is $\rho\lesssim$ 105 g cm-3. In the tracer
particle method, the distributions of the particles tend to be sparse in
expanding and low-density regions. We can also see relatively sparse regions
in polar regions. The blank is just the problem of the tracer particle method
and the region does not affect the nucleosynthesis outcome in the ejecta.
Additionally, we have performed a convergence test of the nucleosynthesis
results inside the jet by changing the number of distributed particles and
confirmed that the results are not changed much by the difference in Paper I.
Particles that appear deep inside the original iron core go through high-
density and high-temperature regions; if the temperature is greater than 1010
K, the compositions are determined under nuclear statistical equilibrium (NSE)
condition: they are obtained from the values of ($\rho$, $T$, $Y_{e}$). Since
these particles suffer from electron captures, we need to calculate the change
in $Y_{e}$ of the ejected tracer particles due to the weak interactions of
$e^{\pm}$ captures and $\beta^{\pm}$ decays until the last stage of NSE before
the network calculation. The change in $Y_{e}$ is given by [81]
$\frac{dY_{e}}{dt}=\sum_{i}(\lambda_{+}-\lambda_{-})y_{i},$ (3)
where $\lambda_{+}$ represents the $\beta^{-}$ and positron capture rates and
$\lambda_{-}$ represents the $\beta^{+}$ and electron capture rates. Figure 6
shows the distribution of the ejected mass in $M_{\odot}$ against the electron
fraction of ejected particles at the end of NSE ($Y_{e,f}$). After the end of
NSE, the nucleosynthesis calculation is done along the Lagrange evolution of
each particle by using a large nuclear reaction network. Since the time of
hydrodynamical simulation is insufficient to follow the decays of radioactive
nuclei, the density and temperature profiles of particles are extrapolated
assuming adiabatic expansion as in Refs. ref:fujimoto_2007 and ref:ono_2009.
We continue the nucleosynthesis calculation of the radioactive decays until
$\sim$ 1010 yr after the explosion. Note that we neglect the feedback of
energy generations due to nuclear processes such as photodisintegrations in
the hydrodynamical calculation because our nucleosynthesis calculations are
just postprocessing. The effects of neutrino absorptions on the evolution of
$Y_{e}$ should be noted here because we neglect the effects in the
nucleosynthesis calculations, but it could be critical for the nucleosynthesis
outcome. The inner edge of the accretion disk of the R51 model has $\rho\sim$
109 g cm-3 and the estimated neutrino luminosity ranges from $\sim$ 1051 to
$\sim$ 1052 erg s-1, which is about one order of magnitude smaller than that
of canonical core-collapse supernovae. Therefore, the neutrino absorptions
could not be effective. Note that the range of the neutrino luminosity does
not differ much from that of other models in Paper I. Additionally, although
the progenitor and specified initial angular momentum and magnetic field
distributions differ from that of our model, recent nucleosynthesis
calculations in a magnetorotationally driven core-collapse supernova model
with an approximate neutrino transport scheme including effects of neutrino
absorptions on $Y_{e}$ have revealed [52] that the peak distribution of
$Y_{e}$ in the ejecta is shifted from $\sim$ 0.17 to $\sim$ 0.15 between with
and without neutrino absorptions, but the results of the nucleosynthesis are
not much affected by the difference. From the above considerations, the
effects of neutrino absorptions on $Y_{e}$ in our model would be small and we
neglect the effects.
Figure 5: Distribution of the tracer particles on the $XZ$-plane at the end of
the simulation ($t_{f}$ = 1.504 s). The particles initially located at the
inner iron core, Si-rich layers and oxygen-rich layers are indicated in red,
green, and blue, respectively.
Figure 6: Ejected masses against electron fraction at the end of NSE stage.
To investigate the heavy-element nucleosynthesis including the p\- and
r-processes, we calculate the nucleosynthesis along some Lagrange tracks of
the ejected particles explained above using the large nuclear reaction network
[81] including 4463 nuclei (up to 292Am), where the reaction rates are mainly
based on the REACLIB database [82, 83] and the adopted theoretical mass
formula is the extended Thomas-Fermi plus Strutinsky integral (ETFSI).[84] It
is noted that the reaction rates in the hydrostatic nucleosynthesis
calculations are based on the JINA REACLIB database, which use the theoretical
mass formula of the finite-range droplet model (FRDM).[85] Therefore, there
could be some inconsistency between the hydrostatic and explosive
nucleosynthesis calculations. However, taking into account the marked
uncertainty of the reaction rates far from the valley of the nuclear
stability, the differences are acceptable in the present purpose.
### 3.3 P\- and r-processes
To investigate the total yield of the ejecta, we combine the results of the
hydrostatic and explosive nucleosyntheses. We assume that all the unshocked
matter located at larger radii than the jet front in the star (including the
hydrogen envelope) of $\theta<$ 15∘ is successfully ejected by the jet,
keeping the precollapse compositions unchanged.
As shown in Fig. 6, the mass distribution of the ejecta tends to decrease as
the electron fraction $Y_{e,f}$ decreases. The lowest value of $Y_{e,f}$ is
0.192 among the ejected particles. We find that the low-$Y_{e,f}$ particles
are ejected from deep inside the disk, which are originally located from the
edge of the iron core to the inner Si-rich layer and fall into near the BH due
to gravitational collapse. The low-$Y_{e,f}$ particles strongly suffer from
electron captures, which reduces the electron fractions of the particles. The
distributions of compositions of the particles are initialized by the final
state in NSE. On the other hand, the particles that do not suffer from nuclear
burning are just pushed up by the inner jet at larger radii. Such particles
maintain the precollapse compositions. Therefore, the precollapse abundances
are crucial in part to determine the total compositions of the ejected matter.
The final overproduction factors $X(i)/X(i)_{\,\odot}$ averaged over the
ejecta against the mass number are shown in Fig. 7. Symbols connected by lines
indicate the isotopes. Note that neutron-rich elements of $45<A<55$ and
$60<A<160$ are highly overproduced relative to the solar values. Figure 8
shows the abundances of ejected particles that have different electron
fractions at the end of the NSE stage. The overproduced elements of
$140<A<200$ originate from the ejected matter, which has a lower $Y_{e,f}$ of
around 0.2 (Fig. 8), where the ejected materials undergo $r$-process
nucleosynthesis. On the other hand, the ejected particles of $Y_{e,f}\sim$ 0.3
produce elements of $60<A<90$. The overproduced elements of $A>90$ are
primarily synthesized in the jet except for 180Ta. The overproduction factors
have a peak at $A=195$ (the neutron magic number of 126). It should be noted
that our jet model cannot considerably produce the elements around the third
peak. In contrast, they are significantly produced in the study by Fujimoto et
al. [1, 2], which is attributed to the different distribution of the lowest
part of $Y_{e,f}$ of ejecta. In Fujimoto et al., the particles with
$Y_{e,f}\sim$ 0.1 are also ejected from their collapsar model of 40
$M_{\odot}$ (see e.g., Fig. 5 in Ref. ref:fujimoto_2008). Since the lowest
$Y_{e,f}$ is about 0.2 in our model, strong r-process and fissions do not
proceed. The difference in the distribution of $Y_{e,f}$ may be ascribed to
those of the progenitors and implemented initial distributions of the angular
momentum and the magnetic field. In particular, we include the initial
toroidal magnetic field, which may inhibit matter to fall deep inside the core
and suffer from strong electron captures.
S-elements of $60<A<90$ are overproduced significantly, which is due to the
weak s-process in the hydrostatic evolutional stage. In contrast, no
s-elements of $A>90$ are overproduced, which might be compensated by the
products of the main s-process in the relatively low mass AGB stars.
The overproduction factors of 35 p-elements are shown in Fig. 9. P-elements
whose overproduction factors are greater than 10 are 74Se, 78Kr, 84Sr, 92Mo,
180Ta, 180W, 184Os, 190Pt, and 196Hg. On the other hand, p-elements of 74Se,
78Kr, 84Sr, 180W, and 180Ta are overproduced in the hydrostatic
nucleosynthesis (see Fig. 3); those of 92Mo, 184Os, 190Pt, and 196Hg are
mainly overproduced in explosive nucleosynthesis. Underproduced p-elements in
the previous study of the p-process in Type II supernovae [15] such as 94Mo
and 96,98Ru are not produced in our explosion model. Rayet et al. [86, 15]
have investigated the p-process (gamma process) in parametrized supernova
explosion models. They concluded that the production of p-elements is very
sensitive to the maximum temperature reached during the explosion, and the
light ($N\lesssim 50$), intermediate ($50\lesssim N\lesssim 82$) and heavy
($N\gtrsim 82$) p-elements are produced in the peak temperature classified as
$T_{\rm 9,\,max}\gtrsim 3$, $3\gtrsim T_{\rm 9,\,max}\gtrsim 2.7$, $2.5\gtrsim
T_{\rm 9,\,max}$, respectively, where $T_{9}=T/\,(10^{9}\,{\rm K})$. The mass
distribution against the peak temperatures of the ejecta initially located in
the oxygen- and neon-rich layers is shown in Fig. 10. The major part of the
ejected matter has $T_{\rm 9,\,max}\lesssim 2.5$. Thereby, heavy p-elements
are considerably produced in the explosive nucleosynthesis (Fig. 9). It is
emphasized that light p-elements of 74Se, 78Kr, 84Sr, and 92Mo are primarily
synthesized in the ejecta of $T_{\rm 9,\,max}\sim 16$ and $Y_{e,f}\sim 0.48$
initially located in Si-rich layers. Since the peak temperature is very high,
protons, neutrons and alpha particles are significantly produced by
photodisintegrations of heavy nuclei, and the light p-elements are produced by
a sequence of neutron captures and subsequent proton captures as described in
Fujimoto et al. [1], the scenario of which was originally proposed by Howard
et al. [87]. Since light p-elements are already produced in the hydrostatic
nucleosynthesis, the increments due to the primary process in the explosive
nucleosynthesis are not prominent except for 92Mo. Note that in Fujimoto et
al. [1], intermediate p-elements such as 133In, 115Sn, and 138La are also
produced by fission only in the ejecta of $Y_{e,f}\sim 0.1$. In our model, all
the ejected matter has $Y_{e,f}\gtrsim 0.2$; therefore, fission reactions are
not effective and intermediate p-elements cannot be produced.
Figure 7: Same as Fig. 1 but for the ejecta with the use of the larger
network. Figure 8: Abundances of ejected particles with $Y_{e,f}$ that have
different electron fractions at the end of NSE stage. Figure 9: Same as Fig.
7 but 35 p-elements. Figure 10: Ejected mass originated from initial oxygen-
and neon-rich layers against the peak temperature $T_{\rm 9,\,max}$
($T_{9}=T/\,(10^{9}{\rm K})$).
## 4 Summary and discussion
We have investigated the nucleosynthesis in a massive star of 32 $M_{\odot}$
helium core with solar metallicity during the stellar evolution and the
jetlike supernova explosion. In this section, we summarize the results and
discuss the uncertainties of the production of elements and contribution to
the chemical evolution of galaxies (§4.1). We also give additional discussion
by comparing with observations in §4.2.
Hydrostatic nucleosynthesis: 1) S-elements of $60<A<90$ are highly
overproduced relative to the solar abundances, which is similar to the weak
s-process scenario proposed in previous studies. [59, 64, 65] 2) Although
photodisintegrations of seed s-elements during oxygen burning produce
p-elements, the produced elements are disintegrated in the later stages except
for 180Ta. 3) Three elements, Sr, Y, and Zr, are not much overproduced
compared with the solar values except for 96Zr.
Explosive nucleosynthesis: 4) Elements of $90<A<160$ are significantly
overproduced relative to the solar values. 5) The overproduced elements of
$140<A<200$ originate from the ejected matter with lower $Y_{e,f}$ around 0.2,
which results in the r-process. 6) The p-process produces mainly heavy
p-elements ($N>50$) because the peak temperatures in the oxygen- and neon-rich
layers are relatively low. 7) Light p-elements are produced as primary ones in
the ejected matter, which has a high peak temperature. 8) Compared with the
previous study of r-process nucleosynthesis calculations in a collapsar model
of 40 $M_{\odot}$ by Fujimoto et al. [1, 2], our jet model cannot considerably
produce both the elements around the third peak of the solar r-elements and
intermediate p-elements. This may be attributed to the differences in the
progenitor and the specified initial angular momentum and magnetic field
distributions.
After all, our supernova explosion model results in neutron-rich elements of
$70<A<140$ and weak s-elements of $60<A<90$. The origin of other underproduced
elements would be ascribed to different explosion mechanisms of supernovae.
Here, we try to deduce the qualitative constraint on the event rate for our
explosion model. Let us assume canonical supernova explosion to be spherical
and/or neutrino-driven aspherical ones such as in Refs. ref:rauscher_2002 and
ref:fujimoto_2011. The overproduction levels for the elements of $60<A<160$
are 1 – 2 orders of magnitude higher than that for those of $20<A<60$ compared
with canonical ones. If we neglect the mass loss, the total ejected mass of
our jetlike explosion model is around 2 $M_{\odot}$, which is about one order
of magnitude less than that of canonical ones. Unless the event rate of our
model is comparable to or one order of magnitude less than that of canonical
ones, the elements of $60<A<160$ are too produced to explain the solar system
abundance pattern. Therefore, the event rate of our model could be one order
of magnitude less than that of canonical ones. The origin of other
underproduced elements would be ascribed to different types of supernovae.
However, the constraint speculated here is not strict.
### 4.1 Uncertainties of production of elements related to the chemical
evolution of galaxies
We discuss the uncertainties concerning overproductions. To investigate the
weak s-process in the hydrostatic nucleosynthesis, we adopt $\beta$-decay
rates of TY87. However, the table of the rates for $\rho$ and $T$ only covers
the temperature and density ranges in the helium burning stage. As noted in
§2.2, we find that productions of neutron-rich elements of $85<A<100$, in
particular, Sr, Y, Z, and Mo, are very sensitive to the adopted $\beta$-decay
rates. Therefore, it is urgent to construct a table of the $\beta$-decay
rates, which covers all evolutional stages.
In the explosive nucleosynthesis calculation, we use the theoretical reaction
rates based on ETFSI mass formula for elements far from the valley of the
nuclear stability, in which no experimental cross sections are available. The
produced abundance pattern should depend on an adopted mass formula.[1]
We assume that all the matter that has larger radii than the jet front in the
star of $\theta<$ 15∘ is successfully ejected by the jet. However, the angle
from which the matter should be ejected is rather uncertain. Although we
neglect the effects of the mass loss, a star of 70 $M_{\odot}$ with solar
metallicity should suffer from a significant mass loss.[36] If the mass loss
is effective, almost all the hydrogen-rich envelope should be ejected as the
stellar wind. If the above uncertainties are taken into account, the
overproduction levels could become one order of magnitude less than those
described in the previous sections. However, the overall abundance patterns do
not change qualitatively except for the overproduction levels.
To investigate the possible contribution of our explosion model to the
chemical evolution of galaxies, we estimate overproduction factors averaged
over progenitor masses weighted on the basis of Salpeter’s stellar initial
mass function (IMF).[88] The IMF averaged overproduction factor of the element
$i$, $f_{\rm IMF}\,(i)$, is defined as
$f_{\rm IMF}\,(i)\equiv\frac{\displaystyle\int_{M_{1}}^{M_{2}}X\,(i,M)\,f_{\rm
ej}(M)\,M\,\phi\,(M)\,dM\,/\,X(i)_{\,\odot}}{\displaystyle\sum_{j}\int_{M_{1}}^{M_{2}}X\,(j,M)\,f_{\rm
ej}(M)\,M\,\phi\,(M)\,dM},$ (4)
where $X(i,M)$ is the mass fraction of the element $i$ in the ejecta, $M$ the
initial progenitor mass, $f_{\rm ej}$ the mass ratio of the ejecta to the
initial mass and $\phi\,(M)\propto M^{-2.35}$ the number of stars within the
mass range between $M$ and $M+dM$, which is Salpeter’s IMF. Let us adopt
spherical postexplosion models from Rauscher et al. [53] for progenitors of
15, 19, 20, 21, 25, 30, 35, and 40 $M_{\odot}$333We take the nucleosynthesis
data of postsupernova models from the web site:
http://homepages.spa.umn.edu/~alex/nucleosynthesis/RHHW02.shtml. Note that the
data of 30, 35, and 40 $M_{\odot}$ are not yet published, and necessary data
for the integration in Eq. (4) are obtained by interpolation. with the solar
metallicity and take our model for 70 $M_{\odot}$. During the integrations in
equation (4), necessary values of $X(i,M)$ and $f_{\rm ej}$ are obtained by
interpolation of sample values. We take $M_{1}$ to be 15 $M_{\odot}$ and
$M_{2}$ to be 70 $M_{\odot}$. For our 70 $M_{\odot}$ model, we assume all of
the hydrogen envelope to be ejected as the stellar wind, because 70
$M_{\odot}$ with the solar metallicity may strongly suffer from the mass
loss.[36] Figure 11 shows the IMF averaged overproduction factors. We also
show the overproduction factors averaged only from 15 $M_{\odot}$ to 40
$M_{\odot}$ in Fig. 12 for reference. We can see relatively high
overproduction of 40K relative to the solar value, which arises from the large
enhancement of the star of 20 $M_{\odot}$ (see Fig. 4 in Ref.
ref:rauscher_2002). We also recognize relatively larger overproduction factors
for elements of $60<A<90$. The overproduction level of $60<A<90$ is slightly
enhanced by a few factors due to the inclusion of our 70 $M_{\odot}$ model
(Fig. 11) compared with that averaged over the range only between 15
$M_{\odot}$ and 40 $M_{\odot}$ (Fig. 12). Recall that the production of
elements of $60<A<90$ results from the weak s-process. Overproduced neutron-
rich elements of $90<A<160$ are attributed to the 70 $M_{\odot}$ model in
which the elements are primarily synthesized in the explosive nucleosynthesis
by the r-process. P-elements of $110<A<200$ are overproduced to the solar
values, which mainly come from the p-process in 15 – 40 $M_{\odot}$ stars.
180Ta is highly overproduced in our 70 $M_{\odot}$ model. Therefore, it is
interesting whether our model could contribute to the solar 180Ta abundance.
If we average the overproduction factors over the mass range only between 15
$M_{\odot}$ and 40 $M_{\odot}$ (Fig. 12), the IMF averaged overproduction
factor of 180Ta is 1.18 in logarithmic scale. On the other hand, if we include
the 70 $M_{\odot}$ model in the integration in Eq. (4), the overproduction
factor becomes 1.23, which corresponds to a 12% increase. Therefore, the
contribution of 40 – 70 $M_{\odot}$ to the solar 180Ta abundance is negligible
in view of uncertainties of the present study. Overall, our model contributes
to the solar weak s-elements of $70<A<90$ and neutron-rich elements of
$90<A<160$. However, we should treat the results with caution because our
jetlike explosion model is only one specific set of parameters of angular
momentum and magnetic field distributions, and the fraction of such an
aspherical explosion is highly uncertain. As suggested by Fujimoto et al., [2]
the r-process does not occur in a less energetic jet and the jet properties
depend on specified parameters of the initial angular momentum and magnetic
field distributions. If the fraction of jetlike explosions among the
progenitors is less than 10%, the contribution of jet-induced nucleosynthesis
above 40 $M_{\odot}$ to the chemical evolution of galaxies would be minor.
Therefore, we should regard the contributions of our model deduced here as the
upper limits. In addition, we investigate only the progenitors with solar
metallicity. As a consequence, some simulations of the chemical evolution of
galaxies are required to ascertain the contribution especially for elements
whose productions depend on the metallicity, which is beyond the scope of this
paper.
Figure 11: Same as Fig. 1 but for averaged over progenitor masses weighed on
the basis of Salpeter’s IMF with the use of spherical explosion models from
Rauscher et al. for 15, 19, 20, 21, 25, 30, 35, and 40 $M_{\odot}$ progenitors
and our explosion model for 70 $M_{\odot}$. Figure 12: Same as Fig. 11 but
for only from 15 to 40 $M_{\odot}$.
### 4.2 Comparison with abundances in metal-poor stars and the possibility
for LEPP
Although our progenitor is assumed to have the solar metallicity, the
production of r-elements, which are primarily synthesized without seeds, does
not depend on the metallicity. Therefore, it is worthwhile to compare our
results with the abundances of extremely metal-poor stars, which are not
affected seriously by the s-process. It is noted that the overproduced
elements of $Z>38$ are primarily produced in the explosive nucleosynthesis.
Abundance ratios relative to Sr against the atomic number are shown in Fig.
13. Solid and dashed lines denote abundance ratios of our model and that of
solar system r-process elements, [89] respectively. Circles, triangles, and
squares indicate the values of observations of very metal-poor stars
CS22892-052 [17], HD88609 [90] and HD122563 [90], respectively. The symbols
without error bars indicate the upper limits. We can see that the abundance
pattern of CS22892-052, which is a typical r-process-rich ([Eu/Fe] $\gtrsim
1$) star, coincides well with that of the solar r-element pattern, although
some exceptions are recognized. The r-process-poor stars ([Eu/Fe] $\lesssim
1$), HD88609 and HD122563, have a clearly decreasing trend as the atomic
number increases. It is noted that although [Eu/Fe] values of r-process-poor
stars are low compared with those of the r-process-rich ones, abundances in
HD88609 ([Fe/H] $\sim-3.0$) and HD122563 ([Fe/H] $\sim-2.7$) should come from
the weak r-process because the sources of the s-process, AGB stars, have not
had sufficient time to evolve before the formation of such metal-poor halo
stars [90]. The abundances of the ejecta of our model show a decreasing trend
in proportion to the decrease in the atomic number, which is similar to the
case of HD88609 and HD122563.[16] This result can be attributed to the
decrease in the ejected masses at lower values of $Y_{e,f}$ (Fig. 6).
The abundance ratios relative to the solar values are summarized in Table 2,
where the observational values are taken from Sneden et al. [17] for
CS22892-052 and Honda et al. [90] for HD88609 and HD122563. The values of
[Sr/Fe], [Y/Fe] and [Zr/Fe] are similarly observed and the ejecta also has the
same tendency. While [Sr/Eu] of the r-process-rich star CS22892-052 ([Sr/Eu]
$\sim-1$) is very small compared with that of the r-process-poor stars HD88609
and HD122563 ([Sr/Eu] $\sim+0.3$), the value of the ejecta ([Sr/Eu]
$\sim-0.4$) is closer to that of the r-process-poor stars than to that of the
r-process-rich stars, which reflects the declining trend of abundances as the
atomic number increases (Fig. 13).
We find that Sr-Y-Zr isotopes are primarily synthesized in the explosive
nucleosynthesis by a similar process of primary synthesis of light p-elements
as described in §3.3. The ejected matter of $T_{\rm 9,\,max}\sim 16$ and
$Y_{e,f}\sim 0.45$ produces most isotopes of Sr-Y-Zr, which is more neutron
rich than that in the case of primary light p-element synthesis. In such high
peak temperature and density, there exist protons, neutrons, and alpha
particles, where a lot of neutrons are produced by electron captures. After
the temperature decreases to 8$\times$109 K, a sequence of neutron captures
and $\beta^{-}$-decays produces slightly neutron-rich Sr-Y-Zr isotopes from
lighter elements. After the exhaustion of neutrons, proton captures and gamma
processes follow. Recall that 96Zr is overproduced relative to the solar value
in the hydrostatic nucleosynthesis (Fig. 1). 88Sr, 89Y, and 91, 92, 94, 96Zr
are highly overproduced relative to the solar values due to the primary
process in the explosive nucleosynthesis. It is noted that 96Zr is more
produced in the explosive nucleosynthesis.
Travaglio et al. [18] have suggested that based on a galactic chemical
evolution (GCE) model, a primary process from massive stars (LEPP) other than
the general s\- and r-processes is needed to explain 8% of the solar abundance
for Sr and 18% of the solar Y and Zr abundances. In their GCE model, the
yields of the s-process have been derived from AGB models and they have also
added a small contribution ($\sim 10$% of the solar ones) from the weak
s-component for Sr. It is emphasized that the contribution from the r-process
has been deduced from the very r-process-rich CS22892-052 [17], that is,
contributions to r-elements from r-process-poor stars like our explosion model
have not been included. In our explosion model, the ejecta has a larger
[Sr/Eu] than that of r-process-rich stars, and Sr, Y, and Zr are mainly
produced by the primary process. Therefore, our explosion model could be one
of the sites of LEPP. However, the calculations are limited to only one model
of the progenitor with the solar metallicity and specific set of parameters of
initial distribution of magnetic field and angular momentum. As suggested in
Ref. ref:fujimoto_2008, ejected masses of r-elements depend on the jet
properties such as the explosion energies. Therefore, the effects of explosion
models on the chemical evolution of galaxies remain uncertain and should be
studied in the future.
Figure 13: Abundance ratios relative to Sr: calculated ejecta (solid line), solar system r-process elements (dashed line) (Simmerer et al. [89]) and three very metal-poor stars.[90, 17] Symbols with error bars indicate the observations of their abundances. Triangles, squares and circles represent HD88609, HD122563, and CS22892-052, respectively. Symbols without error bar indicate the upper limits. Table 2: Abundance ratios relative to the solar values for extremely metal-poor stars and our model indicated by “Ejecta”. | [Fe/H] | [Sr/Fe] | [Y/Fe] | [Zr/Fe] | [Eu/Fe] | [Sr/Eu]
---|---|---|---|---|---|---
CS22892-052 | $-3.1$ | $+0.6$ | $+0.44$ | $+0.78$ | $+1.64$ | $-1.04$
HD88609 | $-3.0$ | $-0.05$ | $-0.12$ | $+0.24$ | $-0.33$ | $+0.28$
HD122563 | $-2.7$ | $-0.27$ | $-0.37$ | $-0.10$ | $-0.52$ | $+0.25$
Ejecta | – | $+1.50$ | $+1.79$ | $+1.65$ | $+1.90$ | $-0.4$
## Acknowledgements
We would like to thank A. Heger and his collaborators for offering their data.
M. Ono thanks N. Nishimura for stimulating discussion. K. Kotake is grateful
to K. Sato for continuous encouragement. This work has been supported in part
by Grants-in-Aid for Scientific Research (Nos. 18540279, 19104006, 20740150,
22540297 and 24540278) from the Ministry of Education, Culture, Sports,
Science and Technology of Japan.
## Appendix A Thermal Population of Ground and Isomeric States of
${}^{180}{\rm Ta}$
180Ta is one of the rarest isotopes in the solar system and it has a long-
lived isomeric state. The isomeric state has $J^{\pi}=9^{-}$ and the half-life
$\tau_{1/2}$ is 1.2$\times 10^{15}$ yr, while the ground state has
$J^{\pi}=1^{+}$ and $\tau_{1/2}\simeq 8.152$ h. We treat specially the
reaction rates concerning 180Ta by a method similar to that described in Ref.
ref:rauscher_2002.
Because of the selection rule for the spin and parity, the isomeric state
(180mTa) cannot directly decay into the grand state (180gTa). However, if the
temperature is sufficiently high, 180mTa can decay into the ground state
through thermally excited states. In stellar interiors during some burning
stages and in supernova explosions, the two states of 180Ta are thermally
populated. In thermal equilibrium, the population ratio $P_{\rm iso}$ of the
isomer relative to the ground state is given as [53]
$P_{\rm iso}=\frac{(2J_{\rm iso}+1)\exp(-E_{\rm iso}/kT)}{(2J_{\rm
gs}+1)}=\frac{19}{3}e^{-0.8738/T_{9}},$ (5)
where $J_{\rm gs}$ and $J_{\rm iso}$ are the spins of the ground and isomeric
states, respectively, and $E_{\rm iso}$ the excitation energy of the isomer,
$T_{9}=T/10^{9}$ K. If the temperature decreases to a critical temperature
$T_{\rm crit}$, the two states no longer interact with each other. In the
explosive burning scenario, the critical temperature is crucial to determine
the amount of 180mTa. Belic et al. [91] have derived the effective decay rate
of 180mTa by photoactivation experiments. We simply assume $T_{\rm crit}$ as
0.35$\times$109 K from the temperature-dependent decay rate (Fig. 4 in Ref.
ref:belic_1999). The effective 180Ta rates of neutron-induced reaction and
$\beta$-decays are given by
$\lambda_{\rm eff}=f_{\rm gs}\lambda_{\rm gs}+f_{\rm iso}\lambda_{\rm iso},$
(6)
where $f_{\rm gs}$ and $f_{\rm iso}$ are the fractions of the ground and
isomeric states, respectively. For this effective decay rate, we derive
$\lambda_{\rm gs}$ from the half-life of the ground state and $\lambda_{\rm
iso}$ from Belic et al. [91] For the neutron capture of 180Ta, $\lambda_{\rm
gs}$ is taken from Ref. ref:rauscher_2002 and $\lambda_{\rm iso}$ is taken
from the JINA REACLIB database.[60] If $T<T_{\rm crit}$, we make all the
180gTa decayed by hand and set $f_{\rm gs}$ to be 0.
## References
* [1] S. Fujimoto, M. Hashimoto, K. Kotake and S. Yamada, 656,2007,382.
* [2] S. Fujimoto, N. Nishimura and M. Hashimoto, 680,2008,1350.
* [3] F.-K. Thielemann, A. Arcones, R. Käppeli, M. Liebendörfer, T. Rauscher, C. Winteler, C. Fröhlich, I. Dillmann, T. Fischer, G. Martinez-Pinedo, K. Langanke, K. Farouqi, K.-L. Kratz, I. Panov and I. K. Korneev, Prog. Part. Nucl. Phys.,66,2011,346.
* [4] E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys.,29,1957,547.
* [5] R. D. Hoffman, S. E. Woosley and Y.-Z. Qian, 482,1997,951.
* [6] K. Otsuki, H. Tagoshi, T. Kajino and S. Wanajo, 533,2000,424.
* [7] S. Wanajo, 666,2007,L77.
* [8] T. Kuroda, S. Wanajo and K. Nomoto, 672,2008,1068.
* [9] T. Fischer, S. C. Whitehouse, A. Mezzacappa, F.-K. Thielemann and M. Liebendörfer, Astron. Astrophys.,517,2010,A80.
* [10] B. D. Metzger, G. Martínez-Pinedo, S. Darbha, E. Quataert, A. Arcones, D. Kasen, R. Thomas, P. Nugent, I. V. Panov and N. T. Zinner, Mon. Not. R. Astron. Soc.,406,2010,2650.
* [11] L. F. Roberts, D. Kasen, W. H. Lee and E. Ramirez-Ruiz, 736,2011,L21.
* [12] S. Goriely, A. Bauswein and H.-T. Janka, 738,2011,L32.
* [13] S. Wanajo and H.-T. Janka, 746,2012,180.
* [14] F. Käppeler, R. Gallino, S. Bisterzo and W. Aoki, Rev. Mod. Phys.,83,2011,157.
* [15] M. Rayet, M. Arnould, M. Hashimoto, N. Prantzos and K. Nomoto, Astron. Astrophys.,298,1995,517.
* [16] I. U. Roederer, J. J. Cowan, A. I. Karakas, K.-L. Kratz, M. Lugaro, J. Simmerer, K. Farouqi and C. Sneden, 724,2010,975.
* [17] C. Sneden, J. J. Cowan, J. E. Lawler, I. I. Ivans, S. Burles, T. C. Beers, F. Primas, V. Hill, J. W. Truran, G. M. Fuller, B. Pfeiffer and K.-L. Kratz, 591,2003,936.
* [18] C. Travaglio, R. Gallino, E. Arnone, J. Cowan, F. Jordan and C. Sneden, 601,2004,864.
* [19] L. Wang, J. C. Wheeler, P. Höflich, A. Khokhlov, D. Baade, D. Branch, P. Challis, A. V. Filippenko, C. Fransson, P. Garnavich, R. P. Kirshner, P. Lundqvist, R. McCray, N. Panagia, C. S. J. Pun, M. M. Phillips, G. Sonneborn and N. B. Suntzeff, 579,2002,671.
* [20] M. Tanaka, K. Maeda, P. A. Mazzali and K. Nomoto, 668,2007,L19.
* [21] S. Woosley and T. Janka, Nat. Phys.,1,2005,147.
* [22] K. Kotake, K. Sato and K. Takahashi, Rep. Prog. Phys.,69,2006,971.
* [23] K. Kotake, arXiv:1110.5107.
* [24] K. Kotake, T. Takiwaki, Y. Suwa, W. Iwakami Nakano, S. Kawagoe, Y. Masada and S. Fujimoto, arXiv:1204.2330.
* [25] K. Kotake, K. Sumiyoshi, S. Yamada, T. Takiwaki, T. Kuroda, Y. Suwa and H. Nagakura, arXiv:1205.6284.
* [26] A. Marek and H.-T. Janka, 694,2009,664.
* [27] Y. Suwa, K. Kotake, T. Takiwaki, S. C. Whitehouse, M. Liebendörfer and K. Sato, Publ. Astron. Soc. Jpn.,62,2010,L49.
* [28] T. Takiwaki, K. Kotake and Y. Suwa, arXiv:1108.3989.
* [29] T. Kuroda, K. Kotake and T. Takiwaki, arXiv:1202.2487.
* [30] T. Fischer, I. Sagert, G. Pagliara, M. Hempel, J. Schaffner-Bielich, T. Rauscher, F.-K. Thielemann, R. Käppeli, G. Martínez-Pinedo and M. Liebendörfer, Astrophys. J. Suppl.,194,2011,39.
* [31] K. Kotake, H. Sawai, S. Yamada and K. Sato, 608,2004,391.
* [32] H. Sawai, K. Kotake and S. Yamada, 631,2005,446.
* [33] M. Shibata, Y. T. Liu, S. L. Shapiro and B. C. Stephens, 74,2006,104026.
* [34] Y. Suwa, T. Takiwaki, K. Kotake and K. Sato, Publ. Astron. Soc. Jpn.,59,2007,771.
* [35] T. Takiwaki, K. Kotake and K. Sato, 691,2009,1360.
* [36] A. Heger, C. L. Fryer, S. E. Woosley, N. Langer and D. H. Hartmann, 591,2003,288.
* [37] A. I. MacFadyen and S. E. Woosley, 524,1999,262.
* [38] S. Harikae, K. Kotake, T. Takiwaki and Y. Sekiguchi, 720,2010,614.
* [39] I. Zalamea and A. M. Beloborodov, Mon. Not. R. Astron. Soc.,410,2011,2302.
* [40] S. E. Woosley, 405,1993,273.
* [41] S. Koide, K. Shibata, T. Kudoh and D. L. Meier, Science,295,2002,1688.
* [42] D. Proga, A. I. MacFadyen, P. J. Armitage and M. C. Begelman, 599,2003,L5.
* [43] Y. Mizuno, S. Yamada, S. Koide and K. Shibata, 615,2004,389.
* [44] S. Fujimoto, K. Kotake, S. Yamada, M. Hashimoto and K. Sato, 644,2006,1040.
* [45] S. Nagataki, R. Takahashi, A. Mizuta and T. Takiwaki, 659,2007,512.
* [46] S. Harikae, T. Takiwaki and K. Kotake, 704,2009,354.
* [47] S. Nagataki, 704,2009,937.
* [48] S. Nagataki, Publ. Astron. Soc. Jpn.,163,2011,1243.
* [49] R. D. Blandford and R. Znajek, Mon. Not. R. Astron. Soc.,179,1977,433.
* [50] S. Nagataki, A. Mizuta, S. Yamada, H. Takabe and K. Sato, 596,2003,401.
* [51] S. Nagataki, A. Mizuta and K. Sato, 647,2006,1255.
* [52] C. Winteler, R. Käppeli, A. Perego, A. Arcones, N. Vasset, N. Nishimura, M. Liebendörfer and F.-K. Thielemann, 750,2012,L22 .
* [53] T. Rauscher, A. Heger, R. D. Hoffman and S. E. Woosley, 576,2002,323.
* [54] S. E. Woosley and T. A. Weaver, Astrophys. J. Suppl.,101,1995,181.
* [55] S. Fujimoto, K. Kotake, M. Hashimoto, M. Ono and N. Ohnishi, 738,2011,61.
* [56] M. Ono, M. Hashimoto, S. Fujimoto, K. Kotake and S. Yamada, 122,2009,755.
* [57] M. Hashimoto, 94,1995,663.
* [58] J. Puls, J. S. Vink and F. Najarro, Astron. Astrophys. Rev.,16,2008,209.
* [59] N. Prantzos, M. Arnould and J.-P. Arcoragi, 315,1987,209.
* [60] R. H. Cyburt, A. M. Amthor, R. Ferguson, Z. Meisel, K. Smith, S. Warren, A. Heger, R. D. Hoffman, T. Rauscher, A. Sakharuk, H. Schatz, F. K. Thielemann and M. Wiescher, Astrophys. J. Suppl.,189,2010,240.
* [61] I. Dillmann, M. Heil, F. Käppeler, R. Plag, T. Rauscher and F.-K. Thielemann, AIP Conf. Ser., Capture Gamma-Ray Spectroscopy and Related Topics, ed. A. Woehr & A. Aprahamian,819,2006,123
* [62] K. Takahashi and K. Yokoi, At. Data Nucl. Data Tables,36,1987,375.
* [63] E. Anders and N. Grevesse, Geochim. Cosmochim. Acta,53,1989,197.
* [64] N. Langer, J.-P. Arcoragi and M. Arnould, Astron. Astrophys.,210,1989,187.
* [65] M. Rayet and M. Hashimoto, Astron. Astrophys.,354,2000,740.
* [66] M. Arnould, Astron. Astrophys.,46,1976,117.
* [67] K. Nomoto and M. Hashimoto, 163,1988,13.
* [68] M. Hashimoto, K. Nomoto and T. Shigeyama, Astron. Astrophys.,210,1989,L5.
* [69] J. M. Stone and M. L. Norman, Astrophys. J. Suppl.,80,1992,753.
* [70] J. M. Stone and M. L. Norman, Astrophys. J. Suppl.,80,1992,791.
* [71] K. Kotake, S. Yamada and K. Sato, 595,2003,304.
* [72] H. Shen, H. Toki, K. Oyamatsu and K. Sumiyoshi, 637,1998,435.
* [73] N. Yasutake, K. Kotake, M. Hashimoto and S. Yamada, 75,2007,084012.
* [74] B. Paczyńsky and P. J. Wiita, Astron. Astrophys.,88,1980,23.
* [75] A. Heger, S. E. Woosley and H. C. Spruit, 626,2005,350.
* [76] S. A. Balbus and J. F. Hawley, Rev. Mod. Phys.,70,1998,1.
* [77] M. Obergaulinger, P. Cerdá-Durán, E. Müller and M. A. Aloy, Astron. Astrophys.,498,2009,241.
* [78] T. Piran, Rev. Mod. Phys.,76,2005,1143.
* [79] J. Granot and E. Ramirez-Ruiz, 609,2004,L9.
* [80] S. Nagataki, M. Hashimoto, K. Sato and S. Yamada, 486,1997,1026.
* [81] S. Nishimura, K. Kotake, M. Hashimoto, S. Yamada, N. Nishimura, S. Fujimoto and K. Sato, 642,2006,410.
* [82] T. Rauscher and F.-K. Thielemann, At. Data Nucl. Data Tables,75,2000,1
* [83] T. Rauscher and F.-K. Thielemann, At. Data Nucl. Data Tables,79,2001,47
* [84] S. Goriely, F. Tondeur and J. M. Pearson, At. Data Nucl. Data Tables,77,2001,311.
* [85] P. Möller, J. R. Nix, W. D. Myers and W. J. Swiatecki, At. Data Nucl. Data Tables,59,1995,185.
* [86] M. Rayet, M. Arnould and N. Prantzos, Astron. Astrophys.,227,1990,271.
* [87] W. M. Howard, B. S. Meyer and S. E. Woosely, 373,1991,L5.
* [88] E. E. Salpeter, 121,1955,161.
* [89] J. Simmerer, C. Sneden, J. J. Cowan, J. Collier, V. M. Woolf and J. E. Lawler, 617,2004,1091.
* [90] S. Honda, W. Aoki, Y. Ishimaru and S. Wanajo, 666,2007,1189.
* [91] D. Belic, C. Arlandini, J. Besserer, J. de Boer, J. J. Carroll, J. Enders, T. Hartmann, F. Käppeler, H. Kaiser, U. Kneissl, M. Loewe, H. J. Maier, H. Maser, P. Mohr, P. von Neumann-Cosel, A. Nord, H. H. Pitz, A. Richter, M. Schumann, S. Volz and A. Zilges, 83,1999,5242.
|
arxiv-papers
| 2012-03-29T11:20:41 |
2024-09-04T02:49:29.164853
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Masaomi Ono, Masa-aki Hashimoto, Shin-ichiro Fujimoto, Kei Kotake,\n Shoichi Yamada",
"submitter": "Masaomi Ono",
"url": "https://arxiv.org/abs/1203.6488"
}
|
1203.6527
|
# Existence and stability of stationary solutions to the full compressible
Navier-Stokes-Korteweg system
Zhengzheng Chen
School of Mathematics and Statistics
Wuhan University, Wuhan 430072, China
Huijiang Zhao
School of Mathematics and Statistics,
Wuhan University, Wuhan 430072, China Corresponding author.E-mail:
hhjjzhao@hotmail.com
###### Abstract
This paper is concerned with the existence, uniqueness and nonlinear stability
of stationary solutions to the Cauchy problem of the full compressible Navier-
Stokes-Korteweg system effected by external force of general form in
$\mathbb{R}^{3}$. Based on the weighted-$L^{2}$ method and some elaborate
$L^{\infty}$ estimates of solutions to the linearized problem, the existence
and uniqueness of stationary solution are obtained by the contraction mapping
principle. The proof of the stability result is given by an elementary energy
method and relies on some intrinsic properties of the full compressible
Navier-Stokes-Korteweg system.
Keywords Navier-Stokes-Korteweg system; Stationary solution; Nonlinear
stability;
AMS Subject Classifications: 35M10, 35Q35
## 1 Introduction
In this paper, we are interested in the following nonisothermal compressible
fluid models of Korteweg type, which can be derived from a Cahn-Hilliard like
free energy( see the pioneering work by Dunn and Serrin [2], and also [3, 4,
5]).
$\displaystyle\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho
v)=G(x),\\\\[5.69054pt] (\rho v)_{t}+\nabla\cdot(\rho v\bigotimes
v)=\nabla\cdot(S+K)+\rho F(x),\\\\[5.69054pt]
\displaystyle\left[\rho\left(e+\frac{v^{2}}{2}\right)\right]_{t}+\nabla\cdot\left[\rho
v\left(e+\frac{v^{2}}{2}\right)\right]=\nabla\cdot(\tilde{\alpha}\nabla\theta)+\nabla\cdot\left((S+K)\cdot
v\right)+\rho v\cdot F(x)+H(x),\end{array}\right.$ (1.4)
Here $(x,t)\in\mathbb{R}^{3}\times\mathbb{R}^{+}$,
$\rho>0,v=(v_{1},v_{2},v_{3}),\theta>0$ and $e$ denote the density, the
velocity, the internal energy and the temperature of the fluids respectively.
$\tilde{\alpha}$ is the heat conduction coefficient.
$F(x)=(F_{1}(x),F_{2}(x),F_{3}(x)),G(x),H(x)$ are the given external force,
mass source and energy source, respectively. The viscous stress tensor $S$ and
the Korteweg stress tensor $K$ are given by
$\displaystyle\left\\{\begin{array}[]{ll}S_{i,j}=(\mu^{\prime}\nabla\cdot
v-P(\rho,e))\delta_{ij}+2\mu d_{ij}(v)\\\\[5.69054pt]
K_{i,j}=\displaystyle\frac{\kappa}{2}(\Delta\rho^{2}-|\nabla\rho|^{2})\delta_{ij}-\kappa\partial_{i}\rho\partial_{j}\rho,\end{array}\right.$
(1.7)
where $d_{ij}(v)=(\partial_{i}v_{j}+\partial_{j}v_{i})/2$ is the strain
tensor, $P$ is the pressure, $\mu$ and $\mu^{\prime}$ are the viscosity
coefficients, and $\kappa$ is the capillary coefficient. Notice that when
$\kappa=0$, system (1.4) is reduced to the compressible Navier-Stokes system.
In this paper, we consider the case of $e=C_{\triangledown}\theta$, where
$C_{\triangledown}$ is the heat capacity at the constant volume. Our basic
assumptions are as follows: $\bar{\rho},\bar{\theta},\kappa,\mu,\mu^{\prime}$
and $\tilde{\alpha}$ are the constants satisfying
$\bar{\rho},\bar{\theta},\kappa,\mu,\tilde{\alpha}>0$ and
$\frac{2}{3}\mu+\mu^{\prime}\geq 0$; $C_{\triangledown}>0$ is a constant and
$P=P(\rho,\theta)>0$ is a smooth function of $\rho,\theta>0$ satisfying
$P_{\rho}(\rho,\theta),P_{\theta}(\rho,\theta)>0$.
The main purpose of this manuscript is to study the nonlinear stability of
stationary solutions to the Cauchy problem of the compressible Navier-Stokes-
Korteweg system (1.1). It is convenient to study the Cauchy problem for the
following form which is equivalent to (1.4) for classical solutions,
$\displaystyle\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho
v)=G(x),\\\\[5.69054pt] \displaystyle\rho
v_{t}+\rho(v\cdot\nabla)v=\displaystyle\mu\Delta
v+(\mu+\mu^{\prime})\nabla(\nabla\cdot v)-\nabla
P(\rho,\theta)+\kappa\rho\nabla\Delta\rho+\rho F(x)-vG(x),\\\\[5.69054pt]
\displaystyle\rho
C_{\triangledown}\left(\theta_{t}+(v\cdot\nabla)\theta\right)+\theta
P_{\theta}(\rho,\theta)\nabla\cdot
v=\tilde{\alpha}\Delta\theta+\Psi(v)+\Phi(\rho,v)+H(x)+\frac{v^{2}}{2}G(x)-C_{\triangledown}G(x)\theta,\end{array}\right.$
(1.11)
with the initial date
$(\rho,v,\theta)(t,x)|_{t=0}=(\rho_{0},v_{0},\theta_{0})(x)\rightarrow(\bar{\rho},0,\bar{\theta})\quad
as\,|x|\rightarrow+\infty.$ (1.12)
Here
$\displaystyle\left\\{\begin{array}[]{ll}\Psi(v)=\mu^{\prime}(\nabla\cdot
v)^{2}+2\mu\mathbb{D}v:\mathbb{D}v,\,\,\mathbb{D}v=(d_{ij}(v))_{i,j=1}^{3},\\\\[5.69054pt]
\Phi(\rho,v)=\kappa\left(\frac{|\nabla\rho|^{2}}{2}+\rho\Delta\rho\right)\nabla\cdot
v-\kappa(\nabla\rho\bigotimes\nabla\rho):\nabla v\end{array}\right.$ (1.15)
The stationary problem corresponding to the initial value problem (1.11),
(1.12) is
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot(\rho
v)=G(x),\\\\[5.69054pt]
\displaystyle(v\cdot\nabla)v=\displaystyle\frac{1}{\rho}\left\\{\mu\Delta
v+(\mu+\mu^{\prime})\nabla(\nabla\cdot v)-\nabla
P(\rho,\theta)\right\\}+\kappa\nabla\Delta\rho+F(x)-\frac{v}{\rho}G(x),\\\\[8.53581pt]
\displaystyle(v\cdot\nabla)\theta+\frac{\theta P_{\theta}(\rho,\theta)}{\rho
C_{\triangledown}}\nabla\cdot v=\frac{1}{\rho
C_{\triangledown}}\left\\{\tilde{\alpha}\Delta\theta+\Psi(v)+\Phi(\rho,v)+H(x)+\frac{v^{2}}{2}G(x)-C_{\triangledown}G(x)\theta\right\\},\end{array}\right.$
(1.19)
Before stating our main results, we explain some notations as follows, which
are borrowed from [6] and [7].
Notations: Throughout this paper, we use the standard notation in vector
analysis. For example, we put for scalar $u$, vectors
$v=(v_{1},v_{2},v_{3}),w=(w_{1},w_{2},w_{3})$ and matrix $f=(f_{ij})_{1\leq
i,j\leq 3}$.
$\Delta u=\sum_{i=1}^{3}\frac{\partial^{2}u}{\partial x_{i}^{2}},\quad\Delta
v=(\Delta v_{1},\Delta v_{2},\Delta
v_{3}),\quad(v\cdot\nabla)u=\sum_{i=1}^{3}v_{i}\frac{\partial u}{\partial
x_{i}},$
$(v\cdot\nabla)w=\left((v\cdot\nabla)w_{1},(v\cdot\nabla)w_{2},(v\cdot\nabla)w_{3}\right),$
$\displaystyle\nabla^{k}u=\left\\{\partial_{x}^{\alpha}u||\alpha|=k\right\\},\quad\nabla^{k}v=\left\\{\partial_{x}^{\alpha}v_{i}||\alpha|=k,i=1,2,3\right\\},$
$\nabla\cdot v=\sum_{i=1}^{3}\frac{\partial v_{i}}{\partial
x_{i}},\quad\nabla\cdot f=\left(\sum_{j=1}^{3}\frac{\partial f_{1j}}{\partial
x_{j}},\sum_{j=1}^{3}\frac{\partial f_{2j}}{\partial
x_{j}},\sum_{j=1}^{3}\frac{\partial f_{3j}}{\partial x_{j}}\right)$
Here $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ is a multi-index,
$|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}$ and
$\partial_{x}^{\alpha}=\partial^{|\alpha|}/\partial x_{1}^{\alpha_{1}}\partial
x_{2}^{\alpha_{2}}\partial x_{3}^{\alpha_{3}}$. Moreover, we also use the
notations
$\nabla u=\left(\frac{{\partial u}}{\partial x_{1}},\frac{{\partial
u}}{\partial x_{2}},\frac{{\partial u}}{\partial x_{3}}\right),\quad\nabla
u=\left(\frac{\partial^{2}u}{\partial x_{j}\partial x_{i}}\right)_{1\leq
i,\,j\leq 3},$
and denote $\frac{\partial u}{\partial x_{i}}$ by $\partial_{i}u$ or
$u_{x_{i}}$ without any confusion.
Next, we introduce some function spaces. Let $L^{p}$ denote the usual $L^{p}$
space, put for scalars $u_{1},u_{2}$ and vectors
$v=(v_{1},v_{2},\cdots,v_{n})$, $w=(w_{1},w_{2},\cdots,w_{n})$,
$\|u_{1}\|_{L^{p}}=\left(\int_{\mathbb{R}^{3}}|u_{1}(x)|^{p}dx\right)^{\frac{1}{p}},\quad\|v\|_{L^{p}}=\left(\sum_{i=1}^{n}\|v_{i}\|^{p}_{L^{p}}\right)^{\frac{1}{p}},\quad(1\leq
p<\infty),$
$\|u_{1}\|_{L^{\infty}}=\sup_{\mathbb{R}^{3}}|u_{1}(x)|,\quad\|v\|_{L^{\infty}}=\max_{1\leq
i\leq n}\|v_{i}(x)\|_{L^{\infty}},\quad\langle
u_{1},u_{2}\rangle=\int_{\mathbb{R}^{3}}u_{1}u_{2}dx,$ $\langle
v,w\rangle=\sum_{i=1}^{n}\langle
v_{i},w_{i}\rangle,\quad\|v\|_{k}=\left(\sum_{0\leq l\leq
k}\|\nabla^{l}v\|^{2}\right)^{\frac{1}{2}}\,with\,\|\cdot\|=\|\cdot\|_{L^{2}}$
$H^{k}=\\{u\in L^{1}_{loc}|\,\|u\|_{k}<\infty\\},\quad\hat{H}^{k}=\\{u\in
L^{1}_{loc}|\,\nabla u\in H^{k-1}\\},$
where $u$ is either a vector or scalar. Further we put
$\mathcal{H}^{k,l}=\big{\\{}(\sigma,v)|\,\sigma\in H^{k},\,\,v\in
H^{l}\big{\\}},\qquad\hat{\mathcal{H}}^{k,l}=\big{\\{}(\sigma,v)|\,\sigma\in\hat{H}^{k},\,\,v\in\hat{H}^{l}\big{\\}},$
$\mathcal{H}^{j,k,l}=\big{\\{}(\sigma,v,\vartheta)|\,\sigma\in H^{j},\,\,v\in
H^{k},\vartheta\in H^{l}\big{\\}},$
$\hat{\mathcal{H}}^{j,k,l}=\big{\\{}(\sigma,v,\vartheta)|\,\sigma\in\hat{H}^{j},\,\,v\in\hat{H}^{k},\vartheta\in\hat{H}^{l}\big{\\}},$
and
$\|(\sigma,v)\|=\|\sigma\|_{k}+\|v\|_{l},\qquad\|(\sigma,v,\vartheta)\|_{j,k,l}=\|\sigma\|_{j}+\|v\|_{k}+\|\vartheta\|_{l}.$
###### Definition 1.1.
$I_{\epsilon}^{k}=\displaystyle\big{\\{}\sigma\in
H^{k}|\,\|\sigma\|_{I^{k}}<\epsilon\big{\\}},\quad
J_{\epsilon}^{k}=\displaystyle\big{\\{}v\in
H^{k}|\,\|v\|_{J^{k}}<\epsilon\big{\\}},\quad
N_{\epsilon}^{k}=\displaystyle\big{\\{}\vartheta\in
H^{k}|\,\|\vartheta\|_{N^{k}}<\epsilon\big{\\}},$
where
$\|\sigma\|_{I^{k}}=\|\sigma\|_{L^{6}}+\sum_{\nu=1}^{k}\left\|(1+|x|)^{\nu}(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma,\nabla^{\nu+2}\sigma)\right\|+\left\|(1+|x|)^{2}\sigma\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}\nabla\sigma\right\|_{L^{\infty}},$
$\|v\|_{J^{k}}=\|v\|_{\hat{J}^{k}}+\sum_{\nu=1}^{k}\left\|(1+|x|)^{\nu-1}\nabla^{\nu}v\right\|,\quad\|\vartheta\|_{N^{k}}=\|\vartheta\|_{\hat{J}^{k}}+\sum_{\nu=1}^{k}\left\|(1+|x|)^{\nu-1}(\nabla^{\nu}\vartheta,\nabla^{\nu+1}\vartheta)\right\|,$
and $\|\cdot\|_{\hat{J}^{k}}$ is defined by
$\|u\|_{\hat{J}^{k}}=\|u\|_{L^{6}}+\sum_{\nu=0}^{1}\left\|(1+|x|)^{\nu+1}\nabla^{\nu}u\right\|_{L^{\infty}}+\displaystyle\left\|(1+|x|)^{2}\nabla^{2}u\right\|_{L^{\infty}}.$
Moreover, we put
$\Lambda_{\epsilon}^{j,k,l}=\big{\\{}(\sigma,v,\vartheta)|\,\sigma\in
I_{\epsilon}^{j},v\in J_{\epsilon}^{k},\vartheta\in
N_{\epsilon}^{l},\|(\sigma,v,\vartheta)\|_{\Lambda^{j,k,l}}<\epsilon\big{\\}},$
$\left\|(\sigma,v,\vartheta)\right\|_{\Lambda^{j,k,l}}=\|\sigma\|_{I^{k}}+\|v\|_{J^{k}}+\|\vartheta\|_{N^{k}},$
$\begin{array}[]{ll}\dot{\Lambda}_{\epsilon}^{j,k,l}=\big{\\{}(\sigma,v,\vartheta)&\in\Lambda_{\epsilon}^{j,k,l}|\,\nabla\cdot
v=\nabla\cdot V_{1}+V_{2}\,for\,some\,V_{1},V_{2}\\\\[5.69054pt]
&\left.\,such\,that\,\left\|(1+|x|)^{3}V_{1}\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}V_{2}\right\|_{L^{1}}\leq\epsilon\right\\},\end{array}$
$\mathcal{L}=\\{U|U=\nabla\cdot
U_{1}+U_{2}\,for\,some\,U_{1},U_{2}\,and\,satisfies\,\|U\|_{\mathcal{L}}<\infty\\},$
where
$\|U\|_{\mathcal{L}}=\sum_{\nu=1}^{3}\left\|(1+|x|)^{\nu+1}\nabla^{\nu}U\right\|+\left\|(1+|x|)^{3}(U,\nabla
U)\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}U_{1}\right\|_{L^{\infty}}+\|U_{2}\|_{L^{1}}.$
In this paper, we consider the case where the mass source $G$, the external
force $F$ and energy source $H$ are given by the following form
$\left(\begin{array}[]{ll}G\\\\[2.84526pt] F\\\\[2.84526pt] H\\\\[2.84526pt]
\end{array}\right)=\nabla\cdot\left(\begin{array}[]{ll}G_{1}\\\\[2.84526pt]
F_{1}\\\\[2.84526pt] H_{1}\\\\[2.84526pt]
\end{array}\right)+\left(\begin{array}[]{ll}G_{2}\\\\[2.84526pt]
F_{2}\\\\[2.84526pt] H_{2}\\\\[2.84526pt] \end{array}\right)$
where $F_{1}=(F_{1,\,ij}(x))_{1\leq i,\,j\leq 3}$,
$F_{2}=(F_{2,\,i}(x))_{1\leq i\leq 3}$; $G_{1}=(G_{1,\,i}(x))_{1\leq i\leq
3}$, $G_{2}=G_{2}(x)$; $H_{1}=(H_{1,\,i}(x))_{1\leq i\leq 3}$,
$H_{2}=H_{2}(x)$.
Now we begin to state our main results. As [7], regarding $\rho$ as a smooth
function $(P,\theta)$, Our first Theorem is concerning the existence of
stationary solution to (1.19), and its weighted-$L^{2}$ and $L^{\infty}$
estimates.
###### Theorem 1.1.
Let $\bar{\rho}$, $\bar{\theta}$ be any positive constants, and set
$\bar{P}=P(\bar{\rho},\bar{\theta})$. There exists small constants $c_{0}>0$
and $\epsilon_{0}>0$ depending on $\bar{\rho}$ and $\bar{\theta}$, such that
if $(G,F,H)\in\mathcal{H}^{4,3,4}$ and satisfies the estimate:
$\|(G,F,H)\|_{\mathcal{L}}+\left\|(1+|x|)^{4}\nabla^{4}(G,H)\right\|+\left\|(1+|x|)^{-1}G\right\|_{L^{1}}\leq
c_{0}\epsilon$
for some positive constant $\epsilon\leq\epsilon_{0}$, then (1.19) admits a
solution of the form: $(P,v,\theta)=(\bar{P}+\sigma,v,\bar{\theta}+\vartheta)$
where $(\sigma,v,\theta)\in\dot{\Lambda}_{\epsilon}^{4,5,5}$. Furthermore the
solution is unique in the following sense: if there is another solution
$(\bar{P}+\sigma_{1},v_{1},\bar{\theta}+\vartheta_{1})$ satisfying (1.19) with
the same $(G,F,H)$, and
$\|(\sigma_{1},v_{1},\vartheta_{1})\|_{{\Lambda}^{4,5,5}}\leq\epsilon$, then
$(\sigma_{1},v_{1},\vartheta_{1})=(\sigma,v,\vartheta)$.
Next, we consider the stability of the stationary solution of (1.19) with
respect to the initial disturbance. Let $(\rho^{*},v^{*},\vartheta^{*})$ be
the stationary solution obtained in Theorem 1.1, then the stability of
$(\rho^{*},v^{*},\vartheta^{*})$ means the solvability of the non-stationary
problem (1.11), (1.12). Let us introduce first the class of functions which
solutions of (1.11), (1.12) belong to.
###### Definition 1.2.
$\mathcal{C}(0,T;\mathcal{H}^{j,k,l})=\left\\{(\sigma,w,\vartheta)(t,x)\left|\begin{array}[]{c}\sigma(t,x)\in
C^{0}(0,T;H^{j})\bigcap C^{1}(0,T;H^{j-2}),\\\\[5.69054pt] w(t,x)\in
C^{0}(0,T;H^{k})\bigcap C^{1}(0,T;H^{k-2}),\\\\[5.69054pt] \vartheta(t,x)\in
C^{0}(0,T;H^{l})\bigcap C^{1}(0,T;H^{l-2})\end{array}\right.\right\\}$
Then, we have the following Theorem.
###### Theorem 1.2.
There exist $C>0$ and $\delta>0$ such that if
$\|(\rho_{0}-\rho^{*},v_{0}-v^{*},\theta_{0}-\vartheta^{*})\|_{4,3,3}\leq\delta$,
then the Cauchy problem (1.11), (1.12) admits a unique solution
$(\rho,v,\theta)=(\rho^{*}+\sigma,v^{*}+w,\theta^{*}+\vartheta)$ globally in
time, where
$(\sigma,w,\vartheta)\in\mathcal{C}(0,\infty;\mathcal{H}^{4,3,3})$,
$\nabla\sigma\in L^{2}(0,\infty;H^{4})$, $\nabla w,\nabla\vartheta\in
L^{2}(0,\infty;H^{3})$. Moreover, the solution $(\sigma,w,\vartheta)$
satisfies the estimate:
$\|(\sigma,w,\vartheta)(t)\|^{2}_{4,3,3}+\int_{0}^{t}\left\|\nabla(\sigma,w,\vartheta)(s)\right\|^{2}_{4,3,3}\,ds\leq
C\|(\sigma,w,\vartheta)(0)\|^{2}_{4,3,3}.$ (1.20)
for any $t>0$ and
$\|(\sigma,v,\vartheta)(t)\|_{L^{\infty}}\rightarrow 0\quad
as\,t\rightarrow\infty.$
The compressible Navier-Stokes-Kortewg system has been attracted many
attentions due to its applications in fluid mechanics as well as mathematical
challenge. A lot of mathematical results on such system have been obtained.
More precisely, Hattori and Li [12, 13] proved the local existence and the
global existence of smooth solutions for the compressible fluid models of
Korteweg type in Sobolev space. Danchin and Desjardins [11] proved existence
and uniqueness results of suitably smooth solutions for the compressible fluid
models of Korteweg type in critical Besov space. Bresch, Desjardins and Lin
[8] showed the global existence of weak solution to the compressible fluid
models of Korteweg type, then Haspot improved their results in [9]. The local
existence of strong solutions for the compressible fluid models of Korteweg
type was proved by M. Kotschote [14]. Wang and Tan [15] established the
optimal $L^{2}$ decay rates of global smooth solutions for the compressible
fluid models of Korteweg type without external force. Recently, Li [17]
discussed the global existence of smooth solution to the following Cauchy
problem of the isothermal compressible fluid models of Korteweg type with
potential external force.
$\displaystyle\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho
u)=0,\\\\[5.69054pt] (\rho u)_{t}+\nabla\cdot(\rho u\bigotimes
u)=\nabla\cdot(S+K)+\rho F(x),\\\\[5.69054pt]
(\rho,u)|_{t=0}=(\rho_{0},u_{0}).\end{array}\right.$ (1.24)
Here $F(x)=-\nabla\phi$ with $\phi$ being a scalar function and $S,K$ are
defined as in (1.7). He proved that there exists a unique stationary solution
$(\tilde{\rho}(x),0)$ to problem (1.24) if $\phi(x)$ satisfies some smallness
condition in the $H^{3}$ norm. The nonlinear stability of the stationary
solution $(\tilde{\rho}(x),0)$ and the optimal $L^{2}$-decay rate of smooth
solutions to (1.24) were also proved in [17]. Motivated by the work Y. Shibata
and K. Tanaka [6] for the study of compressible Navier-Stokes equations, when
the external force is given by the general form $F=\nabla\cdot F_{1}+F_{2}$
and also mass source $G$ appears, it is expect that the stationary solution is
nontrivial in general. On the other hand, all the above results are concerning
about the isothermal compressible fluid models of Korteweg type, for the
nonisothermal compressible fluid models of Korteweg type, fewer results have
been obtained. To our knowledge, the only available result for the
nonisothermal case is [10], where the existence and uniqueness of strong
solutions was proved in critical space. Based on these observations, we
consider in this paper the nonlinear stability of stationary solutions to the
full compressible Navier-Stokes-Korteweg system (1.4).
Now we outline the main ideas used in proving our main results. The proof of
Theorem 1.1 is motivated by the method developed by Y. Shibata and K. Tanaka
[6]. Firstly, as mentioned before, we choose $(P,v,\theta)$ as the independent
variables and regarding $\rho$ as a smooth function of $(P,\theta)$. Then in
the same sprit as [6], we need to establish the corresponding linear theory in
the $L^{2}$-framework for (1.19) by employing the Banach closed range theorem.
Compared with the case of compressible Navier-Stokes system, the appearance of
the third order terms $\nabla\Delta\sigma$ and $\nabla\Delta\vartheta$ in the
velocity equation $(2.7)_{2}$ result in more difficulties when we estimate the
$L^{2}$ norm of the solutions to the approximate problem. In particular, an
additional term $\nabla(\nabla\cdot v)$ appears in the energy estimate. To
close the $L^{2}$ energy type estimate, we frequently use the structures of
the approximate system. Then by choosing some suitably space-weights and
multipliers, the weighted-$L^{2}$ estimate of solutions to the linearized
problem is also obtained. In order to deal with the nonlinear problem, we have
to derive the weighted-$L^{\infty}$ estimates for solutions
$(\sigma,v,\vartheta)$ to the linearized equation (2.110). The
weighted-$L^{\infty}$ estimates for $v$ and $\vartheta$ can be deduced in the
same way as that of compressible Navier-Stokes equations. However, for the
weighted-$L^{\infty}$ estimates of $\sigma$, we need to perform some delicate
estimates related to the Bessel potential(see (2.139) for detail). Moreover,
the highly nonlinear terms $\Psi(\tilde{v})$ and
$\Phi(\tilde{\rho},\tilde{v})$ in $(\ref{2.82})$ are overcome by some delicate
analysis. Having obtained the weighted-$L^{2}$ and weighted-$L^{\infty}$
estimates of solutions to the linearized problem, Theorem 1.1 follows by the
contraction mapping principle. As for the nonlinear stability of the
stationary solution obtained above, the key step is to deduce some certain a
priori estimates for solutions to the initial value problem (3.4),(3.5) in the
$H^{3}$ framework. Based on the properties we obtained on the stationary
solution and some delicate estimates, we can deduce the desired a priori
estimates. It is worth to point out that, for the compressible Navier-Stokes-
Korteweg system (1.7), the appearance of the Korteweg tensor
$\rho\nabla\Delta\rho$ results in more regularity for the density than the
velocity and internal energy (see (1.20)). In fact, we frequently use
integration by parts and the equation $(\ref{3.1})_{1}$ when we estimate the
the terms containing $\nabla\Delta\sigma$. As a result, the Korteweg term is
split into the energy and the terms small in the $L^{2}$ norm.
Another interesting problem is to investigate the convergence rate of the non-
stationary solutions to the stationary solutions when the time goes to
infinity. As mentioned before, this problem has been studied by some authors
for the isothermal compressible Navier-Stokes-Korteweg system with
$G=0,F=-\nabla\phi$ or without any external force(cf.[17], [15, 16]). But to
obtain the convergence rate in our case, it appear to be more delicate since
the stationary solution is nontrivial generally. We will consider this problem
in a forthcoming paper.
Before concluding this section, we also mention that the nonlinear stability
of stationary solution for the compressible Navier-Stokes system has been
studied by many authors. For the non-isentropic case, we refer to [20, 21] for
the stability of constant state $(\bar{\rho},0,\bar{\theta})$ in
$\mathbb{R}^{3}$, [22] for the stability of nontrivial stationary solution
$(\rho^{*}(x),0,\bar{\theta})$ in an exterior domain of $\mathbb{R}^{3}$ and
[7, 23] for the stability of generally nontrivial stationary solution
$(\rho^{*}(x),v^{*}(x),\theta^{*}(x))$ in $\mathbb{R}^{3}$ and an exterior
domain of $\mathbb{R}^{3}$, respectively. For the isentropic case, the
interesting readers are referred to [6, 25, 26] for the stability of generally
nontrivial stationary solution $(\rho^{*}(x),v^{*}(x))$ in $\mathbb{R}^{3}$ or
an exterior domain of $\mathbb{R}^{3}$ and [24] for the stability of
nontrivial stationary solution $(\rho^{*}(x),0)$ in an exterior domain of
$\mathbb{R}^{3}$.
The rest of this paper is organized as follows. In Section 2, we study the
stationary problem. The non-stationary problem will be studied in Section 3.
## 2 Stationary problem
This section is devoted to the stationary problem (1.19). Take any two
constants $\bar{\rho},\bar{\theta}>0$. As mentioned in Section 1, by regarding
$\rho$ as the function of $(P,\theta)$, changing the variables
$(P,v,\theta)\rightarrow(\bar{P}+\sigma,v,\bar{\theta}+\vartheta)$, and
rewriting the third equation by using the first one, (1.19) can be then
reformulated as
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot
v\displaystyle+\frac{\rho_{P}}{\rho}(v\cdot\nabla)\sigma=\displaystyle-\frac{\rho_{\theta}}{\rho}(v\cdot\nabla)\vartheta+\frac{G(x)}{\rho},\\\\[5.69054pt]
\displaystyle-\mu\Delta v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta=-\rho(v\cdot\nabla)v+\hat{f},\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta\vartheta=-\eta_{1}(v\cdot\nabla)\vartheta+\eta_{2}(v\cdot\nabla)\sigma+\Psi(v)+\hat{\Phi}-\eta_{3}G+H+\frac{v^{2}}{2}G(x)-C_{\triangledown}(\vartheta+\bar{\theta})G,\end{array}\right.$
(2.4)
where
$\displaystyle\left\\{\begin{array}[]{ll}\gamma_{1}=\bar{\rho}\bar{\rho}_{P},\,\,\gamma_{2}=\bar{\rho}\bar{\rho}_{\theta},\,\,\bar{\rho}_{P}=\rho_{P}(\bar{P},\bar{\theta}),\,\,\bar{\rho}_{\theta}=\rho_{\theta}(\bar{P},\bar{\theta}),\\\\[5.69054pt]
\displaystyle\eta_{1}=\displaystyle\rho
C_{\triangledown}-\frac{\theta\rho_{\theta}^{2}}{\rho\rho_{P}},\quad\eta_{2}=\displaystyle\frac{\theta\rho_{\theta}}{\rho},\quad\eta_{3}=\displaystyle\frac{\theta\rho_{\theta}}{\rho\rho_{P}},\\\\[5.69054pt]
\hat{f}=\kappa\rho\left(\nabla\sigma\cdot\nabla^{2}\rho_{P}+\nabla\rho_{P}\cdot\nabla^{2}\sigma+\nabla\rho_{P}\Delta\sigma\right)+\kappa\left(\rho\rho_{P}-\bar{\rho}\bar{\rho}_{P}\right)\nabla\Delta\sigma\\\\[5.69054pt]
\qquad+\kappa\rho\left(\nabla\vartheta\cdot\nabla^{2}\rho_{\theta}+\nabla\rho_{\theta}\cdot\nabla^{2}\vartheta+\nabla\rho_{\theta}\Delta\vartheta\right)+\kappa\left(\rho\rho_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\nabla\Delta\vartheta,\\\\[5.69054pt]
\hat{\Phi}=\kappa\left[\frac{1}{2}|\rho_{P}\nabla\sigma+\rho_{\theta}\nabla\vartheta|^{2}+\rho(\nabla\rho_{P}\cdot\nabla\sigma+\rho_{P}\Delta\sigma+\nabla\rho_{\theta}\cdot\nabla\vartheta+\rho_{\theta}\Delta\vartheta)\right]\nabla\cdot
v\\\\[5.69054pt]
\qquad-\kappa\left[(\rho_{P}\nabla\sigma+\rho_{\theta}\nabla\vartheta)\bigotimes(\rho_{P}\nabla\sigma+\rho_{\theta}\nabla\vartheta)\right]:\nabla
v.\end{array}\right.$ (2.11)
Our goal of this section is to prove Theorem 1.1 by application of weighted
$L^{2}$-method to the linearized problem for (2.4).
### 2.1 Weighted $L^{2}$ theory for linearized problem
We shall consider the linearized equation of (2.4):
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot
v\displaystyle+(a\cdot\nabla)\sigma=g,\\\\[5.69054pt] \displaystyle-\mu\Delta
v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta=f,\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta\vartheta=h,\end{array}\right.$ (2.15)
where $a=(a_{1}(x),a_{2}(x),a_{3}(x))$, $(g,f,h)\in\mathcal{H}^{4,3,3}$ are
given. Throughout this subsection, we put
$f=-(b_{1}\cdot\nabla)c_{1}+\tilde{f},\qquad
h=-(b_{2}\cdot\nabla)c_{2}+\tilde{h}.$
and assume that
$a\in\hat{H}^{4},\quad\left\|(1+|x|)a\right\|_{L^{\infty}}+\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu-1}\nabla^{\nu}a\right\|\leq\delta$
(2.16) $\left(g,\tilde{f},\tilde{h}\right)\in\mathcal{H}^{4,3,4},\quad
b_{1},b_{2},c_{1}\in J^{5},c_{2}\in N^{5},$ (2.17)
$\|(1+|x|)(g,\tilde{h})\|+\displaystyle\sum_{\nu=1}^{4}(1+|x|)^{\nu}\nabla^{\nu}(g,\tilde{h})\|+\displaystyle\sum_{\nu=0}^{3}(1+|x|)^{\nu+1}\nabla^{\nu}(\tilde{f},\tilde{h})\|\leq\infty$
(2.18)
#### 2.1.1 Solution to approximate problem
First, we solve the approximate problem:
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot
v\displaystyle+(a\cdot\nabla)\sigma-\epsilon\Delta\sigma+\epsilon\sigma=g,\\\\[5.69054pt]
\displaystyle-\mu\Delta v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta+\epsilon
v=f,\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta\vartheta+\epsilon\vartheta=h,\end{array}\right.$
(2.22)
in $\mathcal{H}^{3,2,3}$. In the following lemma, we prove some fundamental a
priori estimate needed later.
###### Lemma 2.1.
Suppose that $(\sigma,v,\vartheta)\in\mathcal{H}^{3,2,3}$ is a solution to
(2.22). Then there exists two positive constants
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})$
and
$\epsilon_{0}=\epsilon_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})<1$
such that if $\delta$ in (2.16) satisfies $\delta\leq\delta_{0}$ and
$0<\epsilon<\epsilon_{0}$, we have the following estimate:
$\left\|\nabla(\sigma,v,\vartheta)\right\|^{2}_{2,1,2}+\epsilon\left\|(\sigma,v,\vartheta)\right\|^{2}_{2,1,1}\leq
C\epsilon^{-1}\|(g,f,h)\|^{2}+C\|\nabla(g,h)\|^{2}.$ (2.23)
Here, $C>0$ is a constant depending only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
Proof. The proof consists of four steps.
Step 1. Taking the $L^{2}$ inner product with $\sigma$ and $v$ on
$(\ref{2.7})_{1}$, $(\ref{2.7})_{2}$ , respectively, using integration by
parts and canceling the term $\langle\nabla\sigma,v\rangle$ by adding the two
resultant equations together, we have
$\begin{array}[]{rl}&\mu\|\nabla v\|^{2}+(\mu+\mu^{\prime})\|\nabla\cdot
v\|^{2}+\epsilon\|(\sigma,v)\|^{2}_{1,0}\\\\[5.69054pt] &=\langle
g,\sigma\rangle+\langle
f,v\rangle+\kappa\gamma_{1}\langle\nabla\Delta\sigma,v\rangle+\kappa\gamma_{2}\langle\nabla\Delta\vartheta,v\rangle-\langle(a\cdot\nabla)\sigma,\sigma\rangle.\end{array}$
(2.24)
Differentiating $(\ref{2.7})_{1}$ and $(\ref{2.7})_{2}$, and employing the
same argument, we have
$\begin{array}[]{rl}&\mu\|\nabla^{2}v\|^{2}+(\mu+\mu^{\prime})\|\nabla(\nabla\cdot
v)\|^{2}+\epsilon\|\nabla(\sigma,v)\|^{2}_{1,0}\\\\[5.69054pt] &=\langle\nabla
g,\nabla\sigma\rangle+\langle\nabla f,\nabla
v\rangle+\kappa\gamma_{1}\langle\nabla(\nabla\Delta\sigma),\nabla
v\rangle+\kappa\gamma_{2}\langle\nabla(\nabla\Delta\vartheta),\nabla
v\rangle-\langle\nabla((a\cdot\nabla)\sigma),\nabla\sigma\rangle.\end{array}$
(2.25)
Adding (2.25) to (2.24) yields
$\begin{array}[]{rl}&\mu\|\nabla v\|^{2}_{1}+(\mu+\mu^{\prime})\|(\nabla\cdot
v)\|^{2}_{1}+\epsilon\|(\sigma,v)\|^{2}_{2,1}\\\\[5.69054pt]
=&\displaystyle\sum_{\nu=0}^{1}\left\\{\langle\nabla^{\nu}g,\nabla^{\nu}\sigma\rangle+\langle\nabla^{\nu}f,\nabla^{\nu}v\rangle+\kappa\gamma_{1}\langle\nabla^{\nu}(\nabla\Delta\sigma),\nabla^{\nu}v\rangle\right.\\\\[5.69054pt]
&\left.\quad\quad\quad+\kappa\gamma_{2}\langle\nabla^{\nu}(\nabla\Delta\vartheta),\nabla^{\nu}v\rangle-\langle\nabla^{\nu}((a\cdot\nabla)\sigma),\nabla^{\nu}\sigma\rangle\right\\}=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}.\end{array}$
(2.26)
It follows from the Cauchy inequality that
$\displaystyle\begin{array}[]{ll}&I_{1}\leq\displaystyle\frac{\epsilon}{4}\|\sigma\|^{2}+C_{\epsilon}\|g\|^{2}+\eta\|\Delta\sigma\|^{2}+C_{\eta}\|g\|^{2},\\\\[8.53581pt]
&I_{2}\leq\displaystyle\frac{\epsilon}{4}\|v\|^{2}+C_{\epsilon}\|f\|^{2}+\eta\|\nabla(\nabla\cdot
v)\|^{2}+C_{\eta}\|f\|^{2},\\\\[8.53581pt]
&I_{3}\leq\displaystyle\eta(\|\nabla\sigma\|^{2}+\|\nabla\Delta\sigma\|^{2})+C_{\eta}\|\nabla(\nabla\cdot
v)\|^{2},\end{array}$ (2.30)
and
$I_{4}\leq\displaystyle\eta(\|\nabla\vartheta\|^{2}+\|\nabla\Delta\vartheta\|^{2})+C_{\eta}\|\nabla(\nabla\cdot
v)\|^{2}.$ (2.31)
Here and hereafter, $\eta>0$ denotes a sufficiently small constant and
$C_{\epsilon},C_{\eta}$ denote some positive constants depending only on
$\epsilon$ and $\eta$, respectively. Moreover, the Cauchy-Schwartz inequality
and the Hardy inequality imply that
$\begin{array}[]{ll}I_{5}&\leq\displaystyle
C\left(\left|\langle(a\cdot\nabla)\sigma,\sigma\rangle\right|+\left|\langle(a\cdot\nabla)\sigma,\Delta\sigma\rangle\right|\right)\\\\[5.69054pt]
&\leq\displaystyle
C\|(1+|x|)a\|_{L^{\infty}}\displaystyle\left(\|\nabla\sigma\|\left\|\frac{\sigma}{|x|}\right\|+\displaystyle\left\|\frac{\nabla\sigma}{|x|}\right\|\|\Delta\sigma\|\right)\\\\[5.69054pt]
&\leq C\delta\|\nabla\sigma\|_{1}^{2}.\end{array}$ (2.32)
Combining (2.26)-(2.32), we obtain
$\begin{array}[]{ll}&\mu\|\nabla v\|^{2}_{1}+(\mu+\mu^{\prime})\|\nabla\cdot
v\|^{2}_{1}+\epsilon\|(\sigma,v)\|^{2}_{2,1}\\\\[5.69054pt] &\leq
C\eta\|\nabla(\sigma,\vartheta)\|^{2}_{2}+C\delta\|\nabla\sigma\|^{2}_{1}+C_{\eta}\|\nabla(\nabla\cdot
v)\|^{2}+(C_{\epsilon}+C_{\eta})\|(g,f)\|^{2}.\end{array}$ (2.33)
Step 2. Differentiating $(\ref{2.7})_{1}$, we get
$\nabla(\nabla\cdot
v)=-\nabla((a\cdot\nabla)\sigma)+\epsilon\nabla\Delta\sigma-\epsilon\nabla\sigma+\nabla
g.$
which together with the sobolev inequality imply that
$\begin{array}[]{ll}\|\nabla(\nabla\cdot v)\|^{2}&\leq\displaystyle
C\left(\|(a,\nabla
a)\|_{L^{\infty}}\|\nabla\sigma\|_{1}^{2}+\epsilon^{2}\|\nabla\Delta\sigma\|^{2}+\epsilon^{2}\|\nabla\sigma\|^{2}+\|\nabla
g\|^{2}\right)\\\\[5.69054pt] &\leq
C\left(\delta^{2}\|\nabla\sigma\|_{1}^{2}+\epsilon^{2}\|\nabla\Delta\sigma\|^{2}+\epsilon^{2}\|\nabla\sigma\|^{2}+\|\nabla
g\|^{2}\right)\end{array}$ (2.34)
Step 3. Taking the $L^{2}$ inner product with $\nabla\sigma$ on
$(\ref{2.7})_{2}$, we have from the Cauchy inequality that
$\begin{array}[]{ll}&\|\nabla\sigma\|^{2}+\kappa\gamma_{1}\|\Delta\sigma\|^{2}\\\\[5.69054pt]
&=\mu\langle\Delta
v,\nabla\sigma\rangle+(\mu+\mu^{\prime})\langle\nabla(\nabla\cdot
v),\nabla\sigma\rangle+\kappa\gamma_{2}\langle\nabla\Delta\vartheta,\nabla\sigma\rangle-\epsilon\langle
v,\nabla\sigma\rangle+\langle f,\nabla\sigma\rangle\\\\[5.69054pt]
&\leq\displaystyle\frac{1}{2}\|\nabla\sigma\|^{2}+C\left(\|\Delta
v\|^{2}+\|\nabla(\nabla\cdot
v)\|^{2}+\|\nabla\Delta\vartheta\|^{2}+\epsilon^{2}\|v\|^{2}+\|f\|^{2}\right)\end{array}$
(2.35)
Consequently,
$\|\nabla\sigma\|^{2}+\|\Delta\sigma\|^{2}\leq C\left(\|\Delta
v\|^{2}+\|\nabla\Delta\vartheta\|^{2}+\epsilon^{2}\|v\|^{2}+\|f\|^{2}\right).$
(2.36)
On the other hand, it follows from $(\ref{2.7})_{2}$ that
$\|\nabla\Delta\sigma\|^{2}\leq C\left(\|\Delta
v\|^{2}+\|\nabla\Delta\vartheta\|^{2}+\|\nabla\sigma\|^{2}+\epsilon^{2}\|v\|^{2}+\|f\|^{2}\right).$
(2.37)
Therefore, we have from a linear combination of (2.36) and (2.37) that
$\|\nabla\sigma\|^{2}_{2}\leq C\left(\|\Delta
v\|^{2}+\|\nabla\Delta\vartheta\|^{2}+\epsilon^{2}\|v\|^{2}+\|f\|^{2}\right).$
(2.38)
Step 4. By using the same argument as (2.24) and (2.25), one can get from
$(\ref{2.7})_{3}$ that
$\begin{array}[]{ll}\tilde{\alpha}\|\nabla\vartheta\|^{2}_{1}+\epsilon\|\vartheta\|_{1}^{2}&=\langle
h,\vartheta\rangle+\langle\nabla h,\nabla\vartheta\rangle\\\\[5.69054pt]
&\leq\displaystyle\frac{\epsilon}{2}\|\vartheta\|^{2}+\frac{\tilde{\alpha}}{2}\|\nabla^{2}\vartheta\|^{2}+(C_{\epsilon}+C)\|h\|^{2}\end{array}$
(2.39)
which implies
$\tilde{\alpha}\|\nabla\vartheta\|^{2}_{1}+\epsilon\|\vartheta\|_{1}^{2}\leq\displaystyle(C_{\epsilon}+C)\|h\|^{2}$
(2.40)
On the other hand,
$\tilde{\alpha}\|\nabla\Delta\vartheta\|^{2}=\|-\epsilon\nabla\vartheta+\nabla
h\|^{2}\leq\epsilon^{2}\|\nabla\vartheta\|^{2}+\|\nabla h\|^{2}$ (2.41)
Combining (2.40) and (2.41), we obtain
$\|\nabla\vartheta\|^{2}_{2}+\epsilon\|\vartheta\|_{1}^{2}\leq\displaystyle
C_{\epsilon}\|h\|^{2}+C\|\nabla h\|^{2}.$ (2.42)
Thus, by some suitably linear combinations of (2.33), (2.34), (2.38) and
(2.42) and using the smallness of $\epsilon,\eta$ and $\delta$, we can get
(2.23). This completes the proof of Lemma 2.1.
Now, we employ the closed range theorem to prove the existence of solution to
(2.22). We introduce the operator $A$ defined on $D(A)\subset L^{2}$ into
$H^{1}\times L^{2}\times H^{1}$ by
$A(\sigma,v,\vartheta)=\left(A_{1}(\sigma,v,\vartheta),A_{2}(\sigma,v,\vartheta),A_{3}(\sigma,v,\vartheta)\right)$
where $D(A)=\mathcal{H}^{3,2,3}$ and
$\left\\{\begin{array}[]{ll}&A_{1}(\sigma,v,\vartheta)=\nabla\cdot
v\displaystyle+(a\cdot\nabla)\sigma-\epsilon\Delta\sigma+\epsilon\sigma,\\\\[5.69054pt]
&A_{2}(\sigma,v,\vartheta)=-\mu\Delta v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta+\epsilon
v,\\\\[5.69054pt]
&A_{3}(\sigma,v,\vartheta)=-\tilde{\alpha}\Delta\vartheta+\epsilon\vartheta.\end{array}\right.$
Clearly, $A$ is closed operator. Furthermore, Lemma 2.1 implies that for each
$0<\epsilon<\epsilon_{0}$, the range of $A$ is closed.
###### Proposition 2.1.
There exists two positive constants
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})$
and $\epsilon_{0}=\epsilon_{0}(\gamma_{1},\gamma_{2},\kappa,\\\
\mu,\mu^{\prime},\tilde{\alpha})<1$ such that if $\delta$ in (2.16) satisfies
$\delta\leq\delta_{0}$ and $0<\epsilon<\epsilon_{0}$, then (2.22) has a
solution $(\sigma,v,\vartheta)\in\mathcal{H}^{3,2,3}$, which satisfies
$\left\|(\sigma,v,\vartheta)\right\|_{3,2,3}\leq
C(\epsilon)(\|(g,f,h)\|+\|\nabla(g,h)\|).$ (2.43)
where $C(\epsilon)>0$ is a constant depending only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha}$ and $\epsilon$,
and $C(\epsilon)\rightarrow\infty$ as $\epsilon\rightarrow 0$.
Proof. Firstly, for any $(\sigma,v,\vartheta)\in\mathcal{H}^{3,2,3}$ and
$(\sigma_{*},v_{*},\vartheta_{*})\in\mathcal{H}^{\infty,\infty,\infty}$, we
have from integration by parts that
$\displaystyle\begin{array}[]{ll}\langle
A(\sigma,v,\vartheta),(\sigma_{*},v_{*},\vartheta_{*})\rangle&=\langle\nabla\cdot
v\displaystyle+(a\cdot\nabla)\sigma-\epsilon\Delta\sigma+\epsilon\sigma,\sigma_{*}\rangle+\langle-\tilde{\alpha}\Delta\vartheta+\epsilon\vartheta,\vartheta_{*}\rangle\\\\[5.69054pt]
&\quad+\langle-\mu\Delta v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta+\epsilon
v,v_{*}\rangle\\\\[5.69054pt] &=\langle\sigma,-\nabla\cdot
v_{*}-\nabla\cdot(a\sigma_{*})-\epsilon\Delta\sigma_{*}+\epsilon\sigma_{*}+\kappa\gamma_{1}\Delta(\nabla\cdot
v_{*})\rangle\\\\[5.69054pt] &\quad+\langle v,-\mu\Delta
v_{*}-(\mu+\mu^{\prime})\nabla(\nabla\cdot v_{*})-\nabla\sigma_{*}+\epsilon
v_{*}\rangle\\\\[5.69054pt]
&\quad+\langle\vartheta,-\tilde{\alpha}\Delta\vartheta_{*}+\epsilon\vartheta_{*}+\kappa\gamma_{2}\Delta(\nabla\cdot
v_{*})\rangle\end{array}$ (2.49)
Therefore, $D(A^{*})=\mathcal{H}^{2,3,2}$ and for any
$(\sigma_{*},v_{*},\vartheta_{*})\in\mathcal{H}^{2,3,2}$,
$A^{*}(\sigma_{*},v_{*},\vartheta_{*})=\left(A^{*}_{1}(\sigma_{*},v_{*},\vartheta_{*}),A^{*}_{2}(\sigma_{*},v_{*},\vartheta_{*}),A^{*}_{3}(\sigma_{*},v_{*},\vartheta_{*})\right),$
where
$\left\\{\begin{array}[]{ll}&A_{1}^{*}(\sigma_{*},v_{*},\vartheta_{*})=-\nabla\cdot
v_{*}-\nabla\cdot(a\sigma_{*})-\epsilon\Delta\sigma_{*}+\epsilon\sigma_{*}+\kappa\gamma_{1}\Delta(\nabla\cdot
v_{*}),\\\\[5.69054pt] &A_{2}^{*}(\sigma_{*},v_{*},\vartheta_{*})=-\mu\Delta
v_{*}-(\mu+\mu^{\prime})\nabla(\nabla\cdot v_{*})-\nabla\sigma_{*}+\epsilon
v_{*},\\\\[5.69054pt]
&A_{3}^{*}(\sigma_{*},v_{*},\vartheta_{*})=-\tilde{\alpha}\Delta\vartheta_{*}+\epsilon\vartheta_{*}+\kappa\gamma_{2}\Delta(\nabla\cdot
v_{*}).\end{array}\right.$ (2.50)
Employing the same argument as in the proof of Lemma 2.1 and using the
equation:
$\Delta(\nabla\cdot
v_{*})=-\frac{1}{2\mu+\mu^{\prime}}\left(\Delta\sigma_{*}-\epsilon\nabla\cdot
v_{*}+\nabla\cdot A_{2}^{*}\right)$
which follows by taking the divergence ”$\nabla\cdot$ ” on both side of
$(\ref{2.27})_{2}$, one can get
$\|\Delta(\nabla\cdot
v_{*})\|+\|\nabla(\sigma_{*},v_{*},\vartheta_{*})\|_{1}+\epsilon\|(\sigma_{*},v_{*},\vartheta_{*})\|_{2,1,1}\leq
C_{\epsilon}\|(A_{1}^{*},A_{2}^{*},A_{3}^{*},\nabla\cdot A_{2}^{*})\|$ (2.51)
Hence the closed range theorem implies the existence of solution to (2.22).
(2.43) follows directly from (2.23). This completes the proof of Proposition
2.1.
#### 2.1.2 Solution to linearized problem (2.15) and its $L^{2}$ estimate
In the following Lemma, we discuss the estimate for solution to (2.22)
independent of $\epsilon$.
###### Lemma 2.2.
Assume that $(\sigma,v,\vartheta)\in\mathcal{H}^{3,2,3}$ is a solution of the
approximate problem (2.22). Then there exists a constant
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})>0$
such that such that if $\delta$ in (2.16) satisfies $\delta\leq\delta_{0}$, we
have the estimate
$\|\nabla(\sigma,v,\vartheta)\|_{5,4,5}\leq
C\left\\{\|(1+|x|)(g,f,h)\|+\|\nabla(g,f,h)\|_{3,2,3}\right\\},$ (2.52)
where the constant $C>0$ depends only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
Proof. Using the Friedrichs mollifier, we may assume that
$(\sigma,v,\vartheta)\in\mathcal{H}^{\infty,\infty,\infty}$. By the same
argument as in the proof of Lemma 2.1, we have
$\|\nabla(\sigma,v,\vartheta)\|^{2}_{2,1,2}\leq
C_{1}\Big{\\{}\|f\|^{2}+\|\nabla(g,f)\|^{2}+\sum_{\nu=0}^{1}\left(\langle\nabla^{\nu}g,\nabla^{\nu}\sigma\rangle+\langle\nabla^{\nu}f,\nabla^{\nu}v\rangle+\langle\nabla^{\nu}h,\nabla^{\nu}\vartheta\rangle\right)\Big{\\}}$
(2.53)
For the third term on the right hand of (2.53), the Cauchy inequality and the
Hardy inequality imply that
$\begin{array}[]{ll}&\displaystyle\sum_{\nu=0}^{1}\left(\langle\nabla^{\nu}g,\nabla^{\nu}\sigma\rangle+\langle\nabla^{\nu}f,\nabla^{\nu}v\rangle+\langle\nabla^{\nu}h,\nabla^{\nu}\vartheta\rangle\right)\\\\[5.69054pt]
&\quad\quad\leq\displaystyle\frac{1}{2C_{1}}\|\nabla(\sigma,v,\vartheta)\|_{1}^{2}+C\|(1+|x|)(g,f,h)\|^{2}\end{array}$
(2.54)
Consequently,
$\|\nabla(\sigma,v,\vartheta)\|^{2}_{2,1,2}\leq
C\left\\{\|(1+|x|)(g,f,h)\|^{2}+\|\nabla(g,h)\|^{2}\right\\}$ (2.55)
where the constant $C$ depends only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
Moreover, for any multi-index $\alpha$ with $1\leq|\alpha|\leq k-1$, applying
$\partial_{x}^{\alpha}$ to (2.22), we have
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot\partial_{x}^{\alpha}v\displaystyle+(a\cdot\nabla)\partial_{x}^{\alpha}\sigma-\epsilon\Delta\partial_{x}^{\alpha}\sigma+\epsilon\partial_{x}^{\alpha}\sigma=\partial_{x}^{\alpha}g-I_{\alpha},\\\\[5.69054pt]
\displaystyle-\mu\Delta\partial_{x}^{\alpha}v-(\mu+\mu^{\prime})\nabla(\nabla\cdot\partial_{x}^{\alpha}v)+\nabla\partial_{x}^{\alpha}\sigma-\kappa\gamma_{1}\nabla\Delta\partial_{x}^{\alpha}\sigma-\kappa\gamma_{2}\nabla\Delta\partial_{x}^{\alpha}\vartheta+\epsilon\partial_{x}^{\alpha}v=\partial_{x}^{\alpha}f,\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta\partial_{x}^{\alpha}\vartheta+\epsilon\partial_{x}^{\alpha}\vartheta=\partial_{x}^{\alpha}h,\end{array}\right.$
(2.59)
where
$I_{\alpha}=\displaystyle\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left(\partial_{x}^{\alpha-\beta}a\cdot\nabla\right)\partial^{\beta}_{x}\sigma$
with $C^{\alpha}_{\beta}$ being the binomial coefficients corresponding to
multi-indices. Notice that the third term on the right hand of (2.53) can also
be estimated as follows:
$\begin{array}[]{ll}&\displaystyle\sum_{\nu=0}^{1}\left(\langle\nabla^{\nu}g,\nabla^{\nu}\sigma\rangle+\langle\nabla^{\nu}f,\nabla^{\nu}v\rangle+\langle\nabla^{\nu}h,\nabla^{\nu}\vartheta\rangle\right)\\\\[5.69054pt]
&\quad\leq\displaystyle\frac{1}{2C_{1}}\|\nabla^{2}(\sigma,v,\vartheta)\|^{2}+C\left\\{\|(g,f,h)\|^{2}+\|(\sigma,v,\vartheta)\|^{2}\right\\}\end{array}$
(2.60)
Thus, it follows from (2.53) and (2.60) that
$\|\nabla(\sigma,v,\vartheta)\|^{2}_{2,1,2}\leq
C\left\\{\|(g,f,h)\|^{2}+\|\nabla(g,h)\|^{2}+\|(\sigma,v,\vartheta)\|^{2}\right\\}$
(2.61)
Applying (2.61) to (2.59), we have
$\|\nabla\partial_{x}^{\alpha}(\sigma,v,\vartheta)\|^{2}_{2,1,2}\leq
C\left\\{\|\partial_{x}^{\alpha}(g,f,h)\|^{2}+\|\nabla\partial_{x}^{\alpha}(g,h)\|^{2}+\|\partial_{x}^{\alpha}(\sigma,v,\vartheta)\|^{2}+\|I_{\alpha}\|_{1}^{2}\right\\}$
(2.62)
Since
$\|I_{\alpha}\|^{2}_{1}\leq C\delta^{2}\|\nabla\sigma\|^{2}_{|\alpha|}$ (2.63)
as follows from the Sobolev inequality and the assumption (2.16). We get from
(2.62) and (2.63) that
$\begin{array}[]{ll}&\|\nabla^{|\alpha|+3}\sigma,\nabla^{|\alpha|+2}v,\nabla^{|\alpha|+3}\vartheta\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\|\partial_{x}^{\alpha}(g,f,h)\|^{2}+\|\nabla\partial_{x}^{\alpha}(g,h)\|^{2}+\|\partial_{x}^{\alpha}(\sigma,v,\vartheta)\|^{2}+\delta^{2}\|\nabla\sigma\|^{2}_{|\alpha|}\right\\}\end{array}$
(2.64)
Combining (2.55) and (2.64), we obtain (2.52) if $\delta>0$ is small enough.
This completes the proof of Lemma 2.2.
Now, we are ready to show the existence of solution to the linearized problem
(2.15) by using (2.52).
###### Proposition 2.2.
There exists
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})>0$
such that such that if $\delta$ in (2.16) satisfies $\delta\leq\delta_{0}$,
then the linearized problem (2.15) admits a solution
$(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{6,5,6}$ which satisfies the
estimate:
$\|(\sigma,v,\vartheta)\|_{L^{6}}+\|\nabla(\sigma,v,\vartheta)\|_{5,4,5}\leq
C\left\\{\|(1+|x|)(g,f,h)\|+\|\nabla(g,f,h)\|_{3,2,3}\right\\}$ (2.65)
where the constant $C>0$ depends only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
Proof. Set
$K=\|(1+|x|)(g,f,h)\|+\|\nabla(g,f,h)\|_{3,2,3}$
From Proposition 2.1 and Lemma 2.2, it follows that for each
$0<\epsilon<\epsilon_{0}$($\epsilon_{0}$ is given in Proposition 2.1), (2.22)
admits a solution
$(\sigma^{\epsilon},v^{\epsilon},\vartheta^{\epsilon})\in\mathcal{H}^{6,5,6}$
which satisfies
$\|\nabla(\sigma^{\epsilon},v^{\epsilon},\vartheta^{\epsilon})\|_{5,4,5}\leq
CK.$
The Sobolev inequality imply that
$\|(\sigma^{\epsilon},v^{\epsilon},\vartheta^{\epsilon})\|_{L^{6}}\leq
C\|\nabla(\sigma^{\epsilon},v^{\epsilon},\vartheta^{\epsilon})\|\leq CK.$
Choosing an appropriate subsequence, there exist $(\sigma,v,\vartheta)\in
L^{6}$,
$(\hat{\sigma}_{i},\hat{v}_{i},\hat{\vartheta}_{i})\in\mathcal{H}^{5,4,5}$
such that
$(\sigma^{\epsilon},v^{\epsilon},\vartheta^{\epsilon})\rightharpoonup(\sigma,v,\vartheta)\quad
weakly\,\,in\,\,L^{6}$ $\left(\frac{\partial\sigma^{\epsilon}}{\partial
x_{i}},\frac{\partial v^{\epsilon}}{\partial
x_{i}},\frac{\partial\vartheta^{\epsilon}}{\partial
x_{i}}\right)\rightharpoonup\left(\hat{\sigma}_{i},\hat{v}_{i},\hat{\vartheta}_{i}\right)\quad
weakly\,\,in\,\,\mathcal{H}^{5,4,5}.$
as $\epsilon\rightarrow 0$, then one can check easily that
$\left(\frac{\partial\sigma}{\partial x_{i}},\frac{\partial v}{\partial
x_{i}},\frac{\partial\vartheta}{\partial
x_{i}}\right)=\left(\hat{\sigma}_{i},\hat{v}_{i},\hat{\vartheta}_{i}\right),$
and
$\|(\sigma,v,\vartheta)\|+\|\nabla(\sigma,v,\vartheta)\|_{5,4,5}\leq CK.$
On the other hand, we have
$\begin{array}[]{ll}&\nabla\cdot
v^{\epsilon}\displaystyle+(a\cdot\nabla)\sigma^{\epsilon}-\epsilon\Delta\sigma^{\epsilon}+\epsilon\sigma^{\epsilon}\longrightarrow\nabla\cdot
v\displaystyle+(a\cdot\nabla)\sigma,\\\\[5.69054pt] &-\mu\Delta
v^{\epsilon}-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v^{\epsilon})+\nabla\sigma^{\epsilon}-\kappa\gamma_{1}\nabla\Delta\sigma^{\epsilon}-\kappa\gamma_{2}\nabla\Delta\vartheta^{\epsilon}+\epsilon
v^{\epsilon}\\\\[5.69054pt] &\longrightarrow-\mu\Delta
v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta,\\\\[5.69054pt]
&-\tilde{\alpha}\Delta\vartheta^{\epsilon}+\epsilon\vartheta^{\epsilon}\longrightarrow-\tilde{\alpha}\Delta\vartheta.\end{array}$
in distribution sense. This completes the proof of Proposition 2.2.
#### 2.1.3 Weighted $L^{2}$ estimate for solution to the linearized equation
(2.15)
In this subsection, we give the weighted $L^{2}$ estimate for the solution to
(2.15).
###### Lemma 2.3.
Let $(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{6,5,6}$ be a solution to (2.15)
which satisfies (2.65). Then there exists a constant
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})>0$
such that such that if $\delta$ in (2.16) satisfies $\delta\leq\delta_{0}$, we
have for any $1\leq l\leq 4$ that
$\begin{array}[]{ll}&\displaystyle\sum_{\nu=1}^{l}\Big{\\{}\left\|(1+|x|)^{\nu}\left(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma,\nabla^{\nu+2}\sigma\right)\right\|+\left\|(1+|x|)^{\nu}\left(\nabla^{\nu+1}v,\nabla^{\nu+1}\vartheta,\nabla^{\nu+2}\vartheta\right)\right\|\Big{\\}}\\\\[5.69054pt]
&\leq
C\left\\{\|\nabla(\sigma,v,\vartheta)\|+\|b_{1}\|_{J^{5}}\|c_{1}\|_{J^{5}}+\|b_{2}\|_{J^{5}}\|c_{2}\|_{N^{5}}+\displaystyle\sum_{\nu=1}^{l}\left\|(1+|x|)^{\nu}\nabla^{\nu-1}(\tilde{f},\tilde{h})\right\|\right.\\\\[5.69054pt]
&\left.\qquad+\displaystyle\sum_{\nu=1}^{l}\left\|(1+|x|)^{\nu}\nabla^{\nu}(g,\tilde{h})\right\|\right\\}\end{array}$
(2.66)
where $C>0$ is a constant depending only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
Proof. The proof is divided into four steps.
Step 1. Using the Friedrichs mollifier, we may assume that
$(\sigma,v,\vartheta)\in\mathcal{H}^{\infty,\infty,\infty}$. For any multi-
index $\alpha$ with $|\alpha|=l-1$, applying $\partial_{x}^{\alpha}$ to
$(\ref{2.3})_{2}$, then taking the $L^{2}$ inner product with
$(1+|x|)^{2l}\nabla\partial_{x}^{\alpha}\sigma$ on the resultant equation and
summing up $\alpha$, we have
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left|\langle|\nabla^{l+1}v|,(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|+\left|\langle\nabla^{l}\sigma,(1+|x|)^{2l-1}\nabla^{l+1}\sigma\rangle\right|\right.\\\\[5.69054pt]
&\left.\,\,\,\,\,+\left|\langle\nabla^{l}\Delta\vartheta,(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|+\left|\langle\nabla^{l-1}\\{-(b_{1}\cdot\nabla)c_{1}+\tilde{f}\\},(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|\right\\}\end{array}$
(2.67)
From $(\ref{2.3})_{2}$, we also obtain
$\begin{array}[]{ll}\displaystyle\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|^{2}&\leq
C\left\\{\left\|(1+|x|)^{l}\nabla^{l}\sigma\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\right\|^{2}\right.\\\\[5.69054pt]
&\quad+\left|\langle\nabla^{l}\Delta\vartheta+\nabla^{l-1}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l}\nabla^{l+2}\sigma\rangle\right|\bigg{\\}}\end{array}$
(2.68)
Thus, it follows from a linear combination of (2.67) and (2.68) that
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\right\|^{2}+\Big{(}\left|\langle|\nabla^{l+1}v|,(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|\right.\\\\[8.53581pt]
&\quad+\left|\langle\nabla^{l}\sigma,(1+|x|)^{2l-1}\nabla^{l+1}\sigma\rangle\right|+\left|\langle\nabla^{l}\Delta\vartheta,(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|\\\\[8.53581pt]
&\quad+\left|\langle\nabla^{l-1}\tilde{f},(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|+\left|\langle\nabla^{l}\Delta\vartheta,(1+|x|)^{2l}\nabla^{l+2}\sigma\rangle\right|\Big{)}\\\\[8.53581pt]
&\quad+\left|\langle\nabla^{l-1}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l}\nabla^{l+2}\sigma\rangle\right|+\left|\langle\nabla^{l-1}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|\bigg{\\}}\\\\[8.53581pt]
&=C\left\\{\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\right\|^{2}+I_{1}+I_{2}+I_{3}\right\\}.\end{array}$
(2.69)
The cauchy inequality implies that
$\begin{array}[]{ll}I_{1}&\leq\eta\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}+C_{\eta}\Big{\\{}\|(1+|x|)^{l}\nabla^{l+1}v\|^{2}\\\\[5.69054pt]
&\quad+\|(1+|x|)^{l-1}\nabla^{l}\sigma\|^{2}+\|(1+|x|)^{l}\nabla^{l+2}\vartheta\|^{2}+\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\|^{2}\Big{\\}}\end{array}$
(2.70)
For $I_{2}$, notice that
$\left\|(1+|x|)^{|\alpha|+|\beta|}|\partial_{x}^{\alpha}b_{1}||\partial_{x}^{\beta}c_{1}|\right\|\leq
C\|b\|_{J^{5}}\|c\|_{J^{5}}$ (2.71)
for any multi-index $\alpha,\beta$ with $|\alpha|\leq 1$ or $|\beta|\leq 1$
and $|\alpha|,|\beta|\leq 5$. If $1\leq l\leq 3$, since
$(1+|x|)^{l}\nabla^{l-1}\\{(b_{1}\cdot\nabla)c_{1}\\}\in L^{2}$
as follows from (2.71), we have
$I_{2}\leq\eta\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}.$
(2.72)
If $l=4$, we get from integration by parts that
$\begin{array}[]{ll}I_{2}&\leq\left|\langle\nabla^{l}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l}\nabla^{l+1}\sigma\rangle\right|+\left|\langle\nabla^{l-1}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l-1}\nabla^{l+1}\sigma\rangle\right|\\\\[5.69054pt]
&=I_{2,1}+I_{2,2}\end{array}$ (2.73)
Using the Leibniz formula, we have
$\begin{array}[]{ll}I_{2,1}&\leq\displaystyle\sum_{|\alpha|=4}\displaystyle\sum_{\beta\leq\alpha,\,\,|\beta|=1,2}C_{\alpha}^{\beta}\left|\langle(\partial_{x}^{\alpha-\beta}b_{1}\cdot\nabla)\partial_{x}^{\beta}c_{1},(1+|x|)^{2l}\nabla^{l+1}\sigma\rangle\right|\\\\[8.53581pt]
&\quad+\displaystyle\sum_{|\alpha|=4}\displaystyle\sum_{\beta\leq\alpha,\,\,|\beta|=0,3,4}C_{\alpha}^{\beta}\left|\langle(\partial_{x}^{\alpha-\beta}b_{1}\cdot\nabla)\partial_{x}^{\beta}c_{1},(1+|x|)^{2l-1}\nabla^{l+1}\sigma\rangle\right|\\\\[8.53581pt]
&=I^{1}_{2,1}+I^{2}_{2,1}\end{array}$ (2.74)
By integration by parts, $I^{1}_{2,1}$ can be estimated as follows
$\begin{array}[]{ll}I^{1}_{2,1}&\leq C\left\|(1+|x|)^{2}\nabla
b_{1}\right\|_{L^{\infty}}\Big{\\{}\left\|(1+|x|)^{2}\nabla^{3}c_{1}\right\|\left\|(1+|x|)^{4}\nabla^{6}\sigma\right\|\\\\[8.53581pt]
&\quad+\displaystyle\sum_{\nu=1}^{2}\left\|(1+|x|)^{\nu}\nabla^{\nu+2}c_{1}\right\|\left\|(1+|x|)^{4}\nabla^{5}\sigma\right\|\Big{\\}}\\\\[8.53581pt]
&\quad+(\mbox{the same term except for the exchang
of}\,\,b_{1}\,\,and\,\,c_{1})\\\\[8.53581pt]
&\leq\eta\left\|(1+|x|)^{4}(\nabla^{5}\sigma,\nabla^{6}\sigma)\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}\end{array}$
(2.75)
For $I^{2}_{2,1}$, we deduce from (2.71) that
$I^{2}_{2,1}\leq\eta\left\|(1+|x|)^{4}\nabla^{5}\sigma\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}.$
(2.76)
Combining (2.73)-(2.75), we obtain
$I_{2,1}\leq\eta\left\|(1+|x|)^{4}(\nabla^{5}\sigma,\nabla^{6}\sigma)\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}.$
(2.77)
Similarly, we can also get
$I_{2,2}\leq\eta\left\|(1+|x|)^{4}(\nabla^{5}\sigma,\nabla^{6}\sigma)\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}.$
(2.78)
Thus, it follows from (2.72), (2.73), (2.77) and (2.78) that
$I_{2}\leq\eta\left\|(1+|x|)^{l}(\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}.$
(2.79)
Similar to the estimate of $I_{2}$, we have if $1\leq l\leq 3$,
$\displaystyle
I_{3}\leq\eta\left\|(1+|x|)^{l}\nabla^{l}\sigma\right\|^{2}+C_{\eta}\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}},$
(2.80)
and if $l=4$,
$\displaystyle
I_{3}\leq\eta\left\|(1+|x|)^{4}\nabla^{4}\sigma\right\|^{2}+C_{\eta}\left(\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}+\left\|(1+|x|)^{3}\nabla^{3}\sigma\right\|^{2}\right).$
(2.81)
Substituting (2.70), (2.79)-(2.81) into (2.69), we arrive at
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l+1}\vartheta\right\|^{2}\right.\\\\[5.69054pt]
&\left.\quad+\left\|(1+|x|)^{l-1}(\nabla^{l-1}\sigma,\nabla^{l}\sigma)\right\|^{2}+\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}\right\\}.\end{array}$
(2.82)
Step 2. For any multi-index $\alpha$ with $|\alpha|=l$, applying
$\partial_{x}^{\alpha}$ to $(\ref{2.3})_{2}$, then taking the $L^{2}$ inner
product with $(1+|x|)^{2l}\partial_{x}^{\alpha}v$ on the resultant equation,
we have from integration by parts that
$\begin{array}[]{ll}&\mu\|(1+|x|)^{l}\nabla\partial_{x}^{\alpha}v\|^{2}+\mu\langle\nabla\partial_{x}^{\alpha}v,2l(1+|x|)^{2l-1}\frac{x}{|x|}\partial_{x}^{\alpha}v\rangle+(\mu+\mu^{\prime})\|(1+|x|)^{l}\nabla\cdot\partial_{x}^{\alpha}v\|^{2}\\\\[5.69054pt]
&+(\mu+\mu^{\prime})\langle\nabla\cdot\partial_{x}^{\alpha}v,2l(1+|x|)^{2l-1}\frac{x}{|x|}\partial_{x}^{\alpha}v\rangle+\langle\nabla\partial_{x}^{\alpha}\sigma,(1+|x|)^{2l}\partial_{x}^{\alpha}v\rangle\\\\[5.69054pt]
&-\kappa\gamma_{1}\langle\nabla\Delta\partial_{x}^{\alpha}\sigma,(1+|x|)^{2l}\partial_{x}^{\alpha}v\rangle-\kappa\gamma_{2}\langle\nabla\Delta\partial_{x}^{\alpha}\vartheta,(1+|x|)^{2l}\partial_{x}^{\alpha}v\rangle\\\\[5.69054pt]
&=\langle\nabla\Delta\partial_{x}^{\alpha}\\{-(b_{1}\cdot\nabla)c_{1}+\tilde{f}\\},(1+|x|)^{2l}\partial_{x}^{\alpha}v\rangle\end{array}$
(2.83)
Applying $\partial_{x}^{\alpha}$ to $(\ref{2.3})_{1}$, then taking the $L^{2}$
inner product with $(1+|x|)^{2l}\partial_{x}^{\alpha}v$ on the resultant
equation, we have from integration by parts that
$\begin{array}[]{ll}&-\langle\partial_{x}^{\alpha}v,(1+|x|)^{2l}\nabla\partial_{x}^{\alpha}\sigma\rangle-\langle\partial_{x}^{\alpha}v,2l(1+|x|)^{2l-1}\frac{x}{|x|}\partial_{x}^{\alpha}\sigma\rangle\\\\[5.69054pt]
&\hskip
85.35826pt+\langle\nabla\partial_{x}^{\alpha}((a\cdot\nabla)\sigma),(1+|x|)^{2l}\partial_{x}^{\alpha}\sigma\rangle=\langle\partial_{x}^{\alpha}g,(1+|x|)^{2l}\partial_{x}^{\alpha}\sigma\rangle\end{array}$
(2.84)
Canceling the term
$-\langle\partial_{x}^{\alpha}v,(1+|x|)^{2l}\nabla\partial_{x}^{\alpha}\sigma\rangle$
by adding (2.84) to (2.83), and taking summation with respect to $\alpha$, we
obtain
$\begin{array}[]{ll}\displaystyle\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}&\leq
C\left\\{\Big{(}\left|\langle\nabla^{l+1}v,(1+|x|)^{2l}\nabla^{l}v\rangle\right|+\left|\langle\nabla^{l}v,(1+|x|)^{2l-1}\nabla^{l}\sigma\rangle\right|\right.\\\\[5.69054pt]
&\quad+\left|\langle\nabla^{l}g,(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|+\left|\langle\nabla^{l}\tilde{f},(1+|x|)^{2l}\nabla^{l}v\rangle\right|\Big{)}\\\\[5.69054pt]
&\quad+\left|\langle\nabla^{l}\\{(a\cdot\nabla)\sigma\\},(1+|x|)^{2l}\nabla^{l}\sigma\rangle\right|+\left|\langle\nabla^{l}\\{(b_{1}\cdot\nabla)c_{1}\\},(1+|x|)^{2l}\nabla^{l}v\rangle\right|\\\\[5.69054pt]
&\quad+\left|\langle\nabla\Delta\nabla^{l}\sigma,(1+|x|)^{2l}\nabla^{l}v\rangle\right|+\left|\langle\nabla\Delta\nabla^{l}\vartheta,(1+|x|)^{2l}\nabla^{l}v\rangle\right|\Big{\\}}\\\\[5.69054pt]
&=C\\{I_{4}+I_{5}+I_{6}+I_{7}+I_{8}\\}\end{array}$ (2.85)
Integration by parts and the Cauchy inequality imply that
$\begin{array}[]{ll}I_{4}&\leq\eta\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}v)\right\|^{2}\\\\[5.69054pt]
&\quad+C_{\eta}\left\\{\|(1+|x|)^{l-1}\nabla^{l}v\|^{2}+\|(1+|x|)^{l}\nabla^{l}g\|^{2}+\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\|^{2}\right\\}.\end{array}$
(2.86)
Similar to the estimate of $I_{2,1}$, we have
$I_{6}\leq\eta\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}+C_{\eta}\left\\{\|(1+|x|)^{l-1}\nabla^{l}v\|^{2}+\|b_{1}\|^{2}_{J^{k+1}}\|c_{1}\|^{2}_{J^{k+1}}\right\\}.$
(2.87)
For $I_{7}$, we deduce from integration by parts and $(\ref{2.3})_{1}$ that
$\begin{array}[]{ll}I_{7}&\leq\left|\langle\Delta\nabla^{l}\sigma,2l(1+|x|)^{2l-1}\frac{x}{|x|}\cdot\nabla^{l}v\rangle\right|+\left|\langle\Delta\nabla^{l}\sigma,(1+|x|)^{2l}\nabla^{l}(\nabla\cdot
v)\rangle\right|\\\\[5.69054pt] &\leq
2l\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l-1}|\nabla^{l}v|\rangle\right|+\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}\nabla^{l}\\{(a\cdot\nabla)\sigma\\}\rangle\right|\\\\[5.69054pt]
&\quad+\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l-1}\nabla^{l}g|\rangle\right|\\\\[5.69054pt]
&\leq\eta\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|^{2}+C_{\eta}\left(\|(1+|x|)^{l-1}\nabla^{l}v\|^{2}+\|(1+|x|)^{l}\nabla^{l}g\|^{2}\right)+A\end{array}$
(2.88)
where
$\begin{array}[]{ll}A&=\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}\nabla^{l}\\{(a\cdot\nabla)\sigma\\}\rangle\right|\\\\[5.69054pt]
&=\displaystyle\sum_{|\alpha|=l}\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}\partial_{x}^{\alpha}\\{(a\cdot\nabla)\sigma\\}\rangle\right|\\\\[5.69054pt]
&\leq\displaystyle\sum_{|\alpha|=l}\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}(a\cdot\nabla)\partial_{x}^{\alpha}\sigma\rangle\right|\\\\[5.69054pt]
&\quad+\displaystyle\sum_{|\alpha|=l}\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}\partial_{x}^{\alpha-\beta}a\cdot\partial_{x}^{\beta}\nabla\sigma)\rangle\right|=A_{1}+A_{2}.\end{array}$
(2.89)
The Sobolev inequality and the Cauchy inequality imply that
$\begin{array}[]{ll}A_{1}&\leq
C\|a\|_{L^{\infty}}\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|\left\|(1+|x|)^{l}\nabla^{l+1}\sigma\right\|\\\\[5.69054pt]
&\leq
C\delta\left\|(1+|x|)^{l}(\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}\end{array}$
(2.90)
$\begin{array}[]{ll}A_{2}&\leq\displaystyle\sum_{|\alpha|=l}\bigg{\\{}\displaystyle\sum_{\beta<\alpha,\,|\alpha-\beta|\leq\frac{|\alpha|}{2}}+\displaystyle\sum_{\beta<\alpha,\,|\alpha-\beta|>\frac{|\alpha|}{2}}\bigg{\\}}C_{\alpha}^{\beta}\left|\langle\nabla^{l+2}\sigma,(1+|x|)^{2l}\partial_{x}^{\alpha-\beta}a\cdot\partial_{x}^{\beta}\nabla\sigma)\rangle\right|\\\\[8.53581pt]
&\leq\displaystyle\sum_{|\alpha|=l}\displaystyle\sum_{\beta<\alpha,\,|\alpha-\beta|\leq\frac{|\alpha|}{2}}C_{\alpha}^{\beta}\left\|(1+|x|)^{|\alpha-\beta|-1}\partial_{x}^{\alpha}a\right\|_{L^{\infty}}\left\|(1+|x|)^{|\beta|+1}\nabla^{|\beta|+1}\sigma\right\|\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|\\\\[8.53581pt]
&\quad+\displaystyle\sum_{|\alpha|=l}\displaystyle\sum_{\beta<\alpha,\,|\alpha-\beta|>\frac{|\alpha|}{2}}C_{\alpha}^{\beta}\left\|(1+|x|)^{|\alpha-\beta|-1}\partial_{x}^{\alpha}a\right\|\left\|(1+|x|)^{|\beta|+1}\nabla^{|\beta|+1}\sigma\right\|_{L^{\infty}}\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|\\\\[8.53581pt]
&\leq
C\delta\displaystyle\sum_{\nu=l}^{l}\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}.\end{array}$
(2.91)
Combining (2.88)-(2.91), we obtain
$\begin{array}[]{ll}I_{7}&\leq\eta\left\|(1+|x|)^{l}\nabla^{l+2}\sigma\right\|^{2}+C_{\eta}\left(\left\|(1+|x|)^{l-1}\nabla^{l}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l}g\right\|^{2}\right)\\\\[5.69054pt]
&\quad+C\delta\displaystyle\sum_{\nu=1}^{l}\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}.\end{array}$
(2.92)
Finally, similar to the estimate of $A$ and $I_{7}$, respectively, we have
$I_{5}\leq
C\delta\displaystyle\sum_{\nu=1}^{l}\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma)\right\|^{2}.$
(2.93)
and
$\begin{array}[]{ll}I_{8}&\leq\eta\left\|(1+|x|)^{l}\nabla^{l+2}\vartheta\right\|^{2}+C_{\eta}\left(\left\|(1+|x|)^{l-1}\nabla^{l}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l}g\right\|^{2}\right)\\\\[5.69054pt]
&\quad+C\delta\displaystyle\sum_{\nu=1}^{l}\left\|(1+|x|)^{l}(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\vartheta)\right\|^{2}.\end{array}$
(2.94)
Substituting (2.86), (2.87), (2.92)-(2.94) into (2.85), by the smallness of
$\eta$, we arrive at
$\begin{array}[]{ll}\displaystyle\left\|(1+|x|)^{l}\nabla^{l+1}v\right\|^{2}&\leq
C\left\\{\eta\left\|(1+|x|)^{l}\nabla^{l}\sigma\right\|^{2}+(\eta+\delta)\left\|(1+|x|)^{l+2}\nabla^{l+2}(\sigma,\vartheta)\right\|^{2}\right.\\\\[5.69054pt]
&\quad+C_{\eta}\Big{(}\left\|(1+|x|)^{l}\nabla^{l}g\right\|^{2}+\left\|(1+|x|)^{l-1}\nabla^{l}v\right\|^{2}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{f}\right\|^{2}\Big{)}\\\\[5.69054pt]
&\quad+\displaystyle\delta\sum_{\nu=1}^{l}\left\|(1+|x|)^{\nu}(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma)\right\|^{2}\Big{\\}}\end{array}$
(2.95)
Step 3. For any multi-index $\alpha$ with $|\alpha|=l$, applying
$\partial_{x}^{\alpha}$ to $(\ref{2.3})_{3}$, then taking the $L^{2}$ inner
product with $(1+|x|)^{2l}\partial_{x}^{\alpha}\vartheta$ on the resultant
equation, integrating by parts and summing up $\alpha$, we can get
$\begin{array}[]{ll}\tilde{\alpha}\left\|(1+|x|)^{l}\nabla^{l+1}\vartheta\right\|^{2}&\leq
C\left|\langle\nabla^{l+1}\vartheta,(1+|x|)^{2l-1}\nabla^{l}\vartheta|\rangle\right|+C\left|\langle\nabla^{l}\\{-(b_{2}\cdot\nabla)c_{2}+\tilde{h}\\},(1+|x|)^{2l}\nabla^{l}\vartheta\rangle\right|\\\\[5.69054pt]
&=I_{9}+I_{10}.\end{array}$ (2.96)
For $I_{9}$, the Cauchy inequality imply that
$I_{9}\leq\frac{\tilde{\alpha}}{4}\left\|(1+|x|)^{l}\nabla^{l+1}\vartheta\right\|^{2}+C\left\|(1+|x|)^{l-1}\nabla^{l}\vartheta\right\|^{2}.$
(2.97)
Similar to the estimate of $I_{7}$, $I_{10}$ can be estimated as follows
$\begin{array}[]{ll}I_{10}&\leq\displaystyle\frac{\tilde{\alpha}}{4}\left\|(1+|x|)^{l}\nabla^{l+1}\vartheta\right\|^{2}\\\\[5.69054pt]
&\qquad\qquad\qquad\quad+C\left\\{\left\|(1+|x|)^{l-1}\nabla^{l}\vartheta\right\|^{2}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{h}\right\|^{2}\right\\}.\end{array}$
(2.98)
Putting (2.97) and (2.98) into (2.96) gives
$\left\|(1+|x|)^{l}\nabla^{l+1}\vartheta\right\|^{2}\leq
C\left\\{\left\|(1+|x|)^{l-1}\nabla^{l}\vartheta\right\|^{2}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}+\left\|(1+|x|)^{l}\nabla^{l-1}\tilde{h}\right\|^{2}\right\\}.$
(2.99)
On the other hand, we also get from $(\ref{2.3})_{3}$ that
$\left\|(1+|x|)^{l}\nabla^{l+2}\vartheta\right\|^{2}\leq
C\left\|(1+|x|)^{l}\nabla^{l}h\right\|^{2}\leq
C\left(\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}+\left\|(1+|x|)^{l}\nabla^{l}\tilde{h}\right\|^{2}\right).$
(2.100)
Consequently, we deduce from (2.99) and (2.100) that
$\begin{array}[]{ll}&\left\|(1+|x|)^{l}(\nabla^{l+1}\vartheta,\nabla^{l+2}\vartheta)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left\|(1+|x|)^{l-1}\nabla^{l}\vartheta\right\|^{2}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}+\left\|(1+|x|)^{l}(\nabla^{l-1}\tilde{h},\nabla^{l}\tilde{h})\right\|^{2}\right\\}.\end{array}$
(2.101)
Step 4. Now, we begin to prove (2.66) by using the estimates in the above
three steps. We use the method of induction. First, for the case of $l=1$, we
derive from (2.69), (2.70), (2.72) and (2.80) that
$\begin{array}[]{ll}\displaystyle\left\|(1+|x|)(\nabla\sigma,\nabla^{2}\sigma,\nabla^{3}\sigma)\right\|^{2}&\leq
C\left\\{\left\|(1+|x|)\nabla^{2}v\right\|^{2}+\left\|(1+|x|)\tilde{f}\right\|^{2}+\left\|(1+|x|)\nabla^{3}\vartheta\right\|^{2}\right.\\\\[5.69054pt]
&\left.\quad\quad+\left\|\nabla\sigma\right\|^{2}+\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}\right\\}.\end{array}$
(2.102)
which, together with (2.95) and (2.101) with $l=1$ gives
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)(\nabla\sigma,\nabla^{2}\sigma,\nabla^{3}\sigma)\right\|^{2}+\left\|(1+|x|)\nabla^{2}v\right\|^{2}+\left\|(1+|x|)(\nabla^{2}\vartheta,\nabla^{3}\vartheta)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left\|\nabla(\sigma,v,\vartheta)\right\|^{2}+\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}+\left\|(1+|x|)(\tilde{f},\tilde{h},\nabla
g,\nabla\tilde{h})\right\|^{2}\right\\}.\end{array}$ (2.103)
by the smallness of $\eta$ and $\delta$. Thus, we assume for $l\geq 2$ that
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)^{l-1}(\nabla^{l-1}\sigma,\nabla^{l}\sigma,\nabla^{l+1}\sigma)\right\|^{2}+\left\|(1+|x|)^{l-1}\nabla^{l}v\right\|^{2}+\left\|(1+|x|)^{l-1}(\nabla^{l}\vartheta,\nabla^{l+1}\vartheta)\right\|^{2}\\\\[5.69054pt]
&\hskip 28.45274pt\leq
C\left\\{\left\|\nabla(\sigma,v,\vartheta)\right\|^{2}+\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}\right.\\\\[5.69054pt]
&\displaystyle\hskip
56.9055pt+\sum_{\nu=1}^{l-1}\left\|(1+|x|)^{\nu}(\nabla^{\nu-1}\tilde{f},\nabla^{\nu}\tilde{h},\nabla^{\nu}g,\nabla^{\nu-1}\tilde{h})\right\|^{2}\Big{\\}}.\end{array}$
(2.104)
Furthermore, the linear combination
$[M_{1}\times(\ref{2.69})+(\ref{2.56})]+M_{2}\times(\ref{2.75})$ for $M_{1}>0$
and $M_{2}>0$ large enough in turn gives
$\begin{array}[]{ll}&\displaystyle\left\|(1+|x|)^{l}\left(\nabla^{l}\sigma,\nabla^{l+1}\sigma,\nabla^{l+2}\sigma\right)\right\|^{2}+\left\|(1+|x|)^{l}\left(\nabla^{l+1}v,\nabla^{l+1}\vartheta,\nabla^{l+2}\vartheta\right)\right\|^{2}\\\\[5.69054pt]
&\leq
C\left\\{\left\|(1+|x|)^{l-1}(\nabla^{l-1}\sigma,\nabla^{l}\sigma,\nabla^{l}v,\nabla^{l}\vartheta)\right\|^{2}+\|b_{1}\|^{2}_{J^{5}}\|c_{1}\|^{2}_{J^{5}}+\|b_{2}\|^{2}_{J^{5}}\|c_{2}\|^{2}_{N^{5}}\right.\\\\[5.69054pt]
&\quad+\displaystyle\left\|(1+|x|)^{l}\nabla^{l-1}(\tilde{f},\tilde{h})\right\|+\left\|(1+|x|)^{l}\nabla^{l}(g,\tilde{h})\right\|+\delta\sum_{\nu=1}^{l-1}\left\|(1+|x|)^{\nu}\left(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma\right)\right\|^{2}\Big{\\}}\end{array}$
(2.105)
provided that $\eta$ and $\delta$ are small enough. Combining (2.104) with
(2.105), if $\delta>0$ is small enough, we can get (2.66). This completes the
proof of Lemma 2.3.
Combining Proposition 2.2 and Lemma 2.3, we have the following theorem.
###### Theorem 2.1.
There exists
$\delta_{0}=\delta_{0}(\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime},\tilde{\alpha})>0$
such that such that if $\delta$ in (2.16) satisfies $\delta\leq\delta_{0}$,
then (2.15) admits a solution
$(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{6,5,6}$ which satisfies the
estimate:
$\begin{array}[]{ll}&\|(\sigma,v,\vartheta)\|_{L^{6}}+\displaystyle\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu}\left(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma,\nabla^{\nu+2}\sigma\right)\right\|+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\nabla^{\nu}v\right\|\\\\[5.69054pt]
&+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\left(\nabla^{\nu}\vartheta,\nabla^{\nu+1}\vartheta\right)\right\|\\\\[5.69054pt]
&\quad\leq
C\left\\{\|b_{1}\|_{J^{5}}\|c_{1}\|_{J^{5}}+\|b_{2}\|_{J^{5}}\|c_{2}\|_{N^{5}}+\|(1+|x|)(g,\tilde{h})\|\right.\\\\[5.69054pt]
&\qquad\quad\displaystyle+\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu}\nabla^{\nu}(g,\tilde{h})\right\|+\displaystyle\sum_{\nu=0}^{3}\displaystyle\left\|(1+|x|)^{\nu+1}\nabla^{\nu}(\tilde{f},\tilde{h})\right\|\Big{\\}}\end{array}$
(2.106)
where $C>0$ is a constant depending only on
$\gamma_{1},\gamma_{2},\kappa,\mu,\mu^{\prime}$ and $\tilde{\alpha}$.
### 2.2 Proof of Theorem 1.1
In this subsection, we shall construct a solution to (1.19) by the contraction
mapping principle in $\dot{\Lambda}_{\epsilon}^{4,5,5}$. To this end, we
consider the following iteration system
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot
v\displaystyle+\frac{\tilde{\rho}_{P}}{\tilde{\rho}}(\tilde{v}\cdot\nabla)\sigma=g,\\\\[5.69054pt]
\displaystyle-\mu\Delta v-(\mu+\mu^{\prime})\nabla(\nabla\cdot
v)+\nabla\sigma-\kappa\gamma_{1}\nabla\Delta\sigma-\kappa\gamma_{2}\nabla\Delta\vartheta=-\bar{\rho}(\tilde{v}\cdot\nabla)\tilde{v}+\tilde{f},\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta\vartheta=-\bar{\eta}_{1}(\tilde{v}\cdot\nabla)\tilde{\vartheta}+\tilde{h},\end{array}\right.$
(2.110)
where
$\displaystyle\left\\{\begin{array}[]{ll}g=\displaystyle-\frac{\tilde{\rho}_{\theta}}{\tilde{\rho}}(\tilde{v}\cdot\nabla)\tilde{\vartheta}+\frac{G(x)}{\tilde{\rho}},\\\\[5.69054pt]
\tilde{f}=-(\tilde{\rho}-\bar{\rho})(\tilde{v}\cdot\nabla)\tilde{v}+\kappa\tilde{\rho}\left(\nabla\tilde{\sigma}\cdot\nabla^{2}\tilde{\rho}_{P}+\nabla\tilde{\rho}_{P}\cdot\nabla^{2}\tilde{\sigma}+\nabla\tilde{\rho}_{P}\Delta\tilde{\sigma}\right)+\kappa\left(\tilde{\rho}\tilde{\rho}_{P}-\bar{\rho}\bar{\rho}_{P}\right)\nabla\Delta\tilde{\sigma}\\\\[5.69054pt]
\qquad+\kappa\tilde{\rho}\left(\nabla\tilde{\vartheta}\cdot\nabla^{2}\tilde{\rho}_{\theta}+\nabla\tilde{\rho}_{\theta}\cdot\nabla^{2}\tilde{\vartheta}+\nabla\tilde{\rho}_{\theta}\Delta\tilde{\vartheta}\right)+\kappa\left(\tilde{\rho}\tilde{\rho}_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\nabla\Delta\tilde{\vartheta}+\tilde{\rho}F-\tilde{v}G,\\\\[5.69054pt]
\tilde{h}=-(\tilde{\eta}_{1}-\bar{\eta}_{1})(\tilde{v}\cdot\nabla)\tilde{\vartheta}+\tilde{\eta}_{2}(\tilde{v}\cdot\nabla)\tilde{\sigma}\displaystyle+\Psi(\tilde{v})-\tilde{\eta}_{3}G+H+\Phi(\tilde{\rho},\tilde{v})+\frac{\tilde{v}^{2}}{2}G-C_{\triangledown}\tilde{\theta}G,\\\\[5.69054pt]
\displaystyle\tilde{\eta}_{1}=\displaystyle\tilde{\rho}C_{\triangledown}-\frac{\tilde{\theta}\tilde{\rho}_{\theta}^{2}}{\tilde{\rho}\tilde{\rho}_{P}},\quad\tilde{\eta}_{2}=\displaystyle\frac{\tilde{\theta}\tilde{\rho}_{\theta}}{\tilde{\rho}},\quad\tilde{\eta}_{3}=\displaystyle\frac{\tilde{\theta}\tilde{\rho}_{\theta}}{\tilde{\rho}\tilde{\rho}_{P}},\quad\tilde{\theta}=\bar{\theta}+\tilde{\vartheta}.\\\\[5.69054pt]
\end{array}\right.$ (2.116)
Here,
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
is given, and
$\tilde{\rho}_{P}=\rho_{P}(\bar{P}+\tilde{\sigma},\bar{\theta}+\tilde{\vartheta}),\bar{\eta}_{1}=\eta_{1}(\bar{P},\bar{\theta})$,
etc.
#### 2.2.1 Introduction of solution map $T$ for (2.4)
Firstly, we apply Theorem 2.1 to (2.110) to get the weighted $L^{2}$ estimate.
Let
$\displaystyle
a=\displaystyle-\frac{\tilde{\rho}_{P}}{\tilde{\rho}}\tilde{v},\quad
b_{1}=c_{1}={\bar{\rho}}^{\frac{1}{2}}\tilde{v},\quad
b_{2}=\bar{\eta}_{1}\tilde{v},\quad c_{2}=\tilde{\vartheta},$ (2.117)
and $g,\tilde{f},\tilde{h}$ in Theorem 2.1 be defined as in (2.116). We choose
$\epsilon>0$ sufficiently small such that
$\frac{\bar{\rho}}{2}<\tilde{\rho}<2\bar{\rho}$, as follows from the sobolev
inequality. Assume that the assumptions of Theorem 2.1 hold and denote
$K_{0}=\displaystyle\sum_{\nu=0}^{3}\left\|(1+|x|)^{\nu+1}\nabla^{\nu}(G,F,H)\right\|+\left\|(1+|x|)^{4}\nabla^{4}(G,H)\right\|<\infty$
(2.118)
then we can check (2.17) and (2.18) easily and additionally we have
$\|(1+|x|)(g,\tilde{h})\|+\displaystyle\sum_{\nu=1}^{4}(1+|x|)^{\nu}\nabla^{\nu}(g,\tilde{h})\|+\displaystyle\sum_{\nu=0}^{3}(1+|x|)^{\nu+1}\nabla^{\nu}(\tilde{f},\tilde{h})\|\leq
C\left(\epsilon^{2}+K_{0}\right).$
for some constant $C=C(\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa)>0$.
Applying Theorem 2.1 to (2.110), we have the following lemma.
###### Lemma 2.4.
Let $(G,F,H)\in H^{4,3,4}$ satisfy (2.118). Then there exists a constant
$\epsilon_{0}>0$ such that if $\epsilon\leq\epsilon_{0}$, (2.110) with
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{6,5,6}$
admits a solution $(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{6,5,6}$ which
satisfies the estimate:
$\begin{array}[]{ll}&\|(\sigma,v,\vartheta)\|_{L^{6}}+\displaystyle\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu}\left(\nabla^{\nu}\sigma,\nabla^{\nu+1}\sigma,\nabla^{\nu+2}\sigma\right)\right\|+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\nabla^{\nu}v\right\|\\\\[8.53581pt]
&\quad+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\left(\nabla^{\nu}\vartheta,\nabla^{\nu+1}\vartheta\right)\right\|\leq
C\left(\epsilon^{2}+K_{0}\right)\end{array}$ (2.119)
where the constant $C$ depends only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$.
Based on Lemma 2.4, we can define the solution map
$T:\dot{\Lambda}_{\epsilon}^{4,5,5}\rightarrow\hat{\mathcal{H}}^{6,5,6}$ by
$(\sigma,v,\vartheta)=T(\tilde{\sigma},\tilde{v},\tilde{\vartheta})$. Since
the contraction mapping principle will be applied to prove Theorem 1.1, we
have to show that
$T(\tilde{\sigma},\tilde{v},\tilde{\vartheta})=(\sigma,v,\vartheta)\in\dot{\Lambda}_{\epsilon}^{4,5,5}$.
To this end, we first cite the following lemma which will play an important
role when we estimate the solution by the $L^{\infty}$ norm.
###### Lemma 2.5.
([6]) Let $E(x)$ be a scalar function satisfying
$\left|\partial_{x}^{\alpha}E(x)\right|\leq\frac{C_{\alpha}}{|x|^{|\alpha|+1}},\quad|\alpha|=0,1,2.$
(i) If $\phi(x)$ is a smooth scalar function of the form
$\phi=\nabla\cdot\phi_{1}+\phi_{2}$ satisfying
$L_{1}(\phi)\equiv\left\|(1+|x|)^{3}\phi\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}\phi_{1}\right\|_{L^{\infty}}+\|\phi_{2}\|_{L^{1}}<\infty,$
then we have for any multi-index $\alpha$ with $|\alpha|=0,1$
$\left|\partial_{x}^{\alpha}(E\ast\phi)(x)\right|\leq\frac{C_{\alpha}}{|x|^{|\alpha|+1}}L_{1}(\phi).$
(ii) If $\phi(x)$ is a smooth scalar function of the form
$\phi=\phi_{1}\phi_{2}$ satisfying
$L_{2}(\phi)\equiv\left\|(1+|x|)^{2}\phi\right\|_{L^{\infty}}+\left\|(1+|x|)^{3}(\nabla\phi_{1})\phi_{2}\right\|_{L^{\infty}}+\left\|(1+|x|)^{3}\phi_{1}(\nabla\phi_{2})\right\|_{1}<\infty,$
then we have for any multi-index $\alpha$ with $|\alpha|=1,2$
$\left|\partial_{x}^{\alpha}(E\ast\phi)(x)\right|\leq\frac{C_{\alpha}}{|x|^{|\alpha|}}L_{2}(\phi).$
Here $C_{\alpha}$ denotes a constant depending only on $\alpha$.
With the aid of the Helmholtz decomposition and the Fourier transform, the
solution of (2.110) can be formulated as follows, cf. [6].
$v=w+\nabla
p,\quad\displaystyle\sigma-\kappa\gamma_{1}\Delta\sigma=\Phi+\kappa\gamma_{2}\Delta\vartheta+(2\mu+\mu^{\prime})\Delta
p,\quad\vartheta=E\ast\Theta,$ (2.120)
where
$\displaystyle\left\\{\begin{array}[]{ll}w_{j}(x)=\displaystyle\sum_{i=1}^{3}E_{ij}\ast
f_{i}(x),\\\\[5.69054pt] p(x)=E_{0}\ast R(x),\\\\[5.69054pt]
\Phi=E_{0}\ast(\nabla\cdot f).\end{array}\right.$ (2.124)
and
$\displaystyle\left\\{\begin{array}[]{ll}E_{ij}(x)=\displaystyle\frac{1}{8\pi\mu}\left(\frac{\delta_{ij}}{|x|}-\frac{x_{i}x_{j}}{|x|^{3}}\right),\quad
E_{0}=-\frac{1}{4\pi|x|}\\\\[8.53581pt]
f_{i}=-\bar{\rho}(\tilde{v}\cdot\nabla)\tilde{v}_{i}+\tilde{f}_{i},\\\\[5.69054pt]
R(x)=\displaystyle-\frac{\tilde{\rho}_{\theta}}{\tilde{\rho}}(\tilde{v}\cdot\nabla)\sigma+g,\\\\[5.69054pt]
\Theta=\displaystyle\frac{1}{\tilde{\alpha}}\\{\bar{\eta}_{1}(\tilde{v}\cdot\nabla)\tilde{\vartheta}-\tilde{h}\\}.\end{array}\right.$
(2.129)
Now, we shall estimate the $L^{\infty}$ norm of the solution to (2.110) by
using Lemma 2.5.
###### Lemma 2.6.
Let $(G,F,H)\in H^{4,3,4}$ satisfy the following estimate:
$K\equiv\|(G,F,H)\|_{\mathcal{L}}+\left\|(1+|x|)^{4}\nabla^{4}(G,H)\right\|<\infty$
(2.130)
If $(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{6,5,6}$ is a solution to (2.110)
with
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
and satisfies (2.119), then $(\sigma,v,\vartheta)$ satisfies the estimate:
$\begin{array}[]{ll}&\displaystyle\sum_{\nu=0}^{1}\left\|(1+|x|)^{2}\nabla^{\nu}\sigma\right\|_{L^{\infty}}+\displaystyle\sum_{\nu=0}^{1}\left\|(1+|x|)^{\nu+1}\nabla^{\nu}(v,\vartheta)\right\|_{L^{\infty}}\\\\[11.38109pt]
&\qquad+\left\|(1+|x|)^{2}\nabla^{2}(v,\vartheta)\right\|_{L^{\infty}}\leq
C\left(\epsilon^{2}+K\right)\end{array}$ (2.131)
where the constant $C>0$ depends only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$.
Proof. First, we deduce an estimate on $f$. Since
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$,
there exits $\tilde{V}_{1}=(\tilde{V}^{i}_{1})_{1\leq i\leq 3}$ and
$\tilde{V}_{2}$ such that
$\nabla\cdot\tilde{v}=\nabla\cdot\tilde{V}_{1}+\tilde{V}_{2}$, and
$\left\|(1+|x|)^{3}\tilde{V}_{1}\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}\tilde{V}_{2}\right\|_{L^{1}}\leq\epsilon$
Thus we have
$\begin{array}[]{ll}f_{i}&=-\tilde{\rho}(\tilde{v}\cdot\nabla)\tilde{v}_{i}+\kappa\tilde{\rho}[\nabla\tilde{\sigma}\cdot\nabla^{2}\tilde{\rho}_{P}+\nabla\tilde{\rho}_{P}\cdot\nabla^{2}\tilde{\sigma}+\nabla\tilde{\rho}_{P}\Delta\tilde{\sigma}]_{i}+\kappa\left(\tilde{\rho}\tilde{\rho}_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\Delta\tilde{\theta}_{x_{i}}\\\\[5.69054pt]
&\quad+\kappa\tilde{\rho}[\nabla\tilde{\vartheta}\cdot\nabla^{2}\tilde{\rho}_{\theta}+\nabla\tilde{\rho}_{\theta}\cdot\nabla^{2}\tilde{\vartheta}+\nabla\tilde{\rho}_{\theta}\Delta\tilde{\vartheta}]_{i}+\kappa\left(\tilde{\rho}\tilde{\rho}_{P}-\bar{\rho}\bar{\rho}_{P}\right)\Delta\tilde{\sigma}_{x_{i}}+\tilde{\rho}F_{i}-\tilde{v}_{i}G\\\\[5.69054pt]
&=\nabla\cdot\left(-\tilde{\rho}\tilde{v}_{i}\tilde{v}+\tilde{\rho}\tilde{v}_{i}\tilde{V}_{1}+\rho
F_{1,i}\right)+\left\\{-\tilde{\rho}(\tilde{V}_{1}\cdot\nabla)\tilde{v}_{i}-\tilde{v}_{i}(\tilde{V}_{1}\cdot\nabla)\tilde{\rho}+\tilde{\rho}\tilde{v}_{i}\tilde{V}_{2}+\tilde{\rho}F_{2,i}\right.\\\\[5.69054pt]
&\quad+\kappa\tilde{\rho}[\nabla\tilde{\vartheta}\cdot\nabla^{2}\tilde{\rho}_{\theta}+\nabla\tilde{\rho}_{\theta}\cdot\nabla^{2}\tilde{\vartheta}+\nabla\tilde{\rho}_{\theta}\Delta\tilde{\vartheta}]_{i}+\kappa\tilde{\rho}[\nabla\tilde{\sigma}\cdot\nabla^{2}\tilde{\rho}_{P}+\nabla\tilde{\rho}_{P}\cdot\nabla^{2}\tilde{\sigma}+\nabla\tilde{\rho}_{P}\Delta\tilde{\sigma}]_{i}\\\\[5.69054pt]
&\left.\quad+\tilde{v}_{i}(\tilde{v}\cdot\nabla)\tilde{\rho}-\nabla\tilde{\rho}\cdot
F_{1,i}+\kappa\left(\tilde{\rho}\tilde{\rho}_{P}-\bar{\rho}\bar{\rho}_{P}\right)\Delta\tilde{\sigma}_{x_{i}}-\tilde{v}_{i}G+\kappa\left(\tilde{\rho}\tilde{\rho}_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\Delta\tilde{\vartheta}_{x_{i}}\right\\}\\\\[5.69054pt]
&=\nabla\cdot f_{i,1}+f_{i,2}.\end{array}$
Here $[\cdots]_{i}$ denotes the $i-th$ component of the vector $[\cdots]$.
By
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
and (2.119), using the Sobolev inequality and mean value theorem, we have
$\left\|(1+|x|)^{3}f_{i}\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}f_{1,i}\right\|_{L^{\infty}}+\left\|f_{2,i}\right\|_{L^{1}}\leq
C(\epsilon^{2}+K_{1})$
and
$\left\|(1+|x|)^{3}\nabla
f_{i}\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}f_{i}\right\|_{L^{\infty}}\leq
C(\epsilon^{2}+K_{1})$
where
$K_{1}=\left\|(1+|x|)^{3}(F,G,\nabla F,\nabla
G)\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}F_{1}\right\|_{L^{\infty}}+\|F_{2}\|_{L^{1}}.$
Hence, by (i) and (ii) of Lemma 2.5, we obtain
$|x||w_{j}|,\,\,|x|^{2}\left(|\Phi|,|\nabla\Phi|,|\nabla
w_{j}|,|\nabla^{2}w_{j}|\right)\leq C(\epsilon^{2}+K_{1}).$ (2.132)
As for $\nabla p,\nabla^{2}p,\nabla^{3}p$, due to [7],
$|x||\nabla p|,\,\,|x|^{2}\left(|\nabla^{2}p|,|\nabla^{3}p|\right)\leq
C(\epsilon^{2}+K_{0}+K_{2}).$ (2.133)
where
$K_{2}=\left\|(1+|x|)^{2}G\right\|_{L^{\infty}}+\left\|(1+|x|)^{3}(\nabla
G,\nabla^{2}G)\right\|_{L^{\infty}}$
Combining $(\ref{2.86})_{1}$, (2.132) and (2.133) yields
$|x||v|,\,\,|x|^{2}\left(|\nabla v|,|\nabla^{2}v|\right)\leq
C(\epsilon^{2}+K_{0}+K_{1}+K_{2}).$ (2.134)
Next, we turn to estimate $\vartheta$. To this end, we rewrite $\Theta$ as
$\begin{array}[]{ll}\Theta&=\displaystyle\frac{1}{\tilde{\alpha}}\left\\{\tilde{\eta}_{1}(\tilde{v}\cdot\nabla)\tilde{\vartheta}-\tilde{\eta}_{2}(\tilde{v}\cdot\nabla)\tilde{\sigma}\displaystyle-\Psi(\tilde{v})+\tilde{\eta}_{3}G-H-\Phi(\tilde{\rho},\tilde{v})-\frac{\tilde{v}^{2}}{2}G+C_{\triangledown}(\tilde{\vartheta}+\bar{\theta})G\right\\}\\\\[5.69054pt]
&=\displaystyle\frac{1}{\tilde{\alpha}}\nabla\cdot\left\\{(\tilde{\eta}_{1}\tilde{\vartheta}-\tilde{\eta}_{2}\tilde{\sigma})(\tilde{v}-\tilde{V}_{1})+\tilde{\eta}_{3}G_{1}-H_{1}-\frac{\tilde{v}^{2}}{2}G_{1}+C_{\triangledown}\tilde{\vartheta}G_{1}\right\\}\\\\[5.69054pt]
&\quad+\displaystyle\frac{1}{\tilde{\alpha}}\nabla\cdot\left\\{(\tilde{V}_{1}\cdot\nabla)(\eta_{1}\tilde{\vartheta}-\eta_{2}\tilde{\sigma})-(\tilde{\eta}_{1}\tilde{\vartheta}-\tilde{\eta}_{2}\tilde{\sigma})\tilde{V}_{2}-\nabla\tilde{\eta}_{1}\tilde{v}\tilde{\vartheta}+\nabla\tilde{\eta}_{2}\tilde{v}\tilde{\sigma}-\Psi(\tilde{v})-\Phi(\tilde{\rho},\tilde{v})\right.\\\\[5.69054pt]
&\quad\left.-\nabla\tilde{\eta}_{3}\cdot
G_{1}+\tilde{\eta}_{3}G_{2}+H_{2}+\nabla\cdot\tilde{v}\tilde{v}\cdot
G_{1}-\displaystyle\frac{\tilde{v}^{2}}{2}G_{2}+C_{\triangledown}\nabla\tilde{\vartheta}\cdot
G_{1}-C_{\triangledown}G_{2}(\tilde{\vartheta}+\bar{\theta})\right\\}\\\\[5.69054pt]
&=\nabla\cdot\Theta_{1}+\Theta_{2}.\end{array}$
and
$\begin{array}[]{ll}\Theta&=\displaystyle\sum_{i=1}^{3}(-\frac{1}{\tilde{\alpha}}\tilde{\eta}_{2}\tilde{v}_{i})\tilde{\sigma}_{x_{i}}+\frac{1}{\tilde{\alpha}}\left\\{\tilde{\eta}_{1}(\tilde{v}\cdot\nabla)\tilde{\vartheta}\displaystyle-\Psi(\tilde{v})+\tilde{\eta}_{3}G-H-\Phi(\tilde{\rho},\tilde{v})-\frac{\tilde{v}^{2}}{2}G+C_{\triangledown}(\tilde{\vartheta}+\bar{\theta})G\right\\}\\\\[5.69054pt]
&=\displaystyle\sum_{i=1}^{3}\Theta^{i}_{1}\Theta^{i}_{2}+\Theta_{3}\end{array}$
Since
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$,
it follows from (2.119) and the Sobolev inequality that
$\left\|(1+|x|)^{3}\Theta\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}\Theta_{1}\right\|_{L^{\infty}}+\left\|\Theta_{2}\right\|_{L^{1}}\leq
C(\epsilon^{2}+K_{3}),$
$\left\|(1+|x|)^{3}\nabla\Theta_{3}\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}\Theta_{3}\right\|_{L^{\infty}}\leq
C(\epsilon^{2}+K_{3}),$
$\left\|(1+|x|)^{3}\Theta^{i}_{1}\Theta^{i}_{2}\right\|_{L^{\infty}}+\left\|(1+|x|)^{3}(\nabla\Theta^{i}_{1})\Theta^{i}_{2}\right\|_{L^{\infty}}+\left\|(1+|x|)^{3}\Theta^{i}_{1}(\nabla\Theta^{i}_{2})\right\|_{L^{\infty}}\leq
C\epsilon^{2},$
where
$K_{3}=\left\|(1+|x|)^{3}(G,H,\nabla G,\nabla
H)\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}(G_{1},H_{1})\right\|_{L^{\infty}}+\|(G_{2},H_{2})\|_{L^{1}}.$
Thus, it follows from (i) and (ii) of Lemma 2.5 that
$|x||\vartheta|,\,\,|x|^{2}\left(|\nabla\vartheta|,|\nabla^{2}\vartheta|\right)\leq
C(\epsilon^{2}+K_{3}).$ (2.135)
Finally, for the estimate of $\vartheta$, taking the Fourier transform on both
side of $(\ref{2.86})_{2}$, we have
$\begin{array}[]{ll}\sigma(x)&=\displaystyle\frac{1}{(2\pi)^{3/2}}\mathcal{F}^{-1}\left(\frac{1}{1+\kappa\gamma_{1}|\xi|^{2}}\right)\ast
L(\Phi,p,\vartheta)\\\\[8.53581pt]
&=\displaystyle\frac{1}{(4\pi\kappa\gamma_{1})^{3/2}}\int_{\mathbb{R}^{3}}\left(\int_{0}^{\infty}e^{-t-\frac{|y|^{2}}{4\gamma_{1}t}}t^{-\frac{3}{2}}\,dt\right)(\Phi+\kappa\gamma_{2}\Delta\vartheta+(2\mu+\mu^{\prime})\Delta
p)(x-y)\,dy\end{array}$ (2.136)
where
$L(\Phi,p,\vartheta)=\Phi+\kappa\gamma_{2}\Delta\vartheta+(2\mu+\mu^{\prime})\Delta
p$ and we have used the fact that (cf. [18])
$\displaystyle\mathcal{F}^{-1}\left(\frac{1}{1+\kappa\gamma_{1}|\xi|^{2}}\right)=\displaystyle\frac{1}{(2\kappa\gamma_{1})^{3/2}}\int_{0}^{\infty}e^{-t-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}t^{-\frac{3}{2}}\,dt$
(2.137)
Note that the right hand of (2.137) is the so-called Bessel potential. We
deduce from (2.132), (2.133) and (2.135) that
$\begin{array}[]{ll}|x|^{2}|\sigma(x)|&\leq\displaystyle\frac{|x|^{2}}{(4\pi\kappa\gamma_{1})^{3/2}}\int_{\mathbb{R}^{3}}\left(\int_{0}^{\infty}e^{-t-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}t^{-\frac{3}{2}}\,dt\right)\frac{1}{|x-y|^{2}}\,dy\\\\[8.53581pt]
&\quad\times\displaystyle\sup_{(x-y)\in\mathbb{R}^{3}}\left\\{|x-y|^{2}\left|\Phi+\kappa\gamma_{2}\Delta\vartheta+(2\mu+\mu^{\prime})\Delta
p\right|(x-y)\right\\}\\\\[8.53581pt] &\leq C(\epsilon^{2}+K)\cdot
A\end{array}$ (2.138)
where $K$ is defined by $(\ref{2.89})$ and
$\begin{array}[]{ll}A&=\displaystyle\int_{\mathbb{R}^{3}}\left(\int_{0}^{\infty}e^{-t-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}t^{-\frac{3}{2}}\,dt\right)\frac{|x|^{2}}{|x-y|^{2}}\,dy\\\\[8.53581pt]
&=\displaystyle\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{3}}\frac{|x|^{2}}{|x-y|^{2}}e^{-t-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt\\\\[8.53581pt]
&\leq\displaystyle
2\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{3}}e^{-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt+2\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{3}}\frac{|y|^{2}}{|x-y|^{2}}e^{-t-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt\\\\[8.53581pt]
&\leq\displaystyle
16(\kappa\gamma_{1})^{3/2}\int_{0}^{\infty}e^{-t}\,dt\int_{\mathbb{R}^{3}}e^{-|z|^{2}}\,dz+2\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{B(x,\,1)}\frac{|y|^{2}}{|x-y|^{2}}e^{-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt\\\\[8.53581pt]
&\quad\displaystyle+2\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{3}\setminus
B(x,\,1)}\frac{|y|^{2}}{|x-y|^{2}}e^{-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt\\\\[8.53581pt]
&\leq\displaystyle
C+8\kappa\gamma_{1}\int_{0}^{\infty}e^{-t}t^{-\frac{1}{2}}\left(\int_{B(x,\,1)}\frac{1}{|x-y|^{2}}\,dy\right)dt\\\\[8.53581pt]
&\displaystyle\quad+2\int_{0}^{\infty}e^{-t}t^{-\frac{3}{2}}\left(\int_{\mathbb{R}^{3}\setminus
B(x,\,1)}|y|^{2}e^{-\frac{|y|^{2}}{4\kappa\gamma_{1}t}}\,dy\right)dt\\\\[8.53581pt]
&\leq C\end{array}$ (2.139)
Here $B(x,\,1)$ denotes the unit ball in $\mathbb{R}^{3}$. Consequently, it
follows from (2.138) and (2.139) that
$|x|^{2}|\sigma(x)|\leq C(\epsilon^{2}+K)$ (2.140)
Differentiating the equation (2.136) and notice that
$\left\||x|^{2}|\nabla^{3}\vartheta|\right\|_{L^{\infty}}\leq
C\left\|\nabla(|x|^{2}|\nabla^{3}\vartheta|)\right\|_{1}\leq
C(\epsilon^{2}+K_{0}),$
by using the same argument as above, we can also obtain
$|x|^{2}|\nabla\sigma(x)|\leq C(\epsilon^{2}+K)$ (2.141)
Next, we consider the case of $|x|<1$. The Sobolev inequality and (2.119)
imply that
$\begin{array}[]{ll}&\|(\sigma,v,\vartheta)\|_{L^{\infty}}\leq
C\|\nabla(\sigma,v,\vartheta)\|_{1}\leq C(\epsilon^{2}+K_{0})\\\\[5.69054pt]
&\|\nabla^{\nu}(\sigma,v,\vartheta)\|_{L^{\infty}}\leq
C\|\nabla^{\nu+1}(\sigma,v,\vartheta)\|_{1}\leq
C(\epsilon^{2}+K_{0}),\quad\nu=1,\,2.\end{array}$ (2.142)
(2.131) thus follows from (2.134), (2.135), (2.140), (2.141) and (2.142). This
completes the proof of Lemma 2.6.
In the following Proposition, we show that the solution
$(\sigma,v,\vartheta)\in\dot{\Lambda}_{\epsilon}^{4,5,5}$.
###### Proposition 2.3.
There exits $c_{0}>0$ such that for any sufficiently small constant
$\epsilon>0$, if $(G,F,H)\in\mathcal{H}^{4,3,4}$ satisfies
$K+\left\|(1+|x|)^{-1}G\right\|_{L^{1}}\leq c_{0}\epsilon\quad(\mbox{$K$ is
defined in Lemma 2.6})$
then (2.110) with
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
admits a solution
$(\sigma,v,\vartheta)=T(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$.
Proof. By Lemmas 2.4 and 2.6, it follows that (2.110) has a solution
$(\sigma,v,\vartheta)\in\hat{\mathcal{H}}^{4,5,5}$ , which satisfies
$\left\|(\sigma,v,\vartheta)\right\|_{{\Lambda}^{4,5,5}}\leq
C(\epsilon^{2}+K)\leq C(\epsilon^{2}+c_{0}\epsilon),$
where the constant $C>0$ depends only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$. Thus
if we take $c_{0}\leq\frac{1}{2C}$ and $\epsilon>0$ is small enough, it
follows that $(\sigma,v,\vartheta)\in\Lambda_{\epsilon}^{4,5,5}$. Finally, we
define $V_{1}$ and $V_{2}$ by
$V_{1}=-\frac{\tilde{\rho}_{P}}{\tilde{\rho}}\tilde{v}\sigma,\quad
V_{2}=\nabla\cdot\left(\frac{\tilde{\rho}_{P}}{\tilde{\rho}}\tilde{v}\right)\sigma-\frac{\tilde{\rho}_{\theta}}{\tilde{\rho}}\tilde{v}\cdot\nabla\vartheta+\frac{G}{\tilde{\rho}}$
Then it follows from $(\ref{2.81})_{1}$ that
$\nabla\cdot v=\nabla\cdot V_{1}+V_{2}.$
Moreover, by
$(\tilde{\sigma},\tilde{v},\tilde{\vartheta})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
, (2.119) and (2.131), we have form the Sobolev inequality that
$\begin{array}[]{ll}\left\|(1+|x|)^{3}V_{1}\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}V_{2}\right\|_{L^{1}}&\leq
C\left\\{\epsilon^{2}+K+\left\|(1+|x|)^{-1}G\right\|_{L^{1}}\right\\}\\\\[5.69054pt]
&\leq C(\epsilon^{2}+c_{0}\epsilon)\leq\epsilon\end{array}$
if $c_{0}\leq\frac{1}{2C}$ and $\epsilon>0$ is sufficiently small. This
completes the proof of Proposition 2.3.
#### 2.2.2 Contraction of the solution map $T$
In this subsection, we shall show that the solution map $T$ for (2.110) is
contractive. Suppose that
$(\tilde{\sigma}^{j},\tilde{v}^{j},\tilde{\vartheta}^{j})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
and
$(\sigma^{j},v^{j},\vartheta^{j})=T(\tilde{\sigma}^{j},\tilde{v}^{j},\tilde{\vartheta}^{j})$
for $j=1,2$, then we deduce from (2.110) that
$\displaystyle\left\\{\begin{array}[]{ll}\nabla\cdot(v^{1}-v^{2})\displaystyle+\frac{\tilde{\rho}^{1}_{P}}{\tilde{\rho}^{1}}(\tilde{v}^{1}\cdot\nabla)(\sigma^{1}-\sigma^{2})=g,\\\\[5.69054pt]
\displaystyle-\mu\Delta(v^{1}-v^{2})-(\mu+\mu^{\prime})\nabla(\nabla\cdot(v^{1}-v^{2}))+\nabla(\sigma^{1}-\sigma^{2})-\kappa\gamma_{1}\nabla\Delta(\sigma^{1}-\sigma^{2})\\\
-\kappa\gamma_{2}\nabla\Delta(\vartheta^{1}-\vartheta^{2})=-\tilde{\rho}^{2}(\tilde{v}^{1}-\tilde{v}^{2})\cdot\nabla\tilde{v}^{1}-\tilde{\rho}^{2}(\tilde{v}^{2}\cdot\nabla)(\tilde{v}^{1}-\tilde{v}^{2})+\tilde{f},\\\\[5.69054pt]
\displaystyle-\tilde{\alpha}\Delta(\vartheta^{1}-\vartheta^{2})=-\tilde{\eta}_{1}^{2}[(\tilde{v}^{1}-\tilde{v}^{2})\cdot\nabla\tilde{\vartheta}^{1}+(\tilde{v}^{2}\cdot\nabla)(\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})]+\tilde{h},\end{array}\right.$
(2.147)
where
$\displaystyle\left\\{\begin{array}[]{ll}g=\displaystyle-\left(\frac{\tilde{\rho}_{P}^{1}}{\tilde{\rho}^{1}}-\frac{\tilde{\rho}_{P}^{2}}{\tilde{\rho}^{2}}\tilde{v}^{2}\right)\cdot\nabla\sigma^{2}-\left(\frac{\tilde{\rho}_{\theta}^{1}}{\tilde{\rho}^{1}}\tilde{v}^{1}\cdot\nabla\tilde{\vartheta}^{1}-\frac{\tilde{\rho}_{\theta}^{2}}{\tilde{\rho}^{2}}\tilde{v}^{2}\cdot\nabla\tilde{\vartheta}^{2}\right)+\left(\frac{1}{\tilde{\rho}^{1}}-\frac{1}{\tilde{\rho}^{2}}\right)G,\\\\[8.53581pt]
\tilde{f}=-(\tilde{\rho}^{1}-\tilde{\rho}^{2})(\tilde{v}^{1}\cdot\nabla)\tilde{v}^{1}-(\tilde{\rho}^{1}-\tilde{\rho}^{2})F-(\tilde{v}^{1}-\tilde{v}^{2})G+\big{\\{}\kappa\tilde{\rho}^{1}\left(\nabla\tilde{\sigma}^{1}\cdot\nabla^{2}\tilde{\rho}^{1}_{P}+\right.\\\\[5.69054pt]
\qquad\left.+\nabla\tilde{\rho}^{1}_{P}\cdot\nabla^{2}\tilde{\sigma}^{1}+\nabla\tilde{\rho}^{1}_{P}\Delta\tilde{\sigma}^{1}\right)-\kappa\tilde{\rho}^{2}\left(\nabla\tilde{\sigma}^{2}\cdot\nabla^{2}\tilde{\rho}^{2}_{P}+\nabla\tilde{\rho}^{2}_{P}\cdot\nabla^{2}\tilde{\sigma}^{2}+\nabla\tilde{\rho}^{2}_{P}\Delta\tilde{\sigma}^{2}\right)\big{\\}}\\\\[5.69054pt]
\qquad+\kappa\Big{\\{}\tilde{\rho}^{1}\big{(}\nabla\tilde{\vartheta}^{1}\cdot\nabla^{2}\tilde{\rho}^{1}_{\theta}+\nabla\tilde{\rho}^{1}_{\theta}\cdot\nabla^{2}\tilde{\vartheta}^{1}+\nabla\tilde{\rho}^{1}_{\theta}\Delta\tilde{\vartheta}^{1}\big{)}-\tilde{\rho}^{2}\big{(}\nabla\tilde{\vartheta}^{2}\cdot\nabla^{2}\tilde{\rho}^{2}_{\theta}+\nabla\tilde{\rho}^{2}_{\theta}\cdot\nabla^{2}\tilde{\vartheta}^{2}\\\\[5.69054pt]
\qquad+\nabla\tilde{\rho}^{2}_{\theta}\Delta\tilde{\vartheta}^{2}\big{)}\Big{\\}}+\kappa\left\\{\left(\tilde{\rho}^{1}\tilde{\rho}^{1}_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\nabla\Delta\tilde{\vartheta}^{1}-\left(\tilde{\rho}^{2}\tilde{\rho}^{2}_{\theta}-\bar{\rho}\bar{\rho}_{\theta}\right)\nabla\Delta\tilde{\vartheta}^{2}\right\\}\\\\[5.69054pt]
\qquad+\kappa\left\\{\left(\tilde{\rho}^{1}\tilde{\rho}^{1}_{P}-\bar{\rho}\bar{\rho}_{P}\right)\nabla\Delta\tilde{\sigma}^{1}-\left(\tilde{\rho}^{2}\tilde{\rho}^{2}_{P}-\bar{\rho}\bar{\rho}_{P}\right)\nabla\Delta\tilde{\sigma}^{2}\right\\},\\\\[5.69054pt]
\tilde{h}=-(\tilde{\eta}_{1}^{1}-\tilde{\eta}^{2}_{1})(\tilde{v}^{1}\cdot\nabla)\tilde{\vartheta}^{1}+(\tilde{\eta}_{1}^{1}-\tilde{\eta}^{2}_{2})(\tilde{v}^{1}\cdot\nabla)\tilde{\sigma}^{1}+\tilde{\eta}^{2}_{2}\big{(}(\tilde{v}^{1}\cdot\nabla)\tilde{\sigma}^{1}-(\tilde{v}^{2}\cdot\nabla)\tilde{\sigma}^{2}\big{)}\\\\[5.69054pt]
\qquad+\Psi(\tilde{v}^{1})-\Psi(\tilde{v}^{2})+\Phi(\tilde{\rho}^{1},\tilde{v}^{1})-\Phi(\tilde{\rho}^{2},\tilde{v}^{2})+\frac{1}{2}(\tilde{v}^{1}+\tilde{v}^{2})(\tilde{v}^{1}-\tilde{v}^{2})G\\\\[5.69054pt]
\qquad-\left(\tilde{\eta}^{1}_{3}-\tilde{\eta}^{2}_{3}\right)G-C_{\triangledown}(\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})G,\\\\[5.69054pt]
\displaystyle\tilde{\eta}_{1}=\displaystyle\tilde{\rho}^{j}C_{\triangledown}-\frac{\tilde{\theta}^{j}(\tilde{\rho}^{j}_{\theta})^{2}}{\tilde{\rho}^{j}\tilde{\rho}^{j}_{P}},\quad\tilde{\eta}^{j}_{2}=\displaystyle\frac{\tilde{\theta}^{j}\tilde{\rho}^{j}_{\theta}}{\tilde{\rho}^{j}},\quad\tilde{\eta}^{j}_{3}=\displaystyle\frac{\tilde{\theta}^{j}(\tilde{\rho}^{j}_{\theta})^{2}}{\tilde{\rho}^{j}\tilde{\rho}^{j}_{P}},\quad\tilde{\theta}^{j}=\bar{\theta}+\tilde{\vartheta}^{j},\,\,j=1,2.\\\\[5.69054pt]
\end{array}\right.$ (2.158)
Since
$\begin{array}[]{ll}&\left\|(1+|x|)(g,\tilde{h})\right\|\displaystyle+\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu}\nabla^{\nu}(g,\tilde{h})\right\|+\displaystyle\sum_{\nu=0}^{3}\displaystyle\left\|(1+|x|)^{\nu+1}\nabla^{\nu}(\tilde{f},\tilde{h})\right\|\\\\[5.69054pt]
&\quad\leq
C(\epsilon+K)\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\end{array}$
(2.159)
as follows from the Sobolev inequality for $K$ defined in (2.130). Applying
Theorem 2.1 to (2.147), we obtain
$\begin{array}[]{ll}&\|(\sigma^{1}-\sigma^{2},v^{1}-v^{2},\vartheta^{1}-\vartheta^{2})\|_{L^{6}}+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\nabla^{\nu}(v^{1}-v^{2})\right\|\\\\[5.69054pt]
&\quad\quad\quad+\displaystyle\sum_{\nu=1}^{4}\left\|(1+|x|)^{\nu}\left(\nabla^{\nu}(\sigma^{1}-\sigma^{2}),\nabla^{\nu+1}(\sigma^{1}-\sigma^{2}),\nabla^{\nu+2}(\sigma^{1}-\sigma^{2})\right)\right\|\\\\[5.69054pt]
&\qquad\quad\quad\quad+\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}\left(\nabla^{\nu}(\vartheta^{1}-\vartheta^{2}),\nabla^{\nu+1}(\vartheta^{1}-\vartheta^{2})\right)\right\|\\\\[5.69054pt]
&\quad\quad\quad\qquad\qquad\leq
C(\epsilon+K)\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\end{array}$
(2.160)
Similarly, by the same argument as in the proof Lemma 2.6, we can get
$\begin{array}[]{ll}&\displaystyle\sum_{\nu=0}^{1}\left\|(1+|x|)^{2}\nabla^{\nu}(\sigma^{1}-\sigma^{2})\right\|_{L^{\infty}}+\displaystyle\sum_{\nu=0}^{1}\left\|(1+|x|)^{\nu+1}\nabla^{\nu}(v^{1}-v^{2},\,\vartheta^{1}-\vartheta^{2})\right\|_{L^{\infty}}\\\\[11.38109pt]
&\displaystyle\quad+\left\|(1+|x|)^{2}\nabla^{2}(v^{1}-v^{2},\,\vartheta^{1}-\vartheta^{2})\right\|_{L^{\infty}}\\\\[5.69054pt]
&\quad\quad\leq
C(\epsilon+K)\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\\\\[5.69054pt]
&\quad\quad\quad\quad+C\epsilon\left\\{\left\|(1+|x|)^{3}(\tilde{V}_{1}^{1}-\tilde{V}_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(\tilde{V}_{2}^{1}-\tilde{V}_{2}^{2})\right\|_{L^{1}}\right\\}\end{array}$
(2.161)
where $\tilde{V}^{j}_{1},\tilde{V}^{j}_{2},\,j=1,2$ are functions satisfying
$\nabla\cdot\tilde{v}^{j}=\nabla\cdot\tilde{V}^{j}_{1}+\tilde{V}^{j}_{2},\quad\left\|(1+|x|)^{3}\tilde{V}_{1}^{j}\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}\tilde{V}_{2}^{j}\right\|_{L^{1}}\leq\epsilon$
(2.162)
Moreover, if we define $\tilde{V}^{j}_{1},\tilde{V}^{j}_{2},\,j=1,2$ as
$V_{1}^{j}=-\frac{\tilde{\rho}^{j}_{P}}{\tilde{\rho}^{j}}\tilde{v}^{j}\sigma^{j},\quad
V_{2}^{j}=\nabla\cdot\left(\frac{\tilde{\rho}^{j}_{P}}{\tilde{\rho}^{j}}\tilde{v}^{j}\right)\sigma^{j}-\frac{\tilde{\rho}^{j}_{\theta}}{\tilde{\rho}^{j}}\tilde{v}^{j}\cdot\nabla\vartheta^{j}+\frac{G}{\tilde{\rho}^{j}}$
(2.163)
then we get from $(\ref{2.102})_{1}$ that
$\nabla\cdot(v^{1}-v^{2})=\nabla\cdot(V_{1}^{1}-V_{1}^{2})+V_{2}^{1}-V_{2}^{2}$
and
$\begin{array}[]{ll}&\left\|(1+|x|)^{3}(V_{1}^{1}-V_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(V_{2}^{1}-V_{2}^{2})\right\|_{L^{1}}\\\\[5.69054pt]
&\quad\leq
C\left(\epsilon+\left\|(1+|x|)^{-1}G\right\|_{L^{1}}\right)\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\end{array}$
(2.164)
Combining (2.160)-(2.164), we obtain
$\begin{array}[]{ll}&\left\|(\sigma^{1}-\sigma^{2},v^{1}-v^{2},\vartheta^{1}-\vartheta^{2})\right\|_{\Lambda^{4,5,5}}\\\\[5.69054pt]
&\qquad+\left\|(1+|x|)^{3}(V_{1}^{1}-V_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(V_{2}^{1}-V_{2}^{2})\right\|_{L^{1}}\\\\[5.69054pt]
&\quad\leq
C\left(\epsilon+K\right)\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\\\\[5.69054pt]
&\qquad\quad+C\epsilon\left\\{\left\|(1+|x|)^{3}(\tilde{V}_{1}^{1}-\tilde{V}_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(\tilde{V}_{2}^{1}-\tilde{V}_{2}^{2})\right\|_{L^{1}}\right\\}\end{array}$
(2.165)
Therefore, we have the following Proposition.
###### Proposition 2.4.
There exits a constant $c_{0}>0$ such that for any sufficiently small constant
$\epsilon>0$, if $(G,F,H)\in\mathcal{H}^{4,3,4}$ satisfies
$K+\left\|(1+|x|)^{-1}G\right\|_{L^{1}}\leq c_{0}\epsilon\quad(\mbox{$K$ is
defined in Lemma 2.6}),$
the for
$(\tilde{\sigma}^{j},\tilde{v}^{j},\tilde{\vartheta}^{j})\in\dot{\Lambda}_{\epsilon}^{4,5,5}$
and
$(\sigma^{j},v^{j},\vartheta^{j})=T(\tilde{\sigma}^{j},\tilde{v}^{j},\tilde{\vartheta}^{j})$,
j=1,2, we have the following estimates
$\begin{array}[]{ll}&\left\|(\sigma^{1}-\sigma^{2},v^{1}-v^{2},\vartheta^{1}-\vartheta^{2})\right\|_{\Lambda^{4,5,5}}\\\\[5.69054pt]
&\qquad+\left\|(1+|x|)^{3}(V_{1}^{1}-V_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(V_{2}^{1}-V_{2}^{2})\right\|_{L^{1}}\\\\[5.69054pt]
&\quad\leq\displaystyle\frac{1}{2}\left\\{\left\|(\tilde{\sigma}^{1}-\tilde{\sigma}^{2},\tilde{v}^{1}-\tilde{v}^{2},\tilde{\vartheta}^{1}-\tilde{\vartheta}^{2})\right\|_{\Lambda^{4,5,5}}\right.\\\\[5.69054pt]
&\left.\qquad\quad+\left\|(1+|x|)^{3}(\tilde{V}_{1}^{1}-\tilde{V}_{1}^{2})\right\|_{L^{\infty}}+\left\|(1+|x|)^{-1}(\tilde{V}_{2}^{1}-\tilde{V}_{2}^{2})\right\|_{L^{1}}\right\\}\end{array}$
(2.166)
where $(\tilde{V}_{1}^{j},\tilde{V}_{2}^{j}),\,j=1,2$ satisfy (2.162) and
$(V_{1}^{j},V_{2}^{j}),\,j=1,2$ are defined by (2.164).
Hence, by Propositions 2.3 and 2.4, the contraction mapping principle implies
the existence and uniqueness of solution to (1.19). This completes the proof
of Theorem 1.1.
## 3 Non-stationary problem
In this section, we consider the stability of the Stationary solution with
respect to the initial disturbance $(\rho_{0},v_{0},\vartheta_{0})$ . Fix
$\bar{\rho},\bar{\theta}$ to be positive constants and let $F,G,H$ be small in
the sense of Theorem 1.1. We denote the corresponding stationary solution
obtained in Theorem 1.1 by $(P^{*},v^{*},\theta^{*})$ , and set
$\rho^{*}\equiv\bar{\rho}+\sigma^{*}=\rho(P^{*},\theta^{*})$. Then by direct
calculations, we have the following estimate for $\sigma^{*}$:
$\begin{array}[]{ll}\|\sigma^{*}\|_{N^{5}}&\equiv\displaystyle\sum_{\nu=1}^{5}\left\|(1+|x|)^{\nu-1}(\nabla^{\nu}\sigma^{*},\nabla^{\nu+1}\sigma^{*})\right\|+\sum_{\nu=0}^{1}\displaystyle\left\|(1+|x|)^{\nu+1}\nabla^{\nu}\sigma^{*}\right\|_{L^{\infty}}+\left\|(1+|x|)^{2}\nabla^{2}\sigma^{*}\right\|_{L^{\infty}}\\\\[5.69054pt]
&\leq C\epsilon,\end{array}$
where the constant $C>0$ is depending only on $\bar{\rho}$ and $\bar{\theta}$.
Thus, we have
$\left\|(\sigma^{*},v^{*},\theta^{*})\right\|_{\mathcal{F}^{5,5,5}}\equiv\|\sigma^{*}\|_{N^{5}}+\|v^{*}\|_{J^{5}}+\|\vartheta^{*}\|_{N^{5}}\leq(C+1)\epsilon.$
For simplicity, we assume in this section that
$\left\|(\sigma^{*},v^{*},\theta^{*})\right\|_{\mathcal{F}^{5,5,5}}\leq\epsilon$
for $\epsilon$ sufficiently small. Define the new variables
$\sigma(t,x)=\rho(t,x)-\rho^{*},\quad
w(t,x)=v(t,x)-v^{*},\quad\vartheta(t,x)=\theta(t,x)-\theta^{*},$
then the initial value problem (1.11), (1.12) is reformulated as
$\displaystyle\left\\{\begin{array}[]{ll}\sigma_{t}(t)+\nabla\cdot\\{(\rho^{*}+\sigma(t))w(t)\\}=-\nabla\cdot\left(v^{*}\sigma(t)\right),\\\\[5.69054pt]
w(t)_{t}-\frac{1}{\rho^{*}}\left[\mu\Delta
w(t)+(\mu+\mu^{\prime})\nabla(\nabla\cdot
w(t))\right]+A(t)\nabla\sigma(t)-\kappa\nabla\Delta\sigma(t)+B(t)\nabla\vartheta(t)=f(t),\\\\[5.69054pt]
\vartheta_{t}(t)-\tilde{\alpha}D^{*}\Delta\vartheta(t)+E(t)\nabla\cdot
w(t)=h(t),\end{array}\right.$ (3.4)
with initial date
$(\sigma,w,\vartheta)(t,x)|_{t=0}=(\sigma_{0},w_{0},\vartheta_{0})(x)\equiv(\sigma-\sigma^{*},v-v^{*},\theta-\theta^{*})(0,x).$
(3.5)
where
$\begin{array}[]{ll}f(t)&=-(v^{*}\cdot\nabla)w(t)-(w(t)\cdot\nabla)(v^{*}+w(t))-\left(A(t)-A^{*}\right)\nabla\rho^{*}-\left(B(t)-B^{*}\right)\nabla\vartheta^{*}\\\\[5.69054pt]
&\displaystyle\quad-\left(\left(\frac{v}{\rho}\right)(t)-\frac{v^{*}}{\rho^{*}}\right)G-\frac{\sigma(t)}{\rho^{*}(\rho^{*}+\sigma(t))}\left[\mu\Delta(v^{*}+w(t))+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot(v^{*}+w(t))\right)\right],\\\\[8.53581pt]
h(t)&=-(v^{*}\cdot\nabla)\vartheta(t)-\left(w(t)\cdot\nabla\right)(\theta^{*}+\vartheta(t))+\tilde{\alpha}\left(D(t)-D^{*}\right)\Delta(\theta^{*}+\vartheta(t))+\left(D(t)-D^{*}\right)H\\\\[5.69054pt]
&\quad+\left(D(t)-D^{*}\right)\left(\Psi(v^{*})+\Phi(\rho^{*},v^{*})\right)+D(t)\left[\Psi(v)(t)+\Phi(\rho,v)(t)-\Psi(v^{*})-\Phi(\rho^{*},v^{*})\right]\\\\[5.69054pt]
&\quad\displaystyle+\frac{1}{2}\left[D(t)v^{2}(t)-D^{*}v^{*2}\right]G-C_{\triangledown}\left[D(t)\theta(t)-D^{*}\theta^{*}\right]G-(E(t)-E^{*})\nabla\cdot
v^{*},\end{array}$
and
$A(t)=\frac{P_{\rho}(\rho,\theta)}{\rho},\quad
B(t)=\frac{P_{\theta}(\rho,\theta)}{\rho},\quad
D(t)=\frac{1}{C_{\triangledown}\rho},\quad E(t)=\frac{\theta
P_{\theta}(\rho,\theta)}{C_{\triangledown}\rho}$
with
$A^{*}=A(\rho^{*},\theta^{*}),A(t)=A\left(\rho^{*}+\sigma(t),\theta^{*}+\vartheta(t)\right),\,etc.$
Moreover, we set $A_{i}(t),B_{i}(t),E_{i}(t),i=1,2$ and $D_{1}(t)$ to be
functions satisfying:
$A(t)-A^{*}=A_{1}(t)\sigma(t)+A_{2}(t)\vartheta(t),B(t)-B^{*}=B_{1}(t)\sigma(t)+B_{2}(t)\vartheta(t)$,
$E(t)-E^{*}=E_{1}(t)\sigma(t)+E_{2}(t)\vartheta(t)$ and
$D(t)-D^{*}=D_{1}(t)\sigma(t)$, respectively.
The aim of this section is to prove Theorem 1.2. The proof consists of the
following two steps. The first one is the local existence result:
###### Proposition 3.1.
Suppose that $(\sigma,w,\vartheta)(0)\in\mathcal{H}^{4,3,3}$. Then there
exists a constant $t_{0}>0$ such that the initial value problem (3.4)-(3.5)
admits a unique solution
$(\sigma,w,\vartheta)(0)\in\mathcal{C}(0,t_{0};\mathcal{H}^{4,3,3})$.
Moreover, $(\sigma,w,\vartheta)(t)$ satisfies
$\left\|(\sigma,w,\vartheta)(t)\right\|^{2}_{4,3,3}\leq
2\left\|(\sigma,w,\vartheta)(0)\right\|^{2}_{4,3,3}$
for any $t\in[0,t_{0}]$.
And the other is an a priori estimate:
###### Proposition 3.2.
Let $(\sigma,w,\vartheta)(0)\in\mathcal{C}(0,t_{1};\mathcal{H}^{4,3,3})$ be a
solution to the initial value problem (3.4)-(3.5) for some positive constant
$t_{1}$. Then there exists a constant $\epsilon_{0}>0$ such that if
$\epsilon\leq\epsilon_{0}$ and $\sup_{0\leq t\leq
t_{1}}\|(\sigma,w,\vartheta)(t)\|_{4,3,3}$,
$\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon$, then it holds
$\left\|(\sigma,w,\vartheta)(t)\right\|^{2}_{4,3,3}+\int_{0}^{t}\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(s)\right\|^{2}_{4,3,3}\,ds\leq
C\left\|(\sigma,w,\vartheta)(0)\right\|^{2}_{4,3,3}$ (3.6)
for any $t\in[0,t_{1}]$, where the constant $C>0$ is depending only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$.
For the proof of the local existence, we can apply the H. Hattori-D. Li [12]
method directly. So we shall devote the following sections to the proof of
Proposition 3.2.
Before proving the a priori estimate (3.6), let us introduce the absolute
constant $\bar{\epsilon}>0$ such that
$C_{0}\bar{\epsilon}=1/4\min\\{\bar{\rho},\bar{\theta}\\}$, where $C_{0}$ is
the constant which appears in the inequality $\|\cdot\|_{L^{\infty}}\leq
C_{0}\|\cdot\|_{2}$. In the following lemmas and their proofs, the small
constant $\epsilon$ is at least taken in such a way that
$\|(\sigma,w,\vartheta)(t)\|_{4,3,3},\quad\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon\leq\bar{\epsilon}$
so that
$(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})\equiv\left\\{(\rho,\theta)\Big{|}\frac{\bar{\rho}}{2}\leq\rho\leq\frac{3\bar{\rho}}{2},\frac{\bar{\theta}}{2}\leq\rho\leq\frac{3\bar{\theta}}{2}\right\\}.$
### 3.1 Some estimates for $f(t),h(t)$ and their derivatives
###### Lemma 3.1.
Let $(\sigma,w,\vartheta)(t)$ and $(\sigma^{*},w^{*},\vartheta^{*})$ be
satisfying
$\|(\sigma,w,\vartheta)(t)\|_{4,3,3},\quad\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon.$
Then for a multi-index $\alpha$ with $0\leq|\alpha|\leq 3$, we have
(i) If we write $\partial_{x}^{\alpha}f(t)$ of the form
$\partial_{x}^{\alpha}f(t)=-\frac{\sigma(t)}{\rho^{*}(\rho^{*}+\sigma(t))}\left[\mu\Delta\partial_{x}^{\alpha}w(t)+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot\partial_{x}^{\alpha}w(t)\right)\right]+F_{\alpha}(t),$
then $F_{\alpha}(t)$ satisfies the estimate:
$\displaystyle F_{\alpha}(t)\leq C\left\\{\begin{array}[]{ll}|\nabla
v^{*}||w(t)|+(|v^{*}|+|w(t)|)|\nabla
w(t)|+\left(|\nabla\sigma^{*}|+|\nabla\theta^{*}|+|\nabla^{2}v^{*}|\right)|\sigma(t)|\\\\[5.69054pt]
+\left(|\nabla\sigma^{*}|+|\nabla\theta^{*}|\right)|\vartheta(t)|+\left(|w(t)|+|\sigma(t)|\right)|G|,\quad
if\,\alpha=0,\\\\[5.69054pt]
\left|\nabla^{|\alpha|+1}v^{*}\right||w(t)|\displaystyle+\sum_{\nu=1}^{|\alpha|+1}\left|\nabla^{\nu}w(t)\right|+\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left(\left|\nabla^{\nu}\sigma^{*}\right|+\left|\nabla^{\nu}\theta^{*}\right|+\left|\nabla^{\nu+1}v^{*}\right|\right)|\sigma(t)|\\\\[5.69054pt]
+\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left(\left|\nabla^{\nu}\sigma^{*}\right|+\left|\nabla^{\nu}\theta^{*}\right|\right)|\vartheta(t)|+\sum_{\nu=1}^{|\alpha|}\left(\left|\nabla^{\nu}\displaystyle\sigma(t)\right|+\left|\nabla^{\nu}\vartheta(t)\right|\right)\\\\[5.69054pt]
+(|w(t)|+|\sigma(t)|)\displaystyle\sum_{\nu=0}^{|\alpha|}|\nabla^{\nu}G|+R^{|\alpha|}_{F}(t),\quad
if\,|\alpha|=1,2,3.\end{array}\right.$ (3.12)
Here, $R^{k}_{F}(t)=0,k=1,2$ and $R^{3}_{F}(t)$ satisfies
$\|R^{3}_{F}(t)\|_{L^{\frac{3}{2}}}\leq
C\epsilon\left\|(\nabla^{2}\sigma,\nabla^{3}w)(t)\right\|_{1,0}$ (3.13)
(ii) If we write $\partial_{x}^{\alpha}h(t)$ of the form
$\partial_{x}^{\alpha}h(t)=\tilde{\alpha}D_{1}(t)\sigma(t)\Delta\partial_{x}^{\alpha}\vartheta(t)+H_{\alpha}(t),$
then $H_{\alpha}(t)$ satisfies the estimate:
$\displaystyle H_{\alpha}(t)\leq
C\left\\{\begin{array}[]{ll}\left(|\nabla^{3}\theta^{*}|+|\nabla
v^{*}|+|\nabla v^{*}||\nabla^{2}\sigma(t)|+|\nabla
w(t)||\nabla^{2}\sigma^{*}|\right)|\sigma(t)|+|v^{*}||\nabla\vartheta(t)|\\\\[5.69054pt]
+\left(|\nabla\vartheta(t)|+|\nabla\theta^{*}|\right)|w(t)|+\left(|\nabla\sigma^{*}|+|\nabla\sigma(t)|\right)|\nabla\sigma(t)|+|\vartheta(t)||\nabla
v^{*}|\\\\[5.69054pt]
+\left(|\nabla\sigma(t)|+|\nabla^{2}\sigma(t)|+|\nabla\sigma^{*}|+|\nabla^{2}\sigma^{*}|+|\nabla
w(t)|+|\nabla v^{*}|\right)|\nabla w(t)|\\\\[5.69054pt]
+|\sigma(t)||H|+|(\sigma,w,\vartheta)(t)||G|,\quad
if\,\alpha=0,\\\\[5.69054pt]
\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left(\left|\nabla^{\nu+1}\sigma^{*}\right|+\left|\nabla^{\nu+1}\theta^{*}\right|+\left|\nabla^{\nu}v^{*}\right|+\left|\nabla^{\nu-1}H\right|\right)|\sigma(t)|+\displaystyle\sum_{\nu=1}^{|\alpha|+2}\left|\nabla^{\nu}\sigma(t)\right|\\\\[5.69054pt]
+\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left(\left|\nabla^{\nu}\displaystyle
w(t)\right|+\left|\nabla^{\nu}\vartheta(t)\right|\right)+|\vartheta(t)|\displaystyle\sum_{\nu=1}^{|\alpha|+1}|\nabla^{\nu}v^{*}|+|w(t)|\big{|}\nabla^{|\alpha|+1}\theta^{*}\big{|}\\\\[5.69054pt]
+|(\sigma,w,\vartheta)(t)|\displaystyle\sum_{\nu=0}^{|\alpha|}\left|\nabla^{\nu}G\right|+R^{|\alpha|}_{H}(t),\quad
if\,|\alpha|=1,2,3.\end{array}\right.$ (3.21)
Here, $R^{1}_{H}(t)=0$ and $R^{2}_{H}(t)$, $R^{3}_{H}(t)$ satisfies
$\|R^{2}_{H}(t)\|_{L^{\frac{3}{2}}}\leq
C\epsilon\left\|\nabla^{2}w(t)\right\|,\quad\|R^{3}_{H}(t)\|_{L^{\frac{3}{2}}}\leq
C\epsilon\left\|(\nabla\sigma,\nabla w)(t)\right\|_{3,2}$ (3.22)
Proof. By the Leibniz formula and the Sobolev embedding: $H^{2}\hookrightarrow
L^{\infty}$, we can check (3.12), (3.13) with
$\displaystyle\left\\{\begin{array}[]{ll}R^{k}_{F}(t)=0,\quad
if\,k=1,2,\\\\[5.69054pt]
R^{3}_{F}(t)=\left|\nabla^{2}w(t)\right|\left|\nabla^{3}\sigma(t)\right|+\left|\nabla^{2}w(t)\right|^{2}+\left|\nabla^{4}\sigma^{*}\right||\nabla\sigma(t)|,\end{array}\right.$
(3.25)
and
$\displaystyle\left\\{\begin{array}[]{ll}R^{1}_{H}(t)=0,\quad
R^{2}_{H}(t)=\left|\nabla^{2}w(t)\right|^{2},\\\\[5.69054pt]
R^{3}_{H}(t)=\left(\left|\nabla^{2}\vartheta(t)\right|+\left|\nabla^{2}w(t)\right|\right)\left|\nabla^{2}w(t)\right|+\left(\left|\nabla^{3}\sigma(t)\right|+\left|\nabla^{4}\sigma(t)\right|+\left|\nabla^{3}w(t)\right|\right)\left|\nabla^{2}w(t)\right|\\\\[5.69054pt]
+\left|\nabla^{3}\sigma(t)\right|\left|\nabla^{3}w(t)\right|+\left(\left|\nabla^{4}v^{*}(t)\right|+\left|\nabla^{5}\sigma^{*}(t)\right|\right)|\nabla
w(t)|+\left(\left|\nabla\sigma(t)\right|+\left|\nabla^{2}\sigma(t)\right|\right)|\nabla^{4}v^{*}|\end{array}\right.$
(3.29)
For the proof of (3.13) and (3.22), we only give here the estimate of the most
difficult term $R_{H}^{3}$. The others can be dealt with similarly. Using the
Gagliard-Nirenberg inequality, we have
$\begin{array}[]{ll}\|R^{2}_{H}(t)\|_{L^{\frac{3}{2}}}&\leq
C\Big{\\{}\left\|\nabla^{2}\vartheta(t)\right\|_{L^{6}}\left\|\nabla^{2}w(t)\right\|+\left\|\nabla^{2}w(t)\right\|_{L^{6}}\left\|\nabla^{2}w(t)\right\|+\left\|\nabla^{3}\sigma(t)\right\|_{L^{6}}\left\|\nabla^{2}w(t)\right\|\\\\[5.69054pt]
&\qquad+\left\|\nabla^{2}w(t)\right\|_{L^{6}}\left\|\nabla^{3}w(t)\right\|+\left\|\nabla^{2}w(t)\right\|_{L^{6}}\left\|\nabla^{4}\sigma(t)\right\|+\left\|\nabla^{3}\sigma(t)\right\|_{L^{6}}\left\|\nabla^{3}w(t)\right\|\\\\[5.69054pt]
&\qquad+\left\|(\nabla^{4}v^{*},\nabla^{5}\sigma^{*})\right\|_{L^{6}}\left\|\nabla
w(t)\right\|+\left\|(\nabla\sigma,\nabla^{2}\sigma)(t)\right\|\left\|\nabla^{4}v^{*}\right\|_{L^{6}}\Big{\\}}\\\\[5.69054pt]
&\leq C\epsilon\left\|(\nabla\sigma,\nabla w)(t)\right\|_{3,2}\end{array}$
which is the desired estimate $(\ref{3.7})_{2}$. This completes the proof of
Lemma 3.1.
### 3.2 Estimates for $\nabla w(t),\nabla\vartheta(t)$ and their derivatives
up to $\nabla^{4}w(t),\nabla^{4}\vartheta(t)$
###### Lemma 3.2.
Let $(\sigma,w,\vartheta)(t)\in\mathcal{C}(0,t_{1};\mathcal{H}^{4,3,3})$ be a
solution to the Cauchy problem (3.4)-(3.5) for some positive constant $t_{1}$.
Then there exists four constants $\epsilon_{0},\lambda_{0}>0$ and
$d_{1},d_{2}>0$ such that if $0<\epsilon\leq\epsilon_{0}$ and
$\|(\sigma,w,\vartheta)(t)\|_{4,3,3},\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon$,
then it holds
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\left(\left\|\sigma(t)\right\|^{2}+\langle\hat{A}(t)\nabla\sigma(t),\nabla\sigma(t)\rangle+\langle\tilde{A}(t)w(t),w(t)\rangle+\langle\tilde{B}(t)\vartheta(t),\vartheta(t)\rangle\right)\\\\[5.69054pt]
\qquad+d_{1}\|\nabla w(t)\|^{2}+d_{2}\|\nabla\vartheta(t)\|^{2}\leq
C\epsilon\|\nabla\sigma(t)\|_{1}^{2}\end{array}$ (3.30)
and for $1\leq k\leq 3$ and any $\lambda$ with $0<\lambda<\lambda_{0}$,
$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\left(\left\|\nabla^{k}\sigma(t)\right\|^{2}+\langle\hat{A}(t)\nabla^{k+1}\sigma(t),\nabla^{k+1}\sigma(t)\rangle+\langle\tilde{A}(t)\nabla^{k}w(t),\nabla^{k}w(t)\rangle\right.\\\\[5.69054pt]
&\left.\quad+\langle\tilde{B}(t)\nabla^{k}\vartheta(t),\nabla^{k}\vartheta(t)\rangle\right)+d_{1}\|\nabla^{k+1}w(t)\|^{2}+d_{2}\|\nabla^{k+1}\vartheta(t)\|^{2}\\\\[5.69054pt]
&\leq
C(\epsilon+\lambda)\|\nabla(\sigma,w,\vartheta)(t)\|^{2}_{k+1,k-1,k-1}+C\lambda^{-1}\|\nabla^{k}(w,\vartheta)(t)\|^{2}\end{array}$
(3.31)
where the constant $C>0$ is depending only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$.
Setting
$\hat{A}(t)=\frac{\rho}{P_{\rho}(\rho,\theta)},\quad\tilde{A}(t)=\frac{\rho^{2}}{P_{\rho}(\rho,\theta)},\quad\tilde{B}(t)=\frac{C_{\triangledown}\rho^{2}}{\theta
P_{\rho}(\rho,\theta)}$
then
$\hat{A}(t)=\hat{A}(\rho^{*}+\sigma(t),\vartheta^{*}+\vartheta(t)),\tilde{A}(t)=\tilde{A}(\rho^{*}+\sigma(t),\vartheta^{*}+\vartheta(t))$
and $\tilde{B}(t)=\tilde{B}(\rho^{*}+\sigma(t),\vartheta^{*}+\vartheta(t))$.
Proof. Using the Friedrichs mollifier, we may assume that
$(\sigma,w,\vartheta)\in\mathcal{C}(0,t_{1};\mathcal{H}^{\infty,\infty,\infty})$.
For any multi-index $\alpha$ with $0\leq|\alpha|\leq 3$, applying
$\partial_{x}^{\alpha}$ to $(\ref{3.1})_{1}$, $(\ref{3.1})_{2}$,
$(\ref{3.1})_{3}$, then taking the $L^{2}$ inner product of the resultant
equations with $\partial_{x}^{\alpha}\sigma(t)$,
$\tilde{A}(t)\partial_{x}^{\alpha}w(t)$, and
$\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)$, respectively, we have
$\displaystyle\frac{1}{2}\frac{d}{dt}\left\|\partial_{x}^{\alpha}\sigma(t)\right\|^{2}-\langle(\rho^{*}+\sigma(t))\partial_{x}^{\alpha}w(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle=\langle\partial_{x}^{\alpha}(v^{*}\sigma(t))+I_{\alpha}(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle,$
$\begin{array}[]{ll}&\displaystyle\langle\tilde{A}(t)\partial_{x}^{\alpha}w_{t}(t),\partial_{x}^{\alpha}w(t)\rangle-\langle\frac{\tilde{A}(t)}{\rho^{*}}\partial_{x}^{\alpha}\\{\mu\Delta
w(t)+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot
w(t)\right)\\},\partial_{x}^{\alpha}w(t)\rangle\\\\[5.69054pt]
&+\langle(\rho^{*}+\sigma(t))\nabla\partial_{x}^{\alpha}\sigma(t),\partial_{x}^{\alpha}w(t)\rangle-\kappa\langle\nabla\Delta\partial_{x}^{\alpha}\sigma(t),\tilde{A}(t)\partial_{x}^{\alpha}w(t)\rangle+\langle\tilde{A}(t)B(t)\nabla\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}w(t)\rangle\\\\[5.69054pt]
&=\langle\partial_{x}^{\alpha}f(t)+J_{\alpha}(t),\tilde{A}(t)\partial_{x}^{\alpha}w(t)\rangle,\end{array}$
and
$\begin{array}[]{ll}&\displaystyle\langle\tilde{B}(t)\partial_{x}^{\alpha}\vartheta_{t}(t),\partial_{x}^{\alpha}\vartheta(t)\rangle-\tilde{\alpha}\langle
D^{*}\Delta\partial_{x}^{\alpha}\vartheta(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle+\langle\tilde{A}(t)B(t)\nabla\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}(\nabla\cdot
w)(t)\rangle\\\\[5.69054pt]
&=\langle\partial_{x}^{\alpha}h(t)+K_{\alpha}(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle,\end{array}$
where
$\begin{array}[]{ll}&\displaystyle
I_{\alpha}(t)=\sum_{\beta<\alpha}C_{\alpha}^{\beta}\partial_{x}^{\alpha-\beta}\left(\rho^{*}+\sigma(t)\right)\partial_{x}^{\beta}w(t),\\\\[5.69054pt]
&\displaystyle
J_{\alpha}(t)=\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left\\{\left(\partial_{x}^{\alpha-\beta}\frac{1}{\rho^{*}}\right)\partial_{x}^{\beta}(\mu\Delta
w(t)+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot
w(t)\right))-\left(\partial_{x}^{\alpha-\beta}A(t)\right)\nabla\partial_{x}^{\beta}\sigma(t)\right.\\\\[5.69054pt]
&\left.\hskip
85.35826pt-\left(\partial_{x}^{\alpha-\beta}B(t)\right)\nabla\partial_{x}^{\beta}\vartheta(t)\right\\},\\\\[5.69054pt]
&\displaystyle
K_{\alpha}(t)=\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left\\{\tilde{\alpha}\left(\partial_{x}^{\alpha-\beta}D^{*}\right)\Delta\partial_{x}^{\beta}\vartheta(t)-\left(\partial_{x}^{\alpha-\beta}E(t)\right)\nabla\cdot\partial_{x}^{\beta}w(t)\right\\}.\end{array}$
Canceling the terms
$\langle(\rho^{*}+\sigma(t))\nabla\partial_{x}^{\alpha}\sigma(t),\partial_{x}^{\alpha}w(t)\rangle$
and
$\langle\tilde{A}(t)B(t)\nabla\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}w(t)\rangle$
by adding the above three formulas and using the identities
$\begin{array}[]{ll}&\displaystyle\langle\tilde{A}(t)\partial_{x}^{\alpha}w_{t}(t),\partial_{x}^{\alpha}w(t)\rangle=\frac{1}{2}\frac{d}{dt}\langle\tilde{A}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle-\frac{1}{2}\langle\tilde{A}_{t}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle\\\\[5.69054pt]
&\displaystyle\langle\tilde{B}(t)\partial_{x}^{\alpha}\vartheta_{t}(t),\partial_{x}^{\alpha}\vartheta(t)\rangle=\frac{1}{2}\frac{d}{dt}\langle\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha},\vartheta(t)\rangle-\frac{1}{2}\langle\tilde{B}_{t}(t)\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}\vartheta(t)\rangle,\end{array}$
we get from integration by parts that
$\begin{array}[]{ll}&\displaystyle\frac{d}{dt}\left(\left\|\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+\langle\tilde{A}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle+\langle\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}\vartheta(t)\rangle\right)\\\\[8.53581pt]
&\displaystyle+\langle\frac{\tilde{A}(t)}{\rho^{*}}\nabla\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}\nabla
w(t)\rangle+\tilde{\alpha}\langle
D^{*}\nabla\partial_{x}^{\alpha}\vartheta(t),\tilde{B}(t)\nabla\partial_{x}^{\alpha}\vartheta(t)\rangle\\\\[5.69054pt]
&\leq\left|\langle\partial_{x}^{\alpha}(v^{*}\sigma(t)),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|+\tilde{\alpha}\left|\langle\nabla(D^{*}\tilde{B}(t))\cdot\nabla\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}\vartheta(t)\rangle\right|\\\\[5.69054pt]
&\quad+\left|\langle\nabla(\tilde{A}(t)B(t))\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}w(t)\rangle\right|+\kappa\langle\nabla\Delta\partial_{x}^{\alpha}\sigma(t),\tilde{A}(t)\partial_{x}^{\alpha}w(t)\rangle\\\\[5.69054pt]
&\quad+\left[\mu\left|\langle\nabla\left(\frac{\tilde{A}(t)}{\rho^{*}}\right)\partial_{x}^{\alpha}w(t),\nabla\partial_{x}^{\alpha}w(t)\rangle\right|+(\mu+\mu^{\prime})\left|\langle\nabla\left(\frac{\tilde{A}(t)}{\rho^{*}}\right)\cdot\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}(\nabla\cdot
w)(t)\rangle\right|\right]\\\\[5.69054pt]
&\quad+\left|\langle\partial_{x}^{\alpha}f(t),\tilde{A}(t)\partial_{x}^{\alpha}w(t)\rangle\right|+\left|\langle\partial_{x}^{\alpha}h(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle\right|\\\\[5.69054pt]
&\quad+\left[\left|\langle
I_{\alpha}(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|+\left|\langle
J_{\alpha}(t),\tilde{A}(t)\partial_{x}^{\alpha}w(t)\rangle\right|+\left|\langle
K_{\alpha}(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle\right|\right]\\\\[5.69054pt]
&\quad\displaystyle+\frac{1}{2}\left|\langle\tilde{A}_{t}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle\right|\displaystyle+\frac{1}{2}\left|\langle\tilde{B}_{t}(t)\partial_{x}^{\alpha}\vartheta(t),\partial_{x}^{\alpha}\vartheta(t)\rangle\right|\\\\[5.69054pt]
&\displaystyle=I_{1}+I_{2}+\cdots+I_{10}.\end{array}$ (3.32)
Now, we estimate $I_{i},i=1,2,\ldots,10$ term by term. First, if $\alpha=0$,
employing the Hardy inequality, we have
$I_{1}\leq\|(1+|x|)v^{*}\|_{L^{\infty}}\left\|\frac{\sigma(t)}{|x|}\right\|\left\|\nabla\sigma(t)\right\|\leq
C\epsilon\left\|\nabla\sigma(t)\right\|^{2}.$ (3.33)
If $1\leq|\alpha|\leq 3$, using integration by parts and the Sobolev
inequality, we get
$\begin{array}[]{ll}I_{1}&\leq\displaystyle\sum_{\beta\leq\alpha}C_{\alpha}^{\beta}\left|\langle\partial_{x}^{\alpha-\beta}\nabla\cdot
v^{*}\partial_{x}^{\beta}\sigma(t)+\partial_{x}^{\alpha-\beta}v^{*}\cdot\partial_{x}^{\beta}\nabla\sigma(t),\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\leq
C\displaystyle\sum_{\beta\leq\alpha}\left\\{\left\|\partial_{x}^{\alpha-\beta}\nabla\cdot
v^{*}\right\|_{L^{3}}\left\|\partial_{x}^{\beta}\sigma(t)\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}\sigma(t)\right\|+\left\|\partial_{x}^{\alpha-\beta}v^{*}\right\|_{L^{6}}\left\|\partial_{x}^{\beta}\nabla\sigma(t)\right\|\left\|\partial_{x}^{\alpha}\sigma(t)\right\|_{L^{3}}\right\\}\\\\[5.69054pt]
&\leq C\epsilon\left\|\nabla\sigma(t)\right\|_{|\alpha|}^{2}\end{array}$
(3.34)
where we have used the following inequalities (cf.[1, 19]).
$\|u\|_{L^{3}}\leq\|u\|_{L^{2}}+\|u\|_{L^{6}},\quad\|u\|_{L^{6}}\leq C\|\nabla
u\|,\quad\forall u\in H^{1}(\mathbb{R}^{3})$
$I_{2}$ and $I_{3}$ can be estimated as follows
$\begin{array}[]{ll}I_{2}&\leq\displaystyle
C\left\|(\nabla\sigma^{*},\nabla\sigma(t),\nabla\theta^{*},\nabla\vartheta(t))\right\|_{L^{3}}\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|\left\|\partial_{x}^{\alpha}\sigma(t)\right\|_{L^{6}}\\\\[5.69054pt]
&\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|^{2}\end{array}$
(3.35) $\begin{array}[]{ll}I_{3}&\leq\displaystyle
C\left\|(1+|x|)^{2}(\nabla\sigma^{*},\nabla\theta^{*})\right\|_{L^{\infty}}\left\|\frac{\partial_{x}^{\alpha}w(t)}{|x|}\right\|\left\|\frac{\partial_{x}^{\alpha}\vartheta(t)}{|x|}\right\|\\\\[8.53581pt]
&\quad+C\left\|(\nabla\sigma,\nabla\vartheta)(t)\right\|\left\|\partial_{x}^{\alpha}w(t)\right\|_{L^{3}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|_{L^{6}}\\\\[5.69054pt]
&\leq C\epsilon\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|,|\alpha|,|\alpha|}\end{array}$
(3.36)
Similar to the estimate of $I_{2}$, we can get
$I_{5}\leq\displaystyle
C\epsilon\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}.$ (3.37)
Now, we turn to estimate $I_{4}$. We deduce from integration by parts and the
equation $(\ref{3.1})_{1}$ that
$\begin{array}[]{ll}I_{4}&=-\displaystyle\kappa\langle\Delta\partial_{x}^{\alpha}\sigma(t),\nabla\cdot\left(\tilde{A}(t)\partial_{x}^{\alpha}w(t)\right)\rangle\\\\[5.69054pt]
&=-\displaystyle\kappa\langle\Delta\partial_{x}^{\alpha}\sigma(t),(\nabla\hat{A}(t))(\rho^{*}+\sigma(t))\partial_{x}^{\alpha}w(t)+\hat{A}(t)\nabla\cdot\left\\{(\rho^{*}+\sigma(t))\partial_{x}^{\alpha}w(t)\right\\}\rangle\\\\[5.69054pt]
&=-\displaystyle\kappa\langle\Delta\partial_{x}^{\alpha}\sigma(t),(\nabla\hat{A}(t))(\rho^{*}+\sigma(t))\partial_{x}^{\alpha}w(t)\rangle+\kappa\langle\Delta\partial_{x}^{\alpha}\sigma(t),\hat{A}(t)\partial_{x}^{\alpha}\sigma_{t}(t)\rangle\\\\[5.69054pt]
&\quad\displaystyle+\kappa\sum_{\beta<\alpha}C_{\alpha}^{\beta}\langle\Delta\partial_{x}^{\alpha}\sigma(t),\nabla\cdot\left\\{\partial_{x}^{\alpha-\beta}(\rho^{*}+\sigma(t))\partial_{x}^{\beta}w(t)\right\\}\rangle+\kappa\langle\Delta\partial_{x}^{\alpha}\sigma(t),\nabla\cdot\partial_{x}^{\alpha}(\sigma(t)v^{*})\rangle\\\\[5.69054pt]
&=I_{4,1}+I_{4,2}+I_{4,3}+I_{4,4}\end{array}$ (3.38)
$I_{4,1}$ can be estimated as follows
$\begin{array}[]{ll}I_{4,1}&\leq
C\|\Delta\partial_{x}^{\alpha}\sigma(t)\|\|\nabla\hat{A}(t)\|_{L^{3}}\|(\rho^{*},\sigma(t))\|_{L^{\infty}}\|\partial_{x}^{\alpha}w(t)\|_{L^{6}}\\\\[5.69054pt]
&\leq
C\|(\nabla\rho^{*},\nabla\sigma(t),\nabla\theta^{*},\nabla\vartheta(t))\|_{1}\|(\rho^{*},\sigma(t))\|_{L^{\infty}}\|\nabla\partial_{x}^{\alpha}w(t)\|\|\Delta\partial_{x}^{\alpha}\sigma(t)\|\\\\[5.69054pt]
&\leq
C\epsilon\left(\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}+\left\|\nabla^{2}\partial_{x}^{\alpha}\sigma(t)\right\|^{2}\right)\end{array}$
(3.39)
For $I_{4,2}$, using integration by parts and the equation $(\ref{3.1})_{1}$
again, we have
$\begin{array}[]{ll}I_{4,2}&=\displaystyle-\frac{\kappa}{2}\frac{d}{dt}\langle\nabla\partial_{x}^{\alpha}\sigma(t),\hat{A}(t)\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+\frac{\kappa}{2}\langle\nabla\partial_{x}^{\alpha}\sigma(t),\hat{A}_{t}(t)\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\\\\[5.69054pt]
&\quad+\langle\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\hat{A}(t)\nabla\cdot\left\\{(\rho^{*}+\sigma(t))\partial_{x}^{\alpha}w(t)\right\\}\rangle\\\\[5.69054pt]
&\quad\displaystyle+\kappa\sum_{\beta<\alpha}C_{\alpha}^{\beta}\langle\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\hat{A}(t)\nabla\cdot\left\\{\partial_{x}^{\alpha-\beta}(\rho^{*}+\sigma(t))\partial_{x}^{\beta}w(t)\right\\}\rangle\\\\[5.69054pt]
&\quad+\langle\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\hat{A}(t)\nabla\cdot\partial_{x}^{\alpha}(\sigma(t)v^{*})\rangle\\\\[5.69054pt]
&=\displaystyle-\frac{\kappa}{2}\frac{d}{dt}\langle\nabla\partial_{x}^{\alpha}\sigma(t),\hat{A}(t)\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+I_{4,2}^{1}+I_{4,2}^{2}+I_{4,2}^{3}+I_{4,2}^{4}.\end{array}$
(3.40)
To estimate $I_{4,2}^{1}$, we use the equation $(\ref{3.1})_{1}$ and
$(\ref{3.1})_{3}$,
$\begin{array}[]{ll}I_{4,2}^{1}&=\displaystyle\frac{\kappa}{2}\langle\left(\hat{A}_{\rho}(t)\sigma_{t}(t)+\hat{A}_{\theta}(t)\vartheta_{t}(t)\right)\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\\\\[5.69054pt]
&\leq
C\left\|\nabla\cdot\\{(\rho^{*}+\sigma(t))w(t)\\}+\nabla\cdot(v^{*}\sigma(t))\right\|_{L^{\infty}}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}\\\\[5.69054pt]
&\quad+C\left\|h(t)-\tilde{\alpha}(D(t)-D^{*})\Delta\vartheta(t)-E(t)(\nabla\cdot
w)(t)\right\|_{L^{\infty}}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}\\\\[5.69054pt]
&\displaystyle\quad+\frac{\kappa\tilde{\alpha}}{2}\left|\langle\hat{A}_{\theta}(t)D(t)\Delta\vartheta(t)\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+\displaystyle\frac{\kappa\tilde{\alpha}}{2}\left|\langle\nabla\big{(}\hat{A}_{\theta}(t)D(t)\big{)}\cdot\nabla\vartheta(t),|\nabla\partial_{x}^{\alpha}\sigma(t)|^{2}\rangle\right|\\\\[5.69054pt]
&\displaystyle\quad+\kappa\tilde{\alpha}\left|\langle\hat{A}_{\theta}(t)D(t)\nabla\vartheta(t)\cdot\nabla\partial_{x}^{\alpha}\sigma(t),\Delta\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+C\left\|\nabla\vartheta(t)\right\|_{L^{\infty}}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|\left\|\Delta\partial_{x}^{\alpha}\sigma(t)\right\|\\\\[5.69054pt]
&\displaystyle\quad+C\left\|\left(\nabla\sigma^{*},\nabla\sigma(t),\nabla\vartheta^{*},\nabla\vartheta(t)\right)\right\|_{L^{\infty}}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}\\\\[5.69054pt]
&\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|_{1}^{2}.\end{array}$
(3.41)
It follows from the Hölder inequality and the Sobolev inequality that
$\begin{array}[]{ll}I_{4,2}^{2}&\leq\displaystyle
C\left\|\left(\nabla\sigma^{*},\nabla\sigma(t),\nabla\vartheta(t),\nabla\vartheta^{*}\right)\right\|_{L^{6}}\left\|(\nabla\sigma^{*},\nabla\sigma(t))\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}w(t)\right\|_{L^{6}}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|\\\\[5.69054pt]
&\quad+C\left\|(\rho^{*},\sigma(t))\right\|_{L^{\infty}}\left\|\left(\nabla\sigma^{*},\nabla\sigma(t),\nabla\vartheta(t),\nabla\vartheta^{*}\right)\right\|_{L^{\infty}}\left\|\partial_{x}^{\alpha}(\nabla\cdot
w)(t)\right\|\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|\\\\[5.69054pt]
&\leq
C\epsilon\left(\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}\right)\end{array}$
(3.42)
and
$\begin{array}[]{ll}I_{4,2}^{3}+I_{4,2}^{4}&\leq\displaystyle
C\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|\left\|\left(\nabla\sigma^{*},\nabla\sigma(t),\nabla\vartheta(t),\nabla\vartheta^{*}\right)\right\|_{L^{6}}\\\\[5.69054pt]
&\quad\times\left\\{\left\|\partial_{x}^{\alpha-\beta}\nabla(\rho^{*}+\sigma(t))\right\|_{L^{6}}\left\|\partial_{x}^{\beta}w(t)\right\|_{L^{6}}+\left\|\partial_{x}^{\alpha-\beta}(\rho^{*}+\sigma(t))\right\|_{L^{6}}\left\|\nabla\partial_{x}^{\beta}w(t)\right\|_{L^{6}}\right.\\\\[5.69054pt]
&\qquad\left.+\left\|\partial_{x}^{\alpha-\beta}\nabla\sigma(t)\right\|_{L^{6}}\left\|\partial_{x}^{\beta}v^{*}\right\|_{L^{6}}+\left\|\partial_{x}^{\alpha-\beta}\sigma(t)\right\|_{L^{6}}\left\|\partial_{x}^{\beta}(\nabla\cdot
v^{*})\right\|_{L^{6}}\right\\}\\\\[5.69054pt] &\leq
C\epsilon\left(\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}_{1}+\left\|\nabla
w(t)\right\|^{2}_{|\alpha|}\right).\end{array}$ (3.43)
Combining (3.40)-(3.43), we obtain
$I_{4,2}\leq\displaystyle-\frac{\kappa}{2}\frac{d}{dt}\langle\nabla\partial_{x}^{\alpha}\sigma(t),\hat{A}(t)\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+C\epsilon\left(\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}_{1}+\left\|\nabla
w(t)\right\|^{2}_{|\alpha|}\right).$ (3.44)
Similar to the estimate of $I_{1}$, we can also get
$I_{4,3}\leq
C\epsilon\left(\left\|\nabla\sigma(t)\right\|^{2}_{|\alpha|+1}+\left\|\nabla
w(t)\right\|^{2}_{|\alpha|}\right).$ (3.45) $I_{4,4}\leq
C\epsilon\left\|\nabla\sigma(t)\right\|^{2}_{|\alpha|+1}.$ (3.46)
Putting (3.39), (3.44)-(3.46) into (3.38) gives rise to
$I_{4}\leq\displaystyle-\frac{\kappa}{2}\frac{d}{dt}\langle\hat{A}(t)\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+C\epsilon\left(\left\|\nabla\sigma(t)\right\|^{2}_{|\alpha|+1}+\left\|\nabla
w(t)\right\|^{2}_{|\alpha|}\right).$ (3.47)
To estimate $I_{6}$ and $I_{7}$, we use Lemma 3.1. Here, we only give the
detailed estimation of $I_{7}$. $I_{6}$ can be estimated similarly. In fact,
$I_{7}$ can be divided into the following two parts
$\begin{array}[]{ll}I_{7}&\leq\displaystyle\left|\langle
H_{\alpha}(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle\right|+\tilde{\alpha}\left|\langle
D_{1}(t)\sigma(t)\Delta\partial_{x}^{\alpha}\vartheta(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle\right|\\\\[5.69054pt]
&=I_{7,1}+I_{7,2}.\end{array}$ (3.48)
For $I_{7,2}$, using integration by parts, we have
$\begin{array}[]{ll}I_{7,2}&\leq\displaystyle\tilde{\alpha}\left|\langle\nabla(D_{1}(t)\sigma(t))\cdot\nabla\partial_{x}^{\alpha}\vartheta(t),\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\rangle\right|+\tilde{\alpha}\left|\langle
D_{1}(t)\sigma(t)\nabla\partial_{x}^{\alpha}\vartheta(t),\nabla\left(\tilde{B}(t)\partial_{x}^{\alpha}\vartheta(t)\right)\rangle\right|\\\\[5.69054pt]
&\leq\left\|(\nabla\sigma^{*},\nabla\sigma(t),\nabla\vartheta^{*},\nabla\vartheta(t))\right\|_{L^{3}}\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|_{L^{6}}+C\left\|\sigma(t)\right\|_{L^{\infty}}\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|^{2}\\\\[5.69054pt]
&\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|^{2}.\end{array}$
(3.49)
To estimate $I_{7,1}$, we use (3.21). If $\alpha=0$,
$\begin{array}[]{ll}I_{7,1}&\leq\displaystyle
C\bigg{\\{}\left\|(1+|x|)^{2}(\nabla
v^{*},\nabla^{2}\sigma^{*},\nabla\theta^{*},\nabla^{3}\theta^{*},H,G)\right\|_{L^{\infty}}\left\|\frac{(\sigma,w,\vartheta)(t)}{|x|}\right\|\left\|\frac{\vartheta(t)}{|x|}\right\|\\\\[5.69054pt]
&\quad+\left\|(1+|x|)v^{*}\right\|_{L^{\infty}}\left\|\frac{\vartheta(t)}{|x|}\right\|\left\|\nabla\vartheta(t)\right\|+\left\|w(t)\right\|_{L^{3}}\left\|\vartheta(t)\right\|_{L^{6}}\|\left\|\nabla\vartheta(t)\right\|\\\\[5.69054pt]
&\quad+\left\|(\nabla\sigma(t),\nabla^{2}\sigma(t),\nabla
w(t),\nabla\sigma^{*},\nabla^{2}\sigma^{*},\nabla
v^{*})\right\|_{L^{3}}\left\|\nabla
w(t)\right\|\left\|\vartheta(t)\right\|_{L^{6}}\\\\[5.69054pt]
&\quad+\left\|(\nabla\sigma^{*},\nabla\sigma(t))\right\|\left\|\nabla\sigma(t)\right\|_{L^{3}}\left\|\vartheta(t)\right\|_{L^{6}}\bigg{\\}}\\\\[5.69054pt]
&\leq C\epsilon\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{1,0,0}.\end{array}$ (3.50)
and if $1\leq|\alpha|\leq 3$,
$\begin{array}[]{ll}I_{7,1}&\leq\displaystyle
C\left\\{\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left\|(\nabla^{\nu+1}\sigma^{*},\nabla^{\nu}v^{*},\nabla^{\nu+1}\theta^{*},\nabla^{\nu-1}H)\right\|_{L^{3}}\left\|\sigma(t)\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|\right.\\\\[8.53581pt]
&\quad+\displaystyle\sum_{\nu=1}^{|\alpha|+2}\left\|\nabla^{\nu}\sigma(t)\right\|\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|+\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left(\left\|\nabla^{\nu}w(t)\right\|+\left\|\nabla^{\nu}\vartheta(t)\right\|\right)\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|\\\\[5.69054pt]
&\quad+\displaystyle\sum_{\nu=1}^{|\alpha|+1}\left\|\nabla^{\nu}v^{*}\right\|_{L^{3}}\left\|\vartheta(t)\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|+\left\|\nabla^{|\alpha|+1}\theta^{*}\right\|_{L^{3}}\left\|w(t)\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|\\\\[5.69054pt]
&\quad\left.+\displaystyle\sum_{\nu=0}^{|\alpha|}\left\|(\sigma,w,\vartheta)(t)\right\|_{L^{6}}\left\|\nabla^{\nu}G\right\|_{L^{3}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|+\left\|R_{H}^{|\alpha|}(t)\right\|_{L^{3/2}}\left\|\partial_{x}^{\alpha}\vartheta(t)\right\|_{L^{3}}\right\\}\\\\[5.69054pt]
&\leq C(\epsilon+\lambda)\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|+1,|\alpha|,|\alpha|}+C\lambda^{-1}\left\|\nabla^{|\alpha|}\vartheta(t)\right\|^{2}.\end{array}$
(3.51)
Combining (3.48)-(3.51), we have
$I_{7}\leq C\left\\{\begin{array}[]{ll}\epsilon\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{1,0,0},\quad\alpha=0,\\\\[5.69054pt]
(\epsilon+\lambda)\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|+1,|\alpha|,|\alpha|}+\lambda^{-1}\left\|\nabla^{|\alpha|}\vartheta(t)\right\|^{2},\quad
1\leq|\alpha|\leq 3.\end{array}\right.$ (3.52)
By using the same argument as $I_{7}$, one can get
$I_{6}\leq C\left\\{\begin{array}[]{ll}\epsilon\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2},\quad\alpha=0,\\\\[5.69054pt]
(\epsilon+\lambda)\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|-1,|\alpha|,|\alpha|-1}+\lambda^{-1}\left\|\nabla^{|\alpha|}w(t)\right\|^{2},\quad
1\leq|\alpha|\leq 3.\end{array}\right.$ (3.53)
Similar to the estimate of $I_{1}$, it is easy to check that
$I_{8}\leq C\epsilon\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|,|\alpha|,|\alpha|}.$ (3.54)
In order to estimate $I_{9}$ and $I_{10}$, we use the equations
$(\ref{3.1})_{1}$ and $(\ref{3.1})_{2}$ again. In fact for $I_{9}$,
$\begin{array}[]{ll}2I_{9}&\leq
C\displaystyle\left|\langle\tilde{A}_{\rho}(t)\sigma_{t}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle\right|+\left|\langle\tilde{A}_{\theta}(t)\vartheta_{t}(t)\partial_{x}^{\alpha}w(t),\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&=I_{9,1}+I_{9,2}\end{array}$ (3.55)
Using $(\ref{3.1})_{1}$, $(\ref{3.1})_{2}$ and (3.21), $I_{9,1}$ and $I_{9,2}$
can be estimated as follows
$\begin{array}[]{ll}I_{9,1}&=\displaystyle\left|\langle\nabla\cdot\\{(\rho^{*}+\sigma(t))w(t)+v^{*}\sigma(t)\\},\tilde{A}_{\rho}(t)\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&\leq C\left\|(\nabla\sigma^{*},\nabla\sigma(t),\nabla w(t),\nabla
v^{*})\right\|\left\|(w(t),\sigma(t),\rho^{*},v^{*})\right\|_{L^{6}}\left\|\partial_{x}^{\alpha}w(t)\right\|^{2}_{L^{6}}\\\\[5.69054pt]
&\leq C\epsilon\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}\end{array}$
(3.56)
$\begin{array}[]{ll}I_{9,2}&=\displaystyle\left|\langle\tilde{\alpha}D(t)\Delta\vartheta(t)-E(t)(\nabla\cdot
w)(t)+h(t),\tilde{A}_{\theta}(t)\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&\leq\tilde{\alpha}\left|\langle\nabla
D(t)\nabla\vartheta(t),\tilde{A}_{\theta}(t)\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|+\tilde{\alpha}\left|\langle
D(t)\nabla\vartheta(t),\nabla\tilde{A}_{\theta}(t)\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&\quad+2\tilde{\alpha}\left|\langle
D(t)\nabla\vartheta(t),\tilde{A}_{\theta}(t)\nabla\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&\quad+\left|\langle-E(t)(\nabla\cdot
w)(t)+H_{0}(t),\tilde{A}_{\theta}(t)\partial_{x}^{\alpha}w(t)\partial_{x}^{\alpha}w(t)\rangle\right|\\\\[5.69054pt]
&\leq
C\left\\{\left\|\partial_{x}^{\alpha}w(t)\right\|^{2}_{L^{6}}\big{(}\left\|(\nabla^{3}\theta^{*},\nabla
v^{*},\nabla^{2}\sigma(t),\nabla
w(t),\nabla\vartheta(t),\nabla\theta^{*},G,H)\right\|\right.\\\\[5.69054pt]
&\quad\times\left\|(\sigma(t),w(t),\vartheta(t),v^{*})\right\|_{L^{6}}+\left\|(\nabla\sigma(t),\nabla^{2}\sigma(t),\nabla\sigma^{*},\nabla^{2}\sigma^{*},\nabla\vartheta(t),\nabla\theta^{*},\nabla
w(t),\nabla v^{*})\right\|\\\\[5.69054pt]
&\quad\times\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|_{L^{6}}\big{)}+\left\|\nabla\vartheta(t)\right\|_{L^{3}}\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|\left\|\partial_{x}^{\alpha}w(t)\right\|_{L^{6}}\Big{\\}}\\\\[5.69054pt]
&\leq C\epsilon\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}\end{array}$
(3.57)
Consequently,
$I_{9}\leq C\epsilon\left\|\nabla\partial_{x}^{\alpha}w(t)\right\|^{2}.$
(3.58)
Finally, the term $I_{10}$ is estimated in way similar to that of $I_{9}$, and
we have
$I_{10}\leq
C\epsilon\left\|\nabla\partial_{x}^{\alpha}\vartheta(t)\right\|^{2}.$ (3.59)
Combining (3.32)-(3.37), (3.47), (3.52)-(3.54), (3.58) and (3.59), we can
obtain (3.30) and (3.31), if we take
$d_{1}=\min_{(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})}\\{\frac{\mu\tilde{A}(\rho,\theta)}{\rho}\\}$,
$d_{2}=\min_{(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})}(\tilde{\alpha}D(\rho,\theta)\tilde{B}(\rho,\theta)\\}$
and choose $\epsilon$ and $\lambda$ small enough. This completes the proof of
Lemma 3.2.
### 3.3 Estimates for $\nabla\sigma(t)$ and its derivatives up to
$\nabla^{5}\sigma(t)$
###### Lemma 3.3.
Let $(\sigma,w,\vartheta)(t)\in\mathcal{C}(0,t_{1};\mathcal{H}^{4,3,3})$ be a
solution to the Cauchy problem (3.4)-(3.5) for some positive constant $t_{1}$.
Then there exists three constants $\epsilon_{0},\lambda_{0}>0$ and $d_{3}>0$
such that if $\epsilon\leq\epsilon_{0}$ and
$\|(\sigma,w,\vartheta)(t)\|_{4,3,3},\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon$,
then it holds
$\displaystyle\frac{d}{dt}\langle
w(t),\nabla\sigma(t)\rangle+d_{3}\left\|\nabla\sigma(t)\right\|^{2}+\kappa\left\|\nabla^{2}\sigma(t)\right\|^{2}\leq
C\|(\nabla w,\nabla\vartheta)(t)\|^{2}$ (3.60)
and for $1\leq k\leq 3$ and any $\lambda$ with $0<\lambda<\lambda_{0}$,
$\displaystyle\frac{d}{dt}\langle\nabla^{k}w(t),\nabla^{k+1}\sigma(t)\rangle+d_{3}\left\|\nabla^{k+1}\sigma(t)\right\|^{2}+\kappa\left\|\nabla^{k+2}\sigma(t)\right\|^{2}\leq
C\|(\nabla\sigma,\nabla w,\nabla\vartheta)(t)\|^{2}$ (3.61)
where the constant $C>0$ is depending only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$.
Proof. Using the Friedrichs mollifier, we may assume that
$(\sigma,w,\vartheta)\in\mathcal{C}(0,t_{1};\mathcal{H}^{\infty,\infty,\infty})$.
For any multi-index $\alpha$ with $0\leq|\alpha|\leq 3$, applying
$\partial_{x}^{\alpha}$ to $(\ref{3.1})_{2}$, then taking the $L^{2}$ inner
product of the resultant equations with
$\partial_{x}^{\alpha}\nabla\sigma(t)$, we have
$\begin{array}[]{ll}&\langle
A(t)\nabla\partial_{x}^{\alpha}\sigma(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+\kappa\left\|\nabla^{2}\partial_{x}^{\alpha}\sigma(t)\right\|^{2}\\\\[5.69054pt]
&=-\langle\partial_{x}^{\alpha}w_{t}(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+\left|\langle\partial_{x}^{\alpha}\left\\{\frac{1}{\rho^{*}}\left[\mu\Delta
w(t)+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot\right)w(t)\right]\right\\},\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\quad+\displaystyle\sum_{\beta<\alpha}C_{\alpha}^{\beta}\left|\langle\partial_{x}^{\alpha-\beta}A(t)\partial_{x}^{\beta}\nabla\sigma(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\quad+\left|\langle\partial_{x}^{\alpha}(B(t)\nabla\vartheta(t)),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|+\left|\langle\partial_{x}^{\alpha}f(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle\right|\\\\[5.69054pt]
&=I_{1}+I_{2}+\cdots+I_{5}.\end{array}$ (3.62)
For $I_{1}$, we deduce from integration by parts and $(\ref{3.1})_{1}$ that
$\begin{array}[]{ll}I_{1}&=\displaystyle-\frac{d}{dt}\langle\partial_{x}^{\alpha}w(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle-\langle\partial_{x}^{\alpha}(\nabla\cdot
w)(t),\partial_{x}^{\alpha}\sigma_{t}(t)\rangle\\\\[5.69054pt]
&=\displaystyle-\frac{d}{dt}\langle\partial_{x}^{\alpha}w(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+\langle\partial_{x}^{\alpha}(\nabla\cdot
w)(t),\partial_{x}^{\alpha}\nabla\cdot\\{(\rho^{*}+\sigma(t))w(t)\\}\rangle\\\\[5.69054pt]
&\quad+\displaystyle\langle\partial_{x}^{\alpha}(\nabla\cdot
w)(t),\partial_{x}^{\alpha}\nabla\cdot(\sigma(t)v^{*})\rangle\\\\[5.69054pt]
&=\displaystyle-\frac{d}{dt}\langle\partial_{x}^{\alpha}w(t),\nabla\partial_{x}^{\alpha}\sigma(t)\rangle+I_{1,1}+I_{1,2}\end{array}$
(3.63)
By using the way similar to that of (3.34), we have
$I_{1,1}+I_{1,2}\leq
C\epsilon\left\|\nabla^{2}\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+C\epsilon\left(\left\|\nabla
w(t)\right\|^{2}_{|\alpha|+1}+\left\|\nabla\sigma(t)\right\|^{2}_{|\alpha|}\right)$
(3.64)
For $I_{2}$, let $\alpha_{0}\leq\alpha$ be a multi-index with
$|\alpha_{0}|=1$, then it follows from integration by parts and the Cauchy
inequality that
$\begin{array}[]{ll}I_{2}&=\displaystyle\left|\langle\partial_{x}^{\alpha-\alpha_{0}}\left\\{\frac{1}{\rho^{*}}\left[\mu\Delta
w(t)+(\mu+\mu^{\prime})\nabla\left(\nabla\cdot\right)w(t)\right]\right\\},\nabla\partial_{x}^{\alpha+\alpha_{0}}\sigma(t)\rangle\right|\\\\[5.69054pt]
&\leq\lambda\left\|\nabla^{2}\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+C\lambda^{-1}\left\|\nabla
w(t)\right\|^{2}_{|\alpha|}\end{array}$ (3.65)
Using the Cauchy inequality , $I_{3}$ and $I_{4}$ can be estimated as follows
$I_{3}\leq C\epsilon\left\|\nabla\sigma(t)\right\|^{2}_{|\alpha|},$ (3.66)
$I_{4}\leq\lambda\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}+C\lambda^{-1}\left\|\nabla\vartheta(t)\right\|^{2}_{|\alpha|}.$
(3.67)
Finally, similar to the estimate of $I_{7}$ in the proof of Lemma 3.2, we have
$I_{5}\leq
C(\epsilon+\lambda)\left\|\nabla\partial_{x}^{\alpha}\sigma(t)\right\|^{2}_{1}+C\lambda^{-1}\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{|\alpha|-1,|\alpha|,|\alpha|-1}.$ (3.68)
Combining (3.62)-(3.68) and summing up $\alpha$ with $|\alpha|=k$, we can get
(3.61), if we take
$d_{3}=\min_{(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})}\left\\{\frac{P_{\rho}(\rho,\theta)}{\rho}\right\\}$
and choose $\epsilon$ and $\lambda$ small enough. For $\alpha=0$, by using the
same argument as above, we can also get (3.60). This completes the proof of
Lemma 3.3.
### 3.4 Proof of Proposition 3.2
Let $(\sigma,w,\vartheta)(t)\in\mathcal{C}(0,t_{1};\mathcal{H}^{4,3,3})$ be a
solution to the Cauchy problem (3.4)-(3.5) for some positive constant $t_{1}$.
Furthermore, we assume that
$\|(\sigma,w,\vartheta)(t)\|_{4,3,3},\|(\sigma,w,\vartheta)\|_{\mathcal{F}^{5,5,5}}\leq\epsilon$,
where $\epsilon>0$ is small enough such that we can use the results obtained
in Lemmas 3.2-3.3. Set
$\left[\sigma,w,\vartheta\right](t)=\left\|\sigma(t)\right\|^{2}+\langle\hat{A}(t)\nabla\sigma(t),\nabla\sigma(t)\rangle+\langle\tilde{A}(t)w(t),w(t)\rangle+\langle\tilde{B}(t)\vartheta(t),\vartheta(t)\rangle$
where $\langle\hat{A}(t)$, $\tilde{A}(t)$ and $\tilde{B}(t)$ are defined as in
Lemma 3.2.
Multiplying (3.60) with a small constant $\lambda_{0}$, then adding the
resultant equation to (3.30), we have
$\displaystyle\frac{d}{dt}\left\\{a_{0}\left[\sigma,w,\vartheta\right](t)+b_{0}\langle
w(t),\nabla\sigma(t)\rangle\right\\}+\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{1,0,0}\leq 0$ (3.69)
provided that $\epsilon>0$ is small enough. Here and hereafter,
$a_{\nu},b_{\nu}>0,\nu=0,1,\cdots,3$ denote some constants depending only on
$\bar{\rho},\bar{\theta},\mu,\mu^{\prime},\kappa$ and $\tilde{\alpha}$. Then
summing up (3.61), (3.31) with $k=1$ and (3.69), we get
$\displaystyle\frac{d}{dt}\left\\{\sum_{\nu=0}^{1}a_{\nu}\left[\nabla^{\nu}\sigma,\nabla^{\nu}w,\nabla^{\nu}\vartheta\right](t)+\sum_{\nu=0}^{1}b_{\nu}\langle\nabla^{\nu}w(t),\nabla^{\mu+1}\sigma(t)\rangle\right\\}+\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{2,1,1}\leq 0$ (3.70)
Similarly, summing up (3.61), (3.31) with $k=2$ and (3.70) gives
$\displaystyle\frac{d}{dt}\left\\{\sum_{\nu=0}^{2}a_{\nu}\left[\nabla^{\nu}\sigma,\nabla^{\nu}w,\nabla^{\nu}\vartheta\right](t)+\sum_{\nu=0}^{2}b_{\nu}\langle\nabla^{\nu}w(t),\nabla^{\mu+1}\sigma(t)\rangle\right\\}+\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{3,2,2}\leq 0$ (3.71)
and summing up (3.61), (3.31) with $k=3$ and (3.71) gives
$\displaystyle\frac{d}{dt}\left\\{\sum_{\nu=0}^{3}a_{\nu}\left[\nabla^{\nu}\sigma,\nabla^{\nu}w,\nabla^{\nu}\vartheta\right](t)+\sum_{\nu=0}^{3}b_{\nu}\langle\nabla^{\nu}w(t),\nabla^{\mu+1}\sigma(t)\rangle\right\\}+\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(t)\right\|^{2}_{4,3,3}\leq 0$ (3.72)
Integrating (3.72) with respect to $t$ over $[0,t]$, we have
$\displaystyle
N\left[\sigma,w,\vartheta\right](t)+\int_{0}^{t}\left\|(\nabla\sigma,\nabla
w,\nabla\vartheta)(s)\right\|^{2}_{4,3,3}\,ds\leq
N\left[\nabla^{\nu}\sigma,\nabla^{\nu}w,\nabla^{\nu}\vartheta\right](0)$
(3.73)
for any $t\in[0,t_{1}]$, where
$N\left[\sigma,w,\vartheta\right](t)\equiv\sum_{\nu=0}^{3}a_{\nu}\left[\nabla^{\nu}\sigma,\nabla^{\nu}w,\nabla^{\nu}\vartheta\right](t)+\sum_{\nu=0}^{3}b_{\nu}\langle\nabla^{\nu}w(t),\nabla^{\mu+1}\sigma(t)\rangle,\quad
t>0$
Denote
$B_{0}=\min_{(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})}\left\\{\hat{A}(\rho,\theta),\tilde{A}(\rho,\theta),\tilde{B}(\rho,\theta),1\right\\}$
and
$B_{1}=\max_{(\rho,\theta)\in\mathcal{S}(\bar{\rho},\bar{\theta})}\\{\hat{A}(\rho,\theta),\tilde{A}(\rho,\theta),\\\
\tilde{B}(\rho,\theta),1\\}$. Since we may assume without loss of generality
that $a_{\nu}\leq a_{\nu-1}$ and $b_{\nu}\leq a_{\nu}\min\\{B_{0},1\\}/4$ for
$\nu=1,2,3$, it follows from simple calculation that
$\displaystyle\frac{\alpha_{3}}{4}B_{0}\left\|(\sigma,w,\vartheta)(t)\right\|^{2}_{4,3,3}\leq
N\left[\sigma,w,\vartheta\right](t)\leq
2\alpha_{0}\left\|(\sigma,w,\vartheta)(t)\right\|^{2}_{4,3,3}$ (3.74)
for each $t\in[0,t_{1}]$. Combining $(\ref{3.52})$ and $(\ref{3.53})$, we get
(3.6). This completes the proof of Proposition 3.2.
Hence, by Propositions 3.1 and 3.2, we finally arrive at the conclusion of
Theorem 1.2.
Acknowledgment
This work was supported by a grant from the National Natural Science
Foundation of China under contract 10925103 and “the Fundamental Research
Funds for the Central Universities”.
## References
* [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1985.
* [2] J. E. Dunn, J. Serrin, On the thermomechanics of interstital working. Arch. Ration. Mech. Anal., 88 (1985), 95-133.
* [3] D. M. Anderson, G. B. McFadden, G. B. Wheeler, Diffuse-interface methods in fluid mech, Ann. Rev. Fluid Mech., 30 (1998), 139-165.
* [4] J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys., 28 (1998), 258-267.
* [5] M. E. Gurtin, D. Polignone, J. Vinals, Two-phase binary fluids and immiscible fluids describled by an order parameter, Math. Models Methods Appl. Sci., 6 (6) (1996), 815-831.
* [6] Y. Shibata, K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Japan, 55 (3) (2003), 797-826.
* [7] J. Z. Qian, H. Yin, On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance, J. Differential Equations, 237 (2007), 225-256.
* [8] D. Bresch, B. Desjardins, C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.
* [9] B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.
* [10] B. Haspot, Existence of strong solutions for nonisothermal Korteweg system, Ann. Math. Blaise Pascal, 16 (2009) (2), 431-481.
* [11] R. Danchin, B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincare Anal. Nonlinear, 18 (2001), 97-133.
* [12] H. Hattori, D. Li, Solutions for two dimensional systems for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98.
* [13] H. Hattori, D. Li, Golobal solutions of a high dimensional systems for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97.
* [14] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. Henri. Poincare Anal. Nonlinear, 25 (4) (2008), 679-696.
* [15] Y. J. Wang, Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.
* [16] Z. Tan, H. Q. Wang, X. J. Kai, Global existence and optimal $L^{2}$ decay rate for the strong solutions to the compressible fluid models of Korteweg type, J. Math. Anal. Appl. 390 (2012), 181-187.
* [17] Y. P. Li, Global existence and optimal decay rate for the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.
* [18] L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, Volume 19, 1998.
* [19] M. E. Talor, Partial Differential Equations(III), Nonlinear Equation, Springer, 1996.
* [20] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci.,55 (1979), 337-342.
* [21] A. Matsumura, T. Nishida, The initial value problems for the equations of motion of compressible viscous and heat-conductive gas, J. Math. Kyoto Univ., 20 (1980), 67-104.
* [22] A. Matsumura, T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89(4) (1983), 445-464.
* [23] A. Matsumura, T. Nishida, Exterior stationary problems for the equations of motion of compressible viscous and heat-conductive fluids, in: Differntial Equations, Xanthi,1987, in: Lecture Notes in Pure and Appl. Math., 118 Dekker, New York, 1989, 473-479.
* [24] A. Novotny, M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in 3$D$-exterior domains ($v_{\infty}=0$), J. Math. Kyoto Univ., 36 (1996), 389-422.
* [25] A. Novotny, M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in 3$D$-exterior domains ($v_{\infty}\neq 0$), Math. Ann., 308 (1997), 439-489.
* [26] K. Tanaka, A stability of steady flow of compressible viscous fluid with respect to intial disturbance ($v_{\infty}\neq 0$), Math. Methods Appl. Sci., 29 (2006) (12), 1451-1466.
|
arxiv-papers
| 2012-03-29T14:10:48 |
2024-09-04T02:49:29.175655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhengzheng Chen, Huijiang Zhao",
"submitter": "Chen Zhengzheng",
"url": "https://arxiv.org/abs/1203.6527"
}
|
1203.6529
|
# Time periodic solutions of compressible fluid models of Korteweg type
Zhengzheng Chen
School of Mathematics and Statistics
Wuhan University, Wuhan 430072, China
Qinghua Xiao
School of Mathematics and Statistics
Wuhan University, Wuhan 430072, China
Huijiang Zhao
School of Mathematics and Statistics
Wuhan University, Wuhan 430072, China Corresponding author.E-mail:
hhjjzhao@hotmail.com
###### Abstract
This paper is concerned with the existence, uniqueness and time-asymptotic
stability of time periodic solutions to the compressible Navier-Stokes-
Korteweg system effected by a time periodic external force in
$\mathbb{R}^{n}$. Our analysis is based on a combination of the energy method
and the time decay estimates of solutions to the linearized system.
Keywords Navier-Stokes-Korteweg system; Capillary fluids; Time periodic
solution; Energy estimates;
AMS Subject Classifications 2010: 35M10, 35Q35, 35B10.
## 1 Introduction
The compressible Navier-Stokes-Korteweg system for the density $\rho>0$ and
velocity $u=(u_{1},u_{2},\cdots,u_{n})\\\ \in\mathbb{R}^{n}$ is written as :
$\displaystyle\left\\{\begin{array}[]{ll}\rho_{t}+\nabla\cdot(\rho
u)=0,\\\\[5.69054pt] (\rho u)_{t}+\nabla\cdot(\rho u\bigotimes u)+\nabla
P(\rho)-\mu\Delta u-(\nu+\mu)\nabla(\nabla\cdot
u)=\kappa\rho\nabla\Delta\rho+\rho f(t,x).\end{array}\right.$ (1.3)
Here, $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{n}$, $P=P(\rho)$ is the
pressure, $\mu,\nu$ are the viscosity coefficients, $\kappa$ is the capillary
coefficient, and $f(t,x)=(f_{1},f_{2},f_{3})(t,x)$ is a given external force.
System (1.1) can be used to describe the motion of the compressible isothermal
fluids with capillarity effect of materials, see the pioneering work by Dunn
and Serrin [1], and also [2, 3, 4].
In this paper, we consider the problem (1.1) for $(\rho,u)$ around a constant
state $(\rho_{\infty},0)$ for $n\geq 5$, where $\rho_{\infty}$ is a positive
constant. Throughout this paper, we make the following basic assumptions:
(H1): $\mu$, $\nu$ and $\kappa$ are positive constants and satisfying
$\nu+\frac{2}{n}\mu\geq 0$.
(H2): $P(\rho)$ is smooth in a neighborhood of $\rho_{\infty}$ satisfying
$P^{\prime}(\rho_{\infty})>0$.
(H3): $f$ is time periodic with period $T>0$.
The main purpose of this paper is to show that the problem (1.1) admits a time
periodic solution around the constant state $(\rho_{\infty},0)$ which has the
same period as $f$. By combining the energy method and the optimal decay
estimates of solutions to the linearized system, we prove the existence of a
time periodic solution in some suitable function space. Notice that some
similar results have been obtained for the compressible Navier-Stokes
equations and Boltzmann equation, cf. [8, 9, 10, 11].
Precisely, Let $N\geq n+2$ be a positive integer, define the solution space by
$X_{M}(0,T)=\left\\{(\rho,u)(t,x)\left|\begin{array}[]{c}\rho(t,x)\in
C(0,T;H^{N}(\mathbb{R}^{n}))\cap
C^{1}(0,T;H^{N-2}(\mathbb{R}^{n})),\\\\[5.69054pt] u(t,x)\in
C(0,T;H^{N-1}(\mathbb{R}^{n}))\cap
C^{1}(0,T;H^{N-3}(\mathbb{R}^{n})),\\\\[5.69054pt] \nabla\rho(t,x)\in
L^{2}(0,T;H^{N+1}(\mathbb{R}^{n})),\\\\[5.69054pt] \nabla u(t,x)\in
L^{2}(0,T;H^{N}(\mathbb{R}^{n})),|||(\rho,u)|||\leq
M,\end{array}\right.\right\\}$ (1.4)
for some positive constant $M$ and with the norm
$|||(\rho,u)|||^{2}=\sup_{0\leq t\leq
T}\left\\{\|\rho(t)\|_{N}^{2}+\|u(t)\|_{N-1}^{2}\right\\}+\int_{0}^{T}\left(\|\nabla\rho(t)\|_{N+1}^{2}+\|\nabla
u(t)\|_{N}^{2}\right)dt.$ (1.5)
Then the existence of the time periodic solution can be stated as follows.
###### Theorem 1.1.
Let $n\geq 5,N\geq n+2$. Assume the assumptions (H1)-(H3) hold, and $f(t,x)\in
C(0,T;H^{N-1}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n}))$. Then there exists a
small constant $\delta_{0}>0$ and a constant $M_{0}>0$ which are dependent on
$\rho_{\infty}$, such that if
$\sup_{0\leq t\leq T}\|f(t)\|_{H^{N-1}\cap L^{1}}\leq\delta_{0},$ (1.6)
then the problem (1.1) admits a time periodic solution $(\rho^{per},u^{per})$
with period $T$, satisfying
$(\rho^{per}-\rho_{\infty},u^{per})\in X_{M_{0}}(0,T)$
Furthermore the periodic solution is unique in the following sense: if there
is another time periodic solution $(\rho_{1}^{per},u_{1}^{per})$ satisfying
(1.1) with the same $f$, and $(\rho_{1}^{per}-\rho_{\infty},u_{1}^{per})\in
X_{M_{0}}(0,T)$, then $(\rho_{1}^{per},u_{1}^{per})=(\rho^{per},u^{per})$.
To study the stability of the time periodic solution $(\rho^{per},u^{per})$
obtained in Theorem 1.1, we consider the problem (1.1) with the following
initial date
$(\rho,u)(t,x)|_{t=0}=(\rho_{0},u_{0})(x)\rightarrow(\rho_{\infty},0),\quad
as\,\,|x|\rightarrow\infty.$ (1.7)
Here $\rho_{0}(x)$ and $u_{0}(x)$ is a small perturbation of the time periodic
solution $(\rho^{per},u^{per})$. And we have the following stability result.
###### Theorem 1.2.
Under the assumptions of Theorem 1.1, let $(\rho^{per},u^{per})$ be the time
periodic solution thus obtained. If the initial date $(\rho_{0},u_{0})$ be
such that $\|(\rho_{0}-\rho^{per}(0),u_{0}-u^{per}(0)\|_{N-1}$ is sufficiently
small, then the Cauchy problem (1.1), (1.7) has a unique classical solution
$(\rho,u)$ globally in time, which satisfies
$\begin{array}[]{rl}&\rho-\rho^{per}\in
C(0,\infty;H^{N-1}(\mathbb{R}^{n}))\cap
C^{1}(0,\infty;H^{N-3}(\mathbb{R}^{n})),\\\\[5.69054pt] &u-u^{per}\in
C(0,\infty;H^{N-2}(\mathbb{R}^{n}))\cap
C^{1}(0,\infty;H^{N-4}(\mathbb{R}^{n})).\end{array}$ (1.8)
Moreover, there exists a constant $C_{0}>0$ such that
$\begin{array}[]{rl}&\|(\rho-\rho^{per})(t)\|^{2}_{N-1}+\|(u-u^{per})(t)\|^{2}_{N-2}+\displaystyle\int_{0}^{t}\left(\|\nabla(\rho-\rho^{per})(\tau)\|^{2}_{N-1}+\|\nabla(u-u^{per})(\tau)\|^{2}_{N-2}\right)d\tau\\\\[8.53581pt]
&\leq
C_{0}\left(\|\rho_{0}-\rho^{per}(0)\|^{2}_{N-1}+\|u_{0}-u^{per}(0)\|^{2}_{N-2}\right),\end{array}$
(1.9)
for any $t\geq 0$ and
$\displaystyle\|(\rho-\rho^{per},u-u^{per})\|_{L^{\infty}}\rightarrow
0\,\,\,as\,\,t\rightarrow\infty.$ (1.10)
Now we outline the main ingredients used in proving of our main results. For
the proof of Theorem 1.1, thanks to the time decay estimates of solutions to
the linear system (2.14) (see Lemma 2.1 below), we can show the integral in
(4.5) is convergent. Based on this and the elaborate energy estimates given in
Section 3, we prove the existence of time periodic solution by the contraction
mapping principle. Here, similar to the case of compressible Navier-Stokes
equations, Theorem 1.1 is obtained only in the case $n\geq 5$ because of the
convergence of the integral in (4.5). Thus, how to deal with the case $n<5$,
especially, the physical case $n=3$, is still an open problem. Theorem 1.2 is
established by the energy method. The key ingredient in the proof of Theorem
1.2, among other things, is to get the a priori estimates, which can be done
similarly to the estimates in Section 3.
There have been a lot of studies on the mathematical theory of the
compressible Navier-Stokes-Korteweg system. For example, Hattori and Li [12,
13] proved the local existence and the global existence of smooth solutions in
Sobolev space. Danchin and Desjardins [7] studied the existence of suitably
smooth solutions in critical Besov space. Bresch, Desjardins and Lin [5]
considered the global existence of weak solution, then Haspot improved their
results in [6]. The local existence of strong solutions was proven in [14].
Recently, Wang and Tan [15] established the optimal decay rates of global
smooth solutions without external force. Li [16] discussed the global
existence and optimal $L^{2}$-decay rate of smooth solutions with potential
external force.
The rest of the paper is organized as follows. In Section 2, we will
reformulate the problem and give some preliminaries for later use. In Section
3, we give the energy estimates on the linearized system (2.8). The proof of
Theorem 1.1 is given in Section 4. In the last section, we will study the
stability of the time periodic solution.
Notations: Throughout this paper, for simplicity, we will omit the variables
$t,x$ of functions if it does not cauchy any confusion. $C$ denotes a generic
positive constant which may vary in different estimates.
$\langle\cdot,\cdot\rangle$ is the inner product in $L^{2}(\mathbb{R}^{n})$.
The norm in the usual Sobolev Space $H^{s}(\mathbb{R}^{n})$ are denoted by
$\|\cdot\|_{s}$ for $s\geq 0$. When s=0, we will simply use $\|\cdot\|$.
Moreover, we denote $\|\cdot\|_{H^{s}}+\|\cdot\|_{L^{1}}$ by
$\|\cdot\|_{H^{s}\cap L^{1}}$. If $g=(g_{1},g_{2},\cdots,g_{n})$, then
$\|g\|=\displaystyle\sum_{k=1}^{n}(\|g_{k}\|^{2})^{\frac{1}{2}}$.
$\nabla=(\partial_{1},\partial_{2},\cdots,\partial_{n})$ with
$\partial_{i}=\partial_{x_{i}},i=1,2,\cdots,n$ and for any integer $l\geq 0$,
$\nabla^{l}g$ denotes all $x$ derivatives of order $l$ of the function $g$.
Finally, for multi-index $\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$,
it is standard that
$\partial_{x}^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}^{\alpha_{2}}\cdots\partial_{x_{n}}^{\alpha_{n}},\quad|\alpha|=\sum_{i=1}^{n}\alpha_{i}.$
## 2 Reformulated system and preliminaries
We reformulate the system (1.1) in this section. Firstly, set
$\gamma=\sqrt{P^{\prime}(\rho_{\infty})},\quad\kappa^{\prime}=\frac{\rho_{\infty}}{\gamma}\kappa,\quad\mu^{\prime}=\frac{\mu}{\rho_{\infty}},\quad\nu^{\prime}=\frac{\nu+\mu}{\rho_{\infty}},\quad\lambda_{1}=\frac{\gamma}{\rho_{\infty}},\quad\lambda_{2}=\frac{\rho_{\infty}}{\gamma},$
and define the new variables
$\sigma=\rho-\rho_{\infty},\quad v=\lambda_{2}u,$
then the system (1.1) is reformulated as
$\displaystyle\left\\{\begin{array}[]{ll}\sigma_{t}+\gamma\nabla\cdot
v=G_{1}(\sigma,v),\\\\[5.69054pt] v_{t}-\mu^{\prime}\Delta
v-\nu^{\prime}\nabla(\nabla\cdot
v)+\gamma\nabla\sigma-\kappa^{\prime}\nabla\Delta\sigma=G_{2}(\sigma,v)+\lambda_{2}f,\end{array}\right.$
(2.3)
where
$\begin{array}[]{rl}&G_{1}(\sigma,v)=-\lambda_{1}\nabla\cdot(\sigma
v),\\\\[5.69054pt]
&G_{2}(\sigma,v)=\displaystyle-\frac{\sigma}{\rho_{\infty}(\sigma+\rho_{\infty})}(\mu\Delta
v+\nu\nabla(\nabla\cdot
v))-\lambda_{1}(v\cdot\nabla)v-\lambda_{2}\left[\frac{P^{\prime}(\sigma+\rho_{\infty})}{\sigma+\rho_{\infty}}-\frac{P^{\prime}(\rho_{\infty})}{\rho_{\infty}}\right]\nabla\sigma.\end{array}$
Notice that $G_{1}$ and $G_{2}$ have the following properties:
$\begin{array}[]{rl}&G_{1}(\sigma,v)\thicksim\nabla\sigma\cdot
v+\sigma\nabla\cdot v,\\\\[5.69054pt] &G_{2}(\sigma,v)\thicksim\sigma\Delta
v+\sigma\nabla(\nabla\cdot v)+(v\cdot\nabla)v+\sigma\nabla\sigma.\end{array}$
(2.4)
Here $\thicksim$ means that two side are of same order.
Set $U=(\sigma,v)$, $G=(G_{1},G_{2})$, $F=(0,\lambda_{2}f)$ and
$\mathbb{A}=\left(\begin{array}[]{ll}\quad\quad 0\qquad\qquad\qquad\gamma
div\\\\[8.53581pt]
\gamma\nabla-\kappa^{\prime}\nabla\Delta\qquad-\mu^{\prime}\Delta-\nu^{\prime}\nabla
div\end{array}\right),$
then the system (2.1) takes the form
$U_{t}+\mathbb{A}U=G(U)+F.$ (2.5)
We first consider the linearized system of (2.3):
$\displaystyle\left\\{\begin{array}[]{ll}\sigma_{t}+\gamma\nabla\cdot
v=G_{1}(\tilde{U}),\\\\[5.69054pt] v_{t}-\mu^{\prime}\Delta
v-\nu^{\prime}\nabla(\nabla\cdot
v)+\gamma\nabla\sigma-\kappa^{\prime}\nabla\Delta\sigma=G_{2}(\tilde{U})+\lambda_{2}f,\end{array}\right.$
(2.8)
for any given functions $\tilde{U}=(\tilde{\sigma},\tilde{v})$ satisfying
$\tilde{\sigma}\in H^{N+2}(\mathbb{R}^{n}),\quad\tilde{v}\in
H^{N+1}(\mathbb{R}^{n}).$
Notice that the system (2.8) can be written as
$U_{t}+\mathbb{A}U=G(\tilde{U})+F.$ (2.9)
By the Duhamel’s principle, the solution to the system (2.8) can be written in
the mild form as
$U(t)=\displaystyle\mathbb{S}(t,s)U(s)+\int_{s}^{t}\mathbb{S}(t,\tau)(G(\tilde{U})+F)(\tau)d\tau,\quad
t\geq s,$ (2.10)
where $\mathbb{S}(t,s)$ is the corresponding linearized solution operator
defined by
$\mathbb{S}(t,s)=e^{(t-s)\mathbb{A}},\quad t\geq s.$
Indeed, the corresponding homogeneous linear system to (2.8) is
$\displaystyle\left\\{\begin{array}[]{ll}\sigma_{t}+\gamma\nabla\cdot
v=0,\\\\[5.69054pt] v_{t}-\mu^{\prime}\Delta v-\nu^{\prime}\nabla(\nabla\cdot
v)+\gamma\nabla\sigma-\kappa^{\prime}\nabla\Delta\sigma=0,\\\\[5.69054pt]
\sigma|_{t=s}=\sigma_{s}(x),\quad v|_{t=s}=v_{s}(x).\end{array}\right.$ (2.14)
By repeating the argument in the proof of Theorem 1.3 in [15], we can get the
following result for the problem (2.14). The details are omitted here.
###### Lemma 2.1.
Let $l\geq 0$ be an integer. Assume that $(\sigma,v)$ is the solution of the
problem (2.14) with the initial date $\sigma_{s}\in H^{l+1}\cap L^{1}$ and
$v_{s}\in H^{l}\cap L^{1}$, then
$\|\sigma(t)\|\leq\displaystyle
C(1+t)^{-\frac{n}{4}}\left(\|(\sigma_{s},v_{s})\|_{L^{1}}+\|(\sigma_{s},v_{s})\|\right),$
$\|\nabla^{k+1}\sigma(t)\|\leq\displaystyle
C(1+t)^{-\frac{n}{4}-\frac{k+1}{2}}\left(\|(\sigma_{s},v_{s})\|_{L^{1}}+\|(\nabla^{k+1}\sigma_{s},\nabla^{k}v_{s})\|\right),$
$\|\nabla^{k}v(t)\|\leq\displaystyle
C(1+t)^{-\frac{n}{4}-\frac{k}{2}}\left(\|(\sigma_{s},v_{s})\|_{L^{1}}+\|(\nabla^{k+1}\sigma_{s},\nabla^{k}v_{s})\|\right),$
where $k$ is an integer satisfying $0\leq k\leq l$.
## 3 Energy estimates
In this section, we will perform some energy estimates on solutions
$(\sigma,v)$ to problem (2.8). Throughout of this section, we assume that
$f(t,x)\in H^{N-1}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n})$ for all $t\geq
0$. For later use, we list some standard inequalities as follows. cf. [8].
###### Lemma 3.1.
Let $m$ be a positive integer and $u\in H^{[\frac{n}{2}]+1}(\mathbb{R}^{n})$,
then
$\|u\|^{2}_{L^{\infty}}\leq C\|\nabla^{m+1}u\|\|\nabla^{m-1}u\|\quad
for\,\,n=2m,$ $\|u\|^{2}_{L^{\infty}}\leq
C\|\nabla^{m+1}u\|\|\nabla^{m}u\|\quad for\,\,n=2m+1.$
###### Lemma 3.2.
Let $m$ be the integer defined in Lemma 3.1 and $f,g,h\in
H^{[\frac{n}{2}]+1}(\mathbb{R}^{n})$ , then we have
$(i)\left|\int_{\mathbb{R}^{n}}f\cdot g\cdot
h\,dx\right|\leq\epsilon\|\nabla^{m-1}f\|_{2}^{2}+C_{\epsilon}\|g\|^{2}\|h\|^{2},$
$(ii)\left|\int_{\mathbb{R}^{n}}f\cdot g\cdot
h\,dx\right|\leq\epsilon\|f\|_{2}^{2}+C_{\epsilon}\|\nabla^{m-1}g\|^{2}_{2}\|h\|^{2},$
for any $\epsilon>0$. Here and hereafter, $C_{\epsilon}$ denotes a positive
constant depending only on $\epsilon$.
We first give the energy estimate on the low order derivatives of
$(\sigma,v)$.
###### Lemma 3.3.
Let $n\geq 5$, $N\geq n+2$, then there exists two suitably small constants
$d_{0}>0$ and $\epsilon_{0}>0$ such that for $0<\epsilon\leq\epsilon_{0}$, it
holds
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|U(t)\|^{2}+\|\nabla\sigma(t)\|^{2}+d_{0}\langle
v,\nabla\sigma\rangle(t)\right)+\|\nabla
v(t)\|^{2}+\|\nabla\sigma(t)\|^{2}_{1}\\\\[5.69054pt] &\leq\epsilon
C\left(\|\nabla^{3}\sigma(t)\|^{2}_{m-2}+\|\nabla^{2}v(t)\|^{2}_{m-1}\right)+C_{\epsilon}C\left(\|\tilde{U}(t)\|^{2}_{m+1}\|\nabla\tilde{U}(t)\|^{2}_{1}+\|f(t)\|^{2}_{L^{1}\cap
L^{2}}\right),\end{array}$ (3.1)
where $m$ is defined in Lemma 3.1 and $C$ depends only on
$\rho_{\infty},\mu,\nu$ and $\kappa$.
Proof. Multiplying $(\ref{2.3})_{1}$ and $(\ref{2.3})_{2}$ by $\sigma$ and
$v$, respectively, and integrating them over $\mathbb{R}^{n}$, we have from
integrating by parts that
$\begin{array}[]{rl}&\displaystyle\frac{1}{2}\frac{d}{dt}\|U\|^{2}+\mu^{\prime}\|\nabla
v\|^{2}+\nu^{\prime}\|\nabla\cdot v\|^{2}\\\\[5.69054pt] =&\langle
G_{1}(\tilde{U}),\sigma\rangle+\langle
G_{2}(\tilde{U}),v\rangle+\kappa^{\prime}\langle\nabla\Delta\sigma,v\rangle+\lambda_{2}\langle
f,v\rangle\\\\[5.69054pt] =&I_{0}+I_{1}+I_{2}+I_{3}.\end{array}$ (3.2)
From (2.4) and Lemma 3.2, we have
$\begin{array}[]{rl}I_{0}&\leq\epsilon\|\nabla^{m-1}\sigma\|_{2}^{2}+C_{\epsilon}C\left(\|\nabla\tilde{\sigma}\|^{2}\|\tilde{v}\|^{2}+\|\tilde{\sigma}\|^{2}\|\nabla\tilde{v}\|^{2}\right)\\\\[5.69054pt]
&\leq\epsilon\|\nabla^{m-1}\sigma\|_{2}^{2}+C_{\epsilon}C\|\tilde{U}\|^{2}\|\nabla\tilde{U}\|^{2},\end{array}$
(3.3)
and
$I_{1}\leq\epsilon\|\nabla^{m-1}v\|^{2}_{2}+C_{\epsilon}C\|\tilde{U}\|^{2}\|\nabla\tilde{U}\|_{1}^{2}.$
(3.4)
For $I_{2}$, integrating by parts and using $(\ref{2.3})_{1}$, (2.4) and Lemma
3.2, we deduce that
$\begin{array}[]{rl}I_{2}&=\displaystyle-\kappa^{\prime}\langle\Delta\sigma,\nabla\cdot
v\rangle=\frac{\kappa^{\prime}}{\gamma}\langle\Delta\sigma,\sigma_{t}-G_{1}(\tilde{U})\rangle\\\\[8.53581pt]
&=\displaystyle-\frac{\kappa^{\prime}}{2\gamma}\frac{d}{dt}\|\nabla\sigma\|^{2}-\frac{\kappa^{\prime}}{\gamma}\langle\Delta\sigma,G_{1}(\tilde{U})\rangle\\\\[8.53581pt]
&\displaystyle\leq-\frac{\kappa^{\prime}}{2\gamma}\frac{d}{dt}\|\nabla\sigma\|^{2}+\epsilon\|\nabla^{2}\sigma\|^{2}+C_{\epsilon}C\|\nabla\tilde{U}\|^{2}\|\nabla^{m-1}\tilde{U}\|^{2}_{2}.\end{array}$
(3.5)
For $I_{3}$, Lemma 3.1 gives
$I_{3}\leq\epsilon\|\nabla^{m-1}v\|^{2}_{2}+C_{\epsilon}C\|f\|^{2}_{L^{1}}.$
(3.6)
Since $n\geq 5$, $N\geq n+2$, we have $m-1\geq 1$. Substituting (3.3)-(3.6)
into (3.2) yields
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|U\|^{2}+\|\nabla\sigma\|^{2}\right)+\|\nabla
v\|^{2}+\|\nabla\cdot v\|^{2}\\\\[5.69054pt] &\leq\epsilon
C\left(\|\nabla^{m-1}\sigma\|_{2}^{2}+\|\nabla^{2}\sigma\|^{2}\right)+\epsilon
C\|\nabla^{2}v\|^{2}_{m-1}+C_{\epsilon}C\left(\|\tilde{U}\|^{2}_{m+1}\|\nabla\tilde{U}\|^{2}_{1}+\|f\|_{L^{1}}^{2}\right),\end{array}$
(3.7)
provided that $\epsilon$ is small enough, where $C$ depends only on
$\rho_{\infty},\mu,\nu$ and $\kappa$.
Next, we estimate $\|\nabla\sigma\|^{2}$. Taking the $L^{2}$ inner product
with $\nabla\sigma$ on both side of $(\ref{2.3})_{2}$ and then integrating by
parts, we have
$\begin{array}[]{rl}&\displaystyle\gamma\|\nabla\sigma\|^{2}+\kappa^{\prime}\|\nabla^{2}\sigma\|^{2}\\\\[5.69054pt]
&=-\langle v_{t},\nabla\sigma\rangle+\mu^{\prime}\langle\Delta
v,\nabla\sigma\rangle+\nu^{\prime}\langle\nabla(\nabla\cdot
v),\nabla\sigma\rangle+\langle
G_{2}(\tilde{U})+\lambda_{2}f,\nabla\sigma\rangle\\\\[5.69054pt]
&=I_{4}+I_{5}+I_{6}+I_{7}.\end{array}$ (3.8)
Similar to (3.5), the term $I_{4}$ can be controlled by
$\begin{array}[]{rl}I_{4}&=-\displaystyle\frac{d}{dt}\langle
v,\nabla\sigma\rangle-\langle\nabla\cdot v,\sigma_{t}\rangle\\\\[5.69054pt]
&=-\displaystyle\frac{d}{dt}\langle v,\nabla\sigma\rangle-\langle\nabla\cdot
v,-\gamma\nabla\cdot v+G_{1}(\tilde{U})\rangle\\\\[5.69054pt]
&\leq-\displaystyle\frac{d}{dt}\langle
v,\nabla\sigma\rangle+2\gamma\|\nabla\cdot
v\|^{2}+C\|\nabla^{m-1}\tilde{U}\|^{2}_{2}\|\nabla\tilde{U}\|^{2}.\end{array}$
(3.9)
Integrating by parts and using the Cauchy-Schwartz inequality, it is easy to
get
$I_{5}+I_{6}\leq\frac{\kappa^{\prime}}{4}\|\nabla^{2}\sigma\|^{2}+C(\|\nabla
v\|^{2}+\|\nabla\cdot v\|^{2}).$ (3.10)
Finally, (2.4) and the Cauchy-Schwartz inequality imply that
$I_{7}\leq\displaystyle\frac{\gamma}{2}\|\nabla\sigma\|^{2}+C\left(\|\nabla^{m-1}\tilde{U}\|^{2}_{2}\|\nabla\tilde{U}\|^{2}_{1}+\|f\|^{2}\right).$
(3.11)
Combining (3.8)-(3.11), we obtain
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\langle
v,\nabla\sigma\rangle+\|\nabla\sigma\|^{2}+\|\nabla^{2}\sigma\|^{2}\\\\[5.69054pt]
&\leq C(\|\nabla v\|^{2}+\|\nabla\cdot
v\|^{2})+C\left(\|\nabla^{m-1}\tilde{U}\|^{2}_{2}\|\nabla\tilde{U}\|^{2}_{1}+\|f\|^{2}\right).\end{array}$
(3.12)
where the constant $C$ depends only on $\rho_{\infty},\mu,\nu$ and $\kappa$.
Multiplying (3.12) with a small constant $d_{0}>0$ and then adding the
resultant equation to (3.7), one can get (3.1) immediately by the smallness of
$d_{0}$ and $\epsilon$. This completes the proof of Lemma 3.3.
Next, we derive the energy estimate on the high order derivatives of
$(\sigma,v)$. We establish the following lemma.
###### Lemma 3.4.
Let $n\geq 5$, $N\geq n+2$, then there exists two suitably small constants
$d_{1}>0$ and $\epsilon_{1}>0$ such that for $0<\epsilon\leq\epsilon_{1}$, it
holds
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\nabla\sigma(t)\|^{2}_{N}+\|\nabla
v(t)\|^{2}_{N-1}+d_{1}\sum_{|\alpha|=1}^{N}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle(t)\right)+\|\nabla^{2}\sigma(t)\|^{2}_{N}+\|\nabla^{2}v(t)\|^{2}_{N-1}\\\\[11.38109pt]
&\leq\epsilon
C\|\nabla\sigma(t)\|^{2}+C_{\epsilon}C\left(\|\nabla\tilde{U}(t)\|^{2}_{N-2}\|\nabla\tilde{U}(t)\|^{2}_{N}+\|f(t)\|^{2}_{N-1}\right),\end{array}$
(3.13)
where $C$ is depending only on $\rho_{\infty},\mu,\nu$ and $\kappa$.
Proof. For each multi-index $\alpha$ with $1\leq|\alpha|\leq N$, applying
$\partial_{x}^{\alpha}$ to $(\ref{2.3})_{1}$ and $(\ref{2.3})_{2}$ and then
taking the $L^{2}$ inner product with $\partial_{x}^{\alpha}\sigma$ and
$\partial_{x}^{\alpha}v$ on the two resultant equations respectively, we have
from integrating by parts that
$\begin{array}[]{rl}&\displaystyle\frac{1}{2}\frac{d}{dt}\left(\|\partial_{x}^{\alpha}\sigma\|^{2}+\|\partial_{x}^{\alpha}v\|^{2}\right)+\mu^{\prime}\|\partial_{x}^{\alpha}\nabla
v\|^{2}+\nu^{\prime}\|\partial_{x}^{\alpha}\nabla\cdot v\|^{2}\\\\[5.69054pt]
=&\langle\partial_{x}^{\alpha}G_{1}(\tilde{U}),\partial_{x}^{\alpha}\sigma\rangle+\langle\partial_{x}^{\alpha}G_{2}(\tilde{U}),\partial_{x}^{\alpha}v\rangle+\kappa^{\prime}\langle\partial_{x}^{\alpha}\nabla\Delta\sigma,\partial_{x}^{\alpha}v\rangle+\lambda_{2}\langle\partial_{x}^{\alpha}f,\partial_{x}^{\alpha}v\rangle\\\\[5.69054pt]
=&I_{8}+I_{9}+I_{10}+I_{11}.\end{array}$ (3.14)
Now, we estimate $I_{8}$-$I_{11}$ term by term. For $I_{8}$, we deduce from
(2.4) and the Cauchy-Schwartz inequality that
$\begin{array}[]{rl}I_{8}&\leq\epsilon\|\partial_{x}^{\alpha}\sigma\|^{2}+C_{\epsilon}\|\partial_{x}^{\alpha}G_{1}(\tilde{U})\|^{2}\\\\[5.69054pt]
&\leq\epsilon\|\partial_{x}^{\alpha}\sigma\|^{2}+C_{\epsilon}C\left(\|\partial_{x}^{\alpha}(\nabla\tilde{\sigma}\cdot\tilde{v})\|^{2}+\|\partial_{x}^{\alpha}(\tilde{\sigma}\nabla\cdot\tilde{v})\|^{2}\right).\end{array}$
(3.15)
By Leibniz’s formula and Minkowski’s inequality, we get
$\begin{array}[]{rl}\|\partial_{x}^{\alpha}(\nabla\tilde{\sigma}\cdot\tilde{v})\|^{2}\leq&C(\|(\partial_{x}^{\alpha}\nabla\tilde{\sigma})\cdot\tilde{v}\|^{2}+\|\nabla\tilde{\sigma}\cdot\partial_{x}^{\alpha}\tilde{v}\|^{2})+C\displaystyle\sum_{0<|\beta|=|\alpha|-1}C^{\alpha}_{\beta}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\cdot\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}\\\\[5.69054pt]
&+C\displaystyle\sum_{0<|\beta|\leq|\alpha|-2,\,|\alpha-\beta|\leq\frac{N}{2}}C^{\alpha}_{\beta}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\cdot\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}\\\\[5.69054pt]
&+C\displaystyle\sum_{0<|\beta|\leq|\alpha|-2,\,|\alpha-\beta|>\frac{N}{2}}C^{\alpha}_{\beta}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\cdot\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}\\\\[5.69054pt]
=&J_{0}+J_{1}+J_{2}+J_{3}.\end{array}$ (3.16)
Here $C^{\alpha}_{\beta}$ denotes the binomial coefficients corresponding to
multi-indices. For $J_{0}$, lemma 3.1 gives
$\begin{array}[]{rl}J_{0}\leq&C\left(\|\tilde{v}\|^{2}_{L^{\infty}}\|\partial_{x}^{\alpha}\nabla\tilde{\sigma}\|^{2}+\|\nabla\tilde{\sigma}\|^{2}_{L^{\infty}}\|\partial_{x}^{\alpha}\tilde{v}\|^{2}\right)\\\\[5.69054pt]
\leq&C\left(\|\nabla\tilde{v}\|^{2}_{N-5}\|\nabla^{2}\tilde{\sigma}\|^{2}_{N-1}+\|\nabla^{2}\tilde{\sigma}\|^{2}_{N-5}\|\nabla\tilde{v}\|^{2}_{N-1}\right),\end{array}$
(3.17)
where, in the last inequality of (3.17), we have used the fact that $m-1\geq
1$ and $m+1\leq N-4$ due to $N\geq n+2$ and $n\geq 5$. Similarly, it holds
that
$J_{1}\leq
C\displaystyle\sum_{0<|\beta|=|\alpha|-1}\|\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}_{L^{\infty}}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\|^{2}\\\
\leq
C\|\nabla^{2}\tilde{v}\|^{2}_{N-5}\|\nabla^{2}\tilde{\sigma}\|_{N-2}^{2}.$
(3.18)
For the terms $J_{2}$ and $J_{3}$, notice that for any $\beta\leq\alpha$ with
$|\alpha-\beta|\leq\frac{N}{2}$,
$|\alpha-\beta|+m+1\leq\frac{N}{2}+\frac{n}{2}+1\leq\frac{N}{2}+\frac{N}{2}=N,$
and for any $\beta\leq\alpha$ with $|\alpha-\beta|>\frac{N}{2}$,
$|\beta|+m+2=|\alpha|-|\alpha-\beta|+m+2<N-\frac{N}{2}+\frac{n}{2}+2\leq N+1.$
which implies $|\beta|+m+2\leq N$ since $|\beta|$ and $m$ are positive
integers. Hence, we deduce from Lemma 3.1 that
$J_{2}\leq
C\displaystyle\sum_{0<|\beta|\leq|\alpha|-2,\,|\alpha-\beta|\leq\frac{N}{2}}\|\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}_{L^{\infty}}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\|^{2}\leq
C\|\nabla^{2}\tilde{v}\|^{2}_{N-2}\|\nabla^{2}\tilde{\sigma}\|_{N-3}^{2},$
(3.19)
and
$J_{3}\leq
C\displaystyle\sum_{0<|\beta|\leq|\alpha|-2,\,|\alpha-\beta|>\frac{N}{2}}\|\partial_{x}^{\beta}\nabla\tilde{\sigma}\|^{2}_{L^{\infty}}\|\partial_{x}^{\alpha-\beta}\tilde{v}\|^{2}\leq
C\|\nabla^{2}\tilde{v}\|^{2}_{N-3}\|\nabla^{2}\tilde{\sigma}\|_{N-2}^{2}.$
(3.20)
Putting (3.17)-(3.20) into (3.16), we arrive at
$\|\partial_{x}^{\alpha}(\nabla\tilde{\sigma}\cdot\tilde{v})\|^{2}\leq
C\left(\|\nabla\tilde{v}\|^{2}_{N-5}\|\nabla^{2}\tilde{\sigma}\|^{2}_{N-1}+\|\nabla^{2}\tilde{U}\|^{2}_{N-3}\|\nabla\tilde{U}\|^{2}_{N-1}\right).$
(3.21)
Similarly, it holds
$\|\partial_{x}^{\alpha}(\tilde{\sigma}\nabla\cdot\tilde{v})\|^{2}\leq
C\left(\|\nabla\tilde{\sigma}\|^{2}_{N-5}\|\nabla^{2}\tilde{v}\|^{2}_{N-1}+\|\nabla^{2}\tilde{U}\|^{2}_{N-3}\|\nabla\tilde{U}\|^{2}_{N-1}\right).$
(3.22)
Combining (3.15), (3.21) and (3.22) yields
$I_{8}\leq\epsilon\|\partial_{x}^{\alpha}\sigma\|^{2}+C_{\epsilon}C\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}.$
(3.23)
For the term $I_{9}$, let $\alpha_{0}\leq\alpha$ with $|\alpha_{0}|=1$, then
$I_{9}=-\langle\partial_{x}^{\alpha-\alpha_{0}}G_{2},\partial_{x}^{\alpha+\alpha_{0}}v\rangle\leq\epsilon\|\partial_{x}^{\alpha+\alpha_{0}}v\|^{2}+C_{\epsilon}\|\partial_{x}^{\alpha-\alpha_{0}}G_{2}\|^{2}.$
(3.24)
Similar to the estimate of (3.21), we have
$\|\partial_{x}^{\alpha-\alpha_{0}}G_{2}\|^{2}\leq
C\|\tilde{U}\|^{2}_{N-1}\|\nabla\tilde{U}\|^{2}_{N}.$ (3.25)
Thus, it follows from (3.24) and (3.25) that
$I_{9}\leq\epsilon\|\partial_{x}^{\alpha+\alpha_{0}}v\|^{2}+C\|\tilde{U}\|^{2}_{N-1}\|\nabla\tilde{U}\|^{2}_{N}.$
(3.26)
Notice that (3.21) and (3.22) imply
$\|\partial_{x}^{\alpha}G_{1}\|^{2}\leq
C\left(\|\partial_{x}^{\alpha}(\nabla\tilde{\sigma}\cdot\tilde{v})\|^{2}+\|\partial_{x}^{\alpha}(\tilde{\sigma}\nabla\cdot\tilde{v})\|^{2}\right)\leq
C\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}.$ (3.27)
Therefore, we derive from $(\ref{2.3})_{1}$, (3.27) and the Cauchy-Schwartz
inequality that
$\begin{array}[]{rl}I_{10}=&-\displaystyle\frac{\kappa^{\prime}}{\gamma}\langle\partial_{x}^{\alpha}\Delta\sigma,-\partial_{x}^{\alpha}\sigma_{t}+\partial_{x}^{\alpha}G_{1}(\tilde{U})\rangle\\\\[8.53581pt]
=&-\displaystyle\frac{\kappa^{\prime}}{\gamma}\langle\partial_{x}^{\alpha}\nabla\sigma,\partial_{x}^{\alpha}\nabla\sigma_{t}\rangle-\displaystyle\frac{\kappa^{\prime}}{\gamma}\langle\partial_{x}^{\alpha}\Delta\sigma,\partial_{x}^{\alpha}G_{1}(\tilde{U})\rangle\\\\[5.69054pt]
\leq&-\displaystyle\frac{\kappa^{\prime}}{2\gamma}\displaystyle\frac{d}{dt}\|\partial_{x}^{\alpha}\nabla\sigma\|^{2}+\epsilon\|\partial_{x}^{\alpha}\Delta\sigma\|^{2}+C_{\epsilon}C\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}.\end{array}$
(3.28)
Moreover, it holds that
$I_{11}=-\lambda_{2}\langle\partial_{x}^{\alpha+\alpha_{0}}v,\partial_{x}^{\alpha-\alpha_{0}}f\rangle\leq\epsilon\|\partial_{x}^{\alpha+\alpha_{0}}v\|^{2}+C_{\epsilon}C\|f\|^{2}_{N-1}.$
(3.29)
where $\alpha_{0}$ is defined in (3.24). Combining (3.14), (3.23), (3.26),
(3.28) and (3.29), if $\epsilon$ is small enough, we have
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\partial_{x}^{\alpha}\sigma\|^{2}_{1}+\|\partial_{x}^{\alpha}v\|^{2}\right)+\|\partial_{x}^{\alpha}\nabla
v\|^{2}+\|\partial_{x}^{\alpha}\nabla\cdot v\|^{2}\\\\[5.69054pt]
&\leq\epsilon C\|\partial_{x}^{\alpha}\sigma\|^{2}+\epsilon
C\|\partial_{x}^{\alpha}\Delta\sigma\|^{2}+C_{\epsilon}C\left(\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}+\|f\|_{N-1}^{2}\right),\end{array}$
(3.30)
where $C$ depends only on $\rho_{\infty},\mu,\nu$ and $\kappa$.
Now we turn to estimate $\|\partial_{x}^{\alpha}\Delta\sigma\|^{2}$ for
$1\leq|\alpha|\leq N$. As we did for the first order derivative estimate,
applying $\partial_{x}^{\alpha}$ to $(\ref{2.3})_{2}$ and then taking the
$L^{2}$ inner product with $\partial_{x}^{\alpha}\nabla\sigma$ on the
resultant equation, we get from integrating by parts that
$\begin{array}[]{rl}&\displaystyle\kappa^{\prime}\|\partial_{x}^{\alpha}\Delta\sigma\|^{2}+\gamma\|\partial_{x}^{\alpha}\nabla\sigma\|^{2}\\\\[5.69054pt]
&=-\langle\partial_{x}^{\alpha}v_{t},\partial_{x}^{\alpha}\nabla\sigma\rangle+\mu^{\prime}\langle\partial_{x}^{\alpha}\Delta
v,\partial_{x}^{\alpha}\nabla\sigma\rangle+\nu^{\prime}\langle\partial_{x}^{\alpha}\nabla(\nabla\cdot
v),\partial_{x}^{\alpha}\nabla\sigma\rangle\\\\[5.69054pt] &\hskip
11.38109pt+\langle\partial_{x}^{\alpha}G_{2}(\tilde{U}),\partial_{x}^{\alpha}\nabla\sigma\rangle+\lambda_{2}\langle\partial_{x}^{\alpha}f,\partial_{x}^{\alpha}\nabla\sigma\rangle\\\\[5.69054pt]
&=I_{12}+I_{13}+I_{14}+I_{15}+I_{16}.\end{array}$ (3.31)
The first term $I_{12}$ is controlled by
$\begin{array}[]{rl}I_{12}=&-\displaystyle\frac{d}{dt}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle+\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma_{t}\rangle\\\\[8.53581pt]
=&-\displaystyle\frac{d}{dt}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle-\displaystyle\langle\partial_{x}^{\alpha}\nabla\cdot
v,\partial_{x}^{\alpha}(-\gamma\nabla\cdot
v+G_{1}(\tilde{U}))\rangle\\\\[5.69054pt]
\leq&-\displaystyle\frac{d}{dt}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle+2\gamma\|\partial_{x}^{\alpha}\nabla\cdot
v\|^{2}+C\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}.\end{array}$
(3.32)
Here, in the last inequality of (3.32), we have used (3.27). By integrating by
parts, the Cauchy-Schwartz inequality and (3.25), the other terms
$I_{13}$-$I_{15}$ can be estimated as follows.
$I_{13}+I_{14}\leq\frac{\kappa^{\prime}}{4}\|\partial_{x}^{\alpha}\nabla^{2}\sigma\|^{2}+C\left(\|\partial_{x}^{\alpha}\nabla
v\|^{2}+\|\partial_{x}^{\alpha}\nabla\cdot v\|^{2}\right),$ (3.33)
$I_{15}\leq\frac{\kappa^{\prime}}{4}\|\partial_{x}^{\alpha+\alpha_{0}}\nabla\sigma\|^{2}+C\|\tilde{U}\|^{2}_{N-1}\|\nabla\tilde{U}\|^{2}_{N},$
(3.34)
$I_{16}\leq\frac{\kappa^{\prime}}{4}\|\partial_{x}^{\alpha+\alpha_{0}}\nabla\sigma\|^{2}+C\|f\|^{2}_{N-1}.$
(3.35)
where $\alpha_{0}$ is given in (3.24). Combining (3.31)-(3.35), we obtain
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle+\kappa^{\prime}\|\partial_{x}^{\alpha}\Delta\sigma\|^{2}+\gamma\|\partial_{x}^{\alpha}\nabla\sigma\|^{2}\\\\[5.69054pt]
&\leq C\left(\|\partial_{x}^{\alpha}\nabla
v\|^{2}+\|\partial_{x}^{\alpha}\nabla\cdot
v\|^{2}\right)+C\left(\|\tilde{U}\|^{2}_{N-1}\|\nabla\tilde{U}\|^{2}_{N}+\|f\|_{N-1}^{2}\right).\end{array}$
(3.36)
Multiplying (3.36) with a suitably small constant $d_{1}>0$ and then adding
the resultant equation to (3.30) gives
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\partial_{x}^{\alpha}\sigma\|^{2}_{1}+\|\partial_{x}^{\alpha}v\|^{2}+d_{1}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle\right)+\|\partial_{x}^{\alpha}\nabla\sigma\|^{2}_{1}+\|\partial_{x}^{\alpha}\nabla
v\|^{2}\\\\[5.69054pt] &\leq\epsilon
C\|\partial_{x}^{\alpha}\sigma\|^{2}+CC_{\epsilon}\left(\|\nabla\tilde{U}\|^{2}_{N-2}\|\nabla\tilde{U}\|^{2}_{N}+\|f\|_{N-1}^{2}\right),\end{array}$
(3.37)
provided that $d_{1}$ and $\epsilon$ are small enough, where $C$ depends only
on $\rho_{\infty},\mu,\nu$ and $\kappa$. Summing up $\alpha$ with
$1\leq|\alpha|\leq N$ in (3.37), then (3.13) follows immediately by the
smallness of $\epsilon$. This completes the proof of Lemma 3.4.
As a consequence of Lemmas 3.3-3.4, we have the following Corollary.
###### Corollary 3.1.
Let $n\geq 5$, $N\geq n+2$, then there exists two suitably small constants
$d_{0}>0$ and $d_{1}>0$ such that
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\sigma(t)\|^{2}_{N+1}+\|v(t)\|^{2}_{N}+d_{0}\langle
v,\nabla\sigma\rangle(t)+d_{1}\displaystyle\sum_{|\alpha|=1}^{N}\langle\partial_{x}^{\alpha}v,\partial_{x}^{\alpha}\nabla\sigma\rangle(t)\right)+\|\nabla\sigma(t)\|^{2}_{N+1}+\|\nabla
v(t)\|^{2}_{N}\\\\[11.38109pt] &\leq
C\left(\|\tilde{U}(t)\|^{2}_{N-1}\|\nabla\tilde{U}(t)\|^{2}_{N}+\|f(t)\|^{2}_{H^{N-1}\cap{L^{1}}}\right),\end{array}$
(3.38)
where $C$ depends only on $\rho_{\infty},\mu,\nu$ and $\kappa$.
Proof. Notice that, from the fact that $m-1\geq 1$ and $m+1\leq N-4$, we have
$\|\nabla^{3}\sigma\|^{2}_{m-2}+\|\nabla^{2}v\|^{2}_{m-1}\leq
C\|\nabla^{2}\tilde{U}\|_{N-6}^{2},$
and
$\|\tilde{U}\|_{m+1}\leq\|\tilde{U}\|_{N-4}^{2}.$
Adding (3.37) to (3.1), we obtain (3.38) immediately by the smallness of
$\epsilon$. This completes the proof of Corollary 3.1.
## 4 Existence of time periodic solution
In this section, we will combine the linearized decay estimate Lemma 2.1 with
the energy estimates Corollary 3.1 to show the existence of time periodic
solution to (1.1). Now, we are ready to prove Theorem 1.1 as follows.
Proof of Theorem 1.1. The proof is divided into two steps.
Step 1. Suppose that there exists a time periodic solution
$U^{per}(t):=(\sigma^{per}(x,t),v^{per}(x,t)),t\in\mathbb{R}$ of the system
(2.3) with period $T$, and $U^{per}(t)\in X_{M_{0}}(0,T)$ for some constant
$M_{0}>0$. Then it solves (2.5) with initial date $U_{s}=U^{per}(s)$ for any
given time $s\in\mathbb{R}$. Choosing $s=-kT$ for $k\in\mathbb{N}$. Clearly,
$U^{per}(-kT)=U^{per}(0)$, thus (2.5) can be written in the mild form as
$U^{per}(t)=\displaystyle\mathbb{S}(t,-kT)U^{per}(0)+\int_{-kT}^{t}\mathbb{S}(t,\tau)(G(U^{per})(\tau)+F(\tau))d\tau.$
(4.1)
Denote $\mathbb{S}(t,-kT)U^{per}(0):=(\sigma^{per}_{1}(t),v^{per}_{1}(t))$.
Applying Lemma 2.1 to $\mathbb{S}(t,-kT)U^{per}(0)$, we have
$\begin{array}[]{rl}\|\sigma^{per}_{1}(t)\|_{N}\leq&\displaystyle(1+t+kT)^{-\frac{n}{4}}\left(\|(\sigma_{0}^{per},v_{0}^{per})\|_{L^{1}}+\|\sigma_{0}^{per}\|_{N}^{2}+\|v_{0}^{per}\|^{2}_{N-1}\right)\\\\[5.69054pt]
&\longrightarrow 0\quad as\quad k\rightarrow\infty.\end{array}$ (4.2)
and
$\begin{array}[]{rl}\|v_{1}^{per}(t)\|_{N-1}\leq&\displaystyle(1+t+kT)^{-\frac{n}{4}}\left(\|(\sigma_{0}^{per},v_{0}^{per})\|_{L^{1}}+\|\sigma_{0}^{per}\|_{N}^{2}+\|v_{0}^{per}\|^{2}_{N-1}\right)\\\\[5.69054pt]
&\longrightarrow 0\quad as\quad k\rightarrow\infty.\end{array}$ (4.3)
Since $L^{2}\cap L^{1}$ is dense in $L^{2}$, (4.2) and (4.3) still hold for
$U^{per}(0)=(\sigma^{per}_{0},v^{per}_{0})\in H^{N}(\mathbb{R}^{n})\times
H^{N-1}(\mathbb{R}^{n})$. On the other hand, denote
$\mathbb{S}(t,\tau)(G(U^{per})(\tau)+F(\tau)):=(S_{1}(t,\tau),S_{2}(t,\tau)).$
By using Lemma 2.1 again, we get
$\|S_{1}(t,\tau)\|_{N}\leq\displaystyle(1+t-\tau)^{-\frac{n}{4}}K_{0},\quad\|S_{2}(t,\tau)\|_{N-1}\leq\displaystyle(1+t-\tau)^{-\frac{n}{4}}K_{0},$
(4.4)
where
$\begin{array}[]{rl}K_{0}=&\|(G_{1}(U^{per}),G_{2}(U^{per})+\lambda_{2}f)(\tau)\|_{L^{1}}\\\\[5.69054pt]
&+\|G_{1}(U^{per})(\tau)\|_{N}+\|(G_{2}(U^{per})+\lambda_{2}f)(\tau)\|_{N-1}.\end{array}$
Then (4.4) guarantees the convergence of the integral in (4.1) since
$\frac{n}{4}>1$ when $n\geq 5$. Thus, letting $k\rightarrow\infty$ in (4.1),
we obtain
$U^{per}(t)=\displaystyle\int_{-\infty}^{t}\mathbb{S}(t,\tau)(G(U^{per})+F)(\tau)d\tau.$
(4.5)
For any $U=(\sigma,v)\in X_{M_{0}}(0,T)$, define
$\Psi[U](t)=\displaystyle\int_{-\infty}^{t}\mathbb{S}(t,\tau)(G(U)+F)(\tau)d\tau.$
Then (4.5) shows that $U^{per}$ is a fixed point of $\Psi[U]$.
Conversely, suppose that $\Psi$ has a unique fixed point, denoted by
$U_{1}(t)=(\sigma_{1},v_{1})(t)$. We show that $U_{1}(t)$ is time periodic
with period $T$. To this end, setting $U_{2}(t)=U_{1}(t+T)$. Since the period
of $f$ is $T$, the period of $F$ is $T$ too. Thus, we have
$\begin{array}[]{rl}U_{2}(t)&=U_{1}(t+T)=\Psi[U_{1}](t+T)\\\\[8.53581pt]
&=\displaystyle\int^{t+T}_{-\infty}\mathbb{S}(t+T,\tau)(G(U_{1})(\tau)+F(\tau))d\tau\\\\[8.53581pt]
&=\displaystyle\int^{t}_{-\infty}\mathbb{S}(t+T,s+T)\left(G(U_{1})(s+T)+F(s+T)\right)ds\\\\[8.53581pt]
&=\displaystyle\int^{t}_{-\infty}\mathbb{S}(t,s)\left(G(U_{2})(s)+F(s)\right)ds\\\\[8.53581pt]
&=\Psi[U_{2}](t)\end{array}$ (4.6)
where we have used
$\mathbb{S}(t+T,s+T)=\mathbb{S}(t,s).$
Then by uniqueness, $U_{2}=U_{1}$, which proves the periodicity of $U_{1}(t)$.
Since $U_{1}(t)$ is differentiable with respect to $t$, it is the desired
periodic solution of the system (2.1).
Step 2. Now, it remains to show that if (H1)-(H3) hold, and
$\sup_{0\leq t\leq T}\|f(t)\|_{H^{N-1}\cap L^{1}}$
is sufficiently small, then $\Psi$ has a unique fixed point in the space
$X_{M_{0}}(0,T)$ for some appropriate constant $M_{0}>0$. The proof is divided
into two parts.
(i) Assume that $\tilde{U}=(\tilde{\sigma},\tilde{v})$ in the system (2.8) is
time periodic with period $T$. Denote $U=\Psi[\tilde{U}]$ with $U=(\sigma,v)$.
Then by the same argument as (4.6), one can show that $U$ is also time
periodic with period $T$. Notice that $U$ satisfies the system (2.8). Thus,
for $n\geq 5$ and $N\geq n+2$, Corollary 3.1 holds. Integrating (3.38) in $t$
over $[0,T]$ to get
$\begin{array}[]{rl}&\displaystyle\int^{T}_{0}\left(\|\nabla\sigma(t)\|^{2}_{N+1}+\|\nabla
v(t)\|^{2}_{N}\right)dt\\\\[8.53581pt] &\leq
C\displaystyle\int^{T}_{0}\left(\|\tilde{U}(t)\|^{2}_{N-1}\|\nabla\tilde{U}(t)\|^{2}_{N}+\|f(t)\|_{N-1}^{2}+\|f(t)\|^{2}_{L^{1}}\right)dt\\\\[8.53581pt]
&\leq C\displaystyle\sup_{0\leq t\leq
T}\|\tilde{U}(t)\|^{2}_{N-1}\displaystyle\int^{T}_{0}\|\nabla\tilde{U}(t)\|^{2}_{N}dt+\displaystyle\int^{T}_{0}\|f(t)\|^{2}_{H^{N-1}\cap{L^{1}}}dt\\\\[8.53581pt]
&\leq C|||\tilde{U}(t)|||^{4}+CT\displaystyle\sup_{0\leq t\leq
T}\|f(t)\|_{H^{N-1}\cap{L^{1}}}^{2}.\end{array}$ (4.7)
On the other hand, by Lemma 2.1, we have
$\|\sigma(t)\|_{N}\leq\displaystyle\int^{t}_{-\infty}(1+t-\tau)^{-\frac{n}{4}}K_{1}\,d\tau,\quad\|v(t)\|_{N-1}\leq\displaystyle\int^{t}_{-\infty}(1+t-\tau)^{-\frac{n}{4}}K_{1}\,d\tau,$
(4.8)
where
$\begin{array}[]{rl}K_{1}=&\|(G_{1}(\tilde{U}),G_{2}(\tilde{U})+\lambda_{2}f)(\tau)\|_{L^{1}}\\\\[5.69054pt]
&+\|G_{1}(\tilde{U})(\tau)\|_{N}+\|(G_{2}(\tilde{U})+\lambda_{2}f)(\tau)\|_{N-1}.\end{array}$
(4.9)
From (2.4), (3.25) and (3.27), we easily deduce that
$\begin{array}[]{rl}\|(G_{1}(\tilde{U})(\tau)\|_{L^{1}}&\leq
C\|\nabla\tilde{U}(\tau)\|\|\tilde{U}(\tau)\|,\\\\[5.69054pt]
\|(G_{1}(\tilde{U})(\tau)\|_{N}&\leq
C\|\nabla\tilde{U}(\tau)\|_{N-2}\|\nabla\tilde{U}(\tau)\|_{N},\\\\[5.69054pt]
\|G_{2}(\tilde{U})+\lambda_{2}f)(\tau)\|_{L^{1}}&\leq
C\|\nabla\tilde{U}(\tau)\|_{1}\|\tilde{U}(\tau)\|+C\|f(\tau)\|_{L^{1}},\\\\[5.69054pt]
\|G_{2}(\tilde{U})+\lambda_{2}f)(\tau)\|_{N-1}&\leq
C\|\tilde{U}(\tau)\|_{N-1}\|\nabla\tilde{U}(\tau)\|_{N}+C\|f(\tau)\|_{N-1}.\end{array}$
(4.10)
Combining (4.8)-(4.10), we obtain
$\begin{array}[]{rl}\|\sigma(t)\|_{N}\leq&C\displaystyle\int^{t}_{-\infty}(1+t-\tau)^{-\frac{n}{4}}\left(\|\tilde{U}(\tau)\|_{N-1}\|\nabla\tilde{U}(\tau)\|_{N}+\|f(\tau)\|_{H^{N-1}\cap{L^{1}}}\right)d\tau\\\\[5.69054pt]
\leq&C\displaystyle\sum_{j=0}^{\infty}A_{j}+C\int^{t}_{-\infty}(1+t-\tau)^{-\frac{n}{4}}\|f(\tau)\|_{H^{N-1}\cap{L^{1}}}d\tau\\\\[5.69054pt]
\leq&C\displaystyle\sum_{j=0}^{\infty}A_{j}+C\sup_{0\leq t\leq
T}\|f(t)\|_{H^{N-1}\cap{L^{1}}},\end{array}$ (4.11)
where
$\begin{array}[]{rl}A_{j}=&C\displaystyle\int^{t-jT}_{t-(j+1)T}(1+t-\tau)^{-\frac{n}{4}}\|\tilde{U}(\tau)\|_{N-1}\|\nabla\tilde{U}(\tau)\|_{N}d\tau\\\\[5.69054pt]
\leq&C\displaystyle\left(\int^{t-jT}_{t-(j+1)T}(1+t-\tau)^{-\frac{n}{2}}d\tau\right)^{\frac{1}{2}}\left(\int^{t-jT}_{t-(j+1)T}\|\tilde{U}(\tau)\|_{N-1}^{2}\|\nabla\tilde{U}(\tau)\|_{N}^{2}d\tau\right)^{\frac{1}{2}}\\\\[5.69054pt]
\leq&C\displaystyle(1+jT)^{-\frac{n}{4}}\sup_{0\leq\tau\leq
T}\|\tilde{U}(\tau)\|_{N-1}\left(\int^{T}_{0}\|\nabla\tilde{U}(\tau)\|_{N}^{2}d\tau\right)^{\frac{1}{2}}\\\\[8.53581pt]
\leq&C\displaystyle(1+jT)^{-\frac{n}{4}}|||\tilde{U}|||^{2}\end{array}$ (4.12)
Since $\frac{n}{4}>1$ when $n\geq 5$, substituting (4.12) into (4.11) gives
$\|\sigma(t)\|_{N}\leq C|||\tilde{U}|||^{2}+C\sup_{0\leq t\leq
T}\|f(t)\|_{H^{N-1}\cap{L^{1}}}.$ (4.13)
Similarly, it holds that
$\|v(t)\|_{N-1}\leq C|||\tilde{U}|||^{2}+C\sup_{0\leq t\leq
T}\|f(t)\|_{H^{N-1}\cap{L^{1}}}.$ (4.14)
Thus, we deduce from (4.7), (4.13) and (4.14) that
$|||\Psi[\tilde{U}]|||\leq C_{1}|||\tilde{U}|||^{2}+C_{2}\sup_{0\leq t\leq
T}\|f(t)\|_{H^{N-1}\cap{L^{1}}},$ (4.15)
where $C_{1}$ and $C_{2}$ are some positive constants depending only on
$\rho_{\infty},\mu,\nu,\kappa$ and $T$.
(ii) Let $\tilde{U}_{1}=(\tilde{\sigma}_{1},\tilde{v}_{1})$ and
$\tilde{U}_{2}=(\tilde{\sigma}_{2},\tilde{v}_{2})$ be time periodic functions
with period $T$ in the space $X_{M_{0}}(0,T)$, where $M_{0}>0$ will be
determined below. Then similar to (i), we can get
$|||\Psi[\tilde{U}_{1}]-\Psi[\tilde{U}_{2}]|||\leq
C_{3}\left(|||\tilde{U}_{1}|||+|||\tilde{U}_{2}|||\right)|||\tilde{U}_{1}-\tilde{U}_{2}|||,$
(4.16)
where $C_{3}$ is a positive constant depending only on
$\rho_{\infty},\mu,\nu,\kappa$ and $T$. Choose $M_{0}>0$ and a sufficiently
small constant $\delta>0$ such that
$C_{1}M_{0}^{2}+C_{2}\delta\leq M_{0},\quad\mbox{and}\,\,2C_{3}M_{0}<1$ (4.17)
That is,
$\frac{1-\sqrt{1-4C_{1}C_{2}\delta}}{2C_{1}}\leq
M_{0}\leq\min\left\\{\frac{1+\sqrt{1-4C_{1}C_{2}\delta}}{2C_{1}},\frac{1}{2C_{3}},1\right\\}$
(4.18)
Notice that
$\frac{1-\sqrt{1-4C_{1}C_{2}\delta}}{2C_{1}}\longrightarrow 0\quad
as\quad\delta\longrightarrow 0.$
Then there exists a constant $\delta_{0}>0$ depending only on
$\rho_{\infty},\mu,\nu,\kappa$ and $T$ such that if $0<\delta\leq\delta_{0}$,
the set of $M_{0}$ that satisfying (4.18) is not empty. For
$0<\delta\leq\delta_{0}$, when $M_{0}$ satisfies (4.18), $\Psi$ is a
contraction map in the complete space $X_{M_{0}}(0,T)$, thus $\Psi$ has a
unique fixed point in $X_{M_{0}}(0,T)$. This completes the proof of Theorem
1.1.
## 5 Stability of time periodic solution
This section is devoted to proving Theorem 1.2 on the stability of the
obtained time periodic solution. We shall establish the global existence of
smooth solutions to the Cauchy problem (1.1), (1.5).
First, let $(\rho^{per},u^{per})$ be the time periodic solution obtained in
Theorem 1.1 and $(\rho,u)$ be the solution of the Cauchy problem (1.1), (1.5).
Denote
$(\sigma^{per},v^{per})=(\rho^{per}-\rho_{\infty},\lambda_{2}u^{per}),$
$(\sigma,v)=(\rho-\rho_{\infty},\lambda_{2}u).$
Let $(\bar{\sigma},\bar{v})=(\sigma-\sigma^{per},v-v^{per})$, then
$(\bar{\sigma},\bar{v})$ satisfies
$\displaystyle\left\\{\begin{array}[]{ll}\bar{\sigma}_{t}+\gamma\nabla\cdot\bar{v}=G_{1}(\bar{\sigma}+\sigma^{per},\bar{v}+v^{per})-G_{1}(\sigma^{per},v^{per}),\\\\[5.69054pt]
\bar{v}_{t}-\mu^{\prime}\Delta\bar{v}-\nu^{\prime}\nabla(\nabla\cdot\bar{v})+\gamma\nabla\bar{\sigma}-\kappa^{\prime}\nabla\Delta\bar{\sigma}=G_{2}(\bar{\sigma}+\sigma^{per},\bar{v}+v^{per})-G_{2}(\sigma^{per},v^{per}),\end{array}\right.$
(5.3)
with the initial date
$\displaystyle\bar{\sigma}|_{t=0}=\bar{\sigma}_{0}(x)=\rho_{0}(x)-\rho^{per}(0),\quad\bar{v}|_{t=0}=\bar{v}_{0}(x)=\lambda_{2}(u_{0}(x)-u^{per}(0)).$
(5.4)
Define the solution space by $\bar{X}(0,\infty)$, where for $0\leq t_{1}\leq
t_{2}\leq\infty$,
$\bar{X}(t_{1},t_{2})=\left\\{(\bar{\sigma},\bar{v})(t,x)\left|\begin{array}[]{c}\bar{\sigma}(t,x)\in
C(t_{1},t_{2};H^{N-1}(\mathbb{R}^{n}))\cap
C^{1}(t_{1},t_{2};H^{N-3}(\mathbb{R}^{n})),\\\\[5.69054pt] \bar{v}(t,x)\in
C(t_{1},t_{2};H^{N-2}(\mathbb{R}^{n}))\cap
C^{1}(t_{1},t_{2};H^{N-4}(\mathbb{R}^{n})),\\\\[5.69054pt]
\nabla\bar{\sigma}(t,x)\in
L^{2}(t_{1},t_{2};H^{N-1}(\mathbb{R}^{n})),\nabla\bar{v}(t,x)\in
L^{2}(t_{1},t_{2};H^{N-2}(\mathbb{R}^{n})),\end{array}\right.\right\\}$ (5.5)
with the norm
$\|(\bar{\sigma},\bar{v})(t)\nparallel^{2}:=\sup_{t_{1}\leq t\leq
t_{2}}\left\\{\|\bar{\sigma}(t)\|_{N-1}^{2}+\|\bar{v}(t)\|_{N-2}^{2}\right\\}+\int_{t_{1}}^{t_{2}}\left(\|\nabla\bar{\sigma}(t)\|_{N-1}^{2}+\|\nabla\bar{v}(t)\|_{N-2}^{2}\right)dt.$
(5.6)
Notice that $(\sigma^{per},v^{per})\in\bar{X}(0,T)$.
By using the dual argument and iteration technique as [12], one can prove the
following local existence of the Cauchy problem (5.3), (5.4). We omit the
proof here for brevity.
###### Lemma 5.1.
(Local existence) Under the assumptions of Theorem 1.1, suppose that
$(\bar{\sigma}_{0},\bar{v}_{0})\in H^{N-1}(\mathbb{R}^{n})\times
H^{N-2}(\mathbb{R}^{n})$ and $\inf\rho_{0}(x)>0$. Then there exists a positive
constant $T_{0}$ depending only on
$\|(\bar{\sigma}_{0},\bar{v}_{0})\nparallel$ such that the Cauchy problem
(5.3), (5.4) admits a unique classical solution
$(\bar{\sigma},\bar{v})\in\bar{X}(0,T_{0})$ which satisfies
$\|(\bar{\sigma},\bar{v})(t)\nparallel\leq
C_{4}\|(\bar{\sigma}_{0},\bar{v}_{0})\nparallel,$
where $C_{4}$ is a positive constant independent of
$\|(\bar{\sigma}_{0},\bar{v}_{0})\nparallel$.
As usual, the global existence will be obtained by a combination of the local
existence result Lemma 5.1 and the a priori estimate below.
###### Lemma 5.2.
(A priori estimate) Under the assumptions of Lemma 5.1, suppose that the
Cauchy problem (5.3), (5.4) has a unique classical solution
$(\bar{\sigma},\bar{v})\in\bar{X}(0,T_{1})$ for some positive constant
$T_{1}$. Then there exists two small constants $\delta>0$ and $C_{5}>0$ which
are independent of $T_{1}$ such that if
$\sup_{0\leq t\leq T_{1}}\|(\bar{\sigma},\bar{v})(t)\nparallel\leq\delta,$
(5.7)
it holds that
$\|\bar{\sigma}(t)\|^{2}_{N-1}+\|\bar{v}(t)\|^{2}_{N-2}+\int_{0}^{t}\left(\|\nabla\bar{\sigma}(\tau)\|^{2}_{N-1}+\|\nabla\bar{v}(\tau)\|^{2}_{N-2}\right)d\tau\leq
C_{5}\left(\|\bar{\sigma}_{0}\|^{2}_{N-1}+\|\bar{v}_{0}\|^{2}_{N-2}\right)$
(5.8)
for all $t\in[0,T_{1}]$.
Proof. Noticing that some smallness conditions can be imposed on
$(\sigma^{per},v^{per})$, without loss of generality, we may assume
$|||(\sigma^{per},v^{per})|||\leq\epsilon$ with $\epsilon>0$ being
sufficiently small. Then by the similar argument as in the proof of Lemmas
3.3-3.4, we can obtain
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\bar{U}\|^{2}+\|\nabla\bar{\sigma}\|^{2}+d_{2}\langle\bar{v},\nabla\bar{\sigma}\rangle\right)+\|\nabla\bar{v}\|^{2}+\|\nabla\bar{\sigma}\|^{2}_{1}\\\\[5.69054pt]
&\leq\epsilon
C\left(\|\nabla^{3}\sigma\|^{2}_{N-7}+\|\nabla^{2}\bar{v}\|^{2}_{N-6}\right),\end{array}$
(5.9)
and
$\begin{array}[]{rl}&\displaystyle\frac{d}{dt}\left(\|\nabla\bar{\sigma}\|^{2}_{N-2}+\|\nabla\bar{v}\|^{2}_{N-3}+d_{3}\sum_{|\alpha|=1}^{N-2}\langle\partial_{x}^{\alpha}\bar{v},\partial_{x}^{\alpha}\nabla\bar{\sigma}\rangle\right)+\|\nabla^{2}\bar{\sigma}\|^{2}_{N-2}+\|\nabla^{2}\bar{v}\|^{2}_{N-3}\\\\[11.38109pt]
&\leq\epsilon
C\left(\|\nabla\bar{\sigma}\|^{2}+\|\nabla\bar{v}\|^{2}\right),\end{array}$
(5.10)
where $d_{2}>0$ and $d_{3}>0$ are some suitably small constants, and $C$ is a
constant depending only on $\rho_{\infty},\mu,\nu$ and $\kappa$. Adding (5.10)
to (5.9), it holds
$\begin{array}[]{rl}\displaystyle\frac{d}{dt}&\displaystyle\left(\|\bar{\sigma}\|^{2}_{N-1}+\|\bar{v}\|^{2}_{N-2}+d_{2}\langle\bar{v},\nabla\bar{\sigma}\rangle+d_{3}\sum_{|\alpha|=1}^{N-2}\langle\partial_{x}^{\alpha}\bar{v},\partial_{x}^{\alpha}\nabla\bar{\sigma}\rangle\right)\\\\[11.38109pt]
&+\|\nabla\bar{\sigma}\|^{2}_{N-1}+\|\nabla\bar{v}\|^{2}_{N-2}\leq
0,\end{array}$ (5.11)
provided that $\epsilon$ is sufficiently small. Integrating (5.11) in $t$ over
$(0,t)$, one can immediately get (5.8) since
$\|\bar{\sigma}\|^{2}_{N-1}+\|\bar{v}\|^{2}_{N-2}+d_{2}\langle\bar{v},\nabla\bar{\sigma}\rangle+d_{3}\sum_{|\alpha|=1}^{N-2}\langle\partial_{x}^{\alpha}\bar{v},\partial_{x}^{\alpha}\nabla\bar{\sigma}\rangle\thicksim\|\bar{\sigma}\|^{2}_{N-1}+\|\nabla\bar{v}\|^{2}_{N-2}.$
by the smallness of $d_{2}$ and $d_{3}$. This completes the proof of Lemma
5.2.
Proof of Theorem 1.2. By Lemmas 5.1-5.2 and the continuity argument, the
Cauchy problem (5.3), (5.4) admits a unique solution $(\bar{\sigma},\bar{v})$
globally in time, which satisfies (1.8) and (1.9). Then all the statements in
Theorem 1.2 follow immediately. This completes the proof of Theorem 1.2.
## References
* [1] J. E. Dunn, J. Serrin, On the thermomechanics of interstital working, Arch. Rational Mech. Anal., 88 (1985), 95-133.
* [2] D. M. Anderson, G. B. McFadden, G. B. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30 (1998), 139-165.
* [3] J. W. Cahn, J. E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys., 28 (1998), 258-267.
* [4] M. E. Gurtin, D. Polignone, J. Vinals, Two-phase binary fluids and immiscible fluids describled by an order parameter, Math. Models Methods Appl. Sci., 6 (6) (1996), 815-831.
* [5] D. Bresch, B. Desjardins, C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.
* [6] B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.
* [7] R. Danchin, B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincare Anal. Nonlinear, 18 (2001), 97-133.
* [8] H. F. Ma, S. Ukai, T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293.
* [9] S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dynam. Systems, 14 (2006), 579-596.
* [10] S. Ukai, T. Yang, The Boltzmann equation in the sapce $L^{2}\cap L^{\infty}_{\beta}$: global and time periodic solution, Analysis and Applications, 4 (3) (2006), 263-310.
* [11] R. J. Duan, S. Ukai, T. Yang, H. J. Zhao, Optimal decay estimates on the linearized Boltzmann equations with time dependent force and their applications, Comm. Math. Phys., 277 (1) (2008), 189-236.
* [12] H. Hattori, D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98.
* [13] H. Hattori, D. Li, Golobal solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97.
* [14] M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. Henri Poincare Anal. Nonlinear, 25 (4) (2008), 679-696.
* [15] Y. J. Wang, Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271.
* [16] Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232.
|
arxiv-papers
| 2012-03-29T14:18:27 |
2024-09-04T02:49:29.187730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhengzheng Chen, Qinghua Xiao, Huijiang Zhao",
"submitter": "Chen Zhengzheng",
"url": "https://arxiv.org/abs/1203.6529"
}
|
1203.6632
|
# Kelvin Probe Studies of Cesium Telluride Photocathode for AWA Photoinjector
Eric Wisniewski High Energy Physics Division, Argonne National Laboratory,
9700 S. Cass, Lemont, IL 60439 Physics Department, Illinois Institute of
Technology, 3300 South Federal Street, Chicago, IL 60616 Daniel Velazquez
High Energy Physics Division, Argonne National Laboratory, 9700 S. Cass,
Lemont, IL 60439 Physics Department, Illinois Institute of Technology, 3300
South Federal Street, Chicago, IL 60616 Zikri Yusof zyusof@anl.gov High
Energy Physics Division, Argonne National Laboratory, 9700 S. Cass, Lemont, IL
60439 Linda Spentzouris Physics Department, Illinois Institute of
Technology, 3300 South Federal Street, Chicago, IL 60616 Jeff Terry Physics
Department, Illinois Institute of Technology, 3300 South Federal Street,
Chicago, IL 60616 Katherine Harkay Accelerator Science Division, Argonne
National Laboratory, 9700 S. Cass, Lemont, IL 60439
###### Abstract
Cesium telluride is an important photocathode as an electron source for
particle accelerators. It has a relatively high quantum efficiency $(>1\%)$,
is sufficiently robust in a photoinjector, and has a long lifetime. This
photocathode is grown in-house for a new Argonne Wakefield Accelerator (AWA)
beamline to produce high charge per bunch ($\approx$50 nC) in a long bunch
train. Here, we present a study of the work function of cesium telluride
photocathode using the Kelvin Probe technique. The study includes an
investigation of the correlation between the quantum efficiency and the work
function, the effect of photocathode aging, the effect of UV exposure on the
work function, and the evolution of the work function during and after
photocathode rejuvenation via heating.
###### pacs:
41.75.Fr, 41.75.Lx, 73.20.At
Cesium telluride ($\mathrm{Cs}_{2}\mathrm{Te}$) photocathodes are a proven
electron source for particle accelerators. They have a high quantum efficiency
(10% at 4.9 eV photon energy), a long lifetime (months) and are robust in a
high gradient environment Kong _et al._ (1995a). The new RF photocathode
drive gun being commissioned at the Argonne Wakefield Accelerator (AWA) is a
high peak-current electron beam source for the new 75 MeV linear electron
accelerator, to be used to excite wakefields in dielectric-loaded accelerating
(DLA) structures and other novel high-gradient structures Conde _et al._
(2010). A unique requirement of the AWA experimental program is the ability to
produce long trains of high-charge bunches, hence the need for a high quantum
efficiency (QE) photocathode such as $\mathrm{Cs}_{2}\mathrm{Te}$. The AWA is
producing $\mathrm{Cs}_{2}\mathrm{Te}$ photocathodes for use in the new high-
charge, 1.3 GHz photoinjector Conde _et al._ (2011). In particular, an
electron bunch train of 30 bunches with up to 50 nC per bunch is expected to
be produced. The substantial demands on the photocathode necessitate a
thorough understanding of the photocathode and its parameters. The QE at a
particular photon energy and the work function ($\phi$) are two important
parameters of electron emission. Here, we present the results of Kelvin probe
measurements of the work function on $\mathrm{Cs}_{2}\mathrm{Te}$
photocathodes. We examined (i) the correlation between the QE and the work
function; (ii) how QE and the work function evolved with photocathode aging;
(iii) effects of rejuvenation of the photocathode via heating, and (iv) the
effects on the work function upon exposure to UV light.
The Kelvin probe method is a non-contact, non-destructive technique that is
used to measure the potential difference between a sample and the Kelvin probe
tip (reference) when the two are in electrical contact. The tip and the sample
are set in a parallel plate capacitor configuration and the circuit is
completed through ground connection, thus aligning the Fermi levels of tip and
sample. The electrical contact between the tip and the sample causes electron
migration from higher- to lower-Fermi-level material, creating an electric
field between the tip and sample. The potential associated with this electric
field is called the contact potential difference (CPD), which multiplied by
the electron charge results in the difference of the work functions of the
sample and tip. Hence, knowing the work function of the reference tip and
measuring the CPD allows the sample work function to be calculated. The
validity of the work function measurement therefore relies on the calibration
of the tip using a known reference. The theory and details of the method have
been described in detail elsewhere Surplice and D’Arcy (1970).
Fig. 1 shows the band diagram for a p-type semiconductor. The work function is
defined as the energy difference between the vacuum potential level and the
Fermi level which is located in the energy gap between the valence and
conduction bands. On the other hand, the photoemission threshold is defined as
the difference between the vacuum level and the valence band maximum.
Therefore the work function in a semiconductor is not the same as the
photoemission threshold, unlike the case of a metal. In this experiment, what
is measured is the actual work function and not the photoemission threshold.
The photocathodes studied were fabricated in the AWA photocathode laboratory
using a standard recipe and procedure Kong _et al._ (1995b),Michelato _et
al._ (1997). AWA photocathodes are deposited on a molybdenum plug designed to
fit into the back wall of the gun. In preparation for deposition, the plug is
polished and cleaned, placed under vacuum, then heated to $120^{\circ}$C. A 22
nm layer of tellurium is deposited via thermal evaporation. When tellurium
deposition is complete, cesium deposition commences and the photocurrent is
monitored. Deposition continues for several minutes after maximum photocurrent
is achieved. The result of this process is a $\mathrm{Cs}_{2}\mathrm{Te}$ thin
film photocathode on a molybdenum substrate with an effective photocathode
diameter of 31 mm and a typical initial QE of 15%. QE is measured at 4.9 eV
photon energy to closely match the photoinjector laser. All QE values reported
in this paper were measured using 4.9 eV photon energy.
Figure 1: Band diagram for p-type semiconductor. The work function is measured
from the vacuum potential level ($\mathrm{E}_{\mathrm{vac}}$) to the Fermi
level ($\mathrm{E}_{\mathrm{F}}$), while the photoemission threshold $(E_{t})$
is measured from $\mathrm{E}_{\mathrm{vac}}$ to the valence band maximum. In
this experiment, what is measured is the actual work function and not the
photoemission threshold.
The experimental setup is pictured in Fig. 2. It included a large vacuum
chamber where the $\mathrm{Cs}_{2}\mathrm{Te}$ cathodes were fabricated. The
Kelvin probe was housed in the smaller vacuum chamber connected to the back. A
long-stroke actuator holding the cathode plug provided the means to easily
move the plug back and forth from the deposition chamber (for fabrication and
quantum efficiency measurements) to the Kelvin probe chamber (for work
function measurements). All QE and Kelvin probe measurements were made in
situ. $\mathrm{Cs}_{2}\mathrm{Te}$ were fabricated and maintained under ultra-
high vacuum (UHV) conditions with base pressure of $1.5\times 10^{-10}$ Torr.
Figure 2: Top view, schematic (not to scale) of experimental setup, showing
the Kelvin Probe chamber attached to the back of the deposition chamber. The
actuator is used to move the photocathode from one chamber to the other a
distance of 1̃.5 m. Inset: Drawing of the Kelvin probe tip and sample
(photocathode) illustrating the relative orientation as seen from above -
(zoomed view).
The Kelvin Probe system is a McAllister Technical Services KP 6500 which
includes control software and electronics, and data collection via a PCI
National Instruments data acquisition card. The Kelvin probe was positioned in
a port in the smaller chamber oriented at $45^{\circ}$ with respect to the
sample actuator. In order to keep the surfaces of the KP tip and the sample
parallel, the tip was customized to face at $45^{\circ}$ from the longitudinal
axis of the tip (see Fig. 2, inset). The tip can probe a 2 mm diameter
circular area where the work function measurement is the average work function
over the probed surface. The coarse sample to tip distance was varied manually
using a linear translator attached to the Kelvin probe chamber. The fine
adjustment to the sample to tip distance and the tip oscillation along the
longitudinal axis of the Kelvin probe were controlled by means of the
computer-controlled voice coil system. The effect of stray capacitances was
minimized by doing a spectral analysis to find the resonances of the vibrating
probe and subsequently choosing to operate at an off-resonant frequency.
Since the Kelvin probe measures the position of the Fermi level of the sample
relative to the reference tip, calibration of the latter is necessary in order
to obtain the absolute value of the sample’s Fermi level relative to the
vacuum level, and thus to be able to obtain the sample’s work function. In the
setup described here, the tip was made of type 304 stainless steel coated with
nichrome, a non-magnetic alloy of nickel and chromium. Calibration was
performed using three references of known work function: polycrystalline
molybdenum with work function 4.6 eV Michaelson (1977), polycrystalline
tellurium with work function 4.95 eV Michaelson (1977),and highly oriented
pyrolytic graphite (HOPG) with work function 4.6 eV Takahashi _et al._
(1985),Suzuki _et al._ (2000). In the configuration for calibration the
sample played the role of reference and the tip was the material probed, hence
the work function of the tip was calculated by adding the contact potential
difference measured to the work function of the reference sample. The work
function of the tip was taken as the average of the values found in
calibration and the uncertainty taken to be the largest measured. The
resulting tip work function value was $4.6\pm 0.1$ eV.
For this experiment, six $\mathrm{Cs}_{2}\mathrm{Te}$ photocathodes were
fabricated and studied. The initial average value of the QE for the cathodes
in the study was 16.7%, the range was [15.5,18.8%], and standard deviation was
1.3%. The initial average value of the work function was 2.3 eV, the range was
[2.22,2.36], and standard deviation 0.055 eV. The work function and QE were
recorded and tracked for five indexed points on the cathode surface, as shown
in Fig. 3 for a newly-grown photocathode. For a photocathode of this size,
uniformity in QE could be an issue. As can be seen in Fig. 3, the photocathode
that had been fabricated was relatively uniform, both in QE and in the
measured work function. Applying the rigid band picture and using an average
measured value of 2.3 eV as the work function for
$\mathrm{Cs}_{2}\mathrm{Te}$, band gap of 3.3 eV and electron affinity of 0.2
eV Sommer (1980),Powell _et al._ (1973) this places the Fermi level $\approx
0.5$ eV below the middle of the band gap, making this a p-type semiconductor.
While it is uncertain if the rigid band model is accurate to describe the
aging effect of QE and the work function, it is still a useful model to obtain
an initial quantitative comparison on how much the Fermi level may have
changed. Thus, applying the same calculation for the typical cathode after
aging for 2-3 weeks, with an average value of the work function of 2.8, the
position of the Fermi level is now $\approx 1$ eV below the mid-gap, a shift
of 0.5 eV towards the valence band. Certainly, there may be other factors that
can cause a change in the work function that we measured beyond just a shift
in the Fermi level, including an increase in the electron affinity due to
surface contaminants, etc. We intend to investigate this further in future
studies.
Figure 3: Local variation of photocathode quantum efficiency and work
function. The data were taken within one day of cathode fabrication.
It is well-known from previous studies that the QE of
$\mathrm{Cs}_{2}\mathrm{Te}$ diminishes over time Kong _et al._ (1995a),
Michelato _et al._ (1997). We investigated the evolution of QE and work
function. The plot in Fig. 4, top shows the correlation between QE and work
function at a point on the photocathode over a period of 3 weeks. As the value
of the QE dropped, the work function correspondingly increased. The variation
of QE and work function over a period of time can be seen in Fig. 4, bottom.
The value for the QE initially dropped rapidly and then started to level off
after 15 days. The work function followed this pattern inversely, and appeared
to change very little after 20 days. The observed trend of increased work
function with decreasing QE is similar to that observed previously Lederer
_et al._ (2007) for $\mathrm{Cs}_{2}\mathrm{Te}$ photocathode after operation.
A fit of QE vs. work function using a power law for photoemission has been
done for metals and semiconductorsKane (1962), using $QE=A(E_{t}-h\nu)^{P}$;
where $h\nu$ is the photon energy, $E_{t}$ is the photoemission threshold, $P$
is a fit parameter ($P$=2 for metals). Since we probe the work function and
not the photoemission threshold, we make the simplification of $\phi=E_{t}$ in
attempting this fit. The result was not very meaningful, yielding values of P
in the range of 2.5-4.6, outside of what is expected theoretically. While it
is unambiguously clear that there is an inverse relationship between QE and
work function, we are not able to make a quantitative determination of this
exact relationship based on our available data at present. This is something
we intend to investigate further in the future.
After aging, the photocathode was rejuvenated via heating at $120^{\circ}$ C
for 4 days.111One photocathode was not a candidate for rejuvenation due to
poor vacuum. Previous studies on photocathode rejuvenation via heating have
shown a QE recovery up to about 60% of the original value Kong _et al._
(1995b), di Bona _et al._ (1996). In our experiment, five of six
photocathode’s QE went from 30% of the original QE prior to heating to an
average of about 60% of the original value after heating. Curiously, however,
this increase in QE after rejuvenation was accompanied by an increase in the
work function. This was contrary to the pattern seen in Fig. 4, top, where QE
and work function were inversely related during the aging process.
There are many possible explanations for such an observation, including the
possibility that the process of heating has changed the chemistry or nature of
the photocathode, especially on the surface, resulting in an increase in the
photocathode’s electron affinity Yates and Campbell (2011). More studies are
required to determine the cause of this unexpected behavior.
Figure 4: Top: QE vs. work function; typical data. Data taken at two locations
on the cathode over a period of about 3 weeks. Bottom: Time evolution of the
QE and $\phi$. The box represents the time period the cathode was heated to
$120^{\circ}$C to rejuvenate the QE. Data tracks changes observed at 2
different locations on the cathode. The work function measurement was
performed first followed by QE measurement.
Exposure to 4.9 eV light has an effect on the work workfunction of the
photocathode. This is shown in the bottom curve in Fig. 5. We initially
measured the work function to be 2.4 eV. The photocathode was illuminated with
4.9 eV light for 2 minutes inside the deposition chamber. After the light
exposure, the photocathode was transferred to the Kelvin Probe chamber and the
work function was measured. There is a time delay of about 3 minutes from the
end of light exposure to the start of work function measurement. A clear drop
in the value of the work function by at least 150 meV was observed. The work
function appeared to recover its original value over a time period of 30
minutes. When this experiment was repeated using a 3.7 eV light, the work
function showed no obvious effect similar to that of the 4.9 eV light. (see
plot in top curve of Fig. 5). It was found that 3.7 eV light produced
measurable photocurrent with a QE 0.1-0.2%, indicating that 3.7 eV is above
the photoemission threshold, consistent with the literature Sommer
(1980),Powell _et al._ (1973). There is a curious similarity with the result
reported earlier by Sertore et al although changes in the work function were
not reported. They found that QE rejuvenation took place while simultaneously
heating to $300^{\circ}$C AND illuminating with 4.9 eV light, while
illuminating with 3.7 eV light produced no rejuvenation effects Michelato _et
al._ (1997).
Figure 5: Comparison of the effect of exposure to 3.7 eV light and 4.9 eV
light. Light exposure ends in both cases at time t=0 and work function
measurement begins about 3 minutes later. Pre-exposure work function data is
also plotted.
We performed two further investigations on the UV exposure effects. First, the
exposure time using 4.9 eV light was varied, as shown in Fig. 6, top. Longer
exposure time caused a larger drop in the work function and a longer recovery
time. However, it appeared that the exposure time saturates at approximately
20 minutes, whereby longer exposure time did not seem to cause the work
function to drop further.
Figure 6: Top: Work function measured after various exposure times. The
initial $\Delta\phi$ saturates after about 20 minutes exposure. Bottom: Work
function measured after 2 min. exposure at different intensities. There is
still a measurable effect at 28% intensity.
Secondly, the intensity of the 4.9 eV light was varied using neutral density
filters. The cathode was illuminated for 2 minutes at a particular spot, then
the cathode was moved into the Kelvin probe chamber for the work function
measurement. The drop in the work function diminished as the intensity
decreased. We found that the induced work function reduction scaled with the
light intensity (Fig.6, bottom).
A similar observation has been reported on Kelvin probe measurements on Indium
Tin Oxide (ITO) Kim _et al._ (2000). In that work, the drop in the work
function was attributed to either charging effects, or photochemistry Kamat
(1993). We will be conducting further investigation to understand the origin
of this observation in the $\mathrm{Cs}_{2}\mathrm{Te}$ photocathode. As we
discussed earlier, chemical changes can have significant effects on the work
function. Unfortunately to date there have been no significant studies linking
changes in work function to surface chemistry.
In summary, a Kelvin Probe was used to measure the work function of
$\mathrm{Cs}_{2}\mathrm{Te}$ photocathodes grown for the AWA drive-beam
photoinjector. The fresh cathode was found to have an initial work function of
about 2.3 eV increasing to 2.8 eV as the cathode ages and the QE declines. The
QE scaled inversely with the work function over time. The effect of
rejuvenation via heating produced a different correlation whereby both QE and
the work function increased after heating. Exposure to 4.9 eV light produced a
temporary drop in the measured work function, with a recovery time on the
order of 30 minutes. The magnitude of the drop in work function is dependent
upon the exposure time and the intensity of the 4.9 eV light. Exposure to 3.7
eV light produced no noticeable effect.
###### Acknowledgements.
We thank Richard Rosenberg, Wei Gai, and acknowledge valuable discussion with
Karoly Nemeth. This work was funded by the U.S. Dept of Energy Office of
Science under contract number DE-AC02-06CH11357 and the National Science
Foundation under grant number 0969989. The submitted manuscript has been
created by UChicago Argonne, LLC, Operator of Argonne National Laboratory
(”Argonne”). Argonne, a U.S. Department of Energy Office of Science
laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S.
Government retains for itself, and others acting on its behalf, a paid-up
nonexclusive, irrevocable worldwide license in said article to reproduce,
prepare derivative works, distribute copies to the public, and perform
publicly and display publicly, by or on behalf of the Government.
## References
* Kong _et al._ (1995a) S. H. Kong, J. Kinross-Wright, D. C. Nguyen, and R. L. Sheffield, J. Appl. Phys. 77, 6031 (1995a).
* Conde _et al._ (2010) M. Conde, D. Doran, W. Gai, R. Konecny, W. Liu, J. Power, Z. Yusof, S. Antipov, and C. Jing, in _Proceedings of LINAC2010_ (Tsukuba, Japan, 2010) pp. 319–321.
* Conde _et al._ (2011) M. Conde, D. Doran, W. Gai, R. Konecny, W. Liu, J. Power, E. Wisniewski, Z. Yusof, S. Antipov, and C. Jing, Proceedings of PAC2011 (2011).
* Surplice and D’Arcy (1970) N. A. Surplice and R. J. D’Arcy, Jnl. of Phys. E: Scientific Instruments 3, 477 (1970).
* Kong _et al._ (1995b) S. H. Kong, D. C. Nguyen, R. L. Sheffield, and B. A. Sherwood, Nucl. Instr. and Meth. A 358, 276 (1995b).
* Michelato _et al._ (1997) P. Michelato, C. Pagani, D. Sertore, A. di Bona, and S. Valeri, Nucl. Instr. and Meth. A 393, 464 (1997).
* Michaelson (1977) H. B. Michaelson, J. Appl. Phys. 48, 4729 (1977).
* Takahashi _et al._ (1985) T. Takahashi, H. Tokailin, and T. Sagawa, Phys. Rev. B 32, 8317 (1985).
* Suzuki _et al._ (2000) S. Suzuki, C. Bower, Y. Watanabe, and O. Zhou, Appl. Phys. Lett. 76, 4007 (2000).
* Sommer (1980) A. Sommer, _Photoemissive Materials_ (Robert E. Krieger Pub. Co., Huntington, NY, 1980).
* Powell _et al._ (1973) R. A. Powell, W. E. Spicer, G. B. Fisher, and P. Gregory, Phys. Rev. B 8, 3987 (1973).
* Lederer _et al._ (2007) S. Lederer, H. Duerr, J. Han, P. Michelato, L. Monaco, R. Ovsyannikov, C. Paganini, S. Schreiber, D. Sertore, M. Sperling, F. Stephan, and A. Vollmer, in _Proceedings of the 29th International FEL Conference_ (2007) pp. 457–460.
* Kane (1962) E. O. Kane, Phys. Rev. 127, 131 (1962).
* Note (1) One photocathode was not a candidate for rejuvenation due to poor vacuum.
* di Bona _et al._ (1996) A. di Bona, F. Sabary, S. Valeri, P. Michelato, D. Sertore, and G. Suberlucq, J. Appl. Phys. 80, 3024 (1996).
* Yates and Campbell (2011) J. T. Yates and C. T. Campbell, Proceedings of the National Academy of Sciences 108, 911 (2011).
* Kim _et al._ (2000) J. Kim, B. Lagel, E. Moons, N. Johansson, I. Baikie, W. Salaneck, R. Friend, and F. Cacialli, Synthetic Metals 111-112, 311 (2000).
* Kamat (1993) P. V. Kamat, Chemical Reviews 93, 267 (1993).
|
arxiv-papers
| 2012-03-29T19:14:19 |
2024-09-04T02:49:29.196336
|
{
"license": "Public Domain",
"authors": "Eric Wisniewski and Daniel Velazquez and Zikri Yusof and Linda\n Spentzouris and Jeff Terry and Katherine Harkay",
"submitter": "Eric Wisniewski",
"url": "https://arxiv.org/abs/1203.6632"
}
|
1203.6670
|
# Energy levels and extension of the Schrödinger operator.
Y. C. Cantelaube e-mail : yves.cantelaube@univ-paris-diderot.fr
###### Abstract
Although energy levels are often given by solutions of the radial equation
such that u(0) is non zero, and hence by first-order singular functions which
are not eigenfunctions of H, the latter is always considered as the only
operator that gives energy levels. Vibrational levels of diatomic molecules
are a usual example. We show that the operator which has singular
eigenfunctions, or pseudofunctions, that give energy levels, is the operator
whose action on pseudofunctions amounts to the embedding in the distributions
of R3 of their Hamiltonian in R3/{0}. When its eigenfunctions are regular,
this operator amounts to H. Energy levels, which are given by eigenfunctions
of H when u(0) is zero, are thus given in any case by eigenfunctions of this
operator, which is an extension of the Schrödinger operator, but not of the
Hamiltonian.
U.F.R. de Physique, Université Paris Diderot, Bâtiment Condorcet, 75205 Paris
cedex 13, France
## 1 Introduction. Energy levels and radial equation.
In a central potential $\mathcal{V}$(r) energy levels $\mathcal{E}{}_{n}$ are
eigenvalues of the Hamiltonian that belong to normalized eigenfunctions
$\psi_{n}$ =
[$\mathit{w_{n}}\left(r\right)$/r]$Y_{\ell}^{\mu}\left(\theta,\varphi\right)$
which behave at the origin like rℓ$Y_{\ell}^{\mu}\left(\theta,\varphi\right)$,
and hence eigenvalues of the radial equation that belong to normalized
eigensolutions $\mathit{w_{n}}\left(r\right)$ which behave at the origin like
rℓ+1,
$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+\mathcal{V}(r)\right]w_{n}\left(r\right)=\mathcal{E}{}_{n}\mathit{w_{n}\left(r\right)}$
(1)
As physical potentials $\mathcal{V}$(r) are not exactly known, one must
substitute theoretical and then approximate potentials V(r) for which the
Hamiltonian has a discrete set of normalizable eigenfunctions $\Psi_{n}$=
[$\mathit{u_{n}}\left(r\right)$/r]$Y_{\ell}^{\mu}\left(\theta,\varphi\right)$,
and hence the radial equation a discrete set of normalizable analytic
eigensolutions $\mathit{u_{n}}\left(r\right)$which behave at the origin like
rℓ+1,
$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+V\left(r\right)\right]u_{n}\left(r\right)=E_{n}\mathit{u_{n}\left(r\right)}$
(2)
One assumes indeed that if V(r) $\mathcal{\approx V}$(r), $\Psi_{n}$ will be
similar to $\psi_{n}$, $\mathit{u_{n}}\left(r\right)$ to
$\mathit{w_{n}}\left(r\right)$, and $\mathit{E_{n}\approx\mathcal{E}{}_{n}}$.
The solutions of the radial equations, which are written in the form of
series, $\mathit{u_{n}\left(r\right)}$ = $\mathit{r^{\lambda}}$$\sum_{k\geq
0}{\textstyle a_{k}^{n}r^{k}}$, are given in $\textrm{R}^{3}$/{0} by both
roots $\lambda$= $\ell+1$, and $\lambda$= $-\ell$, but in R3 for $\ell$ > 0
only by the root $\lambda$= $\ell+1$ [1]. These solutions are normalizable and
give, when substituted in $\Psi_{n}$, eigenfunctions of H, or solutions of the
Schrödinger equation. For $\ell$ = 0 the radial equations
$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\mathcal{V}\left(r\right)\right]w_{n}\left(r\right)=\mathcal{E}{}_{n}\mathit{w_{n}\left(r\right)}$
(3)
$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+V\left(r\right)\right]u_{n}\left(r\right)=E_{n}\mathit{u_{n}\left(r\right)}$
(4)
have solutions in $\textrm{R}{}^{3}$ given by both roots, that is, $\lambda$ =
1, and $\lambda$= 0. Both roots give normalizable solutions, but only the
solutions given by the root $\lambda$ = 1 give, when substituted in
$\Psi_{n}$, solutions of the Schrödinger equation [1]. Insofar as energy
levels $\mathcal{E}{}_{n}$ are eigenvalues of the Hamiltonian, the solutions
$\mathit{w_{n}}\left(r\right)$ of (3) behave at the origin like r. In order
for $\Psi_{n}$ to be eigenfunctions of H, one must substitute potentials V(r)
for which the following conditions must be satisfied: i) V(r) must be close to
$\mathcal{V}\left(r\right)$, at least in the neighborhood of the minimum; ii)
Eq.(4) must have a discrete set $\mathit{u_{n}}\left(r\right)$ of normalizable
analytic solutions ; iii) these solutions must behave at the origin like r.
Now, explicit analytic solutions of (4) can be derived only for a limited
number of potential energy functions V(r). When we demand that the first two
conditions be satisfied, this number is much more limited. If we moreover
demand that these solutions behave at the origin like r, the problem cannot
necessarily be solved. This is why one substitutes potentials V(r) close to
$\mathcal{V}\left(r\right)$, for which (4) has a discrete set of normalizable
solutions regardless of their value at the origin. One intuitively expects
indeed that if $V\left(r\right)\approx\mathcal{V}\left(r\right)$ and if (4)
has a discrete set of normalizable solutions $\mathit{u_{n}}\left(r\right)$,
moreover demanding that they behave at the origin like r is a supplementary
condition that is not required to obtain eigenvalues $\mathit{E_{n}}$ of (4)
close to the eigenvalues $\mathcal{E}{}_{n}$ of (3). This is shown by the
following example.
## 2 Vibrational levels of diatomic molecules.
The vibrational levels of diatomic molecules are of the form
$\mathcal{E}{}_{n}=-\mathit{V_{m}+(n+\text{\textonehalf)}}\hbar\omega+C_{2}(n+\text{\textonehalf)}^{2}+C_{3}(n+\text{\textonehalf)}^{3}+...\qquad
n=0,1,2...$
In the Born-Oppenheimer approximation the potential energy of interaction
between the two nuclei of a diatomic molecule is a central potential V(r),
attractive at large distances, repulsive at short distances, with a minimum at
r = $\mathit{r_{m}}$. By expanding the potential in powers of r –
$\mathit{r{}_{m}}$ in the neighborhood of the minimum by neglecting the terms
of order $\geq$ 3, it gives
$V\left(r\right)=\text{\textonehalf}m\omega{}^{2}\left(r-r_{m}\right)^{2}-V_{m}\qquad\qquad\omega=[V\,"(r_{m})/m]^{1/2}$
(5)
This parabolic potential is used as a first approximation, which holds in the
neighborhood of the minimum, and then gives the first levels. The radial
equation for $\ell=0$ involving this potential,
$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\text{\textonehalf}m\omega{}^{2}\left(r-r_{m}\right)^{2}-V_{m}\right]u_{n}\left(r\right)=E_{n}\mathit{u_{n}\left(r\right)}$
(6)
has the square integrable solutions and the corresponding eigenvalues
$u_{n}\left(r\right)=N_{n}exp\left[-\text{\textonehalf}\beta^{2}(r-r_{m})^{2}\right]H_{n}\left[\beta(r-r_{m})\right]\qquad\beta^{2}=m\omega/\hbar$
$E_{n}=(n+\text{\textonehalf)}\hbar\omega-V_{m}\qquad\qquad\qquad n=0,1,2...$
(7)
where $\mathit{H_{n}}$ are Hermite polynomials of order n and $\mathit{N_{n}}$
normalization constants. Experimental data show that these eigenvalues give
estimates of the first vibrational levels of most molecules. Now, except for
the case where $\beta r_{m}$ coincides with a zero of $\mathit{H_{n}}$,
$u_{n}\left(0\right)=N_{n}exp(-\text{\textonehalf}\beta^{2}r_{m}{}^{2})H_{n}(\beta
r_{m})\neq 0$ (8)
Analytical forms for V(r) closer to the realistic physical potential have been
proposed, the most frequently used is the Morse potential [2]
$V_{M}(r)=V_{m}\\{exp\left[-2a(r-r_{m})\right]-2exp\left[-a(r-r_{m})\right]\\}$
(9)
For $\ell$ = 0 the square-integrable solutions and the eigenvalues of the
radial equation are [2]
$u_{n}(r)=exp\left[-z(r)/2\right]\left[z(r)\right]^{b/2}L_{n+b}^{b}\left[z(r)\right]$
$z=2d\,exp\left[-a(r-r_{m})\right]\qquad
d=(2mV_{m})^{\text{\textonehalf}}(a\hbar)^{-1}\qquad b=2d-1-2n$
$E_{n}=(n+\text{\textonehalf)}\hbar-\left(n+\text{\textonehalf}\right)^{2}(\hbar^{2}\omega^{2}/4V_{m})-V_{m}\qquad\omega=a(2V_{m}/m)^{\text{\textonehalf}}$
where $\mathit{L_{n+b}^{b}}\left(z\right)$ are generalized Laguerre
polynomials. Experimental data show that these eigenvalues give very accurate
values for the vibrational levels of nearly all molecules, but again the
condition $u{}_{n}$(0) = 0 is not satisfied.
If one considers that by substituting the potential (5) “we are left with the
equation of a one-dimensional harmonic oscillator centered at r =
$\mathit{r_{m}}$” [3,$\textrm{A}_{\textrm{V}}$], no boundary condition is
required at the origin, so that in particular perturbation theory for a one-
dimensional harmonic oscillator is applied to the third-order term.
Otherwise, arguments have been proposed to justify that the fact that
$u{}_{n}(0)\neq 0$ is of no significance. According to Pauling et al. the
solutions of the radial equation must be zero at r = 0 and +$\infty$, whereas
the solutions of the harmonic oscillator must be zero at r =
$\text{\textendash}\infty$ and +$\infty$, but because of the rapid decrease in
the harmonic oscillator functions outside the classically permitted region,
“it does not introduce a serious error to consider that the two sets of
boundary conditions are practically equivalent” [4, See also 5,6,7]. As noted
by Schiff, the eigenvalues of the radial equation are those of a linear
harmonic oscillator “if the domain of r is extended to $-\infty$” [8], which
amounts to substituting a one-dimensional problem for a three-dimensional
problem. Or else, one considers that, as (6) is the radial equation for zero
angular momentum states, the exact solutions must be rigorously zero at r = 0,
but the solutions (7) are “practically zero at the origin”
[3,$\textrm{F}{}_{\textrm{VII}}$]. It should be noted that one often considers
indeed that the exact solutions of (6) can be obtained only if the variable is
taken from $-\infty$ to +$\infty$, or if the solutions vanish at the origin.
In fact, (7) are the exact solutions of (6), that is, the solutions in R3
($r\geq$0), as is easily checked.
Similarly, the value of the Morse potential is sometimes considered as so
large at r = 0 that the Morse eigenfunctions are “effectively zero for r < 0”
[9]. In fact they are non zero at r = 0, but according to Morse since “in
every case rΨ will be extremely small … this discrepancy will not affect the
values of the energy levels” [2, See also 6,7].
But if the two boundary conditions, $u_{n}\left(0\right)$ = 0 and
$u{}_{n}\left(0\right)\approx 0$, are practically equivalent, it merely means
that the former condition is not required. First-order singular functions are
not indeed “practically eigenfunctions” of H, they are eigenfunctions of H in
$\textrm{R}^{3}$/{0}, but not in $\textrm{R}^{3}$. In other words, whether or
not u(0) is very small, if it is non zero, Ψn behaves at the origin like 1/r,
and what these examples show is thus that energy levels are not affected
because that they are not given by eigenfunctions of H. By changing the
condition at the origin, we change indeed the operator.
## 3 The energy levels and the operator $\mathit{H_{d}}$.
We must thus determine the operator which has singular eigenfunctions of the
form $\Psi_{n}$ =
[$\mathit{u_{n}}\left(r\right)$/r]$Y_{\ell}^{\mu}\left(\theta,\varphi\right)$
and corresponding eigenvalues $\mathit{E_{n}}$, such that for $\ell$ = 0
$\mathit{u_{n}\left(r\right)}$ and $\mathit{E_{n}}$ are solutions of (4) with
$u_{n}\left(0\right)\neq 0$. Now, when $\Psi_{n}$ and $u_{n}$(r), which are of
$\textrm{C}^{\infty}(\textrm{R}$3/{0}), are singular at the origin, they
define in R3 distributions called pseudofunctions, denoted by
$\textrm{Pf}.\Psi_{n}$ and $\textrm{Pf}.u_{n}$(r) [10], and the latter are not
a solution of (2) [1]. The radial equation which has in R3 singular solutions
$\textrm{Pf}.u_{n}$(r) given by the root $\lambda$ = $-\ell$ is the extension
in R3 of (2), it is the equation [1]
$\textrm{Pf}.\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+V\left(r\right)\right]u_{n}\left(r\right)=E_{n}\textrm{Pf}.\mathit{u_{n}\left(r\right)}$
(10)
Its solutions, given by both roots $\lambda$ = $\ell+1$ and $\lambda$ =
$-\ell$, are either functions $\mathit{u_{n}}\left(r\right)$ which behave at
the origin like $r{}^{\ell+1}$, or pseudofunctions
$\textrm{Pf.}\mathit{u_{n}}\left(r\right)$ which behave at the origin like
$\textrm{Pf.}r^{-\ell}$. (The symbol Pf. is used when we have either the
former or the latter). When the solutions of (10) are given by the roots
$\lambda$ = $\ell+1$ and $\lambda$ = 0 (i.e. $\lambda$ = $-\mathit{\ell}$ for
$\ell$ = 0), the symbol Pf. is useless and (10) is written in the form (2).
Just as $\Psi_{n}$ defines in R3 the pseudofunction
$\textrm{Pf}.\Psi_{n}=\textrm{Pf.}[\mathit{u_{n}}\left(r\right)/r]Y_{\ell}^{\mu}\left(\theta,\varphi\right)$,
its Hamiltonien in R3/{0}, which is the Hamiltonian in the sense of the
functions
$H\Psi_{n}=\frac{1}{r}\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+V(r)\right]\mathit{u_{n}}(r)Y_{\ell}^{\mu}\left(\theta,\varphi\right)\qquad\qquad
r>0$
defines in R3 the distribution
$\textrm{Pf}.H\Psi_{n}=\textrm{Pf}.\left\\{\frac{1}{r}\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+V(r)\right]\mathit{u_{n}}(r)Y_{\ell}^{\mu}\left(\theta,\varphi\right)\right\\}$
(11)
Let $\mathit{H_{d}}$ the operator whose action on a pseudofunction
Pf.$\Psi_{n}$ amounts to the embedding in the distributions of R3 of the
Hamiltonian of $\Psi_{n}$ in R3/{0},
$\mathit{H_{d}}\textrm{Pf}.\Psi_{n}=\textrm{Pf}.H\Psi_{n}$ (12)
If Pf.$\mathit{u_{n}\left(r\right)}$ is a solution of (10), whether it is
given by the root $\lambda$ = $\mathit{\ell}$ \+ 1, or by the root $\lambda$ =
$-\mathit{\ell}$, by substituting (10) in (12), we obtain the eigenvalue
equation of $\mathit{H_{d}}$,
$H_{d}\textrm{Pf.}\Psi_{n}=E_{n}\textrm{Pf.}\Psi_{n}$ (13)
where $\mathit{E_{n}}$ is the eigenvalue of (10) that belongs to
Pf.$\mathit{u_{n}}\left(r\right)$.
The normalizable solutions are given by the roots $\lambda$ =
$\mathit{\ell}$\+ 1 $\forall\ell$, and $\lambda$ = 0, that is, $\lambda\in N$.
As they are less singular at the origin than 1/$r{}^{3}$, the symbol Pf. can
be dropped [10], and (13) is written
$H_{d}\Psi_{n}=E_{n}\Psi_{n}\qquad\qquad\lambda\in N$ (14)
In particular for $\ell$ = 0, $\Psi_{n}$ = $(4\pi){}^{-1/2}[u_{n}(r)/r]$,
whether $u_{n}\left(0\right)=0$ if $\lambda$ = 1, or $u_{n}\left(0\right)\neq
0$ if $\lambda$ = 0, in both cases
$H_{d}\frac{1}{\sqrt{4\pi}}\frac{u_{n}(r)}{r}=\frac{1}{\sqrt{4\pi}}E_{n}\frac{u_{n}(r)}{r}\qquad\qquad\lambda=0$
(15)
where $\mathit{u_{n}\left(r\right)}$ and $\mathit{E_{n}}$ are solutions of the
radial equation (4). $\mathit{H_{d}}$ is thus the required operator.
In order to compare the operators $\mathit{H_{d}}$ and H, let us recall that
the Hamiltonian of
$\textrm{Pf.}\Psi_{n}{\displaystyle{\textstyle=\textrm{Pf}.r^{\lambda-1}\sum_{k\geq
0}a_{k}^{n}r^{k}}\mathit{Y_{\ell}^{\mu}\left(\theta,\varphi\right)}}$ is the
Hamiltonian in the sense of the distributions given by [1]
$H\textrm{Pf}.\Psi_{n}=\textrm{Pf}.\left\\{\frac{1}{r}\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)\hbar^{2}}{2mr^{2}}+V(r)\right]\mathit{u_{n}}(r)Y_{\ell}^{\mu}\left(\theta,\varphi\right)\right\\}-\frac{\hbar^{2}}{2m}Q_{\lambda,\ell}\left(\delta\right)$
$Q_{\lambda,l}\left(\delta\right)=\sum_{k=0}^{-\lambda}a_{k}\chi_{p(k)}B_{\ell,p(k)}C_{p(k)}\mathit{r{}^{\ell}\mathit{\mathit{Y_{\ell}^{\mu}\left(\theta,\varphi\right)}}}\Delta^{p(k)}\delta\qquad
p=-\frac{k+\lambda-\ell}{2}$
$\chi_{p}=\begin{cases}1\;&if\;p\in N\\\ 0&if\;p\notin N\end{cases}\qquad
B_{\ell,p}=\frac{1-2\ell}{4p+1}\qquad
C_{p}=-\frac{(4p+1)\pi^{3/2}}{2^{2p-1}p!\,\Gamma(p+3/2)}$ (16)
where $\Delta^{p}$ is the iterated Laplacian,$\delta$ the Dirac mass, where
the even, resp. odd, terms of the sum are zero , if $\lambda-\ell$ is odd,
resp. even, and where $\mathit{Q_{\lambda,\ell}\left(\delta\right)}$ $\neq$ 0
if and only if there is at least one coefficient $a_{k}\neq 0$ for which k \+
$\lambda-\ell$ is an even negative integer [1]. As
$(H_{d}-H)\textrm{Pf.}\Psi_{n}=\left[\textrm{Pf}.,H\right]\Psi_{n}=\frac{\hbar^{2}}{2m}Q_{\lambda,\ell}\left(\delta\right)$
(17)
the operators $\mathit{H{}_{d}}$ and H differ when they act on pseudofunctions
for which the operators Pf. and H do not commute, or for which
$\mathit{Q_{\lambda,\ell}\left(\delta\right)}$$\neq$ 0\. Considering that
energy levels are given in any case by the operator H amounts to identify the
operators Hd and H, and hence to take no account of the noncommutation of the
operators Pf. and H. As the operators $\mathit{H{}_{d}}$ and H are equivalent
in $\textrm{R}^{3}$/{0}, and in $\textrm{R}^{3}$ when the Laplacian is the
Laplacian in the sense of the functions, it comes from the confusion between
the Laplacians in $\textrm{R}^{3}$/{0} and in $\textrm{R}^{3}$, or/and between
the Laplacians in the sense of the functions and in the sense of the
distributions [11]. By substituting (10) in (16), we obtain
$H\textrm{Pf}.\Psi_{n}=E_{n}\textrm{Pf}.\Psi_{n}-\frac{\hbar^{2}}{2m}Q_{\lambda,\ell}\left(\delta\right)$
(18)
When the solutions of the radial equation are given by the root $\lambda$ =
$\mathit{\ell}$\+ 1, p = – (k \+ 1)/2 $\notin N$ $\forall k$, so that
$Q_{\ell+}{}_{1,\ell}$ = 0, $\Psi_{n}$ behaves at the origin like
$r{}^{\ell}Y_{\ell}^{\mu}\left(\theta,\varphi\right)$, it is an eigenfunction
of H. When the solutions of the radial are given by the root $\lambda$ =
$-\ell$, as $\mathit{a_{o}\neq}$ 0 and p = $\ell\in N$ for k = 0,
$Q{}_{-\ell,\ell}\neq 0$, $\textrm{Pf}.\Psi_{n}$ behaves at the origin like
$\textrm{Pf}.r{}^{-(\ell+1)}Y_{\ell}^{\mu}\left(\theta,\varphi\right)$, it is
not an eigenfunction of H [1]. As $Q{}_{0,0}=-\sqrt{4\pi}u_{n}(0)\delta$, in
the case of normalizable solutions (18) is written
$H\Psi_{n}=E_{n}\Psi_{n}\qquad\qquad\lambda=\ell+1$
$H\,\frac{1}{\sqrt{4\pi}}\frac{u_{n}(r)}{r}=\frac{1}{\sqrt{4\pi}}E_{n}\frac{u_{n}(r)}{r}+\frac{\hbar^{2}\sqrt{\pi}}{m}u_{n}(0)\delta\qquad\qquad\lambda=0$
(19)
Comparison between (14), (15) and (19) shows that energy levels, which are
given by eigenfunctions of H when $u_{n}$(0) = 0, are given in any case by
eigenfunctions of $\mathit{H_{d}}$.
## 4 The Hamiltonian, the Schrödinger operator and the operator
$\mathit{H_{d}}$.
The differential form
$H=-\frac{\hbar^{2}}{2m}\Delta+V$ (20)
is usually called Hamiltonian or Schrödinger operator. Now, a differential
form such as (20) defines an operator only on a given class of functions, so
that the same differential form can define different operators [See e.g. 12].
Any function Ψ, or pseudofunction Pf.Ψ, belongs to the domain $\mathcal{D}$ on
which the Hamiltonian, denoted by H, is defined by (20). On the other hand,
the operator involved in the Schrödinger equation, or Schrödinger operator,
denoted by $\mathit{H_{s}}$, can be defined on the set $\mathcal{D}{}_{s}$ of
the functions which are any superpositions of eigenfunctions of H. As
$\mathcal{D}{}_{s}$ is a subset of $\mathcal{D}$, and as the two operators are
equivalent on $\mathcal{D}{}_{s}$, that is, $H\Psi\equiv H_{s}\Psi$ if
$\Psi\in\mathcal{D}_{s}$, the Schrödinger operator is a restriction of the
Hamiltonian. As long as one confines oneself to solutions of the Schrödinger
equation, these two operators, which have the same eigenfunctions, are
equivalent, so that the distinction is not necessary, and in fact it is not
made. Nevertheless the distinction can be made with respect to the operator
$\mathit{H}_{d}$.
The latter is defined on the same domain $\mathcal{D}$ as H. Moreover (13) and
(18) show that the eigenfunctions of H, or of $\mathit{H{}_{s}}$, form a
subset of the eigenfunctions of $\mathit{H{}_{d}}$, so that if
$\Psi\in\mathcal{D}_{s}$, then $\mathit{H_{d}}\Psi\equiv H_{s}\Psi$. The
operator $\mathit{H_{d}}$ is thus also an extension of the Schrödinger
operator. But the fact that the eigenvalue equation of $\mathit{H{}_{d}}$ is
an extension of the eigenvalue equation of H to the pseudofunctions such that
$\mathit{Q_{\lambda,\ell}\left(\delta\right)}\neq$ 0 does not mean that
$\mathit{H{}_{d}}$ is an extension of H. Eq.(17) shows indeed that they are
different when $\mathit{Q_{\lambda,\ell}\left(\delta\right)}\neq$ 0, or when
the operators Pf. and H do not commute. It follows that the operators
$\mathit{H}$ and $\mathit{H{}_{d}}$ are two different extensions of the
Schrödinger operator.
The condition for the operator $\mathit{H}$ (and then $\mathit{H}_{s}$) to be
self-adjoint is usually written u(0) = 0. However, according to Merzbacher the
condition that the Hamiltonian must be self-adjoint implies that any two
physically admissible eigensolutions of (2) must satisfy the condition [13]
$\textrm{lim }_{r\rightarrow
0}\left(u_{1}^{*}\frac{du_{2}}{dr}-u_{2}\frac{du_{1}^{*}}{dr}\right)=0$ (21)
This condition is satisfied, either for u(0) = 0 and u ’(0)$\neq$ 0, in which
case $\Psi$ is an eigenfunction of $\mathit{H_{d}}$ and of H, or for u ’(0) =
0 and u(0) $\neq$ 0, in which case $\Psi$ is an eigenfunction of
$\mathit{H_{d}}$, but not of H. It follows that (21) is in fact the condition
for $\mathit{H_{d}}$ to be self-adjoint.
The operator $\mathit{H_{d}}$ can also be defined, by using the linearity of
the operator Pf., from the operator $\textrm{Pf}.\Delta$ introduced in [11].
As
$\textrm{Pf}.H=\textrm{Pf}.(-\frac{\hbar^{2}}{2m}\Delta+V)$
by similarly defining the operator $\Delta_{d}$ as
$\Delta_{d}\textrm{Pf}.\Psi=\textrm{Pf}.\Delta\Psi$ (22)
we have
$H_{d}=-\frac{\hbar^{2}}{2m}\Delta_{d}+V$
## 5 A physically not required condition.
The boundary condition $\mathit{u_{n}}$(0) = 0 is definitely stated as a
condition required to obtain energy levels because it is the condition for
$\Psi_{n}$ to be an eigenfunction of the Hamiltonian, but in practice this
condition is not respected. It can be imposed when the solutions are
determined with the help of numerical or approximation methods, but not when
energy levels are determined with the help of analytic solutions.
In the case of diatomic molecules, in the parabolic like in the Morse
potential, the fact that $\mathit{u_{n}}(0)\neq 0$ is regarded as of no
significance owing to the fact that the origin being some distance outside of
the classically allowed region, $\mathit{u_{n}}$(0) is “very small”. As the
parabolic potential (5) is an expansion of the potential in the neighborhood
of the minimum by neglecting the terms of order higher than 2, it remains
close to the physical potential only in the neighborhood of the minimum. The
Morse potential (9) is closer to the physical potential, and in a region more
extended on both sides of the minimum. Comparison between the measured
vibrational levels $\mathcal{E}{}_{n}$ and the eigenvalues $\mathit{E_{n}}$ of
the radial equations involving these potentials shows that the agreement
between these eigenvalues and the energy levels decreases with n, and that the
number of eigenvalues which are close to energy levels is all the greater as
the region in which V(r) $\approx$ $\mathcal{V}$(r) is more extended. It means
that this agreement essentially depends on $\mathit{u_{n}}$(r) in the region
where V(r) is close to $\mathcal{V}$(r), where one therefore expects that
$\mathit{u_{n}}$(r) is close to $\mathit{w_{n}}$(r), the region where nearly
the whole wave function is concentrated. It follows that if energy levels are
not affected by the fact that $\mathit{u_{n}}$(0) $\neq$ 0, it does not come
from the fact that $\mathit{u_{n}}$(0) is “very small”, but from the fact that
the origin is outside of the region where V(r) $\approx\mathcal{V}(r)$, where
$u_{n}\left(r\right)\approx w_{n}\left(r\right)$, and where the wave function
is concentrated, that is, in a region where $\mathit{u_{n}}$(r) has little or
none influence on the fact that $\mathit{E_{n}}$ is close or not to
$\mathcal{E}{}_{n}$.
Besides, in the case of the potential (5) $\mathit{u_{n}}$(r) always vanishes
inside of the classically allowed region. It means in particular that the
condition $\mathit{u_{n}}$(0) = 0 can be satisfied only inside of this region.
Consider a physical potential $\mathcal{V}{}_{o}\left(r\right)$, and a
supplementary potential, or a perturbation, $\mathcal{W}\left(r\right)$, for
which one substitutes the spherical oscillator
$V_{o}\left(r\right)=\text{\textonehalf}m\omega{}^{2}r^{2}$ and the linear
potential W(r) = – Cr, so that the approximation of the physical potential
$\mathcal{V}\left(r\right)$ = $\mathcal{V}{}_{o}\left(r\right)$ \+
$\mathcal{W}\left(r\right)$ is the potential V(r) = $V_{o}\left(r\right)$ \+
W(r) of the form (5), that is,
$V\left(r\right)=\text{\textonehalf}m\omega{}^{2}r^{2}-Cr=\text{\textonehalf}m\omega{}^{2}\left(r-r_{m}\right)^{2}+V_{m}$
$r_{m}=C/m\omega^{2}\qquad\qquad V_{m}=C^{2}/2m\omega^{2}$ (23)
As C and then $\mathit{r{}_{m}}$ have any values, the origin is or not outside
of the classically permitted region. When $\beta\mathit{r_{m}}$ coincides with
a zero of a Hermite polynomial of order N, that is, $\mathit{H_{N}\left(\beta
r_{m}\right)}$ = 0, then $u_{N}\left(0\right)$ = 0, but $u_{n}\left(0\right)$
$\neq$ 0 for n < N. Insofar as the agreement between the eigenvalues of the
radial equation and the energy levels decreases with n, the eigenvalues
$\mathit{E_{n}}$ with n < N that belong to solutions $u_{n}\left(r\right)$
which do not satisfy the condition $u_{n}\left(0\right)$ = 0 must be closer to
an energy level than $\mathit{E_{N}}$ that belongs to a solution
$u_{N}\left(r\right)$ which satisfies this condition. The latter then cannot
be considered as the condition required to obtain accurate, or better
estimates of energy levels.
In the absence of supplementary potential, or of perturbation, the
approximation of the physical potential is the spherical oscillator,
$\mathcal{V}$(r) $\approx$ $V$(r), with
$V\left(r\right)=\text{\textonehalf}m\omega{}^{2}r^{2}$
$u_{n}\left(r\right)=N_{n}exp(-\text{\textonehalf}\beta^{2}r^{2})H_{n}(\beta
r)\qquad E_{n}=(n+\text{\textonehalf)}\hbar\omega$
$u_{n}\left(0\right)=N_{n}H_{n}(0)$
As Hermite polynomials satisfy the condition $H_{2p}\left(0\right)\neq 0$,
$H_{2p+1}\left(0\right)=0$, half the square-integrable solutions of the radial
equation satisfy the condition $\mathit{u_{n}}$(0) = 0, and hence half the
energy levels are given by eigenvalues of H. If the supplementary potential,
or the perturbation, is a 1D potential $\mathcal{W}$(x) for which one
substitutes the linear potential W(x) = – Cx, the potential V(r) =
$V{}_{o}(r)$ + W(x) used as an approximation of the physical potential
$\mathcal{V}$(r) = $\mathcal{V}{}_{o}$(r) + $\mathcal{W}$(x) is still a
spherical oscillator, that is,
$V\left(\rho\right)=\text{\textonehalf}m\omega{}^{2}r^{2}-Cx=\text{\textonehalf}m\omega{}^{2}\rho^{2}-V_{m}$
$\rho=\left[(x-C/m\omega^{2})^{2}+y^{2}+z^{2}\right]\qquad
V_{m}=C^{2}/2m\omega^{2}$
Half the energy levels are still given by eigenvalues of H. But when the
supplementary potential, or the perturbation, is the 3D potential
$\mathcal{W}$(r) for which one substitutes the linear potential W(r) = – Cr,
the potential used as an approximation of the physical potential is the
potential (23), so that the solutions of the radial equation, which are given
by (7), do not satisfy the condition $\mathit{u_{n}}$(0) = 0, and then that
the energy levels are not given by eigenvalues of H, as it is the case for
diatomic molecules. It shows that whether or not energy levels are given by
eigenvalues of H only depends on the potential used as an approximation of the
physical potential – and hence on the latter.
Perturbation methods, in particular variational methods, apply to the
eigenfunctions of H on which the theory is based. By substituting
$\mathit{H_{d}}$ for H these methods are extended to the eigenfunctions of
$\mathit{H_{d}}$ which behave at the origin like 1/r. It is in agreement with
the fact that energy levels are obtained with the help of such functions.
Similarly, situations which are described with the help of functions which
behave at the origin like 1/r, require, to hold in $\textrm{R}{}^{3}$, and not
only in $\textrm{R}{}^{3}$/{0}), that one substitutes $\mathit{H_{d}}$ for H.
This is the case of ingoing, outgoing and standing spherical waves which are
described with the help of Green functions. As the latter behave at the origin
like 1/r, they are eigenfunctions of H in $\textrm{R}{}^{3}$/{0}), but not in
$\textrm{R}{}^{3}$, so that this description does not hold at the origin,
which is physically unsatisfactory. As Green functions are eigenfunctions of
$\mathit{H_{d}}$ in $\textrm{R}{}^{3}$, by substituting this operator, the
description of spherical waves in terms of Green functions holds in the whole
space.
Moreover, wave functions divergent at the origin, but normalizable, are
encountered in the relativistic theory of the hydrogen atom. As the
relativistic generalization of the Schrödinger equation is the Klein-Gordon
equation which governs the wave function in the absence of electromagnetic
field for a relativistic particle with spin zero, if the Laplacian must be
taken in the sense of the distributions in this equation, one will be
similarly led to substitute for $\Delta$ the operator $\Delta_{d}$ defined in
(22).
## 6 Conclusion.
One usually considers, by putting forward different arguments, that the
resolution of the Schrödinger equation must be supplemented with the boundary
condition $\mathit{u_{n}}$(0) = 0, but when this condition is not satisfied,
one admits, by putting forward other arguments, that this is of no
significance. In fact the resolution of the Schrödinger equation need not be
supplemented with any boundary condition [1], but the fact that the condition
$\mathit{u_{n}}$(0) = 0 is not satisfied is significant, since it means,
whether or not $\mathit{u_{n}}$(0) is very small, that energy levels are not
given by solutions of the Schrödinger equation. As energy levels are obtained
by substituting theoretical and then approximate potentials, it should be
taken for granted that in approximate solutions the boundary condition
$\mathit{u_{n}}$(0) = 0, and hence the operator H, or $\mathit{H_{s}}$, are
not required to obtain energy levels, and hence that the operator
$\mathit{H{}_{d}}$, less stringent, is sufficient. The point is that the only
potentials that are exactly known, and then used to obtain energy levels, are
theoretical and then approximate potentials.
## References
* [1] Y. Cantelaube, Solutions of the Schrödinger equation, boundary condition at the origin, and theory of distributions, arXiv:1203.0551.
* [2] P. M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels, Phys. Rev. 34, 57 (1929).
* [3] C. Cohen-Tannoudji, B. Diu and F. Lalöe, Quantum Mechanics (Hermann and John Wiley and sons, Paris-New-York, 1977).
* [4] L. Pauling and E. Bright Wilson, Introduction to Quantum Mechanics with applications to chemistry (Dover Publications, INC, New-York, 6-415, reprint 1963).
* [5] H. Eyring, J. Walter and G.E. Kimbali, Quantum Chemistry (John Wiley & Sons, INC., Chapman & Hall, LTD., 1944).
* [6] I. N. Levine, Molecular spectroscopy, (John Wiley & Sons, 1975).
* [7] I. N. Levine, Quantum Chemistry, 5th ed. (Prentice Hall, New Jersey, 1991).
* [8] L. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill Book Company, Inc. NY, 1955).
* [9] K. T. Hecht, Quantum Mechanics (Springer, New York, 2000).
* [10] L.Schwartz, Théorie des distributions (Paris, Hermann, 1966).
* [11] Y. Cantelaube and A.L. Khelif, Laplacian in polar coordinates, regular singular function algebra, and theory of distributions, J. Math. Phys. 51, 053518 (2010).
* [12] E. Prugovecki, Quantum Mechanics in Hilbert Space, 2nd ed. (Academic Press, 1981).
* [13] E. Merzbacher, Quantum Mechanics, 3rd ed. (John Wiley and sons, INC., 1998).
|
arxiv-papers
| 2012-03-29T21:03:18 |
2024-09-04T02:49:29.202274
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y.C. Cantelaube",
"submitter": "Yves Cantelaube",
"url": "https://arxiv.org/abs/1203.6670"
}
|
1203.6836
|
# Fast method for quantum mechanical molecular dynamics
Anders M. N. Niklasson111Corresponding Author Email: amn@lanl.gov Marc J.
Cawkwell Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545
###### Abstract
As the processing power available for scientific computing grows, first
principles Born-Oppenheimer molecular dynamics simulations are becoming
increasingly popular for the study of a wide range of problems in materials
science, chemistry and biology. Nevertheless, the computational cost of Born-
Oppenheimer molecular dynamics still remains prohibitively large for many
potential applications. Here we show how to avoid a major computational
bottleneck: the self-consistent-field optimization prior to the force
calculations. The optimization-free quantum mechanical molecular dynamics
method gives trajectories that are almost indistinguishable from an “exact”
microcanonical Born-Oppenheimer molecular dynamics simulation even when low
pre-factor linear scaling sparse matrix algebra is used. Our findings show
that the computational gap between classical and quantum mechanical molecular
dynamics simulations can be significantly reduced.
electronic structure theory, molecular dynamics, Born-Oppenheimer molecular
dynamics, tight-binding theory, self-consistent tight binding theory, self-
consistent-charge density functional tight-binding theory, density matrix,
linear scaling electronic structure theory, Car-Parrinello molecular dynamics,
self-consistent field, extended Lagrangian molecular dynamics
††preprint: LA-UR 12-23991
## I Introduction
The past three decades have witnessed a dramatic increase in the use of the
molecular dynamics simulation method Karplus and McCammon (2002); Marx and
Hutter (2000). While it is unquestionably a powerful and widely used tool, its
ability to calculate physical properties is limited by the quality and the
computational complexity of the interatomic potentials. Among computationally
tractable models, the most accurate are explicitly quantum mechanical with
interatomic forces calculated on-the-fly using a nuclear potential energy
surface that is determined by the electronic ground state within the Born-
Oppenheimer approximation Wang and Karplus (1973); Leforestier (1978); Marx
and Hutter (2000). In Hartree-Fock Roothaan (1951); McWeeny (1960) or density
functional theory Hohenberg and Kohn (1964); Kohn and Sham (1965); Parr and
Yang (1989); Dreizler and Gross (1990), the electronic ground-state density is
given through a self-consistent-field (SCF) optimization procedure, which
involves iterative mixed solutions of the single-particle eigenvalue equations
and accounts for details in the charge distribution. Since the interatomic
forces are sensitive to the electrostatic potential Feynman (1939), molecular
dynamics simulations are often of poor quality without a high degree of self-
consistent-field convergence. This is unfortunate since the iterative self-
consistent-field procedure is computationally expensive and in practice always
approximate.
Recently there have been efforts to reduce the computational cost of the self-
consistent-field optimization without causing any significant deviation from
“exact” Born-Oppenheimer molecular dynamics simulations Pulay and Fogarasi
(2004); Niklasson et al. (2006); Kühne et al. (2006). In this article we go
one step further, and in analogy to time-dependent techniques such as
Ehrenfest molecular dynamics Ehrenfest (1927); Alonso (2008); Jakowski (2009)
or the Car-Parrinello method Car and Parrinello (1985); Marx and Hutter
(2000); Tuckerman (2002); Hartke (1992); Schlegel (1992); Herbert (1992);
Kirchner (2012), we show how the electronic ground state optimization can be
circumvented fully without any noticeable reduction in accuracy in comparison
to “exact” Born-Oppenheimer molecular dynamics.
Our optimization-free dynamics is based on a reformulation of extended
Lagrangian Born-Oppenheimer molecular dynamics Niklasson (2008) in the limit
of vanishing self-consistent-field optimization. The method is presented
within a general free energy formulation that is valid also at finite
electronic temperatures and should be applicable to a broad class of
materials. In addition to the removal of the costly self-consistent-field
optimization we also demonstrate compatibility with low pre-factor linear
scaling electronic structure theory Goedecker (1999); Bowler and Miyazaki
(2012); Niklasson (2002). The combined scheme provides a very efficient,
energy conserving, low-complexity method for performing accurate quantum
molecular dynamics simulations.
## II Fast Quantum Mechanical Molecular Dynamics
### II.1 Born-Oppenheimer molecular dynamics
Born-Oppenheimer molecular dynamics based on density functional theory can be
described by the Lagrangian
${\cal L}^{\rm BO}({\bf
R},{\bf\dot{R}})=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}-U[{\bf R};\rho],$
(1)
where the potential energy,
$\begin{array}[]{l}{\displaystyle U[{\bf R};\rho]=2\sum_{i\in\rm
occ}\varepsilon_{i}-\frac{1}{2}\iint\frac{\rho({\bf r})\rho({\bf
r^{\prime}})}{|{\bf r^{\prime}-r}|}d{\bf r^{\prime}}d{\bf r}}\\\
~{}~{}~{}~{}~{}{\displaystyle-\int V_{\rm xc}[\rho]\rho({\bf r})d{\bf
r}+E_{\rm xc}[\rho]+E_{\rm zz}[{\bf R}],}\end{array}$ (2)
is calculated at the self-consistent electronic ground state density,
$\rho({\bf r})$, for the nuclear configuration ${\bf R}=\\{R_{k}\\}$ Parr and
Yang (1989); Dreizler and Gross (1990). Here, $\varepsilon_{i}$ are the
(doubly) occupied eigenvalues of the effective single-particle Kohn-Sham
Hamiltonian,
$H[\rho]=-\frac{1}{2}\nabla^{2}+V_{\rm n}({\bf R},{\bf r})+\int\frac{\rho({\bf
r^{\prime}})}{|{\bf r^{\prime}-r}|}d{\bf r^{\prime}}+V_{\rm xc}[\rho],$ (3)
where $V_{\rm xc}[\rho]$ is the exchange correlation potential, $V_{\rm
n}({\bf R},{\bf r})$ the external (nuclear) potential, and
$-\frac{1}{2}\nabla^{2}$ the kinetic energy operator. $E_{\rm zz}[{\bf R}]$ is
the electrostatic ion-ion repulsion and $E_{\rm xc}[\rho]$ the exchange
correlation energy.
If the electron density deviates from the ground state density $\rho$ by some
small amount $\delta\rho$, the error in the potential energy is essentially of
the order $\delta\rho^{2}$, depending on the particular formulation used for
calculating $U[{\bf R};\rho+\delta\rho]$ Harris (1985); Sutton et al. (1988);
Foulkes and Haydock (1989). However, since the Hellmann-Feynman theorem is
valid only at the ground state density, we do not have a simple expression for
the forces that avoids calculating derivatives of the electronic density,
$\partial(\rho+\delta\rho)/\partial R_{k}$. In practical calculations, the
accuracy of the potential energy can therefore not be expected to hold also
for the forces and a high degree of self-consistent-field convergence is
therefore typically required.
### II.2 Extended Lagrangian molecular dynamics
Here we outline how we can circumvent the self-consistent-field procedure in
Born-Oppenheimer molecular dynamics. Instead of recalculating the ground state
density before each force evaluation with an iterative optimization procedure,
the idea here is to use an auxiliary density $n({\bf r})$, as in extended
Lagrangian Born-Oppenheimer molecular dynamics Niklasson (2008); Niklasson et
al. (2009); Steneteg et al. (2010); Zheng et al. (2011); Niklasson et al.
(2011), which evolves through a harmonic oscillator centered around the ground
state density $\rho({\bf r})$. Based on a general free energy formulation of
extended Lagrangian Born-Oppenheimer molecular dynamics Niklasson et al.
(2011) in the limit of vanishing self-consistent-field optimization, we define
the extended Lagrangian:
$\begin{array}[]{l}{\displaystyle{\cal L}({\bf
R},{\bf\dot{R}},n,{\dot{n}})=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}-{\cal
U}[{\bf R};n]+T_{e}{\cal S}[{\bf R};n]}\\\
{\displaystyle+\frac{1}{2}\mu\int{\dot{n}}({\bf r})^{2}d{\bf
r}-\frac{1}{2}\mu\omega^{2}\int\left(\rho({\bf r})-n({\bf r})\right)^{2}d{\bf
r}}.\end{array}$ (4)
While the potential and entropy terms, ${\cal U}$ and ${\cal S}$, are well
defined at the ground state density Parr and Yang (1989), i.e. when $n=\rho$,
there are several different options when $n$ deviates from $\rho$, e.g. the
Harris-Foulkes functional Harris (1985); Sutton et al. (1988); Foulkes and
Haydock (1989). In a more general case, the potential energy and entropy term
may therefore also be determined by $n({\bf r})$ implicitly through an
additional function $\sigma[n({\bf r})]$, i.e. ${\cal U}[{\bf R};n]\equiv{\cal
U}[{\bf R};n,\sigma[n]]$ and ${\cal S}[{\bf R};n]\equiv{\cal S}[{\bf
R};n,\sigma[n]]$. Here $\sigma[n({\bf r})]$ is a temperature dependent density
given from the diagonal part of the real-space representation of the (doubly
occupied) density matrix, which is given through a Fermi-operator expansion
Parr and Yang (1989) of the effective single-particle Hamiltonian, $H[n]$,
i.e.
$\sigma({\bf r})\equiv\sigma[n({\bf
r})]=2\left.{\left(e^{\beta(H[n]-\mu_{0}I)}+1\right)^{-1}}\right\rvert_{\bf
r=r^{\prime}}.$ (5)
At zero electronic temperature the Fermi-operator expansion corresponds to a
step function with the step formed at the chemical potential, $\mu_{0}$. In
our Lagrangian above, $\mu$ and $\omega$ are fictitious mass and frequency
parameters of the harmonic oscillator and $\beta$ is the inverse electronic
temperature, i. e. $\beta=1/(k_{\rm B}T_{e})$. The purpose of the entropy-like
term ${\cal S}[{\bf R};n]$ is here to make the derived forces of our dynamics
variationally correct for a given entropy-independent density, $n({\bf r})$,
at any electronic temperature. This approach is different from the regular
formulation where the density is determined by the entropy through the
minimization of the electronic free energy functional Parr and Yang (1989);
Weinert and Davenport (1992); Wentzcovitch et al. (1992); Niklasson (2008b).
#### II.2.1 Equations of motion
The molecular trajectories corresponding to the extended free energy
Lagrangian ${\cal L}$ in Eq. (4) are determined by the Euler-Lagrange
equations of motion,
$\begin{array}[]{l}{\displaystyle
M_{k}{\ddot{R}}_{k}=\left.{-\frac{\partial{\cal U}[{\bf R};n]}{\partial
R_{k}}}\right\rvert_{n}+T_{e}\left.{\frac{\partial{\cal S}[{\bf
R};n]}{\partial R_{k}}}\right\rvert_{n}}\\\ {}{}\\\
{\displaystyle-\left.{\frac{\mu\omega^{2}}{2}\frac{\partial}{\partial
R_{k}}\int\left(\rho({\bf r})-n({\bf r})\right)^{2}d{\bf
r}}\right\rvert_{n}},\\\ \end{array}$ (6)
and
$\begin{array}[]{l}{\displaystyle\mu{\ddot{n}}({\bf
r})=\mu\omega^{2}\left(\rho({\bf r})-n({\bf r})\right)}\\\ {}{}\\\
{\displaystyle-\left.{\frac{\delta{\cal U}[{\bf R};n]}{\delta
n}}\right\rvert_{\bf R}+T_{e}\left.{\frac{\delta{\cal S}[{\bf R};n]}{\delta
n}}\right\rvert_{\bf R}},\end{array}$ (7)
where the partial derivatives are taken with respect to constant density, $n$,
or coordinates, ${\bf R}$. The limit $\mu\rightarrow 0$ gives us the equations
of motion of our extended Lagrangian dynamics,
$M_{k}{\ddot{R}}_{k}=-\left.{\frac{\partial{\cal U}[{\bf R};n]}{\partial
R_{k}}}\right\rvert_{n}+T_{e}\left.{\frac{\partial{\cal S}[{\bf
R};n]}{\partial R_{k}}}\right\rvert_{n}$ (8) ${\ddot{n}}({\bf
r})=\omega^{2}\big{(}\rho({\bf r})-n({\bf r})\big{)},$ (9)
where we have defined ${\cal S}[{\bf R};n]$ such that
$\left.{\frac{\delta{\cal U}[{\bf R};n]}{\delta n}}\right\rvert_{\bf
R}=T_{e}\left.{\frac{\delta{\cal S}[{\bf R};n]}{\delta n}}\right\rvert_{\bf
R}.$ (10)
As is shown in the Appendix (Sec. VII.1), the corresponding property for
$\partial{\cal S}/\partial R_{k}$ is also of importance for the calculation of
the Pulay force in Eq. (8). Notice that these equations still require a full
self-consistent field optimization, since the auxiliary density $n({\bf r})$
evolves around the ground state density $\rho({\bf r})$.
Since the nuclear degrees of freedom do not depend on the mass parameter $\mu$
in Eqs. (8) and (9), the total free energy,
$E_{\rm tot}=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}+{\cal U}[{\bf
R};n]-T_{e}{\cal S}[{\bf R};n],$ (11)
is a constant of motion in the limit of vanishing $\mu$. Moreover, if $E_{\rm
tot}$ is close to the exact ground state free energy for approximate densities
$n({\bf r})$, we can also expect that the forces of the extended Lagrangian
dynamics should be accurate.
The forces in Eq. (8) are calculated at the approximate, unrelaxed, density
$n({\bf r})$ using a Hellmann-Feynman-like expression, where the partial
derivatives are taken with respect to a constant density $n({\bf r})$. This is
possible only because $n({\bf r})$ appears as an independent dynamical
variable. In general, as mentioned above, this can not be assumed, since the
Hellmann-Feynman force expression is formally applicable only at the ground
density. A more detailed derivation of explicit force expressions is given in
the Appendix.
#### II.2.2 Entropy contribution
Depending on the particular functional form chosen for the potential energy
term, ${\cal U}({\bf R};n)$, we may not have access to a simple explicit
expression of ${\cal S}[{\bf R};n]$ that fulfills Eq. (10). In this case an
approximate entropy term has to be used. This has no effect on the dynamics in
Eqs. (8) and (9), since the forces remain exact by definition. An
approximation of the entropy term therefore only affects the estimate of the
constant of motion, $E_{\rm tot}$, in Eq. (11).
We have found that the regular expression for the electronic entropy Parr and
Yang (1989),
${\cal S}[{\bf R};n]=-2k_{\rm
B}\sum_{i}\left\\{f_{i}\ln(f_{i})+(1-f_{i})\ln(1-f_{i})\right\\},$ (12)
which formally is defined only at the ground state density, i.e. when
$n=\rho$, typically provides a highly accurate approximation also for
approximate densities as will be illustrated in the examples below. Here
$f_{i}$ are the occupation numbers of the states, i.e. the eigenvalues of the
density matrix in Eq. (5). These are determined by the Fermi-Dirac
distribution of the single-particle eigenvalues $\varepsilon_{i}$ of the
Hamiltonian $H[n]$, i.e.
$f_{i}=\left[e^{\beta(\varepsilon_{i}-\mu)}+1\right]^{-1}.$ (13)
By comparing the calculation of $E_{\rm tot}$ in Eq. (11) using the
approximate entropy term, ${\cal S}[{\bf R};n]$, in Eq. (12) to “exact”, fully
optimized, Born-Oppenheimer molecular simulations, we can estimate the
accuracy of our dynamics.
### II.3 Fast quantum mechanical molecular dynamics
As in extended Lagrangian Born-Oppenheimer molecular dynamics, the
irreversibility of regular Born-Oppenheimer molecular dynamics that is caused
by the self-consistent-field optimization, can be avoided, since the density
$n({\bf r})$ can be integrated using a reversible geometric integration
algorithm Leimkuhler and Reich (2004); Niklasson (2008); Niklasson et al.
(2009); Odell et al. (2009), e.g. the Verlet algorithm as in Eq. (16) below.
This prevents the unphysical drift in the energy and phase space of regular
Born-Oppenheimer molecular dynamics Pulay and Fogarasi (2004); Niklasson et
al. (2006); Kühne et al. (2006) and our dynamics will therefore exhibit long-
term stability of the free energy $E_{\rm tot}$ in Eq. (11).
A main problem so far is that we still need to calculate the self-consistent
ground state density $\rho({\bf r})$ in the integration of $n({\bf r})$ in Eq.
(9). Fortunately, various geometric integrations of the auxiliary density
$n({\bf r})$ in Eq. (9) are stable also for approximate ground state density
estimates of $\rho({\bf r})$, as long as the approximation of $\rho({\bf r})$
is at least infinitesimally closer to the exact ground state compared to
$n({\bf r})$. Using an integration time step of $\delta t$, this stability
holds if the value of the dimensionless variable $\kappa=\delta
t^{2}\omega^{2}$ is chosen to be appropriately small Niklasson et al. (2009);
Odell et al. (2009, 2011). For energy functionals that are convex in the
vicinity of the ground state density we may therefore replace $\rho({\bf r})$
in Eq. (9) by a linear combination $(1-c)n+c\sigma$ Dederichs and Zeller
(1983), which gives us the approximate equations of motion
$M_{k}{\ddot{R}}_{k}=-\left.{\frac{\partial{\cal U}[{\bf R};n]}{\partial
R_{k}}}\right\rvert_{n}+T_{e}\left.{\frac{\partial{\cal S}[{\bf
R};n]}{\partial R_{k}}}\right\rvert_{n},$ (14)
and
${\ddot{n}}({\bf r})=\omega^{2}\big{(}\sigma({\bf r})-n({\bf r})\big{)},$ (15)
where the constant $\omega^{2}$ has been rescaled by $c$. The Verlet
integration of Eq. (15), including a weak dissipation to avoid an accumulation
of numerical noise Niklasson et al. (2009); Steneteg et al. (2010),
${\displaystyle n_{t+\delta t}=2n_{t}-n_{t-\delta t}+\delta
t^{2}\omega^{2}\left(\sigma_{t}-n_{t}\right)+\alpha\sum_{k=0}^{K}c_{k}n_{t-k\delta
t},}$ (16)
is therefore stable if a sufficiently small positive value of $\kappa=\delta
t^{2}\omega^{2}$ is chosen Niklasson et al. (2009). Thus, without any self-
consistent-field optimization of $\rho({\bf r})$, the previously optimized
values of $\kappa$ in Ref. Niklasson et al. (2009); Odell et al. (2009, 2011)
should be rescaled by a positive factor $\leq 1$. Certain ill behaved (non-
convex) functionals with self-consistent-field instabilities Dederichs and
Zeller (1983) can not be treated in this framework.
The proposed molecular dynamics as given by Eqs. (14) and (15) is the central
result of this paper. The equations of motion do not involve any ground state
self-consistent-field optimization prior to the force evaluations and only one
single diagonalization or density matrix construction is required in each time
step. The frequency $\omega$ of the electronic density is well separated from
the nuclear vibrational oscillations. Using a value of $\delta
t\omega={\sqrt{\kappa}}\approx 1$ and an integration time step $\delta t$,
which is $\sim 1/15$ of the period of the nuclear motion, the frequencies
differ by a factor of 5. As will be demonstrated in the examples below, the
scheme is also fully compatible with linear scaling electronic structure
theory Goedecker (1999); Bowler and Miyazaki (2012). This compatibility is
crucial in order to simulate large systems. The removal of the costly ground
state optimization, in combination with low-complexity linear scaling solvers,
provide a computationally fast quantum mechanical molecular dynamics (Fast-
QMMD) that can match the fidelity and accuracy of regular Born-Oppenheimer
molecular dynamics.
There are several alternative approaches to derive or motivate the equations
of motion of the fast quantum mechanical molecular dynamics, Eqs. (14) and
(15), and details of the dynamics may vary depending on the choice of the
functional form of ${\cal U}({\bf R};n)$. However, the particular derivation
presented here is the most transparent and general approach that we have found
so far.
The equations of motion are given in terms of the electron density, but they
should be generally applicable to a large class of methods, such as Hartree-
Fock theory, which is analyzed in the Appendix (Sec. VII.1), or plane wave
pseudo-potential methods Steneteg et al. (2010). Here we will demonstrate our
fast quantum mechanical molecular dynamics scheme using self-consistent-charge
density functional tight-binding theory Sutton et al. (1988); Finnis et al.
(1998); Elstner et al. (1998); Finnis (2003), as implemented in the electronic
structure code latte Sanville et al. (2010), either with an orthogonal or a
non-orthogonal representation and both at zero and at finite electronic
temperatures. With this method we can easily reach the time and length scales
necessary to establish long-term energy conservation and linear scaling of the
computational cost. Details of the computational method and our particular
choices of ${\cal U}({\bf R};n)$ are given in the Appendix.
Figure 1: Total energy fluctuations, Eq. (11), using “exact” (4 SCF/step)
Born-Oppenheimer molecular dynamics (BOMD), and the fast quantum mechanical
molecular dynamics, Eqs. (14) and (15), (Fast-QMMD), with ($\tau>0$) or
without ($\tau=0$) thresholding applied in the low pre-factor linear scaling
solver Niklasson (2002). The simulations were performed with the molecular
dynamics program latte using self-consistent-charge density functional based
tight-binding theory in an orthogonal formulation at $T_{e}=0$, i.e. as in
Eqs. (57), (58) and (60).
Figure 2: Panel a) shows the x-plane phase space trajectory of a single carbon
atom (C) based on an “exact” (4 SCF/step) Born-Oppenheimer molecular dynamics
(BOMD, dashed line) and the fast quantum mechanical molecular dynamics (Fast-
QMMD, solid line). Panel b) shows the fluctuations in the net auxiliary charge
$n_{i}(t)$ and ground state charge $q_{i}(t)$ for the same carbon atom
($i$=C). The numerical threshold $\tau$ is applied in the linear scaling
solver Niklasson (2002). The simulations were performed with the program latte
using self-consistent-charge tight-binding theory in an orthogonal formulation
at zero electronic temperature, i.e. as in Eqs. (57), (58) and (60).
## III Examples
### III.1 Orthogonal representation
Table 1: Wall clock timings of the fast quantum mechanical molecular dynamics (Fast-QMMD) simulations in comparison to Born-Oppenheimer molecular dynamics (BOMD) (4 SCF/step), without ($\tau=0$) and with ($\tau>0$) a low pre-factor linear scaling solver for the density matrix Niklasson (2002) with threshold tolerance $\tau$. The program (latte in its orthogonal formulation at $T_{e}=0$) was executed on a single core of a 2.66 GHz Quad-Core Intel Xeon processor. Polyethene chain C100H202 | Efficiency
---|---
BOMD ($\tau=0$) | 7.5 s/step
Fast-QMMD ($\tau=0$) | 1.5 s/step
Fast-QMMD ($\tau=10^{-5}$) | 0.61 s/step
Liquid Methane (CH4)100 | Efficiency
BOMD ($\tau=0$) | 12.5 s/step
Fast-QMMD ($\tau=0$) | 2.5 s/step
Fast-QMMD ($\tau=10^{-5}$) | 0.35 s/step
Figure 1 shows the fluctuations in the total energy (kinetic plus potential)
using the fast quantum mechanical molecular dynamics, Eqs. (14) and (15), as
implemented in Eqs. (57), (58) and (60), and an “exact” Born-Oppenheimer
molecular dynamics Niklasson (2008), for liquid methane (density = 0.422
g/cm3) at room temperature. The calculations were performed with the latte
molecular dynamics program using periodic boundary conditions and an
integration time step of $\delta t=0.5$ fs. Since the molecular system is
chaotic, any infinitesimally small deviation will eventually lead to a
divergence between different simulations. However, even after hundreds of time
steps and over 300 fs of simulation time the total energy curves are virtually
on top of each other as is seen in the inset. The same remarkable agreement is
seen in Fig. 2, which shows the projected phase space of an individual carbon
atom and the fluctuations of its net charge. In this case the C atom was
displaced compared to the simulation in Fig. 1 to further enhance the charge
fluctuations.
### III.2 Linear scaling
The fast quantum mechanical molecular dynamics scheme is also stable in
combination with approximate linear scaling sparse matrix algebra Goedecker
(1999); Bowler and Miyazaki (2012). Using the recursive second order spectral
projection method for the construction of the density matrix Niklasson (2002)
with a numerical threshold, $\tau=10^{-5}$, below which all elements are set
to zero after each individual projection, we notice excellent accuracy and
stability in Fig. 1 without any systematic drift in the total energy.
Despite their high efficiency and low computational pre-factor compared to
alternative linear scaling electronic structure methods Rubensson and Rudberg
(2011), it has been argued that recursive purification algorithms are non-
variational and therefore incompatible with forces of a conservative system
Bowler and Miyazaki (2012), which is necessary for long-term energy
conservation. As is evident from Figs. 1 and 2, this is not a problem. The
graphs are practically indistinguishable from “exact” Born-Oppenheimer
molecular dynamics, without any signs of a systematic drift in the total
energy. The corresponding linear scaling compatibility with microcanonical
simulations was recently also demonstrated for self-consistent-field-optimized
extended Lagrangian Born-Oppenheimer molecular dynamics Cawkwell and Niklasson
(2012).
The gain in speed using the fast quantum mechanical molecular dynamics scheme
in comparison to Born-Oppenheimer molecular dynamics is illustrated by the
wall-clock timings shown in Tab. 1.
Figure 3: Total energy fluctuations, Eq. (11), using “exact” (4 SCF/step)
Born-Oppenheimer molecular dynamics (BOMD), and the fast quantum mechanical
molecular dynamics, Eqs. (14) and 15), (Fast-QMMD), with ($\tau=10^{-5}$) or
without ($\tau=0$) thresholding applied in the low pre-factor linear scaling
solver Niklasson (2002). The simulations were performed with the molecular
dynamics program latte in the non-orthogonal formulation at $k_{\rm B}T_{e}=0$
eV, i.e. as implemented in Eqs. (50), (51) and (55) with the entropy term
approximated by ${\cal S}=0$.
Figure 4: Total free energy fluctuations, Eq. (11), using “exact” (4 SCF/step)
Born-Oppenheimer molecular dynamics (BOMD), and the fast quantum mechanical
molecular dynamics, Eqs. (14) and (15), (Fast-QMMD). The simulations were
performed with the molecular dynamics program latte using the non-orthogonal
formulation at an electronic temperature of $k_{\rm B}T_{e}=0.5$ eV, i.e. as
implemented in Eqs. (50), (51) and (55) with the entropy term approximated by
Eq. (56).
### III.3 Non-orthogonal representation
For non-orthogonal representations at finite electronic temperatures, a Pulay
force term and a finite approximate entropy contribution to the total free
energy have to be included. Figures 3 and 4 illustrate the total energy
fluctuations for the fast quantum mechanical molecular dynamics simulations of
a hydrocarbon chain as implemented in latte using Eqs. (50), (51) and (55),
with the approximate entropy term in Eq. (56). The electronic temperature of
the examples in Figure 3 is set to zero, $k_{\rm B}T_{e}=0$ eV, and for the
examples in Fig. 4, $k_{\rm B}T_{e}=2$ eV. In the first time step the initial
nuclear temperature, $T_{\rm init}$, was set to $300$ K using a Gaussian
distribution of the velocities. Despite the approximation of $\rho$ in Eq.
(15) and the approximate estimate of the entropy contribution to the free
energy there is virtually no difference seen between the fast quantum
mechanical and the Born-Oppenheimer molecular dynamics simulations.
As in the orthogonal case, the non-orthogonal formulation of our fast quantum
mechanical molecular dynamics is fully compatible with linear scaling
complexity in the construction of the density matrix at $T_{e}=0$ K. In Fig. 3
the reduced complexity simulation shows no significant deviation from “exact”
Born-Oppenheimer molecular dynamics. At finite electronic temperatures, the
linear scaling construction of the Fermi operator Niklasson (2003, 2008b) has
not yet been implemented.
### III.4 Long-term stability and conservation of the total energy
To assess the long-term energy conservation and the stability we use a test
system comprised of 16 molecules of isocyanic acid, HNCO, at a density of 1.14
g cm-3. The system was first thermalized to a temperature of 300 K over a
simulation time of 12.5 ps by the rescaling of the nuclear velocities. The
simulations used an integration time step, $\delta t$, of 0.25 ps. The
simulations were performed using self-consistent tight-binding theory Sutton
et al. (1988); Finnis et al. (1998); Elstner et al. (1998); Finnis (2003) with
a non-orthogonal basis as implemented in latte, using Eqs. (50), (51) and (55)
with the entropy term approximated by Eq. (56).
Fast quantum mechanical molecular dynamics and “exact” Born-Oppenheimer
molecular dynamics simulations with 4 self-consistent field cycles per time
step were performed over 250,000 time steps (62.5 ps) with $T_{e}=0$ K and
$k_{\text{B}}T_{e}=0.5$ eV. The latter temperature is small with respect to
the HOMO-LUMO gap of HNCO, which is about 6.0 eV, yet the entropy term, Eq.
(12) or Eq. (56), contributes about 0.19 eV to the total energy owing to the
partial occupation of states in the vicinity of the chemical potential.
Trajectories computed at $T_{e}=0$ K with “exact” Born-Oppenheimer molecular
dynamics and the fast quantum mechanical molecular dynamics method without
($\tau=0$) and with ($\tau=10^{-5}$) linear scaling constructions of the
density matrix are presented in Fig. 5. The standard deviation of the
fluctuations of the total energy about its mean and an estimate of the level
of the systematic drift of the total energies are presented in Table 2. These
data show that the fast quantum mechanical molecular dynamics simulations
yield trajectories that are effectively indistinguishable from the “exact”
Born-Oppenheimer trajectories. Moreover, as was seen above, the fast quantum
mechanical molecular dynamics scheme appears to be fully compatible with
linear scaling construction of the density matrix and the resulting
approximate forces, since this trajectory differs from the “exact” Born-
Oppenheimer molecular dynamics trajectory only by a small-amplitude random-
walk of the total energy about its mean Cawkwell and Niklasson (2012). The
systematic drift in energy is several orders of magnitude smaller than in
previous attempts to combine linear scaling solvers with regular Born-
Oppenheimer molecular dynamics Mauri and Galli (1994); Horsfield et al.
(1996); Tushida (2008); Shimojo et al. (2008).
The trajectories computed with an electronic temperature corresponding to
$k_{\text{B}}T_{e}=0.5$ eV differ qualitatively from those computed with zero
electronic temperature. Figure 6 and Table 2 show that while the “exact” Born-
Oppenheimer trajectory conserves the free energy to an extremely high
tolerance over the duration of the simulation, the total free energy in the
fast quantum mechanical molecular dynamics simulation exhibit random-walk
behaviour about the mean value. Although the fast quantum mechanical molecular
dynamics simulations involve an approximate expression for the entropy, we
find that this alone cannot account for the level of fluctuations observed.
Instead, we have found that the rescaling of the $\kappa$ value in the
integration, Eq. (16), affects this random-walk. By changing the rescaling
factor to 3/4, instead of 1/2 as in all the other examples, the amplitude of
the random walk is significantly reduced. Nevertheless, the fast quantum
mechanical molecular dynamics trajectories at finite electronic temperature
exhibit systematic drifts in the total energy that are negligible and the
fluctuations of the total energy about the mean are of the same order as those
that arise from the application of the approximate linear scaling method at
$T_{e}=0$ K.
Figure 5: Total energy versus time for liquid isocyanic acid with a nuclear temperature of 300 K and $T_{e}=0$ K computed with “exact” Born-Oppenheimer MD and the Fast QMMD method with exact and approximate linear scaling density matrix constructions. The numerical threshold $\tau$ is applied in the linear scaling solver Niklasson (2002) below which all matrix elements are set to zero after each iteration. Figure 6: Total free energy versus time for liquid isocyanic acid with a nuclear temperature of 300 K and $k_{\text{B}}T_{e}=0.5$ eV computed with “exact” Born-Oppenheimer molecular dynamics (BOMD) and the fast quantum mechanical molecular dynamics (Fast-QMMD) method with $\kappa$ rescaled by 3/4 instead of 1/2. Table 2: Standard deviation, $\sigma$, of the total energy about its mean value and the upper bound of the systematic drift of the total energy, $E_{\text{drift}}$, computed from “exact” Born-Oppenheimer molecular dynamics (BOMD) and fast quantum mechanical molecular dynamics (Fast-QMMD) simulations of liquid isocyanic acid. The simulation were performed with the latte program in the non-orthogonal formulation, i.e. as implemented in Eqs. (50), (51) and (55) with the entropy term approximated by Eq. (56). $k_{\text{B}}T_{e}$ | | $\sigma$ | $E_{\text{drift}}$
---|---|---|---
(eV) | | ($\mu$eV) | ($\mu$eV/atom/ps)
| Fast-QMMD ($\tau=0$) | 0.315 | $5.10\times 10^{-3}$
0.0 | Fast-QMMD ($\tau=10^{-5}$) | 0.702 | 0.285
| BOMD (4 SCF/step) | 0.358 | $9.94\times 10^{-3}$
| Fast-QMMD $(1/2)\kappa$ | 2.47 | 1.43
0.5 | Fast-QMMD $(3/4)\kappa$ | 0.786 | $7.85\times 10^{-2}$
| BOMD (4 SCF/step) | 0.361 | $8.50\times 10^{-2}$
## IV Convergence properties
Figure 7: The root mean square deviation (RMSD) between the fast quantum
mechanical molecular dynamics, Eqs. (14)-(15), and “exact” (4 SCF/step) Born-
Oppenheimer molecular dynamics, for the nuclear forces, the net Mulliken
charges and the total energy for a Naphthalene molecule at room temperature.
The simulation were performed with the latte molecular dynamics program using
self-consistent-charge tight-binding theory in an orthogonal formulation at
$T_{e}=0$, i.e. as implemented in Eqs. (57), (58) and (60) with ${\cal S}=0$..
The fast quantum mechanical molecular dynamics scheme, Eqs. (14) and (15), can
also be analyzed in terms of the convergence to “exact” Born-Oppenheimer
molecular dynamics as a function of the finite integration time step $\delta
t$. By comparing the deviation in forces, net Mulliken charges, and the total
energy, between the fast quantum mechanical molecular dynamics scheme and an
“exact” Born-Oppenheimer molecular dynamics as a function of $\delta t$ we can
study the consistency between the two methods. Figure 7 shows the difference
between a fully converged “exact” Born-Oppenheimer molecular dynamics
simulation and the fast quantum mechanical molecular dynamics scheme as
measured by the root mean square deviation over 200 fs of simulation time. We
find that the deviation of the nuclear forces, the charges $\\{q_{i}\\}$, as
well as the total energy difference are of the order $\delta t^{2}$ with a
small pre-factor. This convergence demonstrates a consistency between the fast
quantum mechanical scheme and Born-Oppenheimer molecular dynamics using Verlet
integration, where the optimization-free scheme behaves as a well controlled
and tunable approximation. As in “exact” Born-Oppenheimer molecular dynamics,
the dominating error is governed by the local truncation error arising from
the choice of finite integration time step $\delta t$, which is much larger
than any difference between the fast quantum mechanical molecular dynamics and
Born-Oppenheimer molecular dynamics.
## V Summary and Conclusions
Based on a free energy formulation of extended Lagrangian Born-Oppenheimer
molecular dynamics in the limit of vanishing self-consistent-field
optimization, we have derived and demonstrated a fast quantum mechanical
molecular dynamics scheme, Eqs. (14) and (15), which with a high precision can
match the accuracy and fidelity of Born-Oppenheimer molecular dynamics. In
addition to the removal of the self-consistent-field optimization we have also
demonstrated compatibility with low pre-factor linear scaling solvers. The
combined scheme provides a very efficient, energy conserving, low-complexity
method to perform accurate quantum molecular dynamics simulations. Our
findings show how the computational gap between classical and quantum
mechanical molecular dynamics simulations can be reduced significantly.
## VI Acknowledgment
We acknowledge support by the United States Department of Energy Office of
Basic Energy Sciences and the LANL Laboratory Directed Research and
Development Program. Discussions with E. Chisolm, J. Coe, T. Peery, S.
Niklasson, C. Ticknor, C.J. Tymczak, and G. Zheng, as well as stimulating
contributions at the T-Division Ten Bar Java group are gratefully
acknowledged. LANL is operated by Los Alamos National Security, LLC, for the
NNSA of the U.S. DOE under Contract No. DE-AC52-06NA25396.
## VII Appendix
### VII.1 Calculating the forces in Hartree-Fock theory
Here we present some details of the fast quantum mechanical molecular
dynamics, Eqs. (14) and (15), using a simple but general Hartree-Fock
formalism, which should be directly applicable to a broad class of hybrid and
semi-empirical electronic structure schemes. Instead of the auxiliary density
variable $n({\bf r})$ we will here use the more general density matrix $P$. In
this formalism the extended free-energy Lagrangian in Eq. (4) is given by
$\begin{array}[]{l}{\displaystyle{\cal L}({\bf
R},{\bf\dot{R}};P,{\dot{P}})=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}-{\cal
U}[{\bf R};P]+T_{e}{\cal S}[{\bf R};P]}\\\ {\displaystyle+\frac{1}{2}\mu
Tr[{\dot{P}}^{2}]-\frac{1}{2}\mu\omega^{2}Tr[({\mathcal{D}}_{\rm
gs}-P)^{2}],}\\\ \end{array}$ (17)
with the potential energy chosen as
${\cal U}[{\bf R};P]=2Tr[hD(P)]+Tr\\{D(P)G[D(P)]\\},$ (18)
and ground state (gs) density matrix ${\mathcal{D}}_{\rm gs}$. ${\cal S}[{\bf
R};P]$ is an unspecified electronic entropy term, which will be determined by
the requirement to make the derived forces variationally correct, and $T_{e}$
is the electronic temperature. The Fockian (or the effective single-particle
Hamiltonian) is
$F[P]=h+G[P],$ (19)
with the short-hand notation, $G[P]=2J[P]-K[P]$, where $J[P]$ and $K[P]$ are
the conventional Coulomb and exchange matrices, and $h$ is the matrix of the
one-electron part Roothaan (1951); McWeeny (1960). The temperature dependent
density matrix
$D(P)=Z\left(e^{\beta(F^{\perp}[P]-\mu_{0}I)}+1\right)^{-1}Z^{T},$ (20)
which corresponds to $\sigma[n]$ in Eq. (5), is given as a Fermi function of
the orthogonalized Fockian,
$F^{\perp}[P]=Z^{T}F[P]Z.$ (21)
Here $Z$ and its transpose $Z^{T}$ are the inverse Löwdin or Cholesky-like
factors of the overlap matrix, $S$, determined by the relation
$Z^{T}SZ=I.$ (22)
At zero electronic temperature ($T_{e}=0$ K) the Fermi-operator expansion in
Eq. (20) is given by the Heaviside step function, with the step formed at the
chemical potential $\mu_{0}$, separating the occupied from the unoccupied
states.
The Euler-Lagrange equations of motion of ${\cal L}$ in Eq. (17) are given by
$\begin{array}[]{l}{\displaystyle
M_{k}{\ddot{R}}_{k}=-\left.{\frac{\partial{\cal U}}{\partial
R_{k}}}\right\rvert_{P}+T_{e}\left.{\frac{\partial{\cal S}}{\partial
R_{k}}}\right\rvert_{P}}\\\ {}{}\\\
{\displaystyle-\left.{\frac{1}{2}\mu\omega^{2}\frac{\partial}{\partial
R_{k}}Tr[({\mathcal{D}}_{\rm gs}-P)^{2}]}\right\rvert_{P},}\end{array}$ (23)
and
$\mu{\ddot{P}}=\mu\omega^{2}({\mathcal{D}}_{\rm
gs}-P)-\left.{\frac{\partial{\cal U}}{\partial P}}\right\rvert_{\bf
R}+T_{e}\left.{\frac{\partial{\cal S}}{\partial P}}\right\rvert_{\bf R}.$ (24)
A cumbersome but fairly straightforward derivation (see Ref. Niklasson (2008b)
for a closely related example), using the relation and notation
$Z_{R_{k}}=\partial Z/\partial R_{k}=-(1/2)S^{-1}S_{R_{k}}Z$, and defining the
${\cal S}$ term such that
$T_{e}\left.{\frac{\partial{\cal S}}{\partial P_{ij}}}\right\rvert_{\bf
R}=2Tr\left[F^{\perp}[D]D^{\perp}_{P_{ij}}\right]$ (25)
and
$T_{e}\left.{\frac{\partial{\cal S}}{\partial
R_{k}}}\right\rvert_{P}=2Tr\left[F^{\perp}[D]D^{\perp}_{R_{k}}\right],$ (26)
gives the equations of motion
$\begin{array}[]{l}{\displaystyle
M_{k}{\ddot{R}}_{k}=-2Tr[h_{R_{k}}D]-Tr[DG_{R_{k}}(D)]}\\\ {}{}\\\
{\displaystyle+Tr[(DF[D]S^{-1}+S^{-1}F[D]D)S_{R}]}\\\ {}{}\\\
{\displaystyle-\left.{\frac{1}{2}\mu\omega^{2}\frac{\partial}{\partial
R_{k}}Tr[({\mathcal{D}}_{\rm gs}-P)^{2}]}\right\rvert_{P}},\end{array}$ (27)
and
$\mu{\ddot{P}}=\mu\omega^{2}({\mathcal{D}}_{\rm gs}-P).$ (28)
Notice that because of matrix symmetry $P_{ij}$ is not independent form
$P_{ji}$. The partial derivatives of matrix elements $P_{ij}$ are therefore
both over $P_{ij}$ and $P_{ji}$. In the limit $\mu\rightarrow 0$, we get the
final equations of motion for the fast quantum mechanical molecular dynamics
scheme,
$\begin{array}[]{l}{\displaystyle
M_{k}{\ddot{R}}_{k}=-2Tr\left[h_{R_{k}}D\right]-Tr\left[DG_{R_{k}}(D)\right]}\\\
{}{}\\\
{\displaystyle+Tr\left[(DF[D]S^{-1}+S^{-1}F[D]D)S_{R_{k}}\right]},\end{array}$
(29) ${\ddot{P}}=\omega^{2}\left(D(P)-P\right),$ (30)
where we have included the substitution of ${\mathcal{D}}_{\rm gs}$ with
$D(P)$ in the same way as in Eq. (15), i.e. with $\omega^{2}$ rescaled by a
constant $c\leq 1$. The notation for the partial derivative of the two-
electron term is defined as $G_{R_{k}}(D)=\left.{\partial G(D)/\partial
R_{k}}\right\rvert_{D}$, i.e. under the condition of constant density matrix
$D$.
The last term in Eq. (29), which includes the basis-set dependence $S_{R_{k}}$
is the Pulay force term that here is given in a generalized form that is valid
also for non-idempotent density matrices at finite electronic tememperatures
Niklasson (2008b).
### VII.2 Approximate Entropy contribution
The ${\cal S}[{\bf R};P]$ term is defined such that the two conditions in Eqs.
(25) and (26) are fulfilled. At the self-consistent ground state density, i.e.
when $P=D={\mathcal{D}}_{\rm gs}$, both these conditions are automatically
satisfied by the corresponding regular ground state (gs) electronic entropy
contribution to the free energy Parr and Yang (1989),
$\begin{array}[]{l}{\displaystyle{\cal S}_{\rm gs}[{\bf R};P]={\cal S}_{\rm
gs}[{\bf R};D^{\perp}(P)]}\\\ {}{}\\\ {\displaystyle=-2k_{\rm
B}Tr[D^{\perp}\ln(D^{\perp})+(I-D^{\perp})\ln(I-D^{\perp})]},\end{array}$ (31)
where the relation between $D^{\perp}$ and $D$ is given by the congruence
transformation
$D=ZD^{\perp}Z^{T}.$ (32)
A related derivation is given in Ref. Niklasson (2008b). Using the approximate
estimate ${\cal S}_{\rm gs}[{\bf R};P]$ in Eq. (31) when $P$ and $D$ deviate
from the ground state gives,
$\begin{array}[]{l}{\displaystyle T_{e}\left.{\frac{\partial{\cal S}}{\partial
P_{ij}}}\right\rvert_{\bf
R}=2Tr\left[F^{\perp}[P]D^{\perp}_{P_{ij}}\right]},{}{}\\\ {\displaystyle
T_{e}\left.{\frac{\partial{\cal S}}{\partial
R_{k}}}\right\rvert_{P}=2Tr\left[F^{\perp}[P]D^{\perp}_{R_{k}}\right]},\\\
\end{array}$ (33)
which only approximately fulfills the conditions in Eqs. (25) and (26). It is
possible to show that the error is linear in $\delta P=D-P$ by a linearization
of $F^{\perp}[D]$ around $P$. Since $D(P)$ and $P$ can be assumed to be close
to the ground state, $\delta P$ is small. From the scaling result illustrated
in Fig. 7 the error should therefore be quadratic in the integration time
step, i.e. $\sim\delta t^{2}$. We may therefore approximate the total free
energy using ${\cal S}_{\rm gs}[{\bf R};P]$, which is zero at $T_{e}=0$ K.
However, for the exact formulation and derivation of the equations of motion,
Eqs. (29) and (30), the entropy contribution, $T_{e}{\cal S}[{\bf R};P]$, is
unknown, both at finite and zero temperatures. As is seen in the equations of
motion, Eqs. (29) and (30), this does not affect the forces or the dynamics,
only the estimate of the constant of motion,
$E_{\rm tot}=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}+{\cal U}[{\bf
R};P]-T_{e}{\cal S}[{\bf R};P],$ (34)
is approximated. By comparing the approximate $E_{\rm tot}$ to optimized
“exact” Born-Oppenheimer molecular dynamics simulations, the accuracy of the
dynamics can be estimated.
### VII.3 Alternative potential energy forms
As an alternative to the potential energy, ${\cal U}({\bf R};P)$, in Eq. (18)
we may chose other functional forms that are equivalent at the ground state,
i.e. when $P=D={\cal D}_{gs}$. By using the Harris-Foulkes-like relation
Harris (1985); Foulkes and Haydock (1989),
${\displaystyle Tr[DG(D)]\approx Tr[(2D-P)G(P)]},$ (35)
which has an error of second order in $\delta P=D-P$, we may, for example,
choose
${\cal U}[{\bf R};P]=2Tr[hD(P)]+Tr\\{[2D(P)-P]G(P)\\},$ (36)
as our potential energy term. In this case, the equations of motion at
$T_{e}=0$ corresponding to Eqs. (29) and (30) become
$\begin{array}[]{l}{\displaystyle
M_{k}{\ddot{R}}_{k}=-2Tr\left[h_{R_{k}}D\right]-Tr\\{[2D-P]G_{R_{k}}(P)\\}}\\\
{}{}\\\
{\displaystyle+Tr\left[(DF[P]S^{-1}+S^{-1}F[P]D)S_{R_{k}}\right]},\end{array}$
(37)
and
${\ddot{P}}=\omega^{2}\left(D-P\right),$ (38)
with the constant of motion
$\begin{array}[]{l}{\displaystyle E_{\rm
tot}=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}+2Tr[hD]}\\\ {}{}\\\
{\displaystyle+Tr\\{(2D-P)G(P)\\}-T_{e}{\cal S}({\bf R};P)}.\end{array}$ (39)
The entropy term that makes the nuclear forces variationally correct is here
fulfilled by the expression in Eq. (31). With this choice of potential our
dynamics only requires one Fockian or effective single particle Hamitonian
construction per time step. Unfortunately, the error in the Pulay force has
been found to be large compared to Eq. (29). The dynamics in Eqs. (37) and
(38) should therefore be used only for orthogonal representations, i.e. when
the overlap matrix $S=I$.
### VII.4 Self-Consistent-Charge Density Functional Tight-Binding Theory
In self-consistent-charge density functional based tight-binding theory Sutton
et al. (1988); Finnis et al. (1998); Elstner et al. (1998); Finnis (2003) the
continuous electronic density, $\sigma({\bf n})$, or the density matrix,
$D(P)$, in Eq. (18) is replaced by the net Mulliken charges ${\bf q}[{\bf
n}]=\\{q_{i}\\}$ for each atom $i$, where ${\bf n}=\\{n_{i}\\}$ are the
dynamical variables corresponding to $P$. The potential energy functional
${\cal U}$ in Eq. (18) is then reduced to
${\cal U}[{\bf R};{\bf n}]=2\sum_{i\in\rm
occ}\varepsilon_{i}-\frac{1}{2}\sum_{i,j}q_{i}({\bf n})q_{j}({\bf
n})\gamma_{ij}+E_{\rm pair}[{\bf R}].$ (40)
Here $\varepsilon_{i}$ are the (doubly) occupied eigenvalues of the charge
dependent effective single-particle Hamiltonian
$\begin{array}[]{l}{\displaystyle H_{i\alpha,j\beta}[{\bf
n}]=h_{i\alpha,j\beta}}\\\ {}{}\\\
{\displaystyle+(1/2)\sum_{k\beta^{\prime}}\left(S_{i\alpha,k\beta^{\prime}}V^{ee}_{k\beta^{\prime},j\beta}+V^{ee}_{i\alpha,k\beta^{\prime}}S_{k\beta^{\prime},j\beta}\right)}\end{array}$
(41)
where
$V^{ee}_{j\beta,k\beta^{\prime}}=\sum_{l}q_{l}({\bf
n})\gamma_{jl}\delta_{jk}\delta_{\beta^{\prime}\beta},$ (42)
$h_{i\alpha,j\beta}$ is a parameterized Slater-Koster tight-binding
Hamiltonian, $S_{i\alpha,j\beta}$ the overlap matrix, $i$ and $j$ are atomic
indices and $\alpha$ and $\beta$ are orbital labels Sanville et al. (2010).
The net Mulliken charges are given by
$q_{i}[{\bf n}]\ =2\sum_{\alpha\in
i}\left(\varrho^{\perp}_{i\alpha,i\alpha}-{\varrho^{0}}^{\perp}_{i\alpha,i\alpha}\right),$
(43)
with the density matrix
$\varrho^{\perp}=\varrho^{\perp}[{\bf n}]=\left(e^{\beta(H^{\perp}[{\bf
n}]-\mu_{0}I)}+1\right)^{-1},$ (44)
using the orthogonalized Hamiltonian
$H^{\perp}[{\bf n}]=Z^{T}H[{\bf n}]Z.$ (45)
Here ${\varrho^{0}}$ is the density matrix of the corresponding separate non-
interacting atoms. The de-orthogonalized density matrix is
$\varrho=\varrho[{\bf n}]=Z\varrho^{\perp}[{\bf n}]Z^{T},$ (46)
and as above, the congruence transformation factors are defined through
$Z^{T}SZ=I,$ (47)
where $S$ is the basis set overlap matrix.
The electron-electron interaction in Eq. (40) is determined by $\gamma_{ij}$,
which decays like $1/R$ at large distances and equals the Hubbard repulsion
for the on-site interaction. $E_{\rm pair}[{\bf R}]$ is a sum of pair
potentials, $\phi(R)$, that provide short-range repulsion. The radial
dependence, $\zeta(R)$, of the Slater-Koster bond integrals, elements of the
overlap matrix, and the $\phi(R)$ are all represented analytically in latte by
the mathematically convenient form,
$\zeta(R)=A_{0}\prod_{i=1}^{4}\exp{(A_{i}R^{i})},$ (48)
where $A_{0}$ to $A_{4}$ are adjustable parameters that are fitted to the
results of quantum chemical calculations on small molecules. To ensure that
the off-diagonal elements of $h$ and $S$ and the $\phi(R)$ in our self-
consistent tight-binding implementation decay smoothly to zero at a specified
distance, $R_{\text{cut}}$, we replace the $\zeta(R)$ by cut-off tails of the
form,
$t(R)=B_{0}+\Delta R(B_{1}+\Delta R(B_{2}\\\ +\Delta R(B_{3}+\Delta
R(B_{4}+\Delta RB_{5}))))$ (49)
at $R=R_{1}$, where $\Delta R=R-R_{1}$ and $B_{0}$ to $B_{5}$ are adjustable
parameters. The adjustable parameters are parameterized to match the value and
first and second derivatives of $t(R)$ and $\zeta(R)$ at $R=R_{1}$ and to set
the value and first and second derivatives of $t(R)$ to zero at
$R=R_{\text{cut}}$.
#### VII.4.1 Non-orthogonal representation at $T_{e}\geq 0$
The fast quantum mechanical molecular dynamics scheme, Eqs. (29) and (30) or
Eqs. (8) and (9), using self-consistent tight-binding theory in its non-
orthogonal formulation is given by
$\begin{array}[]{l}{\displaystyle M_{k}{\ddot{R}}_{k}=-2Tr\left[\varrho
H_{R_{k}}\right]}\\\ {}{}\\\
{\displaystyle+\frac{1}{2}\sum_{i,j}q_{i}q_{j}\frac{\partial\gamma_{ij}}{\partial
R_{k}}+\sum_{i,j}q_{i}\gamma_{ij}\left.{\frac{\partial q_{j}}{\partial
R_{k}}}\right\rvert_{\varrho}}\\\ {}{}\\\ {\displaystyle+Tr[(S^{-1}H[{\bf
q}]\varrho+\varrho H[{\bf q}]S^{-1})S_{R_{k}}]-\frac{\partial E_{\rm
pair}[{\bf R}]}{\partial R_{k}},}\\\ \end{array}$ (50)
and
${\ddot{n}}_{i}=\omega^{2}\left(q_{i}-n_{i}\right),$ (51)
where
$H_{R_{k}}=\left.{\frac{\partial H}{\partial R_{k}}}\right\rvert_{\varrho}$
(52)
and
$S_{R_{k}}=\frac{\partial S}{\partial R_{k}}.$ (53)
The partial derivatives of $q_{j}$ and $H$ in Eqs. (50) and (52) are with
respect to a constant density matrix $\varrho$ in its non-orthogonal form,
i.e. including an $S$ dependence of $q_{j}$,
$\left.{\frac{\partial q_{j}}{\partial
R_{k}}}\right\rvert_{\varrho}=2\sum_{\alpha\in j}\left(\varrho
S_{R_{k}}\right)_{j\alpha,j\alpha}.$ (54)
The total energy is given by
$\begin{array}[]{l}{\displaystyle E_{\rm
tot}=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}+2\sum_{i\in\rm
occ}\varepsilon_{i}}\\\ {}{}\\\
{\displaystyle-\frac{1}{2}\sum_{i,j}q_{i}q_{j}\gamma_{ij}+E_{\rm pair}[{\bf
R}]-T_{e}{\cal S}[{\bf R};{\bf n}]},\end{array}$ (55)
with the entropy contribution to the free energy approximated by
${\cal S}[{\bf R};n]\approx-2k_{\rm
B}\sum_{i}\left\\{f_{i}\ln(f_{i})+(1-f_{i})\ln(1-f_{i})\right\\}.$ (56)
Here $f_{i}=f_{i}[n]$ are the eigenstates of the Fermi operator expansion
$\varrho^{\perp}[n]$ of $H^{\perp}[{\bf n}]$ in Eq. (44).
#### VII.4.2 Orthogonal representation at $T_{e}=0$
For orthogonal formulations, i.e. when $S=I$, and at zero electronic
temperature, $T_{e}=0$, we will base our dynamics on the equations of motion
in Eqs. (37) and (38). In this case the fast quantum mechanical molecular
dynamics scheme, Eqs. (50)-(51), is given by
$\begin{array}[]{l}{\displaystyle M_{k}{\ddot{R}}_{k}=-2Tr\left[\varrho
H_{R_{k}}\right]+\frac{1}{2}\sum_{i,j}\left(n_{i}n_{j}\frac{\partial\gamma_{ij}}{\partial
R_{k}}\right)}\\\ {\displaystyle-\frac{\partial E_{\rm pair}[{\bf
R}]}{\partial R_{k}}},\end{array}$ (57)
${\ddot{n}}_{i}=\omega^{2}\left(q_{i}-n_{i}\right),$ (58)
where
$\\{H_{R_{k}}[{\bf n}]\\}_{i\alpha,j\beta}=\frac{\partial
h_{i\alpha,j\beta}}{\partial
R_{k}}+\sum_{l}n_{l}\frac{\partial\gamma_{il}}{\partial
R_{k}}\delta_{ij}\delta_{\alpha\beta}.$ (59)
The density matrix is given directly from the step function of the
Hamiltonian, $\varrho=\theta(\mu_{0}I-H[{\bf n}])$, without any de-
orthogonalization that requires the calculation of the inverse factorization
of the overlap matrix, Eq. (47). The constant of motion, $E_{\rm tot}$, is
approximate by
$\begin{array}[]{l}{\displaystyle E_{\rm
tot}=\frac{1}{2}\sum_{k}M_{k}{\dot{R}}_{k}^{2}+2\sum_{i\in\rm
occ}\varepsilon_{i}}\\\ {}{}\\\
{\displaystyle-\frac{1}{2}\sum_{i,j}(2n_{i}-q_{i})n_{j}\gamma_{ij}+E_{\rm
pair}[{\bf R}]}.\end{array}$ (60)
#### VII.4.3 General remarks
Apart from the first few initial molecular dynamics time steps, where we apply
a high degree of self-consistent-field convergence and set ${\bf n=q}$, no
ground state self-consistent-field optimization is required. The density
matrix, $\varrho$, and the Hamiltonian, $H$, necessary in the force
calculations (and for the total energy) are calculated only once per time step
in the orthogonal case with one additional construction of the Hamiltonian
required in non-orthogonal simulations. The numerical integration of the
equations of motion in Eq. (14) is performed with the velocity Verlet scheme
and in Eq. (15) with the modified Verlet scheme in Eq. (16) as described in
Ref. Niklasson et al. (2009). For the examples presented here we used the
modified Verlet scheme including dissipation ($\alpha>0$) with $K=5$ and the
constant $\kappa=\delta t^{2}\omega^{2}$ as given in Ref. Niklasson et al.
(2009) was rescaled by a factor $1/2$ in all examples except for one of the
test cases in Fig. 6.
## References
* Karplus and McCammon (2002) M. Karplus and J. A. McCammon, Nat. Struct. Biol. 9, 646 (2002).
* Marx and Hutter (2000) D. Marx and J. Hutter, _Modern Methods and Algorithms of Quantum Chemistry_ (ed. J. Grotendorst, John von Neumann Institute for Computing, Jülich, Germany, 2000), 2nd ed.
* Wang and Karplus (1973) I. S. Y. Wang and M. Karplus, J. Am. Chem. Soc. 95, 8160 (1973).
* Leforestier (1978) C. Leforestier, J. Chem. Phys. 68, 4406 (1978).
* Roothaan (1951) C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951).
* McWeeny (1960) R. McWeeny, Rev. Mod. Phys. 32, 335 (1960).
* Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B:864 (1964).
* Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. B 140, A1133 (1965).
* Parr and Yang (1989) R. G. Parr and W. Yang, _Density-functional theory of atoms and molecules_ (Oxford University Press, Oxford, 1989).
* Dreizler and Gross (1990) R. M. Dreizler and K. U. Gross, _Density-functional theory_ (Springer Verlag, Berlin Heidelberg, 1990).
* Feynman (1939) R. P. Feynman, Phys. Rev. 56, 367 (1939).
* Pulay and Fogarasi (2004) P. Pulay and G. Fogarasi, Chem. Phys. Lett. 386, 272 (2004).
* Niklasson et al. (2006) A. M. N. Niklasson, C. J. Tymczak, and M. Challacombe, Phys. Rev. Lett. 97, 123001 (2006).
* Kühne et al. (2006) T. D. Kühne, M. Krack, F. R. Mohamed, and M. Parrinello, Phys. Rev. Lett. 98, 066401 (2006).
* Ehrenfest (1927) P. Ehrenfest, Z. Phys. 45, 455 (1927).
* Alonso (2008) J. L. Alonso, X. Andrade, P. Echenique, F. Falceto, D. Prada-Garcia, A. Rubio, Phys. Rev. Lett. 101, 096403 (2008).
* Jakowski (2009) J. Jakowski, and K. Morokuma, J. Chem. Phys. 130, 224106 (2009).
* Car and Parrinello (1985) R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).
* Tuckerman (2002) M. Tuckerman, J. Phys.:Condens. Matter 50, 1297 (2002).
* Hartke (1992) B. Hartke, and E. A. Carter, Chem. Phys. Lett. 189, 358 (1992).
* Schlegel (1992) H. B. Schlegel, J. M. Millam, S. S. Iyengar, G. A. Voth, A. D. Daniels, G. Scusseria, and M. J. Frisch, J. Chem. Phys. 114, 9758 (2001).
* Herbert (1992) J. Herbert, and M. Head-Gordon, J. Chem. Phys. 121, 11542 (2004).
* Kirchner (2012) B. Kirchner J. di Dio Philipp, and J. Hutter, Top. Curr. Chem. 307, 109 Springer Verlag, Berlin Heidelberg, (2012).
* Goedecker (1999) S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999).
* Bowler and Miyazaki (2012) D. R. Bowler and T. Miyazaki, Rep. Prog. Phys. 75, 036503 (2012).
* Niklasson (2002) A. M. N. Niklasson, Phys. Rev. B 66, 155115 (2002).
* Cawkwell and Niklasson (2012) M. J. Cawkwell and A. M. N. Niklasson, J. Chem. Phys. (accepted for publication).
* Niklasson (2003) A. M. N. Niklasson, Phys. Rev. B 68, 233104 (2003).
* Mauri and Galli (1994) F. Mauri, and G. Galli, Phys. Rev. B 50, 4316 (1994).
* Horsfield et al. (1996) A. P. Horsfield, A. M. Bratkovsky, M. Fern, D. G. Pettifor, and M. Aoki, Phys. Rev. B 54, 12694 (1996).
* Tushida (2008) E. Tushida, J. Phys.: Condens. Matter 20, 294212 (2008).
* Shimojo et al. (2008) F. Shimojo, R. K. Kalia, A. Nakono, and P. Vashista, Phys. Rev. B 77, 085103 (2008).
* Harris (1985) J. Harris, Phys. Rev. B 31, 1770 (1985).
* Foulkes and Haydock (1989) W. M. C. Foulkes and R. Haydock, Phys. Rev. B 39, 12520 (1989).
* Niklasson (2008) A. M. N. Niklasson, Phys. Rev. Lett. 100, 123004 (2008).
* Niklasson et al. (2009) A. M. N. Niklasson, P. Steneteg, A. Odell, N. Bock, M. Challacombe, C. J. Tymczak, E. Holmström, G. Zheng, and V. Weber, J. Chem. Phys. 130, 214109 (2009).
* Steneteg et al. (2010) P. Steneteg, I. A. Abrikosov, V. Weber, and A. M. N. Niklasson, Phys. Rev. B 82, 075110 (2010).
* Zheng et al. (2011) G. Zheng, A. M. N. Niklasson, and M. Karplus, J. Chem. Phys. 135, 044122 (2011).
* Niklasson et al. (2011) A. M. N. Niklasson, P. Steneteg, and N. Bock, J. Chem. Phys. 135, 164111 (2011).
* Weinert and Davenport (1992) M. Weinert and J. W. Davenport, Phys. Rev. B 45, R13709 (1992).
* Wentzcovitch et al. (1992) R. M. Wentzcovitch, J. L. Martins, and P. B. Allen, Phys. Rev. B 45, R11372 (1992).
* Niklasson (2008b) A. M. N. Niklasson, J. Chem. Phys. 129, 244107 (2008b).
* Dederichs and Zeller (1983) P. H. Dederichs and R. Zeller, Phys. Rev. B 28, 5262 (1983).
* Leimkuhler and Reich (2004) B. Leimkuhler and S. Reich, _Simulating Hamiltonian Dynamics_ (Cambridge University Press, 2004).
* Odell et al. (2009) A. Odell, A. Delin, B. Johansson, N. Bock, M. Challacombe, and A. M. N. Niklasson, 131, 244106 (2009), J. Chem. Phys.
* Odell et al. (2011) A. Odell, A. Delin, B. Johansson, M. Cawkwell, and A. M. N. Niklasson, 135, 224105 (2011), J. Chem. Phys.
* Sutton et al. (1988) A. P. Sutton, M. W. Finnis, D. G. Pettifor, and Y. Ohta, J. Phys. C: Solid State Phys. 21, 35 (1988).
* Finnis et al. (1998) M. W. Finnis, A. T. Paxton, M. Methfessel, and M. van Schilfgaarde, Phys. Rev. Lett. 81, 5149 (1998).
* Elstner et al. (1998) M. Elstner, D. Poresag, G. Jungnickel, J. Elstner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Phys. Rev. B 58, 7260 (1998).
* Finnis (2003) M. Finnis, _Interatomic forces in condensed matter_ (Oxford University Press, 2003).
* Sanville et al. (2010) E. Sanville, N. Bock, W. M. Challacombe, A. M. N. Niklasson, M. J. Cawkwell, D. M. Dattelbaum, and S. Sheffield, _Proceedings of the Fourteenth International Detonation Symposium_ (Office of Naval Research, Arlington VA, ONR-351-10-185, 2010), pp. 91–101.
* Rubensson and Rudberg (2011) E. H. Rubensson and E. Rudberg, J. Phys.: Condes. Matter 23, 075502 (2011).
|
arxiv-papers
| 2012-03-30T15:32:34 |
2024-09-04T02:49:29.211825
|
{
"license": "Public Domain",
"authors": "Anders M. N. Niklasson and Marc J. Cawkwell",
"submitter": "Anders Niklasson",
"url": "https://arxiv.org/abs/1203.6836"
}
|
1203.6903
|
# Early publications about nonzero cosmological constant
I. Horvath11affiliation: Dept. of Physics, Bolyai Military University,
Budapest, POB 15, H-1581, Hungary
Dept. of Physics, Bolyai Military University, Budapest, POB 15, H-1581,
Hungary horvath.istvan@uni-nke.hu horvath.istvan@uni-nke.hu
###### Abstract
In 2011 the Nobel Prize in Physics was awarded for the 1998 discovery of the
nonzero cosmological constant. This discovery is very important and surely
worth to receive the Nobel Prize. However, years earlier several papers had
been published (Paál, Horváth, & Lukács 1992; Holba et al. 1992, Holba et al.
1994) about a very similar discovery from observational data.
History of astronomy; Cosmology: cosmological parameters — dark energy — large
scale structure of the Universe
ieeetr
## 1 2011 Nobel Prize Winners in Physics
”The Nobel Prize in Physics 2011 was divided, one half awarded to Saul
Perlmutter, the other half jointly to Brian P. Schmidt and Adam G. Riess for
the discovery of the accelerating expansion of the Universe through
observations of distant supernovae.” the nobelprize.org wrote.
111http://www.nobelprize.org/nobelprizes/physics/laureates/2011/
Two research teams, Supernova Cosmology Project (SCP) lead by Saul Perlmutter
and High-z Supernova Search Team (HZT) headed by Brian Schmidt, raced to map
the Universe by locating the most distant supernovae. The two research teams
found over 50 distant supernovae which were dimmer than expected - this was a
sign that the expansion of the Universe was accelerating [1], [2].
Theoretically, this was not new idea since Albert Einstein came out the idea
of cosmological constant (often marked Lambda). Einstein did that because he
was guided by the paradigm of the day that the Universe was static. When he
heard the Edwin Hubble discovery that the Universe was actually expanding he
declared that the inclusion of the cosmological constant was his ”biggest
blunder”. Since there was no observation for cosmological constant after that
most scientists assumed that Lambda is zero. A series of papers were published
in Astrophysics and Space Science in the early 90’s years before the supernova
publications calculated $\Omega_{\Lambda}$ from observed data.
## 2 Earlier Publications About Positive Cosmological Constant
Paál et al. [3] used the so called pencil beam survey [4] to find out whether
the regularity found in the galaxy distribution is quasiperiodical or not. It
was found that $q_{0}$ was preferably negative [3]. Therefore a nonzero Lambda
term was needed. The preferred value was $\Omega_{\Lambda}$ equal 2/3. This is
very close to the value that later was observed by the Nobel Prize winners.
Figure 1: This figure was published in Holba et al. (1994) (figure 8. in that
article). The red line represents the flat cosmological model.
In the second paper [5] a two parameter fit was made. The positive
cosmological constant (negative $q_{0}$) found to be still needed. In the
third paper [6] optical and radio quasars were also used to find the preferred
cosmological parameters. Figure 8. in that paper [6] showed the results (see
figure). As it was written the contour meant 80% confidence level. The
preferred region is similar that the supernovae analisys later suggested. For
comparison please see this www page
http://vizion.galileowebcast.hu/HOI/Comparation.jpg Therefore those earlier
suggestions also support the Nobel Prize winning results.
The author thanks for his supervisors, B. Lukács and G. Paál, to being part of
these researches. Unfortunately, G. Paál died in 1992. This was surely
affected the fact that these results were almost unrecognized.
REFERENCES [1] Perlmutter, S. et al., 1999, ApJ, 517, 565 [2] Riess, A. G. et
al., 1998, AJ, 116, 1009 [3] Paál, G., Horváth, I., & Lukács, B. 1992, Ap&SS,
191, 107 [4] Broadhurst, T. J. et al., 1990, Nature, 343, 726 [5] Holba, A.,
Horváth, I., Lukács, B., & Paál, G. 1992, Ap&SS, 198, 111 [6] Holba, A.,
Horváth, I., Lukács, B., & Paál, G. 1994, Ap&SS, 222, 65
|
arxiv-papers
| 2012-03-28T16:28:02 |
2024-09-04T02:49:29.218725
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "I. Horvath",
"submitter": "Istvan Horvath",
"url": "https://arxiv.org/abs/1203.6903"
}
|
1204.0072
|
Generalized fuzzy rough sets based on fuzzy coverings
Authors: Guangming Langa, Qingguo Lia∗ and Lankun Guob
∗Corresponding author: liqingguoli@yahoo.com.cn
aCollege of Mathematics and Econometrics, Hunan University, Changsha, Hunan
410082, P.R. China
bCollege of Information Science and Engineering, Hunan University, Changsha,
Hunan 410082, P.R. China
Abstract. This paper further studies the fuzzy rough sets based on fuzzy
coverings. We first present the notions of the lower and upper approximation
operators based on fuzzy coverings and derive their basic properties. To
facilitate the computation of fuzzy coverings for fuzzy covering rough sets,
the concepts of fuzzy subcoverings, the reducible and intersectional elements,
the union and intersection operations are provided and their properties are
discussed in detail. Afterwards, we introduce the concepts of consistent
functions and fuzzy covering mappings and provide a basic theoretical
foundation for the communication between fuzzy covering information systems.
In addition, the notion of homomorphisms is proposed to reveal the
relationship between fuzzy covering information systems. We show how large-
scale fuzzy covering information systems and dynamic fuzzy covering
information systems can be converted into small-scale ones by means of
homomorphisms. Finally, an illustrative example is employed to show that the
attribute reduction can be simplified significantly by our proposed approach.
# Generalized fuzzy rough sets based on fuzzy coverings
Guangming Langa Qingguo Lia Lankun Guob
a College of Mathematics and Econometrics, Hunan University
Changsha, Hunan 410082, P.R. China
b College of Information Science and Engineering, Hunan University
Changsha, Hunan 410082, P.R. China Corresponding author. Tel./fax: +86 731
8822855, liqingguoli@yahoo.com.cn(Q. G. Li)
E-mail address: langguangming1984@126.com(G. M. Lang), lankun.guo@gmail.com(L.
K. Guo).
> Abstract. This paper further studies the fuzzy rough sets based on fuzzy
> coverings. We first present the notions of the lower and upper approximation
> operators based on fuzzy coverings and derive their basic properties. To
> facilitate the computation of fuzzy coverings for fuzzy covering rough sets,
> the concepts of fuzzy subcoverings, the reducible and intersectional
> elements, the union and intersection operations are provided and their
> properties are discussed in detail. Afterwards, we introduce the concepts of
> consistent functions and fuzzy covering mappings and provide a basic
> theoretical foundation for the communication between fuzzy covering
> information systems. In addition, the notion of homomorphisms is proposed to
> reveal the relationship between fuzzy covering information systems. We show
> how large-scale fuzzy covering information systems and dynamic fuzzy
> covering information systems can be converted into small-scale ones by means
> of homomorphisms. Finally, an illustrative example is employed to show that
> the attribute reduction can be simplified significantly by our proposed
> approach.
>
> Keywords: Rough set; Fuzzy covering; Information system; Homomorphism;
> Attribute reduction
>
## 1 Introduction
Rough set theory, originally constructed on the basis of an equivalence
relation, was proposed by Pawlak[17] for solving inexact or uncertain
problems. But the condition of the equivalence relation is so restrictive that
the applications of rough sets are limited in many practical problems. To deal
with more complex data sets, many researchers have derived a large number of
generalized models by replacing the equivalence relation with a few
mathematical concepts such as fuzzy relations[5, 6, 19, 20, 22, 14, 1] and
coverings[3, 4, 7, 8, 15, 18, 21, 23, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39]
of the universe of discourse.
Recently, the theory of fuzzy rough sets has become a rapidly developing
research area and got a lot of attention. For example, Dubois et al.[5, 6]
initially provided the rough fuzzy sets and the fuzzy rough sets. Then
Radzikowska et al.[19, 20] defined the fuzzy rough sets (respectively, the
L-fuzzy rough sets) based on fuzzy similarity relations (respectively,
residuated lattices). Afterwards, many researchers[15, 33, 4, 7] investigated
fuzzy rough sets based on fuzzy coverings. In practice, we need to compute the
approximations of fuzzy sets in fuzzy covering approximation spaces. But the
classical approximation operators based on coverings are incapable of
computing the approximations of fuzzy sets in the fuzzy covering approximation
space. It motivates us to extend approximation operators of covering
approximation spaces for fuzzy covering approximation spaces. In addition,
there are a large number of fuzzy coverings for the universal set in general.
To facilitate the computation of fuzzy coverings for fuzzy covering rough
sets, it is interesting to investigate the relationship among the elements of
a fuzzy covering and operations on fuzzy coverings.
Meanwhile, many researches[10, 12, 11, 16, 24, 25, 26, 27, 28, 9, 40, 41, 42]
have been conducted on homomorphisms between information systems with the aim
of attribute reductions. For instance, Grzymala-Busse[10, 12, 11] initially
introduced the concept of information system homomorphisms and investigated
its basic properties. Then Li et al.[16] discussed invariant characters of
information systems under some homomorphisms. Afterwards, Wang et al.[24]
found that a complex massive covering information system could be compressed
into a relatively small-scale one under the condition of a homomorphism, and
their attribute reductions are equivalent to each other. Actually, we often
deal with attribute reductions of large-scale fuzzy covering information
systems in practical situations, and the work of Wang et al. mentioned above
inspires that the attribute reduction of large-scale fuzzy covering
information systems may be conducted by means of homomorphisms. But so far we
have not seen any work on homomorphisms between fuzzy covering information
systems. Additionally, the fuzzy covering information system varies with time
due to the dynamic characteristics of data collection, and the non-incremental
approach to compressing the dynamic fuzzy covering information system is often
very costly or even intractable. For this issue, we attempt to apply an
incremental updating scheme to maintain the compression dynamically and avoid
unnecessary computations by utilizing the compression of the original system.
The purpose of this paper is to investigate further fuzzy coverings based
rough sets. First, we present the notions of the lower and upper approximation
operators based on fuzzy coverings by extending Zhu’s model[37], and examine
their basic properties. Particularly, we find that the upper approximation
based on neighborhoods can not be represented without using the neighborhoods
as the classical covering approximation space[37] in the fuzzy approximation
space. Second, we propose the concepts of fuzzy subcoverings, reducible and
intersectional elements, union and intersection operations and investigate
their basic properties in detail. Third, the theoretical foundation is
established for the communication between fuzzy covering information systems.
Concretely, we construct a consistent function by combining the fuzzy
covering, proposed by Deng et al. [4], with the approach in [24], and explore
its main properties known from the consistent function for the classical
covering approximation space in [24]. We also provide the concepts of fuzzy
covering mappings and study their basic properties in detail. Fourth, the
notion of homomorphisms between fuzzy covering information systems is
introduced for attribute reductions. We find that a large-scale fuzzy covering
information system can be compressed into a relatively small-scale one, and
attribute reductions of the original system and image system are equivalent to
each other under the condition of a homomorphism. In addition, we give the
algorithm to construct attribute reducts and employ an example to illustrate
the efficiency of our approach for attribute reductions of fuzzy covering
information systems. We also discuss how to compress the dynamic fuzzy
covering information system.
The rest of this paper is organized as follows: Section 2 briefly reviews the
basic concepts related to the covering information systems and fuzzy covering
information systems. In Section 3, we put forward some concepts such as the
neighborhood operators, the approximation operators and reducible elements for
fuzzy covering approximation spaces, and investigate their basic properties.
Section 4 is devoted to introducing the concept of consistent functions which
provides the theoretical foundation for the communication between fuzzy
covering information systems. In Section 5, we present the notion of
homomorphisms between fuzzy covering information systems and discuss its basic
properties. We also investigate data compressions of fuzzy covering
information systems and dynamic fuzzy covering information systems. An example
is given to illustrate that how to conduct attribute reductions of the fuzzy
covering information system by means of homomorphisms. We conclude the paper
and set further research directions in Section 6.
## 2 Preliminaries
In this section, we briefly recall some basic concepts related to the covering
information system and fuzzy covering information system. Three examples are
given to illustrate two types of covering information systems.
###### Definition 2.1
[2] Let $U$ be a non-empty set $($the universe of discourse$)$. A non-empty
sub-family $\mathscr{C}\subseteq\mathscr{P}(U)$ is called a covering of $U$ if
$(1)$ every element in $\mathscr{C}$ is non-empty;
$(2)$ $\bigcup\\{C|C\in\mathscr{C}\\}=U$, where $\mathscr{P}(U)$ is the
powerset of $U$.
It is clear that the concept of a covering is an extension of the notion of a
partition. In what follows, $(U,\mathscr{C})$ is called a classical covering
approximation space.
To investigate further coverings based rough sets, Chen et al. proposed the
following concepts on coverings.
###### Definition 2.2
[3] Let $\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ be a covering of $U$,
$C_{x}$=$\bigcap\\{C_{i}|x\in C_{i}\text{ and }C_{i}\in\mathscr{C}\\}$ for any
$x\in U$, and $Cov(\mathscr{C})$=$\\{C_{x}|x\in U\\}$. Then $Cov(\mathscr{C})$
is called the induced covering of $\mathscr{C}$.
Suppose $c$ is an attribute, the domain of $c$ is
$\\{c_{1},c_{2},...,c_{N}\\}$, $C_{i}$ means the set of objects in $U$ taking
a certain attribute value $c_{i}$, and $C_{x}=C_{i}\cap C_{j}$, it implies
that the possible value of $x$ regarding the attribute $c$ is $c_{i}$ or
$c_{j}$, and $C_{x}$ is the minimal set containing $x$ in $Cov(\mathscr{C})$.
###### Definition 2.3
[3] Let $\Delta$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$
be a family of coverings of $U$,
$\Delta_{x}$=$\bigcap\\{C_{ix}|C_{ix}\in\mathscr{C}_{i}$, $1\leq i\leq m\\}$
for any $x\in U$, and $Cov(\Delta)$=$\\{\Delta_{x}|x\in U\\}$. Then
$Cov(\Delta)$ is called the induced covering of $\Delta$.
That is to say, $\Delta_{x}$ is the intersection of all the elements including
$x$ of each $\mathscr{C}_{i}$, and it is the minimal set including $x$ in
$Cov(\Delta)$. Furthermore, $Cov(\Delta)$ is a partition if every covering in
$\Delta$ is a partition. In what follows, $(U,\Delta)$ is called a covering
information system. To illustrate how covering information systems are
constructed, we present two examples which have different application
backgrounds.
###### Example 2.4
Let $U=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ be eight houses,
$C=\\{price,color\\}$ the attribute set, the domains of $price$ and $color$
are $\\{high,middle,low\\}$ and $\\{good,bad\\}$, respectively. To evaluate
these houses, specialists $A$ and $B$ are employed and their evaluation
reports are shown as follows:
$\displaystyle high_{A}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{4},x_{5},x_{7}\\},middle_{A}=\\{x_{2},x_{8}\\},low_{A}=\\{x_{3},x_{6}\\};$
$\displaystyle high_{B}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\},middle_{B}=\\{x_{5}\\},low_{B}=\\{x_{3},x_{6}\\};$
$\displaystyle good_{A}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{3},x_{6}\\},bad_{A}=\\{x_{4},x_{5},x_{7},x_{8}\\};$
$\displaystyle good_{B}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{3},x_{5}\\},bad_{B}=\\{x_{4},x_{6},x_{7},x_{8}\\},$
where $high_{A}$ denotes the houses belonging to high price by the specialist
$A$. The meanings of other symbols are similar. Since their evaluations are of
equal importance, we should consider all their advice. Consequently, we obtain
the following results:
$\displaystyle high_{A\vee B}$ $\displaystyle=$ $\displaystyle high_{A}\cup
high_{B}=\\{x_{1},x_{2},x_{4},x_{5},x_{7},x_{8}\\};$ $\displaystyle
middle_{A\vee B}$ $\displaystyle=$ $\displaystyle middle_{A}\cup
middle_{B}=\\{x_{2},x_{5},x_{8}\\};$ $\displaystyle low_{A\vee B}$
$\displaystyle=$ $\displaystyle low_{A}\cup low_{B}=\\{x_{3},x_{6}\\};$
$\displaystyle good_{A\vee B}$ $\displaystyle=$ $\displaystyle good_{A}\cup
good_{B}=\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\};$ $\displaystyle bad_{A\vee B}$
$\displaystyle=$ $\displaystyle bad_{A}\cup
bad_{B}=\\{x_{4},x_{5},x_{6},x_{7},x_{8}\\}.$
Based on the above statement, we derive a covering information system
$(U,\Delta)$, where $\Delta=\\{\mathscr{C}_{price},$ $\mathscr{C}_{color}\\}$,
$\mathscr{C}_{price}=\\{high_{A\vee B},middle_{A\vee B},low_{A\vee B}\\}$ and
$\mathscr{C}_{color}=\\{good_{A\vee B},bad_{A\vee B}\\}$.
###### Example 2.5
Let Table 1 be an incomplete information system, where $\ast$ stands for the
lost value. According to the interpretation in [11], the lost value is
considered to be similar to any value in the domain of the corresponding
attribute. Consequently, we obtain three coverings of $U$ by the attribute set
as follows:
$\mathscr{C}_{structure}=\\{\\{x_{1},x_{2},x_{4},x_{6}\\},\\{x_{2},x_{3},x_{5},x_{6}\\}\\}$,
$\mathscr{C}_{color}=\\{\\{x_{1},x_{2},x_{5}\\},\\{x_{3},x_{4},x_{5},x_{6}\\}\\}$,
$\mathscr{C}_{price}=\\{\\{x_{1},x_{4},x_{5},x_{6}\\},\\{x_{2},x_{3},x_{4},x_{6}\\}\\}$.
Hence, $(U,\Delta)$ is a covering information system, where
$\Delta=\\{\mathscr{C}_{structure},\mathscr{C}_{color},$
$\mathscr{C}_{price}\\}$.
To conduct the communication between covering information systems, Wang et al.
provided the concept of consistent functions based on coverings.
###### Definition 2.6
[24] Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a covering of $U_{1}$,
$C_{x}$=$\bigcap\\{C_{i}|x\in C_{i}\text{ and }C_{i}\in\mathscr{C}\\}$, and
$[x]_{f}=\\{y\in U_{1}|f(x)=f(y)\\}$. If $[x]_{f}\subseteq C_{x}$ for any
$x\in U_{1}$, then $f$ is called a consistent function with respect to
$\mathscr{C}$.
Based on Definition 2.6, Wang et al. constructed a homomorphism between a
complex massive covering information system and a relatively small-scale
covering information system. It has also been proved that their attribute
reductions are equivalent to each other under the condition of a homomorphism.
Hence, the notion of the consistent function provides the foundation for the
communication between covering information systems.
In order to deal with uncertainty and more complex problems, Zadeh[31]
proposed the theory of fuzzy sets by extending the classical set theory. Let
$U$ be a non-empty universe of discourse, a fuzzy set of $U$ is a mapping
$A:U\longrightarrow[0,1]$. We denote by $\mathscr{F}(U)$ the set of all fuzzy
sets of $U$. For any $A,B\in\mathscr{F}(U)$, we say that $A\subseteq B$ if
$A(x)\leq B(x)$ for any $x\in U$. The union of $A$ and $B$, denoted as $A\cup
B$, is defined by $(A\cup B)(x)=A(x)\vee B(x)$ for any $x\in U$, and the
intersection of $A$ and $B$, denoted as $A\cap B$, is defined by $(A\cap
B)(x)=A(x)\wedge B(x)$ for any $x\in U$. The complement of $A$, denoted as
$-A$, is defined by $(-A)(x)=1-A(x)$ for any $x\in U$. Furthermore, a fuzzy
relation on $U$ is a mapping $R:U\times U\longrightarrow[0,1]$. We denote by
$\mathscr{F}(U\times U)$ the set of all fuzzy relations on $U$.
In practical situations, there exist a lot of fuzzy information systems as a
generalization of crisp information systems, and the investigations of fuzzy
information systems have powerful prospects in applications. To conduct the
communication between fuzzy information systems, Wang et al. proposed a
consistent function with respect to a fuzzy relation.
###### Definition 2.7
[27] Let $U_{1}$ and $U_{2}$ be two universes, $f$ a mapping from $U_{1}$ to
$U_{2}$, $R\in\mathscr{F}(U_{1}\times U_{1})$, $[x]_{f}=\\{y\in
U_{1}|f(x)=f(y)\\}$, and $\\{[x]_{f}|x\in U_{1}\\}$ is a partition on $U_{1}$.
For any $x,y\in U_{1}$, if $R(u,v)=R(s,t)$ for any two pairs
$(u,v),(s,t)\in[x]_{f}\times[y]_{f}$, then $f$ is said to be consistent with
respect to $R$.
Based on the consistent function, Wang et al. constructed a homomorphism
between a large-scale fuzzy information system and a relatively small-scale
fuzzy information system. It has been proved that their attribute reductions
are equivalent to each other under the condition of a homomorphism. In this
sense, the notion of the consistent function provides an approach to studying
the communication between fuzzy information systems.
Recently, Deng et al.[4] proposed the concept of a fuzzy covering.
###### Definition 2.8
[4] A fuzzy covering of $U$ is a collection of fuzzy sets
$\mathscr{C}^{\ast}\subseteq\mathscr{F}(U)$ which satisfies
$(1)$ every fuzzy set $C^{\ast}\in\mathscr{C}^{\ast}$ is non-null, i.e.,
$C^{\ast}\neq\emptyset$;
$(2)$ $\forall x\in U,\bigvee_{C^{\ast}\in\mathscr{C}^{\ast}}C^{\ast}(x)>0$.
Unless stated otherwise, $U$ is a finite universe, and $\mathscr{C}^{\ast}$
consists of finite number of sets in this work. In what follows,
$(U,\mathscr{C}^{\ast})$ is called a fuzzy covering approximation space, and
$(U,\Delta^{\ast})$ is called a fuzzy covering information system, where
$\Delta^{\ast}=\\{\mathscr{C}_{i}^{\ast}|1\leq i\leq m\\}$. Throughout the
paper, we denote the set of all fuzzy coverings of $U$ as $C(U)$ for
simplicity.
In the following, we employ an example to illustrate the fuzzy covering
information system.
###### Example 2.9
Let $U=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ be eight houses,
$C=\\{price,color\\}$ the attribute set, the domains of $price$ and $color$
are $\\{high,middle,low\\}$ and $\\{good,bad\\}$, respectively. To evaluate
these houses, specialists $A$ and $B$ are employed and their evaluation
reports are shown as follows:
$\displaystyle high^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.65}{x_{8}};$
$\displaystyle middle^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.5}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle low^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle good^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle bad^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle high_{B}$ $\displaystyle=$
$\displaystyle\frac{0.9}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.8}{x_{8}};$
$\displaystyle middle^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle low^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle good^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0.8}{x_{1}}+\frac{1}{x_{2}}+\frac{0.9}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle bad^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{0.9}{x_{7}}+\frac{1}{x_{8}},$
where $high_{A}$ is the membership degree of each house belonging to the high
price by the specialist $A$. The meanings of the other symbols are similar.
Based on the above results, we obtain a fuzzy covering information system
$(U,\Delta^{\ast})$, where
$\Delta^{\ast}=\\{\mathscr{C}^{\ast}_{price},\mathscr{C}^{\ast}_{color}\\}$,
$\mathscr{C}^{\ast}_{price}=\\{C_{high},C_{middle},C_{low}\\}$ and
$\mathscr{C}^{\ast}_{color}=\\{C_{good},C_{bad}\\}$.
$\displaystyle C_{high}$ $\displaystyle=$ $\displaystyle high^{\ast}_{A}\cup
high^{\ast}_{B}=\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.8}{x_{8}};$
$\displaystyle C_{middle}$ $\displaystyle=$ $\displaystyle
middle^{\ast}_{A}\cup
middle^{\ast}_{B}=\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle C_{low}$ $\displaystyle=$ $\displaystyle low^{\ast}_{A}\cup
low^{\ast}_{B}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle C_{good}$ $\displaystyle=$ $\displaystyle good^{\ast}_{A}\cup
good^{\ast}_{B}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle C_{bad}$ $\displaystyle=$ $\displaystyle bad^{\ast}_{A}\cup
bad^{\ast}_{B}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}.$
It is obvious that we can construct a fuzzy covering of the universe with an
attribute. Since the fuzzy covering rough set theory is effective to handle
uncertain information, the investigation of this theory becomes an important
task in rough set theory.
## 3 The basic properties of the fuzzy covering approximation space
In this section, we introduce the concepts of neighborhoods, the lower and
upper approximation operators to facilitate the computation of fuzzy sets for
fuzzy covering approximation spaces. Then we propose the concepts of fuzzy
subcoverings, irreducible and reducible elements, non-intersectional and
intersectional elements of fuzzy coverings. Afterwards, the union and
intersection operations on two fuzzy coverings are provided. We also construct
two roughness measures and employ several examples to illustrate the proposed
notions.
### 3.1 The lower and upper approximation operations
Before introducing approximation operators, we present the concepts of
neighborhoods and induced fuzzy coverings based on fuzzy coverings.
###### Definition 3.1
Let $(U,\mathscr{C}^{\ast})$ be a fuzzy covering approximation space, and
$x\in U$. Then $C^{\ast}_{\mathscr{C}^{\ast}x}=\bigcap\\{C^{\ast}|C^{\ast}(x)$
$>0\text{ and }C^{\ast}\in\mathscr{C}^{\ast}\\}$ is called the neighborhood of
$x$ concerning $\mathscr{C}^{\ast}$.
We notice that $C_{x}^{\ast}$ is the intersection of all fuzzy subsets whose
membership degrees of $x\in U$ are not zeroes. Assume that
$C_{1}^{\ast},C_{2}^{\ast}\in\mathscr{C}^{\ast}$,
$C_{1}^{\ast}(x)>0,C_{2}^{\ast}(x)>0$, and $C^{\ast}_{x}=C^{\ast}_{1}\cap
C^{\ast}_{2}$ for $x\in U$, it implies that the membership degree of $x$ in
$C^{\ast}_{x}$ is $min\\{C_{1}^{\ast}(x),C_{2}^{\ast}(x)\\}$. In addition, we
observe that the classical neighborhood of a point $C_{x}=\bigcap\\{C|x\in
C\in\mathscr{C}\\}$ is the same as that in Definition 3.1 if the membership
degree for any $x\in U$ has its value only from the set $\\{0,1\\}$, where
$\mathscr{C}$ is a covering of $U$. For convenience, we denote $C^{\ast}$,
$\mathscr{C}^{\ast}$, $C^{\ast}_{\mathscr{C}_{i}x}$ and $C_{\mathscr{C}x}$ as
$C$, $\mathscr{C}$, $C_{ix}$ and $C_{x}$, respectively.
We present the properties of the neighborhood operator below.
###### Proposition 3.2
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $x,y\in U$.
If $C_{x}(y)>0$, then $C_{y}\subseteq C_{x}.$
Proof. Assume that $\mathscr{A}=\\{C|C\in\mathscr{C},C(x)>0\\}$ and
$\mathscr{B}=\\{C^{\prime}|C^{\prime}\in\mathscr{C},C^{\prime}(y)>0\\}$. Since
$C_{x}(y)>0$, it follows that $C(y)>0$ for any $C\in\mathscr{A}$.
Consequently, $C\in\mathscr{B}$. It implies that
$\\{C|C\in\mathscr{C},C(x)>0\\}\subseteq\\{C^{\prime}|C^{\prime}\in\mathscr{C},C^{\prime}(y)>0\\}.$
Therefore, $C_{y}\subseteq C_{x}.$ $\Box$
###### Proposition 3.3
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $x,y\in U$.
If $C_{x}(y)>0$ and $C_{y}(x)>0$, then $C_{y}=C_{x}.$
Proof. Straightforward from Proposition 3.2. $\Box$
Based on Definition 3.1, we present the concept of a fuzzy covering induced by
the original fuzzy covering.
###### Definition 3.4
Let $\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ be a fuzzy covering of $U$,
$C_{x}$=$\bigcap\\{C_{i}|C_{i}(x)>0\text{ and }C_{i}\in\mathscr{C}\\}$ for any
$x\in U$, and $Cov(\mathscr{C})$=$\\{C_{x}|x\in U\\}$. Then $Cov(\mathscr{C})$
is called the induced fuzzy covering of $\mathscr{C}$.
It is clear that $C_{x}$ has the minimal membership degree of $x$ in
$Cov(\mathscr{C})$, and each element of $Cov(\mathscr{C})$ can not be
represented as the union of other elements of $Cov(\mathscr{C})$. In other
words, $C_{x}$ is the minimal set containing $x$ in $Cov(\mathscr{C})$.
Furthermore, $Cov(\mathscr{C})$ is a fuzzy covering of $U$, and it is easy to
prove that the concept presented in Definition 2.2 is a special case of
Definition 3.4 when the values of membership degree are taken from the set
$\\{0,1\\}$.
An example is employed to illustrate the induced fuzzy covering.
###### Example 3.5
Let $U_{1}=\\{x_{1},x_{2},x_{3},x_{4}\\}$, and
$\mathscr{C}_{1}=\\{C^{\prime}_{1},C^{\prime}_{2},C^{\prime}_{3}\\}$, where
$C^{\prime}_{1}=\frac{1}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}$,
$C^{\prime}_{2}=\frac{0.5}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.6}{x_{4}}$,
and
$C^{\prime}_{3}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0}{x_{3}}+\frac{0.5}{x_{4}}$.
By Definition 3.4, we obtain the induced fuzzy covering
$Cov(\mathscr{C}_{1})=\\{C_{x_{i}}|i=1,2,3,4\\}$, where
$C_{x_{1}}=C_{x_{3}}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
and
$C_{x_{2}}=C_{x_{4}}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0}{x_{3}}+\frac{0.5}{x_{4}}$.
For convenience, we denote $C^{\prime}_{i}$ as $C_{i}$ in the following
examples.
We also propose the notion of a fuzzy covering induced by a family of fuzzy
coverings.
###### Definition 3.6
Let $\Delta$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ be a
family of fuzzy coverings of $U$, $\Delta_{x}$=$\bigcap\\{C_{ix}|C_{ix}\in
Cov(\mathscr{C}_{i})$, $1\leq i\leq m\\}$ for any $x\in U$, and
$Cov(\Delta)$=$\\{\Delta_{x}|x\in U\\}$. Then $Cov(\Delta)$ is called the
induced fuzzy covering of $\Delta$.
In other words, $\Delta_{x}$ is the intersection of all the elements whose
membership degrees of $x$ are not zeroes in each $\mathscr{C}_{i}$, and it is
the set whose membership degree of $x$ is the minimal in $Cov(\Delta)$.
Furthermore, given $x,y\in U$, if $\Delta_{x}(y)>0$, then
$\Delta_{y}\subseteq\Delta_{x}$. Consequently, $\Delta_{x}(y)>0$ and
$\Delta_{y}(x)>0$ imply that $\Delta_{x}=\Delta_{y}$. In addition,
$Cov(\Delta)$ is a fuzzy covering of $U$. Therefore, it is easy to verify that
the notion given in Definition 2.3 is a special case of Definition 3.6 when
the values of membership degree are taken from the set $\\{0,1\\}$.
Next, we give an example to illustrate Definition 3.6.
###### Example 3.7
Let $U_{1}=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\Delta=\\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\\}$,
$\mathscr{C}_{1}=\\{C_{4},C_{5},C_{6}\\}$,
$\mathscr{C}_{2}=\\{C_{7},C_{8},C_{9}\\}$, and
$\mathscr{C}_{3}=\\{C_{10},C_{11},C_{12}\\}$, where
$C_{4}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{5}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.6}{x_{4}}$,
$C_{6}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{7}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$
$C_{8}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.7}{x_{3}}+\frac{0.7}{x_{4}}$,
$C_{9}=\frac{0.6}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{10}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$
$C_{11}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$,
and
$C_{12}=\frac{0.8}{x_{1}}+\frac{0.8}{x_{2}}+\frac{0.7}{x_{3}}+\frac{0.7}{x_{4}}$.
By Definition 3.6, we obtain that
$Cov(\Delta)=\\{\Delta_{x_{i}}|i=1,2,3,4\\}$, where
$\Delta_{x_{1}}=\Delta_{x_{2}}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
and
$\Delta_{x_{3}}=\Delta_{x_{4}}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
In practice, the classical approximation operators based on coverings are not
fit for computing the approximations of fuzzy sets in the fuzzy covering
approximation space. To solve this issue, we propose the concepts of the lower
and upper approximation operators based on fuzzy coverings by extending
approximation operators in [37].
###### Definition 3.8
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the lower and upper approximations of X are defined as
$\displaystyle\underline{X}_{\mathscr{C}}$ $\displaystyle=$
$\displaystyle\bigcup\\{C|C\subseteq X\text{ and }C\in\mathscr{C}\\};$
$\displaystyle\overline{X}_{\mathscr{C}}$ $\displaystyle=$
$\displaystyle\left(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0,x\in
U\\}\right)\cup\underline{X}_{\mathscr{C}}.$
The physical meaning of the lower and upper approximations of $X$ is that we
can approximate $X$ by $\underline{X}_{\mathscr{C}}$ and
$\overline{X}_{\mathscr{C}}$. Particularly, if
$\overline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}=X$, then $X$ can be
understood as a definable set. Otherwise, $X$ is undefinable. It is clear that
the lower and upper approximation operations are the same as those[37] in the
classical covering approximation space if $\mathscr{C}$ is a covering of $U$.
In this sense, the notions given in Definition 3.8 are generalizations of the
classical ones into the fuzzy setting. In the following, we investigate their
basic properties in detail.
###### Proposition 3.9
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$X,Y\subseteq U$. Then
$(1)$ $\overline{\emptyset}_{\mathscr{C}}=\emptyset$,
$\underline{\emptyset}_{\mathscr{C}}=\emptyset$;
$(2)$ $\underline{U}_{\mathscr{C}}\subseteq U$,
$\overline{U}_{\mathscr{C}}\subseteq U$;
$(3)$ $\underline{X}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}$,
$\underline{X}_{\mathscr{C}}\subseteq X$;
$(4)$
$\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}\subseteq\overline{(X\cup
Y)}_{\mathscr{C}}$;
$(5)$ $X\subseteq
Y\Longrightarrow\underline{X}_{\mathscr{C}}\subseteq\underline{Y}_{\mathscr{C}},\overline{X}_{\mathscr{C}}\subseteq\overline{Y}_{\mathscr{C}}$;
$(6)$ $\forall C\in\mathscr{C}$, $\underline{C}=C,\overline{C}=C;$
$(7)$
$\underline{(\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$,
$\overline{X}_{\mathscr{C}}=\overline{(\overline{X}_{\mathscr{C}})}_{\mathscr{C}}$;
$(8)$
$\overline{(\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$,
$\underline{(\overline{X}_{\mathscr{C}})}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}$.
Proof. Straightforward from Definition 3.8. $\Box$
###### Proposition 3.10
The following properties do not hold generally in the fuzzy covering
approximation space:
$(1)$ $\underline{(X\cap
Y)}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}$;
$(2)$ $\underline{(-X)}_{\mathscr{C}}=-(\overline{X}_{\mathscr{C}})$;
$(3)$ $\overline{(-X)}_{\mathscr{C}}=-(\underline{X}_{\mathscr{C}})$;
$(4)$
$\underline{(-\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=-(\underline{X}_{\mathscr{C}})$;
$(5)$
$\overline{(-\overline{X}_{\mathscr{C}})}_{\mathscr{C}}=-(\overline{X}_{\mathscr{C}})$;
$(6)$ $\underline{U}_{\mathscr{C}}=U$, $\overline{U}_{\mathscr{C}}=U$;
$(7)$ $\overline{(X\cup
Y)}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}$;
$(8)$ $X\subseteq\overline{X}_{\mathscr{C}}$.
Example 2 in [35] can illustrate that Proposition 3.10(1-5) does not hold
generally in the fuzzy covering approximation space. Specially, we obtain that
$\underline{U}_{\mathscr{C}}=U$ and $X\subseteq\overline{X}_{\mathscr{C}}$ do
not necessarily hold for any $X\subseteq U$. Consequently, the lower and upper
approximation operations are not interior and closure operators, respectively,
in the fuzzy covering approximation space.
We employ an example to illustrate that Proposition 3.10(6-8) does not hold
generally.
###### Example 3.11
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{13},C_{14},C_{15},$
$C_{16}\\}$, where
$C_{13}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{14}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}},$
$C_{15}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0.4}{x_{4}}$,
and
$C_{16}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}.$
By Definition 3.8, it follows that
$\overline{U}_{\mathscr{C}}=\underline{U}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}\neq
U$. For
$X=\frac{0.4}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.5}{x_{4}}$ and
$Y=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$, we
have that
$\overline{X}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.4}{x_{4}}$,
$\overline{Y}_{\mathscr{C}}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and $\overline{(X\cup
Y)}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
Consequently, $\overline{(X\cup
Y)}_{\mathscr{C}}\neq\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}$
and $X\nsubseteq\overline{X}_{\mathscr{C}}$.
Some relationships among $\underline{X}_{\mathscr{C}}$,
$\overline{X}_{\mathscr{C}}$ and $X$ are explored in the following.
###### Proposition 3.12
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$.
$(1)$ If $\underline{X}_{\mathscr{C}}=X$, then
$\overline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$;
$(2)$ If $\underline{X}_{\mathscr{C}}=X$, then $\overline{X}_{\mathscr{C}}=X$;
$(3)$ $\underline{X}_{\mathscr{C}}=X$ if and only if $X$ is a union of
elements in $\mathscr{C}$;
$(4)$ If $X$ is a union of elements in $\mathscr{C}$, then
$\overline{X}_{\mathscr{C}}=X$.
Next, an example is given to illustrate that the converses of Proposition
3.12(1), (2) and (4) do not hold generally.
###### Example 3.13
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{17},C_{18},C_{19},$
$C_{20}\\}$, where
$C_{17}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{18}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{19}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$,
and
$C_{20}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}.$
For
$X=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}=\overline{X}_{\mathscr{C}}$.
But $X$ is not a union of some subsets in the fuzzy covering $\mathscr{C}$.
For
$Y=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$,
according to Definition 3.1, it follows that
$C_{x_{1}}=C_{x_{2}}=C_{x_{3}}=C_{x_{4}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Then we have that $\overline{Y}=Y$, but $Y$ is not a union of some elements of
$\mathscr{C}$.
From Proposition 3.10, we see that
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ does not hold generally for any $X,Y\subseteq U$ in the
fuzzy covering approximation space. But if
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$, then we can obtain the following
results.
###### Proposition 3.14
If
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$, then $C_{1}\cap C_{2}=\emptyset$
or $C_{1}\cap C_{2}$ is a union of elements of $\mathscr{C}$ for any
$C_{1},C_{2}\in\mathscr{C}$.
Proof. Taking any $C_{1},C_{2}\in\mathscr{C},$ it follows that
$\underline{C_{1}}_{\mathscr{C}}\cap\underline{C_{2}}_{\mathscr{C}}=\underline{(C_{1}\cap
C_{2})}_{\mathscr{C}}=C_{1}\cap C_{2}.$ By Proposition 3.12, we have that
$C_{1}\cap C_{2}=\emptyset$ or $C_{1}\cap C_{2}$ is a union of elements of
$\mathscr{C}$ for any $C_{1},C_{2}\in\mathscr{C}$. $\Box$
This proposition shows that the intersection of two elementary elements in a
fuzzy covering $\mathscr{C}$ can be represented as a union of elements of
$\mathscr{C}$ if
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$.
It is clear that
$\underline{X}_{\mathscr{C}}\subseteq\underline{X}_{Cov(\mathscr{C})}$ and
$\overline{X}_{\mathscr{C}}=\overline{X}_{Cov(\mathscr{C})}$ for any
$X\subseteq U$ in the classical covering approximation space
$(U,\mathscr{C})$. But they do not necessarily hold in the fuzzy covering
approximation space. To illustrate this point, we employ the following
example.
###### Example 3.15
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{21},C_{22},$
$C_{23}\\}$, where
$C_{21}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}},$
$C_{22}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{23}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}.$
According to Definition 3.1, we have that
$C_{x_{1}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}},$
$C_{x_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}},$
$C_{x_{3}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{x_{4}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}.$
Taking
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0}{x_{4}}$,
according to Definition 3.8, it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\underline{X}_{Cov(\mathscr{C})}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$.
Consequently,
$\underline{X}_{\mathscr{C}}\nsubseteq\underline{X}_{Cov(\mathscr{C})}.$
Similarly, we obtain that
$\overline{X}_{\mathscr{C}}\neq\overline{X}_{Cov(\mathscr{C})}.$
It is well known that the upper approximation can be represented with
neighborhoods in the classical covering approximation space. But we do not
have the same result in the fuzzy covering approximation space.
###### Theorem 3.16
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}$ holds for
$X\subseteq U$.
Proof. Taking any $X\subseteq U$, according to Definition 3.8, we see that
$\overline{X}_{\mathscr{C}}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup\underline{X}_{\mathscr{C}}$ and
$\bigcup\\{C_{x}|X(x)>0\\}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup(\bigcup\\{C_{x}|\underline{X}_{\mathscr{C}}(x)>0\\})$.
It follows that there exist $C\in\mathscr{C}$ and $C\subseteq X$ for any $x$
satisfying $\underline{X}(x)>0$. Consequently, $C_{x}\subseteq C\subseteq X$.
Hence, $\bigcup\\{C_{x}|\underline{X}_{\mathscr{C}}(x)>0\\}\subseteq X$.
Therefore, $\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}$
holds for $X\subseteq U$. $\Box$
We see that $\overline{X}_{\mathscr{C}}\subseteq\bigcup\\{C_{x}|X(x)>0\\}$
does not necessarily hold for any $X\subseteq U$. So the upper approximation
can not be represented with neighborhoods only in the fuzzy covering
approximation space. To show this point, we give an example below.
###### Example 3.17
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, and
$\mathscr{C}=\\{C_{17},C_{18},C_{19},$ $C_{20}\\}$. For
$X=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}$,
it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}=\overline{X}_{\mathscr{C}}$.
Furthermore,
$\bigcup\\{C_{x}|X(x)>0\\}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Obviously, $\overline{X}_{\mathscr{C}}\nsubseteq\bigcup\\{C_{x}|X(x)>0\\}$.
We now investigate the relationship between the lower and upper approximation
operators.
###### Theorem 3.18
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$, then
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$.
Proof. By Definition 3.1, we have that
$C_{1x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}_{1},i\in I\\}$ and
$C_{2x}=\bigcap\\{C_{j}|C_{j}(x)>0,C_{j}\in\mathscr{C}_{2},j\in J\\}$. For any
$C_{i},$ where $i\in I$,
$C_{i}=\underline{C}_{i\mathscr{C}_{1}}=\underline{C}_{i\mathscr{C}_{2}}$. So
there exists at least $C_{j}\in\mathscr{C}_{2}$ such that $C_{j}\subseteq
C_{i}$ and $C_{j}(x)>0$. Hence, $C_{2x}\subseteq C_{1x}$. Similarly, we obtain
that $C_{1x}\subseteq C_{2x}$. Therefore,
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$. $\Box$
From Theorem 3.18, we see that the lower and upper approximation operations
are not independent in the fuzzy covering approximation space. Concretely, the
lower approximation operation dominates the upper one.
###### Theorem 3.19
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\underline{C}_{\mathscr{C}_{1}}=\underline{C}_{\mathscr{C}_{2}}$ holds for
any $C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$ if and only if
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$.
Proof. Taking any $X\subseteq U$, by Definition 3.8, it follows that
$\underline{X}_{\mathscr{C}_{1}}=\bigcup\\{C_{i}|C_{i}\subseteq
X,C_{i}\in\mathscr{C}_{1},i\in I\\}$. For any
$C_{i}\subseteq\underline{X}_{\mathscr{C}_{1}}$, we have that
$C_{i}=\underline{C}_{i\mathscr{C}_{1}}=\underline{C}_{i\mathscr{C}_{2}}=\bigcup\\{C_{ij}|C_{ij}\in\mathscr{C}_{2},C_{ij}\subseteq
X,i\in I,j\in J\\}$. It implies that
$\underline{X}_{\mathscr{C}_{1}}\subseteq\underline{X}_{\mathscr{C}_{2}}$.
Analogously, it follows that
$\underline{X}_{\mathscr{C}_{2}}\subseteq\underline{X}_{\mathscr{C}_{1}}$.
Thereby, $\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for
any $X\subseteq U$.
The converse is obvious by Definition 3.8. $\Box$
This result indicates that each elementary set in a fuzzy covering is
definable in the other fuzzy covering if and only if two fuzzy coverings of a
universe give the same lower approximations.
### 3.2 The fuzzy subcovering and its properties
It is well-known that the classical upper approximation based on neighborhoods
can be defined equivalently by using a family of subcoverings. In this
subsection, we propose the notion of fuzzy subcoverings and investigate the
relationship between the upper approximation based on neighborhoods and
subcoverings in the fuzzy covering approximation space.
###### Definition 3.20
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $X\subseteq U$,
and $\mathscr{C}^{\prime}\subseteq\mathscr{C}$. If
$X\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}$ , then
$\mathscr{C}^{\prime}$ is called a fuzzy subcovering of $X$.
In other words, the fuzzy subcovering of $X$ is a subset of $\mathscr{C}$
which covers $X$. Obviously, $\mathscr{C}$ is the maximum fuzzy subcovering
for $X\subseteq U$ if $X\subseteq\bigcup\mathscr{C}$. In this work, we denote
the set of all the fuzzy subcoverings of $X$ as $FC(X)$.
###### Theorem 3.21
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ holds for any $X\subseteq U$.
Proof. Taking any $X\subseteq U$, by Definition 3.8, it follows that
$\overline{X}_{\mathscr{C}}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup\underline{X}_{\mathscr{C}}$. By
Proposition 3.9, it implies that $\underline{X}\subseteq
X\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\in FC(X)\\}$. Evidently,
$C_{x}\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\in FC(X)\\}$ for any $x\in
U$ satisfying $X(x)>0$. Thereby,
$\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ holds for any $X\subseteq U$. $\Box$
However,
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}\subseteq\overline{X}_{\mathscr{C}}$ does not hold generally. That is,
the upper approximation may not be represented with a family of fuzzy
subcoverings of $X$ as the classical covering approximation space, which is
shown by the following example.
###### Example 3.22
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{24},C_{25},C_{26}\\}$, where
$C_{24}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{25}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}$
and
$C_{26}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$.
According to Definition 3.20, we obtain all fuzzy subcoverings of
$X=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$ as
$FC(X)=\\{\\{C_{24}\\},\\{C_{26}\\},\\{C_{24},C_{25}\\},\\{C_{24},C_{26}\\},$
$\\{C_{25},C_{26}\\},$ $\\{C_{24},C_{25},C_{26}\\}\\}$. It follows that
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$,
but
$\overline{X}_{\mathscr{C}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Therefore,
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}\nsubseteq\overline{X}_{\mathscr{C}}$.
According to Theorems 3.16 and 3.21, we have that
$\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ for any $X\subseteq U$.
Sometimes, the fuzzy covering $\mathscr{C}$ of $U$ is a trivial subcovering of
$X\subseteq U$. Specially, we do not take $\mathscr{C}$ into account in the
following situation.
###### Proposition 3.23
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. If $|FC(X)|\geq 2$, where $|FC(X)|$ stands for the cardinality of $FC(X)$,
then
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}=\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)-\\{\mathscr{C}\\}\\}$.
Proof. Straightforward. $\Box$
### 3.3 The irreducible and reducible elements of a fuzzy covering
In this subsection, we provide the concepts of reducible and irreducible
elements to formally investigate the relationship among elementary elements of
a fuzzy covering. Although several theorems in this subsection are special
cases of [33], they don’t give their proofs. To better understand the
following results, we prove them concretely in the following.
###### Definition 3.24
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$C\in\mathscr{C}$. If $C$ can not be written as a union of some sets in
$\mathscr{C}-\\{C\\}$, then $C$ is called an irreducible element. Otherwise,
$C$ is called a reducible element.
It is obvious that the concept of the irreducible element in a fuzzy covering
approximation space is an extension of the notion of the irreducible element
in a covering approximation space, and the irreducible element can be used for
the definition of reducts of fuzzy coverings.
###### Proposition 3.25
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then $\mathscr{C}-\\{C\\}$ is still a
fuzzy covering of $U$.
In other words, a fuzzy covering of a universe deleting all reducible elements
is a fuzzy covering, and the rest elements are irreducible.
###### Definition 3.26
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. If every
element of $\mathscr{C}$ is an irreducible element, then $\mathscr{C}$ is
irreducible. Otherwise, $\mathscr{C}$ is reducible.
Next, we discuss the properties of reducible elements of a fuzzy covering.
###### Theorem 3.27
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $C$ a
reducible element of $\mathscr{C}$, and $C_{0}\in\mathscr{C}-\\{C\\}$. Then
$C_{0}$ is a reducible element of $\mathscr{C}$ if and only if it is a
reducible element of $\mathscr{C}-\\{C\\}$.
Proof. We assume that $C_{0}$ is a reducible element of $\mathscr{C}$. It
follows that we can express $C_{0}$ as a union of subset of
$\mathscr{C}-\\{C_{0}\\}$, denoted as $C_{1},C_{2},...,C_{N}$. If there exists
no set which is equal to $C$ in $\\{C_{1},C_{2},...,C_{N}\\}$, then $C_{0}$ is
a reducible element of $\mathscr{C}-\\{C\\}$. If there is a set which is equal
to $C$ in $\\{C_{1},C_{2},...,C_{N}\\}$, taking $C_{1}=C$, then $C_{1}$ is the
union of some sets $\\{D_{1},D_{2},...,D_{M}\\}$ in $\mathscr{C}-\\{C\\}$.
Consequently, we obtain that $C_{0}=D_{1}\cup D_{2}\cup...D_{M}\cup
C_{2}\cup...\cup C_{N}$. Clearly, $D_{1},D_{2},...,D_{M},C_{2},...,C_{N}$ are
not equal to either $C_{0}$ or $C$. So $C$ is a reducible element of
$\mathscr{C}-\\{C\\}$.
Since $C_{0}$ is a reducible element of $\mathscr{C}-\\{C\\}$, it can be
expressed as a union of some sets in $\mathscr{C}-\\{C,C_{0}\\}$. We can
express it as a union of some sets in $\mathscr{C}-\\{C_{0}\\}$. Therefore,
$C_{0}$ is a reducible element of $\mathscr{C}$. $\Box$
Next, we investigate the relationship between the approximation operations and
the reducible elements in the fuzzy covering approximation space.
###### Theorem 3.28
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then
$\underline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}-\\{C\\}}$ holds for
any $X\subseteq U$.
Proof. Taking any $X\subseteq U$, by Definition 3.24, it follows that
$\underline{X}_{\mathscr{C}-\\{C\\}}\subseteq\underline{X}_{\mathscr{C}}\subseteq
X$. Moreover, there exist $C_{1},C_{2},...,C_{N}$ such that
$\underline{X}_{\mathscr{C}}=C_{1}\cup C_{2}\cup...\cup C_{N}$. If none of
$C_{1},C_{2},...,C_{N}$ is equal to $C$, then they belong to
$\mathscr{C}-\\{C\\}$. Consequently, $C_{1},C_{2},...,C_{N}$ are all the
subsets of $\underline{X}_{\mathscr{C}-\\{C\\}}$. If there is a set which is
equal to $C$, then we take $C=C_{1}$. Since $C$ is a reducible element of
$\mathscr{C}$, $C$ can be expressed as some sets in $\mathscr{C}-\\{C\\}$ such
that $C=D_{1}\cup D_{2}\cup...\cup D_{M}$. Hence,
$\underline{X}_{\mathscr{C}}=D_{1}\cup D_{2}\cup...\cup D_{M}\cup
C_{2}\cup...\cup C_{N}$. It implies that
$\underline{X}_{\mathscr{C}}\subseteq\underline{X}_{\mathscr{C}-\\{C\\}}$.
Therefore, $\underline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}-\\{C\\}}$
holds for any $X\subseteq U$. $\Box$
In other words, the lower approximation of any $X\subseteq U_{1}$ in
$\mathscr{C}$ is the same as that in $\mathscr{C}-\\{C\\}$ if $C$ is
reducible.
###### Corollary 3.29
[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then
$\overline{X}_{\mathscr{C}}=\overline{X}_{\mathscr{C}-\\{C\\}}$ holds for any
$X\subseteq U$.
Proof. Straightforward from Theorems 3.27 and 3.28. $\Box$
In this sequel, we use $RED(\mathscr{C})$ to represent the set of all
irreducible elements of a fuzzy covering $\mathscr{C}$. It is easy to see that
$RED(\mathscr{C})=RED(RED(\mathscr{C}))$. Next, we study the relationship
between $RED(\mathscr{C})$ and the lower and upper approximation operations.
###### Corollary 3.30
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\underline{X}_{\mathscr{C}}=\underline{X}_{RED(\mathscr{C})}$ holds for any
$X\subseteq U$.
Proof. Straightforward from Theorem 3.28. $\Box$
###### Corollary 3.31
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\overline{X}_{\mathscr{C}}=\overline{X}_{RED(\mathscr{C})}$ holds for any
$X\subseteq U$.
Proof. Straightforward from Corollary 3.29. $\Box$
Based on Theorem 3.28, Corollaries 3.29, 3.30 and 3.31, we obtain the
following theorem.
###### Theorem 3.32
Let $U$ be a universe, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ two irreducible
fuzzy coverings of $U$. If
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$. then the two fuzzy coverings are the same.
Proof. Taking any $C\in\mathscr{C}_{1}$, by Definition 3.8, it follows that
$\underline{C}_{\mathscr{C}_{1}}=C=\underline{C}_{\mathscr{C}_{2}}$.
Consequently, $C$ is the union of some sets of $\mathscr{C}_{2}$ such that
$C=C_{1}\cup C_{2}\cup...\cup C_{N}$. Similarly, there exist
$D_{i1},D_{i2},...,D_{iM(i)}\in\mathscr{C}_{1}$ such that $C_{i}=D_{i1}\cup
D_{i2}\cup...\cup D_{iM(i)}.$ Hence, $C=D_{11}\cup D_{12}\cup...\cup
D_{N1}\cup D_{N2}\cup...\cup D_{NM(N)}$. Since $C$ is irreducible, $C=D_{ij}$
for all $i,j$. It implies that $C$ is an element of $\mathscr{C}_{2}$. On the
other hand, any element of $\mathscr{C}_{2}$ is an element of
$\mathscr{C}_{1}$. Therefore, the two fuzzy coverings $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$ are the same. $\Box$
###### Corollary 3.33
Let $U$ be a universe, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ two irreducible
fuzzy coverings of $U$. If
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$, then the two fuzzy coverings are the same.
Proof. The proof is similar to that in Theorem 3.32. $\Box$
###### Theorem 3.34
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ holds for
any $X\subseteq U$ if and only if $RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.
Proof. Since $\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$
for any $X\subseteq U$,
$\underline{C}_{\mathscr{C}_{1}}=\underline{C}_{\mathscr{C}_{2}}$ for any
$C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$. Taking any $C\in
RED(\mathscr{C}_{1})$, it follows that $C=\bigcup\\{C_{i}|C_{i}\in
RED(\mathscr{C}_{2}),i\in
I\\}=\bigcup\\{\bigcup\\{C_{ij}|C_{ij}\in\mathscr{C}_{1},i\in I\\}|j\in J\\}$.
It implies that $C\in\mathscr{C}_{2}$. Hence, $RED(\mathscr{C}_{1})\subseteq
RED(\mathscr{C}_{2})$. Similarly, we obtain that
$RED(\mathscr{C}_{2})\subseteq RED(\mathscr{C}_{1})$. Therefore,
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.
The converse is obvious by Definitions 3.8 and 3.24. $\Box$
It can be seen from Theorem 3.34 that two fuzzy coverings of a universe
generate the same lower approximation if and only if there exist the same
irreducible elements in these fuzzy coverings.
To illustrate Theorem 3.34, we supply the following example.
###### Example 3.35
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{17},C_{18},$
$C_{19},C_{20},C_{27},C_{28}\\}$,
$\mathscr{C}_{2}=\\{C_{17},C_{18},C_{19},C_{20},C_{29},$ $C_{30}\\}$, where
$C_{27}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{28}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{29}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}$
and
$C_{30}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}.$
Obviously, we obtain that
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})=\\{C_{17},C_{18},C_{19},C_{20}\\}$.
###### Corollary 3.36
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$ if and only if $RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.
From Corollary 3.36, we see that they have the same irreducible elements if
and only if two fuzzy coverings of a universe generate the same upper
approximation.
###### Corollary 3.37
Let $(U,\mathscr{C}_{1})$ be a fuzzy covering approximation space,
$\mathscr{C}_{2}=\\{\bigcup_{C\in\mathscr{C}^{\prime}}C|\emptyset\neq\mathscr{C}^{\prime}\subseteq\mathscr{C}_{1}\\}$,
and $X\subseteq U$. Then
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ and
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$.
Proof. By Definition 3.24, we observe that
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$. Therefore,
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ and
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$. $\Box$
We also investigate the relationship between the reducible elements and the
neighborhood operator in the fuzzy covering approximation space.
###### Theorem 3.38
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then $C_{x}$ in $\mathscr{C}-\\{C\\}$ is
the same as that in $\mathscr{C}$ for any $x\in U$.
Proof. By Definitions 3.1 and 3.24, we have that
$C_{x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}\\}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}-\\{C\\}\\}$
for any $x\in U$. Therefore, $C_{x}$ in $\mathscr{C}-\\{C\\}$ is the same as
that in $\mathscr{C}$ for any $x\in U$. $\Box$
That is to say, if we delete some reducible elements in the fuzzy covering,
then it will not change the neighborhood $C_{x}$ for any $x\in U$.
###### Corollary 3.39
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then $C_{x}$ in
$RED(\mathscr{C})$ is the same as that in $\mathscr{C}$ for any $x\in U$.
Proof. Straightforward from Theorem 3.38. $\Box$
Corollary 3.39 indicates that $RED(\mathscr{C})$ and $\mathscr{C}$ generate
the same neighborhood $C_{x}$ for any $x\in U$ in the fuzzy covering
approximation space.
###### Corollary 3.40
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$, then $C_{x}$ in $\mathscr{C}_{1}$
is the same as that in $\mathscr{C}_{2}$ for any $x\in U$.
Proof. Straightforward from Corollary 3.39. $\Box$
By Corollary 3.40, if there exist the same irreducible elements in two fuzzy
coverings $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ of $U$, then they generate
the same neighborhood $C_{x}$ for any $x\in U$.
###### Theorem 3.41
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\overline{C}_{\mathscr{C}_{1}}=\overline{C}_{\mathscr{C}_{2}}$ for any
$C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$, then
$\bigcup\\{C_{1x}|X(x)>0\\}=\bigcup\\{C_{2x}|X(x)>0\\}$ for any $X\subseteq
U$.
Proof. By Definition 3.1, we have that
$C_{1x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}_{1},i\in I\\}$ and
$C_{2x}=\bigcap\\{C_{j}|C_{j}(x)>0,C_{j}\in\mathscr{C}_{2},j\in J\\}$ for any
$x\in U$. Assume that there exists $x\in U$ such that $X(x)>0$ and $C_{1x}\neq
C_{2x}$. Without loss of generality, there is $y\in U$ such that
$(C_{1x})(y)>0$ and $(C_{2x})(y)=0$. Obviously, $y\neq x$. Hence, there exist
$C_{j}(y)=0$ and $C_{j}(x)>0$. But
$C_{j}=\overline{C}_{j\mathscr{C}_{2}}=\overline{C}_{j\mathscr{C}_{1}}\supseteq\bigcup\\{C_{1z}|C_{j}(z)>0\\}\supseteq
C_{1x}$. It implies that $C_{j}(y)>0$, which is a contradiction. Consequently,
$C_{1x}=C_{2x}$ for any $x\in U$. Therefore,
$\bigcup\\{C_{1x}|X(x)>0\\}=\bigcup\\{C_{2x}|X(x)>0\\}$ for any $X\subseteq
U$. $\Box$
Theorem 3.41 shows that two fuzzy coverings of a universe generate the same
neighborhood $C_{x}$ for any $x\in U$ if each elementary element has the same
lower approximation in two fuzzy coverings.
### 3.4 The non-intersectional and intersectional elements of fuzzy
coverings, the union and intersection operations on fuzzy coverings
For any universal set $U$, we denote $CC(U)$ as the set of all coverings of
$U$. It is well-known that the number of possible coverings for a set $U$ of
$n$ elements is
$|CC(U)|=\frac{1}{2}\sum^{n}_{k=0}(\frac{n}{k})2^{2^{n-k}},$
the first few of which are 1, 5, 109, 32297, 2147321017. Since $C(U)$ contains
a larger number of fuzzy coverings than $CC(U)$ in practice, it is of interest
to investigate the relationship between fuzzy coverings. In this subsection,
we introduce several operations on fuzzy coverings and study their basic
properties for facilitating the computation of fuzzy coverings.
###### Definition 3.42
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$C\in\mathscr{C}$. If $C$ can not be written as an intersection of some sets
in $\mathscr{C}-\\{C\\}$, then $C$ is called a non-intersectional element.
Otherwise, $C$ is called an intersectional element.
For simplicity, we use $IS(\mathscr{C})$ to represent the set of all non-
intersectional elements of $\mathscr{C}$. It is easy to see that
$IS(\mathscr{C})=IS(IS(\mathscr{C}))$. Notice that the function
$IS:C(U)\longrightarrow C(U)$ that maps $\mathscr{C}$ to $IS(\mathscr{C})$ is
well-defined. Hence, we may view $IS$ as a unary operator on $C(U)$.
We employ an example to illustrate the non-intersectional and intersectional
elements in the following.
###### Example 3.43
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{17},C_{18},C_{19},C_{20},C_{31}\\}$, where
$C_{31}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
By Definition 3.42, we have that
$IS(\mathscr{C})=\\{C_{17},C_{18},C_{19},C_{20}\\}$.
###### Proposition 3.44
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\bigcup\\{C_{x}|X(x)>0\\}=\bigcup\\{C_{IS(\mathscr{C})x}|$ $X(x)>0\\}$ and
$\underline{X}_{IS(\mathscr{C})}\subseteq\underline{X}_{\mathscr{C}}$ for any
$X\subseteq U$.
Proof. By Definition 3.42, it follows that $C_{x}=C_{IS(\mathscr{C})x}$ for
any $x\in U$. Consequently,
$\bigcup\\{C_{x}|X(x)>0\\}=\bigcup\\{C_{IS(\mathscr{C})x}|X(x)>0\\}$ for any
$X\subseteq U$. Furthermore, since $IS(\mathscr{C})\subseteq\mathscr{C}$, we
have that
$\underline{X}_{IS(\mathscr{C})}\subseteq\underline{X}_{\mathscr{C}}$ for any
$X\subseteq U$. $\Box$
We observe that the neighborhood $C_{x}$ generated in the fuzzy covering
$\mathscr{C}$ is the same as that generated in all non-intersectional elements
of $\mathscr{C}$. On the other hand,
$\overline{X}_{\mathscr{C}}\subseteq\overline{X}_{IS(\mathscr{C})}$ does not
necessarily hold for any $X\subseteq U$. An example is given to illustrate
this point.
###### Example 3.45
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{32},C_{33},$
$C_{34},C_{35},C_{36},C_{37},C_{38},C_{39}\\}$, where
$C_{32}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{33}=\frac{0}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{34}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{35}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0}{x_{4}},$
$C_{36}=\frac{0.4}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0}{x_{4}},$
$C_{37}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{38}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.5}{x_{4}}$
and
$C_{39}=\frac{0}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}.$
Evidently, $IS(\mathscr{C})=\\{C_{36},C_{37},C_{38},C_{39}\\}$. Taking
$X=\frac{0.4}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
according to Definitions 3.8 and 3.42, we obtain that
$\overline{X}_{\mathscr{C}}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{IS(\mathscr{C})}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}}.$
Thereby, $\overline{X}_{\mathscr{C}}\nsubseteq\overline{X}_{IS(\mathscr{C})}$.
Following, we present a theorem for the intersection element of a fuzzy
covering.
###### Theorem 3.46
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $C$ an
intersection element of $\mathscr{C}$, and $C_{0}\in\mathscr{C}-\\{C\\}$. Then
$C_{0}$ is an intersection element of $\mathscr{C}$ if and only if it is an
intersection element of $\mathscr{C}-\\{C\\}$.
Proof. The proof is similar to that in Theorem 3.27. $\Box$
Next, we present the notions of the union and intersection operations on fuzzy
coverings, and investigate their basic properties.
###### Definition 3.47
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\mathscr{C}_{1}\cup\mathscr{C}_{2}=\\{C|C\in\mathscr{C}_{1}\text{ or
}C\in\mathscr{C}_{2}\\},$
then $\mathscr{C}_{1}\cup\mathscr{C}_{2}$ is called the union of
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$.
It is obvious that the union operation is to collect all elementary elements
in each fuzzy covering.
###### Proposition 3.48
Let $U$ be a non-empty universe of discourse,
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$, and $X\subseteq U$. Then
$\underline{X}_{\mathscr{C}_{i}}\subseteq\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$,
where $i=1,2.$
Proof. According to Definition 3.8, we have that
$\underline{X}_{\mathscr{C}_{i}}=\bigcup\\{C\in\mathscr{C}_{i}|C\subseteq
X\\}\subseteq(\bigcup\\{C\in\mathscr{C}_{1}|C\subseteq
X\\})\bigcup(\bigcup\\{C\in\mathscr{C}_{2}|C\subseteq
X\\})=\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$. Thereby,
$\underline{X}_{\mathscr{C}_{i}}\subseteq\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$,
where $i=1,2.$ $\Box$
The following example shows that the converse of Proposition 3.48 does not
hold generally.
###### Example 3.49
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and
$\mathscr{C}_{2}=\\{C_{1x_{1}},C_{1x_{2}},C_{1x_{3}},C_{1x_{4}}\\}$. Taking
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.1}{x_{4}}$.
By Definition 3.8, we have that
$\overline{X}_{\mathscr{C}_{1}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$\overline{X}_{\mathscr{C}_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$,
$\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
Obviously,
$\overline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}\nsubseteq\overline{X}_{\mathscr{C}_{2}}$.
###### Definition 3.50
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\mathscr{C}_{1}\cap\mathscr{C}_{2}=\\{C_{1x}\cap C_{2x}|C_{ix}\in
Cov(\mathscr{C}_{i}),x\in U_{1},i=1,2\\},$
then $\mathscr{C}_{1}\cap\mathscr{C}_{2}$ is called the intersection of
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$.
It is obvious that $C_{1x}=\bigcap\\{C|C(x)>0,C\in\mathscr{C}_{1}\\}$ and
$C_{2x}=\bigcap\\{C^{\prime}|C^{\prime}(x)>0,C^{\prime}\in\mathscr{C}_{2}\\}$
for any $x\in U_{1}$. So $\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$ is a fuzzy
covering of $U_{1}$. Furthermore, if we take the value of membership degree
from the set $\\{0,1\\}$, then Definition 3.50 is the same as that in
Definition 4.2 in [24].
It can be found that
$\overline{X}_{\mathscr{C}_{i}}\subseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$
does not necessarily hold for any $X\subseteq U$ in the fuzzy covering
approximation space, where $i=1,2$. To illustrate this point, we give the
following example.
###### Example 3.51
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and
$\mathscr{C}_{2}=\\{\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}\\}$.
According to Definition 3.1, we have that
$C_{1x_{1}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$,
$C_{1x_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$,
$C_{1x_{3}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{1x_{4}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$.
Taking
$X=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
it implies that
$\underline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$,
$\underline{X}_{\mathscr{C}_{1}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$
and
$\underline{X}_{\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$.
Clearly,
$\overline{X}_{\mathscr{C}_{1}}\nsubseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$
and
$\overline{X}_{\mathscr{C}_{2}}\nsubseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$.
By Definitions 3.4 and 3.50, we present the following proposition.
###### Proposition 3.52
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}=Cov(\mathscr{C}_{1}\bigcup\mathscr{C}_{2})$.
###### Definition 3.53
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If there exists
$C^{\ast}\in\mathscr{C}_{2}$ such that $C\subseteq C^{\ast}$ for any
$C\in\mathscr{C}_{1}$, then $\mathscr{C}_{2}$ is said to be coarser than
$\mathscr{C}_{1}$, denoted as $\mathscr{C}_{1}\leq\mathscr{C}_{2}$.
In other words, there exists $C^{\ast}\in\mathscr{C}_{2}$ such that
$C^{\ast}(x)\leq C(x)$ for each $C\in\mathscr{C}_{1}$ and $x\in U$ if
$\mathscr{C}_{1}\leq\mathscr{C}_{2}$.
###### Example 3.54
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and $\mathscr{C}_{2}=\\{C_{40},C_{41}\\}$, where
$C_{40}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{2}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.5}{x_{4}}$.
It is obvious that $\mathscr{C}_{2}$ is coarser than $\mathscr{C}_{1}$.
###### Proposition 3.55
Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\in C(U)$. Then
$(1)$ $\mathscr{C}_{1}\cup\mathscr{C}_{1}=\mathscr{C}_{1};$
$(2)$ $\mathscr{C}_{1}\cap\mathscr{C}_{1}\leq\mathscr{C}_{1};$
$(3)$ $\mathscr{C}_{1}\cup\mathscr{C}_{2}=\mathscr{C}_{2}\cup\mathscr{C}_{1};$
$(4)$ $\mathscr{C}_{1}\cap\mathscr{C}_{2}=\mathscr{C}_{2}\cap\mathscr{C}_{1};$
$(5)$
$(\mathscr{C}_{1}\cup\mathscr{C}_{2})\cup\mathscr{C}_{3}=\mathscr{C}_{1}\cup(\mathscr{C}_{2}\cup\mathscr{C}_{3});$
$(6)$
$(\mathscr{C}_{1}\cap\mathscr{C}_{2})\cap\mathscr{C}_{3}=\mathscr{C}_{1}\cap(\mathscr{C}_{2}\cap\mathscr{C}_{3});$
$(7)$
$\mathscr{C}_{1}\cup(\mathscr{C}_{1}\cap\mathscr{C}_{2})\leq\mathscr{C}_{1};$
$(8)$
$\mathscr{C}_{1}\cap(\mathscr{C}_{1}\cup\mathscr{C}_{2})\leq\mathscr{C}_{1}.$
Proof. Straightforward from Definitions 3.47, 3.50 and 3.53. $\Box$
###### Proposition 3.56
Let $U$ be a non-empty universe of discourse, and $C(U)$ the set of all fuzzy
coverings of $U$. Then $(C(U),\cap,\cup)$ is a lattice.
Proof. Given any $\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$, it is obvious that
$\mathscr{C}_{1}\cup\mathscr{C}_{2}\in C(U)$ and
$\mathscr{C}_{1}\cap\mathscr{C}_{2}\in C(U)$. Therefore, $(C(U),\cap,\cup)$ is
a lattice. $\Box$
We notice that $\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}\\}$ is
the greatest element of $(C(U),\cap,\cup)$, but $(C(U),\cap,\cup)$ is not a
complete lattice necessarily, which is illustrated by the following example.
###### Example 3.57
Let $U=\\{x_{1},x_{2},x_{3}\\}$,
$C_{0}(U)=\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{n},...\\}\subseteq
C(U)$, and
$\mathscr{C}_{n}=\\{\frac{\frac{1}{n}}{x_{1}}+\frac{\frac{1}{n}}{x_{2}}+\frac{\frac{1}{n}}{x_{3}}\\}$.
By Definition 3.50, it follows that $\bigcap
C_{0}(U)=\\{\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}\\}$. It is obvious
that $\bigcap C_{0}(U)\notin C(U)$. Consequently, $(C(U),\leq)$ is not an
intersection structure.
###### Proposition 3.58
Let $U$ be a non-empty universe of discourse. Then
$(U,C(U)\cup\\{\emptyset\\})$ is a topological space.
Proof. Straightforward from Definitions 3.47 and 3.50. $\Box$
At the end of this subsection, we provide two roughness measures of fuzzy sets
as follows.
###### Definition 3.59
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the roughness measure $\mu_{\mathscr{C}}(X)$ regarding $\mathscr{C}$
is defined as
$\displaystyle\mu_{\mathscr{C}}(X)=1-\frac{|\underline{X}_{\mathscr{C}}|}{|\overline{X}_{\mathscr{C}}|},$
where $|\underline{X}_{\mathscr{C}}|=\sum_{x\in
U}\underline{X}_{\mathscr{C}}(x)$ and $|\overline{X}_{\mathscr{C}}|=\sum_{x\in
U}\overline{X}_{\mathscr{C}}(x)$.
###### Definition 3.60
Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the $\alpha\beta-$roughness measure
$\mu^{\alpha\beta}_{\mathscr{C}}(X)$ with respect to $\mathscr{C}$ is defined
as
$\displaystyle\mu^{\alpha\beta}_{\mathscr{C}}(X)=1-\frac{|\underline{X}^{\alpha}_{\mathscr{C}}|}{|\overline{X}^{\beta}_{\mathscr{C}}|},$
where
$\underline{X}^{\alpha}_{\mathscr{C}}=\\{x|\underline{X}_{\mathscr{C}}(x)>\alpha,x\in
U\\}$,
$\overline{X}^{\beta}_{\mathscr{C}}=\\{x|\overline{X}_{\mathscr{C}}(x)>\beta,x\in
U\\}$ and $|\cdot|$ means the cardinality of the set.
An example is employed to illustrate Definitions 3.59 and 3.60 as follows.
###### Example 3.61
Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{21},C_{22},C_{23}\\}$, and
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.1}{x_{4}}$.
According to Definition 3.8, we have that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
It follows that
$\mu_{\mathscr{C}}(X)=1-\frac{0.2+0.4+0.5}{0.2+0.4+0.5+0.5}=0.3125$.
Furthermore, it is obvious that
$\underline{X}^{\alpha}_{\mathscr{C}}=\\{x_{3}\\}$ and
$\overline{X}^{\beta}_{\mathscr{C}}=\\{x_{2},x_{3},x_{4}\\}$ by taking
$\alpha=0.4$ and $\beta=0.2$. Subsequently, it follows that
$\mu^{\alpha\beta}_{\mathscr{C}}(X)=1-\frac{|\\{x_{3}\\}|}{|\\{x_{2},x_{3},x_{4}\\}|}=\frac{2}{3}$.
## 4 Consistent functions for fuzzy covering information systems
In [24], Wang et al. proposed the concept of consistent functions for
attribute reductions of covering information systems. But so far we have not
seen the similar work on fuzzy covering information systems. In this section,
we introduce the concepts of consistent functions, the fuzzy covering mappings
and inverse fuzzy covering mappings based on fuzzy coverings and examine their
basic properties. Additionally, several examples are employed to illustrate
our proposed notions.
As a generalization of the concept of consistent functions given in Definition
2.6, we introduce the notion of consistent functions for constructing
attribute reducts of fuzzy covering information systems.
###### Definition 4.1
Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$[x]$ the block of $U_{1}/IND(f)$ which contains $x$, where $U_{1}/IND(f)$
stands for the blocks of partition of $U_{1}$ by an equivalence relation
$IND(f)$ based on $f$. If $C_{i}(y)=C_{i}(z)$ $(1\leq i\leq N)$ for any
$y,z\in[x]$, then $f$ is called a consistent function with respect to
$\mathscr{C}$.
Unless stated otherwise, we take the equivalence relation
$IND(f)=\\{(x,y)|f(x)=f(y),x,y\in U_{1}\\}$ and $[x]=\\{y\in
U_{1}|f(x)=f(y),x,y\in U_{1}\\}$ when applying Definition 4.1 in this work.
Particularly, it is clear that our proposed function is the same as the
consistent function in [24] when the membership degree for any $x\in U_{1}$
has its value only from the set $\\{0,1\\}$. Thereby, the proposed model can
be viewed as an extension of that given in [24].
An example is employed to illustrate the concept of consistent functions in
the following.
###### Example 4.2
Consider the fuzzy covering approximation space $(U_{1},\mathscr{C}_{1})$ in
Example 3.5. Then, we take $U_{2}=\\{y_{1},y_{2}\\}$ and define a mapping
$f:U_{1}\longrightarrow U_{2}$ as
$f(x_{1})=f(x_{3})=y_{1};f(x_{2})=f(x_{4})=y_{2}.$
Obviously, $f$ is a consistent function with respect to $\mathscr{C}_{1}$.
Now we investigate the relationship between Definitions 2.7 and 4.1. If
$\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering of $U_{1}$, where
$R_{x}(y)=R(x,y)$ for any $x,y\in U_{1}$, then we can express Definition 2.7
as follows: let $U_{1}$ and $U_{2}$ be two universes, $f$ a mapping from
$U_{1}$ to $U_{2}$, $R\in\mathscr{F}(U_{1}\times U_{1})$, and $[x]_{f}=\\{y\in
U_{1}|f(x)=f(y)\\}$, $\\{[x]_{f}|x\in U_{1}\\}$ a partition on $U_{1}$. For
any $x,y\in U_{1}$, if $R(x,v)=R(x,t)$ for any two pairs
$(x,v),(x,t)\in[x]_{f}\times[y]_{f}$, then $f$ is said to be consistent with
respect to $R$. Consequently, the consistent function given in Definition 2.7
is the same as our proposed model if $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$ and $IND(f)=\\{(x,y)|f(x)=f(y),x,y\in U_{1}\\}$.
In the following, we investigate some conditions under which $\\{R_{x}|x\in
U_{1}\\}$ is a fuzzy covering of $U_{1}$.
###### Corollary 4.3
Let $R$ be a fuzzy relation on $U_{1}$. Then
$(1)$ if $R$ is $\alpha$-reflexive, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$, where $1>\alpha>0$;
$(2)$ if $R$ is reflexive, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering
of $U_{1}$;
$(3)$ if $R$ is a fuzzy similarity, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$;
$(4)$ if $R$ is a fuzzy equivalence, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$.
Proof. $(1)$ If $R$ is $\alpha$-reflexive, then $R_{x}(x)\geq\alpha$ for any
$x\in U_{1}$. It follows that $(\bigcup\\{R_{x}|x\in U_{1}\\})(y)\geq\alpha$
for any $y\in U_{1}$. Therefore, $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering
of $U_{1}$.
$(2)$ If $R$ is reflexive, then $R_{x}(x)=1$ for any $x\in U_{1}$. It implies
that $\bigcup\\{R_{x}|x\in U_{1}\\}=U_{1}$. Therefore, $\\{R_{x}|x\in
U_{1}\\}$ is a fuzzy covering of $U_{1}$.
$(3),(4)$ The proof is similar to that in Corollary 4.3(2). $\Box$
Additionally, we can construct a fuzzy relation by a fuzzy covering.
###### Corollary 4.4
Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$,
$R_{x}=C_{x}$ for any $x\in U_{1}$, and $\alpha=min\\{C_{x}(x)|x\in U_{1}\\}$.
Then
$(1)$ $R$ is a $\alpha$-reflexive relation;
$(2)$ $R$ is symmetric if $C_{x}(x)=C_{y}(y)$ for any $x,y\in U_{1}$;
$(3)$ $R$ is transitive;
$(4)$ $R$ is a fuzzy equivalence relation if $\alpha=1$.
Proof. Straightforward from Definition 3.1. $\Box$
By Corollaries 4.3 and 4.4, it is clear that there exists a relationship
between a fuzzy relation and a fuzzy covering. Since both Wang’s model[24] and
our proposed function are based on a fuzzy relation and a fuzzy covering,
respectively, by Corollaries 4.3 and 4.4, we can establish the relationship
between Definitions 2.7 and 4.1.
By means of Zadeh’s extension principle, we propose the concepts of the fuzzy
covering mapping and inverse fuzzy covering mapping.
###### Definition 4.5
Let $S_{1}=(U_{1},\mathscr{C}_{1})$ and $S_{2}=(U_{2},\mathscr{C}_{2})$ be
fuzzy covering approximation spaces, and $f$ a surjection from $U_{1}$ to
$U_{2}$, $f$ induces a mapping from $\mathscr{C}_{1}\text{ to
}\mathscr{C}_{2}$ and a mapping from $\mathscr{C}_{2}\text{ to
}\mathscr{C}_{1}$, that is
$\hat{f}:\mathscr{C}_{1}\longrightarrow\mathscr{C}_{2},C\mid\rightarrow\hat{f}(C)\in\mathscr{C}_{2},\forall
C\in\mathscr{C}_{1};$
$\hat{f}(C)(y)=\left\\{\begin{array}[]{ccc}\bigvee_{x\in
f^{-1}(y)}C(x),&{\rm}\hfil&f^{-1}(y)\neq\emptyset;\\\
0,&{\rm}\hfil&f^{-1}(y)=\emptyset;\end{array}\right.$
$\hat{f}^{-1}:\mathscr{C}_{2}\longrightarrow\mathscr{C}_{1},T\mid\rightarrow\hat{f}^{-1}(T)\in\mathscr{C}_{1},\forall
T\in\mathscr{C}_{2};$
$f^{-1}(T)(x)=T(f(x)),x\in U_{1}.$
Then $\hat{f}$ and $\hat{f}^{-1}$ are called the fuzzy covering mapping and
the inverse fuzzy covering mapping induced by $f$, respectively. In
convenience, we denote $\hat{f}$ and $\hat{f}^{-1}$ as $f$ and $f^{-1}$,
respectively.
By Definition 4.5, we observe that $\hat{f}$ and $\hat{f}^{-1}$ will be
reduced to Definition 4.1 in [24] if the membership degree takes values from
the set $\\{0,1\\}$. The following theorem discusses the problem of fuzzy set
operations under a consistent function $f$.
###### Theorem 4.6
Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$C_{i},C_{j}\in\mathscr{C}$. Then
$(1)$ $f(C_{i}\cap C_{j})\subseteq f(C_{i})\cap f(C_{j})$;
$(2)$ $f(C_{i}\cup C_{j})=f(C_{i})\cup f(C_{j})$;
$(3)$ If $f$ is a consistent function with respect to $\mathscr{C}$, then
$f(C_{i}\cap C_{j})=f(C_{i})\cap f(C_{j})$.
Proof. (1) By Definition 4.5, we obtain that $f(C_{i}\cap C_{j})(f(x))=0$ when
$(C_{i}\cap C_{j})(x)=0$ for $x\in U_{1}$. Moreover, by Definition 4.5, it
follows that $f(C_{i}\cap C_{j})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}\cap C_{j})(x^{\prime})=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\wedge
C_{j}(x^{\prime}))\leq\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\wedge\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=(f(C_{i})\cap f(C_{j}))(y)$. Consequently,
$f(C_{i}\cap C_{j})\subseteq f(C_{i})\cap f(C_{j})$.
(2) According to Definition 4.5, we have that $f(C_{i}\cup C_{j})(f(x))=0$
when $(C_{i}\cup C_{j})(x)=0$ for $x\in U_{1}$. Furthermore, by Definition
4.5, it follows that $f(C_{i}\cup C_{j})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}\cup C_{j})(x^{\prime})=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\vee C_{j}(x^{\prime}))=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\vee\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=(f(C_{i})\cup f(C_{j}))(y)$. Therefore,
$f(C_{i}\cup C_{j})=f(C_{i})\cup f(C_{j})$.
(3) By Theorem 4.6(1), it is obvious that $f(C_{i}\cap C_{j})\subseteq
f(C_{i})\cap f(C_{j})$. So we only need to prove that $f(C_{i})\cap
f(C_{j})\subseteq f(C_{i}\cap C_{j}).$ Suppose that $y\in U_{2}$, there exists
$x\in U_{1}$ such that $f(x)=y$. Based on Definitions 4.1 and 4.5, we have
that $(f(C_{i})\cap f(C_{j}))(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\wedge\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=C_{i}(x^{\prime})\wedge
C_{j}(x^{\prime})\subseteq\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\wedge C_{j}(x^{\prime}))=f(C_{i}\cap C_{j})(y)$.
Thereby, $f(C_{i}\cap C_{j})=f(C_{i})\cap f(C_{j})$. $\Box$
Theorem 4.6 shows that the mapping $f$ preserves some fuzzy set operations,
especially it preserves the intersection operation of fuzzy sets if $f$ is
consistent.
To illustrate Theorem 4.6, we give an example below.
###### Example 4.7
Consider $S=(U_{1},\mathscr{C}_{1})$ in Example 3.5 and the consistent
function $f$ in Example 4.2. Then we observe that $f(C_{1}\cap
C_{2})=f(C_{1})\cap f(C_{2})$, $f(C_{1}\cap C_{3})=f(C_{1})\cap f(C_{3})$ and
$f(C_{2}\cap C_{3})=f(C_{2})\cap f(C_{3})$.
By Theorem 4.6, we obtain the following corollary.
###### Corollary 4.8
Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f(\bigcap_{i=1}^{N}C_{i})=\bigcap_{i=1}^{N}f(C_{i})$.
Subsequently, we investigate the properties of the inverse mapping of a
consistent function.
###### Theorem 4.9
Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$C_{i}\in\mathscr{C}$. Then
$(1)$ $C_{i}\subseteq f^{-1}(f(C_{i}))$;
$(2)$ If $f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(C_{i}))=C_{i}$.
Proof. (1) According to Definition 4.5, we have that
$f^{-1}(f(C_{i}))(x)=f(C_{i})(f(x))$. Taking $y=f(x)$, it follows that
$f(C_{i})(f(x))=f(C_{i})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\geq C_{i}(x).$ Therefore, $C_{i}\subseteq
f^{-1}(f(C_{i}))$.
(2) By Definition 4.5, we see that $f^{-1}(f(C_{i}(x)))=f(C_{i})(f(x))$.
Assume that $y=f(x)$, it follows that
$f(C_{i})(f(x))=f(C_{i})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime}).$ According to Definitions 4.1 and 4.5, it implies
that $C_{i}(x^{\prime})=C_{i}(x)$ for any $x^{\prime}\in f^{-1}(y)$.
Consequently, $\bigvee_{x^{\prime}\in f^{-1}(y)}C_{i}(x^{\prime})=C_{i}(x)$.
Hence, $f^{-1}(f(C_{i}))(x)=C_{i}(x).$ Thereby, $C_{i}=f^{-1}(f(C_{i}))$.
$\Box$
We give an example to illustrate Theorem 4.9 in the following.
###### Example 4.10
Consider $S=(U_{1},\mathscr{C}_{1})$ in Example 3.5 and the consistent
function $f$ in Example 4.2. Then we see that
$f^{-1}(f(C_{1}))(x_{i})=C_{1}(x_{i}),$
$f^{-1}(f(C_{2}))(x_{i})=C_{2}(x_{i}),$ and
$f^{-1}(f(C_{3}))(x_{i})=C_{3}(x_{i}),i=1,2,3,4.$ Therefore,
$f^{-1}(f(C_{1}))=C_{1},$ $f^{-1}(f(C_{2}))=C_{2},$ and
$f^{-1}(f(C_{3}))=C_{3}.$
By Theorem 4.9, we have the following corollary.
###### Corollary 4.11
Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(\bigcap_{i=1}^{N}C_{i}))=\bigcap_{i=1}^{N}C_{i}$.
We also explore the properties of a consistent function on a family of fuzzy
coverings.
###### Theorem 4.12
Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
${\mathscr{C}_{1},\mathscr{C}_{2}}\in C(U_{1})$. If $f$ is a consistent
function with respect to $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$,
respectively, then $f$ is consistent with respect to
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$.
Proof. Based on Definition 4.1, we have that $C_{ix}(y)=C_{ix}(z)$ for any
$y,z\in[x]$, where $i=1,2$. It follows that $C_{1x}(y)\wedge
C_{2x}(y)=C_{1x}(z)\wedge C_{2x}(z)$ for any $y,z\in[x]$. Hence, $(C_{1x}\cap
C_{2x})(y)=(C_{1x}\cap C_{2x})(z)$ for any $y,z\in[x]$. Therefore, $f$ is
consistent with respect to $\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$. $\Box$
The following example is employed to illustrate Theorem 4.12.
###### Example 4.13
Consider $S=(U_{1},\Delta)$ in Example 3.7. We take $U_{2}=\\{y_{1},y_{2}\\}$
and define a mapping $f:U_{1}\rightarrow U_{2}$ as follows:
$f(x_{1})=f(x_{2})=y_{1},f(x_{3})=f(x_{4})=y_{2}.$
It is obvious that $f$ is a consistent function with respect to
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$, respectively. By Definition 4.1, we
observe that $f$ is consistent with respect to
$\mathscr{C}_{1}\cap\mathscr{C}_{2}$.
Based on Theorem 4.12, we obtain the following corollary.
###### Corollary 4.14
Let $\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\in C(U_{1})$, and $f$
a mapping from $U_{1}$ to $U_{2}$. If $f$ is a consistent function with
respect to any $\mathscr{C}_{i}$ $(1\leq i\leq m)$, then $f$ is consistent
with respect to $\bigcap_{i=1}^{m}\mathscr{C}_{i}$.
Now, we introduce two concepts for fuzzy covering approximation spaces.
###### Definition 4.15
Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}_{1}=\\{C_{11},C_{12},...,C_{1N}\\}\in C(U_{1})$, and
$\mathscr{C}_{2}=\\{T_{21},T_{22},...,T_{2M}\\}\in C(U_{2})$. Then
$f(\mathscr{C}_{1})$ and $f(\mathscr{C}_{2})$ are defined by
$\displaystyle f(\mathscr{C}_{1})$ $\displaystyle=$
$\displaystyle\\{f(C_{1i}),C_{1i}\in\mathscr{C}_{1},1\leq i\leq N\\};$
$\displaystyle f^{-1}(\mathscr{C}_{2})$ $\displaystyle=$
$\displaystyle\\{f^{-1}(T_{2j}),T_{2j}\in\mathscr{C}_{2},1\leq j\leq M\\}.$
###### Theorem 4.16
Let $U$ be a non-empty universe of discourse, and $\mathscr{C}\in C(U)$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(\mathscr{C}))=\mathscr{C}.$
Proof. By Theorem 4.9, it follows that $f^{-1}(f(C_{i}))=C_{i}$ for any
$C_{i}\in\mathscr{C}$. Therefore, $f^{-1}(f(\mathscr{C}))=\mathscr{C}.$ $\Box$
Obviously, Examples 3.5 and 4.2 can illustrate Theorem 4.16. Then we get the
following corollary.
###### Corollary 4.17
Let $\mathscr{C}_{i}\in C(U)$, and $\Delta=\\{\mathscr{C}_{i}|i=1,2,...,m\\}$.
If $f$ is a consistent function with respect to any
$\mathscr{C}_{i}\in\Delta$, then $f^{-1}(f(\bigcap\Delta))=\bigcap\Delta.$
At the end of this section, we discuss the fuzzy covering operations under a
consistent function.
###### Theorem 4.18
Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and $\mathscr{C}_{1}$,
$\mathscr{C}_{2}\in C(U_{1})$. Then we have
$(1)$ $f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})\subseteq
f(\mathscr{C}_{1})\bigcap f(\mathscr{C}_{2})$;
$(2)$ If $f$ is a consistent function with respect to $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$, respectively, then
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})=f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$.
Proof. (1) According to Definitions 4.1 and 4.5, it is obvious that
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})\subseteq f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$.
(2) Evidently, we only need to prove that $f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})\subseteq f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Assume
that $C_{x}$ is the minimal element containing $x$ in
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$, $C_{1x}$ is the minimal element
containing $x$ in $Cov(\mathscr{C}_{1})$, and $C_{2x}$ is the minimal element
containing $x$ in $Cov(\mathscr{C}_{2})$ for any $x\in U_{1}$. By Definition
3.50, it follows that $C_{x}=C_{1x}\bigcap C_{2x}$. According to Theorem 4.12,
it implies that $f$ is a consistent function with respect to
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$. Consequently, we obtain that
$f(C_{x})=f(C_{1x})\cap f(C_{2x})$. By Definition 3.1, we have that
$C_{x}(x)>0$ for any $x\in U_{1}$. It follows that $f(C_{x})(f(x))>0$. Hence,
$(f(C_{1x})\cap f(C_{2x}))(f(x))>0.$ Suppose that $f(C_{1x})\cap f(C_{2x})$ is
not the minimal subset containing $f(x)$ in
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Then there exists $x_{0}\in U_{1}$
such that $f(C_{ix_{0}})(f(x))>0$ and $f(C_{1x})\cap f(C_{2x})\cap
f(C_{ix_{0}})\subset f(C_{1x})\cap f(C_{2x}),$ it means that $(f(C_{1x})\cap
f(C_{2x})\cap f(C_{ix_{0}}))(f(x))>0.$ Thereby, there exist $u,v$ and $w$ such
that $C_{1x}(u)>0,C_{2x}(v)>0,C_{ix_{0}}(w)>0$ and $f(u)=f(v)=f(w)=f(x)$.
According to Theorem 4.6, we have that $f(C_{1x})\cap f(C_{2x})=f(C_{1x}\cap
C_{2x})\subseteq f(C_{ix_{0}})$ and $f(C_{1x})\cap f(C_{2x})\cap
f(C_{ix_{0}})=f(C_{1x})\cap f(C_{2x}),$ it implies that $f(C_{1x})\cap
f(C_{2x})$ is the minimal subset containing $f(x)$ in
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Based on the above statement, it
follows that $f(\mathscr{C}_{1})\bigcap f(\mathscr{C}_{2})\subseteq
f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Therefore,
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})=f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$. $\Box$
Based on Theorem 4.18, we have the following corollary.
###### Corollary 4.19
Let $\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}$ be fuzzy coverings
of $U_{1}$, and $f$ a mapping from $U_{1}$ to $U_{2}$. If $f$ is a consistent
function with respect to
$\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}$, respectively, then
$f(\bigcap_{i=1}^{m}\mathscr{C}_{i})=\bigcap_{i=1}^{m}f(\mathscr{C}_{i})$.
## 5 Data compressions of fuzzy covering information systems and dynamic
fuzzy covering information systems
In this section, we further investigate data compressions of fuzzy covering
information systems and dynamic fuzzy covering information systems.
### 5.1 Data compression of fuzzy covering information systems
In this subsection, the concepts of an induced fuzzy covering information
system and homomorphisms between fuzzy covering information systems are
introduced for data compression of the fuzzy covering information system. Then
the algorithm of constructing attribute reducts of fuzzy covering information
systems is provided. An example is finally employed to illustrate the proposed
concepts and algorithm.
###### Definition 5.1
Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
Then $(U_{1},\Delta_{1})$ is referred to as a fuzzy covering information
system and $(U_{2},f(\Delta_{1}))$ is called the $f$-induced fuzzy covering
information system of $(U_{1},\Delta_{1})$.
Definition 5.1 shows that we can induce a new fuzzy covering information
system under a surjection.
Based on Definitions 4.1 and 5.1, we propose the notion of a homomorphism
between two fuzzy covering information systems.
###### Definition 5.2
Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is consistent with respect to any $\mathscr{C}_{i}\in\Delta_{1}$
$(1\leq i\leq m)$ on $U_{1}$, then $f$ is called a homomorphism from
$(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$.
We provide the concept of reducts of fuzzy covering information systems in the
following.
###### Definition 5.3
Let $(U_{1},\Delta_{1})$ be a fuzzy covering information system, and
$\mathscr{C}_{i}\in\Delta_{1}$ $(1\leq i\leq m)$. If
$\bigcap\\{\Delta_{1}-\mathscr{C}_{i}\\}=\bigcap\Delta_{1}\ $, then
$\mathscr{C}_{i}$ is called superfluous. Otherwise, $\mathscr{C}_{i}$ is
called indispensable. The collection of all indispensable elements in
$\Delta_{1}$, denoted as Core($\Delta_{1}$), is called the core of
$\Delta_{1}$. $P\subseteq\Delta_{1}$ is called a reduct of $\Delta_{1}$ if $P$
satisfies: $\bigcap P=\bigcap\Delta_{1}$ and
$\bigcap\\{P-\mathscr{C}\\}\neq\bigcap\Delta_{1}$ for any $\mathscr{C}\in P.$
Now we present the following theorem which shows that the reducts of fuzzy
covering information systems can be preserved under a homomorphism.
###### Theorem 5.4
Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$,
then $P\subseteq\Delta_{1}$ is a reduct of $\Delta_{1}$ if and only if $f(P)$
is a reduct of $f(\Delta_{1})$.
Proof. Suppose $P$ is a reduct of $\Delta_{1}$. It follows that $\bigcap
P=\bigcap\Delta_{1}$. Hence, $f(\bigcap P)=f(\bigcap\Delta_{1})$. Then we
obtain that $\bigcap f(P)=\bigcap f(\Delta_{1})$ since $f$ is a homomorphism
from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$. Assume that there exists
$\mathscr{C}\in P$ such that $\bigcap(f(P)-f(\mathscr{C}))=\bigcap f(P)$. It
implies that $\bigcap(f(P)-f(\mathscr{C}))=\bigcap f(P-\mathscr{C})$. Hence,
we see that $\bigcap f(\Delta_{1})=\bigcap f(P-\mathscr{C})$. It follows that
$f^{-1}(\bigcap f(\Delta_{1}))=f^{-1}(\bigcap f(P-\mathscr{C}))$. We obtain
that $\bigcap\Delta_{1}=\bigcap(P-\mathscr{C})$, which contradicts that $P$ is
a reduct of $\Delta_{1}$. So $f(P)$ is a reduct of $f(\Delta_{1})$.
On the other hand, we assume that $f(P)$ is a reduct of $f(\Delta_{1})$. It
follows that $\bigcap f(\Delta_{1})=\bigcap f(P)$. Since $f$ is a homomorphism
from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$, we obtain that
$f(\bigcap\Delta_{1})=f(\bigcap P)$. It implies that
$\bigcap\Delta_{1}=\bigcap P$. Assume that there exists $\mathscr{C}\in P$
satisfying $\bigcap\Delta_{1}=\bigcap(P-\mathscr{C})$, it follows that
$f(\bigcap\Delta_{1})=f(\bigcap(P-\mathscr{C}))$. Obviously, $\bigcap
f(\Delta_{1})=\bigcap f(P-\mathscr{C})=\bigcap(f(P)-f(\mathscr{C}))$, which is
a contradiction. Therefore, $P\subseteq\Delta_{1}$ is a reduct of
$\Delta_{1}$. $\Box$
By Theorem 5.4, we obtain the following corollary.
###### Corollary 5.5
Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$,
then
$(1)$ $\mathscr{C}$ is indispensable in $\Delta_{1}$ if and only if
$f(\mathscr{C})$ is indispensable in $f(\Delta_{1})$;
$(2)$ $\mathscr{C}$ is superfluous in $\Delta_{1}$ if and only if
$f(\mathscr{C})$ is superfluous in $f(\Delta_{1})$;
$(3)$ The image of the core of $\Delta_{1}$ is the core of $f(\Delta_{1})$,
and the inverse image of the core of $f(\Delta_{1})$ is the core of the
original image.
Proof. Straightforward from Definition 5.3 and Theorem 5.4. $\Box$
From Corollary 5.5, we see that the attribute reductions of the original fuzzy
covering information system and image system are equivalent to each other
under the condition of a homomorphism.
###### Definition 5.6
Let $(U_{1},\mathscr{C}_{1})$ be a fuzzy covering approximation space, the
equivalence relation $R_{\mathscr{C}_{1}}=\\{(x,y)|C_{x}=C_{y},x,y\in
U_{1}\\}$, and $U_{1}/R_{\mathscr{C}_{1}}=\\{R_{\mathscr{C}_{1}}(x)|x\in
U_{1}\\}$. Then $U_{1}/R_{\mathscr{C}_{1}}$ is called the partition based on
$\mathscr{C}_{1}$.
For the sake of convenience, we denote $U_{1}/R_{\mathscr{C}_{1}}$ as
$U_{1}/\mathscr{C}_{1}$ simply.
Following, we employ Table 2 to show the partition based on each fuzzy
covering for the fuzzy covering information system $(U_{1},\Delta_{1})$, where
$P_{ix_{j}}$ stands for the block containing $x_{j}$ in the partition
$U_{1}/R_{\mathscr{C}_{i}}$. It is easy to see that
$P_{\Delta_{1}x_{j}}=\bigcap_{1\leq i\leq m}P_{ix_{j}}$, where
$P_{\Delta_{1}x_{j}}$ denotes the block containing $x_{j}$ in the partition
based on $\Delta_{1}$.
Subsequently, we propose the main feature of the algorithm to construct
attribute reducts of fuzzy covering information systems. It shows how to
construct a homomorphism and compress a large-scale information system into a
small one under the condition of the homomorphism.
###### Algorithm 5.7
Let $U_{1}=\\{x_{1},...,x_{n}\\}$, and
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},$ $...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$. Then
Step 1. Input the fuzzy covering information system $(U_{1},\Delta_{1})$;
Step 2. Computing the partition $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq m)$ and
obtain $U_{1}/\Delta_{1}=\\{C_{i}|1\leq i\leq K\\}$;
Step 3. Define $f:U_{1}\rightarrow U_{2}$ as follows: $f(x)=y_{l}$, $x\in
C_{l}$, where $1\leq l\leq K$ and $U_{2}=\\{y_{1},y_{2},...,y_{K}\\}$;
Step 4. Compute
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$
and obtain $(U_{2},f(\Delta_{1}))$;
Step 5. Construct attribute reducts of $(U_{2},f(\Delta_{1}))$ and obtain a
reduct
$\\{f(\mathscr{C}_{i1}),f(\mathscr{C}_{i2}),...,f(\mathscr{C}_{ik})\\};$
Step 6. Obtain a reduct
$\\{\mathscr{C}_{i1},\mathscr{C}_{i2},...,\mathscr{C}_{ik}\\}$ of
$(U_{1},\Delta_{1})$ and output the results.
Remark. In Example 5.1[24], Wang et al. obtained the partition
$U_{1}/\Delta_{1}$ by only computing $\Delta_{x}$ for any $x\in U_{1}$. But we
get $U_{1}/\Delta_{1}$ by computing $U_{1}/\mathscr{C}_{i}$ for any
$\mathscr{C}_{i}\in\Delta_{1}$ in Algorithm 5.7. By using the proposed
approach, we can compress the dynamic fuzzy covering information system on the
basis of data compression of the original system with lower time complexity,
which is illustrated in Subsection 5.2.
Now, we employ a car evaluation problem to illustrate Algorithm 5.7.
###### Example 5.8
Suppose that $U_{1}=\\{x_{1},x_{2},...,x_{8}\\}$ is a set of eight cars,
$C_{1}=\\{price,structure,size,appearance\\}$ is a set of attributes. The
domains of $price$, $structure$, $size$ and $appearance$ are $\\{high,middle,$
$low\\}$, $\\{excellent,ordinary,poor\\}$, $\\{big,middle,small\\}$ and
$\\{beautiful,fair,ugly\\}$, respectively. In this example, we do not list
their evaluation reports for simplicity. According to the four specialists’
evaluation reports, we obtain the following fuzzy coverings of $U_{1}$ as
$\Delta_{1}=\\{\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size},\mathscr{C}_{appearance}\\}$,
$\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size}$ and
$\mathscr{C}_{appearance}$ are based on $price$, $structure$, $size$ and
$appearance$, respectively, where
$\displaystyle\mathscr{C}_{price}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{1}{x_{5}}+\frac{1}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{structure}$ $\displaystyle=$
$\displaystyle\\{\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}},\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{size}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{1}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}},\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{appearance}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}\\}.$
By Definition 2.8, we see that $(U_{1},\Delta_{1})$ is a fuzzy covering
information system. Furthermore, according to Definitions 3.1, 3.6 and 5.6, we
obtain the following results:
$\displaystyle U_{1}/\mathscr{C}_{price}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2}\\},\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{structure}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\},\\{x_{3},x_{5}\\},\\{x_{6}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{size}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\},\\{x_{4},x_{7},x_{8}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{appearance}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}\\}.$
The partitions
$U_{1}/\mathscr{C}_{price},U_{1}/\mathscr{C}_{structure},U_{1}/\mathscr{C}_{size}$
and $U_{1}/\mathscr{C}_{appearance}$ are shown in Table 3. Then we obtain that
$U_{1}/\Delta_{1}=\\{\\{x_{1},x_{2}\\},\\{x_{3},x_{5}\\},\\{x_{4},x_{7},x_{8}\\},\\{x_{6}\\}\\}.$
Thus we take $U_{2}=\\{y_{1},y_{2},y_{3},y_{4}\\}$ and define a mapping
$f:U_{1}\longrightarrow U_{2}$ as follows:
$f(x_{1})=f(x_{2})=y_{1};f(x_{3})=f(x_{5})=y_{2};f(x_{4})=f(x_{7})=f(x_{8})=y_{3};f(x_{6})=y_{4}.$
According to the function $f$, we obtain that
$f(\Delta_{1})=\\{f(\mathscr{C}_{price}),f(\mathscr{C}_{structure}),f(\mathscr{C}_{size}),f(\mathscr{C}_{appearance})\\}$,
where
$\displaystyle f(\mathscr{C}_{price})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{0.5}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}},\frac{0}{y_{1}}+\frac{1}{y_{2}}+\frac{0.5}{y_{3}}+\frac{1}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{structure})$ $\displaystyle=$
$\displaystyle\\{\frac{0}{y_{1}}+\frac{1}{y_{2}}+\frac{0}{y_{3}}+\frac{0}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{0.5}{y_{3}}+\frac{0}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{size})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{0}{y_{3}}+\frac{1}{y_{4}},\frac{0.5}{y_{1}}+\frac{1}{y_{2}}+\frac{0.5}{y_{3}}+\frac{0.5}{y_{4}},\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{appearance})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}}\\}.$
According to Definition 5.1, we obtain the $f$-induced fuzzy covering
information system $(U_{2},f(\Delta_{1}))$ of $(U_{1},\Delta_{1})$. Clearly,
the size of $(U_{2},f(\Delta_{1}))$ is relatively smaller than that of
$(U_{1},\Delta_{1})$. Then, by Definitions 5.1, 5.2 and 5.3, we have the
following results:
(1) $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to
$(U_{2},f(\Delta_{1}))$;
(2) $f(\mathscr{C}_{appearance})$ is superfluous in $f(\Delta_{1})$ if and
only if $\mathscr{C}_{appearance}$ is superfluous in $\Delta_{1}$;
(3)
$\\{f(\mathscr{C}_{price}),f(\mathscr{C}_{structure}),f(\mathscr{C}_{size})\\}$
is a reduct of $f(\Delta_{1})$ if and only if
$\\{\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size}\\}$ is a
reduct of $\Delta_{1}$.
From Example 5.8, we see that the image system $(U_{2},f(\Delta_{1}))$ has
relatively smaller size than the original system $(U_{1},\Delta_{1})$. But
their attribute reductions are equivalent to each other under the condition of
a homomorphism.
From the practical viewpoint, it may be difficult to construct attribute
reducts of a large-scale fuzzy covering information system directly. However,
we can compress it into a relatively smaller fuzzy covering information system
under the condition of a homomorphism and conduct the attribute reductions on
the image system. Therefore, the notion of a homomorphism may provide a more
efficient approach to dealing with large-scale fuzzy covering information
systems.
### 5.2 Data compression of dynamic fuzzy covering information systems
In Subsection 5.1, we derive a partition based on each fuzzy covering shown in
Table 2, which is useful for data compression of dynamic fuzzy covering
information systems. In the following, we discuss how to compress two types of
dynamic fuzzy covering information systems by utilizing the compression of the
original fuzzy covering information system.
Type 1: Adding a family of fuzzy coverings. By adding a fuzzy covering
$\mathscr{C}_{m+1}$ to the fuzzy covering information system
$(U_{1},\Delta_{1})$, we obtain the dynamic fuzzy covering information system
$(U_{1},\Delta)$, where $\Delta=\Delta_{1}\cup\\{\mathscr{C}_{m+1}\\}$. There
are three steps to compress the dynamic fuzzy covering information system
$(U_{1},\Delta)$. First, we obtain the partition $U_{1}/\mathscr{C}_{m+1}$ in
the sense of Definition 5.6 and get Table 4 by adding
$U_{1}/\mathscr{C}_{m+1}$ into Table 2. Then we derive the partition
$U_{1}/\Delta$ based on $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq m+1)$.
Afterwards, we define the homomorphism $f$ based on $U_{1}/\Delta$ as Example
5.8 and compress $(U_{1},\Delta)$ into a small-scale fuzzy covering
information system $(f(U_{1}),f(\Delta))$. Furthermore, the same process can
be applied to the dynamic fuzzy covering information system when adding a
family of fuzzy coverings.
Type 2: Deleting a family of fuzzy coverings. We obtain the dynamic fuzzy
covering information system $(U_{1},\Delta)$ when deleting the fuzzy covering
$\mathscr{C}_{k}\in\Delta_{1}$, where
$\Delta=\Delta_{1}-\\{\mathscr{C}_{k}\\}$. To compress the dynamic fuzzy
covering information system $(U_{1},\Delta)$, we first derive Table 5 by
canceling the partition $U_{1}/\mathscr{C}_{k}$ in Table 2\. Then we obtain
the partition $U_{1}/\Delta$ based on $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq
k-1,k+1\leq i\leq m)$ and define the homomorphism $f$ as Example 5.8.
Afterwards, $(U_{1},\Delta)$ is compressed into a small-scale fuzzy covering
information system $(f(U_{1}),f(\Delta))$. Moreover, we can compress the
dynamic fuzzy covering information system when deleting a family of fuzzy
coverings using the same approach.
In practice, it may be very costly or even intractable to construct the
compression of the dynamic fuzzy covering information system as the original
fuzzy covering information system. Thus the proposed approach based on the
compression of the original fuzzy covering information system may provide a
more efficient approach to dealing with data compression of dynamic fuzzy
covering information systems.
## 6 Conclusion and further research
In this paper, we have presented some new operations on fuzzy coverings and
investigated their properties in detail. Particularly, the lower and upper
approximation operations based on fuzzy coverings have been introduced for the
fuzzy covering approximation space. Then we have constructed a consistent
function for the communication between fuzzy covering information systems, and
pointed out that a homomorphism is a special fuzzy covering mapping between
the two fuzzy covering information systems. In addition, we have proved that
attribute reductions of the original system and image system are equivalent to
each other under the condition of a homomorphism. We have also applied the
proposed approach to attribute reductions of fuzzy covering information
systems and dynamic fuzzy covering information systems.
In future, we will further study the fuzzy covering information systems by
extending the covering rough sets and apply the proposed method to feature
selections of fuzzy covering information systems. Furthermore, we will discuss
the data compression of dynamic relation information systems and dynamic fuzzy
relation information systems. Especially, we will apply an incremental
updating scheme to maintain the compression dynamically and avoid unnecessary
computations by utilizing the compression of the original system.
## Acknowledgments
We would like to thank the anonymous reviewers very much for their helpful
comments and valuable suggestions. This work is supported by the National
Natural Science Foundation of China (NO. 11071061) and the National Basic
Research Program of China (NO. 2010CB334706).
## References
* [1] T. Beaubouef and F. E. Petry, Uncertainty modeling for database design using intuitionistic and rough set theory, Journal of Intelligent and Fuzzy Systems 20 (2009) 105-117.
* [2] Z. Bonikowski, E. Bryniarski and U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Information Sciences 107(1-4) (1998) 149-167.
* [3] D. G. Chen, C. Z. Wang and Q. H. Hu, A new approach to attributes reduction of consistent and inconsistent covering decision systems with covering rough sets, Information Sciences 177(17) (2007) 3500-3518.
* [4] T. Q. Deng, Y. M. Chen, W. L. Xu and Q. H. Dai, A novel approach to fuzzy rough sets based on a fuzzy covering, Information Sciences 177(11) (2007) 2308-2326.
* [5] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990) 191-209.
* [6] D. Dubois and H. Prade, Putting fuzzy sets and rough sets together, In: R. Slowinski (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of Rough sets Theory, Kluwer, 1992, pp. 203-232.
* [7] T. Feng, S. P. Zhang, J. S. Mi and Q. Feng, Reductions of a fuzzy covering decision system, International Journal of Modelling, Identification and Control 13(3) 225-233.
* [8] X. Ge, J. J. Li and Y. Ge, Some separations in covering approximation spaces, International Journal of Computational and Mathematical Sciences 4 (2010) 156-160.
* [9] Z. T. Gong and Z. Y. Xiao, Communicating between information systems based on including degrees, International Journal of General Systems 39 (2) (2010) 189-206.
* [10] J. W. Grzymala-Busse, Algebraic properties of knowledge representation systems, In: Proceedings of the ACM SIGART International Symposium on Methodologies for Intelligent Systems, Knoxville, 1986, pp: 432-440.
* [11] J. W. Grzymala-Busse, Rough Set and CART Approaches to Mining Incomplete Data, In: 2010 International Conference of Soft Computing and Pattern Recognition (SoCPaR), Paris, France, 2010, pp: 214-219.
* [12] J. W. Grzymala-Busse and W. A. Sedelow Jr., On rough sets and information system homomorphism, Bulletin of the polish academy of sciences: technical sciences 36 (3) (1988) 233-239.
* [13] R. Jensen and Q. Shen, Semantics-preserving dimensionality reduction: rough and fuzzy-rough-based approaches, IEEE Transactions on Knowledge and Data Engineering 16 (12) (2004) 1457-1471.
* [14] A. de Korvin, C. McKeegan and R. Kleyle, Knowledge acquisition using rough sets when membership values are fuzzy sets, Journal of Intelligent and Fuzzy Systems 6 (1998) 237-244.
* [15] T. J. Li, Y. Leung and W. X. Zhang, Generalized fuzzy rough approximation operators based on fuzzy coverings, International Journal of Approximate Reasoning 48(3) (2008) 836-856.
* [16] D. Y. Li and Y. C. Ma, Invariant characters of information systems under some homomorphisms, Information Sciences 129(1-4) (2000) 211-220.
* [17] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11(5) (1982) 341-356.
* [18] J. A. Pomykala, Approximation operations in approximation space, Bulletin of the Polish Academy of Sciences 35(9-10) (1987) 653-662.
* [19] A. M. Radzikowska and E. E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems 126 (2002) 137-155.
* [20] A. M. Radzikowska and E. E. Kerre, Fuzzy rough sets based on residuated lattices, In: Transactions on Rough Sets II, LNCS 3135, 2004, pp. 278-296.
* [21] Z. H. Shi and Z. T. Gong, The further investigation of covering-based rough sets: uncertainty characterization, similarity measure and generalized models, Information Sciences 180(19) (2010) 3745-3763.
* [22] H. Torabi, B. Davvaz and J. Behboodian, Fuzzy random events in incomplete probability models, Journal of Intelligent and Fuzzy Systems 17 (2006) 183-188.
* [23] E.C.C. Tsang, D. Cheng, J. W. T Lee and D. S. Yeung, On the upper approximations of covering generalized rough sets, In: Proceedings of the Third International Conference on Machine Leaming and Cybernetics, 2004, pp. 4200-4203.
* [24] C. Z. Wang, D. G. Chen, C. Wu and Q. H. Hu, Data compression with homomorphism in covering information systems, International Journal of Approximate Reasoning 52(4) (2011) 519-525.
* [25] C. Z. Wang, D. G. Chen and L. K. Zhu, Homomorphisms between fuzzy information systems, Applied Mathematics Letters 22(7) (2009) 1045-1050.
* [26] C. Z. Wang, C. X. Wu and D. G. Chen, A systematic study on attribute reduction with rough sets based on general binary relations, Information Sciences 178 (9) (2008) 2237-2261.
* [27] C. Z. Wang, C. X. Wu, D. G. Chen and W. J. Du, Some properties of relation information systems under homomorphisms, Applied Mathematics Letters 21(9) (2008) 940-945.
* [28] C. Z. Wang, C. X. Wu, D. G. Chen, Q. H. Hu and C. Wu, Communicating between information systems, Information Sciences 178(16) (2008) 3228-3239.
* [29] T. Yang and Q. G. Li, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning 51(3) (2010) 335-345.
* [30] Y. Y. Yao and B. X. Yao, Covering based rough set approximations, Information Sciences 200 (2012) 91-107.
* [31] L. A. Zadeh, Fuzzy sets, Information and control 8(3) (1965) 338-353.
* [32] W. Zakowski, Approximations in the space $(u,\pi)$, Demonstratio Mathematica 16 (1983) 761-769.
* [33] Z. M. Zhang, Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings, Information Sciences 198 (2012) 186-206.
* [34] Y. L. Zhang and M. K. Luo, On minimization of axiom sets characterizing covering-based approximation operators, Information Sciences 181(14) (2011) 3032-3042.
* [35] W. Zhu, Topological approaches to covering rough sets, Information Sciences 177(6) (2007) 1499-1508.
* [36] W. Zhu, Relationship between generalized rough sets based on binary relation and coverings, Information Sciences 179(3) (2009) 210-225.
* [37] P. Zhu, Covering rough sets based on neighborhoods: An approach without using neighborhoods, International Journal of Approximate Reasoning 52(3) (2011) 461-472.
* [38] W. Zhu and F. Y. Wang, Reduction and axiomization of covering generalized rough sets, Information Sciences 152 (2003) 217-230.
* [39] W. Zhu and F. Y. Wang, The fourth type of covering-based rough sets, Information Sciences 201 (2012) 80-92.
* [40] P. Zhu and Q. Y. Wen, Some improved results on communication between information systems, Information Sciences 180(18) (2010) 3521-3531.
* [41] P. Zhu and Q. Y. Wen, Homomorphisms between fuzzy information systems revisited, Applied Mathematics Letters 24(9) (2011) 1548-1553.
* [42] P. Zhu and Q. Y. Wen, A note on communicating between information systems based on including degree, International Journal of General Systems 40 (8) (2011) 837-840.
Table 1: An incomplete information system. $U$ | $structure$ | $color$ | $price$
---|---|---|---
$x_{1}$ | $bad$ | $good$ | $low$
$x_{2}$ | $\ast$ | $good$ | $high$
$x_{3}$ | $good$ | $bad$ | $high$
$x_{4}$ | $bad$ | $bad$ | $\ast$
$x_{5}$ | $good$ | $\ast$ | $low$
$x_{6}$ | $\ast$ | $bad$ | $\ast$
Table 2: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq m)$ and $\Delta_{1}$, respectively. $U_{1}$ | $\mathscr{C}_{1}$ | $\mathscr{C}_{2}$ | . | . | . | $\mathscr{C}_{m}$ | $\Delta_{1}$
---|---|---|---|---|---|---|---
$x_{1}$ | $P_{1x_{1}}$ | $P_{2x_{1}}$ | . | . | . | $P_{mx_{1}}$ | $P_{\Delta_{1}x_{1}}$
$x_{2}$ | $P_{1x_{2}}$ | $P_{2x_{2}}$ | . | . | . | $P_{mx_{2}}$ | $P_{\Delta_{1}x_{2}}$
. | . | . | . | . | . | . | .
. | . | . | . | . | . | . | .
. | . | . | . | . | . | . | .
$x_{n}$ | $P_{1x_{n}}$ | $P_{2x_{2}}$ | . | . | . | $P_{mx_{n}}$ | $P_{\Delta_{1}x_{n}}$
Table 3: The partitions based on $\mathscr{C}_{prize},\mathscr{C}_{structure},\mathscr{C}_{size},\mathscr{C}_{appearance}$ and $\Delta_{1}$, respectively. $U_{1}$ | $\mathscr{C}_{price}$ | $\mathscr{C}_{structure}$ | $\mathscr{C}_{size}$ | $\mathscr{C}_{appearance}$ | $\Delta_{1}$
---|---|---|---|---|---
$x_{1}$ | $\\{x_{1},x_{2}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{1},x_{2}\\}$
$x_{2}$ | $\\{x_{1},x_{2}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{1},x_{2}\\}$
$x_{3}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{3},x_{5}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{3},x_{5}\\}$
$x_{4}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$
$x_{5}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{3},x_{5}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{3},x_{5}\\}$
$x_{6}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{6}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{6}\\}$
$x_{7}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$
$x_{8}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$
Table 4: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq m+1)$ and $\Delta$, respectively. $U_{1}$ | $\mathscr{C}_{1}$ | $\mathscr{C}_{2}$ | . | . | . | $\mathscr{C}_{m}$ | $\mathscr{C}_{m+1}$ | $\Delta$
---|---|---|---|---|---|---|---|---
$x_{1}$ | $P_{1x_{1}}$ | $P_{2x_{1}}$ | . | . | . | $P_{mx_{1}}$ | $P_{(m+1)x_{1}}$ | $P_{\Delta x_{1}}$
$x_{2}$ | $P_{1x_{2}}$ | $P_{2x_{2}}$ | . | . | . | $P_{mx_{2}}$ | $P_{(m+1)x_{2}}$ | $P_{\Delta x_{2}}$
. | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | .
$x_{n}$ | $P_{1x_{n}}$ | $P_{2x_{2}}$ | . | . | . | $P_{mx_{n}}$ | $P_{(m+1)x_{n}}$ | $P_{\Delta x_{n}}$
Table 5: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq k-1,k+1\leq i\leq m)$ and $\Delta$, respectively. $U_{1}$ | $\mathscr{C}_{1}$ | $\mathscr{C}_{2}$ | . | . | . | $\mathscr{C}_{k-1}$ | $\mathscr{C}_{k+1}$ | . | . | . | $\mathscr{C}_{m}$ | $\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
$x_{1}$ | $P_{1x_{1}}$ | $P_{2x_{1}}$ | . | . | . | $P_{(k-1)x_{1}}$ | $P_{(k+1)x_{1}}$ | . | . | . | $P_{mx_{1}}$ | $P_{\Delta x_{1}}$
$x_{2}$ | $P_{1x_{2}}$ | $P_{2x_{2}}$ | . | . | . | $P_{(k-1)x_{2}}$ | $P_{(k+1)x_{2}}$ | . | . | . | $P_{mx_{2}}$ | $P_{\Delta x_{2}}$
. | . | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | . | .
$x_{n}$ | $P_{1x_{n}}$ | $P_{2x_{n}}$ | . | . | . | $P_{(k-1)x_{n}}$ | $P_{(k+1)x_{n}}$ | . | . | . | $P_{mx_{n}}$ | $P_{\Delta x_{n}}$
Potential referees
(1) Jerzy W. Grzymala-Busse, E-mail address: jerzy@ku.edu, Affiliations:
Department of Electrical Engineering and Computer Science, University of
Kansas, Lawrence, U. S. A.
(2) Yiyu Yao, E-mail address: yyao@cs.uregina.ca, Affiliations: Department of
Computer Science, University of Regina, Regina, Saskatchewan, Canada.
(3) Tsau Young Lin, E-mail address: tylin@cs.sjsu.edu, Affiliations:
Department of Computer Science, San Jose State University, San Jose, U. S. A.
(4) Weizhi Wu, E-mail address: wuwz@zjou.edu.cn, Affiliations: School of
Mathematics, Physics and Information Science, Zhejiang Ocean University,
Zhoushan, Zhejiang 316000, P. R. China.
(5) Wei Yao, E-mail address: yaowei0516@163.com, Affiliations: Department of
Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018,
P. R. China.
(6) Ping Zhu, E-mail address: pzhubupt@gmail.com, Affiliations: School of
Science, Beijing University of Posts and Telecommunications, Beijing 100876,
P. R. China.
|
arxiv-papers
| 2012-03-31T04:42:11 |
2024-09-04T02:49:29.229878
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guangming Lang, Qingguo Li and Lankun Guo",
"submitter": "Guangming Lang",
"url": "https://arxiv.org/abs/1204.0072"
}
|
1204.0079
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-090 LHCb-PAPER-2011-044 31 March 2012
First observation of the decay $B_{c}^{+}\rightarrow
J/\psi\pi^{+}\pi^{-}\pi^{+}$
The LHCb collaboration †††Authors are listed on the following pages.
The decay $B_{c}^{+}\rightarrow J/\psi\pi^{+}\pi^{-}\pi^{+}$ is observed for
the first time, using 0.8 fb-1 of $pp$ collisions at $\sqrt{s}=7$ TeV
collected by the LHCb experiment. The ratio of branching fractions ${\cal
B}(B_{c}^{+}\rightarrow J/\psi\pi^{+}\pi^{-}\pi^{+})/{\cal
B}(B_{c}^{+}\rightarrow J/\psi\pi^{+})$ is measured to be $2.41\pm 0.30\pm
0.33$, where the first uncertainty is statistical and the second systematic.
The result is in agreement with theoretical predictions.
Submitted to Physical Review Letters
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, H. Voss10, R. Waldi55, S. Wandernoth11, J. Wang53,
D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, K.
Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
The $B_{c}^{+}$ meson is the ground state of the $\bar{b}c$ quark pair
system111Charge-conjugate states are implied in this Letter.. Studies of its
properties are important, since it is the only meson consisting of two
different heavy quarks. It is also the only meson in which decays of both
constituents compete with each other. Numerous predictions for $B_{c}^{+}$
branching fractions have been published (for a review see e.g. Ref. [1]). To
date, no measurements exist which would allow to test these predictions, even
in ratios. $B_{c}^{+}$ production rates are about three orders of magnitude
smaller at high energy colliders than for the other $B$ mesons composed of a
$b$ quark and a light quark ($B^{+}$, $B^{0}$ and $B_{s}^{0}$). On the
experimental side, whatever is known about the $B_{c}^{+}$ meson was measured
at the Tevatron. It was discovered by the CDF experiment in the semileptonic
decay, $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}l^{+}\nu X$ [2]. This decay mode was later used to measure the
$B_{c}^{+}$ lifetime [3, 4], which is a factor of three shorter than for the
other $B$ mesons as both $b$ and $c$ quark may decay. Only one hadronic decay
mode of $B_{c}^{+}$ has been observed so far,
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$. It
was utilized by CDF [5] and DØ [6] to measure the $B_{c}^{+}$ mass222We use
mass and momentum units in which $c=1$., $6277\pm 6$ MeV [7].
In this Letter, the first observation of the decay mode
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ is presented using a data sample corresponding to
an integrated luminosity of $0.8$ fb-1 collected in 2011 by the LHCb detector
[8], in $pp$ collisions at the LHC at $\sqrt{s}=7$ TeV. The branching fraction
for this decay is expected to be $1.5-2.3$ times higher than for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ [9,
10]. However, the larger number of pions in the final state results in a
smaller total detection efficiency due to the limited detector acceptance. We
measure the $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ branching fraction relative to that for the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay and test the above theoretical predictions.
The LHCb detector [8] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5 GeV to 0.6% at 100 GeV, and an impact parameter (IP) resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and multiwire proportional chambers. The muon
system, electromagnetic and hadron calorimeters provide the capability of
first-level hardware triggering. The single and dimuon hardware triggers
provide good efficiency for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ events. Here, $\pi^{+}[\pi^{-}\pi^{+}]$
stands for either $\pi^{+}$ or $\pi^{+}\pi^{-}\pi^{+}$ depending on the
$B_{c}^{+}$ decay mode. Events passing the hardware trigger are read out and
sent to an event-filter farm for further processing. Here, a software-based
two-stage trigger reduces the rate from 1 MHz to about 3 kHz. The most
efficient software triggers [11] for this analysis require a charged track
with transverse momentum ($p_{\rm T}$) of more than $1.7$ GeV ($p_{\rm T}>1.0$
GeV if identified as muon) and with an IP to any primary $pp$-interaction
vertex (PV) larger than $100$ $\mu$m. A dimuon trigger requiring $p_{\rm
T}(\mu)>0.5$ GeV, large dimuon mass, $M(\mu^{+}\mu^{-})>2.7$ GeV, and with no
IP requirement complements the single track triggers. At the final stage, we
either require a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ candidate with $p_{\rm T}>2.7$ GeV ($>1.5$
GeV in the first 42% of data) or a muon-track pair with significant IP.
In the subsequent offline analysis of the data,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
candidates are selected with the following criteria: $p_{\rm T}(\mu)>0.9$ GeV,
$p_{\rm T}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})>3.0$ GeV ($>1.5$ GeV
in the first 42% of data), $\chi^{2}$ per degree of freedom of the two muons
forming a common vertex, $\chi^{2}_{\rm vtx}(\mu^{+}\mu^{-})/\hbox{\rm
ndf}<9$, and a mass window $3.04<M(\mu^{+}\mu^{-})<3.14$ GeV. We then find
$\pi^{+}\pi^{-}\pi^{+}$ combinations consistent with originating from a common
vertex with $\chi^{2}_{\rm vtx}(\pi^{+}\pi^{-}\pi^{+})/\hbox{\rm ndf}<9$, with
each pion separated from all PVs by at least three standard deviations
($\chi^{2}_{\rm IP}(\pi)>9$), and having $p_{\rm T}(\pi)>0.25$ GeV. A loose
kaon veto is applied using the particle identification system. A five-track
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}\pi^{+}$ vertex is
formed ($\chi^{2}_{\rm vtx}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+})/\hbox{\rm ndf}<9$). To look for candidates in the
normalization mode, $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$, the criteria $p_{\rm T}(\pi)>1.5$ GeV and $\chi^{2}_{\rm
vtx}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})/\hbox{\rm ndf}<16$
are used. All $B_{c}^{+}$ candidates are required to have $p_{\rm T}>4.0$ GeV
and a decay time of at least $0.25$ ps. When more than one PV is
reconstructed, that which gives the smallest IP significance for the
$B_{c}^{+}$ candidate is chosen. The invariant mass of a
$\mu^{+}\mu^{-}\pi^{+}[\pi^{-}\pi^{+}]$ combination is evaluated after the
muon pair is constrained to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mass and all final state particles are constrained to form a common vertex.
Further background suppression is provided by an event selection based on a
likelihood ratio. In the case of uncorrelated input variables this provides
the most efficient discrimination between signal and background. The overall
likelihood is a product of the probability density functions (PDFs), ${\cal
P}(x_{i})$, for the four sensitive variables ($x_{i}$): smallest
$\chi^{2}_{\rm IP}(\pi)$ among the pion candidates, $\chi^{2}_{\rm
vtx}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}])/\hbox{\rm ndf}$, $B_{c}^{+}$ candidate IP
significance, $\chi^{2}_{\rm IP}(B_{c})$, and cosine of the largest opening
angle between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and pion
candidates in the plane transverse to the beam. The latter peaks at positive
values for the signal as the $B_{c}^{+}$ meson has a high transverse momentum.
Background events that combine particles from two different $B$ mesons peak at
negative values, whilst background events that include random combinations of
tracks are uniformly distributed. The signal PDFs, ${\cal P}_{\rm
sig}(x_{i})$, are obtained from a Monte Carlo simulation of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$ decays. The background PDFs, ${\cal P}_{\rm
bkg}(x_{i})$, are obtained from the data with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$
invariant mass in the range $5.35-5.80$ GeV or $6.80-8.50$ GeV (far-
sidebands). A logarithm of the ratio of the signal and background PDFs is
formed: ${\rm DLL}_{\rm sig/bkg}=-2\sum_{i=1}^{4}\ln({\cal P}_{\rm
sig}(x_{i})/{\cal P}_{\rm bkg}(x_{i}))$. Requirements on the log-likelihood
ratio, ${\rm DLL}_{\rm sig/bkg}<-5$ for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ and ${\rm DLL}_{\rm sig/bkg}<-1$ for
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$,
have been chosen to maximize $N_{\rm sig}/\sqrt{N_{\rm sig}+N_{\rm bkg}}$,
where $N_{\rm sig}$ is the expected
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$ signal yield and the $N_{\rm bkg}$ is the
background yield in the $B_{c}^{+}$ peak region ($\pm 2.5\,\sigma$). The
absolute normalization of $N_{\rm sig}$ and $N_{\rm bkg}$ is obtained from a
fit to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$ invariant mass distribution with ${\rm
DLL}_{\rm sig/bkg}<0$, while their dependence on the ${\rm DLL}_{\rm sig/bkg}$
requirement comes from the signal simulation and the far-sidebands,
respectively. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}[\pi^{-}\pi^{+}]$ mass distributions after applying all
requirements are shown in Fig. 1. To determine the signal yields, a Gaussian
signal shape with position and width as free parameters is fitted to these
distributions on top of a background assumed to be an exponential function
with a second order polynomial as argument. We observe $135\pm 14$
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ and $414\pm 25$
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
signal events. Using different signal and background parameterizations in the
fits, the ratio of the signal yields changes by up to 3%.
Figure 1: Invariant mass distribution of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ (top) and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
(bottom) candidates. The maximum likelihood fits of $B_{c}^{+}$ signals are
superimposed. Figure 2: Invariant mass distribution of the
$\pi^{+}\pi^{-}\pi^{+}$ combinations for the sideband-subtracted
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ data (points) and signal simulation (lines). The
solid blue line corresponds to the BLL simulations, the PH model is shown as a
green dashed line and the PHPOL model is shown as a red dotted line. All error
bars are statistical.
The ratio of event yields is converted into a measurement of the ratio of
branching fractions ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$,
where we rely on the simulation for the determination of the ratio of event
selection efficiencies. The production of $B_{c}^{+}$ mesons is simulated
using the BCVEGPY generator [12, 13] which gives a good description of the
observed transverse momentum and pseudorapidity distributions in our data. The
simulation of the two-body
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
decay takes into account the spins of the particles and contains no
ambiguities. The phenomenological model by Berezhnoy, Likhoded and Luchinsky
[10, 14] (BLL) is used to simulate
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ decays. This model, which is based on amplitude
factorisation into hadronic and weak currents, implements
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}W^{+*}$
axial-vector form-factors and a $W^{+*}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ decay
via the exchange of the virtual $a_{1}^{+}(1260)$ and $\rho^{0}(770)$
resonances. Since it is not possible to identify which of the same-sign pions
originates from the $\rho^{0}$ decay, the two $\rho^{0}$ paths interfere. To
explore the model dependence of the efficiency we also use two phase-space
models, implementing the same decay chain with no interference and with either
no polarization in the decay (PH) or helicity amplitudes of 0.46, 0.87 and
0.20 for $+1$, $0$ and $-1$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
helicities (PHPOL), respectively. For the helicity structure in the PHPOL
model, we use the expectation for the $B^{+}\rightarrow D^{*0}a_{1}^{+}(1260)$
decay based on QCD factorisation [15]. The background-subtracted
distribution333For comparisons between the data and simulation we use the data
within $\pm 2.5\,\sigma$ of the observed peak position in the $B_{c}^{+}$ mass
(signal region). We subtract the background distributions as estimated from
the $\pm(5-30)\,\sigma$ near-sidebands. of the $M(\pi^{+}\pi^{-}\pi^{+})$ mass
for the $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ data shown in Fig. 2 exhibits an
$a_{1}^{+}(1260)$ peak and favours the BLL model. The $\rho^{0}(770)$ peak in
the $M(\pi^{+}\pi^{-})$ mass distribution shown in Fig. 3 is smaller than in
the two phase-space models, but more pronounced than in the BLL model, with
the tail favouring the BLL model. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ helicity angle distribution shown in Fig. 4 disfavours the model with
no polarization. Since the BLL model gives the best overall description of the
data, we choose it to evaluate the central value of the ratio of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ to
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
efficiencies, $0.135\pm 0.004$, and use the phase-space models to quantify the
systematic uncertainty. The phase-space models produce relative efficiencies
different by $-9\%$ (PHPOL) and $+5\%$ (PH). We assign a 9% systematic
uncertainty to the model dependence of
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ efficiency.
Figure 3: Invariant mass distribution of the $\pi^{+}\pi^{-}$ combinations
(two entries per $B_{c}^{+}$ candidate) for the sideband-subtracted
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ data (points) and signal simulation (lines). The
solid blue line corresponds to the BLL simulations, the PH model is shown as a
green dashed line and the PHPOL model is shown as a red dotted line. All error
bars are statistical. Figure 4: Distributions of the cosine of the angle
between the $\mu^{+}$ and $B_{c}^{+}$ boosted to the rest frame of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson for the sideband-
subtracted $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}$ (top) and
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ (bottom) data (points) and signal simulation
(lines). In the bottom plot, the solid blue line corresponds to the BLL
simulations, the PH model is shown as a green dashed line and the PHPOL model
is shown as a red dotted line. All error bars are statistical.
The distribution of the $M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})$ mass has an isolated peak of four events at the
$\psi(2S)$ mass. From the $B_{c}^{+}$ sidebands we expect $0.50\pm 0.25$
background events in this peak. This is consistent with $3.6\pm 0.6$ expected
$B_{c}^{+}\rightarrow\psi(2S)\pi^{+}$ events, assuming ${\cal
B}(B_{c}^{+}\rightarrow\psi(2S)\pi^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+})$
equals to ${\cal B}(B^{+}\rightarrow\psi(2S)\pi^{+})/{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})=0.52\pm 0.07$ [7] after subtracting 10% to account for the
phase-space difference. Since this contribution is only $(2.6\pm 1.5)\%$ of
the $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+}$ signal yield, we do not subtract it and assign a
$2\%$ systematic uncertainty to the ratio of the branching fractions due to
the efficiency difference between the
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}a_{1}(1260)$
and $B_{c}^{+}\rightarrow\psi(2S)\pi^{+}$,
$\psi(2S)\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays, as obtained from the simulation.
Other systematic uncertainties are due to limited knowledge of the $B_{c}^{+}$
lifetime [7] ($4\%$), uncertainty in the simulation of charged tracking
efficiency ($5\%$), trigger ($4\%$) and the kaon veto ($5\%$). Summing all
contributions in quadrature, the total systematic error on the branching
fractions ratio amounts to 14%. As a result, we measure the branching fraction
ratio
$\frac{{\cal B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+})}{{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})}=2.41\pm 0.30\pm 0.33,$
where the first uncertainty is statistical and the second systematic.
The obtained result can be compared to theoretical predictions; these assume
factorisation into $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}W^{+*}$ and $W^{+*}\rightarrow\pi^{+}[\pi^{-}\pi^{+}]$. The
contributions of strong interactions to
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}W^{+*}$ are
included in form-factors which can be calculated in various approaches such as
a non-relativistic quark model or sum rules. The coupling of a single pion to
a $W^{+*}$ is described by the pion decay constant. The coupling of three
pions to a $W^{+*}$ is measured in
$\tau^{-}\rightarrow\nu_{\tau}\pi^{-}\pi^{+}\pi^{-}$ decays, which are
dominated by the $a_{1}(1260)$ resonance. The prediction by Rakitin and
Koshkarev, using the no-recoil approximation in
$B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}W^{+*}$, is
${\cal B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})=1.5$ [9]. Likhoded and Luchinsky used three different
approaches to predict the form factors and obtained ${\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}\pi^{+})/{\cal
B}(B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+})=1.9,2.0$ and $2.3$, respectively [10]. Our result prefers the
latter predictions. It is also consistent with ${\cal
B}(B^{+}\rightarrow\bar{D}^{*0}\pi^{+}\pi^{-}\pi^{+})/{\cal
B}(B^{+}\rightarrow\bar{D}^{*0}\pi^{+})=2.00\pm 0.25$ [7], which is mediated
by similar decay mechanisms, and with a similiar ratio of phase-space factors.
Our result constitutes the first test of theoretical predictions for branching
fractions of $B_{c}^{+}$ decays.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] Quarkonium Working Group, N. Brambilla et al., Heavy quarkonium physics, arXiv:hep-ph/0412158, published as CERN Yellow Report CERN-2005-005
* [2] CDF collaboration, F. Abe et al., Observation of the $B_{c}$ meson in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 81 (1998) 2432, arXiv:hep-ex/9805034
* [3] CDF collaboration, A. Abulencia et al., Measurement of the $B_{c}^{+}$ meson lifetime using $B_{c}^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}e^{+}\nu_{e}$, Phys. Rev. Lett. 97 (2006) 012002, arXiv:hep-ex/0603027
* [4] DØ collaboration, V. M. Abazov et al., Measurement of the lifetime of the $B_{c}^{\pm}$ meson in the semileptonic decay channel, Phys. Rev. Lett. 102 (2009) 092001, arXiv:0805.2614
* [5] CDF collaboration, T. Aaltonen et al., Observation of the decay $B_{c}^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{\pm}$ and measurement of the $B_{c}^{\pm}$ mass, Phys. Rev. Lett. 100 (2008) 182002, arXiv:0712.1506
* [6] DØ collaboration, V. M. Abazov et al., Observation of the $B_{c}$ meson in the exclusive decay $B_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi$, Phys. Rev. Lett. 101 (2008) 012001, arXiv:0802.4258
* [7] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
* [8] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [9] A. Rakitin and S. Koshkarev, Hadronic $B_{c}$ decays as a test of $B_{c}$ cross section, Phys. Rev. D81 (2010) 014005, arXiv:0911.3287
* [10] A. K. Likhoded and A. V. Luchinsky, Light hadron production in $B_{c}\rightarrow J/\psi+X$ decays, Phys. Rev. D81 (2010) 014015, arXiv:0910.3089
* [11] V. Gligorov, C. Thomas, and M. Williams, The HLT inclusive $B$ triggers, LHCb-PUB-2011-016
* [12] C.-H. Chang, C. Driouichi, P. Eerola, and X. G. Wu, BCVEGPY: an event generator for hadronic production of the $B_{c}$ meson, Comput. Phys. Commun. 159 (2004) 192, arXiv:hep-ph/0309120
* [13] C.-H. Chang, J.-X. Wang, and X.-G. Wu, An upgraded version of the generator BCVEGPY2.0 for hadronic production of $B_{c}$ meson and its excited states, Comput. Phys. Commun. 175 (2006) 624, arXiv:hep-ph/0604238
* [14] A. Berezhnoy, A. Likhoded, and A. Luchinsky, BC_NPI module for the analysis of $B_{c}\rightarrow J/\psi+n\pi$ and $B_{c}\rightarrow B_{s}+n\pi$ decays within the EvtGen package, arXiv:1104.0808
* [15] J. L. Rosner, Determination of pseudoscalar-charmed-meson decay constants from B-meson decays, Phys. Rev. D42 (1990) 3732
|
arxiv-papers
| 2012-03-31T06:49:54 |
2024-09-04T02:49:29.239919
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C. Bauer,\n Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea,\n A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P.\n Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B.\n Couturier, G.A. Cowan, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, I.\n De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula,\n P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C.\n Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P.\n Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong,\n R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, M. Kaballo, S.\n Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg,\n M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, S. Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, B.K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R. Plackett, S. Playfer, M. Plo\n Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E.\n Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B.\n Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica,\n S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A.\n Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni,\n G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis,\n M. Veltri, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, R.\n Waldi, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams,\n M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K.\n Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Tomasz Skwarnicki",
"url": "https://arxiv.org/abs/1204.0079"
}
|
1204.0104
|
# Kinetic Monte Carlo model of epitaxial graphene growth
Bartomeu Monserrat Sánchez
Imperial College London
Project code: CMTH–Vvedensky–3
Supervisor: D. D. Vvedensky
Assessor: W. M. C. Foulkes
(May 3, 2011)
###### Abstract
In this thesis we present a kinetic Monte Carlo model for the description of
epitaxial graphene growth. Experimental results suggest a growth mechanism by
which clusters of 5 carbon atoms are an intermediate species necessary for
nucleation and island growth. This model is proposed by experimentally
studying the velocity of growth of islands which is a highly nonlinear
function of adatom concentration. In our simulation we incorporate this
intermediate species and show that it can explain all other experimental
observations: the temperature dependence of the adatom nucleation density, the
equilibrium adatom density and the temperature dependence of the equilibrium
island density. All these processes are described only by the kinematics of
the system.
### Acknowledgments
The research described in this thesis corresponds to the final year Physics
MSci project. I was mentored by Dimitri Vvedensky and benefited from his
knowledge continuously. I am really fortunate to have worked with such an
inspiring supervisor.
I collaborated closely with Jonathan Lloyd-Williams, who contributed
enormously to this work. Many interesting discussions allowed us to progress
steadily. Each of us kept their own code capable of producing all the data
presented in this thesis. However, we split some of the analysis and I thank
Jonathan for providing the data for the $j=2$ and rate equations comparison
plots.
Mireia Crispín Ortuzar, Raphael Houdmont, Josep Monserrat and Kevin Troyano
read the manuscript and their comments and suggestions improved it greatly. I
am really thankful to all of them. Also, they all made my life in London
interesting, amusing and fun, together with Freddie, Daniel, Ryan, Bhavika,
Grace, Helena, Virginia and many others.
Mireia has been my inspiration and has pushed me forward with her love in
every enterprise I have taken. Finally, I would like to thank my parents for
giving me the opportunity to study at Imperial and always supporting me in my
studies.
###### Contents
1. 1 Introduction
1. 1.1 Scope of this thesis
2. 1.2 Thesis outline
2. 2 Graphene and epitaxial growth
1. 2.1 Graphene
1. 2.1.1 Physical properties
2. 2.1.2 Technological applications
2. 2.2 Epitaxial growth
1. 2.2.1 Theoretical description
2. 2.2.2 Experimental techniques
3. 3 Experimental and theoretical results on epitaxial graphene growth
1. 3.1 Experimental results
1. 3.1.1 Kinetics of epitaxial graphene growth
2. 3.1.2 Graphene nanoclusters
2. 3.2 Theoretical results
3. 3.3 Kinetic Monte Carlo model
4. 4 Kinetic Monte Carlo
1. 4.1 Theoretical studies and computer simulations
2. 4.2 Kinetic Monte Carlo
1. 4.2.1 N-fold way algorithm
2. 4.2.2 Use of kinetic Monte Carlo models to describe epitaxial growth
5. 5 Standard kinetic Monte Carlo model of epitaxial growth
1. 5.1 Description
2. 5.2 Results and discussion
1. 5.2.1 Physical lattice
2. 5.2.2 Island size distribution
3. 5.2.3 Adatom density
4. 5.2.4 Island density
5. 5.2.5 Error analysis
6. 6 Epitaxial graphene growth kinetic Monte Carlo model
1. 6.1 Description
2. 6.2 Results and discussion
1. 6.2.1 Adatom density at nucleation
2. 6.2.2 Adatom density at equilibrium
3. 6.2.3 Island density at equilibrium
3. 6.3 Error analysis: island growth velocity
7. 7 Conclusions
1. 7.1 Summary
2. 7.2 Further work
8. A Island growth model
9. B Island size distribution
10. C Rate equations and kinetic Monte Carlo comparison
## Chapter 1 Introduction
### 1.1 Scope of this thesis
Graphene is a 2-dimensional layer of graphite, formed by carbon atoms arranged
in an hexagonal lattice. For a long time, graphene was used as the starting
point for the theoretical study of carbon allotropes, such as carbon
nanotubes, which can be described by the rolling of a graphene sheet into the
form of a cylinder. But it was thought that graphene was merely a theoretical
construct because thermal fluctuations would render the isolated 2-dimensional
sheets unstable. Nonetheless, it was reported by Novoselov and co-workers in
2004 [1] that graphene sheets had been isolated and some of their physical
properties measured. Since then, it has attracted the attention of the physics
and materials communities [2] due to its interesting physical properties and
its potential for technological applications. The 2010 Physics Nobel Prize was
awarded to Andre Geim and Konstantin Novoselov
> for groundbreaking experiments regarding the two-dimensional material
> graphene
as described by the awarding body.
Recent efforts have focused on synthesising graphene sheets by means of
epitaxial growth, one of the most promising routes to large scale and high
quality graphene production. There exist ample studies characterising the
properties of graphene sheets grown on a variety of substrates. However, very
little work has been done on the growth kinetics of the graphene sheets. The
first experimental studies only became available in 2008 [3, 4]. The
remarkable behaviour observed in these studies is very different from that of
known metal-on-metal growth systems. Initial theoretical studies inspired by
the experimental findings have been conducted using a rate equations approach
[5], and these studies have been able to describe the main features of the
experimental results.
In this thesis we present a kinetic Monte Carlo model for theoretical studies
of epitaxial graphene growth kinetics. The model is based on the basic
principles found experimentally and used in the rate equations study, and
provides further insight into the physical processes governing the system.
### 1.2 Thesis outline
Chapter 2 is organised in two parts, and it describes some background
information relevant to the project. First we describe the physical properties
and technological potential of graphene, which have made it an attractive
system for both the physics and materials communities. In the second part we
present epitaxial growth, discussing the physical processes involved together
with the experimental techniques used in the field.
In Chapter 3 we give a detailed overview of the current understanding of
epitaxial graphene growth. We survey the most relevant experimental and
theoretical results describing the kinetics of epitaxial graphene growth, and
abstract the principal conclusions to be incorporated in a model to study the
system further.
In Chapter 4 we introduce in detail the kinetic Monte Carlo (KMC) technique,
focusing on the N-fold way algorithm. We also provide motivation for the use
of this method in the study of epitaxial growth systems.
In Chapter 5 we present a simple standard KMC model for epitaxial growth,
exemplifying the N-fold way algorithm. This simulation will be used as a
starting point for more complex models appropriate for the study of the
graphene system.
In Chapter 6 we present a KMC model that includes the main features observed
in graphene growth experiments. All the experimental observations are
explained physically within this model, identifying the most important
processes governing the kinetics of epitaxial graphene growth.
In Chapter 7 we summarise the most relevant results and discuss lines of
research that can be followed from the present work.
## Chapter 2 Graphene and epitaxial growth
In this chapter we first present the most relevant physical properties of
graphene and discuss its technological potential. We then look at both
theoretical foundations and experimental techniques in epitaxial growth, which
is most probably the method of choice for large scale and high quality
graphene production.
### 2.1 Graphene
#### 2.1.1 Physical properties
The physical properties of graphene were recently reviewed in Ref.[6].
Graphene is a 2-dimensional layer formed by sp2-hybridised carbon atoms
arranged in a hexagonal lattice and separated by $a\simeq$ 1.42 Å. The hybrid
orbitals give rise to $\sigma$ bonds between atoms. The remaining orbital,
$p_{z}$, is perpendicular to the planar structure and forms covalent bonds
between neighbouring carbon atoms, leading to a half filled $\pi$ band. This
band can be described using a tight-binding approach with a single hopping
matrix element between neighbouring atoms $-t$. The resulting band structure,
first calculated by Wallace in 1947 [7], has a dispersion relation
$E(\textbf{k})=\pm
t\sqrt{3+2\cos(\sqrt{3}k_{y}a)+4\cos\left(\frac{\sqrt{3}}{2}k_{y}a\right)\cos\left(\frac{3}{2}k_{x}a\right)}$
(2.1)
for wavevector ($k_{x}$,$k_{y}$).
Figure 2.1: Electronic dispersion relation of graphene obtained with a tight-
binding calculation with nearest neighbour interactions, given by Eq.(2.1).
The energy $E(\textbf{k})$ is in units of $t$.
Graphene is a zero gap semiconductor, where the Fermi surface consists only of
six Fermi points at the edge of the Brillouin zone, as shown in Fig. 2.1.
Expanding about the Fermi points $\textbf{k}_{F}=\textbf{k}-\textbf{q}$ in
reciprocal space we find $E_{\pm}(\textbf{q})\approx\pm
v_{F}|\textbf{q}|+\mathcal{O}[(q/k_{F})^{2}]$ for constant $v_{F}\simeq$ 106
ms-1. This parallels the dispersion relation of ultrarelativistic particles
described by the massless Dirac equation, so the Fermi points are commonly
called Dirac points. It is this dispersion that determines most of the
singular physical properties of graphene and it means that graphene is a
laboratory condensed matter system to test (2+1)–dimensional quantum
electrodynamics [8].
As reported in Ref.[9] the experimental study of some of the physical
properties confirmed the existence of massless Dirac carriers in graphene. For
instance, their cyclotron mass depends on the square root of the density of
states, and the integer quantum Hall effect occurs at half-integer filling
factors, both characteristic of massless Dirac fermion systems.
#### 2.1.2 Technological applications
The physical properties of graphene result in large carrier mobilities that
persist at room temperature, even with the presence of doping species. This
means graphene holds the potential to become a replacement for silicon in the
electronics industry [2], along with a wider variety of applications.
Transistors and diodes need the presence of a band-gap for their operation, so
standard graphene sheets are not appropriate. Researchers have explored
alternative graphene-based structures with the presence of a band-gap, for
instance graphene nanoribbons [10, 11] where the band-gap is proportional to
the ribbon width, or graphene nanomeshes [12] where the graphene sheets are
punched with an array of nanoscale holes. Graphene-made transistors in the GHz
scale were recently reported by IBM researchers [13, 14, 15] with performances
superior to those of similar silicon transistors.
Other examples are studies on the large heat conductivity of graphene [16, 17]
with potential applications in nanoelectronics where large heat dissipation is
needed.
### 2.2 Epitaxial growth
Epitaxial growth is the name given to the process of producing epitaxial thin
films on substrates. It is a widespread technique with applications ranging
from the production of semiconducting devices to nanotechnology and it has
also attracted the scientific community because of the complex atomic
processes involved.
#### 2.2.1 Theoretical description
Epitaxial growth can be classified in three so-called modes, first introduced
in the seminal work by Bauer [18]. Following Ref.[19], the different growth
modes can be understood by thermodynamic arguments. In the layer, or Frank–van
der Merwe mode, the atoms are more strongly attracted to the substrate than to
themselves, and the epitaxial film is formed layer after layer.
Quantitatively, it arises in the deposition of material A on B when the
surface free energies $\gamma_{i}$ for $i=A,B$ obey
$\gamma_{A}+\gamma_{int}<\gamma_{B}$ (2.2)
for interface free energy $\gamma_{int}$ between surfaces A and B. This
follows because the free energy needs to be minimised at equilibrium, and the
inequality requires to maximise the area covered by deposit A. The island, or
Volmer–Weber mode, results when the atoms are more strongly attracted by each
other than to the substrate, and multilayer (3-dimensional) islands form.
Quantitatively, $\gamma_{A}+\gamma_{int}>\gamma_{B}$. A third hybrid mode
termed Stranski–Krastanov growth mode, or layer-plus-island, arises because
the interface energy $\gamma_{int}$ increases as the layer thickness
increases, so island growth starts as a layer mode but turns into an island
mode.
In the present work we only consider submonolayer growth when the first layer
is forming, because we only reach fractional coverages of the lattice and the
system is purely 2-dimensional.
In the atomic regime there are many processes during deposition and growth.
Atoms are deposited at a certain rate on the substrate, and then undergo a
series of processes. They can re-evaporate, diffuse over the substrate or
along island edges, nucleate to form islands, join growing islands or detach
from islands. Temperature is a key parameter because the different atomic
processes are thermally activated. In the present work we incorporate these
atomistic processes into a kinetic Monte Carlo model.
#### 2.2.2 Experimental techniques
Molecular beam epitaxy (MBE) [20] is the most widespread technique used in
epitaxial growth. A beam of atoms or molecules is deposited on a previously
prepared substrate kept at high temperatures to allow the arriving particles
to diffuse over its surface. The deposition is carried out under ultra-high
vacuum conditions in order to minimise impurities. Control over the beam
allows films grown using MBE to be of very high quality and to have the
desired properties. Typical deposition rates are $\sim$1 ML/s, which are high
enough to significantly reduce the incorporation of impurities into the
growing material. Another technique used in epitaxial growth is vapour phase
epitaxy where the substrate is placed in contact with a gas containing the
deposit elements, and reactions between the two lead to epitaxial growth.
The grown sheets can be observed with high accuracy using different microscopy
techniques such as scanning tunneling microscopy or atomic force microscopy.
Furthermore, the growing process can be monitored using low energy electron
microscopy (LEEM). For example, in some experiments [3, 4] relevant for the
work described in this thesis the changes in reflectivity of a LEEM are used
to infer the carbon adatom concentration on the substrate.
## Chapter 3 Experimental and theoretical results on epitaxial graphene
growth
Epitaxial growth is one of the most promising techniques currently being
explored to synthesise high quality graphene sheets with the appropriate
properties for technological applications. Much ongoing research concentrates
in the production of such epitaxial sheets and their characterisation for a
variety of substrates with different physical properties have been reported.
However, studies of the growth kinetics of graphene are just beginning both
experimentally and theoretically and here we present a review of the current
status of such studies.
### 3.1 Experimental results
In this section we present the most relevant experimental results that
motivated the present work. They are the first that look at the kinetics of
the growth of epitaxial graphene.
#### 3.1.1 Kinetics of epitaxial graphene growth
The most relevant experimental observations of the kinetics of epitaxial
graphene growth have been reported by Loginova and co-workers in Refs.[3, 4].
They deposit pure carbon atoms and ethylene molecules on ruthenium Ru(0001)
and iridium Ir(111) substrates under ultra-high vaccum conditions and high
substrate temperatures. Low Energy Electron Microscopy (LEEM) is used to
monitor the growth of graphene islands and the local carbon adatom
concentration on the substrate.
Figure 3.1: Typical adatom density $n$ curve as a function of time found in
epitaxial growth experiments, adapted from Loginova et al. [3].
A typical adatom concentration profile that will appear throughout this work
is as shown in Fig. 3.1. A deposition flux $F$ is turned on at about time $t=$
150 s. Initially the adatom concentration increases almost linearly with time
due to the constant flux, in the example given of $F=$ 0.0023 ML/s, small for
typical epitaxial experiments. At about $t=$ 250 s the adatom density curve
reaches a maximum, called $n_{nuc}$ because it approximately corresponds to
the onset of nucleation. Islands start nucleating and adatoms start
disappearing from the substrate because they attach to islands. This decrease
continues for some 100 s until an equilibrium between islands and adatoms is
reached. Later on, at $t=$ 700 s, the deposition flux is turned off, quickly
leading to a different equilibrium concentration between adatoms and islands,
smaller than the previous equilibrium because adatoms are no longer being
incorporated externally into the system. The adatom density in this regime is
labelled $n_{eq}$.
Note the definition of $n_{nuc}$ can be slightly confusing in systems where
the critical island size is not well-defined. The critical island size $i$ is
an island size above which adatom attachment can be considered as
irreversible, so that nucleation can be defined as the process of an island
going from size $i\rightarrow(i+1)$. Then, in systems with no critical island
size, large clusters that eventually lead to islands can start appearing
before the maximum of the adatom density profile is reached. Throughout this
work we are going to define $n_{nuc}$ as the maximum of the adatom density
profile.
Figure 3.2: Experimental observation of nonlinear island growth velocity as a
function of adatom concentration. In the plot, $c^{nucl}$ corresponds to
$n_{nuc}$ in the text. Taken from Loginova et al. [3].
In their studies, Loginova and co-workers find that the island growth velocity
presents a nonlinear dependence on the carbon adatom concentration as shown in
Fig. 3.2. The island growth velocity $v$ is defined as $v=P^{-1}dA/dt$ for
island perimeter $P$ and area $A$. In most growth systems the island growth
velocity is found to be proportional to the supersaturation of adatoms, i.e.
proportional to the difference in adatom concentration $n$ and adatom
equilibrium concentration $n_{eq}$, $v=C(n-n_{eq})$ for constant $C$. To
explain the nonlinear relationship found for graphene growth, Loginova and co-
workers propose a model in which the energy barrier for monomer attachment to
islands is larger than the barrier for the formation of clusters of $m$ carbon
atoms and their posterior attachment to islands. They assume that the growth
velocity is proportional to the supersaturation of clusters rather than
adatoms. Following Ref.[3], the concentration of $m$-clusters $c^{(m)}$ in a
supersaturated adatom sea has an exponential dependence on the energy
difference between $m$ isolated carbon atoms and the energy needed to form an
$m-$atom cluster $E_{m}$,
$c^{(m)}=e^{(m\mu-
E_{m})/k_{B}T}=\left(\frac{n}{n_{eq}}\right)^{m}e^{-E_{m}/k_{B}T},$ (3.1)
where the sea of carbon adatoms is assumed to be an ideal lattice gas with
carbon chemical potential $\mu=k_{B}T\ln(n/n_{eq})$. The island growth
velocity as a function of adatom concentration is then
$v=C_{m}(c^{(m)}-c^{(m)}_{eq})=B\left[\left(\frac{n}{n_{eq}}\right)^{m}-1\right]$
(3.2)
where $B=C_{m}e^{-E_{m}/k_{B}T}$ for $C_{m}$ the proportionality constant in
the velocity dependence in cluster supersaturation. Fitting the data in Fig.
3.2, Loginova and co-workers find a best estimate of $m\simeq$ 5\. This means
that in their model clusters formed by 5 carbon atoms are an intermediate
species that determines the growth kinetics of epitaxial graphene growth. This
intermediate species could be formed on the graphene free lattice and diffuse
over the substrate to attach to islands as is assumed throughout Ref.[3]. This
model is the one we are going to take in our KMC approach to the description
of the system. However, it is pointed out by Loginova and co-workers that the
5-clusters could instead form only near island edges at the moment of
attachment to graphene.
Figure 3.3: Adatom density at nucleation $n_{nuc}$ ($c^{nucl}$ in the plot)
and at equilibrium $n_{eq}$ ($c^{eq}$ in the plot) as a function of
temperature. The data corresponds to deposition of carbon (filled labels) or
ethylene (emptly labels). Taken from Loginova et al. [4].
Further observations in the graphene growth system that differ from other
known growth scenarios are the temperature dependence of adatom density at
nucleation and the adatom density at equilibrium.
It can be seen in Fig. 3.3 that the nucleation density increases with
increasing temperature. Most growth systems are diffusion limited, so that
increasing the temperature results in higher adatom mobility and earlier
nucleation. Therefore, the nucleation density usually decreases with
increasing temperature unlike for graphene.
It can also be observed in Fig. 3.3 that the nucleation concentration is
roughly twice as large as the equilibrium density $n_{nuc}\sim 2n_{eq}$. This
indicates a small energy barrier to adatom detachment from graphene sheets and
a correspondingly large attachment barrier for adatoms to growing graphene.
This observation is in agreement with the model by which the dominant species
in the kinetics of graphene growth is an intermediate 5-cluster.
Another experimental observation that remains unpublished is the behaviour of
the island density at equilibrium. In agreement with known growth systems
(described below in Chapter 5) the island density decreases with increasing
temperature. However, the decrease in the graphene system is found to be much
larger than in other systems, suggesting the existence of a mechanism that
allows the occurence of a large number of nucleations at low temperatures. We
thank Elena (Loginova) Starodub for providing the data related to island
density.
We finally note that the growth mechanism of graphene is different for other
metal substrates. For instance, for Ru(0001) and Ir(111), the deposition of
molecules containing hydrogen and carbon decompose rapidly and growth is
determined by individual carbon atoms. However, theoretical and experimental
studies [21, 22] indicate that graphene growth on copper is determined by the
deposited molecules, and that only at late stages of the graphene formation
process is hydrogen released from the composites.
#### 3.1.2 Graphene nanoclusters
Another set of experiments [23, 24] looks more carefully at the very initial
steps of graphene growth when nucleation occurs. The studies are both on
Ru(0001) and Ir(111) as the experiments by Loginova and co-workers reported
above. Small carbon nanoclusters of sizes of the order of tens of carbon atoms
are observed on both substrates. They are described as a possible predecessor
for graphene islands, and could be understood as a critical island size for
graphene.
Figure 3.4: Images of coronene and dome-like carbon nanoclusters on Ru(0001),
together with a depiction of the structure of the clusters. Taken from Cui et
al. [23].
In both cases, the observed nanoclusters have a dome-like shape where the
adatoms at the perimeter of the quasi-circular islands are strongly attached
to the substrate and the adatoms in the center are highly detached from it.
This structure can be seen in Fig. 3.4.
Other studies of epitaxial graphene growth on Rhodium Rh(111) reported in
Ref.[25] also indicate the presence of these nanoclusters. The study indicates
that the nanoclusters are mobile on Rh(111), so they would not correspond to
immobile critical islands. However, the experiment is performed on a different
substrate to the above experiments, and the experimental method differs as
well. This means that the results might not be directly comparable to the
above.
### 3.2 Theoretical results
Based on the experiments reported by Loginova and co-workers in Refs.[3, 4],
Zangwill and Vvedensky proposed a rate equations (RE) model [5] to describe
epitaxial graphene growth. RE are described in the next chapter. They
incorporate the intermediate 5-clusters and propose a system evolution
determined by a set of coupled differential equations for the adatom density
$n$, the 5-atom cluster density $c$ and the island density $N$ as follows
$\displaystyle\frac{dn}{dt}$ $\displaystyle=$ $\displaystyle F-iDn^{i}+iKc-
DnN+K^{\prime}N,$ (3.3) $\displaystyle\frac{dc}{dt}$ $\displaystyle=$
$\displaystyle Dn^{i}-Kc-D^{\prime}cN-jD^{\prime}c^{j},$ (3.4)
$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle D^{\prime}c^{j},$
(3.5)
The parameters are carbon atom deposition flux $F$, adatom diffusion rate $D$,
cluster diffusion rate $D^{\prime}$, cluster dissolution rate $K$ and adatom
detachment rate from islands $K^{\prime}$, all assumed to be of the Arrhenius
form because they are thermally activated. The index $i=$ 5 is the cluster
size, and the index $j$ is an unknown for the number of clusters that need to
come together to form an island.
Figure 3.5: Rate equations comparison with LEEM data from Ref.[4] for the
adatom density profile. Adapted from Zangwill and Vvedensky [5].
The above set of equations together with an appropriate choice of values for
the parameters reproduces the main features of the adatom density curve in
Ref.[4] as shown in Fig. 3.5. The temperature dependence of the adatom density
at nucleation is well-reproduced within the RE by introducing the variable $j$
and taking a value $j\geq$ 6, and with the presence of the cluster
dissociation rate. However, the temperature dependence of island density at
equilibrium cannot be explained with the above RE model. Discrepancies with
the experimental data are attributed to spatial effects that are not included
in the RE approach.
### 3.3 Kinetic Monte Carlo model
Based on the above experimental and theoretical results of graphene growth, a
KMC model to describe epitaxial growth of graphene sheets should incorporate
the following basic ingredients: three different species [3, 4] corresponding
to the experimentally observed carbon monomers, carbon 5-clusters, and
graphene sheets; and a large critical nucleus size [23, 24].
These basic components of the model should form the basis for the description
of the experimental observations on the kinetics of epitaxial graphene growth
[3, 4]:
1. 1.
The adatom density at the onset of nucleation increases with increasing
temperature.
2. 2.
The adatom density at equibrium $n_{eq}$ is roughly half of the adatom density
at the onset of nucleation $n_{nuc}$, namely, $n_{nuc}\sim 2n_{eq}$.
3. 3.
The island density at equilibrium decreases rapidly with increasing
temperature.
In the rest of this thesis we will present such a KMC model and describe the
relevant physical processes in epitaxial graphene growth within the model.
## Chapter 4 Kinetic Monte Carlo
In this chapter we present the kinetic Monte Carlo (KMC) method as used in the
study of epitaxial growth. The most widespread algorithm used in the field is
the so-called N-fold way algorithm, which is adopted in this thesis.
### 4.1 Theoretical studies and computer simulations
In the field of many-body dynamics simulations, and in particular epitaxial
growth systems, there are various approaches with different levels of
approximation and capabilities.
A completely deterministic mean-field approach called rate equations (RE) is
commonly used in studies of epitaxial growth. RE are a finite set of coupled
first order differential equations for the densities of the different species
in an epitaxial system and with a rate associated with each possible process.
The strength of the technique is its simplicity, but the pay-off is in
accuracy. Standard RE cannot include any spatial information, but more complex
approaches using capture numbers can encapsulate it. Zangwill and Vvedensky
have used RE to describe the epitaxial graphene growth system [5] as described
above in Section 3.2.
Molecular dynamics (MD) simulations are used for detailed analysis of the
dynamics of a system. The forces between different components are calculated
and the system is evolved according to these. Both classical and quantum
approaches to MD simulations are used, and major advances in the field, for
instance the Car-Parrinello method [26] incorporating density functional
theory into MD, have led to widespread use of the method. However, the
timescales and system sizes that can be explored remain small, restricting the
applicability of the technique.
KMC looks at intermediate time and length scales and has proved appropriate
for the description of epitaxial growth systems.
### 4.2 Kinetic Monte Carlo
In this section we describe the KMC technique. We first present the N-fold way
algorithm used in this thesis and next we discuss why the KMC method can be
used in the description of epitaxial growth processes.
#### 4.2.1 N-fold way algorithm
The KMC method is based on the use of random numbers to simulate the time
evolution of a system in discrete steps, in such a way that each step
represents a move of the entire system from one state to another. We are going
to concentrate on the N-fold way algorithm, first introduced by Bortz, Kalos
and Lebowitz in 1975 [27].
Following Ref.[28], a rate $r_{i}$ is associated with process $i$, and rates
are usually taken to be of the Arrhenius form because they are thermally
activated,
$r_{i}=\nu_{0}e^{-E_{i}/k_{B}T}$ (4.1)
where $\nu_{0}=2k_{B}T/h$ is of the order of the atomic vibrational frequency,
and $E_{i}$ is the energy barrier associated with process $i$. $T$ is the
temperature, $k_{B}$ is Boltzmann’s constant and $h$ is Planck’s constant.
This form for the rates will imply in a KMC model that events that are more
likely to occur will happen more frequently. At each time step, all rates are
calculated for the given configuration of the system.
To select the process that is going to occur at a given time step, a total
transition rate $R$ is constructed as
$R=\sum_{i=1}^{N}r_{i}$ (4.2)
for a total of $N$ possible processes in a given configuration of the system.
A number $\rho_{1}\in[0,1)$ uniformly distributed is calculated, and then the
process $j$ such that
$\sum_{i=1}^{j-1}r_{i}\leq R\rho_{1}<\sum_{i=1}^{j}r_{i}$ (4.3)
is selected to take place in the given time step. This is shown in Fig. 4.1.
$r_{1}$$r_{2}$$r_{3}$$\ldots$$r_{i-1}$$r_{i}$$r_{i+1}$$\ldots$$r_{N}$$R\rho_{1}\mbox{
for random }\rho_{1}\in[0,1)$ Figure 4.1: Schematic of the N-fold way
algorithm selecting the rate $r_{i}$ to occur in the given time step. The size
of the boxes containing the rates is proportional to the rates themselves.
Practically, possible events can be grouped together and the search time for
the event is reduced. For instance, all free adatoms will have equal
probability of diffusing into any of the four nearest-neighbour sites in a
square lattice simulation, and all these events can be put into a subgroup.
Constructing these subgroups reduces the total rate to a set of partial sums,
and the search then proceeds first by identifying the relevant subgroup and
then searching only within this subgroup. This search method is referred to as
the binning method [28] or 2-level scheme as the search is done in two levels.
There are methods available [29] which are up to 7 times faster than the
binning method and are based on a further subdivision of the list of possible
events, refered to as $K$-level schemes for $K$ subdivisions.
The time evolution of the system is based on the assumption that the
probability of an event occuring is independent of the history of the system
and therefore it obeys Poisson statistics. This assumption will be justified
below in Subsection 4.2.2. Following Ref.[27], the probability that an event
$i$ occurs in the infinitessimal time interval $(t,t+dt)$ is
$p_{i}(t)dt=r_{i}dt$ where $r_{i}$ is the rate associated with the event. The
total probability $P(t)$ that an event occurs in time interval $(t,t+dt)$ is
then
$P(t)dt=\sum_{i}\,p_{i}(t)dt=\sum_{i}\,r_{i}dt=Rdt.$ (4.4)
Let $\pi(t)$ be the probability that no event occurs in time interval $(0,t)$.
Then, the probability that no event occurs in time interval $(0,t+dt)$ is
equal to the probability that no event has occured in time interval $(0,t)$
and the probability than no event occurs in time interval $(t,t+dt)$, namely,
$\pi(t+dt)=\pi(t)(1-Rdt)=\pi(t)-\pi(t)Rdt.$ (4.5)
After rearranging and taking the limit $dt\rightarrow 0$, we obtain
$\frac{d\pi(t)}{dt}=\displaystyle\lim_{dt\to
0}\frac{\pi(t+dt)-\pi(t)}{dt}=-\pi(t)R$ (4.6)
which can be easily solved to give
$\pi(t)=\pi(0)e^{-Rt}$ (4.7)
with $\pi(0)=1$ as the probability of no event occuring in time $t=$ 0 is
unity. The physical time of each time step in the simulation is then assumed
to be distributed as Eq.(4.7) and a second random number $\rho_{2}\in(0,1)$ is
chosen such that the physical time $\tau$ is given via $\rho_{2}=e^{-R\tau}$
or
$\tau=-\frac{\ln(\rho_{2})}{R}.$ (4.8)
This expression gives an average time between events equal to $1/R$.
#### 4.2.2 Use of kinetic Monte Carlo models to describe epitaxial growth
In this section we motivate the use of a stochastic Monte Carlo technique for
the description of epitaxial growth systems.
As described in Ref.[30], deposition often occurs by the random impingement of
atoms or molecules on a substrate. Within a KMC simulation, a deposition flux
is included in the model, and the deposition sites are chosen randomly, in
accordance with experimental realisations.
After deposition, atoms are adsorbed on specific sites on the lattice that
correspond to minima in energy for the configuration of the system. A given
configuration of the adatoms on the substrate is called a state of the system.
The adatoms vibrate at frequencies of the order $\nu_{0}\sim 10^{13}$ s-1
about the minima they occupy. However, these vibrations do not change the
state of the system, only the state of the individual adatoms. The state of
the system changes occasionally when an adatom vibration makes the adatom move
from one minimum to another by going past an energy barrier $E$. Such
transitions are thermally activated, so the rate $r$ associated with them is
of the Arrhenius form.
A KMC model to describe epitaxial growth takes advantage of the different time
scales between the change of state of an adatom and of the entire system by
only considering the latter. During the irrelevant vibrations of an adatom
about a given minimum, the system loses memory about the previous
configurations it occupied, and therefore one can use Poisson statistics to
describe a KMC time step.
## Chapter 5 Standard kinetic Monte Carlo model of epitaxial growth
In this chapter we introduce what we call the KMC standard model of epitaxial
growth. The objective is two-fold: first, to provide an example of the N-fold
way algorithm described above; and second, to have a simple model that is
going to serve as a starting point for further studies and as a reference to
compare with the graphene system.
### 5.1 Description
We consider a 200$\times$200 square lattice with periodic boundary conditions.
The model describes two processes: deposition of single atoms and diffusion of
adatoms (adsorbed atoms) over the substrate. This model does not aim to
describe the graphene system, but the studies we will perform will be keeping
in mind the final objective to study epitaxial graphene growth.
Deposition occurs at a constant flux $F$, and atoms can only be deposited at
an unoccupied lattice location. In the case that a deposition attempt is made
on an occupied site, then a deposition process is still forced to happen in
order to keep the deposition flux constant, and a different site is chosen
repeatedly until an unoccupied site is found. The total coverages explored
here are a small fraction of the lattice because we are interested in the
initial stages of epitaxial graphene growth. Then, such events are rare
indicating that the above modelling of deposition does not introduce an
undesired effect in the model.
Diffusion occurs between nearest-neighbour sites, and an adatom can only
diffuse to an unoccupied site. The diffusion rate $D$ is given by the
Arrhenius form
$D=\nu_{0}e^{-(E_{a}+nE_{b})/k_{B}T}$ (5.1)
where $\nu_{0}=2k_{B}T/h$ for Boltzmann constant $k_{B}$, temperature $T$ and
Planck constant $h$. Furthermore, $E_{a}$ is the energy barrier associated
with the diffusion of an adatom, and $E_{b}$ is an extra energy barrier
associated with a bond to a nearest neighbour. The variable $n$ is the number
of nearest neighbours of the diffusing adatom, and the above bond-counting
model is typically used in epitaxial growth simulations [30]. Note this rate
depends on the local configuration around the diffusing adatom, where nearest-
neighbours result in a penalty energy for diffusion. That is, adatoms that
share bonds with others are less likely to diffuse away and this is the key
ingredient to get island formation.
Temperature is the parameter we tune in order to explore the different regimes
of the system. The energy barrier for adatom diffusion is kept fixed at
$E_{a}=$ 1.00 eV, while the bond energy barrier $E_{b}$ is varied. The flux is
set to $F=$ 1 ML/s and the total coverage reached is $\theta=$ 0.1 ML. The
units of coverage are monolayers (ML) where one monolayer corresponds to a
layer of epitaxial material grown on the substrate.
The model presented here belongs to a class of models referred to as generic
models [30]. These generic models do not have any specifications of critical
island size, that is, no island size above which attachment is irreversible is
defined. Instead, a set of local rules such as the bond-counting scheme above
is defined and the system is left to evolve according to these only.
### 5.2 Results and discussion
In this section we analyse the physical processes involved in epitaxial growth
based on the KMC model described above. We start with a qualitative look at
the growth behaviour, and then we move on to a quantitative analysis that will
serve as a basis for the later study of the graphene system.
#### 5.2.1 Physical lattice
In Fig. 5.1 we can see the physical lattice for different temperatures. All
instances are after the last adatom of a total coverage of $\theta=$ 0.1 ML
has been deposited, corresponding to a 10% coverage of the lattice. The bond
energy barrier used is $E_{b}=$ 0.20 eV.
Figure 5.1: Physical lattice at temperatures (a) $T=$ 550 K, (b) $T=$ 600 K,
(c) $T=$ 650 K, (d) $T=$ 700 K. In all cases, $E_{a}=1.00$ eV, $E_{b}=0.20$
eV, $F=1$ ML/s and $\theta=0.1$ ML. Black squares represent occupied sites and
white squares unoccupied sites.
It can be seen from Fig. 5.1 that as temperature increases the islands that
are formed become larger in size and fewer in number. Adatoms at the edges of
islands are more weakly bound to the island because they have a smaller number
of bonds, so minimising their number is thermodynamically favourable. This
leads to large and circular islands being the most stable configuration for
the system because it minimises the interface free energy.
In Fig. 5.1, larger temperatures present this thermodynamically favourable
state. This is caused by the high thermal energies that lead to large adatom
mobility and the system can easily reach the most favourable configuration.
Under these conditions, islands that are not very large can break up and only
a few nucleations will eventually lead to stable islands, which will be fewer
in number. Also, the islands will be larger in size because there will be less
competition for adatom incorporation due to the presence of a smaller number
of competing islands.
When the flux is turned off all the different cases depicted in Fig. 5.1 will
tend to configurations minimising the free energy for large times, even when
the thermal energies do not facilitate it. A common mechanism that leads to
coarsening in submonolayer epitaxial systems is Ostwald ripening [31, 32].
(a) Adatom density (b) Island density
Figure 5.2: Adatom density $n$ and island density $N$ as a function of time.
The system corresponds to the standard KMC model with bond energy $E_{b}=$
0.20 eV and $T=$ 650 K.
These features can be studied quantitatively by looking at the island size
distribution and at the time and temperature dependence of adatom and island
densities as described in the following subsections. For clarity, we plot in
Fig. 5.2 the time dependence of both adatom and island densities for the
parameters $E_{b}=$ 0.20 eV and $T=$ 650 K. Such profiles are used in the
various analyses presented below for the island size distribution, the adatom
density at nucleation $n_{nuc}$, the adatom density at equilibrium $n_{eq}$
and the island density at equilibrium $N_{eq}$. In the instances shown in Fig.
5.2, the deposition flux is turned off at a coverage of $\theta=$ 0.10 ML
(corresponding to $t=$ 0.10 s for $F=$ 1 ML/s) and the system is then allowed
to relax for a further 0.05 seconds. The main features discussed in the
previous chapter for such profiles are clearly observed.
#### 5.2.2 Island size distribution
The island size distribution is a quantity readily available in KMC studies,
but difficult to obtain in mean field approaches such as rate equations.
Figure 5.3: Island size distribution for the standard KMC model with bond
energy $E=$ 0.20 eV.
In Fig. 5.3 we plot the island size distribution as it is at a coverage of
$\theta$ = 0.1 ML, for bond energy $E_{b}=$ 0.20 eV and for a range of
temperatures. The distributions are calculated when the system has been
allowed to relax after turning the deposition flux off. The island sizes are
normalised in such a manner that the area under the curve is unity, and the
horizontal axis represents $s/s_{av}$ for island size $s$ and average island
size
$s_{av}=\frac{\sum_{s}sN_{s}}{\sum_{s}N_{s}},$ (5.2)
where $N_{s}$ is the number of islands of size $s$.
It can be seen in Fig. 5.3 that as temperature increases the distribution
becomes narrower. At large temperatures, islands can break up more easily than
at low temperatures, so we expect that there is some correction in island
sizes and all islands approach an ideal size. This analysis quantifies the
discussion above based on snapshots of the lattice in Fig. 5.1.
To exemplify this island size correction at high temperatures consider a
system where two islands form nearby. At low temperatures, the islands will
grow by attachment of adatoms, but because they are nearby they compete for
free adatoms in the surface, so both islands will in general be smaller than
the average island size. However, the same situation at high temperatures can
result in the break-up of one of the two islands, and the incorporation of its
adatoms into the other, so that only one island is left, and its size is
closer an average size. Note that in the same manner, if an island is formed
in a region where it is isolated, for low temperatures it will incorporate all
adatoms in its vicinity and thus grow to larger than average sizes. However,
at large temperatures the detachment of adatoms from the island will be more
significant and the size of the island will again be closer to that of others.
The data for island size distribution is not as clear as the data for adatom
or island concentrations, specially for large temperatures. This is because
for high temperatures very few islands are formed on the lattice, in Fig. 5.1
only 10 islands for $T=$ 700K. Furthermore, the islands formed are larger in
size, so the range of possible island sizes is broader. These two
characteristics mean that to obtain useful data for island size distributions
we need to calculate an average over many different instances of the system
evolution. For instance, in the case depicted in Fig. 5.3 for $T=$ 700K, up to
10,000 instances were used.
#### 5.2.3 Adatom density
The experimental results reported by Loginova and co-workers [3, 4] represent
some of the cleanest data on adatom density as a function of time for
epitaxial growth experiments. As described above, the adatom density presents
interesting features, and we will concentrate on its values at the onset of
nucleation, $n_{nuc}$, and at equilibrium with islands, $n_{eq}$.
The behaviour of $n_{nuc}$ is highly dependent on the strength of the bond
between adatoms as shown in Fig. 5.4. For low bond energy the critical island
size is large, that is, the size required for a cluster of atoms to eventually
become an island is large because small clusters will break-up with large
probability due to the weakness of the bonds keeping the atoms together.
Furthermore, as temperature increases bonds break more often, so we expect
that with increasing temperature nucleation occurs at larger adatom
concentrations. In contrast, for large bond energies the thermal vibrations
cannot easily break the bonds, and the critical island size is reduced to a
few atoms even for high temperatures. Then the limiting process for nucleation
is adatom diffusion because as soon as adatoms become nearest neighbours
nucleations occur. Therefore, as temperature is increased, the larger
diffusion of adatoms means that the adatom nucleation occurs sooner and
$n_{nuc}$ decreases with temperature.
Figure 5.4: Adatom nucleation density $n_{nuc}$ as a function of temperature
for nearest neighbour bond strengths $E_{b}=$ 0.10, 0.20, 0.30 eV.
In Fig. 5.4 we plot $n_{nuc}$ as a function of temperature, and for different
adatom bond energies $E_{b}$. Each data point has a statistical error
associated with it that is of order $\sim$0.01 of the value plotted so it is
not depicted. We can observe both the low bond energy limit with $E_{b}=$ 0.10
eV and the large bond energy limit with $E_{b}=$ 0.30 eV. With bond energy
$E_{b}=$ 0.20 eV we can see an intermediate scenario where the lower
temperature range is diffusion dominated and the adatom nucleation
concentration decreases with temperature, but at larger temperatures island
break-up becomes dominant and a larger adatom concentration is required for
nucleation because the critical island size increases. Note also that as
expected, for lower bond energies $n_{nuc}$ is much larger than for large bond
energies.
Figure 5.5: Adatom equilibrium density $n_{eq}$ as a function of temperature
for nearest neighbour bond strengths $E_{b}=$ 0.10, 0.20, 0.30 eV.
The adatom density at equilibrium is shown in Fig. 5.5, plotted as a function
of temperature and for different $E_{b}$. In all cases the equilibrium
concentration increases with temperature because detachment from islands
dominates at larger temperatures. For low binding energy $E_{b}=$ 0.10 eV,
$n_{eq}$ is large because detachment rates are very large. In contrast, for
higher bond energies the adatoms do not detach from islands with such large
rates and $n_{eq}$ almost vanishes. For the lower temperatures and higher
binding energies equilibrium is only reached after a long time because low
diffusion over the substrate means adatoms do not attach to islands
frequently, but strong bonds lead to large islands, so the system takes a long
time to equilibrate.
#### 5.2.4 Island density
In this subsection we study the equilibrium island density $N_{eq}$ for the
standard model. Experimental data on island density is available for the
graphene system, and the RE approach fails to describe it. Therefore, a
careful analysis of island density within KMC could deepen our understanding
of the growth kinetics of graphene.
In the bond-counting scheme used for the standard model, a critical island
size is not well-defined. Therefore, the definition of island is not very
clear. For instance, for low bond energies between nearest neighbours, dimers
are very unstable and break-up frequently. However, for large bond energies,
dimers are very stable and lead to large islands in most cases. In the
standard model, clusters of size larger than 10 are treated as islands. This
choice might not be strictly accurate, but the trends to be observed in the
data arise clearly, therefore simplicity has prevailed in the choice.
Figure 5.6: Equilibrium island density $N_{eq}$ as a function of temperature
for nearest neighbour bond strengths $E_{b}=$ 0.10, 0.20, 0.30 eV.
In Fig. 5.6 we plot the equilibrium island density as a function of
temperature. For the stronger bond cases $E_{b}=$ 0.20, 0.30 eV, a clear
decreasing trend for $N_{eq}$ with temperature is observed. This is in
agreement with the physical lattices above. Note also that the concentration
of islands is larger for the case with stronger binding. This is caused by the
higher difficulty of small clusters to break up if bonds are strong, and
therefore more nucleations are expected in this regime. The low bond energy
case $E_{b}=$ 0.10 eV presents a different behaviour. For the temperature
increase from 550 K to 600 K, a decrease in $N_{eq}$ is observed as expected.
However, further temperature increases do not lead to a clear decrease,
instead the island concentration remains approximately constant. This
behaviour is very particular of the low bond energy case where islands cannot
grow very large due to the high adatom detachment rates. The observed
behaviour of island density means that already at $T\sim$ 600 K the adatom
density on the substrate determines the minimum island density attainable with
the high adatom detachment rate present. Then, further temperature increases
do not lead to a smaller island concentration. This behaviour is confirmed by
the physical lattice in this regime shown in Fig. 5.7, where temperatures
higher than $T=$ 600 K do not change the aspect of the system as significantly
as for the case depicted above in Fig. 5.1. The islands decrease slightly in
size due to the higher temperatures causing more frequent bond breaking.
Figure 5.7: Physical lattice at temperatures (a) $T=$ 550 K, (b) $T=$ 600 K,
(c) $T=$ 650 K, (d) $T=$ 700 K for a bond energy of $E_{b}=$ 0.10 eV. Black
squares represent occupied sites and white squares unoccupied sites.
The behaviour of the system suggests that we use a nearest neighbour bond
energy of the order $E_{b}\sim$ 0.20 eV or above for further studies of
graphene. This is because lower binding energies will not lead to growth of
large islands.
#### 5.2.5 Error analysis
The different values presented above have a statistical uncertainty associated
with them due to the stochastic nature of the KMC method. These statistical
errors are minimised by averaging the results over a large number of
experiments, and this has been done in a manner that the errors of a given
quantity are about 1% of the value of the quantity throughout this work. These
small statistical errors are not shown in the plots because they are too small
to be seen appropriately. In the quantity $N_{eq}$ the statistical errors can
be up to 10% of the value quoted, but even in these circumstances the trends
discussed can be clearly observed.
Another error consideration comes in when evaluating the correctness of the
model and its implementation. The model described here is a simple bond-
counting epitaxial growth model that is widely used in such studies. Its
implementation has been partially checked by finding the expected results for
such bond-counting model. In the next chapter we consider this point further
for the KMC model to describe epitaxial graphene growth.
## Chapter 6 Epitaxial graphene growth kinetic Monte Carlo model
In this chapter we present a KMC model for the study of the kinetics of
epitaxial graphene growth. The experimental observations reported about the
system [3, 4] are explained with the model, and the relevant physical
processes determining the observed behaviour are identified.
However, direct comparison with experiment is not possible for two reasons.
First, the values of the energy barriers associated with the different
processes are not available yet. The present work determines the relevant
physical processes to include in the study of epitaxial graphene growth.
Therefore, the next step is to determine the exact energetics of these
identified processes, for instance using ab initio techniques. Second, the
deposition flux used experimentally is very low, resulting in time scales that
are computationally inaccessible to the present KMC model.
### 6.1 Description
We introduce a KMC model for epitaxial graphene growth in a 200$\times$200
square lattice with periodic boundary conditions. We include adatoms, islands
and an intermediate species formed by four adatoms arranged in a square which
we call tetramers. Experiments indicate that the intermediate species in the
graphene system are 5-atom clusters, but for convenience with the underlaying
square lattice we choose the simpler symmetric tetramer. The presence of
5-clusters in graphene growth is most probably determined by the stability of
such structures on the substrates (observed in grand canonical Monte Carlo
simulations for Niquel Ni(111) [33]), but we postulate that the kinetics are
determined only by the presence of this intermediate species rather than by
its details. This justifies the use of tetramers which represent the stable
symmetric structure in our model. The validity of this assumption can be
confirmed later with the results obtained. In a similar manner, the use of a
square lattice rather than a triangular lattice can be justified for
simplicity of the model.
Deposition occurs as described above for the standard model via individual
atoms. Experimentally, both carbon atoms and ethylene molecules are deposited,
but the behaviour of the growth system is independent of the deposited species
[4] and for simplicity we only consider individual atom deposition.
Adatoms diffuse over the substrate in the same manner as for the standard
model. They only interact to form tetramers and to attach to islands. A
tetramer is formed whenever four adatoms come together in a square
configuration on the lattice. From that moment onwards, the tetramer is
treated as an individual species formed by four occupied sites in a square
configuration. Attachment of adatoms to islands occurs as described above for
the standard model whenever an adatom diffuses to a site with nearest
neighbours belonging to an island. After an adatom has attached to an island,
it is still treated as an individual atom, but subject to the bond-counting
scheme described above. We stress this only occurs for adatoms that diffuse to
sites with nearest neighbours belonging to islands: if a tetramer is a nearest
neighbour, there is no interaction with the adatom based on the fact that
tetramers are a very stable and independent species.
(a)(b)(c)tetramer Figure 6.1: Schematic of possible nucleation configurations
for (a) $j=$ 2, (b) $j=$ 4 and (c) $j=$ 6, where $j$ is the number of
tetramers. For the asymmetric cases $j=$ 2, 6; a $\pi/2$ rotation with respect
to the cases depicted is also a valid nucleation configuration. Note that in
the diagram a black circle represents four adatoms in a square configuration
and a square lattice site represents four square lattice sites in the actual
simulation.
Tetramers can diffuse over the substrate, in a manner analogous to adatom
diffusion. The only difference is that the four adatoms forming the tetramer
move at the same time and in the same direction as an individual unit.
Tetramers only interact with other tetramers to nucleate and with islands to
attach to them. Nucleation occurs when a number $j$ of tetramers come together
on the lattice in a symmetric manner as indicated in Fig. 6.1, where three
possible values for $j$ are shown. As soon as a nucleation occurs, the
tetramers lose their identity and the adatoms forming them become members of
an island and are subject to nearest neighbour interactions via the bond-
counting scheme. Tetramers can also attach to existing islands, and such
events occur whenever the two atoms of a side of the tetramer become nearest
neighbours to sites occupied by islands, in such a manner that a side of the
tetramer square is in contact with an island. This is schematised in Fig. 6.2.
When a tetramer attaches to an island, it again loses its identity and the
individual atoms become subject to nearest neighbour interactions.
$\implies$adatom belonging to tetrameradatom belonging to island Figure 6.2:
Schematic of tetramer diffusion and attachment to islands.
We tune the temperature whilst keeping the energy barriers shown in Table 6.1
at fixed values. Note the tetramer diffusion barrier is kept lower than the
adatom diffusion barrier to promote tetramers to the dominant species in
graphene growth kinetics.
Description | Parameter | Value
---|---|---
Coverage | $\theta$ | 0.10 ML
Adatom diffusion barrier | $E_{a}$ | 1.00 eV
Tetramer diffusion barrier | $E_{t}$ | 0.80 eV
Adatom bond energy | $E_{b}$ | 0.20 eV
Table 6.1: Values of the basic parameters used in the KMC model to describe
epitaxial graphene growth
The model presented here belongs to the generic models class. A possible
variation of this model for the simulation of epitaxial graphene growth is
described in Appendix A.
In the following section we investigate how different processes superimposed
on top of this basic model determine the rich behaviour observed
experimentally for the kinetics of epitaxial graphene growth.
### 6.2 Results and discussion
In this section we study the three experimental observations by Loginova and
co-workers on the epitaxial graphene growth system [3, 4]. For convenience, we
repeat them here:
1. 1.
The adatom density at the onset of nucleation increases with increasing
temperature.
2. 2.
The adatom density at equibrium $n_{eq}$ is roughly half of the adatom density
at the onset of nucleation $n_{nuc}$, namely, $n_{nuc}\sim 2n_{eq}$.
3. 3.
The island density at equilibrium decreases rapidly with increasing
temperature.
In the following subsections we study them individually. There is only
preliminary data for the island size distribution, and it is presented in
Appendix B.
#### 6.2.1 Adatom density at nucleation
It was discussed in Section 3.1 that the adatom density at nucleation,
$n_{nuc}$, is experimentally found to increase as a function of temperature
for the graphene growth system. In standard growth systems, we found in
Chapter 5 that $n_{nuc}$ could present different behaviours with temperature
depending on the binding energy between adatoms. In this section we will
investigate this effect on the KMC graphene model and find a more complex
behaviour.
Inspired by experiments [23, 24] and theoretical studies [5], it appears that
for graphene the critical nucleus sizes are of the order of tens of adatoms.
In our KMC model we have incorporated this observation by allowing nucleations
to occur only when a certain number $j$ of tetramers come together in specific
configurations. Because islands only start appearing with these large sizes,
then the analysis on the standard model with low binding energy might not be
applicable for graphene. This is because even for low bond energies, the
incipient islands are so large that dissociation might be very unlikely and
the growth process would be diffusion limited. Such a scenario would lead to a
decrease in $n_{nuc}$ with increasing temperature, even for low energy bonds.
Furthermore, it was concluded in the previous chapter that binding energies
which are too low do not result in the appearence of large islands.
Even though we might have a diffusion limited growth system, epitaxial
graphene growth is dominated by an intermediate stable species, and this
difference with other growth systems can explain the increase of $n_{nuc}$
with temperature, even with the presence of relatively large incipient
islands. Physically, a process is needed by which the concentration of
clusters giving nucleation is decreased with increasing temperature to
compensate for the larger cluster formation and diffusion rates. Such a
process could be temperature dependent cluster break-up, included in the KMC
model with a rate in the usual Arrhenius form subject to a cluster break-up
energy barrier $E_{k}$.
(a) Standard model (b) 2-tetramer nucleation
(c) 4-tetramer nucleation (d) 6-tetramer nucleation
Figure 6.3: Adatom density at the onset of nucleation as a function of
temperature for (a) standard model, (b) nucleation with $j=$ 2, (c) nucleation
with $j=$ 4, (d) nucleation with $j=$ 6\. In the standard model different
nearest neighbour bond energies are considered, and in the tetramer model
different tetramer break-up energies are considered.
In Fig. 6.3 we show the temperature dependence of $n_{nuc}$ for different
nucleation sizes $j$ and for a range of cluster break-up energies $E_{k}$. We
also plot the corresponding quantities for the standard model discussed above.
For $j=$ 2 we find that for all cluster break-up energy values, $n_{nuc}$
decreases with increasing temperature. This indicates that, as discussed
above, an incipient island formed by $N=$ 8 adatoms is large enough not to
dissociate, so that a low binding energy between adatoms cannot lead to an
increase of $n_{nuc}$ with temperature as for the standard model even at high
temperatures. Furthermore, cluster break-up is also found not to be sufficient
to appropriately describe the temperature behaviour of $n_{nuc}$ of graphene
growth. Even though the tetramer population is reduced as temperature
increases, for $j=$ 2 nucleation is still a frequent event and the break-up
rate is not high enough to act as a significant deterrent for nucleation. This
means the system behaves as a typical diffusion limited growth system and
$n_{nuc}$ decreases with temperature.
For $j=$ 4 more interesting behaviour starts appearing. For zero or very small
cluster break-up, we still find a decreasing trend in $n_{nuc}$. Again, this
is due to the behaviour as a typical diffusion limited growth system. However,
as $E_{k}$ is decreased, an increase in $n_{nuc}$ is observed for the larger
temperature ranges. With $j=$ 4, nucleation events are less likely than with
$j=$ 2, so at large temperatures where cluster break-up is a significant
effect, it can counteract the larger tetramer formation and diffusion rates.
For a critical concentration of tetramers needed for nucleation, the high
tetramer break-up rate means that larger adatom concentrations are needed to
lead to the necessary tetramer concentration for nucleation. This leads to the
increase of $n_{nuc}$ with temperature. It is not observed for low
temperatures because diffusion still prevails over tetramer break-up in that
regime.
For $j=$ 6, with the appropriate choice of $E_{k}$, an increase of $n_{nuc}$
with temperature is observed in the entire temperature range. This is the
experimentally observed behaviour of epitaxial graphene growth. The large
value of $j$ means that the tetramer concentrations needed for nucleation are
so large that tetramer break-up can play a significant role even for low
temperatures. From now on and unless otherwise stated we are going to use
$j=6$ and $E_{k}=1.40$ eV for all further studies.
The above reasoning depends on the fact that the density of clusters at the
onset of nucleation is independent of the cluster break-up energy barrier.
This is indeed the case as shown in Fig. 6.4 for $j=4,6$ and similar behaviour
occurs for $j=2$.
(a) 4-tetramer nucleation (b) 6-tetramer nucleation
Figure 6.4: Tetramer density at the onset of nucleation as a function of
temperature for (a) nucleation with $j=$ 4 and (b) nucleation with $j=$ 6.
In agreement with the rate equations study by Zangwill and Vvedenksy [5], both
cluster break-up and large incipient island sizes are found to be necessary to
explain the increase of $n_{nuc}$ with temperature.
#### 6.2.2 Adatom density at equilibrium
Adatom density at equilibrium is experimentally found to be roughly half of
the adatom density at the onset of nucleation, $n_{nuc}\sim 2n_{eq}$. Such
large $n_{eq}$ require the presence of a large attachment barrier for adatoms
to graphene. The presence of such a barrier initially prompted the proposal of
a 5-cluster attachment mechanism by Loginova and co-workers [3]. In the KMC
model presented in this thesis, two physical processes determine the
predominant growth of graphene by cluster attachment. The first one is a
larger diffusion for tetramers than for adatoms, and the second is the
presence of an adatom attachment barrier. It is only the second that
determines the adatom concentration at equilibrium, and this effect is
discussed here.
We consider a temperature dependent attachment barrier for adatoms to
graphene. Computationally, it is incorporated in the KMC model via a Boltzmann
factor $e^{-E_{p}/k_{B}T}$ such that when an adatom moves to a site with
nearest neighbours belonging to an island, it will attach to the island with
probability $p<e^{-E_{p}/k_{B}T}$. This condition makes adatom attachment more
difficult at low temperatures than at high temperatures. Tuning the energy
barrier $E_{p}$ results in different values for $n_{eq}$.
Figure 6.5: Ratio of equilibrium adatom concentration $n_{eq}$ over adatom
concentration at the onset of nucleation $n_{nuc}$ as a function of
temperature. Different adatom attachment barriers are considered.
In Fig. 6.5 we plot the ratio $n_{eq}/n_{nuc}$ as a function of temperature
for a range of adatom attachment barriers. We calculate $n_{eq}$ from the KMC
model by turning the deposition flux off and allowing the system to relax,
that is, by allowing the system to reach equilibrium between adatoms,
tetramers and islands. As can be seen in Fig. 6.5, the value of $E_{p}$ tunes
the adatom equilibrium concentration, and typical experimental values
$n_{nuc}\sim 2n_{eq}$ can be reached in our model for $E_{p}=$ 0.30 eV. The
ratio is also seen to be roughly constant for the temperature range studied,
although there is a slight increase with temperature. From now on unless
otherwise stated we are going to take $E_{p}=0.30$ eV.
Recalling $n_{eq}$ for the standard model in the previous chapter, we find
significant differences. For the case with adatom bond energy of $E_{b}=$ 0.20
eV as used here, the standard model resulted in vanishingly small equilibrium
concentrations. Therefore, it is clear that an adatom attachment barrier is
essential in obtaining the correct behaviour.
The RE approach by Zangwill and Vvedensky [5] also describes $n_{eq}$
accurately. However, in their model there is no adatom attachment barrier, so
the only process that can also explain this behaviour is a large adatom
detachment rate from islands. The lack of spatial information in the RE means
that the capture zone of islands is smaller than in the KMC model, so adatoms
attach to islands less frequently. This leads to higher values for $n_{eq}$ in
the RE than in KMC models without an adatom attachment barrier.
#### 6.2.3 Island density at equilibrium
Island density at equilibrium is experimentally found to decrease rapidly with
increasing temperature. In the KMC model used above for the description of
graphene growth, as soon as a nucleation occurs, tetramers are more likely to
attach to the existing island than to give rise to further nucleations.
Therefore, to have a significant number of nucleations, the density of
tetramers needs to be kept high by some mechanism. In order to try to describe
this, we incorporate a tetramer attachment barrier $E_{q}$ in the same manner
as an adatom attachment barrier above.
Figure 6.6: Equilibrium island concentration $N_{eq}$ as a function of
temperature. Different tetramer attachment barriers are considered.
In Fig. 6.6 we plot $N_{eq}$ as a function of temperature for a range of
tetramer attachment barriers $E_{q}$. It can be seen that, by tuning the
tetramer attachment barrier, a large decrease in island density can be
engineered into the model. For the standard model discussed above, $N_{eq}$
decreased slowly as a function of temperature. For instance, for a bond energy
$E_{b}=$ 0.20 eV, an increase of 100 K gave a decrease by a factor of 4. In
the present case, an increase of 100 K for $E_{q}=0.30$ eV gives an order of
magnitude decrease in $N_{eq}$.
Note that in order to get a large decrease in island density at nucleation,
tetramer attachment barriers of the same order as adatom attachment barriers
need to be used. In this scenario, the only property of the system that makes
tetramer attachment a predominant effect in graphene growth is the larger
diffusion of tetramers as compared to adatoms. In the next section we are
going to confirm that under these circumstances tetramer attachment is still
the dominant effect.
### 6.3 Error analysis: island growth velocity
In this section we present an error analysis on the KMC model described in
this thesis. Statistical errors due to the stochastic nature of the method are
already discussed in the context of the standard model above, and that
discussion is not repeated here. Instead, we concentrate in the evaluation of
the model used and its computational implementation. We look at the velocity
of island growth and find that the KMC model is consistent with the
$m$-cluster model proposed experimentally for the growth of epitaxial
graphene.
In Ref.[3] the island growth velocity is found to be a nonlinear function of
the adatom concentration, ultimately leading to the proposal of a growth model
for graphene with a 5-cluster intermediate species. As discussed in Section
3.1, postulating that the energy barrier for adatom attachment to islands is
larger than the energy barriers for clusters of size $m$ to form and later to
join islands, it can be shown that the island growth velocity obeys
$v=B\left[\left(\frac{n}{n_{eq}}\right)^{m}-1\right]$ (6.1)
where $n$ is the adatom concentration, $n_{eq}$ is the adatom concentration at
equilibrium and $B$ is a temperature dependent constant. Experimentally, it is
found that $m\simeq$ 5, and the intermediate species are clusters formed by 5
carbon atoms. In our KMC model, where tetramers play the role of the
intermediate species, we expect to have a power law with exponent $m\simeq$ 4
in Eq.(6.1). This result will provide a consistency check for the model used
in this work.
The island growth velocity $v$ is defined as [3]
$v=\frac{1}{P}\frac{dA}{dt}$ (6.2)
for island perimeter $P$ and area $A$. For simplicity, we consider circular
islands with $P=2\pi R$ and $A=\pi R^{2}$ for radius $R$. The circular island
approximation is expected to be accurate at high temperatures where we have
seen that the system tends to large circular islands trying to minimise the
edge free energy. With a total area $A=N$ for $N$ carbon atoms in the island
and an island radius $R=\sqrt{N/\pi}$, then
$P=2\sqrt{\pi N}\hskip 28.45274pt\mbox{and}\hskip 28.45274ptA=N.$ (6.3)
The estimate for the island growth velocity $v(t)$ at time $t$ in the circular
island approximation is then
$v(t)\simeq\frac{1}{P}\frac{\Delta A}{\Delta t}\simeq\frac{1}{2\sqrt{\pi
N(t)}}\frac{N(t+\Delta t)-N(t)}{\Delta t}.$ (6.4)
(a) $E_{q}=0.10$ eV (b) $E_{q}=0.30$ eV
Figure 6.7: Island growth velocity as a function of adatom coverage at $T=$
700K for tetramer attachment barriers (a) $E_{q}=0.10$ eV and (b) $E_{q}=0.30$
eV. A fit to Eq.(6.1) is also shown.
In Figure 6.7 we plot the $v$ as a function of $n$ for tetramer attachment
barriers $E_{q}=0.10,0.30$ eV. The other parameters are those in Table 6.1 and
$E_{k}=1.40$ eV, $E_{p}=0.30$ eV. The data corresponds to $T=700$ K, the large
temperature limit discussed above, and it is then found that $n_{eq}=0.032$ ML
for $E_{q}=0.10$ eV and $n_{eq}=0.039$ ML for $E_{q}=0.30$ eV.
| $E_{q}=0.10$ eV | $E_{q}=0.30$ eV
---|---|---
$B$ | $299.6\pm 88.3$ | $94.9\pm 43.7$
$m$ | $4.09\pm 0.43$ | $4.44\pm 0.80$
Table 6.2: Least-squares fit values of the parameters $B$ and $m$ in Eq.(6.1)
for tetramer attachment barriers $E_{q}=0.10$ eV and $E_{q}=0.30$ eV.
The fit has been done using the Mathematica package with the model given by
Eq.(6.1) and leaving as free parameters to fit $B$ and $m$. The least-squares
fit parameters are shown in Table 6.2 where the quoted errors are the standard
error. We note that the fit parameter $m$ is highly dependent on the value of
$n_{eq}$ which needs to be determined with high accuracy.
In both cases the fit value for $m$ indicates that the implementation of the
model is consistent with the $m$-cluster model proposed by Loginova and co-
workers, in our case with $m\simeq 4$ because we used tetramers as the
intermediate species. In the model described by Eq.(6.1), the attachment to
graphene is via $m$-clusters. In the present study, the growth is via both
tetramers and adatoms so we expect some discrepancies with Eq.(6.1). For
$E_{q}=0.10$ eV, $m$-cluster attachment is more predominant over adatoms than
for the case with $E_{q}=0.30$ eV, and indeed we observe a better fit.
However, the discussion above in Subsection 6.2.3 about $N_{eq}$ suggests that
the appropriate value for tetramer attachment barrier is $E_{q}=0.30$ eV. Even
though the fit is not as accurate as for $E=0.10$ eV, the velocity analysis
here suggests that $E_{q}=0.30$ eV is still consistent with Eq.(6.1). This
indicates that the larger tetramer diffusivity still makes this species the
predominant one in graphene growth.
Computationally, island sizes at time intervals of $\Delta t=2.5$ ms have been
used to obtain the results shown in Fig. 6.7. Large islands can break-up or
coalesce during growth, and then $v$ is not evaluated as the values do not
correspond to growth by attachment of adatoms or tetramers only. Also, a
deposition flux $F=10$ ML/s has been used to explore the larger values of the
density $n$.
Different approaches to check the validity of the model and its implementation
can be considered outside of the results obtained within the model. In
Appendix C we present a RE comparison with the present KMC model that further
validates it.
## Chapter 7 Conclusions
### 7.1 Summary
In this thesis we have introduced a KMC model to study the physical processes
governing the kinetics of epitaxial graphene growth on metal substrates. All
the experimental observations have been explained within the model, and the
relevant kinetic paths determining them have been identified.
The behaviour of adatom density at the onset of nucleation has been explained
by the inclusion of both a large critical island size and a tetramer break-up
rate. These conclusions are in agreement with previous RE studies.
The adatom density at equilibrium has been found to be highly dependent on the
adatom attachment barrier to islands. With no such barrier, the adatom density
at equilibrium is vanishingly small, but tuning the energetics of such a
process can lead to the desired concentration of adatoms at equilibrium.
The island density at equilibrium has been found to be dependent on the
attachment barrier of tetramers to islands. Without such barrier, few
nucleations occur because tetramers are more likely to attach to an existing
island than to nucleate new islands. Only with the inclusion of a large
barrier the desired behaviour in the island density at equilibrium is
obtained. This means that the dominance of tetramers in graphene growth is via
their larger diffusivity rather than due to larger attachment barriers for
adatoms than for tetramers.
### 7.2 Further work
The present work has allowed the determination of the relevant physical
processes governing the particular behaviour of the kinetics of epitaxial
graphene growth. The next natural step is to use this knowledge to construct
an accurate KMC model to be compared directly with experiment. First, the
various energy barriers in the problem need to be determined. A standard way
of accomplishing this is by electronic structure calculations using well-
established first-principles methods such as density functional theory. Also,
both the metal substrate (lattice) and the graphene structure need to be
hexagonal. Finally, an exact intermediate 5-cluster species instead of the
more convenient tetramers needs to be included. Difficulties in this model
might arise when comparing with experiment because high temperatures and large
time intervals need to be studied, and this might be computationally too
demanding. The code can be further optimised, but some approximations might
still be necessary to reach the scales of interest.
The detemination of the relevant energy barriers for epitaxial graphene growth
by means of ab initio techniques can also be used in similar studies such as
lattice-free KMC.
Words in text: 9986.
## Appendix A Island growth model
In this appendix we consider the role of islands in the KMC model presented in
Chapter 6.
The model presented in this thesis falls within the generic model class for
the treatment of islands, but the rules on the existence of tetramers and
nucleation of islands make it different from a pure bond-counting model.
Generic models [30] are those with local rules such as bond counting according
to which the system evolves, and are different from the so-called tailored
models. In the latter, some specifications on the system evolution are made
based on the knowledge already existing about the system. For instance, a
critical island size can be defined above which attachment of adatoms to
islands is irreversible. Another possibility is the prescribed island growth
sequence model [30] that we will discuss here in the context of graphene.
In a prescribed island growth sequence model, adatoms that attach to an island
are instantenously moved to a prescribed position on the island independently
of the attachment position. This is equivalent to having an infinite edge
diffusitivity to the most stable sites. It is used in systems known to grow in
certain shapes, for instance in growth on metal (100) where the islands are
known to be nearly squares and each new attached adatom is immediately
incorporated in the sequence to give the desired square shape.
For graphene growth via 5-clusters, the structure of the hexagonal lattice
makes the 5-clusters have the appropriate size to represent an attachment in
which growth only occurs via the formation of new complete hexagons. In Fig.
A.1 we can see such configurations schematically, where we can note that
3-clusters would also lead to such hexagon by hexagon growth and 5-clusters
actually lead to double hexagon growth. This indicates that graphene grows in
a very prescribed manner, and the tetramer attachment and bond counting scheme
used in this thesis might not be entirely accurate in describing it. A
prescribed island growth sequence, by which tetramer attachment leads to an
immediate relaxation into a more stable configuration might be more
appropriate.
The island shape observed in graphene and shown in Fig. A.2 also suggests a
very specific growth sequence, possibly the result of this attachment
mechanism.
Figure A.1: Schematic of 5-cluster attachment to islands. Taken from [3]
Figure A.2: LEEM image of a Ru(0001) substrate with grown graphene islands on
it. Taken from [3]
## Appendix B Island size distribution
In this appendix we present the preliminary data for the island size
distribution of the KMC model for the description of epitaxial growth. For the
reasons discussed in Chapter 5, data for island size distribution is very
noisy, and to get significant results many instances of the system need to be
used. Time constraints have limited the results to the case of $T=500$ K for
the graphene model so far. These are shown in Fig. B.1 for the parameters in
Table 6.1 and $E_{k}=1.40$ eV, $E_{p}=0.30$ eV and $E_{q}=0.30$ eV as usual.
The shape observed is to be expected for the island size distribution.
However, any detailed analysis will require the behaviour of island size
distribution with temperature.
Figure B.1: Island size distribution for the graphene KMC model at $T=500$ K.
## Appendix C Rate equations and kinetic Monte Carlo comparison
In this appendix we compare the KMC model presented in this thesis with the RE
approach by Zangwill and Vvedensky [5]. The objective of the comparison is to
validate the model used in this work.
| RE | KMC
---|---|---
$E_{a}$ | 1.00 eV | 1.00 eV
$E_{t}$ | 0.80 eV | 0.80 eV
$E_{k}$ | 1.40 eV | 1.40 eV
$E_{b}$ | – | 0.20 eV
$E_{i}$ | 1.60 eV | –
Table C.1: Least-squares fit values of the parameters $B$ and $m$ in Eq.(6.1)
for tetramer attachment barriers $E_{q}=0.10$ eV and $E_{q}=0.30$ eV.
The rate equations model used is the one presented by Zangwill and Vvedensky
[5] as described in Sectin 3.2 above. This RE model includes parameters to
describe adatom and tetramer diffusion, tetramer break-up and adatom
detachment $E_{i}$ from islands. The KMC model has corresponding parameters
for the first three quantities, but for the latter it is the adatom bond
energy between adatoms in an island that maps the effect. In order to carry
out the comparison, RE have been used with $i=4$ and $j=6$ and KMC has been
used with $E_{p}=E_{q}=0$. The rest of parameters are shown in Table C.1. A
temperature of $T=650$ K is used with a flux $F=1$ ML/s. The total coverage
reached is $\theta=0.2$ ML corresponding to 0.2 seconds, and the system is
allowed to relax for a further 0.3 seconds.
(a) Adatom density (b) Tetramer density
Figure C.1: Comparison between RE and KMC model for (a) adatom density and (b)
tetramer density.
Figure C.1 shows the adatom concentration $n$ and the tetramer concentration
$c$ as a function of temperature. In both cases, KMC and RE coincide up to the
nucleation point at about $t\sim 0.1$ s, but a clear disagreement appears
after this. The RE overestimate the concentration of both adatoms and
tetramers. In the KMC approach, islands have spatial extent, so their adatom
and tetramer capture zones are large. However, in the RE the islands are
point-like, and their capture zones are reduced, thus leading to a smaller
decrease in the concentration of adatoms and tetramers.
Figure C.2: Comparison between RE and KMC model for the island density.
In Fig. C.2 we show the comparison between RE and KMC for the island density.
Unlike the above, this quantity is badly reproduced by the rate equations as
already discussed in the work by Zangwill and Vvedensky. The island
concetration is underestimated by the rate equations, and again this is
attributed due to the lack of spatial information [5].
## Bibliography
* [1] K. S. Novoselov et al., Science 306, 666 (2004).
* [2] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
* [3] E. Loginova, N. C. Bartelt, P. J. Feibelman, and K. F. McCarty, New J. Phys. 10, 093026 (2008).
* [4] E. Loginova, N. C. Bartelt, P. J. Feibelman, and K. F. McCarty, New J. Phys. 11, 063046 (2009).
* [5] A. Zangwill and D. D. Vvedensky, arXiv:1011.4861v1.
* [6] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
* [7] P. R. Wallace, Phys. Rev. 71, 622 (1947).
* [8] G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).
* [9] K. S. Novoselov et al., Nature 438, 197 (2005).
* [10] Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 98, 216803 (2006).
* [11] M. Y. Yan, B. Özyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. 98, 206805 (2007).
* [12] J. Bai, X. Zhong, S. Jiang, Y. Huang, and X. Duan, Nature Nanotech. 5, 190 (2010).
* [13] Y.-M. Lin et al., Nano Lett. 9, 422 (2009).
* [14] Y.-M. Lin et al., Science 327, 662 (2010).
* [15] Y. Wu et al., Nature 472, 74 (2011).
* [16] R. Prasher, Science 328, 185 (2010).
* [17] J. H. Seol et al., Science 328, 213 (2010).
* [18] E. Bauer, Zeitschrift für Kristallographier 110, 372 (1958).
* [19] J. A. Venables, Introduction to Surface and Thin Film Processes (Cambridge University Press, 2000).
* [20] J. R. Arthur, Surface Science 500, 189 (2002).
* [21] W. Zhang, P. Wu, Z. Li, and J. Yang, (2011), arXiv:cond-mat.mtrl-sci/1101.3851v1.
* [22] M. Treier et al., Nature Chemistry 3, 61 (2010).
* [23] Y. Cui, Q. Fu, H. Zhang, and X. Bao, Chem. Commun. 47, 1470 (2011).
* [24] P. Lacovig et al., Phys. Rev. Lett. 103, 166101 (2009).
* [25] B. Wang, X. Ma, M. Caffio, R. Schaub, and W.-X. Li, Nano Letters 11, 424 (2011), http://pubs.acs.org/doi/pdf/10.1021/nl103053t.
* [26] R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).
* [27] A. B. Bortz, M. H. Kalos, and J. L. Lebowitz, J. Comput. Phys. 17, 10 (1975).
* [28] P. A. Maksym, Semicond. Sci. Technol. 3, 594 (1988).
* [29] J. L. Blue, I. Beichl, and F. Sullivan, Phys. Rev. E 51, R867 (1995).
* [30] J. W. Evans, P. A. Thiel, and M. C. Bartelt, Surface Science Reports 61, 1 (2006).
* [31] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961).
* [32] M. Petersen, A. Zangwill, and C. Ratsch, Surface Science 536, 55 (2003).
* [33] H. Amara, C. Bichara, and F. Ducastelle, Phys. Rev. B 73, 113404 (2006).
|
arxiv-papers
| 2012-03-31T14:09:59 |
2024-09-04T02:49:29.247857
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "B. Monserrat",
"submitter": "Bartomeu Monserrat",
"url": "https://arxiv.org/abs/1204.0104"
}
|
1204.0178
|
# On the Clausius formulation of the second law in stationary chemical
networks through the theorems of the alternative.
Daniele De Martino Dipartimento di Fisica, Sapienza Università di Roma,p.le
A. Moro 2, 00185 Roma (Italy)
###### Abstract
In this article the Gordan theorem is applied to the thermodynamics of a
chemical reaction network at steady state. From a theoretical viewpoint it is
equivalent to the Clausius formulation of the second law for the out of
equilibrium steady states of chemical networks, i.e. it states that the
exclusion (presence) of closed reactions loops makes possible (impossible) the
definition of a thermodynamic potential and vice versa. On the computational
side, it reveals that calculating reactions free energy and searching
infeasible loops in flux states are dual problems whose solutions are
alternatively inconsistent. The relevance of this result for applications is
discussed with an example in the field of constraints-based modeling of
cellular metabolism where it leads to efficient and scalable methods to afford
the energy balance analysis.
###### pacs:
05.70.Ln,82.60.-s,87.18.Nq,89.75.Hc
## Introduction
The non-equilibrium thermodynamics of chemical reaction networks has been a
subject of great interest and research efforts in recent years, from the
pioneering works on network thermodynamicsG.Oster et al. (1971) to the recent
study of their statistical grounds through fluctuation theoremsP.Gaspard
(2004); Beard and Qian (2007). The quest for general results or underlying
variational principlesP.B.Warren and J.L.Jones (2007) is open, as it is for
out of equilibrium thermodynamics in generalD.Kondepudi and Prigogine (1998).
Regarding its applications, a chemical reaction network at steady state under
the hypothesis of local equilibrium and well-mixing is the simplest (and
standard) model of cell metabolism. This model has been applied to large
(genome-) scale systems only in recent times, where it was successful in
describing and predicting, even at a quantitative level, cellular metabolic
processesVarma and Palsson (1994). In general, the stochiometry of the network
defines linear mass-balanced equations for the reaction fluxes and thus a
feasible solution space for the steady states. In particular, putting forward
a functional hypothesis, one solution can be selected by maximizing a linear
function that describes e.g. the biomass or ATP production. In this way is
possible to use powerful methods and tools from linear programming, a
framework that goes under the name of flux balance analysis (FBA)Orth et al.
(2010). More recently, the question of the implementation of the thermodynamic
constraints has been posed, a framework has been called energy balance
analysis (EBA)Beard et al. (2002). This problem is connected on one hand to
the calculation of reactions free energy, metabolites chemical potential and
concentrationKummel et al. (2006); Henry et al. (2007), on the other to the
removal of infeasible loops from flux configurationsPrice et al. (2002). All
through the literature there is the intuition of the infeasibility of closed
reaction loops, since it is easy to show that their existence renders
impossible to define a thermodynamic potential. So, their exclusion is
necessary, but is it _sufficient_ to guarantee the thermodynamical
consistency? In this article a positive aswer to this question will be given,
showing that it comes from the application of the Gordan theorem, the oldest
theorem of the alternatives and a key result in optimizationSchrijver (1986).
The application of the Gordan theorem in this context states shortly that,
under the hypothesis of local equilibrium for well-mixed systems at constant
pressure and temperature, a consistent thermodynamic potential can be defined
_if and only if_ there are no closed loops. This is substantially equivalent
to the Clausius formulation of the second law in networks where closed
reaction loops play the role of _perpetuum mobile of the second kind_ and its
demonstration leads to address more efficiently the energy balance analysis of
metabolic networks. Infact, more in details, the theorem reveals that finding
infeasible reactions loops and/or calculating reactions free energies are dual
problems whose solutions are alternatively inconsistent. For instance, current
methods apt to find and eliminate infeasible loops suffers of scalability
issuesSchellenberger et al. (2011). Then, this difficulty can be circumvented,
by dealing with the easier dual problem of calculating reactions free energy,
that consists in solving a system of linear inequalities. In the following we
will give an elementary demonstration of this theorem and discuss its
applications with an example in the field of constraints-based modeling of
cellular metabolism where it gives efficient methods to afford the energy
balance analysis.
## I Results
Suppose the stochiometry of a chemical reaction network is given in terms of a
matrix $\mathbb{S}\equiv(S_{i}^{\mu})$, where $S_{i}^{\mu}$ stands for the
stochiometric coefficient of the chemical species $\mu=1\dots M$ in the
reaction $i=1\dots N$. A flux vector ${\bf v}\equiv(v_{i})$ is by
definitionBeard and Qian (2008) _thermodynamically feasible_ provided the
existence of a free energy vector ${\bf\Delta G}\equiv(\Delta G_{i})$ such
that
###### Definition 1 (Thermodynamic feasibility).
$\displaystyle v_{i}\Delta G_{i}<0\qquad\forall i$ $\displaystyle{\bf
r}\cdot{\bf\Delta G}=0\qquad\forall{\bf r}\in Ker(\mathbb{S}).$ (1)
Writing the free energy change in terms of the chemical potentials ${\bf
g}\equiv(g_{\mu})$
$\Delta G_{i}=\sum_{\mu}S_{i}^{\mu}g_{\mu},$ (2)
and defining for sake of semplicity in notations
$\xi_{i}^{\mu}=-sign(v_{i})S_{i}^{\mu}$, we have a system of linear
inequalities equivalent to (1)
$\sum_{\mu}\xi_{i}^{\mu}g_{\mu}>0.$ (3)
Now, the existence of solutions of the system (3) is ruled by the Gordan
theorem: the system (3) has solution _if and only if_ the dual system
###### Definition 2 (Infeasible loops).
$\displaystyle\sum_{i}\xi_{i}^{\mu}k_{i}=0\qquad\forall\mu$ $\displaystyle
k_{i}\geq 0,\quad{\bf k}\neq{\bf 0}$ (4)
has no solution, where the solutions of (2) define closed reaction loops. In
other terms,
###### Theorem (Gordan theorem).
One and only one of the systems (3) and (2) have solution.
###### Proof.
(2) has solution $\Rightarrow$ (3) has no solution.
Suppose to have a solution ${\bf k^{*}}$ of (2). Take any vector ${\bf g}$,
multiply it component by component with the equations of the system (2) and
sum over them. Exchanging the indeces $\mu$ and $i$ in the sums and given the
positivity of ${\bf k^{*}}$, it is straightforward to conclude that no vector
${\bf g}$ can satisfy the system (3).
###### Proof.
(3) has no solution $\Rightarrow$ (2) has solution.
The demonstration is given by induction in the number $M$ of unknows.
The statment is true for $M=1$. Infact, for one unkonwn, the system (3) is
inconsistent if and only if there is at least one couple $i$, $j$ such that
$\xi_{i}^{1}\xi_{j}^{1}=-1/c<0$, and in this case $k_{i}=1$, $k_{j}=c$ and
$k_{l}=0\quad\forall l\neq i,j$ is a solution of (2).
Let’s consider a system of the type (3) with $M$ unknows and suppose it is
inconsistent. We will prove that the dual system has solutions supposing the
theorem true for systems with $M-1$ unknows. We have $\forall
i\quad\sum_{\mu=1}^{M-1}\xi_{i}^{\mu}g_{\mu}>-\xi_{i}^{M}g_{M}$, if
$\xi_{i}^{M}\neq 0$, we can define
$\tilde{\xi}_{i}^{\mu}=-\xi_{i}^{\mu}/\xi_{i}^{M}$, and we have:
$\displaystyle\sum_{\mu}^{M-1}\tilde{\xi}_{i}^{\mu}g_{\mu}=P_{i}>g_{M}\qquad\forall
i:\quad\xi_{i}^{M}<0$
$\displaystyle\sum_{\mu}^{M-1}\tilde{\xi}_{j}^{\mu}g_{\mu}=Q_{j}<g_{M}\qquad\forall
j:\quad\xi_{j}^{M}>0$
$\displaystyle\sum_{\mu}^{M-1}\xi_{l}^{\mu}g_{\mu}=R_{l}>0\qquad\forall
l:\quad\xi_{l}^{M}=0.$ (5)
Writing the system in this form, we can pass to the following system in $M-1$
unknows:
$\displaystyle P_{i}>Q_{j}\qquad\forall
i,j:\quad\xi_{i}^{M}<0\quad\xi_{j}^{M}>0$ $\displaystyle R_{l}>0\qquad\forall
l:\quad\xi_{l}^{M}=0.$ (6)
Now, also this system is inconsistent.
Suppose infact there is a solution ${\bf g^{*}}=(g_{\mu}^{*})$, $\mu=1\dots
M-1$.
We could add to it any $g_{M}^{*}$ such that $max_{j}Q_{j}({\bf
g^{*}})<g_{M}^{*}<min_{i}P_{i}({\bf g^{*}})$ and we will have a solution for
the original system as well, against the hypothesis.
By induction hypothesis, the theorem is true for systems with $M-1$ unknows.
Then, referring to (I), there are $\tilde{k}_{ij}\geq 0$, $k_{l}\geq 0$ with
at least one positive, such that
$\sum_{ij}\tilde{k}_{ij}(\tilde{\xi}_{i}^{\mu}-\tilde{\xi}_{j}^{\mu})+\sum_{l}k_{l}\xi_{l}^{\mu}=0\quad\forall\mu$.
From this we have finally a solution for the system (2):
$\displaystyle k_{i}=-\sum_{j}\tilde{k}_{ij}/\xi_{i}^{M}\qquad\forall
i:\quad\xi_{i}^{M}<0$ $\displaystyle
k_{j}=\sum_{i}\tilde{k}_{ij}/\xi_{j}^{M}\qquad\forall i:\quad\xi_{j}^{M}>0$
$\displaystyle k_{l}\qquad\forall l:\quad\xi_{l}^{M}=0,$ (7)
and the theorem is proven. For sake of simplicity the case in which some of
the fluxes are null is neglected, but it follows along the same lines as an
application of the Motzkin theoremSchrijver (1986).
Apart from its inherent theoretical interest, this result can be used to
perform efficiently the energy balance analysis of metabolic networks, because
it suggests to deal with the dual system (3) if we want to find and correct
flux configurations from infeasible loops (calculating and/or correcting a
consistent chemical potential vector as a by-product).
An algorithm was provided inDe Martino et al. (2012) complementing standard
relaxation methodSchrijver (1986) (or MinOver schemeKrauth and Mezard (1987))
with an exaustive search, in the spirit of recent constructive demonstrations
of the Farkas LemmaD.Avis and B.Kaluzny (2004), the most famous extension of
the Gordan theorem. The algorithm works by correcting step by step the least
unsatisfied constraint, $\alpha$ being constant (Minover) or proportional to
the amount by which the constraint is violated (Relaxation), i.e. a series is
defined:
$\displaystyle i_{l}=min_{i}\sum_{\mu}\xi_{i}^{\mu}g_{\mu}(l)$ (8)
$\displaystyle g_{\mu}(l+1)=g_{\mu}(l)+\alpha\xi_{il}^{\mu},$ (9)
till a solution is found, if any, in polynomial timeDe Martino et al. (2012).
If the algorithm doesn’t converge, by the theorem there should be infeasible
loops. Then, looking at the series of distinct least unsatisfied constraints,
that unless of very bad cases is not large, it is possible to find them. The
Minover scheme alone was already applied to the dual part of the stochiometric
matrix of metabolic networks in order to calculate the production
profileMartelli et al. (2009). In De Martino et al. (2012) this algorithm was
applied to calculate the list of infeasible loops in the recent model of
E.coli metabolism iAF1260, here, as a further example, it is reported in Table
1 a list of the possible unfeasible loops founded with this method in the
model of E.Coli metabolic network iJR904Reed et al. (2003). This network is
composed of $931$ reactions ($245$ of which are reversible) among $761$
metabolites. One of this loop is depicted in fig.1, from which it is clear
that e.g. the putative reversible reaction PPAKr sohuld operate in the
backward direction in order to guarantee thermodynamic feasibility. The
thermodynamic feasibility of flux configurations in which the presence of
these loops is removed from the beginning was tested over $10^{6}$ random
instances. The small number of such loops should be ascribed to the detailed
prior thermodynamic information on reaction reversibility usually provided
with the network models, whose assessement comes from a careful estimate of
the chemical potentialsFleming et al. (2009).
Cycle ID | Lenght | Formula
---|---|---
1 | 3 | ACCOAL $+$ PTA2 $-$ PPAKr(R)
2 | 3 | GLUABUTt7(R) $-$ GLUt2r(R) $+$ ABUTt2
3 | 3 | ACCOAL $-$ SUCOAS(R) $+$ PPCSCT
4 | 3 | VALTA $-$ ALATA_L(R) $+$ VALTA
5 | 3 | ADK3(R) $-$ ADK1(R) $+$ NDPK1(R)
6 | 3 | ADK1(R) $-$ ADK3(R) $-$ NDPK1(R)
7 | 3 | NAt3_1(R) $-$ THRt2(R) $+$ THRt4
8 | 3 | NAt3_1(R) $-$ PROt2r(R) $-$ PROt4
9 | 3 | NAt3_1(R) $-$ GLUt2r(R) $+$ GLUt4
10 | 3 | NAt3_1(R) $-$ SERt2r(R) $+$ SERt4
Table 1: The 10 unfeasible cycles identified from thermodynamic feasibility
analysis of $10^{6}$ different randomly generated flux configurations for the
Escherichia coli metabolic reaction network iJR904. Plus (resp. minus) signs
indicate that the reaction participates in the cycle in its forward (resp.
backward) direction. (R) indicates that the corresponding reaction is
putatively reversible according to Reed et al. (2003). The directions of the
putatively reversible fluxes PPAKr and SOCOAS turn out to be in fact
constrained. Figure 1: An infeasible loop that can be present in the metabolic
network model iJ904 if the putative reversible reaction PPAKr is considered to
work in the backward direction.
## Conclusions
In this article was given a rigourous proof that reaction fluxes in a chemical
reaction network at steady state are thermodynamically feasible if and only if
there are no closed reaction loops. It was showed that this is not a trivial
statment but an application of the Gordan theorem. Infact, it states in detail
that calculating free energies and/or chemical potentials and assessing the
thermodynamic feasibility of flux configurations through the search for
infeasible loops are dual problems whose solutions are mutually inconsistent.
This in turn provides efficient ways to perform the energy balance analysis of
metabolic networks and a genome-scale example was given. From a theoretical
viewpoint, the result could be extended to a more general set of processes to
define the underlying thermodynamic variational principle itself: since there
are no loops, a potential can be defined whose changes have a definite sign.
From this point of view the theorem extends the Clausius formulation of the
second law in a network context, that could be an interesting result for the
emerging field of network scienceA.L.Barabasi (2012) as well. Further, it
would be interesting to investigate the statistical basis of this theorem.
Regarding applications, apart from thermodynamic feasibility, another
important problem related to flux analysis in constraint based modeling
concerns the link between the network structure and its productive
capabilitiesDe Martino and Marinari (2010). In a recently developed framework
inspired by economic systems modeling De Martino et al. (2004), the fluxes are
calculated from the minimal constraint that for each chemical species at
stationarity the overall consumption cannot exceed the total supply (Von
Neumann constraints), and the aim is to calculate producibility profiles and
to infer the biomass coefficients directly from the stochiometryMartelli et
al. (2009). The application of the the Farkas lemma in this context, i.e. to
the dual part of the stochiometric matrix, has been partially shown in
M.Imielinski and Halasz (2006) in order to study the growth media for E.Coli,
revealing the connection between producibility and conserved metabolic pools.
However, similarly to this work, these theorems of the alternative could
provide also interesting insights for the calculation itself of the
producibility profile toghether with the conserved metabolic pools, an aspect
that is leaved for future investigations.
## Acknowledgments
This work is supported by the DREAM Seed Project of the Italian Institute of
Technology (IIT), by the IIT Computational Platform and by the joint
IIT/Sapienza Lab “Nanomedicine”. The author thanks A. De Martino, M. Figliuzzi
and E. Marinari for useful suggestions, fruitful discussions and
encouragement.
## References
* G.Oster et al. (1971) G.Oster, A. Perelson, and A. Katchalsky, Nature p. 393 (1971).
* P.Gaspard (2004) P.Gaspard, journal of chemical physics 120, 8898 (2004).
* Beard and Qian (2007) D. Beard and H. Qian, PLoS ONE 2, e144 (2007), ISSN 1932-6203.
* P.B.Warren and J.L.Jones (2007) P.B.Warren and J.L.Jones, Physical review letters 99, 108101 (2007).
* D.Kondepudi and Prigogine (1998) D.Kondepudi and I. Prigogine, _Modern thermodynamics: from heat engines to dissipative structures_ (Wiley Chichester, 1998).
* Varma and Palsson (1994) A. Varma and B. Palsson, Nature Biotechnology 12, 994 (1994), ISSN 1087-0156.
* Orth et al. (2010) J. Orth, I. Thiele, and B. Palsson, Nature Biotechnology 28, 245 (2010), ISSN 1087-0156.
* Beard et al. (2002) D. Beard, S. Liang, and H. Qian, Biophysical Journal 83, 79 (2002), ISSN 00063495.
* Kummel et al. (2006) A. Kummel, S. Panke, and M. Heinemann, Molecular Systems Biology 2, 2006.0034 (2006), ISSN 1744-4292.
* Henry et al. (2007) C. Henry, L. Broadbelt, and V. Hatzimanikatis, Biophysical Journal 92, 1792 (2007), ISSN 00063495.
* Price et al. (2002) N. Price, I. Famili, D. Beard, and B. Palsson, Biophysical Journal 83, 2879 (2002), ISSN 00063495.
* Schrijver (1986) A. Schrijver, _Theory of linear and integer programming_ (Wiley Online, 1986), 1st ed., ISBN 0-471-90854-1.
* Schellenberger et al. (2011) J. Schellenberger, N. Lewis, and B. Palsson, Biophysical Journal 100, 544 (2011), ISSN 1542-0086.
* Beard and Qian (2008) D. Beard and H. Qian, _Chemical Biophysics: Quantitative Analysis of Cellular Systems (Cambridge Texts in Biomedical Engineering)_ (Cambridge University Press, 2008), 1st ed., ISBN 0521870704.
* De Martino et al. (2012) D. De Martino, M. Figliuzzi, A. De Martino, and E.Marinari, Forthcoming in Plos computational biology (2012).
* Krauth and Mezard (1987) W. Krauth and M. Mezard, Journal of Physics A: Mathematical and General 20, L745 (1987).
* D.Avis and B.Kaluzny (2004) D.Avis and B.Kaluzny, The American Mathematical Monthly 111, 152 (2004).
* Martelli et al. (2009) C. Martelli, A. De Martino, E. Marinari, M. Marsili, and I. Perez Castillo, Proceedings of the National Academy of Sciences of the USA 106, 2607 (2009), ISSN 1091-6490.
* Reed et al. (2003) J. Reed, T. Vo, C. Schilling, and B. Palsson, Genome Biology 4, R54 (2003), ISSN 1465-6914.
* Fleming et al. (2009) R. Fleming, I. Thiele, and H. Nasheuer, Biophysical Chemistry 145, 47 (2009), ISSN 03014622.
* A.L.Barabasi (2012) A.L.Barabasi, Nature physics 8, 14 (2012).
* De Martino and Marinari (2010) A. De Martino and E. Marinari, Journal of Physics Conference Series 233, 012019 (2010).
* De Martino et al. (2004) A. De Martino, M. Marsili, and I. Perez Castillo, Journal of Statistical Mechanics: Theory and Experiment 2004, P04002 (2004).
* M.Imielinski and Halasz (2006) H. M.Imielinski, C.Belta and A. Halasz, Biophysical journal 93, 704 (2006).
|
arxiv-papers
| 2012-04-01T08:25:54 |
2024-09-04T02:49:29.258032
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele De Martino",
"submitter": "Daniele De Martino",
"url": "https://arxiv.org/abs/1204.0178"
}
|
1204.0226
|
∎
11institutetext: M. Zohar 22institutetext: M. Auslender 33institutetext: S.
Hava 44institutetext: Department of Electrical and Computer Engineering, Ben
Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva
Tel.:+972-8-6461583
Fax: +972-8-6472949
44email: zoharmo@ee.bgu.ac.il 55institutetext: L. Faraone 66institutetext:
School of Electrical, Electronic and Computer Engineering, The University of
Western Australia, M018 35 Stirling Highway Crawley WA 6009, Australia
# New resonant cavity-enhanced absorber structures for mid-infrared detector
applications
Moshe Zohar Mark Auslender Lorenzo Faraone Shlomo Hava
(Received: date / Accepted: date)
###### Abstract
A new dielectric Fabry-Perot cavity was designed for a resonant enhancing
optical absorption by a thin absorber layer embedded into the cavity. In this
cavity, the front mirror is a subwavelength grating with $\sim 100$%
retroreflection. For a HgCdTe absorber in a matching cavity of the new type,
the design is shown to meet the combined challenges of increasing the
absorbing efficiency of the entire device up to $\sim 100$ % and reducing its
size and overall complexity, compared to a conventional resonant cavity
enhanced HgCdTe absorber, while maintaining a fairly good tolerance against
the grating’s fabrication errors.
###### Keywords:
Optical resonant cavity Photodetectors HgCdTe Gratings
## 1 Introduction
It is well known, that incorporating a photosensitive or an optically active
layer into a Fabry-Perot (FP) cavity enhances the efficiency of detection by
or emission from the layer, as a consequence of the multiple reflections that
occur between the cavity’s mirrors. The enhancement is maximized if the round-
trip phase, i.e. the phase difference between each succeeding reflection,
satisfies the resonance condition
$\delta_{0}\equiv\delta\left(\lambda_{0}\right)=\frac{4\pi n_{\rm
c}\left(\lambda_{0}\right)t_{c}}{\lambda_{0}}+\varphi_{\rm
f}\left(\lambda_{0}\right)+\varphi_{\rm b}\left(\lambda_{0}\right)=2\pi n.$
(1)
Here $\lambda_{0}$ is a resonance wavelength $n_{\rm c}$ and $t_{\rm c}$ is
the refractive index (RI) and length of the FP cavity, respectively,
$\varphi_{\rm f}$ and $\varphi_{\rm b}$ are the reflection phase of the mirror
that is further away from and adjacent to the illuminated side (hereafter
referred to as the front and back mirror, respectively), and $n$ is an
integer. Eq.(1) is thumb rule accurate regardless of the type of flat-surface
mirror being employed, whether it consist of thin metallic films, or is e.g. a
distributed Bragg reflector (DBR) that is a stack of quarter-wave pairs of
high/low (HL) RI dielectric layers heavens ; knittl .
Optical communication, interconnection and information processing systems
require high-efficiency photodetectors (PDs). In response to this demand, the
field of resonant-cavity-enhanced (RCE) PDs has steadily maturated over past
two decades unlu ; deen . For thermal PDs, the optical absorbance $A$ is an
accurate measure of the efficiency, while for photodiodes and photo-conductors
the key figure is the quantum efficiency $\eta$. For initial design of the RCE
PDs, the adoption of $\eta\approx A$ has a widespread use unlu ; deen since
thus the analysis is framed into optics alone rather than being restricted to
a specific PD technology. It appears unlu that for obtaining $\eta\approx
100\%$ it is at least necessary to have $\sim 100$% resonant reflectivity from
the front mirror, which is problematic for a DBR of any practical thickness
because of the tight material requirements for epitaxial growth. Thus, it is
relevant to consider an alternative replacement for the DBR mirror by a thin
dielectric micro-optical component which can be $\sim 100$ % reflective in a
broad spectral band. A device that can exhibit such a reflection anomaly with
a proper design havaus ; chang-a is a dielectric subwavelength grating
structure.
This paper presents a theoretical study to incorporate a subwavelength grating
as the front mirror in the FP cavity, for PD applications in the mid-wave
infrared (MWIR) range. For a vertical-cavity surface-emitting laser, using an
air-bridge subwavelength grating as a suspended top mirror, has been proposed
before bissaillon ; kim . Only limited work on RCE PDs for operation in the
MWIR range has been reported thus far. Currently this is an area of great
interest due to the potential for obtaining the uncooled PDs.
## 2 Design considerations and simulation tools
We adopted $\eta=A$ [see Sec. 1], so RCE PDs with an absorber layer
characterized by a complex RI, $n_{\rm a}+ik_{\rm a}$, were analyzed only
optically. To compute the reflectance, transmittance and $A$ spectra of RCE
PDs including a grating, we used the rigorous coupled wave analysis recast
with an in-layer S-matrix propagation algorithm aushava1 . For DBR based PDs,
a limiting case at zero grating groove width, which coincides with the
impedance form knittl of transfer matrix method, was employed. The optical
spectra were computed in the range $4.215\leq\lambda\leq 4.615\;\mu{\rm m}$.
We put emphasis on the two issues: (i) optimizing the peak $A$ at
$\lambda_{0}=4.415\;\mu$m and $\lambda_{0}=4.500\;\mu$m, which are kept fixed,
while fulfilling Eq.(1); (ii) seeking the designs with a high tolerance of the
efficiency to errors in the grating fabrication. Interim design parameters
serving as trial ones for computer aided optimization, will be obtained using
the reflection phases [see Eq.(1)] of stand-alone mirrors unlu , which means
the mirrors put on an infinite medium of the same material as the cavity’s
material (CdTe in our designs) and irradiated from the substrate.
## 3 Conventional RCE PD structures
The reported RCE HgCdTe-PDs faraone1 ; faraone2 ; faraone3 have a standard
configuration in which the CdTe FP cavity embeds a thin Hg0.71Cd0.29Te layer
between two DBR mirrors [see in Fig.1(a)]. The HgCdTe/CdTe bilayers provide
good lattice matching but the small difference in RIs of Hg0.56Cd0.44Te and
CdTe requires $m=30$, i.e. $\sim 20\;\mu$m thick DBR mirror, in order to
achieve the desired reflectivity faraone3 . Apart from enlarging the cavity,
the deposition of such a thick multilayer inevitably produces growth defects
which deteriorate the reflectance. In practice, such DBRs have been fabricated
with $m=10$ kanev and $m=7$ faraone3 and optimized with $m=8$ for high
temperature-operation RCE PDs sioma .
Figure 1: RCE HgCdTe-absorber structures with the: (a) (HL)kH DBR and (b)
grating front mirror; the back mirror in both is a (HL)mH DBR; the irradiation
is from a CdZnTe substrate.
We adopted a mid value of $m=15$, while due to faraone1 ; faraone2 ; faraone3
$k=2$, $t_{\rm a}=75\;{\rm nm}$. For a standard placement of the absorbing
layer in the middle of the cavity ($d_{\rm f}=d_{\rm b}$), combining
semianalytical calculation and manual optimization, we found the optimal
cavity length for which $A=81.9\%$ at $\lambda_{0}=4.415\;\mu\rm m$ [RCE-S
structure in Table 2]. The automatic optimization results in the same $t_{\rm
c}$ as obtained manually. Further optimization, allowing for $d_{\rm f}\neq
d_{\rm b}$, yields RCE-Oa and RCE-Ob structures with $\lambda_{0}=4.415$ and
$\lambda_{0}=4.500\;\mu\rm m$, respectively [see in Fig.3 and Table 2]. Figure
3 shows the $A$ spectra of the designed structures along with the reflectance
($R$) from the stand-alone front DBR mirror.
Table 1: The refractive indexes $N=n+ik$ used in the simulations Wavelengths | Ge | SiO | CdTe | Hg0.71Cd0.29Te | Hg0.56Cd0.44Te | CdZeTe
---|---|---|---|---|---|---
$4.415\;\mu\rm m$ | $3.9332$ | $1.78$ | $2.6695$ | $3.4826+i0.1477$ | $2.9709$ | 2.6896
$4.500\;\mu\rm m$ | $3.9325$ | $1.78$ | $2.6691$ | $3.4665+i\;0.1425$ | $2.9697$ | 2.6895
## 4 Grating mirror based RCE PD structures
Figure 1(b) presents a modified RCE HgCdTe-PD, in which the front mirror is
replaced by a one-dimensional (1-D) dielectric grating with period $\Lambda$,
groove width $W$ and depth $t_{\rm g}$, while the absorber, back mirror and
irradiation scheme remain the same as those of the structure in Sec. 3. The
grating is designed as the stand-alone $\sim 100$ % reflectivity mirror. Our
design of the RCE absorber as a whole is done for the backside irradiation
from CdZnTe substrate and is not suitable for the front irradiation from air
[see Fig.1], because the grating mirror breaks down the irradiation symmetry
inherent to the DBR based FP cavities.
### 4.1 Polarization-selective RCE HgCdTe-absorber structures
Earlier work havaus and a recent paper chang-a on designing grating mirrors
have considered optical incidence from air, but here the stand-alone grating
mirror is irradiated from CdTe. To get rid of the grating reflection orders,
we imposed over all the computed wavelength range the subwavelength grating
restriction $n_{\rm CdTe}\Lambda<\lambda$, which is tighter than for the
incidence from air. Because of polarization sensitivity of the 1-D gratings,
we first maximized the $R$ for the incidence from CdTe half-space, separately
for the TE and TM polarization.
Figure 2: The spectra of reflectance from the front DBR mirrors and of
absorbance for the RCE-Oa and RCE-Ob structures.
Figure 3: The spectra of reflectance from the front grating mirrors and of
absorbance for the RCE-TEa and RCE-TEb structures.
Other trial dimensions were estimated semianalytically using the $\varphi_{\rm
f}$ and $\varphi_{\rm b}$ as computed numerically for the stand-alone grating
and DBR, respectively, and then refined by the optimization. The manual design
was performed under the constraints: $d_{\rm f}=d_{\rm b}$, $w=W/\Lambda=0.5$
and $t_{\rm a}=0.075\;\mu{\rm m}$, by controlling the $t_{\rm c}$ which
allowed us to attain the desired $\delta_{0}$. To remind, the optimization
goal is to achieve maximal polarized $A(\lambda_{0})$, minimal changes of
which due to variations in the grating fabrication process are being very
desirable. A fully computer-aided optimization procedure easily allows for
designing the structures with $d_{\rm f}\neq d_{\rm b}$ and $w\neq 0.5$. The
results of such an optimization are the structures RCE-TEa and RCE-TMa
($\lambda_{0}=4.415\;\mu$m), RCE-TEb and RCE-TMb ($\lambda_{0}=4.500\;\mu$m)
for operation with a TE and TM polarized radiation [see Table 2].
The TE and TM polarized $R$ spectra from the stand-alone grating mirrors and
the related $A$ spectra of the RCE-TEa and RCE-TEb structures are shown in
Fig. 3 and 5, respectively. The designed gratings maintain the polarized
$R\geq 99\%$ in a wide band around $\lambda_{0}$. For the TM polarization, as
seen in Fig. 5, the high-$R$ band width greatly exceeds $0.3\;\mu$m, in a good
agreement with the paper chang-a . For the TE polarization, however, this band
is narrower which does not prevent from achieving the $A\approx 100$% peak
[see Fig. 3] and a superior tolerance to a variation in the grating
fabrication process, as discussed in Subsect. 4.2 below.
### 4.2 Design tolerances
An important issue for the practicality of the proposed RCE HgCdTe-PDs is the
sensitivity to variations in the grating fabrication process. In this paper,
we report the tolerance of the value and position of the $A$ peak to the
grating groove duty cycle $w$ variations. The tolerance was found to be very
good for the structures with the optimal $w\neq 0.5$, especially in the TE
case. For example, the RCE-TEa structure still maintains high values of
$A(\lambda_{0})=93.8$% and $98.8$% with the initially prescribed $\lambda_{0}$
and $w=0.528$ and $w=0.728$, respectively. The small drop in $A(\lambda_{0})$,
should not be treated as a deterioration of the detection performance since
the peak $A=99$% appears to be shifted by at most 3 nm from the $\lambda_{0}$,
which indicates an excellent tolerance including a small change of the peak
value and a minute shift of the peak position. The RCE-TEb structure exhibits
similar tolerance properties.
Figure 4: The reflectance from the stand-alone grating mirrors and the
absorbance of the RCE-TMa and RCE-TMb structures.
Figure 5: The peak absorbance and wavelength versus $w$ for the RCE-TMb
structure.
The RCE-TMa,b structures withstand errors in $w$ of $\pm 0.05$ as related to
peak $A$, which indicates a lower but still practical tolerance. However, the
peak wavelength shift proves notably larger as compared to RCE-TEa,b
structures. For example, as $w$ is shifted to $0.287$ the RCE-TMa structure
exhibits a $24$ nm red shift of the absorption peak and a drop of the peak $A$
to $97.5\%$. In this case, we found numerically that the peak wavelength,
within a reasonable range, depends on $w$ linearly, as shown in Fig. 5.
## 5 Conclusion
The RCE HgCdTe-PDs, including new ones in which the front mirror is a $\sim
100\%$ reflectance grating, for applications in MWIR range, were designed and
simulated using semianalytic and computer-aided tools such as a scrutinized
control of the resonant round-trip phase $\delta_{0}$, and the grating
reflection amplitude and phase. The results show that the $\delta_{0}$ control
is crucial for optimizing the efficiency, as was known for the conventional
RCE PDs unlu ; faraone1 ; faraone2 ; faraone3 . In our case, $n_{\rm a}$ and
$n_{\rm c}$ differ significantly, but the reflections from the absorbing layer
have a relatively small effect due to $t_{\rm a}\ll t_{\rm c}$. We also proved
that a non-standard placement of the absorbing layer is capable of increasing
the efficiency of the conventional RCE HgCdTe-PDs, which still remains below
$85.2$%.
It was decisively shown that for a linearly polarized light, the grating
mirror based RCE HgCdTe-PDs can be designed to achieve $\sim 100\%$
efficiency, thus highly outperforming the conventional RCE HgCdTe-PDs, and
simultaneously is highly tolerant to variations in the grating groove duty
cycle $w$. For the TM designs, a linear dependence was found between the $w$
varied around its optimal value and the peak wavelength shifting, in a
response, around the prescribed $\lambda_{0}$.
Table 2: The designed and simulated RCE HgCdTe-absorber structures. The dimensional parameters and $\lambda_{\rm{R}}$ are in microns. lowercase a and b refers to a wavelengths of $4.415\;\mu$m and $4.500\;\mu$m respectively. Structures | $d_{\rm{b}}$ | $t_{\rm{a}}$ | $d_{\rm{f}}$ | $t_{\rm{Ge}}$ | $t_{\rm{g}}$ | $\Lambda$ | $w$ | Peak $A$ | $\lambda_{\rm{R}}$
---|---|---|---|---|---|---|---|---|---
RCE-S | 1.177 | 0.075 | 1.177 | - | - | - | - | 0.819 | -
RCE-Oa | 0.272 | 0.075 | 0.433 | - | - | - | - | 0.831 | -
RCE-Ob | 0.292 | 0.075 | 0.429 | - | - | - | - | 0.828 | -
RCE-TEa | 0.209 | 0.075 | 0.535 | 0.331 | 0.844 | 1.461 | 0.628 | 0.998 | 3.901
RCE-TEb | 0.228 | 0.075 | 0.534 | 0.335 | 0.861 | 1.492 | 0.629 | 0.999 | 3.984
RCE-TMa | 0.259 | 0.075 | 0.504 | 0.628 | 0.921 | 1.456 | 0.337 | 0.999 | 3.888
RCE-TMb | 0.286 | 0.075 | 0.495 | 0.650 | 0.930 | 1.468 | 0.334 | 0.999 | 3.920
## References
* (1) Heavens O. S.: Optical Properties of Thin Solid Films, Dover Publ., New York (1991).
* (2) Knittl Z.: Optics of Thin Films (an Optical Multilayer Theory, Wiley, New York (1976).
* (3) Ünlü M. S. and Strite S.: Resonant cavity enhanced photonic devices, J. Appl. Phys., 78, 607-639 (1995).
* (4) El-Batawy Y. M. and Deen M. J.: Resonant cavity enhanced photodetectors (RCE-PDs): structure, material analysis and optimization, Proc. SPIE, 4999, 363-378 (2003).
* (5) S. Hava and M. Auslender, Silicon grating-based mirror for 1.3-$\mu$m polarized beams: MATLAB-aided design, Appl. Opt., 34, 1053-1058 (1995).
* (6) C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, Ultrabroadband mirror using low-index cladded subwavelength grating, IEEE Photonics Technol. Lett., 16, 518-520 (2004).
* (7) E. Bissaillon, D. Tan, B. Faraji, A. G. Kirk, L. Chrostowski, and D. V. Plant, “High reflectivity air-bridge subwavelength grating reflector and Fabry-Perot cavity in AlGaAs/GaAs, Opt. Express, 14, 2573-2582 (2006).
* (8) J. H. Kim, L. Chrostowski, E. Bisaillon, and D. V. Plant DBR, subwavelength grating, and photonic crystal slab Fabry-Perot cavity design using phase analysis by FDTD, Opt. Express, 15, 10330-10339 (2007).
* (9) M. Auslender and S. Hava, Scattering-matrix propagation algorithm in full-vectorial optics of multilayer grating structures, Opt. Lett., 21, 1765-1767 (1996).
* (10) J. G. A. Wehner, C. A. Musca, R. H. Sewell, J. M. Dell, and L. Faraone, Mercury cadmium telluride resonant-cavity-enhanced photoconductive infrared detectors, Appl. Phys. Lett., 87, 211104:1-3 (2005).
* (11) J. G. A. Wehner, C. A. Musca, R. H. Sewell, J. M. Dell, and L. Faraone, Responsivity and lifetime of resonant-cavity-enhanced HgCdTe detectors, Solid-State Electron., 50, 1640-1648 (2006).
* (12) J. G. A. Wehner, R. H. Sewell, J. Antoszewski, C. A. Musca, J. M. Dell, and L. Faraone, Mercury cadmium telluride/cadmium telluride distributed Bragg reflectors for use with resonant cavity-enhanced detectors, J. Electron. Mater., 34, 710-715 (2005).
* (13) J. Kaniewski, J. Muszalski and J. Piotrowski, Resonant microcavity enhanced infrared photodetectors, Optica Applicata, 17, 405-413 (2007).
* (14) M. Sioma and J. Piotrowski, Modelling and optimisation of high temperature detectors of long wavelength infrared radiation with optically resonant cavity, Opto-Electron. Rev. 12, 157-160 (2004).
|
arxiv-papers
| 2012-04-01T15:48:34 |
2024-09-04T02:49:29.264226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moshe Zohar, Mark Auslender, Lorenzo Faraone and Shlomo Hava",
"submitter": "Mark Auslender",
"url": "https://arxiv.org/abs/1204.0226"
}
|
1204.0246
|
# Evidence for the Direct Two-Photon Transition from $\psi(3686)$ to $J/\psi$
M. Ablikim1, M. N. Achasov5, D. J. Ambrose40, F. F. An1, Q. An41, Z. H. An1,
J. Z. Bai1, R. B. Ferroli18, Y. Ban27, J. Becker2, N. Berger1, M. B.
Bertani18, J. M. Bian39, E. Boger20,a, O. Bondarenko21, I. Boyko20, R. A.
Briere3, V. Bytev20, X. Cai1, A. C. Calcaterra18, G. F. Cao1, J. F. Chang1, G.
Chelkov20,a, G. Chen1, H. S. Chen1, J. C. Chen1, M. L. Chen1, S. J. Chen25, Y.
Chen1, Y. B. Chen1, H. P. Cheng14, Y. P. Chu1, D. Cronin-Hennessy39, H. L.
Dai1, J. P. Dai1, D. Dedovich20, Z. Y. Deng1, A. Denig19, I. Denysenko20,b, M.
Destefanis44, W. M. Ding29, Y. Ding23, L. Y. Dong1, M. Y. Dong1, S. X. Du47,
J. Fang1, S. S. Fang1, L. Fava44,c, F. Feldbauer2, C. Q. Feng41, C. D. Fu1, J.
L. Fu25, Y. Gao36, C. Geng41, K. Goetzen7, W. X. Gong1, W. Gradl19, M.
Greco44, M. H. Gu1, Y. T. Gu9, Y. H. Guan6, A. Q. Guo26, L. B. Guo24, Y. P.
Guo26, Y. L. Han1, X. Q. Hao1, F. A. Harris38, K. L. He1, M. He1, Z. Y. He26,
T. Held2, Y. K. Heng1, Z. L. Hou1, H. M. Hu1, J. F. Hu6, T. Hu1, B. Huang1, G.
M. Huang15, J. S. Huang12, X. T. Huang29, Y. P. Huang1, T. Hussain43, C. S.
Ji41, Q. Ji1, X. B. Ji1, X. L. Ji1, L. K. Jia1, L. L. Jiang1, X. S. Jiang1, J.
B. Jiao29, Z. Jiao14, D. P. Jin1, S. Jin1, F. F. Jing36, N. Kalantar-
Nayestanaki21, M. Kavatsyuk21, W. Kuehn37, W. Lai1, J. S. Lange37, J. K. C.
Leung35, C. H. Li1, Cheng Li41, Cui Li41, D. M. Li47, F. Li1, G. Li1, H. B.
Li1, J. C. Li1, K. Li10, Lei Li1, N. B. Li24, Q. J. Li1, S. L. Li1, W. D. Li1,
W. G. Li1, X. L. Li29, X. N. Li1, X. Q. Li26, X. R. Li28, Z. B. Li33, H.
Liang41, Y. F. Liang31, Y. T. Liang37, G. R. Liao36, X. T. Liao1, B. J. Liu1,
B. J. Liu34, C. L. Liu3, C. X. Liu1, C. Y. Liu1, F. H. Liu30, Fang Liu1, Feng
Liu15, H. Liu1, H. B. Liu6, H. H. Liu13, H. M. Liu1, H. W. Liu1, J. P. Liu45,
Kun Liu27, Kai Liu6, K. Y. Liu23, P. L. Liu29, S. B. Liu41, X. Liu22, X. H.
Liu1, Y. B. Liu26, Y. Liu1, Z. A. Liu1, Zhiqiang Liu1, Zhiqing Liu1, H.
Loehner21, G. R. Lu12, H. J. Lu14, J. G. Lu1, Q. W. Lu30, X. R. Lu6, Y. P.
Lu1, C. L. Luo24, M. X. Luo46, T. Luo38, X. L. Luo1, M. Lv1, C. L. Ma6, F. C.
Ma23, H. L. Ma1, Q. M. Ma1, S. Ma1, T. Ma1, X. Y. Ma1, Y. Ma11, F. E. Maas11,
M. Maggiora44, Q. A. Malik43, H. Mao1, Y. J. Mao27, Z. P. Mao1, J. G.
Messchendorp21, J. Min1, T. J. Min1, R. E. Mitchell17, X. H. Mo1, C. Morales
Morales11, C. Motzko2, N. Yu. Muchnoi5, Y. Nefedov20, C. Nicholson6, I. B..
Nikolaev5, Z. Ning1, S. L. Olsen28, Q. Ouyang1, S. P. Pacetti18,d, J. W.
Park28, M. Pelizaeus38, K. Peters7, J. L. Ping24, R. G. Ping1, R. Poling39, E.
Prencipe19, C. S. J. Pun35, M. Qi25, S. Qian1, C. F. Qiao6, X. S. Qin1, Y.
Qin27, Z. H. Qin1, J. F. Qiu1, K. H. Rashid43, G. Rong1, X. D. Ruan9, A.
Sarantsev20,e, J. Schulze2, M. Shao41, C. P. Shen38,f, X. Y. Shen1, H. Y.
Sheng1, M. R. Shepherd17, X. Y. Song1, S. Spataro44, B. Spruck37, D. H. Sun1,
G. X. Sun1, J. F. Sun12, S. S. Sun1, X. D. Sun1, Y. J. Sun41, Y. Z. Sun1, Z.
J. Sun1, Z. T. Sun41, C. J. Tang31, X. Tang1, E. H. Thorndike40, H. L. Tian1,
D. Toth39, M. U. Ulrich37, G. S. Varner38, B. Wang9, B. Q. Wang27, K. Wang1,
L. L. Wang4, L. S. Wang1, M. Wang29, P. Wang1, P. L. Wang1, Q. Wang1, Q. J.
Wang1, S. G. Wang27, X. F. Wang12, X. L. Wang41, Y. D. Wang41, Y. F. Wang1, Y.
Q. Wang29, Z. Wang1, Z. G. Wang1, Z. Y. Wang1, D. H. Wei8, P. Weidenkaff19, Q.
G. Wen41, S. P. Wen1, M. W. Werner37, U. Wiedner2, L. H. Wu1, N. Wu1, S. X.
Wu41, W. Wu26, Z. Wu1, L. G. Xia36, Z. J. Xiao24, Y. G. Xie1, Q. L. Xiu1, G.
F. Xu1, G. M. Xu27, H. Xu1, Q. J. Xu10, X. P. Xu32, Y. Xu26, Z. R. Xu41, F.
Xue15, Z. Xue1, L. Yan41, W. B. Yan41, Y. H. Yan16, H. X. Yang1, T. Yang9, Y.
Yang15, Y. X. Yang8, H. Ye1, M. Ye1, M. H. Ye4, B. X. Yu1, C. X. Yu26, J. S.
Yu22, S. P. Yu29, C. Z. Yuan1, W. L. Yuan24, Y. Yuan1, A. A. Zafar43, A. Z.
Zallo18, Y. Zeng16, B. X. Zhang1, B. Y. Zhang1, C. C. Zhang1, D. H. Zhang1, H.
H. Zhang33, H. Y. Zhang1, J. Zhang24, J. G. Zhang12, J. Q. Zhang1, J. W.
Zhang1, J. Y. Zhang1, J. Z. Zhang1, L. Zhang25, S. H. Zhang1, T. R. Zhang24,
X. J. Zhang1, X. Y. Zhang29, Y. Zhang1, Y. H. Zhang1, Y. S. Zhang9, Z. P.
Zhang41, Z. Y. Zhang45, G. Zhao1, H. S. Zhao1, J. W. Zhao1, K. X. Zhao24, Lei
Zhao41, Ling Zhao1, M. G. Zhao26, Q. Zhao1, S. J. Zhao47, T. C. Zhao1, X. H.
Zhao25, Y. B. Zhao1, Z. G. Zhao41, A. Zhemchugov20,a, B. Zheng42, J. P.
Zheng1, Y. H. Zheng6, Z. P. Zheng1, B. Zhong1, J. Zhong2, L. Zhou1, X. K.
Zhou6, X. R. Zhou41, C. Zhu1, K. Zhu1, K. J. Zhu1, S. H. Zhu1, X. L. Zhu36, X.
W. Zhu1, Y. M. Zhu26, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, B. S. Zou1, J. H.
Zou1, J. X. Zuo1 (BESIII Collaboration) 1 Institute of High Energy Physics,
Beijing 100049, P. R. China
2 Bochum Ruhr-University, 44780 Bochum, Germany
3 Carnegie Mellon University, Pittsburgh, PA 15213, USA
4 China Center of Advanced Science and Technology, Beijing 100190, P. R. China
5 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
6 Graduate University of Chinese Academy of Sciences, Beijing 100049, P. R.
China
7 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
8 Guangxi Normal University, Guilin 541004, P. R. China
9 GuangXi University, Nanning 530004,P.R.China
10 Hangzhou Normal University, Hangzhou 310036, P. R. China
11 Helmholtz Institute Mainz, J.J. Becherweg 45,D 55099 Mainz,Germany
12 Henan Normal University, Xinxiang 453007, P. R. China
13 Henan University of Science and Technology, Luoyang 471003, P. R. China
14 Huangshan College, Huangshan 245000, P. R. China
15 Huazhong Normal University, Wuhan 430079, P. R. China
16 Hunan University, Changsha 410082, P. R. China
17 Indiana University, Bloomington, Indiana 47405, USA
18 INFN Laboratori Nazionali di Frascati , Frascati, Italy
19 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, 55099
Mainz, Germany
20 Joint Institute for Nuclear Research, 141980 Dubna, Russia
21 KVI/University of Groningen, 9747 AA Groningen, The Netherlands
22 Lanzhou University, Lanzhou 730000, P. R. China
23 Liaoning University, Shenyang 110036, P. R. China
24 Nanjing Normal University, Nanjing 210046, P. R. China
25 Nanjing University, Nanjing 210093, P. R. China
26 Nankai University, Tianjin 300071, P. R. China
27 Peking University, Beijing 100871, P. R. China
28 Seoul National University, Seoul, 151-747 Korea
29 Shandong University, Jinan 250100, P. R. China
30 Shanxi University, Taiyuan 030006, P. R. China
31 Sichuan University, Chengdu 610064, P. R. China
32 Soochow University, Suzhou 215006, China
33 Sun Yat-Sen University, Guangzhou 510275, P. R. China
34 The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.
35 The University of Hong Kong, Pokfulam, Hong Kong
36 Tsinghua University, Beijing 100084, P. R. China
37 Universitaet Giessen, 35392 Giessen, Germany
38 University of Hawaii, Honolulu, Hawaii 96822, USA
39 University of Minnesota, Minneapolis, MN 55455, USA
40 University of Rochester, Rochester, New York 14627, USA
41 University of Science and Technology of China, Hefei 230026, P. R. China
42 University of South China, Hengyang 421001, P. R. China
43 University of the Punjab, Lahore-54590, Pakistan
44 University of Turin and INFN, Turin, Italy
45 Wuhan University, Wuhan 430072, P. R. China
46 Zhejiang University, Hangzhou 310027, P. R. China
47 Zhengzhou University, Zhengzhou 450001, P. R. China
a also at the Moscow Institute of Physics and Technology, Moscow, Russia
b on leave from the Bogolyubov Institute for Theoretical Physics, Kiev,
Ukraine
c University of Piemonte Orientale and INFN (Turin)
d Currently at INFN and University of Perugia, I-06100 Perugia, Italy
e also at the PNPI, Gatchina, Russia
f now at Nagoya University, Nagoya, Japan
###### Abstract
The two-photon transition $\psi(3686)\to\gamma\gamma J/\psi$ is studied in a
sample of 106 million $\psi(3686)$ decays collected by the BESIII detector.
The branching fraction is measured to be $(3.1\pm
0.6(\mathrm{stat})^{+0.8}_{-1.0}(\mathrm{syst}))\times 10^{-4}$ using
$J/\psi\to e^{+}e^{-}$ and $J/\psi\to\mu^{+}\mu^{-}$ decays, and its upper
limit is estimated to be $4.5\times 10^{-4}$ at the 90% conference level. This
work represents the first measurement of a two-photon transition among
charmonium states. The orientation of the $\psi(3686)$ decay plane and the
$J/\psi$ polarization in this decay are also studied. In addition, the product
branching fractions of sequential $E1$ transitions
$\psi(3686)\to\gamma\chi_{cJ},\chi_{cJ}\to\gamma J/\psi(J=0,1,2)$ are
reported.
###### pacs:
14.40.Pq, 13.20.Gd, 14.40.-n
The XYZ Brambilla:2010cs particles, which do not fit potential model
expectations in QCD theory, have been a key challenge to the QCD description
of charmonium-like states Asner:2008nq . To fully understand those states, it
is necessary to consider the coupling of a charmonium state to a $D\bar{D}$
meson pair. These coupled-channel effects, which also play an important role
in the charmonium transitions of low lying states (_i_._e_., from $\psi(3686)$
to $J/\psi$), are especially relevant for the radiative transition processes
Eichten:2004uh . In the well-known electric dipole transitions, the strength
of coupled-channel effects will likely be hard to establish, since the
accompanying relativistic corrections may be more important Li:2009zu .
However, the two-photon transition $\psi(3686)\to\gamma\gamma J/\psi$ is more
sensitive to the coupled-channel effect and thus provides a unique opportunity
to investigate these issues He:2010pb .
Two-photon spectroscopy has been a very powerful tool for the study of the
excitation spectra of a variety of systems with a wide range of sizes, such as
molecules, atomic hydrogen and positronium Pachucki:1996jw . Studying the
analogous process in quarkonium states is a natural extension of this work, in
order to gain insight into non-perturbative QCD phenomena. But so far, two-
photon transitions in quarkonia have eluded experimental observation
Bai:2004cg ; Adam:2005uh ; :2008kb . For example, in a study of
$\psi(3686)\to\gamma\chi_{cJ}(J=0,1,2)$ reported by CLEO-c :2008kb , the upper
limit for $\mathcal{B}(\psi(3686)\to\gamma\gamma J/\psi)$ was estimated to be
$1\times 10^{-3}$.
This Letter presents the first evidence for the two-photon transition
$\psi(3686)\to\gamma\gamma J/\psi$, as well as studies of the orientation of
the $\psi(3686)$ decay plane and the $J/\psi$ polarization in the decay. The
branching fractions of double $E1$ transitions $\psi(3686)\to\gamma(\gamma
J/\psi)_{\chi_{cJ}}$ through $\chi_{cJ}$ intermediate states are also
reported. The data analyzed were obtained by the BESIII experiment :2009vd
viewing electron-positron collisions at the BEPCII collider. An integrated
luminosity of 156.4 $\rm{pb}^{-1}$ was obtained at a center-of-mass energy
$\sqrt{s}=M(\psi(3686))=3.686\,\mathrm{GeV}$. The number of $\psi(3686)$
decays in this sample is estimated to be $(1.06\pm 0.04)\times 10^{8}$
Ablikim:2010zn . In addition, 42.6 $\rm{pb}^{-1}$ of continuum data were taken
below the $\psi(3686)$, at $\sqrt{s}=3.65\,\mathrm{GeV}$, to evaluate the
potential backgrounds from non-resonant events.
The upgraded BEPCII Zhang:2010zz at Beijing is a two-ring electron-positron
collider. The BESIII detector :2009vd is an approximately cylindrically
symmetric detector which covers $93\%$ of the solid angle around the collision
point. In order of increasing distance from the interaction point, the sub-
detectors include a 43-layer main wire drift chamber (MDC), a time-of-flight
(TOF) system with two layers in the barrel region and one layer for each end-
cap, and a 6240 cell CsI(Tl) crystal electro-magnetic calorimeter (EMC) with
both barrel and endcap sections. The barrel components reside within a
superconducting solenoid magnet providing a 1.0 T magnetic field aligned with
the beam axis. Finally, there is a muon chamber consisting of nine layers of
resistive plate chambers within the return yoke of the magnet. The momentum
resolution for charged tracks in the MDC is $0.5\%$ for transverse momenta of
$1\,\mathrm{GeV}/c$. The energy resolution for showers in the EMC is $2.5\%$
for $1\,\mathrm{GeV}$ photons.
Figure 1: Up (a): distributions of $M_{\gamma\gamma-\mathrm{recoil}}$ in data
(points) and in the combined dataset (solid line) of MC simulation of
$\psi(3686)$ decays (shaded histogram) and continuum backgrounds (dashed
line), before the KF is applied. The arrows indicate the window to select a
$J/\psi$ candidate; Down: scatter plots of
$M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}$ versus $M_{\gamma\gamma}$ for the
$\gamma\gamma e^{+}e^{-}$ channel, in data (b), continuum data (c), MC
simulated signal (d), after applying the KF constrain and the
$M_{\gamma\gamma-\mathrm{recoil}}$ window. The corresponding plots for the
$\gamma\gamma\mu^{+}\mu^{-}$ channel are very similar.
This work studies $\psi(3686)\to\gamma\gamma J/\psi$ followed by
$J/\psi\to\ell^{+}\ell^{-}$ ($\ell$ denotes $e$ or $\mu$), which is referred
to as the signal process. Events selected contain exactly two oppositely
charged good tracks in the MDC tracking system, corresponding to the dilepton
from $J/\psi$ decay. The requirements to judge a track as good include
$|\cos\theta|<0.93$ ($\theta$ is the polar angle with respect to the beam
direction), and the minimum distance of approach between the track and the
production vertex less than 10 cm along the beam axis and less than 1 cm
projected in the perpendicular plane. The lepton is identified with the ratio
of EMC shower energy to MDC track momentum, $E/p$, which must be larger than
0.7 for an electron, or smaller than 0.6 for a muon. To suppress non-$J/\psi$
decay leptons, we require the momentum of each lepton to be larger than
$0.8\,\mathrm{GeV}/c$. A vertex fit (VF) constrains the production vertex,
which is updated run-by-run, and the tracks of the dilepton candidates to a
common vertex; only events with $\chi^{2}_{\mathrm{VF}}/\mathrm{d.o.f.}<20$
are accepted.
Reconstructed EMC showers unmatched to either charged track and with an energy
larger than $25\,\mathrm{MeV}$ in the barrel region ($|\cos\theta|<0.80$) or
larger than $50\,\mathrm{MeV}$ in the end-caps ($0.86<|\cos\theta|<0.92$) are
used as photon candidates. To reject bremsstrahlung photons, showers matching
the initial momentum of either lepton within $10^{\circ}$ are also discarded.
Showers from noise, not originating from the beam collision, are suppressed by
requiring the EMC cluster time to lie within a 700 ns window near the event
start time.
Events are required to have only two photon candidates. A kinematic fit (KF)
constrains the vertexed dilepton to the nominal mass of the intermediate
$J/\psi$, and the resulting $J/\psi$ and photon candidates to the known
initial four-momentum of the $\psi(3686)$. The KF fit quality
$\chi^{2}_{\mathrm{KF}}$ is required to be
$\chi^{2}_{\mathrm{KF}}/\mathrm{d.o.f.}<12$. For convenience, we use
$\gamma_{\mathrm{lg}}$ ($\gamma_{\mathrm{sm}}$) to denote the larger (smaller)
energy photon. As indicated in Fig. 1(a), $J/\psi$ candidates are identified
with the requirement that the recoil mass of the two photons,
$M_{\gamma\gamma-\mathrm{recoil}}$, is within
$(3.08,3.14)\,\mathrm{GeV}/c^{2}$.
Scatter plots of recoiling mass $M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}$
from the lower energy photon $\gamma_{\mathrm{sm}}$ versus invariant mass of
two photons $M_{\gamma\gamma}$ are shown in Fig. 1, where clear resonance
bands are seen from the decays $\psi(3686)\to\gamma\chi_{cJ}(J=0,1,2)$ (three
horizontal bands) and $\psi(3686)\to\pi^{0}(\eta)J/\psi$ (two vertical bands).
As indicated in Fig. 1(c), the continuum backgrounds are most dominant at the
top of the plots, of which the primary sources include the bhabha scattering,
the dimuon process and the ISR production of $J/\psi$. These backgrounds are
excluded by discarding events with
$M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}>3.6\,\mathrm{GeV}/c^{2}$. To
suppress backgrounds from $\psi(3686)\rightarrow\pi^{0}(\eta)J/\psi$, the
diphoton invariant mass $M_{\gamma\gamma}$ is required to be larger than
$0.15\,\mathrm{GeV}/c^{2}$ and the recoil momentum of the diphoton must be
larger than $0.25\,\mathrm{GeV}/c$.
Figure 2: (color online) Plot a: unbinned maximum likelihood fit to the
distribution of $M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}$ in data with
combination of the two $J/\psi$-decay modes. Thick lines are the sum of the
fitting models and long-dashed lines are the $\chi_{cJ}$ shapes. Short-dashed
lines represent the two-photon signal processes. Shaded histograms are
$\psi(3686)$-decay backgrounds (yellow) and non-$\psi(3686)$ backgrounds
(green), with the fixed amplitude and shape taken from MC simulation and
continuum data. Plot b: the number of standard deviations, $n_{\sigma}$, of
data points from the fitted curves in plot a. The rates of the signal process
and sequential $\chi_{cJ}$ processes are derived from these fits. Plot c:
distributions of $M_{\gamma\gamma-\mathrm{recoil}}$ in data (signals and known
backgrounds) with the kinematic requirement
$3.44\,\mathrm{GeV}/c^{2}<M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}<3.48\,\mathrm{GeV}/c^{2}$
and with the removal of $\chi^{2}_{\mathrm{KF}}$ and
$M_{\gamma\gamma-\mathrm{recoil}}$ restrictions. Plot d: stacked histograms of
the three $\chi_{cJ}$ components in plot c.
Monte Carlo (MC) simulations of $\psi(3686)$ decays are used to understand the
backgrounds and also to estimate the detection efficiency. At BESIII, the
simulation includes the beam energy spread and treats the initial-state
radiation with KKMC Jadach:2000ir . Specific decay modes from the PDG pdg2012
are modeled with EVTGEN ref:bes3gen , and the unknown decay modes with
Lundcharm Chen:2000tv . The detector response is described using GEANT4
ref:geant4 . For the $\psi(3686)\to\gamma\gamma J/\psi$ channel, the momenta
of decay particles are simulated according to the measured polarization
structure in this work. Generic $\psi(3686)$ decay samples serve for
understanding the background channels; dominant backgrounds were generated
with high statistics. Angular distributions of the cascade $E1$ transitions
$\psi(3686)\to\gamma\chi_{cJ}\to\gamma\gamma J/\psi$ are assumed to follow the
formulae in Ref. Karl:1975jp . Note that the $\chi_{cJ}$ line shapes were
simulated with the Breit-Wigner distributions weighted with
$E^{3}_{\gamma^{*}_{1}}E^{3}_{\gamma^{*}_{2}}$ to account for the double $E1$
transitions, and extended out to $\pm 200\,\mathrm{MeV}/c^{2}$ away from the
nominal masses, using masses and widths in PDG pdg2012 . Here,
$E_{\gamma^{*}_{1}}$($E_{\gamma^{*}_{2}}$) is the energy of the radiative
photon $\gamma^{*}_{1}$($\gamma^{*}_{2}$) in the rest frame of the mother
particle $\psi(3686)$($\chi_{cJ}$).
The yield of the signal process $\psi(3686)\to\gamma\gamma J/\psi$, together
with those of the cascade $E1$ transition processes, is estimated by a global
fit to the spectrum of $M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}$. The fit
results are shown in Fig. 2. The shape and magnitude of $\psi(3686)$-decay
backgrounds were fixed based on MC simulation. Non-$\psi(3686)$ decay
backgrounds are estimated in continuum data, scaling by luminosity and the
$1/s$ dependence of the cross sections. This scaling is verified by the good
description of the $J/\psi$ backgrounds in the
$M_{\gamma\gamma-\mathrm{recoil}}$ distribution shown in Fig. 1(a). The
distributions of the signal process and the cascade $E1$ process are taken
from the reconstructed shapes in MC simulation of the modes and smeared with
an asymmetric Gaussian with free parameters, which is used to compensate for
the difference in line shape between MC and data. By taking the MC shape,
detector resolution and wrong assignment of the $E1$ photon are taken into
account. The quality of goodness-of-fit test,
$\chi^{2}/$d.o.f.$=108.0/94=1.15$ in the $\gamma\gamma e^{+}e^{-}$ mode and
$124.8/94=1.33$ in the $\gamma\gamma\mu^{+}\mu^{-}$ mode. The observed signal
yields are given in Table 1. The $\psi(3686)\to\gamma\gamma J/\psi$ transition
is observed with a statistical significance of 6.6$\sigma$, as determined by
the ratio of the maximum likelihood value and the likelihood value for a fit
with null-signal hypothesis. When the systematic uncertainties are taken into
account with the assumption of Gaussian distributions, the significance is
evaluated to be 3.8$\sigma$, which corresponds to a probability of a
background fluctuation to the observed signal yield of $7.2\times 10^{-5}$.
The upper limit for $\mathcal{B}(\psi(3686)\to\gamma\gamma J/\psi)$ is
estimated to be $4.5\times 10^{-4}$ at the 90% confidence level including
systematic uncertainties.
In calculating $\mathcal{B}(\gamma\gamma J/\psi)$, a correction factor is
included due to the interferences among $\chi_{cJ}$ states. This effect was
checked by the variations of the observed signals in the global fit with
inclusion of a floating interference component, which is modeled by the
detector-smeared shape of a theoretical calculation He:2010pb . It is found
that relative changes on the signal yields are negative with lower bound of
$-10\%$. Hence, a correction factor $0.95$ is assigned and $5\%$ is taken as
systematic uncertainty.
A cross-check on our procedures is performed with the
$M_{\gamma\gamma-\mathrm{recoil}}$ spectrum for the events in the region
$3.44\,\mathrm{GeV}/c^{2}<M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}<3.48\,\mathrm{GeV}/c^{2}$
without restrictions on $\chi^{2}_{\mathrm{KF}}$ and
$M_{\gamma\gamma-\mathrm{recoil}}$, as shown in Fig. 2(c). An excess of data
above known backgrounds can be seen around the $J/\psi$ nominal mass, which is
expected from the sought-after two-photon process. With the inclusion of the
estimated yields of the signal process, the excess is well understood. The
high-mass peak above the $J/\psi$ peak comes from the backgrounds of
$\psi(3686)\to\pi^{0}\pi^{0}J/\psi$ decays. This satellite peak can be well
described in MC simulation. In Fig. 2(d), the three $\chi_{cJ}$ tails show
distinguishable distributions; the small left bump is from the $\chi_{c1}$
tail, while the $\chi_{c0}$ tail is dominant at the right side. The
distribution in data in Fig. 2(c) can only be well described by the simulated
$\chi_{cJ}$ shapes.
Table 1: For different channels: the number of observed signals $n_{e}$ ($n_{\mu}$) and detection efficiency $\epsilon_{e}$ ($\epsilon_{\mu}$) in $\gamma\gamma e^{+}e^{-}$ ($\gamma\gamma\mu^{+}\mu^{-}$) mode; the absolute branching fractions. On the bottom, the relative branching fractions $R_{MN}\equiv\mathcal{B}_{\chi_{cM}}/\mathcal{B}_{\chi_{cN}}$, where $\mathcal{B}_{\chi_{cJ}}\equiv\mathcal{B}(\psi(3686)\to\gamma(\gamma J/\psi)_{\chi_{cJ}})$ are listed. Here the first errors are statistical and the second are systematic. Channels | $n_{e}$ | $\epsilon_{e}$(%) | $n_{\mu}$ | $\epsilon_{\mu}$(%) | $\mathcal{B}$($\times 10^{-4}$)
---|---|---|---|---|---
$\gamma\gamma J/\psi$ | 564$\pm$116 | 22.4 | 536$\pm$128 | 30.0 | $3.1\pm 0.6^{+0.8}_{-1.0}$
$\gamma(\gamma J/\psi)_{\chi_{c0}}$ | 1801$\pm$60 | 19.3 | 2491$\pm$69 | 26.0 | $15.1\pm 0.3\pm 1.0$
$\gamma(\gamma J/\psi)_{\chi_{c1}}$ | 59953$\pm$253 | 28.5 | 81922$\pm$295 | 38.2 | $337.7\pm 0.9\pm 18.3$
$\gamma(\gamma J/\psi)_{\chi_{c2}}$ | 32171$\pm$187 | 27.5 | 44136$\pm$219 | 37.1 | $187.4\pm 0.7\pm 10.2$
$R_{21}\equiv\frac{\mathcal{B}_{\chi_{c2}}}{\mathcal{B}_{\chi_{c1}}}(\%)$ | $R_{01}\equiv\frac{\mathcal{B}_{\chi_{c0}}}{\mathcal{B}_{\chi_{c1}}}(\%)$ | $R_{02}\equiv\frac{\mathcal{B}_{\chi_{c0}}}{\mathcal{B}_{\chi_{c2}}}(\%)$
$55.47\pm 0.26\pm 0.11$ | $4.45\pm 0.09\pm 0.18$ | $8.03\pm 0.17\pm 0.33$
The angle of the normal axis of the $\psi(3686)$ decay plane with respect to
the $\psi(3686)$ polarization vector (aligned to the beam axis), $\beta$, can
be determined in our data. The event rate may be expressed, to leading order,
as $\frac{d\,N}{d\,\cos\beta}\propto 1+a\cos^{2}\beta$. The measurement was
carried out in the rest frame of the $\psi(3686)$ and the decay plane of the
$\psi(3686)$ was determined with the momenta of the two decay particles
$J/\psi$ and $\gamma_{\mathrm{lg}}$. The signal yields in each angular bin
were extracted by the global fit to the corresponding dataset following the
aforementioned procedure. After correction of the extracted signal yields with
the detection efficiency, Fig. 3(a) shows the fit to the distribution of
$|\cos\beta|$ for the sum of the two dilepton modes; we obtain $a=0.53\pm
0.68$.
Table 2: Summary of the systematic uncertainties on the measurement of $\mathcal{B}_{\mathrm{sig}}$ of $\gamma\gamma J/\psi$ signal process, $\mathcal{B}_{\chi_{cJ}}$ for $\chi_{cJ}$ intermediate processes and the relative branching fractions $R_{MN}$, following the notation convention in Table 1. The tot systematic uncertainty is the square root of the sum. A dash (–) means the uncertainty is negligible. Values inside the parentheses are for the $\gamma\gamma\mu^{+}\mu^{-}$ mode, while values outside are for the $\gamma\gamma e^{+}e^{-}$ mode. Numbers without brackets represent uncertainties that are common to both modes. systematic uncertainty(%) | ${\footnotesize\mathcal{B}_{\mathrm{sig}}}$ | $\mathcal{B}_{\chi_{c0}}$ | $\mathcal{B}_{\chi_{c1}}$ | $\mathcal{B}_{\chi_{c2}}$ | ${\footnotesize R_{01}}$ | ${\footnotesize R_{02}}$ | ${\footnotesize R_{21}}$
---|---|---|---|---|---|---|---
lepton track | $2(2)$ | $2(2)$ | $2(2)$ | $2(2)$ | | |
photon shower | 2 | 2 | 2 | 2 | | |
number of photons | 10(3) | 1(1) | 1(1) | 1(1) | 2(–) | 2(–) | –(–)
KF, $\chi^{2}_{\mathrm{KF}}$ requirement | 2(2) | 2(2) | 2(2) | 2(2) | | |
$\chi_{cJ}$ widths | ${}^{+15}_{-25}$ | 3 | – | – | 4 | 4 | 0.2
$M_{\gamma_{\mathrm{sm}}-\mathrm{recoil}}$ resolution | 4(5) | –(–) | –(–) | –(–) | –(–) | –(–) | –(–)
other background | 4(2) | 1(1) | –(–) | –(–) | –(–) | –(–) | –(–)
$\chi_{cJ}$ interference | 5 | 1 | – | – | 1 | 1 | –
fitting | 8(5) | 1(1) | –(–) | –(–) | 1(1) | 1(1) | –(–)
spin-structure | 20 | 1 | – | – | 1 | 1 | –
number of $\psi(3686)$ | 4 | 4 | 4 | 4 | | |
$\mathcal{B}({\tiny J/\psi\to\ell^{+}\ell^{-}})$ | 1 | 1 | 1 | 1 | | |
total | correlated | 14(8) | 3(3) | 3(3) | 3(3) | 2(1) | 2(1) | –(–)
uncorrelated | ${}^{+25}_{-33}$ | 6 | 5 | 5 | 4 | 4 | 0.2
Figure 3: (a) The corrected distribution of the normal angle $\beta$ of the
$\psi(3686)$ decay plane, and (b) the helicity angle $\theta_{\ell}$ of
$J/\psi$ decays. The curves in (a) and (b) present the fits of functions
$P_{0}(1+a\cos^{2}\beta)$ and $P_{0}(1+\alpha\cos^{2}\theta_{\ell})$,
respectively.
The polarization of $J/\psi$ should be helpful in understanding the mechanism
of the transition process Artoisenet:2007xi . The polarization parameter
$\alpha$ can be evaluated from the angular distribution of the decay rate,
expressed as $\frac{d\,N}{d\,\cos\theta_{\ell}}\propto
1+\alpha\cos^{2}\theta_{\ell}$. Here,
$\alpha=\frac{\Gamma_{\mathrm{T}}-2\Gamma_{\mathrm{L}}}{\Gamma_{\mathrm{T}}+2\Gamma_{\mathrm{L}}}$
(with $\Gamma_{\mathrm{T}}$ and $\Gamma_{\mathrm{L}}$ being the transversely
and longitudinally polarized decay widths, respectively) and the helicity
angle $\theta_{\ell}$ is defined as the angle of the lepton in the $J/\psi$
rest frame with respect to the $J/\psi$ boost direction in the laboratory
frame. For fully transverse (longitudinal) polarization, $\alpha=+1(-1)$.
Figure 3(b) shows the distribution of $|\cos\theta_{\ell}|$ for the sum of the
two dilepton modes, after correcting the signal yields for the detection
efficiency and the lepton final state radiation effect. Our fit result is
$\alpha=0.08\pm 0.42$.
Sources of systematic errors on the measurement of branching fractions are
listed in Table 2. Uncertainties associated with the efficiency of the lepton
tracking and identification were studied with a selected control sample of
$\psi(3686)\to\pi^{+}\pi^{-}(\ell^{+}\ell^{-})_{J/\psi}$. The potential bias
due to limiting the maximum number of photon candidates was studied by varying
the limit. Throughout the photon energy region in this work, detection and
energy resolution of photon are well-modeled within a 1% uncertainty
Ablikim:2010zn ; Collaboration:2010rc . Detector resolution of the $\chi_{cJ}$
tails is taken into account up to the accuracy of the MC simulation. The
corresponding systematic uncertainty is evaluated by scanning the sizes of
smearing parameters within their errors. For the signal process, the dominant
uncertainties are from the description of $\chi_{cJ}$ line shapes, _e_._g_.,
$\chi_{cJ}$ widths. The sensitivity to the $\chi_{cJ}$ widths is studied by a
comparison of the signal yields based on different settings of the $\chi_{cJ}$
widths in modeling the $\chi_{cJ}$ resonances within the current world-average
uncertainties. Relative changes of the signal detection efficiencies are
assigned as 20%, by varying the input spin-structure within the measurement
uncertainties and weighting the efficiencies in the Dalitz-like plot of Fig.
1(d).
Many sources of systematic uncertainties in Table 2 cancel out when extracting
the $\psi(3686)$ decay plane parameter $a$ and the $J/\psi$ polarization
parameter $\alpha$. The quadrature sum of the remaining systematic
uncertainties are ${}^{+0.68}_{-0.27}$ and ${}^{+0.07}_{-0.14}$ for $a$ and
$\alpha$, respectively.
To summarize, the first measurement of the two-photon transition
$\psi(3686)\to\gamma\gamma J/\psi$ was carried out at the BESIII experiment.
The branching fraction is given in Table 1, as well as those of the cascade
$E1$ transitions. The measurement of the two-photon process is consistent with
the upper limit obtained in Ref. :2008kb . The results for the signal process
are presented without considering the possible interferences between the
direct transition and the $\chi_{cJ}$ states, due to a lack of theoretical
guidance. The distribution of the normal angle of the $\psi(3686)$ decay plane
is characterized by the parameter $a=0.53\pm
0.68(\mathrm{stat})^{+0.68}_{-0.27}(\mathrm{syst})$, indicating a preference
for a positive value. The $J/\psi$ polarization parameter $\alpha$ was
evaluated as $0.08\pm 0.42(\mathrm{stat})^{+0.07}_{-0.14}(\mathrm{syst})$,
demonstrating a competitive mixing of the longitudinal and transverse
components. These results will help constrain the strength of the coupled-
channel effect in future theoretical calculation. The reported branching
fractions $\mathcal{B}(\psi(3686)\to\gamma(\gamma J/\psi)_{\chi_{cJ}})$ are
consistent with the world average results pdg2012 . The reported relative
branching fractions of $\mathcal{B}_{\chi_{cJ}}$ are obtained with the world’s
best precision.
X.-R. Lu thanks Zhi-Guo He and De-Shan Yang for useful suggestions. We thank
the staff of BEPCII and the computing center for their hard efforts. We are
grateful for support from our institutes and universities and from the
agencies: Ministry of Science and Technology of China, National Natural
Science Foundation of China, Chinese Academy of Sciences, Istituto Nazionale
di Fisica Nucleare, U.S. Department of Energy, U.S. National Science
Foundation, and National Research Foundation of Korea.
## References
* (1) N. Brambilla et al., Eur. Phys. J. C 71, 1534 (2011).
* (2) D. M. Asner et al., Int. J. Mod. Phys. A 24, Suppl. 1, (2009).
* (3) E. J. Eichten, K. Lane and C. Quigg, Phys. Rev. D 69, 094019 (2004); X. Liu, Phys. Lett. B 680, 137 (2009); T. Barnes, AIP Conf. Proc. 1257, 11 (2010).
* (4) B. Q. Li and K. T. Chao, Phys. Rev. D 79, 094004 (2009).
* (5) Z. G. He et al., Phys. Rev. D 83, 054028 (2011).
* (6) K. Pachucki et al., J. Phys. B 29, 177 (1996); A. Quattropani and F. Bassani, Phys. Rev. Lett. 50, 1258 (1983).
* (7) J. Z. Bai et al. [BES Collaboration], Phys. Rev. D 70, 012006 (2004); M. Ablikim et al. [BES Collaboration], Phys. Rev. D 71, 092002 (2005).
* (8) N. E. Adam et al. [CLEO Collaboration], Phys. Rev. Lett. 94, 232002 (2005).
* (9) H. Mendez et al. [CLEO Collaboration], Phys. Rev. D 78, 011102(R) (2008).
* (10) M. Ablikim et al. [BESIII Collaboration], Nucl. Instrum. Meth. A 614, 345 (2010).
* (11) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D 81, 052005 (2010).
* (12) C. Zhang, Sci. China G 53, 2084 (2010).
* (13) S. Jadach et al., Phys. Rev. D 63, 113009 (2001).
* (14) J. Beringer et al. [Particle Data Group], Phys. Rev. D86, 010001 (2012).
* (15) R. G. Ping, Chinese Phys. C 32, 599 (2008).
* (16) J. C. Chen et al., Phys. Rev. D 62, 034003 (2000).
* (17) S. Agostinelli et al. [GEANT Collaboration], Nucl. Instrum. Meth. A 506, 250 (2003); J. Allison et al., IEEE Trans. Nucl. Sci. 53, 270 (2006).
* (18) G. Karl, S. Meshkov and J. L. Rosner, Phys. Rev. D 13, 1203 (1976); M. A. Doncheski, H. Grotch and K. J. Sebastian, Phys. Rev. D 42, 2293 (1990).
* (19) P. Artoisenet, J. P. Lansberg and F. Maltoni, Phys. Lett. B 653, 60 (2007).
* (20) M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 104, 132002 (2010).
|
arxiv-papers
| 2012-04-01T17:18:25 |
2024-09-04T02:49:29.269671
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "BESIII Collaboration: M. Ablikim, M. N. Achasov, D. J. Ambrose, F. F.\n An, Q. An, Z. H. An, J. Z. Bai, R. B. Ferroli, Y. Ban, J. Becker, N. Berger,\n M. B. Bertani, J. M. Bian, E. Boger, O. Bondarenko, I. Boyko, R. A. Briere,\n V. Bytev, X. Cai, A. C. Calcaterra, G. F. Cao, J. F. Chang, G. Chelkov, G.\n Chen, H. S. Chen, J. C. Chen, M. L. Chen, S. J. Chen, Y. Chen, Y. B. Chen, H.\n P. Cheng, Y. P. Chu, D. Cronin-Hennessy, H. L. Dai, J. P. Dai, D. Dedovich,\n Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis, W. M. Ding, Y. Ding, L. Y.\n Dong, M. Y. Dong, S. X. Du, J. Fang, S. S. Fang, L. Fava, F. Feldbauer, C. Q.\n Feng, C. D. Fu, J. L. Fu, Y. Gao, C. Geng, K. Goetzen, W. X. Gong, W. Gradl,\n M. Greco, M. H. Gu, Y. T. Gu, Y. H. Guan, A. Q. Guo, L. B. Guo, Y.P. Guo, Y.\n L. Han, X. Q. Hao, F. A. Harris, K. L. He, M. He, Z. Y. He, T. Held, Y. K.\n Heng, Z. L. Hou, H. M. Hu, J. F. Hu, T. Hu, B. Huang, G. M. Huang, J. S.\n Huang, X. T. Huang, Y. P. Huang, T. Hussain, C. S. Ji, Q. Ji, X. B. Ji, X. L.\n Ji, L. K. Jia, L. L. Jiang, X. S. Jiang, J. B. Jiao, Z. Jiao, D. P. Jin, S.\n Jin, F. F. Jing, N. Kalantar-Nayestanaki, M. Kavatsyuk, W. Kuehn, W. Lai, J.\n S. Lange, J. K. C. Leung, C. H. Li, Cheng Li, Cui Li, D. M. Li, F. Li, G. Li,\n H. B. Li, J. C. Li, K. Li, Lei Li, N. B. Li, Q. J. Li, S. L. Li, W. D. Li, W.\n G. Li, X. L. Li, X. N. Li, X. Q. Li, X. R. Li, Z. B. Li, H. Liang, Y. F.\n Liang, Y. T. Liang, G. R. Liao, X. T. Liao, B. J. Liu, B. J. Liu, C. L. Liu,\n C. X. Liu, C. Y. Liu, F. H. Liu, Fang Liu, Feng Liu, H. Liu, H. B. Liu, H. H.\n Liu, H. M. Liu, H. W. Liu, J. P. Liu, Kun Liu, Kai Liu, K. Y. Liu, P. L. Liu,\n S. B. Liu, X. Liu, X. H. Liu, Y. B. Liu, Y. Liu, Z. A. Liu, Zhiqiang Liu,\n Zhiqing Liu, H. Loehner, G. R. Lu, H. J. Lu, J. G. Lu, Q. W. Lu, X. R. Lu, Y.\n P. Lu, C. L. Luo, M. X. Luo, T. Luo, X. L. Luo, M. Lv, C. L. Ma, F. C. Ma, H.\n L. Ma, Q. M. Ma, S. Ma, T. Ma, X. Y. Ma, Y. Ma, F. E. Maas, M. Maggiora, Q.\n A. Malik, H. Mao, Y. J. Mao, Z. P. Mao, J. G. Messchendorp, J. Min, T. J.\n Min, R. E. Mitchell, X. H. Mo, C. Morales Morales, C. Motzko, N. Yu. Muchnoi,\n Y. Nefedov, C. Nicholson, I. B. Nikolaev, Z. Ning, S. L. Olsen, Q. Ouyang, S.\n P. Pacetti, J. W. Park, M. Pelizaeus, K. Peters, J. L. Ping, R. G. Ping, R.\n Poling, E. Prencipe, C. S. J. Pun, M. Qi, S. Qian, C. F. Qiao, X. S. Qin, Y.\n Qin, Z. H. Qin, J. F. Qiu, K. H. Rashid, G. Rong, X. D. Ruan, A. Sarantsev,\n J. Schulze, M. Shao, C. P. Shen, X. Y. Shen, H. Y. Sheng, M. R. Shepherd, X.\n Y. Song, S. Spataro, B. Spruck, D. H. Sun, G. X. Sun, J. F. Sun, S. S. Sun,\n X. D. Sun, Y. J. Sun, Y. Z. Sun, Z. J. Sun, Z. T. Sun, C. J. Tang, X. Tang,\n E. H. Thorndike, H. L. Tian, D. Toth, M. U. Ulrich, G. S. Varner, B. Wang, B.\n Q. Wang, K. Wang, L. L. Wang, L. S. Wang, M. Wang, P. Wang, P. L. Wang, Q.\n Wang, Q. J. Wang, S. G. Wang, X. F. Wang, X. L. Wang, Y. D. Wang, Y. F. Wang,\n Y. Q. Wang, Z. Wang, Z. G. Wang, Z. Y. Wang, D. H. Wei, P. Weidenkaff, Q. G.\n Wen, S. P. Wen, M. W. Werner, U. Wiedner, L. H. Wu, N. Wu, S. X. Wu, W. Wu,\n Z. Wu, L. G. Xia, Z. J. Xiao, Y. G. Xie, Q. L. Xiu, G. F. Xu, G. M. Xu, H.\n Xu, Q. J. Xu, X. P. Xu, Y. Xu, Z. R. Xu, F. Xue, Z. Xue, L. Yan, W. B. Yan,\n Y. H. Yan, H. X. Yang, T. Yang, Y. Yang, Y. X. Yang, H. Ye, M. Ye, M. H. Ye,\n B. X. Yu, C. X. Yu, J. S. Yu, S. P. Yu, C. Z. Yuan, W. L. Yuan, Y. Yuan, A.\n A. Zafar, A. Z. Zallo, Y. Zeng, B. X. Zhang, B. Y. Zhang, C. C. Zhang, D. H.\n Zhang, H. H. Zhang, H. Y. Zhang, J. Zhang, J. G. Zhang, J. Q. Zhang, J. W.\n Zhang, J. Y. Zhang, J. Z. Zhang, L. Zhang, S. H. Zhang, T. R. Zhang, X. J.\n Zhang, X. Y. Zhang, Y. Zhang, Y. H. Zhang, Y. S. Zhang, Z. P. Zhang, Z. Y.\n Zhang, G. Zhao, H. S. Zhao, J. W. Zhao, K. X. Zhao, Lei Zhao, Ling Zhao, M.\n G. Zhao, Q. Zhao, S. J. Zhao, T. C. Zhao, X. H. Zhao, Y. B. Zhao, Z. G. Zhao,\n A. Zhemchugov, B. Zheng, J. P. Zheng, Y. H. Zheng, Z. P. Zheng, B. Zhong, J.\n Zhong, L. Zhou, X. K. Zhou, X. R. Zhou, C. Zhu, K. Zhu, K. J. Zhu, S. H. Zhu,\n X. L. Zhu, X. W. Zhu, Y. M. Zhu, Y. S. Zhu, Z. A. Zhu, J. Zhuang, B. S. Zou,\n J. H. Zou, J. X. Zuo",
"submitter": "Xiao-Rui Lu",
"url": "https://arxiv.org/abs/1204.0246"
}
|
1204.0291
|
Charles University in Prague
Faculty of Mathematics and Physics
DOCTORAL THESIS
Tomáš Málek
General Relativity in Higher Dimensions
Institute of Theoretical Physics
Supervisor: Mgr. Vojtěch Pravda, Ph.D.
Branch of study: F–1 Theoretical physics, astronomy and astrophysics
Prague 2012
First of all, I would like to thank my supervisor Vojtěch Pravda for constant
guidance, valuable comments and helpful suggestions he provided throughout my
doctoral studies. I am grateful to Alena Pravdová and Jan Novák for careful
reading the manuscript and pointing out numerous misprints.
I also truly appreciate my girlfriend Pavla for the endless patience she
showed especially during the completion of this thesis.
Finally, I acknowledge the financial support by the projects SVV 261301 and
SVV 263301 of the Charles University in Prague.
I declare that I carried out this doctoral thesis independently, and only with
the cited sources, literature and other professional sources.
I understand that my work relates to the rights and obligations under the Act
No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that
the Charles University in Prague has the right to conclude a license agreement
on the use of this work as a school work pursuant to Section 60 paragraph 1 of
the Copyright Act.
Prague, 12 January 2012 Tomáš Málek
Název práce: Obecná relativita ve vyšších dimenzích
Autor: Tomáš Málek
Ústav: Ústav teoretické fyziky
Vedoucí disertační práce: Mgr. Vojtěch Pravda, PhD., Matematický ústav
Akademie věd ČR, vvi.
Abstrakt: V první části této práce analyzujeme Kerrovy–Schildovy a rozšířené
Kerrovy–Schildovy metriky v kontextu vícerozměrné obecné relativity. Pomocí
zobecnění Newmanova–Penroseova formalizmu a algebraické klasifikace Weylova
tensoru, založené na existenci a násobnosti jeho vlastních nulových směrů, do
vyšších dimenzí jsou studovány geometrické vlastnosti Kerrových–Schildových
kongruencí, určeny kompatibilní algebraické typy a v expandujících případech
diskutována přítomnost singularit. Uvedeme také známá přesná řešení, která lze
převést na Kerrův–Schildův tvar metriky a zkonstruujeme nová řešení pomocí
Brinkmannova ”warp produktu“. V druhé části této práce uvažujeme vliv
kvantových korekcí sestávajících se z kvadratických invariantů křivosti na
Einsteinovu–Hilbertovu akci a studujeme přesná řešení těchto kvadratických
teorií gravitace v libovolné dimenzi. Nalezneme třídy Einsteinových
prostoročasů a prostoročasů s nulovým zářením splňující vakuové polní rovnice
a uvedeme příklady těchto metrik.
Klíčová slova: algebraická klasifikace, gravitace ve vyšších dimenzích,
Kerrovy–Schildovy metriky, kvadratická teorie gravitace
Title: General relativity in higher dimensions
Author: Tomáš Málek
Institute: Institute of Theoretical Physics
Supervisor: Mgr. Vojtěch Pravda, PhD., Institute of Mathematics of the
Academy of Sciences of the Czech Republic
Abstract: In the first part of this thesis, Kerr–Schild metrics and extended
Kerr–Schild metrics are analyzed in the context of higher dimensional general
relativity. Employing the higher dimensional generalizations of the
Newman–Penrose formalism and the algebraic classification of spacetimes based
on the existence and multiplicity of Weyl aligned null directions, we
establish various geometrical properties of the Kerr–Schild congruences,
determine compatible Weyl types and in the expanding case discuss the presence
of curvature singularities. We also present known exact solutions admitting
these Kerr–Schild forms and construct some new ones using the Brinkmann warp
product. In the second part, the influence of quantum corrections consisting
of quadratic curvature invariants on the Einstein–Hilbert action is considered
and exact vacuum solutions of these quadratic gravities are studied in
arbitrary dimension. We investigate classes of Einstein spacetimes and
spacetimes with a null radiation term in the Ricci tensor satisfying the
vacuum field equations of quadratic gravity and provide examples of these
metrics.
Keywords: algebraic classification, higher dimensional gravity, Kerr–Schild
metrics, quadratic gravity
###### Contents
1. 1 Introduction
1. 1.1 Newman–Penrose formalism
2. 1.2 Algebraic classification of the Weyl tensor
3. 1.3 Goldberg–Sachs theorem
4. 1.4 Brinkmann warp product
5. 1.5 New results of this thesis
2. 2 Kerr–Schild spacetimes
1. 2.1 General Kerr–Schild vector field
1. 2.1.1 Kerr–Schild congruence in the background spacetime
2. 2.2 Geodetic Kerr–Schild vector field
1. 2.2.1 Ricci tensor
2. 2.2.2 Riemann tensor and algebraic type of the Weyl tensor
3. 2.3 Brinkmann warp product of Kerr–Schild spacetimes
4. 2.4 Einstein Kerr-Schild spacetimes
5. 2.5 Non-expanding Kerr–Schild spacetimes
1. 2.5.1 Examples of non-expanding Einstein generalized Kerr–Schild spacetimes
2. 2.5.2 Warped Einstein Kundt generalized Kerr–Schild spacetimes
6. 2.6 Expanding Kerr–Schild spacetimes
1. 2.6.1 Optical constraint
2. 2.6.2 Algebraic type
3. 2.6.3 $r$-dependence of the Weyl tensor
4. 2.6.4 Singularities
5. 2.6.5 Example of expanding Einstein generalized Kerr–Schild spacetime: Kerr–(A)dS
6. 2.6.6 Expanding generalized Kerr–Schild spacetimes with null radiation
7. 2.6.7 Warped expanding Einstein generalized Kerr–Schild spacetimes
7. 2.7 Other generalizations of the Kerr–Schild ansatz
3. 3 Extended Kerr–Schild spacetimes
1. 3.1 General Kerr–Schild vector field
1. 3.1.1 Kerr–Schild congruence in the background
2. 3.1.2 Relation of the vector fields $\boldsymbol{k}$ and $\boldsymbol{m}$
2. 3.2 Geodetic Kerr–Schild vector field
1. 3.2.1 Ricci tensor
2. 3.2.2 Riemann tensor and algebraic type of the Weyl tensor
3. 3.3 Kundt extended Kerr–Schild spacetimes
1. 3.3.1 Explicit example
2. 3.3.2 Not all vacuum higher dimensional pp -waves belong to the class of Ricci-flat xKS spacetimes
4. 3.4 Expanding extended Kerr–Schild spacetimes
4. 4 Quadratic gravity
1. 4.1 Einstein spacetimes
1. 4.1.1 Type N Einstein spacetimes
2. 4.1.2 Type III Einstein spacetimes
3. 4.1.3 Comparison with other classes of spacetimes
2. 4.2 Spacetimes with aligned null radiation
1. 4.2.1 Explicit solutions of Weyl type N
5. 5 Conclusions and outlook
## Chapter 1 Introduction
Almost a century ago, it was Einstein’s great insight that gravity as a
universal force could be described by a curvature of spacetime consisting of
one time and three spatial dimensions that has led him to formulate the famous
Einstein field equations of general relativity. Although since then the
validity of general relativity has been confirmed by many experiments, it
breaks down at the Planck scale and is expected to emerge as a low energy
limit of a full theory of quantum gravity, whatever that is.
In recent years, growing interest in higher dimensional general relativity and
black hole solutions within this theory [1] has been influenced by several
fields including string theory which contains general relativity and
consistency of which requires an appropriate number of extra dimensions. Let
us also mention higher dimensional supergravity theories, the AdS/CFT
correspondence relating string theory in an $n$-dimensional anti-de Sitter
bulk spacetime with conformal field theory on the lower dimensional boundary
and various brane-world scenarios considering that our four-dimensional
universe lies on a brane embedded in a higher dimensional spacetime.
However, motivations for studying higher dimensional general relativity also
come from this theory itself since it turns out that it exhibits much more
richer dynamics then in the four-dimensional case. Let us point out, for
instance, the existence of black rings or other black objects with various
non-spherical horizon topologies leading to the violation of the four-
dimensional black hole uniqueness theorem in higher dimensions.
In $n$ dimensions, general relativity can be described by the Einstein–Hilbert
action with a Lagrangian $\mathcal{L}_{\text{matter}}$ of matter fields
appearing in the theory
$S=\int\mathrm{d}^{n}x\,\sqrt{-g}\left(\frac{1}{\kappa}\left(R-2\Lambda\right)+\mathcal{L}_{\text{matter}}\right),$
(1.1)
where $\Lambda$ is a cosmological constant and $\kappa=\frac{8\pi G}{c^{4}}$
is Einstein’s constant given by Newton’s gravitational constant $G$ and the
speed of light $c$ so that the non-relativistic limit in four dimensions
yields Newton’s gravity. Using geometrized units $G=c=1$, the variation of the
action (1.1) with respect to the metric leads to the Einstein field equations
$R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=8\pi T_{ab},$ (1.2)
where the energy–momentum tensor $T_{ab}$ is given by the variation of the
matter field Lagrangian $\mathcal{L}_{\text{matter}}$
$T_{ab}=\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\text{matter}})}{\delta
g^{ab}}.$ (1.3)
From the trace of (1.2), one may express the Ricci scalar as
$R=\frac{2n\Lambda}{n-2}-\frac{16\pi}{n-2}T^{c}_{\phantom{c}c}$ (1.4)
and then substituting back, the Einstein field equations can be rewritten to
the equivalent form
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab}+8\pi
T_{ab}-\frac{8\pi}{n-2}T^{c}_{\phantom{c}c}g_{ab}.$ (1.5)
In this thesis, we are mainly focused on Einstein spacetimes for which the
Ricci tensor is proportional to the metric and with regard to (1.5) we define
them by
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab}.$ (1.6)
Occasionally, we also consider spacetimes with null radiation whose
energy–momentum tensor $T_{ab}$ is of the form
$T_{ab}=\frac{\Phi}{8\pi}\ell_{a}\ell_{b},$ (1.7)
where $\boldsymbol{\ell}$ is a null vector.
The increasing attention to higher dimensional general relativity has also led
to the effort to generalize methods successfully applied in four dimensions
such as the Newman–Penrose formalism and algebraic classification briefly
reviewed in the following sections.
### 1.1 Newman–Penrose formalism
In this thesis, we frequently employ the higher dimensional generalization of
the Newman–Penrose (NP) formalism developed in [2, 3] along with the algebraic
classification of the Weyl tensor based on the existence of Weyl aligned null
directions (WANDs) and their multiplicity outlined in the following section
1.2. In this section, let us mention the basic concepts of the NP formalism
and list some necessary definitions and relations.
In four dimensions, the NP formalism provides a useful tool that has been
successfully applied to construct exact solutions or prove theorems of general
relativity. Although the number of equations is greater than in the standard
coordinate approach with the Einstein field equations, these differential
equations are only of the first order and some are redundant. Moreover, the
advantages of the NP formalism arise in connection with the algebraic
classification if one assumes a special algebraic type or if one works in a
frame reflecting some symmetry. It then leads to considerable simplifications
since many of the components vanish.
First of all, let us mention the convention for indices. Throughout the thesis
we mainly use two types of them. Latin lower case indices $a,b,\ldots$ going
from 0 to $n-1$ mostly denoting the vector components and Latin lower case
indices $i,j,\ldots$ that range from 2 to $n-1$ mostly numbering the spacelike
frame vectors. Especially in chapter 3, we also employ indices
$\tilde{\imath}$, $\tilde{\jmath}$ denoted by tilde running from 3 to $n-2$ to
indicate that $\boldsymbol{m}^{(\tilde{\imath})}$ does not include
$\boldsymbol{m}^{(2)}$. Only in exceptional cases when we construct a metric
in $(n+1)$ dimensions, the indices $\tilde{\imath}$, $\tilde{\jmath}$ range
from 2 to $n$ as will be properly emphasized.
A complex null tetrad consisting of two real null vectors $\boldsymbol{\ell}$,
$\boldsymbol{n}$ and two complex conjugate null vectors $\boldsymbol{m}$,
$\bar{\boldsymbol{m}}$ plays a key role in the four-dimensional NP formalism.
However, such a complex frame cannot be constructed in odd dimensions and
thus, in $n$-dimensional spacetimes, it is convenient to introduce a real null
frame consisting of two null vectors
$\boldsymbol{n}\equiv\boldsymbol{m}^{(0)}$,
$\boldsymbol{\ell}\equiv\boldsymbol{m}^{(1)}$ and $n-2$ spacelike orthonormal
vectors $\boldsymbol{m}^{(i)}$ obeying
$\displaystyle
n^{a}n_{a}=\ell^{a}\ell_{a}=n^{a}m^{(i)}_{a}=\ell^{a}m^{(i)}_{a}=0,\qquad
n^{a}\ell_{a}=1,\qquad m^{(i)a}m^{(j)}_{a}=\delta_{ij}.$ (1.8)
Although only two of the frame vectors $\boldsymbol{m}^{(a)}$ are null, we
often refer to such a frame (1.8) as a null frame. Due to the constraints
(1.8) the metric can be expressed in terms of the frame vectors as
$g_{ab}=2n_{(a}\ell_{b)}+\delta_{ij}m^{(i)}_{a}m^{(j)}_{b}.$ (1.9)
The relations (1.8) remain valid and consequently the form of the metric (1.9)
is preserved if one performs Lorentz transformations of the frame. An
arbitrary action of the Lorentz group on the vectors $\boldsymbol{m}^{(a)}$
can be described in terms of null rotations with $\boldsymbol{n}$ or
$\boldsymbol{\ell}$ fixed, boosts in the plane spanned by $\boldsymbol{n}$ and
$\boldsymbol{\ell}$ and spatial rotations as follows [2]. Under null rotations
with $\boldsymbol{\ell}$ fixed the frame is transformed as
$\hat{\boldsymbol{\ell}}=\boldsymbol{\ell},\qquad\hat{\boldsymbol{n}}=\boldsymbol{n}+z_{i}\boldsymbol{m}^{(i)}-\frac{1}{2}z^{2}\boldsymbol{\ell},\qquad{\hat{\boldsymbol{m}}}^{(i)}=\boldsymbol{m}^{(i)}-z_{i}\boldsymbol{\ell},$
(1.10)
where $z_{i}$ are real functions and $z^{2}\equiv z_{i}z^{i}$, whereas null
rotations with $\boldsymbol{n}$ fixed are obtained just by interchanging
$\boldsymbol{\ell}\leftrightarrow\boldsymbol{n}$. Boosts of the frame can be
expressed as
$\hat{\boldsymbol{\ell}}=\lambda\boldsymbol{\ell},\qquad\hat{\boldsymbol{n}}=\lambda^{-1}\boldsymbol{n},\qquad\hat{\boldsymbol{m}}^{(i)}=\boldsymbol{m}^{(i)},$
(1.11)
where $\lambda$ is a real function. Spatial rotations are generated by
$(n-2)\times(n-2)$ orthogonal matrices $X^{i}_{\phantom{i}j}$
$\hat{\boldsymbol{\ell}}=\boldsymbol{\ell},\qquad\hat{\boldsymbol{n}}=\boldsymbol{n},\qquad\hat{\boldsymbol{m}}^{(i)}=X^{i}_{\
j}\boldsymbol{m}^{(j)}.$ (1.12)
The Ricci rotation coefficients $L_{ab}$, $N_{ab}$ and
$\stackrel{{\scriptstyle i}}{{M}}_{ab}$ are defined as frame components of
covariant derivatives of the frame vectors
$\ell_{a;b}=L_{cd}\,m_{a}^{(c)}m_{b}^{(d)},\qquad
n_{a;b}=N_{cd}\,m_{a}^{(c)}m_{b}^{(d)},\qquad
m_{a;b}^{(i)}=\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{cd}$}\,m_{a}^{(c)}m_{b}^{(d)}.$ (1.13)
From (1.8) it follows that some of the Ricci rotation coefficients are related
to others or even vanish
$\displaystyle L_{0a}$ $\displaystyle=0,\qquad L_{1a}=-N_{0a},\qquad
L_{ia}=-\mbox{$\stackrel{{\scriptstyle i}}{{M}}_{0a}$},$ (1.14) $\displaystyle
N_{1a}$ $\displaystyle=0,\qquad N_{ia}=-\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{1a}$},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{ja}$}=-\mbox{$\stackrel{{\scriptstyle j}}{{M}}_{ia}$}$
and therefore we consider only $L_{10}$, $L_{11}$, $L_{1i}$, $L_{i0}$,
$L_{i1}$, $L_{ij}$, $N_{i0}$, $N_{i1}$, $N_{ij}$, $\stackrel{{\scriptstyle
i}}{{M}}_{j0}$, $\stackrel{{\scriptstyle i}}{{M}}_{j1}$ and
$\stackrel{{\scriptstyle i}}{{M}}_{jk}$ as independent. Note that, in $n$
dimensions, the number of independent components is $n^{2}(n-1)/2$. In four
dimensions, these Ricci rotation coefficients correspond to the twelve complex
spin coefficients denoted by Greek letters.
However, in certain special cases, the number of independent coefficients can
be further reduced by an appropriate choice of the frame vectors. For
instance, since
$\ell_{a;b}\ell^{b}=L_{10}\ell_{a}+L_{i0}m_{a}^{(i)},$ (1.15)
the coefficients $L_{i0}$ vanish if the frame vector $\boldsymbol{\ell}$ is
geodetic, i.e. $\ell_{a;b}\ell^{b}\propto\ell_{a}$. Moreover, if
$\boldsymbol{\ell}$ is also affinely parametrized then $\ell_{a;b}\ell^{b}=0$
and consequently $L_{10}=0$. In the case that $\boldsymbol{\ell}$ is geodetic
and affinely parametrized, one may still perform boosts and spins to transform
the frame vectors $\boldsymbol{n}$, $\boldsymbol{m}^{(i)}$ to be parallelly
transported along the geodesics $\boldsymbol{\ell}$ and thus $N_{i0}$ and
$\stackrel{{\scriptstyle i}}{{M}}_{j0}$ vanish. Alternatively, in Kundt
spacetimes, appropriate boost and spins always lead to
$L_{[1i]}=0,\qquad L_{12}\neq 0,\qquad L_{1\tilde{\imath}}=0,$ (1.16)
as it is shown in section 2.5.
The directional derivatives along the frame vectors are denoted as
$\mathrm{D}\equiv\ell^{a}\nabla_{a},\qquad\mbox{$\bigtriangleup$}\equiv
n^{a}\nabla_{a},\qquad\delta_{i}\equiv m_{(i)}^{a}\nabla_{a}$ (1.17)
and one can straightforwardly show that the commutators of these derivatives
satisfy [4]
$\displaystyle\mbox{$\bigtriangleup$}\mathrm{D}-\mathrm{D}\mbox{$\bigtriangleup$}$
$\displaystyle=L_{11}\mathrm{D}+L_{10}\mbox{$\bigtriangleup$}+L_{i1}\delta_{i}-N_{i0}\delta_{i},$
(1.18) $\displaystyle\delta_{i}\mathrm{D}-\mathrm{D}\delta_{i}$
$\displaystyle=(L_{1i}+N_{i0})\mathrm{D}+L_{i0}\mbox{$\bigtriangleup$}+(L_{ji}-\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$})\delta_{j},$
$\displaystyle\delta_{i}\mbox{$\bigtriangleup$}-\mbox{$\bigtriangleup$}\delta_{i}$
$\displaystyle=N_{i1}\mathrm{D}+(L_{i1}-L_{1i})\mbox{$\bigtriangleup$}+(N_{ji}-\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j1}$})\delta_{j},$
$\displaystyle\delta_{i}\delta_{j}-\delta_{j}\delta_{i}$
$\displaystyle=(N_{ij}-N_{ji})\mathrm{D}+(L_{ij}-L_{ji})\mbox{$\bigtriangleup$}+(\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{ki}$}-\mbox{$\stackrel{{\scriptstyle i}}{{M}}_{kj}$})\delta_{k}.$
Lorentz transformations of the frame act on the Ricci rotation coefficients in
the following way [3]. Under null rotations with $\boldsymbol{\ell}$ fixed
(1.10), the Ricci rotation coefficients transform as
$\displaystyle\hat{L}_{11}$
$\displaystyle=L_{11}+z_{i}(L_{1i}+L_{i1})+z_{i}z_{j}L_{ij}-\frac{1}{2}z^{2}L_{10}-\frac{1}{2}z^{2}z_{i}L_{i0},$
$\displaystyle\hat{L}_{10}$
$\displaystyle=L_{10}+z_{i}L_{i0},\qquad\hat{L}_{1i}=L_{1i}-z_{i}L_{10}+z_{j}L_{ji}-z_{i}z_{j}L_{j0},\phantom{\frac{1}{2}}$
$\displaystyle\hat{L}_{i1}$
$\displaystyle=L_{i1}+z_{j}L_{ij}-\frac{1}{2}z^{2}L_{i0},\qquad\hat{L}_{i0}=L_{i0},\qquad\hat{L}_{ij}=L_{ij}-z_{j}L_{i0},$
$\displaystyle\hat{N}_{i1}$
$\displaystyle=N_{i1}+z_{j}N_{ij}+z_{i}L_{11}+z_{j}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{i1}$}-\frac{1}{2}z^{2}(N_{i0}+L_{i1})+z_{i}z_{j}(L_{1j}+L_{j1})$
$\displaystyle\qquad+z_{j}z_{k}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{ik}$}-\frac{1}{2}z^{2}(z_{i}L_{10}+z_{j}L_{ij}+z_{j}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{i0}$})+z_{i}z_{j}z_{k}L_{jk}$
$\displaystyle\qquad+\frac{1}{2}z^{2}\left(-z_{i}z_{j}L_{j0}+\frac{1}{2}z^{2}L_{i0}\right)+\mbox{$\bigtriangleup$}z_{i}+z_{j}\delta_{j}z_{i}-\frac{1}{2}z^{2}\mathrm{D}z_{i},$
(1.19) $\displaystyle\hat{N}_{i0}$
$\displaystyle=N_{i0}+z_{i}L_{10}+z_{j}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{i0}$}+z_{i}z_{j}L_{j0}-\frac{1}{2}z^{2}L_{i0}+\mathrm{D}z_{i},$
$\displaystyle\hat{N}_{ij}$
$\displaystyle=N_{ij}+z_{i}L_{1j}-z_{j}N_{i0}+z_{k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{ij}$}-z_{j}(z_{i}L_{10}+z_{k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{i0}$})+z_{i}z_{k}L_{kj}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\frac{1}{2}z^{2}L_{ij}-z_{i}z_{j}z_{k}L_{k0}+\frac{1}{2}z^{2}z_{j}L_{i0}+\delta_{j}z_{i}-z_{j}\mathrm{D}z_{i},$
$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{j1}$
$\displaystyle=\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j1}$}+2z_{[j}L_{i]1}+z_{k}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jk}$}+2z_{k}z_{[j}L_{i]k}-\frac{1}{2}z^{2}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}-z^{2}z_{[j}L_{i]0},$ $\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{j0}$ $\displaystyle=\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}+2z_{[j}L_{i]0},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{jk}$}=\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jk}$}+2z_{[j}L_{i]k}-z_{k}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}+2z_{k}z_{[i}L_{j]0},$
whereas null rotations with $\boldsymbol{n}$ fixed are obtained by
interchanging $L\leftrightarrow N$ and $0\leftrightarrow 1$. Under boosts
(1.11), we obtain [3]
$\displaystyle\hat{L}_{11}$
$\displaystyle=\lambda^{-1}L_{11}+\lambda^{-2}\mbox{$\bigtriangleup$}\lambda,\qquad\hat{L}_{10}=\lambda
L_{10}+\mathrm{D}\lambda,\qquad\hat{L}_{i1}=L_{i1},$ (1.20)
$\displaystyle\hat{L}_{1i}$
$\displaystyle=L_{1i}+\lambda^{-1}\delta_{i}\lambda,\qquad\hat{L}_{i0}=\lambda^{2}L_{i0},\qquad\hat{L}_{ij}=\lambda
L_{ij},\phantom{\mbox{$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{ii}$}}$
$\displaystyle\hat{N}_{i1}$
$\displaystyle=\lambda^{-2}N_{i1},\qquad\hat{N}_{i0}=N_{i0},\qquad\hat{N}_{ij}=\lambda^{-1}N_{ij},\phantom{\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{ii}$}}$ $\displaystyle\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{j1}$}$
$\displaystyle=\lambda^{-1}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j1}$},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{j0}$}=\lambda\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{jk}$}=\mbox{$\stackrel{{\scriptstyle i}}{{M}}_{jk}$}.$
Finally, spatial rotations (1.12) transform the Ricci rotation coefficients as
[3]
$\displaystyle\hat{L}_{11}$
$\displaystyle=L_{11},\qquad\hat{L}_{10}=L_{10},\qquad\hat{L}_{1i}=X^{i}_{\
j}L_{1j},$ $\displaystyle\hat{L}_{i1}$ $\displaystyle=X^{i}_{\
j}L_{j1},\qquad\hat{L}_{i0}=X^{i}_{\ j}L_{j0},\qquad\hat{L}_{ij}=X^{i}_{\
k}X^{j}_{\ l}L_{kl},\phantom{\mbox{$\stackrel{{\scriptstyle
i}}{{\hat{M}}}_{ii}$}}$ $\displaystyle\hat{N}_{i0}$ $\displaystyle=X^{i}_{\
j}N_{j0},\qquad\mbox{$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{j1}$}=X^{i}_{\
k}X^{j}_{\ l}\mbox{$\stackrel{{\scriptstyle k}}{{M}}_{l1}$}+X^{j}_{\
k}\mbox{$\bigtriangleup$}X^{i}_{\ k},$ (1.21) $\displaystyle\hat{N}_{ij}$
$\displaystyle=X^{i}_{\ k}X^{j}_{\
l}N_{kl},\qquad\mbox{$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{j0}$}=X^{i}_{\
k}X^{j}_{\ l}\mbox{$\stackrel{{\scriptstyle k}}{{M}}_{l0}$}+X^{j}_{\
k}\mathrm{D}X^{i}_{\ k},$ $\displaystyle\hat{N}_{i1}$ $\displaystyle=X^{i}_{\
j}N_{j1},\qquad\mbox{$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{jk}$}=X^{i}_{\
l}X^{j}_{\ m}X^{k}_{\ n}\mbox{$\stackrel{{\scriptstyle
l}}{{M}}_{mn}$}+X^{j}_{\ m}X^{k}_{\ n}\delta_{n}X^{i}_{\ m}.$
For geodetic $\boldsymbol{\ell}$, it follows from (1.19) that the optical
matrix $L_{ij}$ is invariant under null rotations with $\boldsymbol{\ell}$
fixed. Thus, $L_{ij}$ has a special geometrical meaning and can be split into
three quantities, its trace $\theta$, traceless symmetric part $\sigma_{ij}$
and anti-symmetric part $A_{ij}$
$S_{ij}\equiv L_{(ij)}=\sigma_{ij}+\theta\delta_{ij},\qquad A_{ij}\equiv
L_{[ij]},$ (1.22)
that are related to the expansion, shear and twist of the geodetic null
congruence $\boldsymbol{\ell}$, respectively. The corresponding expansion,
shear and twist scalars are defined as
$\theta\equiv\frac{1}{n-2}L_{ii},\qquad\sigma^{2}\equiv\sigma_{ij}\sigma_{ij},\qquad\omega^{2}\equiv
A_{ij}A_{ij}.$ (1.23)
If the geodetic null congruence $\boldsymbol{\ell}$ is also affinely
parametrized, the following identities could be useful
$\ell^{a}_{\phantom{a};a}=L_{ii},\qquad\ell_{a;b}\,\ell^{a;b}=L_{ij}L_{ij},\qquad\ell_{a;b}\,\ell^{b;a}=L_{ij}L_{ji}$
(1.24)
and therefore
$\theta=\frac{1}{n-2}\ell^{a}_{\phantom{a};a},\qquad\sigma^{2}=\ell_{(a;b)}\,\ell^{a;b}-\frac{1}{n-2}(\ell^{a}_{\phantom{a};a})^{2},\qquad\omega^{2}=\ell_{[a;b]}\,\ell^{a;b}.$
(1.25)
The NP formalism can be completed expressing the frame components of the Ricci
identities
$v_{a;bc}-v_{a;cb}=R^{d}_{\phantom{d}abc}v_{d},$ (1.26)
applied on all frame vectors
$v^{a}=\ell^{a},n^{a},m^{a}_{(2)},\dots,m^{a}_{(n-1)}$, and the frame
components of the Bianchi identities
$R_{ab[cd;e]}=0.$ (1.27)
However, we do not list them all since, in this way, one obtains many lengthy
expressions which can be found in [3] and [2], respectively. Here we present
only one of the possible contractions of the Ricci identities with the frame
vectors, namely
$(\ell_{a;bc}-\ell_{a;cb})\,m_{(i)}^{a}\,\ell^{b}\,m_{(j)}^{c}=R^{d}_{\phantom{d}abc}\,\ell_{d}\,m_{(i)}^{a}\,\ell^{b}\,m_{(j)}^{c},$
(1.28)
leading to [3]
$\displaystyle\mathrm{D}L_{ij}-\delta_{j}L_{i0}$
$\displaystyle=L_{10}L_{ij}-L_{i0}(2L_{1j}+N_{j0})-L_{i1}L_{j0}+2L_{k[0|}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{i|j]}$}$ $\displaystyle\qquad-
L_{ik}(L_{kj}+\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j0}$})-C_{0i0j}-\frac{1}{n-2}R_{00}\delta_{ij}.$ (1.29)
If the null vector field $\boldsymbol{\ell}$ is geodetic and affinely
parametrized then (1.29) reduces to the Sachs equation that for spacetimes of
algebraic type I or more special, i.e. $C_{0i0j}=0$ see table 1.1, with
$R_{00}=0$ including e.g. Einstein spaces takes the simple form
$\mathrm{D}L_{ij}=-L_{ik}L_{kj}.$ (1.30)
In chapter 2, this equation allows us to express an explicit form of the
optical matrix $L_{ij}$ for expanding Einstein generalized Kerr–Schild
spacetimes.
### 1.2 Algebraic classification of the Weyl tensor
In this section, we outline one of the basic tools used in the following
chapters, namely the algebraic classification of the Weyl tensor in higher
dimensions based on the existence of preferred null directions and their
multiplicity. Further details can be found in [5, 6, 7], see also [8] for an
introductory review.
In $n>3$ dimensions, the Weyl tensor is defined as
$C_{abcd}=R_{abcd}-\frac{2}{n-2}\left(g_{a[c}R_{d]b}-g_{b[c}R_{d]a}\right)+\frac{2}{(n-1)(n-2)}Rg_{a[c}g_{d]b}$
(1.31)
and inherits all the symmetries of the Riemann tensor, moreover, it is
completely traceless. Obviously, the Weyl tensor contains all information
about the curvature in Ricci-flat spacetimes and therefore it is considered as
the part of the Riemann tensor describing pure gravitational field. Note also
that two spacetimes related by a conformal transformation have the same Weyl
tensors.
One of the equivalent approaches to algebraic classification of the Weyl
tensor in four dimensions is based on the properties of the principal null
directions (PNDs). Every four-dimensional spacetime admits exactly four
discrete PNDs and the Petrov type is determined by their multiplicity. Other
equivalent classifications can be formulated in terms of two-forms, spinors or
scalar invariants, see e.g. [9].
Only the algebraic classification of the Weyl tensor generalizing the four-
dimensional Petrov classification based on the existence of preferred null
directions developed in [5] has been successfully formulated in arbitrary
dimension. Note also that the spinorial approach in five dimensions
established in [10] is not equivalent to the null directions approach neither
to the spinorial classification in four dimensions [11].
First, let us introduce the following definitions. If some quantity $q$
transforms under boosts (1.11) as
$\hat{q}=\lambda^{w}q$ (1.32)
we say that $q$ has a boost weight $w$. Therefore, the null frame vectors
$\boldsymbol{\ell}$, $\boldsymbol{n}$ have boost weight 1 and $-1$,
respectively, and the spacelike frame vectors $\boldsymbol{m}^{(i)}$ are of
boost weight zero. It then follows that every index $0$ contributes to the
boost weight of the given frame component by one, whereas each index $1$
decreases the boost weight by one, for instance, the components $C_{010i}$
have boost weight 1. It is also convenient to define the following operation
reflecting the symmetries of the Riemann tensor
$T_{\\{pqrs\\}}\equiv\frac{1}{2}\left(T_{[ab][cd]}+T_{[cd][ab]}\right),$
(1.33)
which allows us to express the frame components of the Weyl tensor more
compactly. Such symmetries immediately imply that the maximal boost weight of
the Weyl tensor components is 2 and the minimal boost weight is $-2$.
Next, using (1.33), we decompose the Weyl tensor into the frame components and
sort them according to their boost weight
$\displaystyle C_{abcd}$
$\displaystyle=\overbrace{4C_{0i0j}\,n_{\\{a}m^{(i)}_{b}n_{c}m^{(j)}_{d\\}}}^{\text{boost
weight 2}}$
$\displaystyle\qquad+\overbrace{8C_{010i}\,n_{\\{a}\ell_{b}n_{c}m^{(i)}_{d\\}}+4C_{0ijk}\,n_{\\{a}m^{(i)}_{b}m^{(j)}_{c}m^{(k)}_{d\\}}}^{\text{boost
weight 1}}$ $\displaystyle\qquad\\!\begin{aligned}
&+4C_{0101}\,n_{\\{a}\ell_{b}n_{c}\ell_{d\\}}+4C_{01ij}\,n_{\\{a}\ell_{b}m^{(i)}_{c}m^{(j)}_{d\\}}\\\
&+8C_{0i1j}\,n_{\\{a}m^{(i)}_{b}\ell_{c}m^{(j)}_{d\\}}+C_{ijkl}\,m^{(i)}_{\\{a}m^{(j)}_{b}m^{(k)}_{c}m^{(l)}_{d\\}}\end{aligned}\Bigg{\\}}\text{\scriptsize
boost weight 0}$ (1.34)
$\displaystyle\qquad+\underbrace{8C_{101i}\,\ell_{\\{a}n_{b}\ell_{c}m^{(i)}_{d\\}}+4C_{1ijk}\,\ell_{\\{a}m^{(i)}_{b}m^{(j)}_{c}m^{(k)}_{d\\}}}_{\text{boost
weight $-1$}}$
$\displaystyle\qquad+\underbrace{4C_{1i1j}\,\ell_{\\{a}m^{(i)}_{b}\ell_{c}m^{(j)}_{d\\}}}_{\text{boost
weight $-2$}}.$
However, some of these components are redundant due to the remaining
symmetries $C_{a[bcd]}$ and tracelessness of the Weyl tensor leading to the
additional relations
$\displaystyle C_{0i0i}=0,\qquad C_{010j}=C_{0iji},\qquad
C_{0[ijk]}=0,\qquad\phantom{\frac{1}{2}}$ (1.35) $\displaystyle
C_{0101}=C_{0i1i},\qquad C_{i[jkl]}=0,\qquad
C_{0i1j}=-\frac{1}{2}C_{ikjk}+\frac{1}{2}C_{01ij},$ $\displaystyle
C_{011j}=-C_{1iji},\qquad C_{1[ijk]}=0,\qquad
C_{1i1i}=0,\phantom{\frac{1}{2}}$
which reduce number of independent frame components of the Weyl tensor.
Choosing the vector $\boldsymbol{\ell}$ such that as many as possible leading
terms in (1.31) vanish, the highest boost weight of the remaining components
determines the primary algebraic type and we say that $\boldsymbol{\ell}$ is a
Weyl aligned null direction (WAND). Effectively, one may choose an arbitrary
null frame and if none of the null frame vectors is just a desired WAND
perform null rotations with $\boldsymbol{n}$ fixed (1.10) to find
$\boldsymbol{\ell}$ that transforms away the highest possible number of the
components.
If there is no WAND $\boldsymbol{\ell}$, i.e. there is no frame with all boost
weight 2 components $C_{0i0j}$ vanishing, the spacetime is of general Weyl
type G. If all boost weight 2 components vanish and some of the boost weight 1
components are non-zero the spacetime is of type I. Types II, III and N are
determined by vanishing components of all corresponding boost weights up to 1,
0, or $-1$, respectively, and $\boldsymbol{\ell}$ is then referred to as a
multiple WAND. Type O denotes conformally flat spacetimes with the Weyl tensor
completely vanishing. Using the relations (1.35), the conditions on the frame
components of the Weyl tensor determining primary algebraic type are
summarized in table 1.1.
Table 1.1: The conditions on the frame components of the Weyl tensor determining primary algebraic type of a spacetime. Weyl type | vanishing components of the Weyl tensor
---|---
G | $\exists i,j:C_{0i0j}\neq 0$
I | $C_{0i0j}=0$
II | $C_{0i0j}=C_{0ijk}=0$
III | $C_{0i0j}=C_{0ijk}=C_{ijkl}=C_{01ij}=0$
N | $C_{0i0j}=C_{0ijk}=C_{ijkl}=C_{01ij}=C_{1ijk}=0$
Note that the Bel–Deveber criteria generalized to higher dimensions in [12]
are equivalent conditions to those ones given in table 1.1 involving only the
null vector $\boldsymbol{\ell}$ and thus construction of complete null frame
in not required.
Although in [5] the notion algebraically special spacetimes means spacetimes
of Weyl type I or more special, in subsequent papers and also in this thesis,
spacetimes are denoted as algebraically special if they admit a multiple WAND,
i.e. spacetimes of Weyl type II or more special similarly as in the four
dimensional case.
Having established the primary type, similarly we introduce secondary types.
Using null rotations with the WAND $\boldsymbol{\ell}$ fixed (1.10) one may
find a WAND $\boldsymbol{n}$ such that as many as possible trailing terms in
(1.31) vanish. The lowest boost weight of the remaining components determine
the secondary algebraic type. If $\boldsymbol{n}$ is a simple WAND we denote
by Ii, IIi and IIIi the subtypes of the corresponding types I, II and III,
respectively. Type D is defined as a subtype of the primary type II with
$\boldsymbol{n}$ being also a multiple WAND and thus only the boost weight
zero components of the Weyl tensor are non-vanishing.
Note that an additional subclasses can be defined, for instance, type III(a)
is a subclass of type III with $C_{101i}=0$ and thus the Weyl tensor takes the
form
$C_{abcd}=4C_{1ijk}\,\ell_{\\{a}m^{(i)}_{b}m^{(j)}_{c}m^{(k)}_{d\\}}+4C_{1i1j}\,\ell_{\\{a}m^{(i)}_{b}\ell_{c}m^{(j)}_{d\\}}.$
(1.36)
In chapter 4, we introduce other subclasses of type III, namely types III(A)
and III(B) depending on whether the quantity
$\frac{1}{2}C_{1ijk}C_{1ijk}-C_{101i}C_{101i}$ is non-vanishing or vanishing,
respectively. Obviously, type III(a) is a subclass of type III(A).
Finally, let us recall the differences of the algebraic classification based
on the existence of preferred null directions and their multiplicity in four
and higher dimensions. In four dimensions, every spacetime admits exactly four
discrete PNDs. The most general algebraic type is type I and if two or more
PNDs coincide we say that a spacetime is algebraically special. On the other
hand, in higher dimensions, a spacetime admits no, a finite number or a
continuous family of WANDs and thus new general type G arises. Types I, II and
III in four dimensions correspond to types Ii, IIi and IIIi in higher
dimensions, respectively.
### 1.3 Goldberg–Sachs theorem
Let us briefly comment on the validity of the Goldberg–Sachs theorem in higher
dimensions. In four dimensional general relativity, the Goldberg–Sachs theorem
states that an Einstein spacetime is algebraically special if and only if it
admits a congruence of non-shearing null geodesics. Such congruence then
corresponds to the PND. In four dimensions, this theorem is useful when
searching for new exact solutions and, for instance, it has led to the
discovery of the Kerr black hole. However, the statement of the Goldberg–Sachs
theorem cannot be straightforwardly generalized to higher dimensions.
It has been shown that a multiple WAND of Weyl type III and N Ricci-flat
spacetimes is geodetic [2]. This result holds also in the case of Einstein
spacetimes. A multiple WAND of “generic” type II and D Einstein spacetimes is
also geodetic, nevertheless there exit type D spacetimes that admit a non-
geodetic multiple WAND [13]. However, it was shown in [14] that an Einstein
spacetime admitting a non-geodetic multiple WAND also admits a geodetic
multiple WAND.
It also turns out that there exist shearing multiple WANDs. An example of such
spacetime, namely the Kerr–(anti-)de Sitter black hole, is discussed in
section 2.6.5. This fact suggests that the shear-free condition should be
weakened. Let us point out, for instance, the result of chapter 2 that the
optical matrix $L_{ij}$ of expanding Einstein generalized Kerr–Schild
spacetimes takes the block-diagonal form (2.144) with $2\times 2$ blocks
(2.145) and therefore there is no shear in any planes spanned by pairs of the
spacelike frame vectors corresponding to the $2\times 2$ blocks of the optical
matrix. Note also that the optical matrices of all type N Einstein spacetimes
are of this form with just one $2\times 2$ block [2].
### 1.4 Brinkmann warp product
Solving the Einstein field equations is a rather complicated task, especially
in higher dimensions. It would be convenient to have a method for generating
new solutions from the already existing ones. The Brinkmann warp product [15]
is one of such methods allowing construction of $n$-dimensional Einstein
spacetimes from known $(n-1)$-dimensional Ricci-flat or Einstein metrics.
Although gravity in higher dimensions exhibits much richer dynamics due to the
existence of extended black objects with a same mass and angular momentum but
with different horizon topologies such as black strings, black rings, black
Saturns, etc., forbidden by the no-hair theorem in four dimensions, some of
the known four-dimensional exact solutions still have no higher dimensional
analogue which may be obtained using the Brinkmann warp product.
For instance, a higher dimensional C-metric is unknown unlike the four
dimensional case and moreover it cannot belong to the Robinson–Trautman class
[16] of spacetimes admitting expanding, non-shearing and non-twisting geodetic
null congruence. However, an example of five-dimensional C-metric is presented
in [17] using the Brinkmann warp product.
We employ the Brinkmann warp product in section 2.5.2 not only to generate
solutions with one extra dimension but also in order to introduce a
cosmological constant to Ricci-flat solutions. Thus we construct examples of
higher dimensional type N Einstein Kundt spacetimes from higher dimensional
type N Ricci-flat Kundt metrics belonging to the class of spacetimes with
vanishing scalar invariants and from four-dimensional type N Einstein Kundt
metrics. In section 2.6.7, we also present a few examples of expanding warped
metrics such as a rotating black string constructed from the higher
dimensional Kerr–(A)dS black hole. Taking the Minkowski and Schwarzschild
metric as a seed one may also obtain Randall–Sundrum brane model and
Chamblin–Hawking–Reall black hole on a brane, respectively. Unfortunately,
application of the Brinkmann warp product has certain limits. As we mention
below, the sign of cosmological constant of the warped metric is not entirely
arbitrary and, in some cases, a naked singularity may be introduced to the
spacetime.
It has been shown in [15] that using an $(n-1)$-dimensional Einstein metric as
a seed $\mathrm{d}\tilde{s}^{2}$, we can construct an $n$-dimensional Einstein
metric $\mathrm{d}s^{2}$
$\mathrm{d}s^{2}=\frac{1}{f(z)}\mathrm{d}z^{2}+f(z)\mathrm{d}\tilde{s}^{2}$
(1.37)
with the warp factor $f(z)$ given by
$f(z)=-\lambda z^{2}+2dz+b,$ (1.38)
where the cosmological constant $\Lambda$ of the $n$-dimensional warped
Einstein metric is introduced via $\lambda=\frac{2\Lambda}{(n-1)(n-2)}$ and
$b$, $d$ are constant parameters subject to
$\tilde{R}=(n-1)(n-2)(\lambda b+d^{2}),$ (1.39)
where $\tilde{R}$ is the Ricci scalar of the $(n-1)$-dimensional seed metric
$\mathrm{d}\tilde{s}^{2}$. Note that in the case $\tilde{R}=R=0$, the warp
product reduces to the trivial direct product of a seed metric with a one-
dimensional flat space $\mathrm{d}z^{2}$.
Since we consider only Lorentzian metrics the warp factor $f(z)$ has to be
positive. Note that the Ricci scalar $\tilde{R}$ of the seed is proportional
to the discriminant of the quadratic equation $f(z)=0$ and the Ricci scalar
$R=\frac{2n\Lambda}{n-2}=n(n-1)\lambda$ of the warped metric is proportional
to the quadratic term of $f(z)=0$ with the opposite sign. Therefore, it is
obvious [17] that only the combinations of signs of the Ricci scalars $R$ and
$\tilde{R}$ listed in table 1.2 are allowed.
Table 1.2: Allowed combinations of signs of the Ricci scalar $\tilde{R}$ corresponding to the seed metric $\mathrm{d}\tilde{s}^{2}$ and of the Ricci scalar $R$ corresponding to the warped metric $\mathrm{d}s^{2}$ (1.37). | $R<0$ | $R=0$ | $R>0$
---|---|---|---
$\tilde{R}<0$ | ✓ | $\times$ | $\times$
$\tilde{R}=0$ | ✓ | ✓ | $\times$
$\tilde{R}>0$ | ✓ | ✓ | ✓
Several useful statements about Weyl types of the warped metrics have been
given in [17]. It turns out that if the Weyl tensor of the seed metric
$\mathrm{d}\tilde{s}^{2}$ is algebraically special, i.e. of Weyl type II or
more special, then the warped metric $\mathrm{d}s^{2}$ is of the same Weyl
type. On the other hand, seed spacetimes of the general Weyl type G lead to
warped spacetimes of types G, Ii or D and finally a seed of the Weyl type I
yields warped metrics of types I or Ii. It was also shown in [17] that the
Brinkmann warp product introduces a curvature or parallelly propagated
singularity to warped spacetimes $\mathrm{d}s^{2}$ at any point where the warp
factor vanishes $f(z)=0$. Only in the cases when both metrics
$\mathrm{d}\tilde{s}^{2}$ and $\mathrm{d}s^{2}$ are Ricci-flat $\tilde{R}=0$,
$R=0$ or Einstein with negative cosmological constants $\tilde{R}<0$, $R<0$
the warp factor $f(z)$ does not admit roots and therefore such spacetimes are
free from this kind of singularities.
Let us conclude this brief summary of properties of the Brinkmann warp product
by listing a few expressions that are employed in sections 2.5.2 and 2.6.7.
The warped metric (1.37) can be rewritten by an appropriate coordinate
transformation to a different form that could be more convenient in certain
cases. Two such forms are given in [17] and both depend on the combination of
the signs of the Ricci scalars $\tilde{R}$ and $R$. Using one of the possible
transformations, one may put the metric (1.37) to the form conformal to a
direct product
$\displaystyle\lambda>0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\cosh^{-2}(\sqrt{\lambda}x)(\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2})$
$\displaystyle\tilde{R}$ $\displaystyle>0,$ (1.40)
$\displaystyle\lambda=0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2}$
$\displaystyle\tilde{R}$ $\displaystyle=0,$ (1.41)
$\displaystyle\mathrm{d}s^{2}=2e^{2x}(\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2})$
$\displaystyle\tilde{R}$ $\displaystyle>0,$ (1.42)
$\displaystyle\lambda<0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\cos^{-2}(\sqrt{-\lambda}x)(\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2})$
$\displaystyle\tilde{R}$ $\displaystyle<0,$ (1.43)
$\displaystyle\mathrm{d}s^{2}=(-\lambda
x^{2})^{-1}(\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2})$ $\displaystyle\tilde{R}$
$\displaystyle=0,$ (1.44)
$\displaystyle\mathrm{d}s^{2}=\sinh^{-2}(\sqrt{-\lambda}x)(\mathrm{d}x^{2}+\mathrm{d}\tilde{s}^{2})$
$\displaystyle\tilde{R}$ $\displaystyle>0,$ (1.45)
where $\tilde{R}$ and $\lambda$ are related by
$|\tilde{R}|=(n-1)(n-2)|\lambda|$ apart from (1.42) where
$\tilde{R}=(n-1)(n-2)$. A different form can be obtained from (1.37) employing
the transformation $\mathrm{d}z^{2}f(z)^{-1}=\mathrm{d}y^{2}$ that leads to
metrics with a flat extra dimension
$\displaystyle\lambda>0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}y^{2}+\cos^{2}(\sqrt{\lambda}y)\,\mathrm{d}\tilde{s}^{2}$
$\displaystyle\tilde{R}$ $\displaystyle>0,$ (1.46)
$\displaystyle\lambda=0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}y^{2}+y^{2}\,\mathrm{d}\tilde{s}^{2}$
$\displaystyle\tilde{R}$ $\displaystyle>0,$ (1.47)
$\displaystyle\lambda<0:\qquad$
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}y^{2}+\cosh^{2}(\sqrt{-\lambda}y)\,\mathrm{d}\tilde{s}^{2}$
$\displaystyle\tilde{R}$ $\displaystyle<0,$ (1.48)
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}y^{2}+e^{2\sqrt{-\lambda}y}\,\mathrm{d}\tilde{s}^{2}$
$\displaystyle\tilde{R}$ $\displaystyle=0,$ (1.49)
$\displaystyle\mathrm{d}s^{2}=\mathrm{d}y^{2}+\sinh^{2}(\sqrt{-\lambda}y)\,\mathrm{d}\tilde{s}^{2}\qquad$
$\displaystyle\tilde{R}$ $\displaystyle>0.$ (1.50)
The case $R=0$, $\tilde{R}=0$ when the Brinkmann warp product reduces just to
a direct product is same as (1.41).
### 1.5 New results of this thesis
The Kerr–Schild ansatz has turned out to be an effective tool for finding
exact solutions of general relativity in four dimensions. In chapter 2, we
study higher dimensional generalized Kerr–Schild (GKS) metrics having an
(anti-)de Sitter background. We obtain the necessary and sufficient condition
under which the null Kerr-Schild vector field is geodetic. It is shown that
non-expanding Einstein GKS spacetimes are of Weyl type N and belong to the
Kundt class. Using the Brinkmann warp product, some new explicit non-expanding
solutions are constructed. In the case of expanding Einstein GKS spacetimes,
the compatible Weyl types are D or II and the corresponding optical matrices
take a special block-diagonal form satisfying the so-called optical
constraint. This allows us to determine the dependence of various geometric
quantities on the affine parameter $r$ along the geodetic Kerr–Schild
congruence and to discuss presence of curvature singularities at the origin
$r=0$. We also express the optical matrix of the five-dimensional
Kerr–(anti-)de Sitter black hole in order to compare this important example of
expanding GKS spacetime with our general results, namely the forms of the
optical matrices and the Kerr–Schild scalar functions and presence of
singularities.
In a similar way, we analyze the extended Kerr–Schild (xKS) ansatz in chapter
3. It is a further extension of the GKS ansatz where, in addition to the null
Kerr–Schild vector, a spacelike vector field appears in the metric. We are
motivated by the known fact that a straightforward generalization of the KS
form of the Kerr–Newman black hole to higher dimensions has failed and,
moreover, that the CCLP solution of charged rotating black hole in five-
dimensional minimal gauged supergravity takes the xKS form. In contrast to the
GKS case, we obtain in general only the necessary condition under which the
Kerr–Schild vector is geodetic. However, it is shown that this condition
becomes sufficient if we appropriately restrict the geometry of the null and
spacelike vectors appearing in the metric. It turns out that xKS spacetimes
with a geodetic Kerr–Schild vector field are of Weyl type I or more special.
In the case of Kundt xKS spacetimes, the compatible Weyl types are further
restricted depending on the form of the energy–momentum tensor. A few examples
of such spacetimes are also briefly discussed. For an example of an expanding
xKS spacetime, namely the CCLP black hole, we express the optical matrix which
interestingly satisfies the optical constraint obtained in the case of
expanding GKS spacetimes.
In chapter 4, we focus on quadratic gravity (QG) in arbitrary dimension, i.e.
a generalization of the Einstein theory with a Lagrangian containing all
possible polynomial curvature invariants as quantum corrections up to the
second order in the Riemann tensor. We show that all higher dimensional
Einstein spacetimes of the Weyl type N with an appropriately chosen effective
cosmological constant $\Lambda$ depending on the particular parameters of the
theory are exact solutions to QG. We refer to explicitly known metrics within
this class and construct some new ones using the Brinkmann warp product. In
the case of type III Einstein spacetimes, it is shown that the field equations
of QG impose an additional constraint on the Weyl tensor and some examples of
such type III solutions are given. It turns out that not all spacetimes with
vanishing scalar invariants (VSI) solve QG since type III pp -waves do not
satisfy this constraint. We also study a wider class of spacetimes admitting a
null radiation term in the Ricci tensor aligned with a WAND. We show that such
spacetimes of type N and certain subclass of type III solve the source-free
field equations of QG and, in contrast to the Einstein case, the optical
properties of the null geodetic congruence are restricted so that these
solutions belong to the Kundt class. Examples of such type N metrics are also
given explicitly.
## Chapter 2 Kerr–Schild spacetimes
This chapter is mainly based on the original results published in [18, 19].
First, we include a cosmological constant to the Kerr–Schild ansatz in an
appropriate way. Without any additional assumptions, we show under which
conditions the Kerr–Schild vector $\boldsymbol{k}$ is geodetic and
consequently a multiple WAND. For Einstein spacetimes which then have
$\boldsymbol{k}$ necessarily geodetic, the Einstein field equations naturally
lead to the splitting of our analysis to the non-expanding and expanding case.
It will be shown that non-expanding Einstein Kerr–Schild spacetimes are only
of Weyl type N, whereas the expanding spacetimes are of Weyl types II or D. In
the expanding case, we determine the $r$-dependence of the optical matrix and
of the boost weight zero components of the Weyl tensor. This then allows us to
discuss the presence of curvature singularities. Some known examples of both
non-expanding and expanding Kerr–Schild spacetimes will be presented and a few
new solutions will be also given using the Brinkmann warp product.
The complexity of the Einstein field equations, which are a system of
quasilinear partial differential equations of the second order for an unknown
metric, has led to the development of many approaches that simplify solving of
these equations. This is motivated by the fact that a direct attack on the
equations is hopeless, especially in the case of higher dimensions. Since most
of the known exact solutions of four and higher dimensional general gravity
are algebraically special, one can use an appropriate formalism, for instance
the Newman–Penrose formalism briefly summarized in section 1.1, and assume a
special algebraic type of the spacetime under consideration to reduce the
number and simplify the form of the field equations. In fact, this method will
be employed in chapter 4 in the context of quadratic theory of gravity.
Another approach is to reduce the number of unknown independent metric
components by considering an appropriate special form of the metric. For
instance, this form may follow from an assumption of some kind of spacetime
symmetries as in the case of the discovery of the static Schwarzschild black
hole using spherical symmetry. Alternatively, one may directly propose a
convenient form of the unknown metric in order to simplify subsequent
calculations. An important example representing this approach, which has been
successfully applied for finding exact solutions, is the Kerr–Schild (KS)
ansatz
$g_{ab}=\eta_{ab}-2\mathcal{H}k_{a}k_{b},$ (2.1)
where $\mathcal{H}$ is a scalar function and $\boldsymbol{k}$ is a null vector
with respect to both the Minkowski background metric $\eta_{ab}$ and full
metric $g_{ab}$.
The KS ansatz proposed by Kerr and Schild in 1965 [20] has led to the
rediscovery of the four-dimensional rotating solution known as the Kerr black
hole [21]. The reason why they considered metrics in the KS form (2.1) is that
the inverse metric is simply given by
$g^{ab}=\eta^{ab}+2\mathcal{H}k^{a}k^{b}$ which also means that the full
metric corresponds exactly to its linear approximation around the flat
background.
Twenty years later, Myers and Perry managed to generalize the KS form of the
Kerr solution so that they obtained a metric of a rotating black hole in
arbitrary dimension [22]. The situation differs significantly from the four-
dimensional case where the black hole rotates in one rotation plane. Since in
$n$ dimensions, the black hole may rotate arbitrarily in $p$ independent
rotation planes, where $p$ is given by
$p\equiv\left\lfloor\frac{n-1}{2}\right\rfloor.$ (2.2)
This exhibits the fact that the rotation group $SO(n-1)$ has a Cartan subgroup
$U(1)^{p}$.
Despite the simplicity of the Kerr–Schild ansatz (2.1), which makes analytic
calculations tractable, this class of metrics contains physically interesting
solutions such as the above-mentioned Kerr black hole and the Myers–Perry
black hole, both being of Weyl type D, but it also contains radiative
spacetimes of Weyl type N represented, for instance, by pp -waves.
Let us mention that various non-vacuum exact solutions in four dimensions also
admit the Kerr–Schild form, see e.g. [9], and some of them can be
straightforwardly generalized to higher dimensions.
Vaidya’s radiating star [23] and the corresponding higher dimensional analogue
[24] are slight modifications of the spherically symmetric Schwarzschild black
hole where the constant mass parameter appearing in the Kerr–Schild function
$\mathcal{H}$ is replaced by a function $m=m(u)$ depending on the retarded
time $u$. As a consequence, the additional term in the Ricci tensor
corresponding to a null radiation appears due to the change of mass $m(u)$.
Kinnersley’s photon rocket [25] and its higher dimensional counterparts [26]
are other examples of non-vacuum solutions of the Einstein field equations
describing the gravitational field of an accelerating object anisotropically
emitting null radiation.
A well-known example of electro-vacuum solutions, the Kerr–Newman black hole,
which has not yet been successfully generalized to higher dimensions as
opposite to its static limit, the Reissner–Nordstöm black hole, can be also
cast to the Kerr–Schild form. In this case, the Maxwell field is aligned with
the geodetic null congruence $\boldsymbol{k}$, i.e. the vector potential is
proportional to the Kerr–Schild vector.
All the above examples of spacetimes admitting the Kerr–Schild form are
solutions of the Einstein field equations without cosmological constant.
General properties of such Ricci-flat Kerr–Schild metrics have been studied
recently in [27]. The aim of this chapter is to introduce cosmological
constant to the Kerr–Schild ansatz and generalize the results from the Ricci-
flat case to Einstein spaces. Moreover, we will also briefly discuss solutions
with possible cosmological constant which also contain aligned matter fields.
One of the simplest higher dimensional Ricci-flat solutions admitting
Kerr–Schild form is the Schwarzschild–Tangherlini black hole [28]. In the
standard Schwarzschild coordinates, cosmological constant $\Lambda$ enters the
metric in a simple way, i.e. the (A)dS–Schwarzschild–Tangherlini metric [29]
takes the form
$\mathrm{d}s^{2}=-U(r)\,\mathrm{d}t^{2}+U(r)^{-1}\,\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)},$
(2.3)
where
$\displaystyle U(r)$ $\displaystyle=1-\lambda
r^{2}-\left(\frac{2m}{r}\right)^{n-3},$ (2.4) $\displaystyle\lambda$
$\displaystyle=\frac{2\Lambda}{(n-1)(n-2)}$ (2.5)
and $\mathrm{d}\Omega_{(n-2)}^{2}$ is a metric on the $(n-2)$-dimensional
sphere $S^{n-2}$ of unit radius
$\mathrm{d}\Omega_{1}^{2}=\mathrm{d}\varphi^{2},\qquad\mathrm{d}\Omega_{i+1}^{2}=\mathrm{d}\theta_{i}^{2}+\sin^{2}\theta_{i}\,\mathrm{d}\Omega_{i}^{2}\quad(i\in\mathbb{N}),$
(2.6)
with the standard angular coordinates $\varphi\in\langle 0,2\pi\rangle$,
$\theta_{i}\in\langle 0,\pi\rangle$. Using the transformation
$\mathrm{d}t^{\prime}=\mathrm{d}t-\frac{1-U(r)}{U(r)}\,\mathrm{d}r,$ (2.7)
the metric (2.3) can be rewritten as
$\mathrm{d}s^{2}=-\mathrm{d}t^{\prime
2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)}+\left[\lambda
r^{2}+\left(\frac{2m}{r}\right)^{n-3}\right](\mathrm{d}t^{\prime}-\mathrm{d}r)^{2}.$
(2.8)
The first three terms in (2.8) correspond to the Minkowski metric in the
spherical coordinates and the last term is a multiple of a null vector,
therefore, the (A)dS–Schwarzschild–Tangherlini metric (2.8) takes the
Kerr–Schild form with a flat background (2.1).
Note that, setting $m=0$ in (2.8), one directly obtains the (Anti-)de Sitter
metric in the Kerr–Schild form with a flat background
$\mathrm{d}s^{2}=-\mathrm{d}t^{\prime
2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)}+\frac{2\Lambda}{(n-1)(n-2)}r^{2}(\mathrm{d}t^{\prime}-\mathrm{d}r)^{2}.$
(2.9)
The metric (2.3) can be also expressed in another form. One may start with the
higher dimensional Kerr–(A)dS metric [30] and set all the rotation parameters
to zero. Thus, we arrive at the (A)dS–Schwarzschild–Tangherlini metric as a
static limit in the form
$\mathrm{d}s^{2}=-(1-\lambda
r^{2})\,\mathrm{d}\tilde{t}^{2}+\frac{\mathrm{d}r^{2}}{1-\lambda
r^{2}}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)}+\frac{2m}{r^{n-1}}\left(\mathrm{d}\tilde{t}-\frac{\mathrm{d}r}{1-\lambda
r^{2}}\right)^{2}.$ (2.10)
The first three terms now represent the (Anti-)de Sitter metric in the
spherical coordinates and the last term is again a multiple of a null vector.
Hence, the metric (2.10) takes the generalized Kerr–Schild (GKS) form
$g_{ab}=\bar{g}_{ab}-2\mathcal{H}k_{a}k_{b},$ (2.11)
where the background metric $\bar{g}_{ab}$ corresponds to an (A)dS spacetime,
$\mathcal{H}$ is a scalar function and the Kerr–Schild vector $\boldsymbol{k}$
is null with respect to both the full metric $g_{ab}$ and background metric
$\bar{g}_{ab}$.
Therefore, the (A)dS–Schwarzschild–Tangherlini metric can be cast to both the
KS form (2.1) with the flat background and the GKS form (2.11) with an
(anti-)de Sitter background. It implies that the cosmological constant
$\Lambda$ can be included either to the scalar function $\mathcal{H}$ in the
KS form or to the background metric $\bar{g}_{ab}$ in the GKS form. However,
these two possibilities how the cosmological constant enters the metric occur
only in exceptional cases where the Kerr–Schild vector field $\boldsymbol{k}$
in the KS form of the (A)dS–Schwarzschild–Tangherlini metric (2.8) and in the
KS form of the (A)dS metric (2.9) has the same geometrical properties. Namely,
if $\boldsymbol{k}$ is non-twisting and shear-free with the same expansion. It
follows that Robinson–Trautman solutions with a non-vanishing cosmological
constant admitting the KS form can be transformed to the GKS form as has been
shown in the case of the higher dimensional Vaidya metric in section 2.6.6.
However, in general, Einstein GKS spacetimes do not admit the KS form with a
flat background and therefore we adopt the GKS form as a generalization of the
original KS ansatz to the cases of spacetimes with a non-vanishing
cosmological constant.
In fact, it was shown first by Carter in 1968 [31] that his solution of four-
dimensional rotating black hole with a non-vanishing cosmological constant
$\Lambda$ can be cast to the GKS form (2.11). Later, Hawking et al. in 1999
[32] generalized Carter’s solution to five dimensions, but without detailed
derivation. Moreover, their solution is not given in the GKS form. Finally,
Gibbons et al. in 2004 [30] inspired by the previous works employed the GKS
form and special ellipsoidal coordinates which allowed them to construct a
solution representing a rotating black hole with a cosmological constant in
arbitrary dimension.
Throughout the thesis, we will assume that the $n$-dimensional background
(anti-)de Sitter metric with a cosmological constant $\Lambda$ takes the
conformally flat form
$\bar{g}_{ab}=\Omega\eta_{ab},$ (2.12)
where the conformal factor $\Omega$ is given by
$\begin{split}\Omega_{\text{AdS}}&=\frac{(n-2)(n-1)}{2\Lambda t^{2}},\\\
\Omega_{\text{dS}}&=-\frac{(n-2)(n-1)}{2\Lambda{x_{1}}^{2}},\end{split}$
(2.13)
respectively, and the Minkowski metric $\eta_{ab}$ is in the canonical form
$\eta_{ab}=-\mathrm{d}t^{2}+\mathrm{d}x_{1}^{2}+\ldots+\mathrm{d}x_{n-1}^{2},$
(2.14)
which is convenient for the following calculations.
The vacuum Einstein field equations for the conformally flat metric (2.12)
imply that the conformal factor $\Omega$ satisfies
$\frac{\Omega_{,ab}}{\Omega}=\frac{3}{2}\frac{\Omega_{,a}\Omega_{,b}}{\Omega^{2}},\qquad-\frac{1}{4}\frac{\Omega_{,a}\Omega_{,b}}{\Omega^{2}}\bar{g}^{ab}=\frac{2}{(n-2)(n-1)}{\Lambda},$
(2.15)
with both possible signs of the cosmological constant $\Lambda$. Note that the
Minkowski limit $\Lambda=0$ can be obtained by setting $\Omega=1$.
### 2.1 General Kerr–Schild vector field
We will study general properties of the GKS metric (2.11) with an (anti-)de
Sitter background $\bar{g}_{ab}$ where $\mathcal{H}$ is a scalar function and
the Kerr–Schild vector $\boldsymbol{k}$ is null with respect to the full
metric. However, if $\boldsymbol{k}$ is null with respect to the full metric
$g_{ab}$ then it follows that it is also null with respect to the background
metric $\bar{g}_{ab}$ and vice versa. Consequently, the inverse metric to
(2.11) takes the simple form
$g^{ab}=\bar{g}^{ab}+2\mathcal{H}k^{a}k^{b},$ (2.16)
where $\bar{g}^{ab}=\Omega^{-1}\eta^{ab}$. This implies that one may use both
metrics for raising or lowering index of the Kerr–Schild vector
$\boldsymbol{k}$
$k_{a}\equiv g_{ab}k^{b}=\bar{g}_{ab}k^{b},\qquad k^{a}\equiv
g^{ab}k_{b}=\bar{g}^{ab}k_{b}.$ (2.17)
Our choice of the canonical form for the background metric $\bar{g}_{ab}$
(2.12), (2.14) allows us to express Christoffel symbols
$\begin{split}\Gamma^{a}_{bc}&=-\left(\mathcal{H}k^{a}k_{b}\right)_{,c}-\left(\mathcal{H}k^{a}k_{c}\right)_{,b}+g^{as}\left(\mathcal{H}k_{b}k_{c}\right)_{,s}\\\
&\qquad+\frac{1}{2}\frac{\Omega_{,c}}{\Omega}\delta^{a}_{b}+\frac{1}{2}\frac{\Omega_{,b}}{\Omega}\delta^{a}_{c}-\frac{1}{2}\frac{\Omega_{,s}}{\Omega}g^{as}\bar{g}_{bc}.\end{split}$
(2.18)
The first crucial step in our study of properties of the GKS metric (2.11) is
to show under which conditions the Einstein field equations imply that the KS
vector field $\boldsymbol{k}$ is geodetic.
Obviously, due to the form of the GKS ansatz (2.11) it will be convenient to
employ the higher dimensional Newman–Penrose frame formalism, briefly
summarized in section 1.1, and naturally identify the Kerr–Schild vector
$\boldsymbol{k}$ with the null frame vector $\boldsymbol{\ell}$ (1.8). From
now on, we will denote both vectors as $\boldsymbol{k}$, whereas the
corresponding Ricci rotation coefficients as $L_{ab}$.
Since the KS vector $\boldsymbol{k}$ appears in many terms of the Christoffel
symbols $\Gamma^{a}_{bc}$ (2.18), the simplest component of the Ricci tensor
is the boost weight two component $R_{00}=R_{ab}k^{a}k^{b}$. After quite
involved calculations using
$k_{a}k^{a}=k_{a,b}k^{a}=k^{a}_{\phantom{a},b}k_{a}=0$ we end up with a
remarkably simple result
$R_{00}=2\mathcal{H}k_{c;a}k^{a}k^{c}_{\phantom{c};b}k^{b}-\frac{1}{2}(n-2)\left(\frac{\Omega_{,ab}}{\Omega}-\frac{3}{2}\frac{\Omega_{,a}\Omega_{,b}}{\Omega^{2}}\right)k^{a}k^{b},$
(2.19)
for an arbitrary conformal factor $\Omega$. Therefore, for the conformal
factor of an (anti-)de Sitter background metric (2.13) obeying (2.15) we
obtain
$R_{00}=2\mathcal{H}k_{c;a}k^{a}k^{c}_{\phantom{c};b}k^{b}=2\mathcal{H}L_{i0}L_{i0}.$
(2.20)
It now follows from the Einstein field equations that $L_{i0}=0$, i.e.
$\boldsymbol{k}$ is geodetic, if and only if $T_{00}=0$.
###### Proposition 1
The Kerr–Schild vector $\boldsymbol{k}$ in the generalized Kerr–Schild metric
(2.11) is geodetic if and only if the boost weight 2 component of the
energy–momentum tensor $T_{00}=T_{ab}k^{a}k^{b}$ vanishes.
Proposition 1 implies that the Kerr–Schild vector $\boldsymbol{k}$ is geodetic
not only in Einstein GKS spacetimes, where the energy–momentum tensor is
absent, but also in spacetimes with aligned matter content such as aligned
Maxwell field $F_{ab}k^{a}\propto k_{b}$ or aligned pure radiation
$T_{ab}\propto k_{a}k_{b}$.
Moreover, if the Kerr–Schild vector $\boldsymbol{k}$ is geodetic then we may
assume, without loss of generality, that it is also affinely parametrized
since we are still able to rescale $\boldsymbol{k}$ by an appropriate scalar
factor. Subsequently, this factor can be included to the KS function
$\mathcal{H}$ and hence the GKS form of the original metric remains unchanged.
In order to preserve the normalization (1.8) of an already chosen null frame,
one has to rescale $\boldsymbol{k}$ by preforming the boost (1.11)
$\hat{\boldsymbol{k}}=\lambda\boldsymbol{k},\qquad\hat{\boldsymbol{n}}=\lambda^{-1}\boldsymbol{n}.$
(2.21)
Recall that the Kerr–Schild vector $\boldsymbol{k}$ is geodetic if
$k_{a;b}k^{b}=L_{10}k_{a}$ and since we require $\hat{\boldsymbol{k}}$ to be
affinely parametrized, therefore, $\hat{L}_{10}=0$. From the transformation
properties of the Ricci rotation coefficient $L_{10}$ under the boost (1.20),
it follows
$\mathrm{D}\lambda=-\lambda L_{10},$ (2.22)
which determines the necessary scalar factor $\lambda$. Finally, we denote
$\hat{\mathcal{H}}\equiv\lambda^{-2}\mathcal{H}$, therefore, the Kerr–Schild
term $\mathcal{H}k_{a}k_{b}$ transforms under the boost as a quantity with
boost weight 0, i.e.
$\hat{\mathcal{H}}\hat{k}_{a}\hat{k}_{b}=\mathcal{H}k_{a}k_{b}$, and the GKS
form is not affected by this operation.
Thus, in the following sections, the geodetic Kerr–Schild vector
$\boldsymbol{k}$ is assumed to be affinely parametrized. This will lead to a
significant simplification of the necessary calculations.
#### 2.1.1 Kerr–Schild congruence in the background
spacetime
One may compare the geodesicity and the optical properties of the null
Kerr–Schild congruence $\boldsymbol{k}$ in the full GKS spacetime and in the
background (A)dS spacetime. It will be shown that there is a close relation
between the congruences in both spacetimes.
Quantities constructed from the background metric are easily obtained from
quantities constructed from the full GKS metric simply by setting
$\mathcal{H}$ to zero. For instance, using the Christoffel symbols (2.18), it
is straightforward to show that
$\begin{split}k_{a;b}k^{b}&=k_{a,b}k^{b}=k_{a\overline{;}b}k^{b},\\\
k^{a}_{\phantom{a};b}k^{b}&=k^{a}_{\phantom{a},b}k^{b}+\frac{\Omega_{,b}}{\Omega}k^{a}k^{b}=k^{a}_{\phantom{a}\overline{;}b}k^{b},\end{split}$
(2.23)
where “ $\overline{;}$ ” in the expression $k_{a\overline{;}b}$ denotes the
covariant derivative with respect to the background (A)dS metric
$\bar{g}_{ab}$. Thus, one can immediately see that the Kerr–Schild vector
$\boldsymbol{k}$ is geodetic in the full GKS metric if and only if it is
geodetic in the (A)dS background $\bar{g}_{ab}$.
The geometrical properties of the Kerr–Schild congruence $\boldsymbol{k}$ in
the full GKS spacetime are encoded in the optical matrix $L_{ij}$ (1.13).
Following [27], we define a null frame $\bar{\boldsymbol{n}}$,
$\boldsymbol{k}$, $\boldsymbol{m}^{(i)}$ in the background spacetime
$\bar{g}_{ab}$, where
$\bar{n}_{a}=n_{a}+\mathcal{H}k_{a},$ (2.24)
and the remaining frame vectors $\boldsymbol{k}$, $\boldsymbol{m}^{(i)}$ are
same as in the full spacetime. This choice guarantees
$\bar{g}_{ab}=2k_{(a}\bar{n}_{b)}+\delta_{ij}m^{(i)}_{a}m^{(j)}_{b}$ (2.25)
and allows us to express the optical matrix $\bar{L}_{ij}$ in the background
spacetime, which can be then compared with $L_{ij}$. Using (2.18), it follows
$L_{ij}\equiv
k_{a;b}m^{(i)a}m^{(j)b}=k_{a\overline{;}b}m^{(i)a}m^{(j)b}\equiv{\bar{L}_{ij}}.$
(2.26)
Therefore, the optical matrices of the Kerr–Schild congruence $\boldsymbol{k}$
with respect to the full GKS metric $g_{ab}$ and the (A)dS background metric
$\bar{g}_{ab}$ are equal, i.e. the corresponding expansion, shear and twist
scalars have the same values in both spacetimes. Note that for
$\boldsymbol{k}$ being geodetic, $L_{ij}$ does not depend on our particular
choice (2.24) since in such case $L_{ij}$ is invariant under null rotations
with $\boldsymbol{k}$ fixed (1.19).
It should be emphasized that the index of the background frame covector
$\bar{n}_{a}$ (2.24) is raised by the full metric as $\bar{n}^{a}\equiv
g^{ab}\bar{n}_{b}=n^{a}+\mathcal{H}k^{a}$, whereas the contraction with the
background metric gives $\bar{g}^{ab}\bar{n}_{b}=n^{a}-\mathcal{H}k^{a}$.
Similarly as for the vector $\boldsymbol{k}$ (2.17), the vector indices of
$\boldsymbol{m}^{(i)}$ may be raised and lowered by both metrics since
$m^{a}_{(i)}\equiv g^{ab}m^{(i)}_{b}=\bar{g}^{ab}m^{(i)}_{b}$.
One may also compare the remaining Ricci rotational coefficients with respect
to the full GKS spacetime and the background spacetime. Thus, we obtain
$\begin{split}&L_{i0}=\bar{L}_{i0},\qquad L_{10}=\bar{L}_{10},\qquad
L_{1i}=\bar{L}_{1i}-\mathcal{H}\bar{L}_{i0},\qquad
L_{i1}=\bar{L}_{i1},\phantom{\frac{1}{2}}\\\
&L_{11}=\bar{L}_{11}-\mathrm{D}\mathcal{H}-\mathcal{H}\bar{L}_{10}+\mathcal{H}\frac{\Omega_{,a}}{\Omega}k^{a},\qquad
N_{i0}=\bar{N}_{i0},\\\
&N_{i1}=\bar{N}_{i1}+2\mathcal{H}\bar{L}_{1i}-\mathcal{H}\bar{L}_{i1}+\mathcal{H}\bar{N}_{i0}-\mathcal{H}^{2}\bar{L}_{i0}+\delta_{i}\mathcal{H}+\mathcal{H}\frac{\Omega_{,a}}{\Omega}m_{(i)}^{a},\\\
&N_{ij}=\bar{N}_{ij}+\mathcal{H}\bar{L}_{ji}-\mathcal{H}\frac{\Omega_{,a}}{\Omega}k^{a}\delta_{ij},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}=\mbox{$\stackrel{{\scriptstyle i}}{{\bar{M}}}_{j0}$},\\\
&\mbox{$\stackrel{{\scriptstyle i}}{{M}}_{jk}$}=\mbox{$\stackrel{{\scriptstyle
i}}{{\bar{M}}}_{jk}$},\qquad\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j1}$}=\mbox{$\stackrel{{\scriptstyle
i}}{{\bar{M}}}_{j1}$}+\mathcal{H}\left(\bar{L}_{ij}-\bar{L}_{ji}\right)+\mathcal{H}\mbox{$\stackrel{{\scriptstyle
i}}{{\bar{M}}}_{j0}$}.\end{split}$ (2.27)
Let us mention that for $\boldsymbol{k}$ being geodetic and affinely
parametrized the following relations hold
$n_{a;b}k^{b}=\bar{n}_{a\bar{;}b}k^{b},\qquad
m^{(i)}_{a;b}k^{b}=m^{(i)}_{a\bar{;}b}k^{b}.$ (2.28)
In other words, the frame vectors $\boldsymbol{n}$, $\boldsymbol{m}^{(i)}$ are
parallelly transported along $\boldsymbol{k}$ in the full GKS spacetime if and
only if the frame vectors $\bar{\boldsymbol{n}}$, $\boldsymbol{m}^{(i)}$ are
parallelly transported along $\boldsymbol{k}$ in the background spacetime.
This can help us in finding a parallelly propagated frame in the full GKS
spacetime, which can be a nontrivial task. Instead, it could be easier to find
such a frame in the background (A)dS spacetime and then use (2.24) relating
these two frames.
In fact, we partially employ this procedure in section 2.6.5 since the full
GKS metric of the five dimensional Kerr–(A)dS spacetime is quite complicated
and non-diagonal, whereas the background (A)dS metric is diagonal and the
calculations are not so involved.
### 2.2 Geodetic Kerr–Schild vector field
So far, we discussed the properties of the GKS metric (2.11) with an arbitrary
null Kerr–Schild vector field $\boldsymbol{k}$ without any additional
assumptions. In the rest of this chapter, we will consider Einstein GKS
spacetimes and GKS spacetimes with aligned matter fields. Then it follows from
proposition 1 that the Kerr–Schild vector $\boldsymbol{k}$ is geodetic as
discussed in section 2.1. We also assume an affine parametrization of
$\boldsymbol{k}$. Using higher dimensional Newman–Penrose formalism, we employ
the Einstein field equations and analyze the conditions imposed on the GKS
ansatz.
Assuming geodesicity of $\boldsymbol{k}$, we arrive at the convenient
expressions for the contracted Christoffel symbols frequently used in the
following calculations
$\begin{split}\Gamma^{a}_{bc}k^{b}&=-\mathrm{D}\mathcal{H}k^{a}k_{c}+\frac{1}{2}\frac{\Omega_{,c}}{\Omega}k^{a}+\frac{1}{2}\frac{\Omega_{,b}}{\Omega}k^{b}\delta^{a}_{c}-\frac{1}{2}\frac{\Omega_{,b}}{\Omega}\bar{g}^{ab}k_{c},\\\
\Gamma^{a}_{bc}k_{a}&=\mathrm{D}\mathcal{H}k_{b}k_{c}+\frac{1}{2}\frac{\Omega_{,c}}{\Omega}k_{b}+\frac{1}{2}\frac{\Omega_{,b}}{\Omega}k_{c}-\frac{1}{2}\frac{\Omega_{,a}}{\Omega}k^{a}\bar{g}_{bc}.\end{split}$
(2.29)
#### 2.2.1 Ricci tensor
Despite the simple form of the GKS metric (2.11) along with the assumption
that $\boldsymbol{k}$ is geodetic, expressing the Ricci tensor is a quite
complicated task since hundreds of terms appear during the derivation. Hence,
we were able to perform this and some of the following tedious calculations
only by means of the computer algebra system Cadabra [33, 34]. Fortunately,
after many operations, we obtain the Ricci tensor in the compact form
$\begin{split}R_{ab}&=\left(\mathcal{H}k_{a}k_{b}\right)_{;cd}g^{cd}-\left(\mathcal{H}k^{s}k_{a}\right)_{;bs}-\left(\mathcal{H}k^{s}k_{b}\right)_{;as}+\frac{2{\Lambda}}{n-2}\bar{g}_{ab}\\\
&\qquad-2\mathcal{H}\left(\mathrm{D}^{2}\mathcal{H}+L_{ii}\mathrm{D}\mathcal{H}+2\mathcal{H}\omega^{2}\right)k_{a}k_{b},\end{split}$
(2.30)
which for $\Lambda=0$ reduces to the result of [27] where, at first sight, the
sign before the last term is opposite. However, it can be easily shown that
both results are in full accordance, if one rewrites the covariant derivatives
in (2.30) in terms of the partial derivatives.
From (2.30) it follows that the Kerr–Schild vector $\boldsymbol{k}$ is an
eigenvector of the Ricci tensor
$R_{ab}k^{b}=-\left[\mathrm{D}^{2}\mathcal{H}+(n-2)\theta\mathrm{D}\mathcal{H}+2\mathcal{H}\omega^{2}-\frac{2\Lambda}{n-2}\right]k_{a}$
(2.31)
and one can immediately see that the positive boost weight frame components of
the Ricci tensor vanish
$R_{00}=0,\qquad R_{0i}=0.$ (2.32)
The non-vanishing frame components of the Ricci tensor can be
straightforwardly obtained from (2.30) by performing appropriate contractions
with the corresponding frame vectors
$\displaystyle R_{01}$
$\displaystyle=-\mathrm{D}^{2}\mathcal{H}-(n-2)\theta\mathrm{D}\mathcal{H}-2\mathcal{H}\omega^{2}+\frac{2\Lambda}{n-2},$
(2.33) $\displaystyle R_{ij}$
$\displaystyle=2\mathcal{H}L_{ik}L_{jk}-2\left(\mathrm{D}\mathcal{H}+(n-2)\theta\mathcal{H}\right)S_{ij}+\frac{2{\Lambda}}{n-2}\delta_{ij},$
(2.34) $\displaystyle R_{1i}$
$\displaystyle=-\delta_{i}(\mathrm{D}\mathcal{H})+2L_{[i1]}\mathrm{D}\mathcal{H}+2L_{ij}\delta_{j}\mathcal{H}-S_{jj}\delta_{i}\mathcal{H}+2\mathcal{H}\bigg{(}\delta_{j}A_{ij}$
$\displaystyle\qquad+A_{ij}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{kk}$}-A_{jk}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jk}$}-L_{1i}S_{jj}+3L_{ij}L_{[1j]}+L_{ji}L_{(1j)}\bigg{)},$ (2.35)
$\displaystyle R_{11}$
$\displaystyle=\delta_{i}(\delta_{i}\mathcal{H})+\left(N_{ii}-2\mathcal{H}S_{ii}\right)\mathrm{D}\mathcal{H}+\left(4L_{1i}-2L_{i1}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\mathcal{H}$ $\displaystyle\qquad-
S_{ii}\Delta\mathcal{H}+2\mathcal{H}\bigg{(}2\delta_{i}L_{[1i]}+4L_{1i}L_{[1i]}+L_{i1}L_{i1}-L_{11}S_{ii}$
$\displaystyle\qquad+2L_{[1i]}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}-2A_{ij}N_{ij}-2\mathcal{H}\omega^{2}\bigg{)}+\frac{4\mathcal{H}{\Lambda}}{n-1},$
(2.36)
where the components are sorted by their boost weight and some of them were
further simplified using the Ricci identities [3].
#### 2.2.2 Riemann tensor and algebraic type of the Weyl
tensor
In this section, we point out that GKS spacetimes (2.11) with a geodetic
Kerr–Schild vector $\boldsymbol{k}$ are algebraically special and
$\boldsymbol{k}$ is a multiple WAND of the Weyl tensor. First, we express the
frame components of the Riemann tensor. As in the case of the Ricci tensor,
the positive boost weight components of the Riemann tensor identically vanish
$R_{0i0j}=0,\qquad R_{010i}=0,\qquad R_{0ijk}=0.$ (2.37)
The non-vanishing frame components of the Riemann tensor sorted by their boost
weight read
$\displaystyle R_{0101}$
$\displaystyle=\mathrm{D}^{2}\mathcal{H}-\frac{2\Lambda}{(n-2)(n-1)},$ (2.38)
$\displaystyle R_{01ij}$
$\displaystyle=-2A_{ij}\mathrm{D}\mathcal{H}+4\mathcal{H}S_{k[j}A_{i]k},\phantom{\frac{1}{()}}$
(2.39) $\displaystyle R_{0i1j}$
$\displaystyle=-L_{ij}\mathrm{D}\mathcal{H}+2\mathcal{H}A_{ik}L_{kj}+\frac{2\Lambda}{(n-2)(n-1)}\delta_{ij},$
(2.40) $\displaystyle R_{ijkl}$
$\displaystyle=4\mathcal{H}\left(A_{ij}A_{kl}+A_{l[i}A_{j]k}+S_{l[i}S_{j]k}\right)\phantom{\frac{1}{()}}$
$\displaystyle\qquad+\frac{2\Lambda}{(n-2)(n-1)}\left(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\right),$
(2.41) $\displaystyle R_{011i}$
$\displaystyle=-\delta_{i}\left(\mathrm{D}\mathcal{H}\right)+2L_{[i1]}\mathrm{D}\mathcal{H}+L_{ji}\delta_{j}\mathcal{H}+2\mathcal{H}\left(L_{1j}L_{ji}-L_{j1}S_{ij}\right),\phantom{\frac{1}{()}}$
(2.42) $\displaystyle R_{1ijk}$
$\displaystyle=2L_{[j|i}\delta_{|k]}\mathcal{H}+2A_{jk}\delta_{i}\mathcal{H}+4\mathcal{H}\bigg{(}\delta_{[k}S_{j]i}+\mbox{$\stackrel{{\scriptstyle
l}}{{M}}_{[jk]}$}S_{il}-\mbox{$\stackrel{{\scriptstyle
l}}{{M}}_{i[j}$}S_{k]l}$
$\displaystyle\qquad+L_{1i}A_{jk}+L_{1[k}A_{j]i}\bigg{)},$ (2.43)
$\displaystyle R_{1i1j}$
$\displaystyle=\delta_{i}(\delta_{j}\mathcal{H})+\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{(ij)}$}\delta_{k}\mathcal{H}+4L_{1(i}\delta_{j)}\mathcal{H}-2L_{(i|1}\delta_{j)}\mathcal{H}+N_{(ij)}\mathrm{D}\mathcal{H}\phantom{\frac{1}{()}}$
$\displaystyle\qquad-
S_{ij}\Delta\mathcal{H}+2\mathcal{H}\bigg{(}\delta_{(i}L_{1|j)}-\Delta
S_{ij}-2L_{1(i}L_{j)1}+2L_{1i}L_{1j}$ $\displaystyle\qquad-
L_{k(i}N_{k|j)}+L_{1k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{(ij)}$}-2\mathcal{H}L_{k(i}A_{j)k}-2\mathcal{H}A_{ik}A_{jk}\phantom{\frac{1}{()}}$
$\displaystyle\qquad-L_{k(i}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j)1}$}-L_{(i|k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j)1}$}\bigg{)}.$ (2.44)
Note that the boost weight zero components of the Riemann tensor $R_{0101}$,
$R_{01ij}$, $R_{0i1j}$ and $R_{ijkl}$ are given only by the Kerr–Schild
function $\mathcal{H}$, the optical matrix $L_{ij}$ and the cosmological
constant $\Lambda$. Actually, in section 2.6.3, this fact allows us to
explicitly determine the dependence of these components on an affine parameter
$r$ along the geodesics of the Kerr–Schild congruence $\boldsymbol{k}$.
It is also convenient to express explicitly the frame components of the Weyl
tensor (1.31) for a general Ricci tensor that will be occasionally employed
throughout the thesis, explicitly
$\displaystyle C_{0i0j}$
$\displaystyle=R_{0i0j}-\frac{1}{n-2}R_{00}\delta_{ij},\phantom{\frac{1}{(n-1)(n-2)}}$
(2.45) $\displaystyle C_{010i}$
$\displaystyle=R_{010i}+\frac{1}{n-2}R_{0i},\phantom{\frac{1}{(n-1)(n-2)}}$
(2.46) $\displaystyle C_{0ijk}$
$\displaystyle=R_{0ijk}+\frac{1}{n-2}(R_{0k}\delta_{ij}-R_{0j}\delta_{ik}),\phantom{\frac{1}{(n-1)(n-2)}}$
(2.47) $\displaystyle C_{0101}$
$\displaystyle=R_{0101}+\frac{2}{n-2}R_{01}-\frac{1}{(n-1)(n-2)}R,$ (2.48)
$\displaystyle C_{01ij}$
$\displaystyle=R_{01ij},\phantom{\frac{1}{(n-1)(n-2)}}$ (2.49) $\displaystyle
C_{0i1j}$
$\displaystyle=R_{0i1j}-\frac{1}{n-2}(R_{ij}+R_{01}\delta_{ij})+\frac{1}{(n-1)(n-2)}R\delta_{ij},$
(2.50) $\displaystyle C_{ijkl}$
$\displaystyle=R_{ijkl}-\frac{2}{n-2}(R_{j[l}\delta_{k]i}-R_{i[l}\delta_{k]j})+\frac{2}{(n-1)(n-2)}R\delta_{i[k}\delta_{l]j},$
(2.51) $\displaystyle C_{011i}$
$\displaystyle=R_{011i}-\frac{1}{n-2}R_{1i},\phantom{\frac{1}{(n-1)(n-2)}}$
(2.52) $\displaystyle C_{1ijk}$
$\displaystyle=R_{1ijk}+\frac{1}{n-2}(R_{1k}\delta_{ij}-R_{1j}\delta_{ik}),\phantom{\frac{1}{(n-1)(n-2)}}$
(2.53) $\displaystyle C_{1i1j}$
$\displaystyle=R_{1i1j}-\frac{1}{n-2}R_{11}\delta_{ij}.\phantom{\frac{1}{(n-1)(n-2)}}$
(2.54)
Since the positive boost weight frame components of the Ricci tensor (2.32)
and the Riemann tensor (2.37) identically vanish it means that this holds also
for the corresponding components of the Weyl tensor (2.45)–(2.47), i.e.
$C_{0i0j}=0,\qquad C_{010i}=0,\qquad C_{0ijk}=0,$ (2.55)
and therefore
###### Proposition 2
Generalized Kerr–Schild spacetimes (2.11) with a geodetic Kerr–Schild vector
$\boldsymbol{k}$ are algebraically special with $\boldsymbol{k}$ being the
multiple WAND.
In other words, GKS spacetimes (2.11) with a geodetic Kerr–Schild vector
$\boldsymbol{k}$ are of Weyl type II or more special. Let us remind that the
Kerr–Schild vector $\boldsymbol{k}$ is geodetic if and only if the boost
weight 2 component of the energy–momentum tensor vanish $T_{00}=0$ as follows
from proposition 1. Consequently, Einstein GKS spacetimes and GKS spacetimes
with an aligned matter field are algebraically special.
It can be shown [13] that in static spacetimes admitting a WAND
$\boldsymbol{\ell}=(\ell_{t},\ell_{A})$ with the metric not depending on the
direction of time one may always construct a distinct WAND
$\boldsymbol{n}=(-\ell_{t},\ell_{A})$ with the same order of alignment. Thus,
static spacetimes are compatible only with Weyl types G, Ii, D or O.
A similar statement also holds for stationary spacetimes with the metric
remaining unchanged under reflection symmetry, for instance, the symmetry
$t\rightarrow-t$, $\varphi_{i}\rightarrow-\varphi_{i}$ of the higher
dimensional Kerr–(A)dS metric [30] expressed in the Boyer–Lindquist
coordinates. If $\ell$ is a WAND then, again due to the symmetry, there is a
distinct WAND $\boldsymbol{n}$. Additional assumption has to be imposed,
namely, one has to require non-vanishing “divergence scalar” corresponding to
the expansion scalar $\theta$ in case of geodetic WAND. This ensures that both
WANDs do not represent the same null direction
$\boldsymbol{\ell}\neq-\boldsymbol{n}$ and hence such spacetimes are of Weyl
types G, Ii, D or conformally flat. See [13] for further details.
Proposition 2 along with the above statements for static and stationary
spacetimes lead to
###### Corollary 3
Generalized Kerr–Schild spacetimes (2.11) with a geodetic Kerr–Schild vector
$\boldsymbol{k}$ which are
1. (a)
either static
2. (b)
or stationary with reflection symmetry and non-vanishing expansion
are of Weyl type D or conformally flat.
Note that these results immediately imply that the Kerr–(A)dS metrics in all
dimensions [30] are of Weyl type D, as was shown previously in [35] by
explicit calculation of the Weyl tensor.
### 2.3 Brinkmann warp product of Kerr–Schild
spacetimes
The Brinkmann warp product introduced in section 1.4 is a convenient method
for generating new $n$-dimensional solutions of the vacuum Einstein field
equations from known $(n-1)$-dimensional Ricci-flat or Einstein metrics.
Naturally, this warp product can be also applied to Einstein GKS spacetimes.
Thus, let us consider the seed metric of the GKS form (2.11)
$\mathrm{d}\tilde{s}^{2}=\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}-2\mathcal{H}k_{a}k_{b}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}.$
(2.56)
Then the warped metric $\mathrm{d}s^{2}$ is given by (1.37)
$\mathrm{d}s^{2}=\frac{1}{f}\,\mathrm{d}z^{2}+f\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}-2f\mathcal{H}k_{a}k_{b}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}.$
(2.57)
Since the seed background metric $\bar{g}_{ab}$ represents Einstein space of
Weyl type O, i.e. conformally flat, and moreover the warp product preserves
the Weyl type of algebraically special spacetimes, as mentioned in section
1.4, the new warped background metric
${f}^{-1}\,\mathrm{d}z^{2}+f\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
describes necessarily an (anti-)de Sitter or Minkowski spacetime. The
remaining term $2f\mathcal{H}k_{a}k_{b}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$ in
(2.57) is again a multiple of two null vectors since the original Kerr–Schild
vector $\boldsymbol{k}$ is obviously null with respect to the new warped
metric as well. Therefore, the warped metric (2.57) is also an Einstein GKS
metric and thus the Brinkmann warp product preserves the GKS form.
If the $(n-1)$-dimensional seed background metric $\bar{g}_{ab}$ takes the
canonical form (2.12), (2.14) with the conformal factor $\tilde{\Omega}$ given
by the cosmological constant of the seed (1.39) as
$\displaystyle\tilde{\Omega}_{\text{AdS}}=\frac{1}{(\lambda b+d^{2})t^{2}},$
(2.58) $\displaystyle\tilde{\Omega}_{\text{dS}}=-\frac{1}{(\lambda
b+d^{2}){x_{1}}^{2}},$ (2.59)
respectively, then the $n$-dimensional warped background metric can be also
put to the corresponding canonical form
$\displaystyle
f\bar{g}_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+f^{-1}\mathrm{d}z^{2}$
$\displaystyle=f\tilde{\Omega}(-\mathrm{d}t^{2}+\mathrm{d}x_{1}^{2}+\ldots+\mathrm{d}x_{n-2}^{2})+f^{-1}\mathrm{d}z^{2}$
$\displaystyle=\Omega(-\mathrm{d}\hat{t}^{2}+\mathrm{d}\hat{x}_{1}^{2}+\ldots+\mathrm{d}\hat{x}_{n-2}^{2}+\mathrm{d}\hat{z}^{2}),$
(2.60)
where $\Omega$ is defined exactly as in (2.13), the warp factor $f(z)$ is
given by (1.38) and the new and old coordinates are related by an appropriate
transformation depending on the signs of the Ricci scalars $\tilde{R}$ and $R$
of the seed and the warped metric, respectively. Recall that, as discussed in
section 1.4, not all combinations of the signs of $\tilde{R}$ and $R$ are
possible. In the trivial case $\tilde{R}=0$, $R=0$ corresponding to the direct
product, just one extra flat dimension is added to the $(n-1)$-dimensional
Minkowski metric. In other cases, the following coordinate transformations has
to be performed in order to cast the warped background metric to the canonical
form
$\mathrm{AdS}_{n-1}\Rightarrow\mathrm{AdS}_{n}:$
$t=\hat{t},\qquad x_{1}^{2}=\hat{x}_{1}^{2}+\hat{z}^{2},\qquad
x_{\tilde{\imath}}=\hat{x}_{\tilde{\imath}},\qquad
z=\frac{\sqrt{-\left(d^{2}+\lambda
b\right)}\hat{z}}{\lambda\hat{x}_{1}}+\frac{d}{\lambda},$ (2.61)
$\mathrm{dS}_{n-1}\Rightarrow\mathrm{AdS}_{n}:$
$t^{2}=\hat{t}^{2}-\hat{z}^{2},\qquad x_{i}=\hat{x}_{i},\qquad
z=\frac{\sqrt{d^{2}+\lambda b}\,\hat{t}}{-\lambda\hat{z}}+\frac{d}{\lambda},$
(2.62)
$\mathrm{dS}_{n-1}\Rightarrow\mathrm{dS}_{n}:$
$\displaystyle t^{2}=\hat{t}^{2}-\hat{z}^{2},\qquad x_{i}=\hat{x}_{i},\qquad
z=\frac{\sqrt{d^{2}+\lambda b}\,\hat{z}}{\lambda\hat{t}}+\frac{d}{\lambda},$
(2.63)
$\mathrm{M}_{n-1}\Rightarrow\mathrm{AdS}_{n}:$
$\displaystyle t=\hat{t},\qquad x_{i}=\hat{x}_{i},\qquad
z=\frac{1}{\lambda\hat{z}}+\frac{d}{\lambda}.$ (2.64)
We have to emphasize that in the above expressions (2.61)–(2.64),
exceptionally, the index $i$ goes from 1 to $n-2$ and the index
$\tilde{\imath}$ ranges from 2 to $n-2$.
### 2.4 Einstein Kerr-Schild spacetimes
In section 2.2, we discussed the algebraic properties and expressed the Ricci
and Riemann tensors of the GKS metric with the only additional assumption that
the Kerr–Schild vector $\boldsymbol{k}$ is geodetic. As follows from
proposition 1, this includes a wide class of spacetimes, for instance,
Einstein spaces or spacetimes with an aligned matter field. In this section,
we will restrict our analysis to the simplest case, namely, Einstein GKS
spacetimes. Thus, we employ the vacuum Einstein field equations in order to
study their implications for the GKS metric.
In $n$ dimensions, the vacuum Einstein field equations can be rewritten as
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab}.$ (2.65)
Since the Kerr–Schild vector $\boldsymbol{k}$ is geodetic in the case of
Einstein GKS spacetimes, we can substitute the Ricci tensor (2.30) to (2.65).
The term containing the cosmological constant $\Lambda$ on the right hand side
of (2.65) is multiplied by the full GKS metric $g_{ab}$, whereas the term with
the cosmological constant in the expression of the Ricci tensor (2.30) is
multiplied by the background metric $\bar{g}_{ab}$. If we assume that both
cosmological constants are equal, the difference between these terms is
proportional to the Kerr–Schild term $2\mathcal{H}k_{a}k_{b}$ and we arrive at
the Einstein field equations for Einstein GKS metrics in the form
$\begin{split}&\left(\mathcal{H}k_{a}k_{b}\right)_{;cd}g^{cd}-\left(\mathcal{H}k^{s}k_{a}\right)_{;bs}-\left(\mathcal{H}k^{s}k_{b}\right)_{;as}\\\
&\qquad-2\mathcal{H}\left(\mathrm{D}^{2}\mathcal{H}+L_{ii}\mathrm{D}\mathcal{H}+2\mathcal{H}\omega^{2}-\frac{2\Lambda}{n-2}\right)k_{a}k_{b}=0.\end{split}$
(2.66)
Using the frame components of the Ricci tensor (2.32)–(2.36), one may express
the frame components of the Einstein field equations (2.66) as
$\displaystyle\mathrm{D}^{2}\mathcal{H}+(n-2)\theta\mathrm{D}\mathcal{H}+2\,\mathcal{H}\omega^{2}=0,\phantom{\frac{1}{2}}$
(2.67) $\displaystyle
2\mathcal{H}L_{ik}L_{jk}-2\left(\mathrm{D}\mathcal{H}+(n-2)\theta\mathcal{H}\right)S_{ij}=0,\phantom{\frac{1}{2}}$
(2.68)
$\displaystyle\delta_{i}(\mathrm{D}\mathcal{H})-2L_{[i1]}\mathrm{D}\mathcal{H}-2L_{ij}\delta_{j}\mathcal{H}+S_{jj}\delta_{i}\mathcal{H}-2\mathcal{H}\bigg{(}\delta_{j}A_{ij}$
$\displaystyle\qquad+A_{ij}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{kk}$}-A_{jk}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jk}$}-L_{1i}S_{jj}+3L_{ij}L_{[1j]}+L_{ji}L_{(1j)}\bigg{)}=0,$ (2.69)
$\displaystyle\delta_{i}(\delta_{i}\mathcal{H})+\left(N_{ii}-2\mathcal{H}S_{ii}\right)\mathrm{D}\mathcal{H}+\left(4L_{1i}-2L_{i1}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\mathcal{H}$ $\displaystyle\qquad-
S_{ii}\Delta\mathcal{H}+2\mathcal{H}\bigg{(}2\delta_{i}L_{[1i]}+4L_{1i}L_{[1i]}+L_{i1}L_{i1}-L_{11}S_{ii}$
$\displaystyle\qquad+2L_{[1i]}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}-2A_{ij}N_{ij}-2\mathcal{H}\omega^{2}\bigg{)}+\frac{4\mathcal{H}{\Lambda}}{n-1}=0.$
(2.70)
Note that the cosmological constant appears only in the boost weight $-2$
frame component (2.70) corresponding to the contraction of the Einstein field
equations (2.66) with two frame vectors $\boldsymbol{n}$. It may seem that the
terms containing $\Lambda$ in (2.66) and (2.70) are not in accordance, but the
additional term with $\Lambda$ appears when one rewrites the derivatives of
the Ricci rotation coefficients in (2.66) by means of the Ricci identities
[3].
Following [27], we rewrite the trace of (2.68) as
$2\mathcal{H}L_{ij}L_{ij}-2\left(\mathrm{D}\mathcal{H}+(n-2)\theta\mathcal{H}\right)(n-2)\theta=0,$
(2.71)
using $\mathcal{H}^{-1}\mathrm{D}\mathcal{H}=\mathrm{D}\log\mathcal{H}$ and
the decomposition of the optical matrix $L_{ij}$ (1.22), as
$\displaystyle(n-2)\theta(\mathrm{D}\log\mathcal{H})$
$\displaystyle=L_{ij}L_{ij}-(n-2)^{2}\theta^{2}$
$\displaystyle=\sigma^{2}+\omega^{2}-(n-2)(n-3)\theta^{2}.$ (2.72)
Obviously, $\mathcal{H}$ appears in (2.72) only if $\theta\neq 0$, therefore,
it is natural to study non-expanding GKS spacetimes, where $\theta=0$, and
expanding GKS spacetimes, where $\theta\neq 0$, separately. This will be done
in the following sections 2.5 and 2.6, respectively.
The positive boost weight frame components of the Weyl tensor vanish
identically, as was shown in general for GKS spacetimes with a geodetic
Kerr–Schild vector field $\boldsymbol{k}$ (2.55). Substituting the Ricci
tensor of Einstein spaces (2.65) to the definition of the Weyl tensor (1.31),
one obtains
$C_{abcd}=R_{abcd}-\frac{4\Lambda}{(n-1)(n-2)}g_{a[c}g_{d]b}$ (2.73)
and the non-trivial frame components of the Weyl tensor (2.48)–(2.54) for
Einstein GKS spacetimes read
$\displaystyle C_{0101}$ $\displaystyle=R_{0101}+\frac{2\Lambda}{(n-2)(n-1)},$
(2.74) $\displaystyle C_{01ij}$
$\displaystyle=R_{01ij},\phantom{\frac{1}{()}}$ (2.75) $\displaystyle
C_{0i1j}$ $\displaystyle=R_{0i1j}-\frac{2\Lambda}{(n-2)(n-1)}\delta_{ij},$
(2.76) $\displaystyle C_{ijkl}$
$\displaystyle=R_{ijkl}-\frac{2\Lambda}{(n-2)(n-1)}\left(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\right),$
(2.77) $\displaystyle C_{011i}$ $\displaystyle=R_{011i},\qquad
C_{1ijk}=R_{1ijk},\qquad C_{1i1j}=R_{1i1j},\phantom{\frac{1}{()}}$ (2.78)
where the frame components of the Riemann tensor are given in (2.38)–(2.44).
Note that the terms containing cosmological constant $\Lambda$ in
(2.74)–(2.78) cancel the corresponding terms in the Riemann tensor
(2.38)–(2.44) and therefore the cosmological constant actually does not enter
the frame components of the Weyl tensor.
### 2.5 Non-expanding Kerr–Schild spacetimes
In this section, we will consider the simplest subclass of Einstein GKS
spacetimes where the null Kerr–Schild congruence $\boldsymbol{k}$ is non-
expanding ($\theta=0$). Occasionally, we will also admit an additional aligned
radiation term in the Ricci tensor.
Substituting $\theta=0$ to the trace (2.72) of one of the frame component of
the vacuum Einstein field equations, we immediately see that the sum of
squares of the shear $\sigma$ and twist $\omega$ scalars has to vanish
$\sigma^{2}+\omega^{2}=0$ (2.79)
and therefore $\sigma=\omega=0$. In other words, a non-expanding Kerr–Schild
vector field $\boldsymbol{k}$ is also non-shearing and non-twisting. In fact,
this means that non-expanding Einstein GKS spacetimes belong to the Kundt
class of solutions and the optical matrix vanishes
$L_{ij}=0.$ (2.80)
Vanishing of the optical matrix along with the already applied assumption that
$\boldsymbol{k}$ is geodetic and affinely parametrized, i.e. $L_{a0}=0$,
significantly simplifies our calculations. For instance, the vacuum Einstein
field equations (2.67)–(2.70) reduce to
$\displaystyle\mathrm{D}^{2}\mathcal{H}=0,\phantom{\frac{1}{2}}$ (2.81)
$\displaystyle\delta_{i}(\mathrm{D}\mathcal{H})-2L_{[i1]}\mathrm{D}\mathcal{H}=0,\phantom{\frac{1}{2}}$
(2.82)
$\displaystyle\delta_{i}(\delta_{i}\mathcal{H})+N_{ii}\mathrm{D}\mathcal{H}+\left(4L_{1i}-2L_{i1}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\mathcal{H}$
$\displaystyle\qquad+2\mathcal{H}\left(2\delta_{i}L_{[1i]}+4L_{1i}L_{[1i]}+L_{i1}L_{i1}+2L_{[1i]}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)+\frac{4\mathcal{H}{\Lambda}}{n-1}=0.$ (2.83)
The considerable simplification occurs also when we express the frame
components of the Weyl tensor (2.74)–(2.78) for the non-expanding case
$\displaystyle C_{0i0j}$ $\displaystyle=0,\qquad C_{010i}=0,\qquad
C_{0ijk}=0,\phantom{\frac{1}{2}}$ (2.84) $\displaystyle C_{0101}$
$\displaystyle=\mathrm{D}^{2}\mathcal{H},\qquad C_{01ij}=0,\qquad
C_{0i1j}=0,\phantom{\frac{1}{2}}$ (2.85) $\displaystyle C_{ijkl}$
$\displaystyle=0,\qquad
C_{011i}=-\delta_{i}\left(\mathrm{D}\mathcal{H}\right)+2L_{[i1]}\mathrm{D}\mathcal{H},\qquad
C_{1ijk}=0,\phantom{\frac{1}{2}}$ (2.86) $\displaystyle C_{1i1j}$
$\displaystyle=\delta_{i}(\delta_{j}\mathcal{H})+\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{(ij)}$}\delta_{k}\mathcal{H}+4L_{1(i}\delta_{j)}\mathcal{H}-2L_{(i|1}\delta_{j)}\mathcal{H}+N_{(ij)}\mathrm{D}\mathcal{H}\phantom{\frac{1}{()}}$
$\displaystyle\qquad+2\mathcal{H}\bigg{(}\delta_{(i}L_{1|j)}-2L_{1(i}L_{j)1}+2L_{1i}L_{1j}+L_{1k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{(ij)}$}\bigg{)},$ (2.87)
where the only nontrivial boost weight 0 component $C_{0101}$ and boost weight
$-1$ component $C_{011i}$ vanish due to the Einstein field equations (2.81)
and (2.82). Thus, we can conclude that
###### Proposition 4
Einstein generalized Kerr–Schild spacetimes (2.11) with a non-expanding
Kerr–Schild congruence $\boldsymbol{k}$ are of Weyl type N with
$\boldsymbol{k}$ being the multiple WAND. Twist and shear of the Kerr–Schild
congruence $\boldsymbol{k}$ necessarily vanish and therefore these solutions
belong to the Kundt class of Weyl type N Einstein spacetimes.
Let us point out the relation of non-expanding GKS spacetimes with the VSI and
CSI classes of spacetimes defined and discussed in [36, 4, 37, 38, 39]. It was
shown that a spacetime is VSI, i.e. all curvature invariants of all orders
constructed from the Riemann tensor and its covariant derivatives vanish, if
and only if there exists a non-expanding, non-shearing and non-twisting
congruence of null geodesics along which only the negative boost weight frame
components of the Riemann tensor are non-zero. In fact, the Kundt class of
spacetimes of Weyl types III, N, or O with the Ricci tensor of algebraic types
III, N or O is equivalent to the VSI class and all metrics within these
classes are presented in [38]. On the other hand, the CSI class is defined as
spacetimes for which all scalar invariants constructed from the Riemann tensor
and its covariant derivatives are constant and, as was conjectured in [37], a
CSI spacetime is either locally homogeneous or belongs to the Kundt class.
Obviously, non-expanding Ricci-flat GKS metrics (2.11), i.e. non-expanding KS
metrics (2.1) with the flat background, thus belong to the VSI class of
spacetimes since the Riemann tensor is given exactly by the Weyl tensor (2.73)
which has only the boost weight $-2$ components (2.87) along the Kerr–Schild
vector field $\boldsymbol{k}$ representing a non-expanding, non-shearing and
non-twisting congruence as follows from proposition 4.
Unlike the Ricci-flat case, some of the boost weight zero components
(2.38)–(2.41) of the Riemann tensor of non-expanding Einstein GKS spacetimes
are proportional to the cosmological constant $\Lambda$. Therefore, all
curvature invariants either vanish or are constants depending on $\Lambda$ and
thus these spacetimes belong to the CSI class.
The arbitrariness of the choice of the frame vectors $\boldsymbol{n}$,
$\boldsymbol{m}^{(i)}$ can be used to show that, without loss of generality,
one may set $L_{[1i]}=0$ in the case of Kundt spacetimes $L_{ij}=L_{i0}=0$.
Note that all the following Lorentz transformations preserve both $L_{ij}=0$
and $L_{i0}=0$. Let us assume that $\boldsymbol{k}$ is affinely parametrized.
First, we perform a boost (1.20) to break the affine parametrization, namely
$\hat{L}_{10}=\mathrm{D}\lambda$, in such an appropriate way that will be
clear from the final step. Next, we employ null rotations with
$\boldsymbol{k}$ fixed (1.19) under which $L_{10}$ remains unchanged and
$L_{[1i]}$ transform as $\hat{L}_{[1i]}=L_{[1i]}-\frac{1}{2}z_{i}L_{10}$. This
simply determines the functions $z_{i}$ to set all $\hat{L}_{[1i]}$ to zero.
Furthermore, we are still able to align the spacelike frame vectors
$\boldsymbol{m}^{(i)}$ to $L_{1i}$ by spatial rotations (1.21), where
$\hat{L}_{1i}=X_{ij}L_{1j}$ with $X_{ij}$ being an orthogonal matrix, so that
$L_{1i}$ has just one component, let say $L_{12}=L_{21}\neq 0$,
$L_{1\tilde{\imath}}=L_{\tilde{\imath}1}=0$. Again, $L_{10}$ is not affected
by this operation and therefore we can finally perform the first step
reversely, i.e. $\lambda^{\prime}=\lambda^{-1}$, to recover the affine
parametrization of $\boldsymbol{k}$. But since
$\hat{L}_{[1i]}=L_{[1i]}+\frac{1}{2}\lambda^{\prime-1}\delta_{i}\lambda^{\prime}$,
we now require $\delta_{i}\lambda^{\prime}=0$.
Note that the natural frame (2.95) of VSI metrics (2.94) or even the natural
frame of general Kundt metrics in the canonical form [40] are examples of such
frames with $L_{[1i]}=0$ since $\boldsymbol{\ell}=\mathrm{d}u$ are constant
one-forms.
In the case that $L_{[1i]}=0$, the Einstein field equations (2.81)–(2.83)
further simplify to
$\displaystyle\mathrm{D}^{2}\mathcal{H}=0,\qquad\delta_{i}(\mathrm{D}\mathcal{H})=0,\phantom{\frac{1}{2}}$
(2.88)
$\displaystyle\delta_{i}(\delta_{i}\mathcal{H})+N_{ii}\mathrm{D}\mathcal{H}+\left(2L_{1i}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\mathcal{H}+2\mathcal{H}L_{i1}L_{i1}+\frac{4\mathcal{H}{\Lambda}}{n-1}=0.$
(2.89)
One may integrate (2.88) to determine the $r$-dependence of the Kerr–Schild
function $\mathcal{H}$
$\mathcal{H}=f^{(0)}r+g^{(0)},$ (2.90)
where $f^{(0)}$ and $g^{(0)}$ are functions not depending on $r$ subject to
$\delta_{i}f^{(0)}=0$ and it remains only to satisfy (2.89). Recall that $r$
is an affine parameter along the null geodesics $\boldsymbol{k}$.
Note also that the above statement of proposition 4 remains valid if we admit
an additional aligned null radiation term in the Ricci tensor
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab}+\Phi k_{a}k_{b}.$ (2.91)
Since the Ricci tensor (2.91) differs from the case of Einstein spaces (2.65)
just in the frame component $R_{11}$, the aligned null radiation term appears
only on the right hand side of the frame component (2.83) of the Einstein
field equations
$\begin{split}&\delta_{i}(\delta_{i}\mathcal{H})+N_{ii}\mathrm{D}\mathcal{H}+\left(4L_{1i}-2L_{i1}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\mathcal{H}\\\
&\qquad+2\mathcal{H}\left(2\delta_{i}L_{[1i]}+4L_{1i}L_{[1i]}+L_{i1}L_{i1}+2L_{[1i]}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)+\frac{4\mathcal{H}{\Lambda}}{n-1}=\Phi\end{split}$
(2.92)
and as one can immediately see from (2.45)–(2.54), only the boost weight $-2$
frame components of the Weyl tensor $C_{1i1j}$ depend on $R_{11}$. Therefore,
the null radiation term does not affect the derivation of proposition 4.
#### 2.5.1 Examples of non-expanding Einstein generalized
Kerr–Schild spacetimes
Let us recall the statement of proposition 4 that all non-expanding Einstein
GKS spacetimes belong to the Kundt class. As mentioned above, this also holds
for non-expanding GKS spacetimes with null radiation aligned with the
Kerr–Schild vector $\boldsymbol{k}$. Geometrically, the Kundt class of
solutions is defined as spacetimes admitting a geodetic, non-expanding, non-
shearing and non-twisting null congruence generated by a null vector field
that will be represented by the Kerr–Schild vector $\boldsymbol{k}$ in the
case of Kundt GKS spacetimes.
In four dimensions, the Goldberg–Sachs theorem implies that Kundt spacetimes
possibly with a cosmological constant or aligned matter fields are
algebraically special, i.e. of Petrov type II or more special, and the
geodetic non-expanding, non-shearing and non-twisting null congruence
corresponds with the PND [9].
Analogically, it was shown in [3] that higher dimensional Kundt spacetimes
with the vanishing positive boost weight frame components of the Ricci tensor,
$R_{00}=R_{0i}=0$, that admit cosmological constant $\Lambda$ and aligned
matter content, are of Weyl type II or more special again with the geodetic
null congruence being the WAND. In fact, $R_{00}$ vanishes identically for a
Kundt metric and if $R_{0i}\neq 0$, then the spacetime is of Weyl type I [40].
The metric of general $n$-dimensional Kundt spacetimes can be expressed in the
canonical form [38, 40]
$\mathrm{d}s^{2}=2\mathrm{d}u\left[\mathrm{d}v+H(u,v,x^{k})\,\mathrm{d}u+W_{i}(u,v,x^{k})\,\mathrm{d}x^{i}\right]+g_{ij}(u,x^{k})\,\mathrm{d}x^{i}\,\mathrm{d}x^{j},$
(2.93)
where the coordinate $v$ corresponds to an affine parameter along the
geodesics of the non-expanding, non-shearing and non-twisting null congruence
$\boldsymbol{k}=\partial_{v}$ and the transverse metric $g_{ij}$ does not
depend on $v$. It should be emphasized that we denote the transverse spatial
coordinates as $x^{2},\ldots,x^{n-1}$ since in our convention $i$, $j$ range
from 2 to $n-1$ in contrast with [38].
In general, Kundt spacetimes do not admit the GKS form (2.11). This follows
directly from the fact that, without any conditions on the Ricci tensor, the
Kundt metrics are of Weyl type I or more special, whereas the GKS metrics with
geodetic $\boldsymbol{k}$ are of Weyl type II or more special. Even in the
case of Einstein spaces, there exist, for instance, type III Einstein Kundt
spacetimes which are, by proposition 4, incompatible with the GKS form.
However, it can be shown that all Weyl type N VSI metrics written in
appropriate coordinates with a flat transverse space as [38]
$\mathrm{d}s^{2}=2\mathrm{d}u\left[\mathrm{d}v+H(u,v,x^{k})\,\mathrm{d}u+W_{i}(u,v,x^{k})\,\mathrm{d}x^{i}\right]+\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j},$
(2.94)
where $H$ and $W_{i}$ satisfy certain conditions, admit the KS form (2.1). In
fact, the class of type N VSI metrics is equivalent to the class of Weyl type
N Ricci-flat Kundt spacetimes. Recall that we use the different convention for
indices of $W_{i}$ and $x^{k}$. Whereas in [38] indices $i,j,\ldots$ ranges
from $1$ to $n-2$, throughout the thesis we consistently use $i,j,\ldots$
running from $2$ to $n-1$.
Obviously, one may introduce a natural null frame in the VSI spacetime (2.94)
$\displaystyle\ell_{a}\,\mathrm{d}x^{a}=\mathrm{d}u,\qquad
n_{a}\,\mathrm{d}x^{a}=\mathrm{d}v+H\,\mathrm{d}u+W_{i}\,\mathrm{d}x^{i},\qquad
m^{(i)}_{a}\,\mathrm{d}x^{a}=\mathrm{d}x^{i},$ (2.95)
$\displaystyle\ell^{a}\,\partial_{a}=\partial_{v},\qquad
n^{a}\,\partial_{a}=\partial_{u}-H\,\partial_{v},\qquad
m_{(i)}^{a}\,\partial_{a}=\partial_{i}-W_{i}\,\partial_{v}.$
Then, it immediately follows from (2.94) and (2.95) that
$L_{1i}=\frac{1}{2}W_{i,v},\qquad L_{[1i]}=0,\qquad L_{11}=H_{,v}$ (2.96)
and all other components of $L_{ab}$ are zero. Using the notation of [38], the
VSI class can be divided into two distinct subclasses with vanishing
($\epsilon=0$) and non-vanishing ($\epsilon=1$) quantity $L_{1i}L_{1i}$,
respectively. The canonical choice of the functions $W_{i}$ is
$W_{2,v}=-2\frac{\epsilon}{x^{2}},\qquad W_{\tilde{\imath},v}=0.$ (2.97)
One may express the constraints imposed on the undetermined functions $H$ and
$W_{i}$ following from the Einstein field equations and from the condition on
the form of the Weyl tensor. Namely, for Ricci-flat spacetimes of Weyl type N,
in the case $\varepsilon=0$, we obtain [38]
$\begin{split}&W_{2}=0,\qquad
W_{\tilde{\imath}}=x^{2}C_{\tilde{\imath}}(u)+x^{\tilde{\jmath}}B_{\tilde{\jmath}\tilde{\imath}}(u),\phantom{\frac{1}{2}}\\\
&H=H^{0}(u,x^{i}),\qquad\Delta H^{0}-\frac{1}{2}\sum
C^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}=0,\end{split}$
(2.98)
whereas, in the case $\varepsilon=1$, one gets
$\begin{split}&W_{2}=-\frac{{{2}}v}{x^{2}},\qquad
W_{\tilde{\imath}}=C_{\tilde{\imath}}(u)+x^{\tilde{\jmath}}B_{\tilde{\jmath}\tilde{\imath}}(u),\qquad
H=\frac{v^{2}}{2(x^{2})^{2}}+H^{0}(u,x^{i}),\\\
&x^{2}\Delta\left(\frac{H^{0}}{x^{2}}\right)-\frac{1}{(x^{2})^{2}}\sum
W^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}=0,\end{split}$
(2.99)
where $B_{[\tilde{\imath}\tilde{\jmath}]}=0$ in both cases and
$\tilde{\imath}$, $\tilde{\jmath}=3,\dots,n-1$. Similarly, one may also find
$H^{0}(u,v,x^{k})$ such that the VSI metric (2.94) with $H=H^{0}$ is flat [38]
$\displaystyle\epsilon=0:\qquad H^{0}_{\text{flat}}$
$\displaystyle=\frac{1}{2}x^{1}x^{\tilde{\imath}}(C_{\tilde{\imath},u}+B_{\tilde{\imath}\tilde{\jmath}}C_{\tilde{\jmath}})+\frac{1}{2}B_{\tilde{\imath}\tilde{k}}B_{\tilde{\jmath}\tilde{k}}x^{\tilde{\imath}}x^{\tilde{\jmath}}+x^{i}F_{i}(u)$
$\displaystyle\qquad+\frac{1}{8}\left(\sum
C^{2}_{\tilde{\imath}}(x^{2})^{2}+\sum_{\tilde{\imath}\leq\tilde{\jmath}}C_{\tilde{\imath}}C_{\tilde{\jmath}}x^{\tilde{\imath}}x^{\tilde{\jmath}}\right),$
(2.100) $\displaystyle\epsilon=1:\qquad H^{0}_{\text{flat}}$
$\displaystyle=\frac{1}{2}\sum
W_{m}^{2}-\frac{1}{16}+x^{1}F_{0}(u)+x^{1}x^{i}F_{i}(u),$ (2.101)
where $F_{0}(u)$, $F_{i}(u)$ are arbitrary functions of $u$ and $W_{i}$ are
given as for type N (2.98), (2.99). The one-form $\mathrm{d}u$ is associated
with the geodetic null vector field $\partial_{v}$ and therefore all Weyl type
N VSI metrics (2.94) can be written in the KS form (2.1) as
$\mathrm{d}s^{2}=\mathrm{d}s^{2}_{\text{flat}}+\left(H^{0}-H^{0}_{\text{flat}}\right)\mathrm{d}u^{2}.$
(2.102)
Since a metric of general higher dimensional type N Einstein Kundt spacetimes
has not been given explicitly in the literature yet, we cannot simply follow
the above procedure in the case of such metrics. This prevents us from
answering the question whether the implication stated in proposition 4 is
valid also in the opposite direction, i.e. whether there is an equivalency
between the classes of type N Einstein Kundt spacetimes and non-expanding
Einstein GKS spacetimes.
However, we are still able to show, using the results of [41, 42], that at
least all four-dimensional type N Einstein Kundt metrics admit the GKS form.
The metric of type N Kundt spacetimes admitting cosmological constant and
possibly containing pure radiation can be expressed as [41]
$\begin{split}\mathrm{d}s^{2}&=-2\frac{Q^{2}}{P^{2}}\,\mathrm{d}u\,\mathrm{d}v+\left(2k\frac{Q^{2}}{P^{2}}v^{2}-\frac{\left(Q^{2}\right)_{,u}}{P^{2}}v-\frac{Q}{P}H\right)\mathrm{d}u^{2}\\\
&\qquad+\frac{1}{P^{2}}\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}\right),\end{split}$
(2.103)
where
$\begin{split}&P=1+\frac{\Lambda}{12}(x^{2}+y^{2}),\\\
&k=\frac{\Lambda}{6}\alpha(u)^{2}+\frac{1}{2}\left(\beta(u)^{2}+\gamma(u)^{2}\right),\\\
&Q=\left(1-\frac{\Lambda}{12}(x^{2}+y^{2})\right)\alpha(u)+\beta(u)x+\gamma(u)y,\end{split}$
(2.104)
with $\alpha(u)$, $\beta(u)$ and $\gamma(u)$ being arbitrary functions of the
coordinate $u$ and $H=H(x,y,u)$. These spacetimes are Einstein if
$P^{2}(H_{,xx}+H_{,yy})+\frac{2}{3}\Lambda H=0,$ (2.105)
which has a general solution [42]
$H=2f_{1,x}-\frac{\Lambda}{3P}(xf_{1}+yf_{2}),$ (2.106)
where the functions $f_{1}=f_{1}(u,x,y)$ and $f_{2}=f_{2}(u,x,y)$ are subject
to $f_{1,x}=f_{2,y}$, $f_{1,y}=-f_{2,x}$. It can be shown that the Einstein
metrics (2.103), (2.106) are conformally flat for
$H(x,y,u)=\frac{1}{P}\left(A\left(1-\frac{\Lambda}{12}(x^{2}+y^{2})\right)+Bx+Cy\right),$
(2.107)
where $A(u)$, $B(u)$ and $C(u)$ are arbitrary functions of $u$. Therefore, all
four-dimensional type N Kundt metrics (2.103) differ from the conformally flat
case only by a factor of $\mathrm{d}u^{2}$ and thus such metrics take the GKS
form (2.11).
#### 2.5.2 Warped Einstein Kundt generalized Kerr–Schild
spacetimes
In the previous section 2.5.1, we have presented explicitly known Weyl type N
Einstein Kundt metrics in order to show that they can be cast to the GKS form.
However, these metrics are either only four-dimensional but admitting
cosmological constant or arbitrary dimensional but only Ricci-flat. In this
section, we employ these metrics again in order to construct examples of
higher dimensional Einstein Kundt spacetimes belonging to the GKS class with
an almost arbitrary cosmological constant by means of the Brinkmann warp
product.
As discussed in sections 1.4 and 2.3, the Brinkmann warp product (1.37) allows
us to generate new $n$-dimensional Einstein GKS metrics $\mathrm{d}s^{2}$
(2.57) from known $(n-1)$-dimensional Einstein GKS seed metrics
$\mathrm{d}\tilde{s}^{2}$ in the form (2.56). The cosmological constant of the
warped metric $\mathrm{d}s^{2}$ is not completely arbitrary since its sign
depends on the sign of the cosmological constant of the seed metric
$\mathrm{d}\tilde{s}^{2}$. The allowed combinations of these signs were
discussed in section 1.4. Let us also recall that the Brinkmann warp product
preserves the Weyl type of algebraically special spacetimes.
First, let us choose the $n$-dimensional type N Ricci-flat Kundt metric
(2.94), (2.98) and (2.99), i.e. type N subclass of VSI spacetimes, as a seed
$\mathrm{d}\tilde{s}^{2}$. The sign of the Ricci scalar $R$ of the warped
metric $\mathrm{d}s^{2}$ may be zero or negative. Omitting the trivial case of
the direct product, we thus construct $(n+1)$-dimensional type N Einstein GKS
metrics with a negative cosmological constant. One may use (1.44) to cast such
metrics to the form conformal to the direct product
$\begin{split}\mathrm{d}s^{2}&=\frac{1}{-\lambda\tilde{z}^{2}}\Big{(}2\,\mathrm{d}u\left[\mathrm{d}v+H(u,v,x^{k})\,\mathrm{d}u+W_{i}(u,v,x^{k})\,\mathrm{d}x^{i}\right]\\\
&\qquad+\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}+\mathrm{d}\tilde{z}^{2}\Big{)},\phantom{\frac{1}{2}}\end{split}$
(2.108)
where $i,j=2,\dots,n-1$. Performing the coordinate transformation
$v=-\lambda\tilde{v}\tilde{z}^{2}$, we easily put the above metric to the
canonical Kundt form (2.93)
$\mathrm{d}s^{2}=2\,\mathrm{d}u\left[\mathrm{d}\tilde{v}+\tilde{H}\,\mathrm{d}u+\tilde{W}_{\tilde{\imath}}\,\mathrm{d}x^{\tilde{\imath}}\right]+\frac{1}{-\lambda\tilde{z}^{2}}\delta_{\tilde{\imath}\tilde{\jmath}}\,\mathrm{d}x^{\tilde{\imath}}\,\mathrm{d}x^{\tilde{\jmath}},$
(2.109)
with $\tilde{\imath},\tilde{\jmath}=2,\dots,n$ and
$\begin{split}&\tilde{H}=\frac{1}{-\lambda\tilde{z}^{2}}H(u,v,x^{k}),\qquad\mathrm{d}x^{n}=\mathrm{d}\tilde{z},\\\
&\tilde{W}_{i}=\frac{1}{-\lambda\tilde{z}^{2}}W_{i}(u,v,x^{k}),\qquad\tilde{W}_{n}=\frac{2\tilde{v}}{\tilde{z}}.\end{split}$
(2.110)
In fact, metrics (2.108), (2.109) were already discussed in [37, 39] in the
context of CSI spacetimes and supergravity.
Although one may apply the Brinkmann warp product multiple times to obtain
further and further solutions it does not lead to new results in this case.
Since the $(n+1)$-dimensional metric (2.108) is of the form
$\mathrm{d}s^{2}=\frac{1}{-\lambda
z^{2}}\left(\mathrm{d}z^{2}+g^{(0)}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}\right),$
(2.111)
with
$g^{(0)}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}=2\,\mathrm{d}u\left[\mathrm{d}v+H\,\mathrm{d}u+W_{i}\,\mathrm{d}x^{i}\right]+\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}$
not depending on the coordinate $z$. The second application of the warp
product (1.37) gives
$\mathrm{d}s^{\prime
2}=\frac{\mathrm{d}w^{2}}{f(w)}+f(w)\left(\frac{1}{-\lambda
z^{2}}\left(\mathrm{d}z^{2}+g^{(0)}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}\right)\right),$
(2.112)
where the function $f(w)$ is defined in (1.38). The seed metric (2.111) has a
negative cosmological constant and therefore the only possibility is that the
cosmological constant of the warped metric (2.112) is negative as well. Then
the coordinate transformation (2.61), where we replace $x_{1}\rightarrow z$,
$z\rightarrow w$ and $\lambda\rightarrow\lambda^{\prime}$, along with the
relation $\lambda=\lambda^{\prime}b+d^{2}$ following from (1.39) yields
$\displaystyle\frac{f(w)}{-\lambda
z^{2}}=\frac{1}{-\lambda^{\prime}\tilde{z}^{2}},$ (2.113)
$\displaystyle\frac{\mathrm{d}w^{2}}{f(w)}+\frac{f(w)}{-\lambda
z^{2}}\mathrm{d}z^{2}=\frac{1}{-\lambda^{\prime}\tilde{z}^{2}}\left(\mathrm{d}\tilde{w}^{2}+\mathrm{d}\tilde{z}^{2}\right).$
(2.114)
Substituting (2.113) and (2.114) to (2.112) immediately leads to
$\mathrm{d}s^{\prime
2}=\frac{1}{-\lambda^{\prime}\tilde{z}^{2}}\left(g^{(0)}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}+\mathrm{d}\tilde{w}^{2}+\mathrm{d}\tilde{z}^{2}\right)$
(2.115)
and using $v=-\lambda^{\prime}\tilde{v}\tilde{z}^{2}$, we finally arrive at
(2.109) with (2.110) where moreover
$\begin{split}W_{(n+1)}=0,\qquad\mathrm{d}x^{(n+1)}=\mathrm{d}\tilde{w}.\end{split}$
(2.116)
Therefore, we obtained the same class of $(n+2)$-dimensional metrics as in the
case of applying the warp product on the subclass of $(n+1)$-dimensional
metrics (2.94) with $W_{n}=0$ and then swaps $n\leftrightarrow(n+1)$.
So far we have used only the type N Ricci-flat VSI metrics (2.94) as a seed
and thus we constructed higher dimensional type N Einstein Kundt GKS metrics
with a negative cosmological constant. However, one can also warp the four-
dimensional Einstein Kundt metrics (2.103). In this case, there are more
possible combinations of the signs of the Ricci scalars $\tilde{R}$ and $R$ of
the seed and warped metric, respectively. Recall that only the case with both
Ricci scalars being zero or negative is free from curvature or parallelly
propagated singularities at a point where $f(z)=0$.
Such five-dimensional warped metrics can be expressed, for instance, either in
the form conformal to a direct product using (1.40)–(1.45) as in the previous
case or directly in the GKS form using (2.57), (2.60) and the coordinate
transformations (2.61)–(2.64). Here, we use the latter approach.
First, we have to split the four-dimensional type N Kundt metric (2.103) into
the background (anti-)de Sitter or Minkowski metric $\bar{g}_{ab}$ and the
Kerr–Schild term $\mathcal{H}k_{a}k_{b}$. The background metric can be
obtained as a weak-field limit of (2.103) [43]
$\mathrm{d}s^{2}=-2\frac{Q^{2}}{P^{2}}\,\mathrm{d}u\,\mathrm{d}v+2k\frac{Q^{2}}{P^{2}}v^{2}\,\mathrm{d}u^{2}+\frac{1}{P^{2}}\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}\right).$
(2.117)
Therefore, the four-dimensional type N Kundt metric (2.103) can be
straightforwardly cast to the GKS form simply by reordering the terms
$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-2\frac{Q^{2}}{P^{2}}\,\mathrm{d}u\,\mathrm{d}v+2k\frac{Q^{2}}{P^{2}}v^{2}\,\mathrm{d}u^{2}+\frac{1}{P^{2}}\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}\right)$
(2.118)
$\displaystyle\qquad-\left(\frac{\left(Q^{2}\right)_{,u}}{P^{2}}v+\frac{Q}{P}\mathcal{H}\right)\mathrm{d}u^{2},$
where the functions $P$, $Q$ and $k$ are given in (2.104) and, obviously, the
last term corresponds to the Kerr-Schild term $2\mathcal{H}k_{a}k_{b}$. These
seed metrics with a four-dimensional cosmological constant denoted as
$\tilde{\Lambda}$ can be split to several geometrically distinct classes [43].
Depending on whether $\tilde{\Lambda}$ and $k$ is positive, negative or
vanishing, we will denote such possible subclasses as
KN($\tilde{\Lambda}^{+}$, $k^{+}$), KN($\tilde{\Lambda}^{0}$, $k^{+}$),
KN($\tilde{\Lambda}^{0}$, $k^{0}$), KN($\tilde{\Lambda}^{-},k^{+}$),
KN($\tilde{\Lambda}^{-},k^{-}$) and KN($\tilde{\Lambda}^{-},k^{0}$),
respectively.
Five-dimensional Ricci-flat metrics obtained from KN($\tilde{\Lambda}^{+}$,
$k^{+}$) KN($\tilde{\Lambda}^{0}$, $k^{+}$) and KN($\tilde{\Lambda}^{0}$,
$k^{0}$) belong to the VSI class and the warped metrics constructed using the
Ricci-flat seeds from KN($\tilde{\Lambda}^{0}$, $k^{+}$) and
KN($\tilde{\Lambda}^{0}$, $k^{0}$) are already contained in the class of
warped VSI metrics (2.108). Therefore, we restrict ourselves to the cases with
non-vanishing cosmological constants of the seed and warped metrics. Here, we
do not present the warped metrics explicitly, instead, we give the coordinate
transformations putting the corresponding background metric (2.117) to the
canonical form. Then, one may perform the warp product (2.57) and employ
(2.61)–(2.64) to cast the warped background metric (2.60) back to the
canonical form.
##### Generalized Kundt waves KN($\tilde{\Lambda}^{-},k^{+}$)
The Kundt metric (2.103) with the canonical choice $\alpha=0$,
$\beta=\sqrt{2}$, $\gamma=0$, where the functions $Q$ and $k$ are given by the
functions $\alpha$, $\beta$ and $\gamma$ via (2.104)
$Q={\sqrt{2}}x,\qquad k=1,$ (2.119)
represents generalized Kundt waves KN($\tilde{\Lambda}^{-},k^{+}$). One may
put the anti-de Sitter background metric to the canonical form by means of the
coordinate transformation
$\displaystyle u$
$\displaystyle=\frac{Y\mp\sqrt{T^{2}-X^{2}-Z^{2}}}{a},\qquad$ $\displaystyle
T$ $\displaystyle=\frac{a^{2}\left(2-P\right)}{2xv},$ (2.120) $\displaystyle
v$ $\displaystyle=\pm\frac{a}{2\sqrt{T^{2}-X^{2}-Z^{2}}},$ $\displaystyle X$
$\displaystyle=\frac{a^{2}P}{2xv},$ $\displaystyle x$
$\displaystyle=\pm\frac{2a\sqrt{T^{2}-X^{2}-Z^{2}}}{X+T},$ $\displaystyle Y$
$\displaystyle=\frac{a\left(1+2uv\right)}{2v},$ $\displaystyle y$
$\displaystyle=\frac{2aZ}{X+T},$ $\displaystyle Z$
$\displaystyle=\frac{ay}{2xv},$
where $a=\sqrt{-3/\tilde{\Lambda}}$.
##### Generalized pp -waves KN($\tilde{\Lambda}^{-},k^{-}$)
The subclass KN($\tilde{\Lambda}^{-},k^{-}$) which generalizes pp -waves can
be described by the canonical choice $\alpha=1$, $\beta=0$, $\gamma=0$ leading
to
$Q=1-\frac{\tilde{\Lambda}}{12}(x^{2}+y^{2}),\qquad
k=\frac{\tilde{\Lambda}}{6}.$ (2.121)
In this case, the anti-de Sitter background metric can be cast to the
canonical form using the coordinate transformation
$\displaystyle u$
$\displaystyle=\sqrt{2}\left(\pm\sqrt{X^{2}+Y^{2}+Z^{2}}-T\right),\qquad$
$\displaystyle T$ $\displaystyle=\sqrt{2}\frac{a^{2}-uv}{2v},$ (2.122)
$\displaystyle v$
$\displaystyle=\pm\frac{a^{2}}{\sqrt{2}\sqrt{X^{2}+Y^{2}+Z^{2}}},$
$\displaystyle X$ $\displaystyle=\frac{a^{2}P}{\sqrt{2}Qv},$ $\displaystyle x$
$\displaystyle=\frac{2aZ}{X\pm\sqrt{X^{2}+Y^{2}+Z^{2}}},$ $\displaystyle Y$
$\displaystyle=\frac{ax}{\sqrt{2}Qv},$ $\displaystyle y$
$\displaystyle=\frac{2aY}{X\pm\sqrt{X^{2}+Y^{2}+Z^{2}}},$ $\displaystyle Z$
$\displaystyle=\frac{ay}{\sqrt{2}Qv},$
where again $a=\sqrt{-3/\tilde{\Lambda}}$.
##### Generalized Siklos waves KN($\tilde{\Lambda}^{-},k^{0}$)
The last subclass with a negative cosmological constant $\tilde{\Lambda}$,
generalized Siklos waves KN($\tilde{\Lambda}^{-},k^{0}$), is determined by the
canonical choice $\alpha=1$, $\beta=\sqrt{-\Lambda/3}\cos\theta$ and
$\gamma=\sqrt{-\Lambda/3}\sin\theta$. The coordinate transformation
$\displaystyle u$
$\displaystyle=\frac{-T^{2}+X^{2}+Y^{2}+Z^{2}}{\sqrt{2}\left(T+Y\right)},\qquad$
$\displaystyle T$
$\displaystyle=\frac{1}{\sqrt{2}v}\left(a^{2}-uv-\frac{ax}{Q}\right),$ (2.123)
$\displaystyle v$ $\displaystyle=\frac{a^{2}}{\sqrt{2}\left(T+Y\right)},$
$\displaystyle X$ $\displaystyle=\frac{a^{2}P}{\sqrt{2}Qv},$ $\displaystyle x$
$\displaystyle=\sqrt{2}a\frac{(T+Y)^{2}-X^{2}-Z^{2}}{(T+X+Y)^{2}+Z^{2}},$
$\displaystyle Y$ $\displaystyle=\frac{u}{\sqrt{2}}+\frac{ax}{\sqrt{2}Qv},$
$\displaystyle y$ $\displaystyle=2\sqrt{2}a\frac{Z(T+Y)}{(T+X+Y)^{2}+Z^{2}},$
$\displaystyle Z$ $\displaystyle=\frac{ay}{\sqrt{2}Qv},$
with $a=\sqrt{-3/\tilde{\Lambda}}$, brings the background metric to the
canonical form. Note that, in the special subcase when $\theta$ is independent
of $u$, such metrics are equivalent to the Siklos metric [43]
$\mathrm{d}s^{2}=\frac{3}{-\Lambda
x^{2}}\left(\mathrm{d}u\,\mathrm{d}r+H\,\mathrm{d}u^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}\right).$
(2.124)
Obviously, since the Siklos metric (2.124) is conformal to pp -waves, one may
obtain the same five-dimensional metric either by warping the Siklos metric
(2.124) and using (2.61) or by warping pp -waves and using (2.64). Thus, one
may perhaps conjecture that appropriate seeds from KN($\tilde{\Lambda}^{-}$,
$k^{0}$) and KN($\tilde{\Lambda}^{0}$, $k^{0}$) may lead to the same warped
metric.
##### Generalized Kundt and pp -waves KN($\Lambda^{+}$, $k^{+}$)
In the case with a positive cosmological constant $\tilde{\Lambda}$, there is
only one canonical subclass KN($\Lambda^{+}$, $k^{+}$). The canonical choice
is either $\alpha=0$, $\beta=\sqrt{2}$, $\gamma=0$ or $\alpha=1$, $\beta=0$,
$\gamma=0$, where both choices are equivalent and correspond to generalized
Kundt or pp -waves. The functions $Q$ and $k$ are given in (2.119) and
(2.121), respectively. The de Sitter background metric can be cast to the
canonical form using the transformation
$\displaystyle u$
$\displaystyle=\frac{X\mp\sqrt{T^{2}-Y^{2}-Z^{2}}}{a},\qquad$ $\displaystyle
T$ $\displaystyle=\frac{a^{2}P}{2xv},$ (2.125) $\displaystyle v$
$\displaystyle=\pm\frac{a}{2\sqrt{T^{2}-Y^{2}-Z^{2}}},$ $\displaystyle X$
$\displaystyle=\frac{a\left(1+2uv\right)}{2v},$ $\displaystyle x$
$\displaystyle=\pm\frac{2a\sqrt{T^{2}-Y^{2}-Z^{2}}}{T+Z},$ $\displaystyle Y$
$\displaystyle=\frac{ay}{2xv},$ $\displaystyle y$
$\displaystyle=\frac{2aY}{T+Z},$ $\displaystyle Z$
$\displaystyle=\frac{a^{2}\left(2-P\right)}{2xv},$
in the case of generalized Kundt waves or
$\displaystyle u$
$\displaystyle=\sqrt{2}\left(Z\mp\sqrt{T^{2}-X^{2}-Y^{2}}\right),\qquad$
$\displaystyle T$ $\displaystyle=\frac{a^{2}P}{\sqrt{2}Qv},$ (2.126)
$\displaystyle v$
$\displaystyle=\pm\frac{a^{2}}{\sqrt{2}\sqrt{T^{2}-X^{2}-Y^{2}}},$
$\displaystyle X$ $\displaystyle=\frac{ax}{\sqrt{2}Qv},$ $\displaystyle x$
$\displaystyle=\frac{2aX}{T\pm\sqrt{T^{2}-X^{2}-Y^{2}}},$ $\displaystyle Y$
$\displaystyle=\frac{ay}{\sqrt{2}Qv},$ $\displaystyle y$
$\displaystyle=\frac{2aY}{T\pm\sqrt{T^{2}-X^{2}-Y^{2}}},$ $\displaystyle Z$
$\displaystyle=\sqrt{2}\frac{a^{2}+uv}{2v},$
in the case of generalized pp -waves, respectively, where
$a=\sqrt{3/\tilde{\Lambda}}$.
### 2.6 Expanding Kerr–Schild spacetimes
In the previous section 2.5, we have discussed the consequences of the
Einstein field equations for non-expanding ($\theta=0$) Einstein GKS
spacetimes. In that case, the Kerr–Schild function $\mathcal{H}$ does not
enter the trace (2.72) of the components (2.68) of the Einstein field
equations, which then implies that such spacetimes belong to the Kundt class.
In the expanding case ($\theta\neq 0$), on the other hand, we can immediately
express $\mathrm{D}\log\mathcal{H}$ from the trace (2.72) as
$\mathrm{D}\log\mathcal{H}=\frac{L_{ik}L_{ik}}{\theta(n-2)}-(n-2)\theta.$
(2.127)
#### 2.6.1 Optical constraint
Substituting (2.127) back to (2.68) eliminates the Kerr–Schild function
$\mathcal{H}$ and therefore we obtain purely geometrical condition on the
geodetic Kerr–Schild congruence $\boldsymbol{k}$
$L_{ik}L_{jk}=\frac{L_{lk}L_{lk}}{(n-2)\theta}S_{ij},$ (2.128)
referred to as “the optical constraint” [27]. Suppressing indices in matrix
notation, the optical constraint (2.128) reads
$\mathbf{L}\mathbf{L}^{T}=\alpha\mathbf{S}.$ (2.129)
It can be easily shown that $\mathbf{L}$ satisfying the optical constraint is
a normal matrix since (2.129) can be rewritten as [44]
$\left(\mathbf{1}-2\alpha^{-1}\mathbf{L}\right)\left(\mathbf{1}-2\alpha^{-1}\mathbf{L}^{T}\right)=\mathbf{1},$
(2.130)
where, obviously, the matrix $\mathbf{1}-2\alpha^{-1}\mathbf{L}$ is unitary,
one may swap the terms on the left hand side of (2.130) to obtain an
equivalent relation which then leads to the optical constraint in the form
$\mathbf{L}^{T}\mathbf{L}=\alpha\mathbf{S}.$ (2.131)
Therefore, comparing (2.129) with (2.131), the optical matrix $\mathbf{L}$
commutes with its transpose and indeed it is a normal matrix.
A real matrix $\mathbf{M}$ is normal if and only if there is a real orthogonal
matrix $\mathbf{O}$ such that [45]
$\mathbf{O}^{T}\mathbf{M}\mathbf{O}=\begin{pmatrix}\mathcal{M}_{1}&&&\\\
&\mathcal{M}_{2}&&\\\ &&\ddots&\\\ &&&\mathcal{M}_{k}\end{pmatrix},$ (2.132)
where $\mathcal{M}_{i}$ is either a real $1\times 1$ matrix or a real $2\times
2$ matrix of the form
$\mathcal{M}_{i}=\begin{pmatrix}s_{i}&a_{i}\\\ -a_{i}&s_{i}\end{pmatrix}.$
(2.133)
This implies that the optical matrix $L_{ij}$ can be put into the block-
diagonal form (2.132), (2.133) by appropriate spins (1.12) of the frame.
Furthermore, such a canonical frame is compatible with parallel transport
along the geodetic null congruence $\boldsymbol{k}$ [46].
The sparse structure of the optical matrix $L_{ij}$ considerably simplifies
the determination of its dependence on the affine parameter $r$ along null
geodesics $\boldsymbol{k}$ from the Sachs equation (1.30). Due to the block-
diagonal form (2.132), (2.133) of the optical matrix $L_{ij}$ in the canonical
frame, one may integrate (1.30). In the case of a block corresponding to the
$1\times 1$ matrix, we get
$L_{(i)(i)}=\frac{1}{r+b^{0}_{i}},$ (2.134)
whereas, for a block consisting of the $2\times 2$ matrix, the Sachs equation
implies
$\begin{split}L_{i,i+1}&=-L_{i+1,i}=\frac{(a^{0}_{i})^{2}}{(r+b^{0}_{i})^{2}+(a^{0}_{i})^{2}},\\\
L_{(i)(i)}&=L_{(i+1)(i+1)}=\frac{r+b^{0}_{i}}{(r+b^{0}_{i})^{2}+(a^{0}_{i})^{2}},\end{split}$
(2.135)
where $a^{0}$ and $b^{0}_{i}$ are arbitrary functions not depending on $r$.
Putting (2.134) and (2.135) to the optical constraint (2.129), we obtain
$\alpha(r+b^{0}_{i})=1,$ (2.136)
for all values of the index $i$. Therefore, all functions $b^{0}_{i}$ are
equal and, without loss of generality, we may set them to zero. Thus, we may
conclude that the optical matrix $L_{ij}$ of expanding Einstein GKS spacetimes
in a canonical frame takes the block-diagonal form
$\displaystyle
L_{ij}=\left(\begin{array}[]{cccc}\framebox{$\mathcal{L}_{(1)}$}&&&\\\
&\ddots&&\\\ &&\framebox{$\mathcal{L}_{(p)}$}&\\\
&&&\framebox{$\begin{array}[]{ccc}&&\\\ &\tilde{\mathcal{L}}&\\\
&&\end{array}$}\end{array}\right),$ (2.144)
with $2\times 2$ blocks $\mathcal{L}_{(1)},\dots,\mathcal{L}_{(p)}$ of the
form
$\mathcal{L}_{(\mu)}=\left(\begin{array}[]{cc}s_{(2\mu)}&A_{2\mu,2\mu+1}\\\
-A_{2\mu,2\mu+1}&s_{(2\mu)}\end{array}\right),\qquad\mu=1,\ldots,p,$ (2.145)
where the corresponding symmetric diagonal and anti-symmetric anti-diagonal
parts are given by
$s_{(2\mu)}=\frac{r}{r^{2}+(a^{0}_{(2\mu)})^{2}},\qquad
A_{2\mu,2\mu+1}=\frac{a^{0}_{(2\mu)}}{r^{2}+(a^{0}_{(2\mu)})^{2}},$ (2.146)
respectively. The remaining block $\tilde{\mathcal{L}}$ is
$(n-2-2p)\times(n-2-2p)$ diagonal matrix
$\tilde{\mathcal{L}}=\frac{1}{r}\text{diag}(\underbrace{1,\ldots,1}_{(m-2p)},\underbrace{0,\ldots,0}_{(n-2-m)}),$
(2.147)
where $m$ and $n-2$ denote the rank and dimension of the optical matrix
$L_{ij}$ and $p$ corresponds to the number of $2\times 2$ blocks. Clearly,
these quantities are subject to the relation $0\leq 2p\leq m\leq n-2$.
Now, we are able to express the expansion, shear and twist scalars defined in
(1.23) as
$\displaystyle\theta=\frac{1}{n-2}\left(2\sum_{\mu=1}^{p}\frac{r}{r^{2}+(a^{0}_{(2\mu)})^{2}}+\frac{m-2p}{r}\right),$
(2.148)
$\displaystyle\sigma^{2}=2\sum_{\mu=1}^{p}\frac{r^{2}}{(r^{2}+(a^{0}_{(2\mu)})^{2})^{2}}+\frac{m-2p}{r^{2}}-(n-2)\theta^{2},$
(2.149)
$\displaystyle\omega^{2}=2\sum_{\mu=1}^{p}\frac{(a^{0}_{(2\mu)})^{2}}{(r^{2}+(a^{0}_{(2\mu)})^{2})^{2}}$
(2.150)
and the quantity $L_{ij}L_{ij}$ reads
$L_{ik}L_{ik}=(n-2)\theta\frac{1}{r}.$ (2.151)
Using the above results, we can determine the $r$-dependence of the
Kerr–Schild function $\mathcal{H}$ by integrating (2.127)
$\mathcal{H}=\frac{\mathcal{H}_{0}}{r^{m-2p-1}}\prod_{\mu=1}^{p}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}},$
(2.152)
which is identical to the case with vanishing $\Lambda$ discussed in [27].
Note that $\mathcal{H}$ behaves as
$\mathcal{H}\approx\frac{\mathcal{H}_{0}}{r^{m-1}}$ for large $r$ and
therefore Einstein GKS spacetimes (2.11) are asymptotically (anti-)de Sitter
or Minkowski depending on the cosmological constant of the background metric
$\bar{g}_{ab}$ as one approaches the null infinity along a null geodesic of
the Kerr–Schild congruence $\boldsymbol{k}$. The behaviour near the origin
$r=0$ will be investigated in section 2.6.4.
#### 2.6.2 Algebraic type
In section 2.2.2, we have already shown that GKS spacetimes (2.11) with a
geodetic Kerr–Schild vector $\boldsymbol{k}$ which include Einstein spaces are
of Weyl type II or more special as follows from proposition 1. Moreover, non-
expanding Einstein GKS spacetimes are necessarily only of Weyl type N by
proposition 4.
Now, we generalize the argument given in [27] for the Ricci-flat case to
Einstein spaces in order to show that expanding Einstein GKS spacetimes are
incompatible with Weyl types III and N and therefore such spacetimes are only
of types II, D or conformally flat.
The boost weight zero components of the Weyl tensor vanish, by definition, for
Weyl types III and N. In particular, the vanishing frame components $C_{0i1j}$
of Einstein GKS spacetimes given by (2.40) and (2.76) imply
$L_{ij}\mathrm{D}\mathcal{H}=2\mathcal{H}A_{ik}L_{kj}.$ (2.153)
Multiplying the above equation by $L_{lj}$ and using the optical constraint
(2.128), one obtains
$S_{il}\mathrm{D}\mathcal{H}=2\mathcal{H}A_{ik}S_{kl}.$ (2.154)
Taking the trace of (2.154) and eliminating the constant factor $(n-2)$ gives
$\theta\,\mathrm{D}\mathcal{H}=0.$ (2.155)
Next, we employ the Einstein field equations. Substituting
$\mathrm{D}\mathcal{H}=0$ to (2.67) leads to $\omega=0$, therefore, the
optical matrix is symmetric $L_{ij}=S_{ij}$ and one may rewrite (2.68) as
$S_{ik}S_{jk}=(n-2)\theta S_{ij}.$ (2.156)
In a frame of the eigenvectors, $S_{ij}$ takes the form
$S_{ij}=\text{diag}(s_{(2)},s_{(3)},\ldots,s_{(n-1)})$ and (2.156) reduces to
$s^{2}_{(i)}=s_{(i)}\sum_{j}s_{(j)}$ which has the only solution
$L_{ij}=\text{diag}(s,0,\ldots,0).$ (2.157)
Indeed, substituting $\mathrm{D}\mathcal{H}=0$ along with (2.157) to the Weyl
tensor cancels the remaining non-vanishing boost weight zero components
$C_{01ij}$ (2.39), (2.75) and $C_{ijkl}$ (2.41), (2.77).
The canonical form of the optical matrix $L_{ij}$ for Ricci-flat spacetimes of
type N and non-twisting subclass of type III was determined in [2] using the
Bianchi identities and the fact that $C_{abcd}=R_{abcd}$. The same result can
be also obtained for Einstein spaces since the additional terms in the Weyl
tensor of such spacetimes (2.73) are proportional to the cosmological constant
$\Lambda$ and the metric tensor $g_{ab}$ which does not affect the Bianchi
identities, i.e. $C_{ab[cd;e]}=R_{ab[cd;e]}$. Therefore, in the case of non-
twisting Einstein spacetimes of types III and N, the canonical form of the
optical matrix is
$L_{ij}=\text{diag}(s,s,0,\ldots,0).$ (2.158)
The form of the optical matrix of expanding Einstein GKS spacetimes with
$\mathrm{D}\mathcal{H}=0$ (2.157) is not compatible with the form of the
optical matrix for general non-twisting Einstein spacetimes of Weyl types III
and N (2.158). Consequently, expanding Einstein GKS solutions of types III and
N do not exist and we can conclude that
###### Proposition 5
Einstein generalized Kerr–Schild spacetimes (2.11) with an expanding geodetic
Kerr–Schild congruence $\boldsymbol{k}$ are of Weyl types II, D or conformally
flat.
Note that the conformally flat case occurs only if we admit the cosmological
constants of the full and background spacetimes not to be equal, otherwise
$\mathcal{H}$ has to vanish and $g_{ab}$ is given just by the background
metric $\bar{g}_{ab}$.
Conversely, one can immediately see that $\mathrm{D}\mathcal{H}$ has to be
non-vanishing in expanding Einstein GKS spacetimes and thus the Kerr–Schild
function $\mathcal{H}$ has to depend on an affine parameter along the null
geodesics $\boldsymbol{k}$. If we compare this result with the $r$-dependence
of $\mathcal{H}$ (2.152), it is obvious that $m\neq 1$. Therefore, in
accordance with the Goldberg–Sachs theorem, the optical matrix of expanding
Einstein GKS spacetimes is necessarily non-shearing in the case $n=4$ since it
consists either of one $2\times 2$ block (2.145) or of the unit matrix
multiplied by $r^{-1}$. The optical constraint can thus be essentially
considered as a higher dimensional generalization of the Goldberg–Sachs
theorem for Einstein GKS spacetimes which states that there is no shear in two
dimensional planes spanned by two spacelike frame vectors
$\boldsymbol{m}^{(i)}$, $\boldsymbol{m}^{(j)}$ corresponding to the $2\times
2$ blocks in the optical matrix.
Now, we are able to summarize all possible algebraic types of Einstein GKS
spacetimes. Omitting more general types G and I, excluded by proposition 2,
and the trivial conformally flat case, the allowed combinations of the Weyl
types and values of the expansion scalar are depicted in table 2.1.
Table 2.1: Weyl types compatible with Einstein generalized Kerr–Schild spacetimes depending on the values of the expansion scalar $\theta$. | Weyl type
---|---
Expansion | II | D | III | N
$\theta=0$ | $\times$ | $\times$ | $\times$ | ✓
$\theta\neq 0$ | ✓ | ✓ | $\times$ | $\times$
Let us conclude that Einstein GKS spacetimes of Weyl type N admit only a non-
expanding Kerr–Schild congruence $\boldsymbol{k}$ and belong to the Kundt
class. Type III is incompatible with the GKS ansatz in the case of Einstein
spaces, whereas types II and D imply that such spacetimes are expanding.
Note also that the higher dimensional Robinson–Trautman class contains only
solutions of Weyl type D [16] and although there is an intersection with type
D Einstein GKS spacetimes, such as the Schwarzschild–Tangherlini black hole,
in general higher dimensional Robinson–Trautman metrics do not admit the GKS
form. However, it was shown in [16] that general Robinson–Trautman metric
$\mathrm{d}s^{2}=\frac{r^{2}}{P^{2}}\gamma_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}-2\mathrm{d}u\,\mathrm{d}r-2H\mathrm{d}u^{2}$
(2.159)
is conformally flat and Einstein if $\gamma_{ij}$ is of constant curvature
which then implies
$\gamma_{ij}=\delta_{ij},\qquad
P=a(u)+b_{i}(u)x^{i}+c(u)\delta_{ij}x^{i}x^{j},$ (2.160)
where $a(u)$, $b_{i}(u)$ and $c(u)$ are arbitrary functions of the coordinate
$u$. Therefore, every Einstein Robinson–Trautman metric (2.159) possibly
admitting aligned null radiation in the Ricci tensor with $\gamma_{ij}$ and
$P$ of the form (2.160) differs from the corresponding (anti-)de Sitter or
Minkowski background by the factor
$2\mathcal{H}k_{a}k_{b}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}=2(H-H^{0})\,\mathrm{d}u^{2}=-\frac{\mu(u)}{r^{n-3}}\,\mathrm{d}u^{2}.$
(2.161)
If the function $\mu(u)$ does not depend on $u$, such metrics describe static
(A)dS–Schwarzschild–Tangherlini black holes. Otherwise, a null radiation term
appears in the Ricci tensor and the metric corresponds to the Vaidya solution.
#### 2.6.3 $r$-dependence of the Weyl tensor
The boost weight zero components of the Weyl tensor of Einstein GKS spacetimes
(2.74)–(2.77), (2.38)–(2.41) are given only in terms of the optical matrix
$L_{ij}$, the Kerr–Schild function $\mathcal{H}$ and its first and second
derivatives $\mathrm{D}\mathcal{H}$ and $\mathrm{D}^{2}\mathcal{H}$,
respectively. This now allows us to determine easily the $r$-dependence of
these components of the Weyl tensor using the $r$-dependence of the quantities
$L_{ij}$ and $\mathcal{H}$ derived already in section 2.6.1. We will employ
these results later in order to discuss the presence of curvature
singularities in expanding Einstein GKS spacetimes in section 2.6.4.
It is convenient to adopt more compact notation for the boost weight zero
components of the Weyl tensor [13]
$\Phi_{ij}\equiv
C_{0i1j},\qquad\Phi=C_{0101},\qquad\Phi^{S}_{ij}=-\frac{1}{2}C_{ikjk},\qquad\Phi^{A}_{ij}=\frac{1}{2}C_{01ij}.$
(2.162)
First, we express the derivatives of the function $\mathcal{H}$ from (2.152)
$\displaystyle\mathrm{D}\mathcal{H}$
$\displaystyle=-\frac{\mathcal{H}_{0}}{r^{m-2p-2}}\left(\frac{m-2p-1}{r^{2}}+2\sum^{p}_{\mu=1}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}\right)\prod^{p}_{\nu=1}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}},$
(2.163) $\displaystyle\mathrm{D}^{2}\mathcal{H}$
$\displaystyle=\frac{\mathcal{H}_{0}}{r^{m-2p-3}}\Bigg{(}\frac{(m-2p-1)(m-2p)}{r^{4}}$
$\displaystyle\qquad+2\frac{2m-4p-3}{r^{2}}\sum^{p}_{\mu=1}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}+4\sum^{p}_{\mu=1}\frac{1}{(r^{2}+(a^{0}_{(2\mu)})^{2})^{2}}$
$\displaystyle\qquad+4\sum^{p}_{\mu=1}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}\sum^{p}_{\rho=1}\frac{1}{r^{2}+(a^{0}_{(2\rho)})^{2}}\Bigg{)}\prod^{p}_{\nu=1}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}}.$
(2.164)
Substituting the $r$-dependence of the optical matrix $L_{ij}$ (2.144)–(2.147)
and the function $\mathcal{H}$ (2.152) and its derivatives (2.163), (2.164) to
the expressions for the corresponding boost weight zero components of the Weyl
tensor (2.38)–(2.41), (2.74)–(2.77), we immediately obtain the $r$-dependence
of $\Phi_{ij}$
$\displaystyle\Phi_{2\mu,2\mu}$
$\displaystyle=\Phi_{2\mu+1,2\mu+1}=-\mathrm{D}\mathcal{H}s_{(2\mu)}-2\mathcal{H}A^{2}_{2\mu,2\mu+1}\phantom{\frac{1}{2}}$
$\displaystyle=\frac{\mathcal{H}_{0}}{r^{m-2p-3}}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}\Bigg{(}\frac{m-2p-2}{r^{2}}+\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}$
$\displaystyle\qquad+2\sum^{p}_{\nu=1}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}}\Bigg{)}\prod^{p}_{\rho=1}\frac{1}{r^{2}+(a^{0}_{(2\rho)})^{2}},$
(2.165) $\displaystyle\Phi_{2\mu,2\mu+1}$
$\displaystyle=\Phi^{A}_{2\mu,2\mu+1}=-\mathrm{D}(\mathcal{H}A_{2\mu,2\mu+1})\phantom{\frac{1}{2}}$
$\displaystyle=\frac{\mathcal{H}_{0}}{r^{m-2p-2}}\frac{a^{0}_{(2\mu)}}{r^{2}+(a^{0}_{(2\mu)})^{2}}\Bigg{(}\frac{m-2p-1}{r^{2}}+\frac{2}{r^{2}+(a^{0}_{(2\mu)})^{2}}$
$\displaystyle\qquad+2\sum^{p}_{\nu=1}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}}\Bigg{)}\prod^{p}_{\rho=1}\frac{1}{r^{2}+(a^{0}_{(2\rho)})^{2}},$
(2.166) $\displaystyle\Phi_{\alpha\beta}$
$\displaystyle=-r^{-1}\delta_{\alpha\beta},\qquad\Phi=\mathrm{D}^{2}\mathcal{H}.\phantom{\frac{1}{2}}$
(2.167)
Hence, $\Phi_{ij}$ reproduces the block diagonal structure of the optical
matrix $L_{ij}$, where $\mu,\nu,\ldots=1,\dots,p$ number the $2\times 2$
blocks, whereas the elements of the diagonal block are indexed by
$\alpha,\beta,\ldots=2p+2,\dots,n-1$. Similarly, one may determine the
$r$-dependence of the remaining non-vanishing boost weight zero components
$C_{ijkl}$
$\displaystyle C_{2\mu,2\mu+1,2\mu,2\mu+1}$
$\displaystyle=2\mathcal{H}\left(3A^{2}_{2\mu,2\mu+1}-s^{2}_{(2\mu)}\right)\phantom{\frac{1}{2}}$
$\displaystyle=-2\frac{\mathcal{H}_{0}}{r^{m-2p-1}}\frac{r^{2}-3(a^{0}_{(2\mu)})^{2}}{\left(r^{2}+(a^{0}_{(2\mu)})^{2}\right)^{2}}\prod^{p}_{\nu=1}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}},$
(2.168) $\displaystyle C_{2\mu,2\mu+1,2\nu,2\nu+1}$
$\displaystyle=2C_{2\mu,2\nu,2\mu+1,2\nu+1}=-2C_{2\mu,2\nu+1,2\mu+1,2\nu}\phantom{\frac{1}{2}}$
$\displaystyle=4\mathcal{H}A_{2\mu,2\mu+1}A_{2\nu,2\nu+1}\phantom{\frac{1}{2}}$
$\displaystyle=4\frac{\mathcal{H}_{0}}{r^{m-2p-1}}\frac{a^{0}_{(2\mu)}}{r^{2}+(a^{0}_{(2\mu)})^{2}}\frac{a^{0}_{(2\nu)}}{r^{2}+(a^{0}_{(2\nu)})^{2}}\prod^{p}_{\rho=1}\frac{1}{r^{2}+(a^{0}_{(2\rho)})^{2}},$
(2.169) $\displaystyle C_{2\mu,2\nu,2\mu,2\nu}$
$\displaystyle=C_{2\mu,2\nu+1,2\mu,2\nu+1}=-2\mathcal{H}s_{(2\mu)}s_{(2\nu)}\phantom{\frac{1}{2}}$
$\displaystyle=-2\frac{\mathcal{H}_{0}}{r^{m-2p-3}}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}}\frac{1}{r^{2}+(a^{0}_{(2\nu)})^{2}}\prod^{p}_{\rho=1}\frac{1}{r^{2}+(a^{0}_{(2\rho)})^{2}},$
(2.170) $\displaystyle C_{(\alpha)(i)(\alpha)(i)}$
$\displaystyle=-2\mathcal{H}s_{(i)}r^{-1}\phantom{\frac{1}{2}}$
$\displaystyle=-2\frac{\mathcal{H}_{0}}{r^{m-2p-1}}\frac{1}{r^{2}+(a^{0}_{(i)})^{2}}\prod^{p}_{\mu=1}\frac{1}{r^{2}+(a^{0}_{(2\mu)})^{2}},$
(2.171)
where $\mu\neq\nu$.
#### 2.6.4 Singularities
Let us briefly discuss curvature singularities of expanding Einstein GKS
metrics. It is obvious from the $r$-dependence of the Kerr–Schild function
$\mathcal{H}$ (2.152) that it may diverge for $r\to 0$. Namely, $\mathcal{H}$
and consequently the full GKS metric blows up at $r=0$ in the following cases:
* •
in the “generic” case when neither $2p=m$ nor $2p=m-1$, or
* •
in the special cases when $2p=m$ (for $m$ even) or $2p=m-1$ (for $m$ odd) if
at least one of the functions $a^{0}_{(2\mu)}$, not depending on $r$, admits a
real root at $x=x_{0}$.
One may express the Kretschmann scalar at the singular point to verify whether
there is a real curvature singularity. Omitting the trivial conformally flat
case, expanding Einstein GKS spacetimes are of Weyl types D or II as follows
from proposition 5. Thus, the positive boost weight components of the Weyl
tensor and consequently the corresponding components of the Riemann tensor
vanish. Note that the negative boost weight components do not have the
appropriate counterparts in the following contractions. For instance, the term
$R_{011i}n_{a}k_{b}k_{c}m_{d}^{(i)}$ may give a non-zero contribution only
with the term $R_{100i}k^{a}n^{b}n^{c}m^{d}_{(i)}$. Therefore, the Kretschmann
scalar is determined only by the boost weight zero components of the Riemann
tensor
$R_{abcd}R^{abcd}=4\left(R_{0101}\right)^{2}-4R_{01ij}R_{01ij}+8R_{0i1j}R_{0j1i}+R_{ijkl}R_{ijkl}.$
(2.172)
Expressing the frame components of the Riemann tensor in terms of the Weyl
tensor from (2.74)–(2.77) and using the notation (2.162), we can rewrite the
Kretschmann scalar as
$\displaystyle R_{abcd}R^{abcd}$
$\displaystyle=4\Phi^{2}+8\Phi^{S}_{ij}\Phi^{S}_{ij}-24\Phi^{A}_{ij}\Phi^{A}_{ij}+C_{ijkl}C_{ijkl}$
(2.173) $\displaystyle\qquad+\frac{8n}{(n-1)(n-2)^{2}}\,\Lambda^{2}.$
The only additional term in comparison with the Ricci-flat case [27] is the
last constant term proportional to $\Lambda^{2}$ which clearly cannot
influence the divergence of the Kretschmann scalar and thus the presence of
singularities.
Let us discuss behavior of (2.173) for $r\to 0$ in the above mentioned
singular cases. The Kretschmann scalar consists of a sum of squares, except
the third term which is negative, and, as we will see, it is sufficient to
compare only the first and the third term.
In the “generic” case $2p\neq m$, $2p\neq m-1$, one may determine the
behaviour of the following quantities near $r=0$ from the $r$-dependence of
$\mathcal{H}$ (2.152) and $A_{2\mu,2\mu+1}$ (2.146)
$\displaystyle\mathcal{H}\sim r^{-(m-2p-1)},\qquad\mathrm{D}\mathcal{H}\sim
r^{-(m-2p)},\qquad\mathrm{D}^{2}\mathcal{H}\sim r^{-(m-2p+1)},$ (2.174)
$\displaystyle A_{2\mu,2\mu+1}\sim 1\quad\text{if $a^{0}_{(2\mu)}\neq
0$},\qquad A_{2\mu,2\mu+1}=0\quad\text{if $a^{0}_{(2\mu)}=0$ at $x=x_{0}$}.$
Substituting (2.174) to (2.166) and (2.167), we obtain that $\Phi\sim
r^{-(m-2p+1)}$ and either $\Phi^{A}_{2\mu,2\mu+1}\sim r^{-(m-2p)}$ for
$a^{0}_{(2\mu)}\neq 0$ or $\Phi^{A}_{2\mu,2\mu+1}=0$ for $a^{0}_{(2\mu)}=0$.
Therefore, the first term dominates over the third term in (2.173) and the
Kretschmann scalar diverges. Thus, in the “generic” case, a curvature
singularity is always located at $r=0$.
Note that this case also includes all non-twisting expanding Einstein GKS
solutions, where $p=0$, such as the higher dimensional
(A)dS–Schwarzschild–Tangherlini black holes. A five-dimensional example of
these metrics will be presented in section 2.6.5 as a static limit of the
(A)dS–Kerr metric (2.186) with $m=3$, $p=0$, where the corresponding optical
metric $L_{ij}$ and the function $\mathcal{H}$ will be also given explicitly.
Similarly, one may analyze the special cases $2p=m$ (for $m$ even), $2p=m-1$
(for $m$ odd) where some of the functions $a^{0}_{(2\mu)}$ have real roots.
Let $q$ $(q\geq 1)$ denotes the number of such vanishing $a^{0}_{(2\mu)}$.
Now, from (2.146) it follows that if $a^{0}_{(2\mu)}$ has no root then
$A_{2\mu,2\mu+1}\sim 1$ near $r=0$, whereas if $a^{0}_{(2\mu)}$ admits a root
$x=x_{0}$ then $A_{2\mu,2\mu+1}=0$ at this point. The Kerr–Schild function
$\mathcal{H}$ (2.152) and its derivatives behave for $2p=m$ as
$\mathcal{H}\sim r^{-2q+1}\,,\qquad\mathrm{D}\mathcal{H}\sim
r^{-2q}\,,\qquad\mathrm{D}^{2}\mathcal{H}\sim r^{-2q-1}\,,$ (2.175)
therefore, $\Phi\sim r^{-2q-1}$ and $\Phi^{A}_{ij}\Phi^{A}_{ij}\sim
2(p-q)r^{-4q}$. In the case $2p=m-1$ one gets
$\mathcal{H}\sim r^{-2q}\,,\qquad\mathrm{D}\mathcal{H}\sim
r^{-2q-1}\,,\qquad\mathrm{D}^{2}\mathcal{H}\sim r^{-2q-2}$ (2.176)
and consequently $\Phi\sim r^{-2q-2}$ and $\Phi^{A}_{ij}\Phi^{A}_{ij}\sim
2(p-q)r^{-2(2q+1)}$. Therefore, in both special cases $2p=m$ and $2p=m-1$, the
first term in (2.173) dominates again over the third term and if any
$a^{0}_{(2\mu)}$ has a real root at $x=x_{0}$ then a curvature singularity is
located at $r=0$, $x=x_{0}$. Note that this corresponds, for instance, to the
well-known ring shaped singularity of the Kerr black hole.
As will be shown in section 2.6.5, these special cases are represented, e.g.,
by the five-dimensional Kerr–(A)dS metric (2.186), where the optical matrix
$L_{ij}$ of rank $m=3$ has, in the rotating case, one $2\times 2$ block, i.e.
$p=1$. The function $a^{0}_{(2)}$ is given by the spins $a$, $b$ as
$(a^{0}_{(2)})^{2}=a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta$. If one of the
spins is zero, $a^{0}_{(2)}$ admits a root and, indeed, this corresponds to
the special case ($2p=m-1$, $m$ odd) with a curvature singularity located at
$r=0$. In the case with both spins being non-zero, this metric belongs neither
to the “generic” case nor to the special cases since $a^{0}_{(2)}$ never
vanishes and therefore no singularity is present at $r=0$.
#### 2.6.5 Example of expanding Einstein generalized
Kerr–Schild spacetime: Kerr–(A)dS
In this section, we compare our results obtained for general expanding
Einstein GKS spacetimes with an explicit physically interesting example,
namely, the higher dimensional Kerr–(A)dS metric. Considering such metrics in
five dimensions, we find a parallelly transported frame which allows us to
express the optical matrix $L_{ij}$ in the block-diagonal form. Subsequently,
we compare this optical matrix and the Kerr–Schild function $\mathcal{H}$ with
the $r$-dependence of the corresponding quantities of general expanding
Einstein GKS spacetimes derived in the previous sections and discuss the
presence of curvature singularities.
The higher dimensional Kerr–(A)dS metric is an example of an expanding
Einstein GKS spacetime describing a black hole rotating in
$\lfloor(n-1)/2\rfloor$ independent planes with a possible cosmological
constant $\Lambda$. In this sense, it is a generalization of the Myers–Perry
black hole, which can be obtained by taking the limit $\Lambda\rightarrow 0$.
The Kerr–(A)dS metric in arbitrary dimension was derived in [30] in the GKS
form (2.11) using the spheroidal coordinates consisting of the radial
coordinate $r$, time coordinate $t$, $\lfloor(n-1)/2\rfloor$ azimuthal angular
coordinates $\phi_{i}$ and $\lfloor n/2\rfloor$ coordinates $\mu_{i}$ subject
to
$\sum^{\lfloor n/2\rfloor}_{i=1}\mu_{i}^{2}=1.$ (2.177)
The background metric $\bar{g}_{ab}$, Kerr–Schild vector $\boldsymbol{k}$ and
function $\mathcal{H}$ in $n=2k+1$ dimensions are given by
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=-W(1-\lambda
r^{2})\,\mathrm{d}t^{2}+F\,\mathrm{d}r^{2}+\sum^{k}_{i=1}\frac{r^{2}+a_{i}^{2}}{1+\lambda
a_{i}^{2}}(\mathrm{d}\mu_{i}^{2}+\mu_{i}^{2}\,\mathrm{d}\phi_{i}^{2})$
$\displaystyle\qquad+\frac{\lambda}{W(1-\lambda
r^{2})}\left(\sum^{k}_{i=1}\frac{(r^{2}+a_{i}^{2})\mu_{i}\,\mathrm{d}\mu_{i}}{1+\lambda
a_{i}^{2}}\right)^{2},$ (2.178) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=W\,\mathrm{d}t+F\,\mathrm{d}r-\sum^{k}_{i=1}\frac{a_{i}\mu_{i}^{2}}{1+\lambda
a_{i}^{2}}\,\mathrm{d}\phi_{i},$ (2.179) $\displaystyle\mathcal{H}$
$\displaystyle=-\frac{M}{\sum^{k}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\prod^{k}_{j=1}(r^{2}+a_{j}^{2})},$
(2.180)
whereas in $n=2k$ dimensions
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=-W(1-\lambda
r^{2})\,\mathrm{d}t^{2}+F\,\mathrm{d}r^{2}+\sum^{k}_{i=1}\frac{r^{2}+a_{i}^{2}}{1+\lambda
a_{i}^{2}}\,\mathrm{d}\mu_{i}^{2}+\sum^{k-1}_{i=1}\frac{r^{2}+a_{i}^{2}}{1+\lambda
a_{i}^{2}}\mu_{i}^{2}\,\mathrm{d}\phi_{i}^{2}$
$\displaystyle\qquad+\frac{\lambda}{W(1-\lambda
r^{2})}\left(\sum^{k}_{i=1}\frac{(r^{2}+a_{i}^{2})\mu_{i}\,\mathrm{d}\mu_{i}}{1+\lambda
a_{i}^{2}}\right)^{2},$ (2.181) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=W\,\mathrm{d}t+F\,\mathrm{d}r-\sum^{k-1}_{i=1}\frac{a_{i}\mu_{i}^{2}}{1+\lambda
a_{i}^{2}}\,\mathrm{d}\phi_{i},$ (2.182) $\displaystyle\mathcal{H}$
$\displaystyle=-\frac{M}{r\sum^{k}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\prod^{k-1}_{j=1}(r^{2}+a_{j}^{2})},$
(2.183)
where the functions $W$ and $F$ are defined as
$W=\sum^{k}_{i=1}\frac{\mu_{i}^{2}}{1+\lambda a_{i}^{2}},\qquad
F=\frac{r^{2}}{1-\lambda
r^{2}}\sum^{k}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}$ (2.184)
and $\lambda$ is related to the cosmological constant $\Lambda$ via
$\lambda=\frac{2\Lambda}{(n-1)(n-2)}.$ (2.185)
In five dimensions, one may choose the coordinates $\mu_{i}$ as
$\mu_{1}=\sin\theta$, $\mu_{2}=\cos\theta$ that clearly satisfy (2.177) and
denote the azimuthal coordinates as $\phi_{1}=\phi$, $\phi_{2}=\psi$ and the
rotation parameters as $a_{1}=a$, $a_{2}=b$. The background metric
$\bar{g}_{ab}$, Kerr–Schild vector $\boldsymbol{k}$ and function $\mathcal{H}$
are then given by
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=-\frac{(1-\lambda r^{2})\Delta}{(1+\lambda a^{2})(1+\lambda
b^{2})}\,\mathrm{d}t^{2}+\frac{r^{2}\rho^{2}}{(1-\lambda
r^{2})(r^{2}+a^{2})(r^{2}+b^{2})}\,\mathrm{d}r^{2}$
$\displaystyle\qquad+\frac{\rho^{2}}{\Delta}\,\mathrm{d}\theta^{2}+\frac{(r^{2}+a^{2})\sin^{2}\theta}{1+\lambda
a^{2}}\,\mathrm{d}\phi^{2}+\frac{(r^{2}+b^{2})\cos^{2}\theta}{1+\lambda
b^{2}}\,\mathrm{d}\psi^{2},$ (2.186) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\frac{\Delta}{(1+\lambda a^{2})(1+\lambda
b^{2})}\,\mathrm{d}t+\frac{r^{2}\rho^{2}}{(1-\lambda
r^{2})(r^{2}+a^{2})(r^{2}+b^{2})}\,\mathrm{d}r$
$\displaystyle\qquad-\frac{a\sin^{2}\theta}{1+\lambda
a^{2}}\,\mathrm{d}\phi-\frac{b\cos^{2}\theta}{1+\lambda
b^{2}}\,\mathrm{d}\psi,$ (2.187) $\displaystyle\mathcal{H}$
$\displaystyle=-\frac{M}{\rho^{2}},$ (2.188)
where the angular coordinate ranges are as usual $\theta\in\langle
0,\pi\rangle$, $\phi\in\langle 0,2\pi)$, $\psi\in\langle 0,2\pi)$ and the
functions $\rho$, $\Delta$ and $\nu$ are defined as
$\rho^{2}=r^{2}+\nu^{2},\qquad\Delta=1+\lambda\nu^{2},\qquad\nu=\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}.$
(2.189)
In agreement with propositions 1 and 2, it can be shown by a straightforward
calculation that the Kerr–Schild vector $\boldsymbol{k}$ (2.187) is the
geodetic multiple WAND. Moreover, $\boldsymbol{k}$ is already scaled to be
affinely parametrized.
Now we construct a parallelly transported null frame, with $\boldsymbol{k}$
being one of the vectors, such that the optical matrix $L_{ij}$ takes the
block-diagonal form (2.144). However, first of all we find an arbitrary null
frame satisfying the constraints (1.8) and then we can transform it to the
desired form using null rotations and spins which preserve $\boldsymbol{k}$.
Note that, unlike the full Kerr–(A)dS metric, the background metric
$\bar{g}_{ab}$ (2.186) is diagonal and much more simpler. Therefore, it is
easier to construct a null frame in the background spacetime $\boldsymbol{k}$,
$\bar{\boldsymbol{n}}$, $\boldsymbol{m}^{(i)}$ which is related to the frame
in the full spacetime $\boldsymbol{k}$, $\boldsymbol{n}$,
$\boldsymbol{m}^{(i)}$ by (2.24). One may immediately express the inverse
background metric as
$\displaystyle\bar{g}^{ab}\frac{\partial}{\partial
x_{a}}\frac{\partial}{\partial x_{b}}$ $\displaystyle=-\frac{(1+\lambda
a^{2})(1+\lambda b^{2})}{(1-\lambda
r^{2})\Delta}\left(\frac{\partial}{\partial
t}\right)^{2}+\frac{\Delta}{\rho^{2}}\left(\frac{\partial}{\partial\theta}\right)^{2}$
(2.190) $\displaystyle\qquad+\frac{1+\lambda
a^{2}}{(r^{2}+a^{2})\sin^{2}\theta}\left(\frac{\partial}{\partial\phi}\right)^{2}+\frac{1+\lambda
b^{2}}{(r^{2}+b^{2})\cos^{2}\theta}\left(\frac{\partial}{\partial\psi}\right)^{2}$
$\displaystyle\qquad+\frac{(1-\lambda
r^{2})(r^{2}+a^{2})(r^{2}+b^{2})}{r^{2}\rho^{2}}\left(\frac{\partial}{\partial
r}\right)^{2}.$
Since $\bar{g}^{ab}$ is diagonal we can easily choose the covector
$\bar{\boldsymbol{n}}$ same as $\boldsymbol{k}$ with the only difference that
we change the sign of the “$\mathrm{d}t$” component to ensure that
$\bar{\boldsymbol{n}}$ is also null and then we multiply it by an appropriate
factor to satisfy the frame normalization $\bar{g}^{ab}k_{a}\bar{n}_{b}=1$
$\displaystyle\bar{n}_{a}\mathrm{d}x^{a}$ $\displaystyle=-\frac{1-\lambda
r^{2}}{2}\,\mathrm{d}t+\frac{(1+\lambda a^{2})(1+\lambda
b^{2})r^{2}\rho^{2}}{2\Delta(r^{2}+a^{2})(r^{2}+b^{2})}\,\mathrm{d}r$ (2.191)
$\displaystyle\qquad-\frac{a(1+\lambda b^{2})(1-\lambda
r^{2})\sin^{2}\theta}{2\Delta}\,\mathrm{d}\phi$
$\displaystyle\qquad-\frac{b(1+\lambda a^{2})(1-\lambda
r^{2})\cos^{2}\theta}{2\Delta}\,\mathrm{d}\psi.$
Consequently, the corresponding frame vector $\boldsymbol{n}$ in the full
spacetime is given by (2.24) as
$\displaystyle n_{a}\mathrm{d}x^{a}$
$\displaystyle=\left(\frac{M\Delta}{\rho^{2}(1+\lambda a^{2})(1+\lambda
b^{2})}-\frac{1-\lambda r^{2}}{2}\right)\mathrm{d}t$ (2.192)
$\displaystyle\qquad+\frac{r^{2}}{(r^{2}+a^{2})(r^{2}+b^{2})}\bigg{(}\frac{(1+\lambda
a^{2})(1+\lambda b^{2})\rho^{2}}{2\Delta}$
$\displaystyle\qquad+\frac{M}{(1-\lambda
r^{2})}\bigg{)}\,\mathrm{d}r-a\sin^{2}\theta\bigg{(}\frac{(1+\lambda
b^{2})(1-\lambda r^{2})}{2\Delta}$
$\displaystyle\qquad+\frac{M}{\rho^{2}(1+\lambda
a^{2})}\bigg{)}\,\mathrm{d}\phi-b\cos^{2}\theta\bigg{(}\frac{(1+\lambda
a^{2})(1-\lambda r^{2})}{2\Delta}$
$\displaystyle\qquad+\frac{M}{\rho^{2}(1+\lambda
b^{2})}\bigg{)}\,\mathrm{d}\psi.$
It remains to determine three spacelike frame vectors. A general spacelike
vector is of the form
$m_{a}\,\mathrm{d}x^{a}=A\,\mathrm{d}t+B\,\mathrm{d}r+C\,\mathrm{d}\theta+E\,\mathrm{d}\phi+F\,\mathrm{d}\psi$.
The orthogonality conditions
$\bar{g}^{ab}k_{a}m_{b}=\bar{g}^{ab}\bar{n}_{a}m_{b}=0$ imply that $A=0$ and
$B=E\frac{a}{r^{2}+a^{2}}+F\frac{b}{r^{2}+b^{2}}$. Then from
$\bar{g}^{ab}m_{a}m_{b}=1$ it follows that
$C^{2}\frac{\Delta}{\rho^{2}}+E^{2}\left[\frac{1+\lambda
a^{2}\cos^{2}\theta}{\rho^{2}\sin^{2}\theta}+\frac{b^{2}}{r^{2}\rho^{2}}\right]+2EF\left[\frac{ab(1-\lambda
r^{2})}{r^{2}\rho^{2}}\right]\\\ +F^{2}\left[\frac{1+\lambda
b^{2}\sin^{2}\theta}{\rho^{2}\cos^{2}\theta}+\frac{a^{2}}{r^{2}\rho^{2}}\right]=1.$
(2.193)
Obviously, the simplest solution is $C=\rho/\sqrt{\Delta}$, $E=F=0$ which
determines the first of the spacelike frame vectors $\boldsymbol{m}^{(2)}$
$m^{(2)}_{a}\,\mathrm{d}x^{a}=\frac{\rho}{\sqrt{\Delta}}\,\mathrm{d}\theta.$
(2.194)
Expressing $\bar{g}^{ab}m^{(2)}_{a}m_{b}=0$, one obtains $C=0$ and therefore
the “$\mathrm{d}\theta$” component of two remaining spacelike vectors has to
vanish, i.e.
$m^{(\kappa)}_{a}\mathrm{d}x^{a}=\left(E_{\kappa}\frac{a}{r^{2}+a^{2}}+F_{\kappa}\frac{b}{r^{2}+b^{2}}\right)\mathrm{d}r+E_{\kappa}\,\mathrm{d}\phi+F_{\kappa}\,\mathrm{d}\psi,\qquad\kappa=3,4.$
(2.195)
The four unknown functions $E_{\kappa}$, $F_{\kappa}$ have to satisfy three
equations following from the remaining constraints
$\bar{g}^{ab}m^{(3)}_{a}m^{(4)}_{b}=0$,
$\bar{g}^{ab}m^{(3)}_{a}m^{(3)}_{b}=\bar{g}^{ab}m^{(4)}_{a}m^{(4)}_{b}=1$,
which are rather complicated to solve at this moment. Since there is an
arbitrariness, we can try to find $\boldsymbol{m}^{(4)}$ to be parallelly
transported along $\boldsymbol{k}$, i.e. to satisfy the condition
$m^{(4)}_{a;b}k^{a}=0$. Only the contraction $m^{(4)}_{a;b}k^{a}m_{(2)}^{b}$
is sufficiently simple to express and we obtain
$E_{2}=F_{2}\frac{b\sin^{2}\theta}{a\cos^{2}\theta}$ relating the functions
$E_{2}$, $F_{2}$. Putting this expression to the normalization
$\bar{g}^{ab}m^{(4)}_{a}m^{(4)}_{b}=1$ yields
$F_{2}=\frac{ar\cos^{2}\theta}{\nu}$ and therefore
$\displaystyle m^{(4)}_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\frac{abr}{\nu}\left(\frac{\sin^{2}\theta}{r^{2}+a^{2}}+\frac{\cos^{2}\theta}{r^{2}+b^{2}}\right)\mathrm{d}r$
(2.196)
$\displaystyle\qquad+\frac{br\sin^{2}\theta}{\nu}\,\mathrm{d}\phi+\frac{ar\cos^{2}\theta}{\nu}\,\mathrm{d}\psi.$
The remaining unknown functions $E_{1}$, $F_{1}$ can be determined from
$\bar{g}^{ab}m^{(4)}_{a}m^{(3)}_{b}=0$ leading to $E_{1}=-F_{1}\frac{a}{b}$
and subsequently $\bar{g}^{ab}m^{(3)}_{a}m^{(3)}_{b}=1$ implies
$F_{1}=\frac{b\rho\sin\theta\cos\theta}{\sqrt{\Delta}\nu}$. Finally, we thus
obtain
$m^{(3)}_{a}\,\mathrm{d}x^{a}=\frac{\rho\sin\theta\cos\theta}{\sqrt{\Delta}\nu}\left[\left(\frac{b^{2}}{r^{2}+b^{2}}-\frac{a^{2}}{r^{2}+a^{2}}\right)\mathrm{d}r-a\,\mathrm{d}\phi+b\,\mathrm{d}\psi\right].$
(2.197)
So far we have constructed the null frame consisting of $\boldsymbol{k}$
(2.187), $\boldsymbol{n}$ (2.192), $\boldsymbol{m}^{(2)}$ (2.194),
$\boldsymbol{m}^{(3)}$ (2.197) and $\boldsymbol{m}^{(4)}$ (2.196) which can be
expressed in the contravariant form as
$\displaystyle\boldsymbol{k}$ $\displaystyle=-\frac{1}{1-\lambda
r^{2}}\frac{\partial}{\partial t}+\frac{\partial}{\partial
r}-\frac{a}{r^{2}+a^{2}}\frac{\partial}{\partial\varphi}-\frac{b}{r^{2}+b^{2}}\frac{\partial}{\partial\psi},$
(2.198) $\displaystyle\boldsymbol{n}$
$\displaystyle=\left(\frac{1}{2}\frac{(1+\lambda a^{2})(1+\lambda
b^{2})(1-\lambda r^{2})}{\Delta}-\frac{M}{\rho^{2}}\right)\boldsymbol{k}$
$\displaystyle\qquad+\frac{(1+\lambda a^{2})(1+\lambda
b^{2})}{\Delta}\frac{\partial}{\partial t},$
$\displaystyle\boldsymbol{m}^{(2)}$
$\displaystyle=\frac{\sqrt{\Delta}}{\rho}\frac{\partial}{\partial\theta},$
$\displaystyle\boldsymbol{m}^{(3)}$
$\displaystyle=\frac{\rho\sin\theta\cos\theta}{\sqrt{\Delta}\nu}\bigg{[}\frac{(b^{2}-a^{2})(1-\lambda
r^{2})}{\rho^{2}}\frac{\partial}{\partial r}-\frac{a(1+\lambda
a^{2})}{(r^{2}+a^{2})\sin^{2}\theta}\frac{\partial}{\partial\varphi}$
$\displaystyle\qquad+\frac{b(1+\lambda
b^{2})}{(r^{2}+b^{2})\cos^{2}\theta}\frac{\partial}{\partial\psi}\bigg{]},$
$\displaystyle\boldsymbol{m}^{(4)}$
$\displaystyle=\frac{abr}{\nu}\left[\frac{1-\lambda
r^{2}}{r^{2}}\frac{\partial}{\partial r}+\frac{1+\lambda
a^{2}}{a(r^{2}+a^{2})}\frac{\partial}{\partial\varphi}+\frac{1+\lambda
b^{2}}{b(r^{2}+b^{2})}\frac{\partial}{\partial\psi}\right].$
One may show that although this frame is not parallelly transported along the
geodetic Kerr–Schild vector $\boldsymbol{k}$, it is already adapted to the
block-diagonal form of the optical matrix $L_{ij}$, where the $2\times 2$
block corresponds to the plane spanned by the frame vectors
$\boldsymbol{m}^{(2)}$ and $\boldsymbol{m}^{(3)}$.
Recall that the condition that a frame is parallelly transported along a
geodetic ($L_{i0}=0$) and affinely parametrized ($L_{10}=0$) null congruence
$\boldsymbol{k}$ can be written using Ricci rotation coefficients as
$N_{i0}=\mbox{$\stackrel{{\scriptstyle i}}{{M}}_{j0}$}=0.$ (2.199)
Considering our chosen frame (2.198), we have to transform away the following
non-zero coefficients
$\displaystyle N_{20}$
$\displaystyle=\frac{\lambda(a^{2}-b^{2})\sin\theta\cos\theta}{\rho\sqrt{\Delta}},\qquad
N_{30}=-\frac{\lambda(a^{2}-b^{2})r\sin\theta\cos\theta}{\rho\nu\sqrt{\Delta}},$
(2.200) $\displaystyle N_{40}$ $\displaystyle=\frac{\lambda
ab}{\nu},\qquad\mbox{$\stackrel{{\scriptstyle
2}}{{M}}_{30}$}=\frac{\nu}{\rho^{2}},\qquad\mbox{$\stackrel{{\scriptstyle
3}}{{M}}_{20}$}=-\frac{\nu}{\rho^{2}},$
by appropriate Lorentz transformations. First, it is convenient to perform
spins to set $\stackrel{{\scriptstyle i}}{{M}}_{j0}$ to zero since this
simplifies the equations determining the parameters $z_{i}$ of an appropriate
null rotation around $\boldsymbol{k}$ setting $N_{i0}$ to zero.
Note that null rotations with $\boldsymbol{k}$ fixed and spins in any plane
spanned by $\boldsymbol{m}^{(i)}$ and $\boldsymbol{m}^{(j)}$ corresponding to
a $2\times 2$ block in the canonical form of the optical matrix $L_{ij}$
preserve the block-diagonal form of this matrix.
Therefore, we assume the transformation matrix of appropriate spins (1.21) in
the form
$X_{ij}=\begin{pmatrix}\cos\varepsilon(r)&-\sin\varepsilon(r)&0\\\
\sin\varepsilon(r)&\cos\varepsilon(r)&0\\\ 0&0&1\end{pmatrix}.$ (2.201)
Requiring $\mbox{$\stackrel{{\scriptstyle i}}{{\hat{M}}}_{j0}$}=0$ and using
the orthogonality $\mathbf{X}^{-1}=\mathbf{X}^{T}$ of the transformation
matrix, it follows from (1.21) that
$\mathrm{D}\mathbf{X}=-\mathbf{X}\mathbf{M}.$ (2.202)
This is satisfied if
$\mathrm{D}\varepsilon(r)\equiv\frac{\mathrm{d}\varepsilon(r)}{\mathrm{d}r}=\frac{\nu}{r^{2}+\nu^{2}}$
which has a solution $\varepsilon(r)=\tan^{-1}\frac{r}{\nu}$ and therefore
$X_{ij}=\begin{pmatrix}\frac{\nu}{\rho}&-\frac{r}{\rho}&0\\\
\frac{r}{\rho}&\frac{\nu}{\rho}&0\\\ 0&0&1\end{pmatrix}.$ (2.203)
Thus, the spin represented by the transformation matrix (2.203) ensures that
all $\stackrel{{\scriptstyle i}}{{M}}_{j0}$ vanish and $N_{i0}$ transform as
$\displaystyle\hat{N}_{20}$
$\displaystyle=\frac{\nu}{\rho}N_{20}-\frac{r}{\rho}N_{30}=\frac{\lambda(a^{2}-b^{2})\sin\theta\cos\theta}{\sqrt{\Delta}\nu},$
(2.204) $\displaystyle\hat{N}_{30}$
$\displaystyle=\frac{r}{\rho}N_{20}+\frac{\nu}{\rho}N_{30}=0,\qquad\hat{N}_{40}=N_{40}=\frac{\lambda
ab}{\nu}$
and now we set the remaining $\hat{N}_{20}$, $\hat{N}_{40}$ to zero using an
appropriate null rotation with $\boldsymbol{k}$ fixed (1.19) which leads to
$\hat{\hat{N}}_{i0}=\hat{N}_{i0}+\mathrm{D}z_{i}.$ (2.205)
Requiring $\hat{\hat{N}}_{i0}=0$ and substituting (2.204) to (2.205), we can
integrate over $r$ to obtain the functions $z_{i}$
$z_{2}=-\frac{\lambda(a^{2}-b^{2})r\sin\theta\cos\theta}{\sqrt{\Delta}\nu},\qquad
z_{4}=-\frac{\lambda abr}{\nu}.$ (2.206)
Therefore, we have found a parallelly transported frame $\boldsymbol{k}$,
$\hat{\hat{\boldsymbol{n}}}$, $\hat{\hat{\boldsymbol{m}}}^{(2)}$,
$\hat{\hat{\boldsymbol{m}}}^{(3)}$, $\hat{\hat{\boldsymbol{m}}}^{(4)}$ which
is related to the null frame $\boldsymbol{k}$, $\boldsymbol{n}$,
$\boldsymbol{m}^{(2)}$, $\boldsymbol{m}^{(3)}$, $\boldsymbol{m}^{(4)}$ (2.198)
by
$\displaystyle\hat{\hat{\boldsymbol{n}}}$
$\displaystyle=\boldsymbol{m}+z_{2}\left(\frac{\nu}{\rho}\boldsymbol{m}^{(2)}-\frac{r}{\rho}\boldsymbol{m}^{(3)}\right)+z_{4}\boldsymbol{m}^{(4)}$
(2.207)
$\displaystyle\qquad-\frac{1}{2}\frac{\lambda^{2}r^{2}}{\nu^{2}}\left[a^{2}b^{2}+\frac{(a^{2}-b^{2})^{2}\sin^{2}\theta\cos^{2}\theta}{\Delta}\right]\boldsymbol{k},$
$\displaystyle\hat{\hat{\boldsymbol{m}}}^{(2)}$
$\displaystyle=\frac{\nu}{\rho}\boldsymbol{m}^{(2)}-\frac{r}{\rho}\boldsymbol{m}^{(3)}-z_{2}\boldsymbol{k},$
$\displaystyle\hat{\hat{\boldsymbol{m}}}^{(3)}$
$\displaystyle=\frac{r}{\rho}\boldsymbol{m}^{(2)}+\frac{\nu}{\rho}\boldsymbol{m}^{(3)},\qquad\hat{\hat{\boldsymbol{m}}}^{(4)}=\boldsymbol{m}^{(4)}-z_{4}\boldsymbol{k}.$
The optical matrix $L_{ij}$ of the five-dimensional Kerr–(A)dS metric
(2.186)–(2.188) can be straightforwardly expressed in the parallelly
transported frame (2.207) as
$L_{ij}=\begin{pmatrix}\frac{r}{\rho^{2}}&\frac{\nu}{\rho^{2}}&\phantom{a}0\phantom{a}\\\
-\frac{\nu}{\rho^{2}}&\frac{r}{\rho^{2}}&0\\\ 0&0&\frac{1}{r}\end{pmatrix},$
(2.208)
which is obviously of rank $m=3$ and contains one 2 $\times$ 2 block, i.e.
$p=1$, where
$s_{(2)}=\frac{r}{r^{2}+\nu^{2}},\qquad A_{2,3}=\frac{\nu}{r^{2}+\nu^{2}}.$
(2.209)
One may compare this particular optical matrix $L_{ij}$ (2.208) and the
Kerr–Schild function $\mathcal{H}$ (2.188) of the five-dimensional Kerr–(A)dS
metric with the corresponding quantities $L_{ij}$ (2.144)
$L_{ij}=\begin{pmatrix}s_{(2)}&A_{2,3}&\phantom{ab}0\phantom{ab}\\\
-A_{2,3}&s_{(2)}&0\\\ 0&0&\frac{1}{r}\end{pmatrix}$ (2.210)
and $\mathcal{H}$ (2.152)
$\mathcal{H}=\mathcal{H}_{0}\frac{1}{r^{2}+(a^{0}_{(2)})^{2}}$ (2.211)
of general expanding Einstein GKS spacetimes with the same parameters $n=5$,
$m=3$, $p=1$, where
$s_{(2)}=\frac{r}{r^{2}+(a^{0}_{(2)})^{2}},\qquad
A_{2,3}=\frac{a^{0}_{(2)}}{r^{2}+(a^{0}_{(2)})^{2}}.$ (2.212)
The presented optical matrices (2.208), (2.210) and the functions
$\mathcal{H}$ (2.188), (2.211) are in accordance and obviously
$a^{0}_{(2)}=\nu\equiv\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta},\qquad\mathcal{H}_{0}=-M.$
(2.213)
Now we employ the results of section 2.6.4 to investigate the presence of
curvature singularities at $r=0$ in five-dimensional Kerr–(A)dS spacetimes.
According to number of non-zero rotation parameters $a$, $b$ our discussion
can be split into three distinct cases.
In the first case where both rotation parameters are non-zero $a\neq 0$,
$b\neq 0$, the optical matrix $L_{ij}$ has one $2\times 2$ block, i.e. $p=1$,
and therefore $2p=m-1$. This corresponds to the special cases in section 2.6.4
when a curvature singularity is presented if $a^{0}_{(2)}$ (2.213) admits a
real root. Since $a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta$ never vanish there
is no curvature singularity .
If we set one of the rotation parameters to zero, for instance $a\neq 0$,
$b=0$, then the optical matrix $L_{ij}$ has still one $2\times 2$ block and
therefore $2p=m-1$. However, now $a^{0}_{(2)}=a\cos\theta$ has a real root at
$\theta=\frac{\pi}{2}$ and this corresponds to the ring-shaped singularity
known from the four-dimensional Kerr solution.
Whereas in the previous cases the presence of a curvature singularity depends
on the behaviour of the function $a^{0}_{(2)}$, in the last static case
$a=b=0$, the Kerr–(A)dS metric reduces to the (A)dS–Schwarzschild–Tangherlini
black hole where the optical matrix $L_{ij}$ is obviously diagonal, i.e.
$p=0$. Since neither $2p=m-1$ nor $2p=m$, this corresponds to the “generic”
case in section 2.6.4 where a curvature singularity is located at $r=0$.
#### 2.6.6 Expanding generalized Kerr–Schild spacetimes with null radiation
In this section, we present two explicit examples of expanding GKS spacetimes
with a null radiation term in the Ricci tensor which is aligned with the
Kerr–Schild vector $\boldsymbol{k}$ as in (2.91). Namely, the Kinnersley
photon rocket and the Vaidya shining star. Both solutions belong to the
Robinson–Trautman class of spacetimes admitting an expanding, non-shearing and
non-twisting geodetic null congruence.
First, let us point out the differences between four and higher dimensional
cases. In four dimensions, the Robinson–Trautman class contains solutions of
all algebraically special Petrov types II, D, III and N, see, e.g., [9]. On
the other hand, in higher dimensions, the form of the optical matrix of
spacetimes with non-zero expansion and vanishing shear and twist
$L_{ij}=\theta\delta_{ij}$ excludes Weyl types III and N. It follows directly
from the fact that all type N and non-twisting type III Einstein spacetimes
have the optical matrix of rank 2 which, in an appropriate frame, takes the
form $L_{ij}=\text{diag}(s,s,0,\ldots,0)$ [46] and therefore, omitting the
Kundt class, these types III and N spacetimes are non-shearing only in
dimension $n=4$. Indeed, it was shown in [16] that the higher dimensional
Robinson–Trautman class is not so rich as in the four-dimensional case and
contains only solutions of Weyl type D.
##### Kinnersley photon rocket
The Kinnersley photon rocket [25] is a four-dimensional solution of the
Einstein field equations with a null radiation term on the right-hand side
representing arbitrarily moving test particle in the Minkowski background due
to the back-reaction caused by emitted photons. The metric of such spacetime
admitting the KS form (2.1) can be generalized by adding a cosmological
constant $\Lambda$ and is of Petrov type D [43]. This suggests that it could
have a higher dimensional analogue.
Such a solution in arbitrary dimension was found as a special case in [26],
where Kerr–Schild metrics with an acceleration due to null radiation were
studied in the context of the Einstein–Maxwell theory. Subsequently, a
cosmological constant $\Lambda$ was taken into account in [47] and enters the
metric via the Kerr–Schild function $\mathcal{H}$
$\mathcal{H}=-\frac{m(u)}{r^{n-3}}-\frac{\Lambda}{(n-2)(n-1)}r^{2},\\\ $
(2.214)
in the KS form (2.1) with the flat background metric expressed in the
Cartesian-like coordinates as
$\eta_{ab}\,\mathrm{d}x^{a}\,\,dx^{b}=\frac{r^{2}}{P^{2}}\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}-2\mathrm{d}u\,\mathrm{d}r-\left(1-2r\left(\log
P\right)_{,u}\right)\mathrm{d}u^{2},$ (2.215)
where
$P=\left(\dot{z}^{0}-\dot{z}^{1}\right)-\delta_{ij}\dot{z}^{j}x^{i}+\frac{1}{4}\left(\dot{z}^{0}+\dot{z}^{1}\right)\delta_{ij}x^{i}x^{j}$
(2.216)
and the Kerr–Schild vector $\boldsymbol{k}$ is given by
$k^{a}\,\partial_{a}=\partial_{r}.$ (2.217)
The limit $m=0$ represents an (anti-)de Sitter spacetime. Therefore, the
metric can be cast to the GKS form (2.11) since we can split
$\mathcal{H}=\mathcal{H}_{m}+\mathcal{H}_{\Lambda}$ to the parts containing
only $m(u)$ or $\Lambda$, respectively, and then denote
$\bar{g}_{ab}=\eta_{ab}-2\mathcal{H}_{\Lambda}k_{a}k_{b}$ which is obviously
an (anti-)de Sitter background metric and where $\mathcal{H}_{m}k_{a}k_{b}$
now corresponds to the Kerr–Schild term.
The Ricci tensor takes the form (2.91) with a null radiation term aligned with
the Kerr–Schild vector $\boldsymbol{k}$, where $\Phi$ is given by
$\Phi=\frac{n-2}{r^{n-2}}\left((n-1)m\left(\log P\right)_{,u}-m_{,u}\right).$
(2.218)
As already mentioned above, this higher dimensional solution is of the Weyl
type D with an expanding, non-twisting and non-shearing null congruence
corresponding to the Kerr–Schild vector $\boldsymbol{k}$. An arbitrary motion
of a photon rocket can be prescribed by an appropriate choice of the functions
$z^{a}(u)$ defining a timelike worldline in the flat background metric with
$u$ being the proper time. Such various trajectories and corresponding
radiation patterns of emitted photons and mass loss of the rocket were studied
in [48].
Note that, for a rocket at rest, photons are radiated isotropically and the
Kinnersley metric reduces to the spherically symmetric Vaidya solution.
##### Vaidya shining star
The four-dimensional Vaidya solution originally found in [49] is a non-static
generalization of the Schwarzschild metric representing a spherically
symmetric star losing mass which is carried away by emitted radiation. The
higher dimensional Vaidya metric obtained in the Ricci-flat case [24] and
subsequently in the case of Einstein spaces [50] can be transformed to the KS
form (2.1) with the flat background metric as
$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)}$
(2.219) $\displaystyle\qquad+2\left[\frac{m(u)}{r^{n-3}}+\frac{\Lambda
r^{2}}{(n-2)(n-1)}\right]\left(\mathrm{d}t-\mathrm{d}r\right)^{2}$
and differs from the Schwarzschild–Tangherlini metric (2.8) in the way that
the mass $m(u)$ may vary and depends on the retarded time $u=t-r$. This leads
to the null radiation term
$\Phi=\frac{(n-2)}{r^{n-2}}m_{,u},$ (2.220)
appearing in the Ricci tensor of the form (2.91). The metric (2.219) can be
transformed to the GKS form (2.11) with an (A)dS background by introducing a
new time coordinate $t^{\prime}$
$t^{\prime}=t+\int\frac{\lambda r^{2}}{1-\lambda r^{2}}\,\mathrm{d}r,$ (2.221)
where $\lambda$ is given by the cosmological constant $\Lambda$ via (2.185),
or explicitly for a positive and negative cosmological constant
$t^{\prime}=\begin{cases}t-r+\frac{1}{\sqrt{\lambda}}\tanh^{-1}\sqrt{\lambda}r&\quad\lambda>0,\\\
t-r+\frac{1}{\sqrt{-\lambda}}\tan^{-1}\sqrt{-\lambda}r&\quad\lambda<0,\end{cases}$
(2.222)
respectively. Therefore, the cosmological constant $\Lambda$ moves from the
Kerr–Schild function $\mathcal{H}$ to the background metric and one thus
obtains
$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=-(1-\lambda
r^{2})\,\mathrm{d}t^{\prime 2}+\frac{\mathrm{d}r^{2}}{1-\lambda
r^{2}}+r^{2}\,\mathrm{d}\Omega^{2}_{(n-2)}$ (2.223)
$\displaystyle\qquad+\frac{2m(u)}{r^{n-3}}\left(\mathrm{d}t^{\prime}-\frac{\mathrm{d}r}{1-\lambda
r^{2}}\right)^{2},$
where the first three terms represent the corresponding (anti-)de Sitter
background metric in the spherical coordinates.
Note that some other generalizations of the Vaidya metric are known. Four-
dimensional charged Vaidya metrics in the electro-vacuum case were
investigated in [51]. It was shown there that the four-dimensional
Reissner–Nordström–Vaidya metric is of Petrov type D and the four-dimensional
Kerr–Newman–Vaidya solution is of Petrov type II unlike the Kerr–Newman black
hole which is of type D. Vaidya solutions were studied also in the
Einstein–Maxwell theory with sources, for instance, a four-current
proportional to the Kerr–Schild vector $\boldsymbol{k}$ was considered in
[52].
#### 2.6.7 Warped expanding Einstein generalized
Kerr–Schild spacetimes
In section 2.5.2, we employed the Brinkmann warp product, presented in section
1.4, in order to generate examples of higher dimensional non-expanding
Einstein GKS metrics from known four-dimensional Einstein Kundt metrics and
higher dimensional VSI metrics. In this way we obtained solutions with one
extra dimension and this also allowed us to introduce cosmological constant to
Ricci-flat metrics.
Naturally, one may use this method also in the case of expanding Einstein GKS
spacetimes. We already know that the Brinkmann warp product (1.37) preserves
the Weyl type of algebraically special spacetimes. Furthermore, as we have
shown in section 2.3, it preserves the GKS form as well. Therefore, starting
with an $(n-1)$-dimensional expanding Einstein GKS seed metric, which is of
Weyl type II or D by proposition 5, we construct an $n$-dimensional Einstein
GKS metric of the same Weyl type as the seed. This implies that the warped
metric has non-zero expansion.
More precisely, the optical matrices of general seed and warped metrics in
parallelly propagated frames satisfy [46]
$L_{ij}=\tilde{L}_{ij},\qquad L_{i,n-1}=L_{n-1,j}=L_{n-1,n-1}=0,$ (2.224)
where $i,j=2,\ldots,n-2$. Obviously, the optical matrix $L_{ij}$ of the warped
metric is degenerate and the corresponding optical scalars read
$\theta=\frac{n-3}{n-2}\tilde{\theta},\qquad\sigma^{2}=\tilde{\sigma}^{2}+\frac{n-3}{n-2}\tilde{\theta}^{2},\qquad\omega^{2}=\tilde{\omega}^{2}.$
(2.225)
The zeros in the last row and column of the optical matrix $L_{ij}$ lead to
the fact that an expanding warped metric is shearing even if the seed metric
is shear-free. Thus, for instance, metrics belonging to the Robinson–Trautman
class do not remain within this class after application of the Brinkmann warp
product.
The simplest example of a warped expanding GKS metric is an anti-de Sitter
spacetime which has a non-trivial Kerr–Schild term with expanding
$\boldsymbol{k}$ if it is rewritten in the KS form as in (2.9). It can be
obtained by using (1.49), i.e. the case $\tilde{R}=0$, $R<0$, and taking the
Minkowski spacetime as a seed $\mathrm{d}\tilde{s}^{2}$. Then the warp product
leads to the metric expressed in the form
$\mathrm{d}s^{2}=\mathrm{d}y^{2}+e^{2\sqrt{-\lambda}y}\,\eta_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}.$
(2.226)
In fact, these coordinates cover only a part of the complete AdS spacetime and
a similar five-dimensional metric was already considered to solve the
hierarchy problem in the Randall–Sundrum braneworld scenario [53, 54]
$\mathrm{d}s^{2}=\mathrm{d}y^{2}+e^{-2\sqrt{-\lambda}|y|}\,\eta_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b},$
(2.227)
where the absolute value causes a jump discontinuity in the first derivatives
of the metric leading subsequently to a delta function in the second
derivatives. Effectively, there appears a term proportional to $\delta(y)$ in
the Ricci tensor corresponding to an energy–momentum tensor lying on the brane
$y=0$.
Of course, one may choose any Einstein spacetime as a seed in (2.227) instead
of the flat Minkowski metric $\eta_{ab}$. For instance, substituting the
Schwarzschild metric as a slice $\mathrm{d}\tilde{s}^{2}$, we obtain the
Chamblin–Hawking–Reall black hole on a brane
$\mathrm{d}s^{2}=\mathrm{d}y^{2}+e^{-2\sqrt{-\lambda}|y|}\,\mathrm{d}\tilde{s}^{2},$
(2.228)
discussed in [55].
In general, choosing expanding Einstein GKS metrics representing black holes
as a seed, one obtains black string solutions. It can be easily seen directly
from the form of the warp product with a flat extra dimension (1.46)–(1.50)
that an $(n-1)$-dimensional slice $\mathrm{d}\tilde{s}^{2}$ of the full
$n$-dimensional metric $\mathrm{d}s^{2}$ is multiplied by a conformal factor
depending on the extra dimension coordinate $y$ and therefore these black
strings are in general non-homogeneous. They are homogeneous only in the case
of direct product $\tilde{R}=0,R=0$.
Thus, one may generate various black string solutions by warping appropriate
black holes, for instance, taking the Kerr–(A)dS metric as a seed, we get
spinning black string. Nevertheless, let us remind that these warped
spacetimes suffer from naked singularity at points where the warp factor
vanishes, i.e. $f(z)=0$, except the cases $\tilde{R}=0,R=0$ and
$\tilde{R}<0,R<0$ when $f(z)$ does not admit a root.
Here we present only an example of the latter case, i.e. $n$-dimensional black
string with a negative cosmological constant $\Lambda$ sliced by an
$(n-1)$-dimensional Kerr–AdS metric (2.179)–(2.183). Using the form conformal
to a direct product (1.43), the warped metric can be expressed in the GKS form
(2.11) with the background metric $\bar{g}_{ab}$, Kerr–Schild vector
$\boldsymbol{k}$ and function $\mathcal{H}$ given by
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=\frac{1}{\cos^{2}(\sqrt{-\lambda}x)}\Bigg{[}\mathrm{d}x^{2}-W(1-\lambda
r^{2})\,\mathrm{d}t^{2}+F\,\mathrm{d}r^{2}$
$\displaystyle\qquad+\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{r^{2}+a_{i}^{2}}{1+\lambda
a_{i}^{2}}\,\mathrm{d}\mu_{i}^{2}+\sum^{\lfloor
n/2-1\rfloor}_{i=1}\frac{r^{2}+a_{i}^{2}}{1+\lambda
a_{i}^{2}}\mu_{i}^{2}\,\mathrm{d}\phi_{i}^{2}$
$\displaystyle\qquad+\frac{\lambda}{W(1-\lambda
r^{2})}\left(\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{(r^{2}+a_{i}^{2})\mu_{i}\,\mathrm{d}\mu_{i}}{1+\lambda
a_{i}^{2}}\right)^{2}\Bigg{]},$ (2.229) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\frac{1}{\cos(\sqrt{-\lambda}x)}\Bigg{[}W\,\mathrm{d}t+F\,\mathrm{d}r-\sum^{\lfloor
n/2-1\rfloor}_{i=1}\frac{a_{i}\mu_{i}^{2}}{1+\lambda
a_{i}^{2}}\,\mathrm{d}\phi_{i}\Bigg{]},$ (2.230) $\displaystyle\mathcal{H}$
$\displaystyle=\begin{cases}-\frac{M}{r\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\prod^{\lfloor
n/2-1\rfloor}_{j=1}(r^{2}+a_{j}^{2})}&\text{if $n$ is even,}\\\
-\frac{M}{\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\prod^{\lfloor(n-1)/2\rfloor}_{j=1}(r^{2}+a_{j}^{2})}&\text{if
$n$ is odd,}\end{cases}$ (2.231)
where the functions $W$ and $F$ are defined as
$W=\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{\mu_{i}^{2}}{1+\lambda
a_{i}^{2}},\qquad F=\frac{r^{2}}{1-\lambda
r^{2}}\sum^{\lfloor(n-1)/2\rfloor}_{i=1}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}},$
(2.232)
$\lambda<0$ is related to the cosmological constant $\Lambda$ via (2.185) and
the coordinate $x$ is subject to
$-\frac{\pi}{2}<\sqrt{-\lambda}x<\frac{\pi}{2}$. In the case $n=5$, such
metrics have been already constructed in [17].
Let us remark that if we set $\lambda=0$, the black string metric
(2.230)–(2.231) reduces to the direct product of an $(n-1)$-dimensional
Myers–Perry black hole with a flat extra dimension since, in this limit, the
form of a warped metric (1.43) for $\tilde{R}<0,R<0$ smoothly transforms to
the form (1.41) for $\tilde{R}=0,R=0$ and again a naked singularity is not
present.
### 2.7 Other generalizations of the Kerr–Schild
ansatz
Apart from the GKS metric (2.11), there are other possible generalizations of
the original Kerr–Schild ansatz (2.1) that also lead to exact solutions. One
of such examples, referred to as the extended Kerr–Schild ansatz, was proposed
in [56]. In fact, it was shown that the CCLP solution [57] representing
charged rotating black holes in five-dimensional minimal gauged supergravity
previously found by the trial and error method can be put into this form.
Besides an (anti-)de Sitter background metric and the null Kerr–Schild
congruence appearing in the GKS form, this ansatz also consists of an
additional spacelike vector field and will be studied in more detail in
chapter 3.
Another extension of the Kerr–Schild ansatz was also discovered by rewriting
an already known solution. If one performs a Wick rotation of the higher
dimensional Kerr–NUT–(A)dS spacetime [58] in order to change the signature
$(1,n-1)$ of the metric either to the signature $(k,k+1)$ in odd dimension
$n=2k+1$ or to the signature $(k,k)$ in even dimension $n=2k$, respectively,
then the resulting metric can be cast to the multi-Kerr–Schild form [59]
$g_{ab}=\bar{g}_{ab}-2\sum^{k}_{\mu=1}\mathcal{H}_{\mu}k^{(\mu)}_{a}k^{(\mu)}_{b}.$
(2.233)
In odd dimensions, the (A)dS background metric $\bar{g}_{ab}$, the Kerr–Schild
vectors $\boldsymbol{k}^{(\mu)}$ and the functions $\mathcal{H}_{\mu}$ of the
Kerr–NUT–(A)dS spacetime are given by
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=\frac{W}{\prod^{n}_{i=1}\Xi_{i}}\,\mathrm{d}\tau^{2}+\sum_{\mu=1}^{n}\frac{U_{\mu}}{\bar{X}_{\mu}}\,\mathrm{d}x_{\mu}^{2}-\sum_{i=1}^{n}\frac{\gamma_{i}}{\Xi_{i}\prod_{j\neq
i}(a_{i}^{2}-a_{j}^{2})}\,\mathrm{d}\varphi_{i}^{2},$ (2.234) $\displaystyle
k^{(\mu)}_{a}\,\mathrm{d}x^{a}$ $\displaystyle=\frac{W}{1+\lambda
x_{\mu}^{2}}\frac{\mathrm{d}\tau}{\prod^{k}_{i=1}\Xi_{i}}-\frac{U_{\mu}\,\mathrm{d}x_{\mu}}{\bar{X}_{\mu}}$
$\displaystyle\qquad-\sum_{i=1}^{k}\frac{a_{i}\gamma_{i}\,\mathrm{d}\varphi_{i}}{(a_{i}^{2}-x_{\mu}^{2})\Xi_{i}\prod_{j\neq
i}(a_{i}^{2}-a_{j}^{2})},$ (2.235) $\displaystyle\mathcal{H}_{\mu}$
$\displaystyle=-\frac{b_{\mu}}{U_{\mu}},\qquad\bar{X}_{\mu}=\frac{(1+\lambda
x_{\mu}^{2})}{x_{\mu}^{2}}\prod_{i-1}^{k}(a_{i}^{2}-x_{\mu}^{2}),$ (2.236)
whereas, in even dimensions, they can be expressed as
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=\frac{W}{\prod^{k-1}_{i=1}\Xi_{i}}\,\mathrm{d}\tau^{2}+\sum_{\mu=1}^{k}\frac{U_{\mu}}{\bar{X}_{\mu}}\,\mathrm{d}x_{\mu}^{2}-\sum_{i=1}^{k-1}\frac{\gamma_{i}}{a_{i}^{2}\Xi_{i}\prod_{j\neq
i}(a_{i}^{2}-a_{j}^{2})}\,\mathrm{d}\varphi_{i}^{2},$ (2.237) $\displaystyle
k^{(\mu)}_{a}\,\mathrm{d}x^{a}$ $\displaystyle=\frac{W}{1+\lambda
x_{\mu}^{2}}\frac{\mathrm{d}\tau}{\prod^{k-1}_{i=1}\Xi_{i}}-\frac{U_{\mu}\,\mathrm{d}x_{\mu}}{\bar{X}_{\mu}}$
$\displaystyle\qquad-\sum_{i=1}^{k-1}\frac{\gamma_{i}\,\mathrm{d}\varphi_{i}}{(a_{i}^{2}-x_{\mu}^{2})a_{i}\Xi_{i}\prod_{j\neq
i}(a_{i}^{2}-a_{j}^{2})},$ (2.238) $\displaystyle\mathcal{H}_{\mu}$
$\displaystyle=-\frac{b_{\mu}x_{\mu}}{U_{\mu}},\qquad\bar{X}_{\mu}=-(1+\lambda
x_{\mu}^{2})\prod_{i=1}^{k-1}(a_{i}^{2}-x_{\mu}^{2}),$ (2.239)
where $\lambda$ is given as in (2.185), $b_{\mu}$ corresponds to the mass and
NUT parameters and $\Xi_{i}$, $\gamma_{i}$, $W$, $U_{\mu}$ are defined as
$\displaystyle\Xi_{i}$ $\displaystyle=1+\lambda a_{i}^{2},$
$\displaystyle\qquad\gamma_{i}$
$\displaystyle=\prod_{\nu=1}^{k}(a_{i}^{2}-x_{\nu}^{2}),$ (2.240)
$\displaystyle W$ $\displaystyle=\prod_{\nu=1}^{k}(1+\lambda x_{\nu}^{2}),$
$\displaystyle\qquad U_{\mu}$
$\displaystyle=\prod_{\nu\neq\mu}(x_{\nu}^{2}-x_{\mu}^{2}).$
Each of the scalar functions $\mathcal{H}_{\mu}$ contains just one of the mass
or NUT parameters which appears in $\mathcal{H}_{\mu}$ linearly. Therefore,
the NUT parameters enter the metric of this form in an analogous way as the
mass parameter. The vectors $\boldsymbol{k}^{(\mu)}$ are linearly independent,
mutually orthogonal and tangent to the corresponding affinely parametrized
geodetic null congruences.
The last presented example of possible generalizations of the Kerr–Schild
ansatz has appeared recently in [60] where the higher dimensional charged
rotating Kaluza–Klein AdS black hole has been obtained in the form
$g_{ab}=H^{\frac{1}{D-2}}\left(\bar{g}_{ab}+\frac{2m}{UH}k_{a}k_{b}\right),\qquad
A_{a}=\frac{2ms}{UH}k_{a},\qquad\Phi=-\frac{\ln(H)}{D-2},$ (2.241)
as a solution of the Einstein–Maxwell–dilaton theory with the Lagrangian
$\displaystyle\mathcal{L}$
$\displaystyle=\sqrt{-g}\bigg{(}R-\frac{1}{4}(n-1)(n-2)(\partial\Phi)^{2}-\frac{1}{4}e^{-(n-1)\Phi}\mathcal{F}^{2}$
$\displaystyle\qquad+g^{2}(n-1)\left((n-3)e^{\Phi}+e^{-(n-3)\Phi}\right)\bigg{)}.$
(2.242)
The background metric $\bar{g}_{ab}$ represents again an AdS spacetime, but
now $\boldsymbol{k}$ is a timelike vector field
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=-\left(1+g^{2}r^{2}\right)W\,\mathrm{d}\bar{t}^{2}+\sum_{i=1}^{N+\epsilon}\frac{r^{2}+a_{i}^{2}}{\chi_{i}}\,\mathrm{d}\mu_{i}^{2}+\sum_{i=1}^{N}\frac{r^{2}+a_{i}^{2}}{\chi_{i}}\mu_{i}^{2}\,\mathrm{d}\bar{\phi}_{i}^{2}$
$\displaystyle\qquad+F\,\mathrm{d}r^{2}-\frac{g^{2}}{\left(1+g^{2}r^{2}\right)W}\left(\sum_{i=1}^{N+\epsilon}\frac{r^{2}+a_{i}^{2}}{\chi_{i}}\mu_{i}\,\mathrm{d}\mu_{i}\right)^{2},$
(2.243) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=cW\,\mathrm{d}\bar{t}+\sqrt{f(r)}F\,\mathrm{d}r-\sum_{i=1}^{N}\frac{a_{i}\sqrt{\Xi_{i}}}{\chi_{i}}\mu_{i}^{2}\,\mathrm{d}\bar{\phi}_{i},$
(2.244)
where the functions $H$, $U$, $W$, $F$, $f(r)$, $\Xi_{i}$ and $\chi_{i}$ are
defined as
$\displaystyle H$ $\displaystyle=1+\frac{2ms^{2}}{U},\qquad
U=r^{\epsilon}\sum_{i=1}^{N+\epsilon}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\prod_{j=1}^{N}\left(r^{2}+a_{j}^{2}\right),$
(2.245) $\displaystyle F$
$\displaystyle=\frac{r^{2}}{1+g^{2}r^{2}}\sum_{i=1}^{N+\epsilon}\frac{\mu_{i}^{2}}{r^{2}+a_{i}^{2}},\qquad
f(r)=c^{2}-s^{2}\left(1+g^{2}r^{2}\right),$ $\displaystyle W$
$\displaystyle=\sum_{i=1}^{N+\epsilon}\frac{\mu_{i}^{2}}{\chi_{i}},\qquad\Xi_{i}=c^{2}-s^{2}\chi_{i},\qquad\chi_{i}=1-g^{2}a_{i}^{2}$
and $c\equiv\cosh\delta$, $s\equiv\sinh\delta$ with $\delta$ being a charge.
The timelike vector field $\boldsymbol{k}$ is geodetic and its norm with
respect to the background metric depends on the charge
$\bar{g}_{ab}k^{a}k^{b}=-s^{2}$. Note that in the uncharged case, the metric
(2.242) reduces to the GKS form (2.11) and the vector $\boldsymbol{k}$ becomes
null.
## Chapter 3 Extended Kerr–Schild spacetimes
The aim of this chapter is to investigate general properties of an extension
of the original Kerr–Schild ansatz, referred to as the extended Kerr–Schild
ansatz, in a similar way as it is performed for GKS spacetimes in chapter 2.
Later, we may find the results of such analysis useful in finding new exact
solutions in this form. In fact, this extension has been already studied in
[61] using the method of perturbative expansion. It has been shown that the
vacuum field equations truncate beyond a certain low order in an expansion
around the background metric similarly as in the case of GKS spacetimes. Here
we employ the Newman–Penrose formalism which allows us to formulate some
statements about geodesicity and optical properties of the null congruence and
Weyl types of extended Kerr–Schild spacetimes. However, these results have not
been published yet since the analysis is not completed and some aspects can be
further investigated.
We define the extended Kerr–Schild ansatz (xKS) as a metric of the form
$g_{ab}=\bar{g}_{ab}-2\mathcal{H}k_{a}k_{b}-2\mathcal{K}k_{(a}m_{b)},$ (3.1)
where the background metric $\bar{g}_{ab}$ represents an (anti-)de Sitter or
Minkowski spacetime, $\mathcal{H}$ and $\mathcal{K}$ are scalar functions,
$\boldsymbol{k}$ is a null vector and $\boldsymbol{m}$ is a spacelike unit
vector both with respect to the full metric
$k^{a}k_{a}\equiv g_{ab}k^{a}k^{b}=0,\qquad k^{a}m_{a}\equiv
g_{ab}k^{a}m^{b}=0,\qquad m^{a}m_{a}\equiv g_{ab}m^{a}m^{b}=1.$ (3.2)
From the form of the xKS metric (3.1) it then follows that the same holds also
with respect to the background metric
$\bar{g}_{ab}k^{a}k^{b}=0,\qquad\bar{g}_{ab}k^{a}m^{b}=0,\qquad\bar{g}_{ab}m^{a}m^{b}=1$
(3.3)
and the inverse metric is simply given by
$g^{ab}=\bar{g}^{ab}+\left(2\mathcal{H}-\mathcal{K}^{2}\right)k^{a}k^{b}+2\mathcal{K}k^{(a}m^{b)}.$
(3.4)
Note that our definition of the xKS ansatz (3.1) slightly differs from those
ones in [56, 61] by the factors since we follow the notation of the GKS form
(2.11) and also by the fact that we consider the spacelike vector
$\boldsymbol{m}$ to be normalized to unity since we will identify it with one
of the frame vectors.
First, let us provide a motivation for studying xKS spacetimes. As will be
shown in section 3.2, one of the reasons why to consider xKS ansatz (3.1) is
that such metrics cover more general algebraic types than the GKS metrics
(2.11). Recall the results of chapter 2 that GKS spacetimes with a geodetic
Kerr–Schild vector $\boldsymbol{k}$ without any further assumptions are of
Weyl type II or more special. Expanding Einstein GKS spacetimes are compatible
only with types II or D unless conformally flat and non-expanding Einstein GKS
spacetimes are of type N and belong to the Kundt class. Therefore, expanding
Einstein xKS spacetimes could include black hole solutions of a more general
Weyl type than II, for instance black rings [62] that are of type Ii [63].
Non-expanding Einstein xKS spacetimes could contain Kundt metrics of a more
general type than N.
Furthermore, unlike the static charged black hole, rotating charged black hole
as an exact solution of higher dimensional Einstein–Maxwell theory is unknown.
The four-dimensional Kerr–Newman black hole can be cast to the KS form (2.1)
with the background metric $\eta_{ab}$, the Kerr–Schild vector
$\boldsymbol{k}$ and the function $\mathcal{H}$ given by
$\displaystyle\eta_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}=-\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2},\phantom{\frac{1}{2}}$
(3.5) $\displaystyle
k_{a}\,\mathrm{d}x^{a}=\mathrm{d}t+\frac{rx+ay}{r^{2}+a^{2}}\,\mathrm{d}x+\frac{ry-
ax}{r^{2}+a^{2}}\,\mathrm{d}y+\frac{z}{r}\,\mathrm{d}z,$
$\displaystyle\mathcal{H}=-\frac{r^{2}}{r^{4}+a^{2}z^{2}}\left(Mr-\frac{Q^{2}}{2}\right),$
where $M$, $Q$ are mass and charge of the black hole, respectively, the
coordinate $r$ satisfy
$\frac{x^{2}+y^{2}}{r^{2}+a^{2}}+\frac{z^{2}}{r^{2}}=1$ (3.6)
and the vector potential is proportional to the Kerr–Schild vector
$\boldsymbol{k}$
$A=\frac{Qr^{3}}{r^{4}+a^{2}z^{2}}\boldsymbol{k}.$ (3.7)
However, the attempt to generalize this ansatz to higher dimensions using the
KS form of the Myers–Perry black hole has failed [22]. It has turned out that
a vector potential proportional to $\boldsymbol{k}$ cannot simultaneously
satisfy the corresponding Einstein and Maxwell field equations. An open
question is whether the xKS ansatz may resolve these problems.
Another significant reason is that some of the known exact solutions can be
cast to the xKS form. For instance, VSI metrics are examples of such xKS
spacetimes as will be pointed out in section 3.3. Some expanding xKS
spacetimes representing black holes are also known, namely CCLP solution [57]
discussed in section 3.4. In fact, the investigation of the CCLP metric has
led to the xKS ansatz introduced in [56].
Lastly, the xKS metric seems to be still sufficiently simple to be treated
analytically. As in the case of GKS spacetimes in chapter 2, we assume that
the background metric $\bar{g}_{ab}$ representing an (anti-)de Sitter or
Minkowski spacetime takes the conformally flat form (2.12) with a conformal
factor (2.13) and with the flat Minkowski metric in Cartesian coordinates
(2.14). Note that throughout this chapter we do not discuss the limit
$\mathcal{K}=0$ when the xKS ansatz (3.1) reduces to the GKS metric (2.11)
studied in chapter 2.
### 3.1 General Kerr–Schild vector field
As in the case of the GKS ansatz, it can be seen directly from the form of the
xKS metric (3.1) and its inverse (3.4) that the index of the Kerr–Schild
vector $\boldsymbol{k}$ can be raised and lowered by both the full metric
$g_{ab}$ and the background metric $\bar{g}_{ab}$. The form of the xKS ansatz
also implies that it is convenient to employ the higher dimensional
Newman–Penrose formalism reviewed in section 1.1 and identify the null and
spacelike vectors $\boldsymbol{k}$, $\boldsymbol{m}$ appearing in the xKS
metric (3.1) with the vectors $\boldsymbol{\ell}$, $\boldsymbol{m}^{(2)}$ of
the null frame (1.8), respectively. The corresponding Ricci rotation
coefficients will be denoted as $L_{ab}$ and
$M_{ab}\equiv\mbox{$\stackrel{{\scriptstyle 2}}{{M}}_{ab}$}$.
Despite the fact that the xKS metric (3.1) has only few additional terms with
respect to the GKS metric (2.11), the Christoffel symbols contain several
times more terms and it is convenient or in fact necessary to use the computer
algebra system Cadabra [33, 34]. Although the calculations are much more
involved than in the case of GKS spacetimes the boost weight 2 component of
the Ricci tensor $R_{00}=R_{ab}k^{a}k^{b}$ contains again many vanishing
contractions with the null Kerr–Schild vector $\boldsymbol{k}$ leading to the
simple expression
$\displaystyle R_{00}$
$\displaystyle=2\mathcal{H}L_{i0}L_{i0}-\frac{1}{2}\mathcal{K}^{2}L_{\tilde{\imath}0}L_{\tilde{\imath}0}+2\mathcal{K}L_{i(i}L_{2)0}+\mathcal{K}L_{\tilde{\imath}0}M_{\tilde{\imath}0}+2\mathrm{D}\mathcal{K}L_{20}$
(3.8)
$\displaystyle\qquad+\mathcal{K}\mathrm{D}L_{20}-\frac{1}{2}(n-2)\left(\frac{\Omega_{,ab}}{\Omega}-\frac{3}{2}\frac{\Omega_{,a}\Omega_{,b}}{\Omega^{2}}\right)k^{a}k^{b},$
where apart from the indices $i,j,\ldots=2,\ldots,n-1$ employed throughout the
thesis, we define new indices
$\tilde{\imath},\tilde{\jmath},\ldots=3,\ldots,n-1$ denoted by tilde such that
the vector $\boldsymbol{m}$ is excluded in the notation
$\boldsymbol{m}^{(\tilde{\imath})}$.
Since the conformal factor $\Omega$ of an (anti-)de Sitter background
satisfies (2.15) or $\Omega=1$ in the case of Minkowski background, the last
term in (3.8) is identically zero. Moreover, the component $R_{00}$ completely
vanishes if $L_{i0}=0$ and therefore
###### Proposition 6
If the Kerr–Schild vector $\boldsymbol{k}$ in the extended Kerr–Schild metric
(3.1) is geodetic then the boost weight 2 component of the energy–momentum
tensor $T_{00}=T_{ab}k^{a}k^{b}$ vanishes.
Note that if $\mathcal{K}=0$, the xKS metric (3.1) reduces to the GKS form
(2.11) and the implication of proposition 6 also holds in the opposite
direction as follows from proposition 1. However, in the case of xKS
spacetimes, we obtain only a sufficient condition and it is possible to set
$R_{00}$ to zero also for non-geodetic $\boldsymbol{k}$ by a special choice of
$\mathcal{H}$, $\mathcal{K}$ and $\boldsymbol{m}$.
Inspired by the relation (3.88) valid in CCLP spacetimes, we can formulate an
additional covariant condition in order to restrict the geometry of the
vectors $\boldsymbol{k}$ and $\boldsymbol{m}$ so that the implication of
proposition 6 becomes an equivalence. It turns out that such an convenient
condition is $\mathcal{L}_{\boldsymbol{k}}m_{a}\propto m_{a}$,
$\mathcal{L}_{\boldsymbol{m}}k_{a}=0$ since then $L_{20}=0$,
$M_{\tilde{\imath}0}=-L_{2\tilde{\imath}}=-L_{\tilde{\imath}2}$ as will be
shown in section 3.1.2 and therefore all terms apart from the first two in
(3.8) vanish
$R_{00}=\left(2\mathcal{H}-\frac{1}{2}\mathcal{K}^{2}\right)L_{\tilde{\imath}0}L_{\tilde{\imath}0},$
(3.9)
which ensures that the implication holds in both directions and we can
conclude that
###### Corollary 7
In the special case when $\mathcal{L}_{\boldsymbol{k}}m_{a}\propto m_{a}$ and
$\mathcal{L}_{\boldsymbol{m}}k_{a}=0$, the Kerr–Schild vector $\boldsymbol{k}$
in the extended Kerr–Schild metric (3.1) is geodetic if and only if the boost
weight 2 component of the energy–momentum tensor $T_{00}=T_{ab}k^{a}k^{b}$
vanishes.
#### 3.1.1 Kerr–Schild congruence in the background
Following section 2.1.1, the full metric $g_{ab}$ can be expressed in the null
frame (1.8) simply as
$g_{ab}=2k_{(a}n_{b)}+m_{a}m_{b}+\delta_{\tilde{\imath}\tilde{\jmath}}m_{a}^{(\tilde{\imath})}m_{b}^{(\tilde{\jmath})}$
(3.10)
and can be compared with the xKS form (3.1). Obviously, if one chooses
$\bar{n}_{a}=n_{a}+\mathcal{H}k_{a}+\mathcal{K}m_{a}$ (3.11)
then $\bar{\boldsymbol{n}}$, $\boldsymbol{k}$, $\boldsymbol{m}$,
$\boldsymbol{m}^{(\tilde{\imath})}$ form a null frame in the background metric
$\bar{g}_{ab}$. Although the indices of the vectors $\boldsymbol{k}$ and
$\boldsymbol{m}^{(\tilde{\imath})}$ can be raised and lowered by both the full
metric $g_{ab}$ and the background metric $\bar{g}_{ab}$ one has to operate
with the vectors $\bar{\boldsymbol{n}}$ and $\boldsymbol{m}$ carefully since
$\bar{n}^{a}\equiv
g^{ab}\bar{n}_{b}=n^{a}+\mathcal{H}k^{a}+\mathcal{K}m^{a},\qquad m^{a}\equiv
g^{ab}m_{a},$ (3.12)
whereas
$\bar{g}^{ab}\bar{n}_{a}=n^{b}-\mathcal{H}k^{b},\qquad\bar{g}^{ab}m_{a}=m^{b}-\mathcal{K}k^{b}.$
(3.13)
Similarly as for GKS spacetimes in section 2.1.1, the covariant derivative
compatible with the background metric $\bar{g}_{ab}$ can be easily obtained
from the covariant derivative compatible with the full metric $g_{ab}$ by
setting $\mathcal{H}=\mathcal{K}=0$. Thus, we can compare the Ricci rotation
coefficients constructed in the full spacetime using the frame
$\boldsymbol{k}$, $\boldsymbol{n}$, $\boldsymbol{m}$,
$\boldsymbol{m}^{(\tilde{\imath})}$ with those ones in the background
spacetime denoted by barred letters and expressed in terms of the frame
$\boldsymbol{k}$, $\bar{\boldsymbol{n}}$, $\boldsymbol{m}$,
$\boldsymbol{m}^{(\tilde{\imath})}$
$\displaystyle L_{i0}$ $\displaystyle=\bar{L}_{i0},\qquad
L_{10}=\bar{L}_{10},\qquad
L_{ij}=\bar{L}_{ij}-\mathcal{K}\delta_{2[i}\bar{L}_{j]0},\phantom{\frac{1}{2}}$
(3.14) $\displaystyle L_{1i}$
$\displaystyle=\bar{L}_{1i}-\mathcal{H}\bar{L}_{i0}-\frac{1}{2}\mathcal{K}(\bar{L}_{2i}+\bar{M}_{i0})+\frac{1}{2}\mathcal{K}\left(\bar{L}_{10}-\frac{\mathrm{D}\mathcal{K}}{\mathcal{K}}+\frac{\mathrm{D}\Omega}{\Omega}\right)\delta_{2i},$
(3.15) $\displaystyle L_{i1}$
$\displaystyle=\bar{L}_{i1}-\frac{1}{2}\mathcal{K}(\bar{L}_{2i}+\bar{M}_{i0})-\frac{1}{2}\mathcal{K}\left(\bar{L}_{10}+\frac{\mathrm{D}\mathcal{K}}{\mathcal{K}}-\frac{\mathrm{D}\Omega}{\Omega}\right)\delta_{2i},$
(3.16) $\displaystyle L_{11}$
$\displaystyle=\bar{L}_{11}-\mathcal{H}\bar{L}_{10}-\mathrm{D}\mathcal{H}+\mathcal{H}\frac{\mathrm{D}\Omega}{\Omega}-\mathcal{K}(L_{21}-N_{20}),$
(3.17) $\displaystyle M_{\tilde{\imath}0}$
$\displaystyle=\bar{M}_{\tilde{\imath}0}+\frac{1}{2}\mathcal{K}\bar{L}_{\tilde{\imath}0},\qquad
M_{\tilde{\imath}\tilde{\jmath}}=\bar{M}_{\tilde{\imath}\tilde{\jmath}}+\mathcal{K}\bar{S}_{\tilde{\imath}\tilde{\jmath}}-\frac{1}{2}\mathcal{K}\frac{\mathrm{D}\Omega}{\Omega}\delta_{\tilde{\imath}\tilde{\jmath}},$
(3.18) $\displaystyle M_{\tilde{\imath}2}$
$\displaystyle=\bar{M}_{\tilde{\imath}2}+\mathcal{K}(\bar{L}_{2\tilde{\imath}}+\bar{M}_{\tilde{\imath}0}),\phantom{\frac{1}{2}}$
(3.19) $\displaystyle M_{\tilde{\imath}1}$
$\displaystyle=\bar{M}_{\tilde{\imath}1}-\mathcal{H}\bar{L}_{\tilde{\imath}2}+(\mathcal{H}-\frac{1}{2}\mathcal{K})(\bar{L}_{2\tilde{\imath}}+\bar{M}_{\tilde{\imath}0})-\frac{1}{2}\mathcal{H}\mathcal{K}\bar{L}_{\tilde{\imath}0}+\mathcal{K}\bar{L}_{(1\tilde{\imath})}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\bar{M}_{\tilde{\imath}2}+\frac{1}{2}\delta_{\tilde{\imath}}\mathcal{K},$
(3.20) $\displaystyle N_{i0}$
$\displaystyle=\bar{N}_{i0}-\frac{1}{2}\mathcal{K}(\bar{L}_{2i}+\bar{M}_{i0})-\frac{1}{2}\left(\mathcal{K}\bar{L}_{10}+\mathrm{D}\mathcal{K}\right)\delta_{2i},$
(3.21) $\displaystyle N_{21}$
$\displaystyle=\bar{N}_{21}+\delta_{2}\mathcal{H}+2\mathcal{H}\bar{L}_{12}-\mathcal{H}\mathcal{K}\bar{L}_{22}-\mathcal{H}^{2}\bar{L}_{20}+\mathcal{H}\bar{N}_{20}-\mathcal{H}\bar{L}_{21}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\mathcal{K}\bar{N}_{22}+\frac{1}{2}\mathcal{K}(\mathcal{H}+\frac{1}{2}\mathcal{K}^{2})\frac{\mathrm{D}\Omega}{\Omega}-\frac{1}{2}\mathcal{K}\frac{\mbox{$\bigtriangleup$}\Omega}{\Omega}+\frac{3}{2}\mathcal{K}^{2}\frac{\delta_{2}\Omega}{\Omega}+\mathcal{H}\mathcal{K}\bar{L}_{10}$
$\displaystyle\qquad+\mathcal{K}^{2}(\bar{L}_{21}-\bar{N}_{20})-\mbox{$\bigtriangleup$}\mathcal{K},\phantom{\frac{1}{2}}$
(3.22) $\displaystyle N_{i1}$
$\displaystyle=\bar{N}_{i1}+\delta_{i}\mathcal{H}+2\mathcal{H}\bar{L}_{1i}-\mathcal{H}\mathcal{K}(\bar{L}_{2i}+\bar{M}_{i0})-\mathcal{H}^{2}\bar{L}_{i0}+\mathcal{H}\bar{N}_{i0}-\mathcal{H}\bar{L}_{i1}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\mathcal{K}\bar{M}_{i1}-\mathcal{K}\bar{N}_{2i}+\frac{1}{2}\mathcal{K}^{2}(1-\mathcal{K})(\bar{L}_{2i}+\bar{M}_{i0}),$
(3.23) $\displaystyle N_{i2}$
$\displaystyle=\bar{N}_{i2}+\mathcal{H}\bar{L}_{2i}-\frac{1}{2}\mathcal{H}\mathcal{K}\bar{L}_{i0}+\frac{1}{2}\delta_{i}\mathcal{K}+\mathcal{K}\bar{L}_{[1i]}-\frac{1}{2}\mathcal{K}^{2}(\bar{L}_{2i}+\bar{M}_{i0})$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\bar{M}_{i2}+\mathcal{K}\bar{N}_{i0},$
(3.24) $\displaystyle N_{22}$
$\displaystyle=\bar{N}_{22}-\frac{1}{2}\mathcal{K}\frac{\delta_{2}\Omega}{\Omega}+\mathcal{H}\bar{L}_{22}-(\mathcal{H}+\frac{1}{2}\mathcal{K}^{2})\frac{\mathrm{D}\Omega}{\Omega}-\mathcal{K}(\bar{L}_{21}-\bar{N}_{20}),$
(3.25) $\displaystyle N_{2i}$
$\displaystyle=\bar{N}_{2i}+\mathcal{H}\bar{L}_{i2}+\frac{1}{2}\mathcal{H}\mathcal{K}\bar{L}_{i0}-\frac{1}{2}\delta_{i}\mathcal{K}-\mathcal{K}\bar{L}_{(1i)}+\frac{1}{2}\mathcal{K}^{2}(\bar{L}_{2i}+\bar{M}_{i0})$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\bar{M}_{i2},$ (3.26)
$\displaystyle N_{ij}$
$\displaystyle=\bar{N}_{ij}+\mathcal{H}\bar{L}_{ji}-\frac{1}{2}\mathcal{K}\bar{M}_{ij}+\frac{1}{2}\mathcal{K}\bar{M}_{ji}-\mathcal{H}\frac{\mathrm{D}\Omega}{\Omega}\delta_{ij}-\frac{1}{2}\mathcal{K}\frac{\delta_{2}\Omega}{\Omega}\delta_{ij},$
(3.27) $\stackrel{{\scriptstyle\tilde{\imath}}}{{M}}_{\tilde{\jmath}0}$
$\displaystyle=\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{\bar{M}}}_{\tilde{\jmath}0}$},\qquad\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{M}}_{\tilde{\jmath}k}$}=\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{\bar{M}}}_{\tilde{\jmath}k}$}+\mathcal{K}\left(\bar{L}_{[ij]}+\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{\bar{M}}}_{\tilde{\jmath}0}$}\right)\delta_{2k},$
(3.28) $\stackrel{{\scriptstyle\tilde{\imath}}}{{M}}_{\tilde{\jmath}1}$
$\displaystyle=\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{\bar{M}}}_{\tilde{\jmath}1}$}+2\mathcal{H}\bar{L}_{[\tilde{\imath}\tilde{\jmath}]}+\mathcal{H}\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{\bar{M}}}_{\tilde{\jmath}0}$}+\mathcal{K}\bar{M}_{[\tilde{\imath}\tilde{\jmath}]}.\phantom{\frac{1}{2}}$
(3.29)
From (3.14) it follows that the Kerr–Schild vector $\boldsymbol{k}$ is
geodetic with respect to the full spacetime $g_{ab}$ if and only if it is
geodetic with respect to the background spacetime $\bar{g}_{ab}$. Moreover, an
affine parametrization in the former spacetime corresponds to the affine
parametrization in the latter spacetime and vice versa. Note also that a
geodetic $\boldsymbol{k}$ has the same optical properties in the full and
background spacetimes since the corresponding optical matrices are equal
$L_{ij}=\bar{L}_{ij}$.
#### 3.1.2 Relation of the vector fields $\boldsymbol{k}$ and
$\boldsymbol{m}$
Motivated by the observation that the congruences $\boldsymbol{k}$ and
$\hat{\boldsymbol{m}}$ of the CCLP black hole obey (3.88), let us study the
consequences of the relation
$(m_{a;b}-m_{b;a})k^{b}=-\frac{\mathrm{D}\zeta}{\zeta}m_{a},\qquad(k_{a;b}-k_{b;a})m^{b}=0,$
(3.30)
for general xKS spacetimes. The contractions of the first equation in (3.30)
with the vectors $\boldsymbol{n}$, $\boldsymbol{m}$ and
$\boldsymbol{m}^{(\tilde{\imath})}$ give
$L_{21}-N_{20}=0,\qquad L_{22}=-\frac{\mathrm{D}\zeta}{\zeta},\qquad
L_{2\tilde{\imath}}+M_{\tilde{\imath}0}=0,$ (3.31)
respectively. Similarly, the contractions of the second equation in (3.30)
with $\boldsymbol{k}$, $\bar{\boldsymbol{n}}$ and
$\boldsymbol{m}^{(\tilde{\imath})}$ lead to
$L_{20}=0,\qquad L_{[12]}=0,\qquad L_{[2\tilde{\imath}]}=0,$ (3.32)
respectively. We can also express the relations (3.31) and (3.32) for the
Ricci rotation coefficients in the background spacetime using (3.14)–(3.29)
$\displaystyle\bar{L}_{20}=L_{20},\qquad\bar{L}_{21}-\bar{N}_{20}=L_{21}-N_{20}-\frac{1}{2}\mathcal{K}\frac{\mathrm{D}\Omega}{\Omega},$
(3.33)
$\displaystyle\bar{L}_{22}=L_{22},\qquad\bar{L}_{[12]}=L_{[12]}+\frac{1}{2}\mathcal{H}L_{20}-\frac{1}{2}\mathcal{K}L_{10},$
$\displaystyle\bar{L}_{[2\tilde{\imath}]}=L_{[2\tilde{\imath}]}+\frac{1}{2}\mathcal{K}L_{\tilde{\imath}0},\qquad\bar{L}_{2\tilde{\imath}}+\bar{M}_{\tilde{\imath}0}=L_{2\tilde{\imath}}+M_{\tilde{\imath}0}.$
Obviously, since $\boldsymbol{k}$ is geodetic and the conformal factor reads
$\Omega=1$ for the CCLP metric with the Minkowski background, from (3.33) it
follows that (3.30) also holds in the background metric where the covariant
derivatives “;” reduce to the ones “$\bar{;}$” compatible with the background
metric. Then substituting $\hat{m}_{a}=\zeta m_{a}$ where $\zeta$ is the norm
of $\hat{\boldsymbol{m}}$, we obtain (3.88). In other words, (3.30) is indeed
equivalent to (3.88) for the CCLP black hole.
Let us point out that the relation (3.30) can be expressed in terms of the Lie
derivative in the full spacetime $g_{ab}$. Note that if $X^{a}\omega_{a}=0$,
then the Lie derivative of a one-form $\omega$ along a vector field $X$ in
coordinate notation reads
$\displaystyle(\mathcal{L}_{X}\omega)_{a}=\omega_{a;b}X^{b}+X^{b}_{\phantom{b};a}\omega_{b}=(\omega_{a;b}-\omega_{b;a})X^{b}$
(3.34)
and therefore (3.30) is equivalent to
$\mathcal{L}_{\boldsymbol{k}}m_{a}\propto
m_{a},\qquad\mathcal{L}_{\boldsymbol{m}}k_{a}=0.$ (3.35)
Note also that the Lie bracket of the vectors $\boldsymbol{k}$ and
$\boldsymbol{m}$ in terms of the Ricci rotation coefficients reads
$[\boldsymbol{k},\boldsymbol{m}]^{a}=L_{20}\,n^{a}+(L_{12}+N_{20})k^{a}+(L_{i2}-M_{i0})m^{a}_{(i)}.$
(3.36)
If $\boldsymbol{k}$ and $\boldsymbol{m}$ satisfy the relation (3.30) then
$[\boldsymbol{k},\boldsymbol{m}]^{a}=2L_{12}\,k^{a}+L_{22}\,m^{a}+2L_{2\tilde{\imath}}\,m^{a}_{(\tilde{\imath})}$
(3.37)
and therefore the vector fields $\boldsymbol{k}$ and $\boldsymbol{m}$ are
surface-forming provided that $L_{2\tilde{\imath}}=0$.
To conclude this section, let us point out the compatibility of the relation
(3.30) with the form of the optical matrix (2.144) satisfying the optical
constraint (2.128) since it holds also for the CCLP black hole as it is shown
in section 3.4. Comparing $L_{[2\tilde{\imath}]}=0$ (3.32) with the optical
matrix (2.144), it follows that the vector $\boldsymbol{m}$ must not lie in
any plane spanned by two spacelike frame vectors corresponding to a $2\times
2$ block with non-vanishing twist of the null geodetic congruence
$\boldsymbol{k}$ and therefore, omitting the degenerate case $L_{22}=0$,
$\boldsymbol{m}$ lies in a $1\times 1$ block of the optical matrix, i.e.
$L_{22}=\frac{1}{r}$. Then one may integrate (3.31) to obtain
$\zeta=\frac{C}{r}$, where $C$ is an arbitrary function not depending on the
affine parameter $r$ along null geodesics $\boldsymbol{k}$. In the case of the
CCLP metric, the function $C$ corresponds to $\nu$.
### 3.2 Geodetic Kerr–Schild vector field
From now on, we assume the null Kerr–Schild vector field $\boldsymbol{k}$ to
be geodetic. Proposition 6 then implies that $T_{00}=0$ which holds not only
if the energy–momentum tensor is absent, i.e. in Einstein spaces, but also for
spacetimes with aligned matter content such as aligned Maxwell field
$F_{ab}k^{b}\propto k_{a}$ or aligned null radiation $T_{ab}\propto
k_{a}k_{b}$. As commented in section 2.1, without loss of generality, we also
assume that $\boldsymbol{k}$ is affinely parametrized which further simplifies
the following calculations. Note that in the case of GKS spacetimes, i.e. if
$\mathcal{K}=0$, the assumption of Einstein spaces or spacetimes with aligned
matter fields is a sufficient condition for the Kerr–Schild vector
$\boldsymbol{k}$ to be geodetic as stated in proposition 1.
#### 3.2.1 Ricci tensor
Now we express the frame components of the Ricci tensor for xKS spacetimes
(3.1) with the Kerr–Schild vector $\boldsymbol{k}$ being geodetic and affinely
parametrized. Apart from $R_{00}$, these components are much more complicated
then for GKS spacetimes and $R_{0i}$ no longer vanishes identically. In
addition, we present only the components $R_{01}$ and $R_{ij}$ which have been
crucial in the analysis of GKS spacetimes in chapter 2
$\displaystyle R_{00}$ $\displaystyle=0,\phantom{\frac{1}{2}}$ (3.38)
$\displaystyle R_{0i}$
$\displaystyle=-\frac{1}{2}\left(\mathrm{D}^{2}\mathcal{K}+L_{jj}\mathrm{D}\mathcal{K}+2\mathcal{K}\omega^{2}\right)\delta_{2i}+2\mathcal{K}S_{2j}S_{ij}-2\mathcal{K}L_{2j}L_{ij}-S_{2i}\mathrm{D}\mathcal{K}$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\mathrm{D}\\!\left(L_{2i}-M_{i0}\right)+\left(\mathrm{D}\mathcal{K}+\frac{1}{2}\mathcal{K}L_{jj}\right)\left(L_{2i}-M_{i0}\right)$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\left(L_{ij}-\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}\right)\left(L_{2j}-M_{j0}\right),$ (3.39) $\displaystyle
R_{01}$
$\displaystyle=-\mathrm{D}^{2}\mathcal{H}-L_{ii}\mathrm{D}\mathcal{H}-2\mathcal{H}\omega^{2}+\frac{1}{2}\mathcal{K}\mathrm{D}^{2}\mathcal{K}+\frac{1}{2}(\mathrm{D}\mathcal{K})^{2}-\frac{1}{2}\mathcal{K}\left(L_{2i}+M_{i0}\right)\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}$
$\displaystyle\qquad+\mathcal{K}^{-1}\mathrm{D}\\!\left(\mathcal{K}^{2}N_{20}\right)-\frac{1}{2}\delta_{i}\\!\left(\mathcal{K}L_{2i}+\mathcal{K}M_{i0}\right)+\frac{1}{2}\left(\mathcal{K}L_{ii}-M_{ii}\right)\mathrm{D}\mathcal{K}-A_{2i}\delta_{i}\mathcal{K}$
$\displaystyle\qquad+\frac{1}{2}\mathrm{D}\\!\left(\mathcal{K}^{2}L_{22}\right)+\mathcal{K}\left(L_{i1}S_{2i}-L_{1i}A_{2i}-A_{ij}M_{ij}+L_{ii}N_{20}+M_{i0}N_{i0}\right)$
$\displaystyle\qquad-\frac{1}{2}\delta_{2}\mathrm{D}\mathcal{K}-\mathcal{K}^{2}\left(L_{2i}+M_{i0}\right)A_{2i}+\frac{1}{2}\mathcal{K}^{2}L_{22}L_{ii}+\frac{2\Lambda}{n-2},$
(3.40) $\displaystyle R_{ij}$
$\displaystyle=-2S_{ij}\mathrm{D}\mathcal{H}+2\mathcal{H}L_{ik}L_{jk}-2\mathcal{H}L_{kk}S_{ij}-\mathrm{D}\\!\left(\mathcal{K}M_{(ij)}\right)+2\mathcal{K}S_{ij}\mathrm{D}\mathcal{K}-\delta_{2}\left(\mathcal{K}S_{ij}\right)\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\frac{1}{2}\left(L_{j2}+M_{j0}\right)\delta_{i}\mathcal{K}-\frac{1}{2}\left(L_{i2}+M_{i0}\right)\delta_{j}\mathcal{K}+\mathcal{K}\bigg{[}\left(L_{21}+N_{20}\right)S_{ij}$
$\displaystyle\qquad-\frac{1}{2}\left(L_{j2}+M_{j0}\right)L_{1i}-\frac{1}{2}\left(L_{i2}+M_{i0}\right)L_{1j}+\left(A_{2j}+M_{j0}\right)L_{i1}$
$\displaystyle\qquad+\left(A_{2i}+M_{i0}\right)L_{j1}+\frac{1}{2}\left(L_{2i}-M_{i0}\right)N_{j0}+\frac{1}{2}\left(L_{2j}-M_{j0}\right)N_{i0}-S_{ij}M_{kk}$
$\displaystyle\qquad+2A_{(i|k}M_{j)k}+L_{ik}M_{[jk]}+L_{jk}M_{[ik]}-L_{kk}M_{(ij)}-2S_{k(i}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j)2}$}+M_{(ik)}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{k0}$}\phantom{\frac{1}{2}}$
$\displaystyle\qquad+M_{(jk)}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{k0}$}\bigg{]}+\mathcal{K}^{2}\bigg{[}A_{2i}A_{2j}+\frac{1}{2}\left(L_{2i}+M_{i0}\right)\left(L_{2j}+M_{j0}\right)-2L_{(i|2}M_{j)0}$
$\displaystyle\qquad-
L_{ik}S_{jk}-L_{ki}A_{jk}+L_{kk}S_{ij}\bigg{]}+\bigg{[}2\mathcal{K}A_{2j}\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{j}\mathrm{D}\mathcal{K}+L_{j1}\mathrm{D}\mathcal{K}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\left(L_{j2}-M_{j0}\right)\mathrm{D}\mathcal{K}+L_{jk}\delta_{k}\mathcal{K}-\frac{1}{2}L_{kk}\delta_{j}\mathcal{K}-\frac{1}{2}\mathcal{K}L_{kk}L_{1j}+\frac{1}{2}\mathcal{K}L_{1k}L_{jk}$
$\displaystyle\qquad+\mathcal{K}L_{[k1]}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j0}$}+\frac{1}{2}\mathcal{K}L_{kk}L_{j1}-\frac{1}{2}\mathcal{K}L_{k1}\left(2L_{ik}-L_{ki}\right)+\mathcal{K}A_{jk}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{ll}$}$
$\displaystyle\qquad-\mathcal{K}L_{kl}\mbox{$\stackrel{{\scriptstyle
j}}{{M}}_{[kl]}$}-\mathcal{K}A_{jk}N_{k0}-\mathcal{K}\mathrm{D}L_{[1j]}+\mathcal{K}\delta_{k}A_{jk}+\mathcal{K}^{2}A_{2j}L_{kk}+\mathcal{K}^{2}A_{k2}L_{jk}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\mathcal{K}^{2}\frac{1}{2}L_{2k}L_{kj}-\frac{1}{2}\mathcal{K}^{2}L_{k2}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{j0}$}+\frac{1}{2}\mathcal{K}^{2}L_{kj}M_{k0}-\frac{1}{2}\mathcal{K}^{2}\mathrm{D}L_{j2}\bigg{]}\delta_{2i}$
$\displaystyle\qquad+\bigg{[}2\mathcal{K}A_{2i}\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{i}\mathrm{D}\mathcal{K}+L_{i1}\mathrm{D}\mathcal{K}-\frac{1}{2}\mathcal{K}\left(L_{i2}-M_{i0}\right)\mathrm{D}\mathcal{K}+L_{ik}\delta_{k}\mathcal{K}$
$\displaystyle\qquad-\frac{1}{2}L_{kk}\delta_{i}\mathcal{K}-\frac{1}{2}\mathcal{K}L_{kk}L_{1i}+\frac{1}{2}\mathcal{K}L_{1k}L_{ik}+\mathcal{K}L_{[k1]}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{i0}$}+\frac{1}{2}\mathcal{K}L_{kk}L_{i1}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}L_{k1}\left(2L_{ik}-L_{ki}\right)+\mathcal{K}A_{ik}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{ll}$}-\mathcal{K}L_{kl}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{[kl]}$}-\mathcal{K}A_{ik}N_{k0}-\mathcal{K}\mathrm{D}L_{[1i]}$
$\displaystyle\qquad+\mathcal{K}\delta_{k}A_{ik}+\mathcal{K}^{2}A_{2i}L_{kk}+\mathcal{K}^{2}A_{k2}L_{ik}-\mathcal{K}^{2}\frac{1}{2}L_{2k}L_{ki}-\frac{1}{2}\mathcal{K}^{2}L_{k2}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{i0}$}$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}^{2}L_{ki}M_{k0}-\frac{1}{2}\mathcal{K}^{2}\mathrm{D}L_{i2}\bigg{]}\delta_{2j}+\bigg{[}\frac{1}{2}(\mathrm{D}\mathcal{K})^{2}-\mathcal{K}^{2}\omega^{2}\bigg{]}\delta_{2i}\delta_{2j}$
$\displaystyle\qquad+\frac{2\Lambda}{n-2}\delta_{ij}.$ (3.41)
In the case of Einstein GKS spacetimes, the constraints following from the
Einstein field equations involving the components $R_{ij}$ imply that non-
expanding Einstein GKS spacetimes belong to the Kundt class and that the
optical matrices of expanding Einstein GKS spacetimes satisfy the optical
constraint (2.128). Moreover, $R_{01}$ has been also employed to show that
expanding Einstein GKS spacetimes are not compatible with Weyl types III and
N.
Unfortunately, in the case of xKS spacetimes these components of the Ricci
tensor are rather complicated and it seems that we cannot straightforwardly
obtain similar results. However, if one considers a restrictive assumption on
the geometry of the vectors $\boldsymbol{k}$ and $\boldsymbol{m}$, for
instance, such as (3.35), then (3.31) and (3.32) obviously lead to the
simplification of the Ricci tensor components and this analysis is left for
future work.
#### 3.2.2 Riemann tensor and algebraic type of the Weyl
tensor
The frame components of the Riemann tensor for xKS spacetimes with a geodetic
Kerr–Schild vector field $\boldsymbol{k}$ are also more complex than those
ones of GKS spacetimes. Therefore, we list only the non-negative boost weight
components which are be employed in the following sections
$\displaystyle R_{0i0j}$ $\displaystyle=0,\phantom{\frac{1}{2}}$ (3.42)
$\displaystyle R_{010i}$
$\displaystyle=\frac{1}{2}\mathrm{D}^{2}\mathcal{K}\,\delta_{2i}+\frac{1}{2}\mathrm{D}\\!\left(\mathcal{K}L_{2i}+\mathcal{K}M_{i0}\right)+\frac{1}{2}M_{i0}\mathrm{D}\mathcal{K}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\left(L_{2j}+M_{j0}\right)\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$},$ (3.43) $\displaystyle R_{0ijk}$
$\displaystyle=L_{i[j}\delta_{k]2}\,\mathrm{D}\mathcal{K}-\left(L_{[jk]}\mathrm{D}\mathcal{K}+2\mathcal{K}S_{l[j}A_{k]l}\right)\delta_{2i}-2\mathcal{K}A_{il}L_{l[j}\delta_{k]2}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\mathcal{K}\left(L_{2i}+M_{i0}\right)L_{[jk]}+\mathcal{K}L_{2[j}L_{k]i}-\mathcal{K}M_{[j|0}L_{i|k]},\phantom{\frac{1}{2}}$
(3.44) $\displaystyle R_{0101}$
$\displaystyle=\mathrm{D}^{2}\mathcal{H}-\frac{1}{4}(\mathrm{D}\mathcal{K})^{2}-\mathcal{K}\left(L_{2i}+M_{i0}\right)N_{i0}+\mathrm{D}\\!\left(\mathcal{K}L_{21}-\mathcal{K}N_{20}\right)$
$\displaystyle\qquad-
N_{20}\mathrm{D}\mathcal{K}-\frac{1}{2}\mathcal{K}L_{22}\mathrm{D}\mathcal{K}-\frac{1}{4}\mathcal{K}^{2}\left(L_{2i}+M_{i0}\right)\left(L_{2i}+M_{i0}\right)$
$\displaystyle\qquad-\frac{2\Lambda}{(n-2)(n-1)},$ (3.45) $\displaystyle
R_{01ij}$
$\displaystyle=-2A_{ij}\mathrm{D}\mathcal{H}-4\mathcal{H}S_{k[i}A_{j]k}-M_{[ij]}\mathrm{D}\mathcal{K}-M_{[i|0}\,\delta_{j]}\mathcal{K}+\delta_{[i}\mathrm{D}\mathcal{K}\,\delta_{j]2}\phantom{\frac{1}{2}}$
$\displaystyle\qquad+\mathcal{K}L_{2[i}L_{1|j]}+\mathcal{K}L_{2[i}L_{j]1}-2\mathcal{K}A_{ij}\left(L_{21}-N_{20}\right)-\mathcal{K}L_{k[i}M_{j]k}\phantom{\frac{1}{2}}$
$\displaystyle\qquad-\mathcal{K}\left(L_{2k}+M_{k0}\right)\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{[ij]}$}+\mathcal{K}L_{k[i}M_{k|j]}-\frac{1}{2}\mathcal{K}\,\delta_{j}\\!\left(L_{2i}+M_{i0}\right)$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\,\delta_{i}\\!\left(L_{2j}+M_{j0}\right)+\mathcal{K}^{2}L_{2[i}M_{j]0}-L_{k[i}\delta_{j]2}\,\delta_{k}\mathcal{K}+\mathcal{K}L_{2[i}\delta_{j]2}\,\mathrm{D}\mathcal{K}$
$\displaystyle\qquad-2\mathcal{K}L_{[1k]}L_{k[i}\delta_{j]2},\phantom{\frac{1}{2}}$
(3.46) $\displaystyle R_{0i1j}$
$\displaystyle=-\mathrm{D}\mathcal{H}L_{ij}+2\mathcal{H}A_{ik}L_{kj}-\frac{1}{2}\delta_{j}\mathrm{D}\mathcal{K}\,\delta_{2i}+\frac{1}{4}(\mathrm{D}\mathcal{K})^{2}\delta_{2i}\delta_{2j}+\frac{1}{2}L_{2j}\delta_{i}\mathcal{K}$
$\displaystyle\qquad+\frac{1}{2}\left(2L_{(i|1}\delta_{2|j)}-M_{ij}+\mathcal{K}\left(L_{(ij)}+A_{2i}\delta_{2j}-S_{j2}\delta_{2i}\right)\right)\mathrm{D}\mathcal{K}$
$\displaystyle\qquad+\frac{1}{2}L_{kj}\delta_{k}\mathcal{K}\delta_{2i}+\frac{1}{4}\mathcal{K}\left(L_{2i}+M_{i0}\right)\mathrm{D}\mathcal{K}\delta_{2j}+\frac{1}{4}\mathcal{K}\left(L_{2j}+M_{j0}\right)\mathrm{D}\mathcal{K}\delta_{2i}$
$\displaystyle\qquad-\frac{1}{2}\delta_{j}\\!\left(\mathcal{K}L_{2i}+\mathcal{K}M_{i0}\right)+\frac{1}{2}\mathcal{K}\left(L_{2i}+M_{i0}\right)L_{j1}+\frac{1}{2}\mathcal{K}\left(L_{2j}+M_{j0}\right)L_{i1}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\left(L_{2k}+M_{k0}\right)\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{ij}$}-\mathcal{K}\left(L_{21}-N_{20}\right)L_{ij}-\mathcal{K}L_{(1i)}L_{2j}$
$\displaystyle\qquad+\mathcal{K}L_{kj}M_{[ik]}+\mathcal{K}L_{21}S_{ij}-\mathcal{K}L_{k1}A_{ik}\delta_{2j}+\frac{1}{2}\mathcal{K}^{2}L_{22}S_{ij}$
$\displaystyle\qquad-\frac{1}{4}\mathcal{K}^{2}\left(L_{2j}-M_{j0}\right)\left(L_{2i}+M_{i0}\right)+\mathcal{K}^{2}\left(L_{2k}+M_{k0}\right)A_{k(i}\delta_{j)2}$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\left(L_{1k}L_{kj}-L_{k1}L_{jk}\right)\delta_{2i}+\frac{2\Lambda}{(n-2)(n-1)}\delta_{ij}.$
(3.47)
Unlike for the GKS spacetimes, the boost weight 1 components $R_{010i}$ (3.43)
and $R_{0ijk}$ (3.44) of xKS spacetimes do not vanish identically. However,
since the boost weight 2 components $R_{0i0j}$ (3.42) and $R_{00}$ (3.38) of
the Riemann and Ricci tensor, respectively, are identically zero, from (2.45)
it follows that the same also holds for the Weyl tensor
$C_{0i0j}=0$ (3.48)
and therefore
###### Proposition 8
Extended Kerr–Schild spacetimes (3.1) with a geodetic Kerr–Schild vector
$\boldsymbol{k}$ are of the Weyl type I or more special with $\boldsymbol{k}$
being the WAND.
This proposition confirms one of our motivations for considering the xKS form
of metrics that such spacetimes with a geodetic $\boldsymbol{k}$ may cover
more general algebraic types than Einstein GKS spacetimes.
Note also that if we assume that the vectors $\boldsymbol{k}$ and
$\boldsymbol{m}$ obey (3.35) the frame components of the Riemann tensor are
dramatically simplified as in the case of the Ricci tensor.
### 3.3 Kundt extended Kerr–Schild spacetimes
In this section, we study xKS spacetimes with a non-expanding, non-shearing
and non-twisting geodetic null congruence of integral curves of the
Kerr–Schild vector $\boldsymbol{k}$, i.e. the subclass of Kundt metrics
admitting the xKS form (3.1). First, substituting $L_{ij}=0$, we rewrite the
components of the Ricci tensor (3.38)–(3.41) of general xKS spacetimes with a
geodetic $\boldsymbol{k}$ for the Kundt case
$\displaystyle R_{00}$ $\displaystyle=0,\phantom{\frac{1}{2}}$ (3.49)
$\displaystyle R_{0i}$
$\displaystyle=-\frac{1}{2}\mathrm{D}^{2}\mathcal{K}\delta_{2i}-\frac{1}{2}\mathcal{K}\mathrm{D}M_{i0}-M_{i0}\mathrm{D}\mathcal{K}+\frac{1}{2}\mathcal{K}M_{j0}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$},$ (3.50) $\displaystyle R_{01}$
$\displaystyle=-\mathrm{D}^{2}\mathcal{H}+\frac{1}{2}\mathcal{K}\mathrm{D}^{2}\mathcal{K}+\frac{1}{2}(\mathrm{D}\mathcal{K})^{2}-\frac{1}{2}\delta_{2}\mathrm{D}\mathcal{K}+\mathcal{K}^{-1}\mathrm{D}\\!\left(\mathcal{K}^{2}N_{20}\right)$
$\displaystyle\qquad-\frac{1}{2}M_{ii}\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{i}\\!\left(\mathcal{K}M_{i0}\right)+\mathcal{K}M_{i0}N_{i0}-\frac{1}{2}\mathcal{K}M_{i0}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}+\frac{2\Lambda}{n-2},$ (3.51) $\displaystyle R_{22}$
$\displaystyle=\mathcal{K}M_{k2}M_{k0}-\delta_{2}\mathrm{D}\mathcal{K}+2\mathrm{D}(\mathcal{K}L_{21})-2\mathcal{K}L_{[k1]}M_{k0}-2\mathcal{K}\mathrm{D}L_{(12)}\phantom{\frac{1}{2}}$
$\displaystyle\qquad+\frac{1}{2}(\mathrm{D}\mathcal{K})^{2}+\frac{2\Lambda}{n-2},$
(3.52) $\displaystyle R_{\tilde{\imath}2}$
$\displaystyle=-\frac{1}{2}\delta_{\tilde{\imath}}\mathrm{D}\mathcal{K}-\frac{1}{2}\mathrm{D}\\!\left(\mathcal{K}M_{\tilde{\imath}2}\right)-\frac{1}{2}M_{\tilde{\imath}0}\delta_{2}\mathcal{K}+L_{\tilde{\imath}1}\mathrm{D}\mathcal{K}+\frac{1}{2}\mathcal{K}M_{\tilde{\imath}0}\mathrm{D}\mathcal{K}$
$\displaystyle\qquad-\frac{1}{2}\mathcal{K}\bigg{[}\left(L_{12}-2L_{21}+N_{20}\right)M_{\tilde{\imath}0}-M_{\tilde{\imath}k}M_{k0}-M_{k\tilde{\imath}}M_{k0}$
$\displaystyle\qquad+M_{k2}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\imath}0}$}\bigg{]}+\mathcal{K}L_{[k1]}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\imath}0}$}-\mathcal{K}\mathrm{D}L_{[1\tilde{\imath}]},$
(3.53) $\displaystyle R_{\tilde{\imath}\tilde{\jmath}}$
$\displaystyle=-\mathrm{D}\\!\left(\mathcal{K}M_{(\tilde{\imath}\tilde{\jmath})}\right)-M_{(\tilde{\imath}|0}\delta_{\tilde{\jmath})}\mathcal{K}-\mathcal{K}\bigg{[}\frac{1}{2}\left(L_{1\tilde{\jmath}}-2L_{\tilde{\jmath}1}+N_{\tilde{\jmath}0}\right)M_{\tilde{\imath}0}$
$\displaystyle\qquad+\frac{1}{2}\left(L_{1\tilde{\imath}}-2L_{\tilde{\imath}1}+N_{\tilde{\imath}0}\right)M_{\tilde{\jmath}0}+M_{(\tilde{\imath}|k}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\jmath})0}$}+M_{k(\tilde{\imath}}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\jmath})0}$}\bigg{]}$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}^{2}M_{\tilde{\imath}0}M_{\tilde{\jmath}0}+\frac{2\Lambda}{n-2}\delta_{\tilde{\imath}\tilde{\jmath}}.$
(3.54)
Similarly, the components of the Riemann tensor (3.42)–(3.47) reduce to
$\displaystyle R_{0i0j}$ $\displaystyle=0,\qquad
R_{0ijk}=0,\phantom{\frac{1}{2}}$ (3.55) $\displaystyle R_{010i}$
$\displaystyle=\frac{1}{2}\mathrm{D}^{2}\mathcal{K}\,\delta_{2i}+\frac{1}{2}\mathcal{K}\mathrm{D}M_{i0}+M_{i0}\mathrm{D}\mathcal{K}-\frac{1}{2}\mathcal{K}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{j0}$}M_{j0},$ (3.56) $\displaystyle R_{0101}$
$\displaystyle=\mathrm{D}^{2}\mathcal{H}-\frac{1}{4}(\mathrm{D}\mathcal{K})^{2}-\mathcal{K}M_{i0}N_{i0}+\mathrm{D}\\!\left(\mathcal{K}L_{21}-\mathcal{K}N_{20}\right)-N_{20}\mathrm{D}\mathcal{K}$
$\displaystyle\qquad-\frac{1}{4}\mathcal{K}^{2}M_{i0}M_{i0}-\frac{2\Lambda}{(n-2)(n-1)},$
(3.57) $\displaystyle R_{01\tilde{\imath}2}$
$\displaystyle=\frac{1}{2}\delta_{\tilde{\imath}}\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{2}\\!\left(\mathcal{K}M_{\tilde{\imath}0}\right)-\frac{1}{2}M_{\tilde{\imath}2}\mathrm{D}\mathcal{K}-\mathcal{K}M_{k0}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{[\tilde{\imath}2]}$},$ (3.58) $\displaystyle
R_{01\tilde{\imath}\tilde{\jmath}}$
$\displaystyle=\delta_{[\tilde{\imath}}\\!\left(\mathcal{K}M_{\tilde{\jmath}]0}\right)-M_{[\tilde{\imath}\tilde{\jmath}]}\mathrm{D}\mathcal{K}-\mathcal{K}M_{k0}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{[\tilde{\imath}\tilde{\jmath}]}$},\phantom{\frac{1}{2}}$ (3.59)
$\displaystyle R_{0212}$
$\displaystyle=-\frac{1}{2}\delta_{2}\mathrm{D}\mathcal{K}+\frac{1}{4}(\mathrm{D}\mathcal{K})^{2}+L_{21}\mathrm{D}\mathcal{K}+\frac{1}{2}\mathcal{K}M_{k0}M_{k2}+\frac{2\Lambda}{(n-2)(n-1)},$
(3.60) $\displaystyle R_{021\tilde{\imath}}$
$\displaystyle=-\frac{1}{2}\delta_{\tilde{\imath}}\mathrm{D}\mathcal{K}+\frac{1}{4}\left(2L_{\tilde{\imath}1}+\mathcal{K}M_{\tilde{\imath}0}\right)\mathrm{D}\mathcal{K}+\frac{1}{2}\mathcal{K}M_{\tilde{\imath}0}L_{21}+\frac{1}{2}\mathcal{K}M_{k0}M_{k\tilde{\imath}},$
(3.61) $\displaystyle R_{0\tilde{\imath}12}$
$\displaystyle=\frac{1}{4}\left(2L_{\tilde{\imath}1}-2M_{\tilde{\imath}2}+\mathcal{K}M_{\tilde{\imath}0}\right)\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{2}\\!\left(\mathcal{K}M_{\tilde{\imath}0}\right)$
$\displaystyle\qquad+\frac{1}{2}\mathcal{K}\left(M_{\tilde{\imath}0}L_{21}-M_{k0}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\imath}2}$}\right),$ (3.62) $\displaystyle
R_{0\tilde{\imath}1\tilde{\jmath}}$
$\displaystyle=-\frac{1}{2}M_{\tilde{\imath}\tilde{\jmath}}\mathrm{D}\mathcal{K}-\frac{1}{2}\delta_{\tilde{\jmath}}\\!\left(\mathcal{K}M_{\tilde{\imath}0}\right)+\mathcal{K}L_{(\tilde{\imath}|1}M_{\tilde{\jmath})0}-\frac{1}{2}\mathcal{K}M_{k0}\mbox{$\stackrel{{\scriptstyle
k}}{{M}}_{\tilde{\imath}\tilde{\jmath}}$}$
$\displaystyle\qquad+\frac{1}{4}\mathcal{K}^{2}M_{\tilde{\imath}0}M_{\tilde{\jmath}0}+\frac{2\Lambda}{(n-2)(n-1)}\delta_{\tilde{\imath}\tilde{\jmath}}.$
(3.63)
As has been shown in section 3.1, a general xKS metric (3.1) with a geodetic
Kerr–Schild vector $\boldsymbol{k}$ is of Weyl type I or more special. In the
case of Kundt xKS spacetimes, the only non-trivial boost weight 1 frame
components of the Riemann tensor satisfy $R_{010i}=-R_{0i}$ as one can
directly see from (3.50) and (3.56). Putting this relation to (2.45) and
$R_{0ijk}=0$ (3.55) to (2.46), we obtain the boost weight 1 components of the
Weyl tensor
$C_{0ijk}=\frac{1}{n-2}(R_{0k}\delta_{ij}-R_{0j}\delta_{ik}),\qquad
C_{010i}=\frac{3-n}{n-2}R_{0i}$ (3.64)
and thus Kundt xKS spacetimes are of Weyl type II or more special if and only
if $T_{0i}=0$. Note that this statement holds even for general Kundt metrics
[3] not necessarily of the xKS form. Assuming $T_{0i}=0$, the Einstein field
equations $R_{02}=0$ and $R_{0\tilde{\imath}}=0$, where $R_{0i}$ is given by
(3.50), can be written as
$\displaystyle\mathrm{D}^{2}\mathcal{K}=\mathcal{K}M_{\tilde{\jmath}0}M_{\tilde{\jmath}0},$
(3.65)
$\displaystyle\mathrm{D}\\!\left(\mathcal{K}^{2}M_{\tilde{\imath}0}\right)=\mathcal{K}^{2}M_{\tilde{\jmath}0}\mbox{$\stackrel{{\scriptstyle\tilde{\imath}}}{{M}}_{\tilde{\jmath}0}$},$
(3.66)
respectively. The trivial solution $\mathcal{K}=0$ corresponds to the GKS
limit when $T_{0i}=0$ is a necessary condition in order to satisfy the
Einstein field equations since in this case the components $R_{0i}$
identically vanish as was shown in section 2.2.1. The Ricci rotation
coefficients $\stackrel{{\scriptstyle i}}{{M}}_{ja}$ are antisymmetric in the
indices $i$ and $j$, therefore by multiplying (3.66) with
$2\mathcal{K}^{2}M_{\tilde{\imath}0}$ we eliminate the term on the right-hand
side and the remaining term can be combined as
$0=2\mathrm{D}\\!\left(\mathcal{K}^{2}M_{\tilde{\imath}0}\right)\mathcal{K}^{2}M_{\tilde{\imath}0}=\mathrm{D}\\!\left(\mathcal{K}^{4}M_{\tilde{\imath}0}M_{\tilde{\imath}0}\right).$
(3.67)
Consequently,
$\mathcal{K}^{4}M_{\tilde{\imath}0}M_{\tilde{\imath}0}=(c^{0})^{2}$, where the
function $c^{0}$ does not depend on the affine parameter $r$ along the null
geodesics of the Kerr–Schild congruence $\boldsymbol{k}$. Obviously, since we
assume $\mathcal{K}$ to be non-zero, $c^{0}$ vanishes if and only if all
$M_{i0}$ vanish. Substituting the result of (3.67) to (3.65), we obtain a
differential equation $\mathcal{K}^{3}\mathrm{D}^{2}\mathcal{K}=(c^{0})^{2}$
determining the $r$-dependence of the function $\mathcal{K}$ which has two
distinct branches of solutions
$\displaystyle\mathcal{K}$
$\displaystyle=\operatorname{sgn}d^{0}\sqrt{\big{(}d^{0}(r+b^{0})\big{)}^{2}+\frac{(c^{0})^{2}}{(d^{0})^{2}}}$
$\displaystyle\text{if $c^{0}\neq 0$},$ (3.68) $\displaystyle\mathcal{K}$
$\displaystyle=f^{0}r+e^{0}$ $\displaystyle\text{if $c^{0}=0$},$ (3.69)
where $b^{0}$, $d^{0}$, $e^{0}$ and $f^{0}$ are arbitrary functions not
depending on $r$. Finally, we can conclude that
###### Proposition 9
For Kundt extended Kerr–Schild spacetimes with $\mathcal{K}\neq 0$ solving the
Einstein field equations the following statements are equivalent
* (i)
the boost weight 1 components $T_{0i}\equiv T_{ab}k^{a}m^{b}_{(i)}=0$ of the
energy–momentum tensor vanish,
* (ii)
the spacetime is of Weyl type II or more special,
* (iii)
the function $\mathcal{K}$ takes the form (3.68) or (3.69).
Moreover, $\mathcal{K}$ is a linear function (3.69) of an affine parameter $r$
along the null geodesics of the Kerr–Schild congruence $\boldsymbol{k}$ if and
only if $M_{\tilde{\imath}0}=0$.
Note that in Kundt spacetimes the relation between the vectors
$\boldsymbol{k}$ and $\boldsymbol{m}$ (3.30) holds if and only if
$L_{21}=N_{20},\qquad D\zeta=0,\qquad M_{\tilde{\imath}0}=0,\qquad
L_{[12]}=0.$ (3.70)
Obviously, (3.70) is not satisfied for $\mathcal{K}$ of the form (3.68).
#### 3.3.1 Explicit example
Now we present explicit examples of Ricci-flat Kundt xKS spacetimes, namely
the class of spacetimes with vanishing scalar invariants (2.94). Such VSI
metrics can be written as
$\mathrm{d}s^{2}=2\mathrm{d}u\,\mathrm{d}r+\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}+2H(u,r,x^{k})\,\mathrm{d}u^{2}+2W_{i}(u,r,x^{k})\,\mathrm{d}u\,\mathrm{d}x^{i},$
(3.71)
which better exhibits its xKS form (3.1) with the flat background metric
described by the first two terms in (3.71). Obviously, one may identify the
null one-form $\mathrm{d}u$ with the Kerr–Schild vector $\boldsymbol{k}$
$k_{a}\,\mathrm{d}x^{a}=\mathrm{d}u.$ (3.72)
Therefore, the function $\mathcal{H}$ appearing in the xKS ansatz (3.1) is
given by
$\mathcal{H}=-H(u,r,x^{k})$ (3.73)
and the remaining term
$\mathcal{K}\boldsymbol{m}=-W_{i}(u,r,x^{k})\,\mathrm{d}x^{i}$ can be split to
the function $\mathcal{K}$ and the vector $\boldsymbol{m}$ as
$\mathcal{K}=-\sqrt{W_{i}W_{i}},\qquad\boldsymbol{m}=\frac{W_{i}\,\mathrm{d}x^{i}}{\sqrt{W_{j}W_{j}}},$
(3.74)
so that $\boldsymbol{m}$ is a unit vector. It turns out that all VSI metrics
(3.71) admit the xKS form (3.1).
Recall that xKS spacetimes with a geodetic Kerr–Schild vector field
$\boldsymbol{k}$ are of Weyl type I or more special as follows from
proposition 6. Furthermore, proposition 9 implies that Kundt xKS spacetimes
with $R_{0i}=0$ are of Weyl type II or more special. On the other hand, it is
known [4] that VSI spacetimes are of Weyl types III, N or O with the Ricci
tensor of type III or more special, i.e. $R_{00}=R_{0i}=R_{01}=R_{ij}=0$, and
all VSI metrics with the Ricci tensor of types N and O have been given
explicitly in [38]. As has been already mentioned in section 2.5.1, the VSI
class can be divided into two distinct subclasses with vanishing and non-
vanishing quantity $L_{1i}L_{1i}$ denoted as $\epsilon=0$ and $\epsilon=1$,
respectively. The canonical choices of the functions $W_{i}$ in these
subclasses are given in (2.97). The functions $W_{i}$ and $H$ can be further
constrained employing the Einstein field equations and assuming an appropriate
special algebraic type of the spacetime. Note also that for VSI spacetimes the
statements (i) and (ii) of proposition 9 are clearly satisfied and therefore
the function $\mathcal{K}$ takes one of the forms (3.68) or (3.69) depending
on the functions $W_{i}(u,r,x^{k})$ which differ in the subclasses
$\epsilon=0$ and $\epsilon=1$.
##### Case $\epsilon=0$
In the subclass $\epsilon=0$ of VSI spacetimes, the functions
$W_{i}(u,r,x^{k})$ are given by [38]
$W_{2}=0,\qquad W_{\tilde{\imath}}=W_{\tilde{\imath}}^{0}(u,x^{k}),$ (3.75)
where the superscript 0 denotes that the quantity is independent on the
coordinate $r$ corresponding to an affine parameter along the geodesics of the
Kerr–Schild congruence $\boldsymbol{k}$. Then from (3.74) it follows that
$\mathcal{K}$ is of the form (3.69) with $f^{0}=0$ and does not depend on $r$
$\mathcal{K}=-\sqrt{W_{\tilde{\imath}}^{0}W_{\tilde{\imath}}^{0}}=e^{0}.$
(3.76)
Therefore $c^{0}=0$ and thus all the Ricci rotation coefficients $M_{i0}$
vanish. Moreover, if also $N_{20}=0$, then the vector $\boldsymbol{m}$ is
parallelly transported along the null geodesics of the congruence
$\boldsymbol{k}$. On the other hand, if $N_{20}$ is non-vanishing, it can be
transformed to zero by a null rotation with $\boldsymbol{k}$ fixed (1.19)
setting $\mathrm{D}z_{2}=-N_{20}$, however, this Lorentz transformation
changes the vector $\boldsymbol{m}$.
##### General case $\epsilon=1$
In the subclass $\epsilon=1$, the functions $W_{i}(u,r,x^{k})$ read [38]
$W_{2}=-\frac{2}{x^{2}}r,\qquad W_{\tilde{\imath}}=W_{\tilde{\imath}}^{0}.$
(3.77)
If at least one of $W_{\tilde{\imath}}$ is non-zero, the function
$\mathcal{K}$ (3.74) takes the form (3.68)
$\mathcal{K}=-\sqrt{\frac{4}{(x^{2})^{2}}r^{2}+W_{\tilde{\imath}}^{0}W_{\tilde{\imath}}^{0}}.$
(3.78)
Comparing (3.78) with (3.68), it immediately follows that $b^{0}=0$,
$d^{0}=-\frac{2}{x^{2}}$ and
$(c^{0})^{2}=\frac{(x^{2})^{2}}{4}W^{0}_{\tilde{\imath}}W^{0}_{\tilde{\imath}}$
. Since
$M_{\tilde{\imath}0}M_{\tilde{\imath}0}=\mathcal{K}^{-4}(c^{0})^{2}\neq 0$ the
vector $\boldsymbol{m}$ is not parallelly transported along $\boldsymbol{k}$.
##### Special case $\epsilon=1$
In the special case of the subclass $\epsilon=1$ of VSI spacetimes where all
$W^{0}_{\tilde{\imath}}$ in (3.77) vanish, the function $\mathcal{K}$ is of
the form (3.69) with $e^{0}=0$
$\mathcal{K}=-\frac{2}{|x^{2}|}r=f^{0}r.$ (3.79)
As in the case $\epsilon=0$, $c^{0}$ and consequently $M_{\tilde{\imath}0}$
vanish and if $N_{20}$ is zero, the vector $\boldsymbol{m}$ is parallelly
transported along $\boldsymbol{k}$.
#### 3.3.2 Not all vacuum higher dimensional pp -waves belong to the class of
Ricci-flat xKS spacetimes
In the previous section, we have shown that all VSI metrics admit the xKS form
(3.1). The question is whether also all vacuum pp -waves belong to the class
of xKS spacetimes. As mentioned in section 4.1.3, higher dimensional pp -waves
are of Weyl type II or more special and Ricci-flat if not supported by an
appropriate matter field. It is also known that vacuum pp -waves of type III
or more special belong to the VSI class. Therefore, it remains to investigate
the situation of the Weyl type II pp -waves.
All Ricci-flat pp -wave metrics can be written in the form
$\mathrm{d}s^{2}=2\mathrm{d}u\left[\mathrm{d}v+H(u,x^{k})\,\mathrm{d}u+W_{i}(u,x^{k})\,\mathrm{d}x^{i}\right]+g_{ij}(u,x^{k})\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}$
(3.80)
and it does not seem that for general transversal metric $g_{ij}$, pp -wave
metrics (3.80) can be cast to the xKS form. However, one may use the result of
[40] that the transversal Riemannian metric $g_{ij}$ of vacuum pp -waves is
Ricci-flat and the statement that the transversal Riemannian space is locally
homogeneous in the case of CSI Kundt metrics as has been shown in [37]. Since
a Ricci-flat locally homogeneous Riemannian space is flat [64], we can
conclude that Ricci-flat CSI pp -wave metrics can be written in the form
(3.80) with a flat transversal space, i.e. $g_{ij}=\delta_{ij}$, which is
obviously in the xKS form with the Minkowski background. However, note that
there could exist non-CSI Ricci-flat pp -waves and therefore we cannot decide
whether all higher dimensional Ricci-flat pp -waves belong to the class of xKS
spacetimes.
Recall also that type N Ricci-flat VSI spacetimes and consequently vacuum type
N pp -waves can be cast to the KS form (2.1) as has been shown in [27] and as
also immediately follows from the results of section 2.5.1. All the mentioned
facts about higher dimensional vacuum pp -waves are summarized in table 3.1.
Table 3.1: Properties of higher dimensional Ricci-flat pp -waves. Weyl type | KS | xKS | VSI
---|---|---|---
N | ✓ | ✓ | ✓
III | $\times$ | ✓ | ✓
II | $\times$ | only CSI | $\times$
Finally, note that in four dimensions, all vacuum pp -wave metrics are only of
Weyl type N, belong to the VSI class and take the KS form.
### 3.4 Expanding extended Kerr–Schild
spacetimes
In this section, we construct a null frame in the CCLP spacetime [57] which
then allows us to express the optical matrix, show that it obeys the optical
constraint and determine the algebraic type. The CCLP solution represents a
charged rotating black hole in five-dimensional minimal gauged supergravity or
equivalently in the Einstein–Maxwell–Chern–Simons theory with a negative
cosmological constant $\Lambda$ and the Chern–Simons coefficient $\chi=1$
described by the field equations
$\displaystyle
R^{a}_{b}-2(F_{bc}F^{ac}-\frac{1}{6}\delta^{a}_{b}F_{cd}F^{cd})+\frac{2}{3\Lambda}\delta^{a}_{b}=0,$
(3.81)
$\displaystyle\nabla_{b}F^{ab}+\frac{\chi}{2\sqrt{3}\sqrt{-g}}\epsilon^{abcde}F_{bc}F_{de}=0$
(3.82)
and it is an example of expanding xKS spacetime. In fact, the xKS ansatz has
been proposed in [56] where it has been shown that the CCLP black hole can be
cast to the form
$g_{ab}=\bar{g}_{ab}-2\mathcal{H}k_{a}k_{b}-2\hat{\mathcal{K}}k_{(a}\hat{m}_{b)},$
(3.83)
where we put hat over the $\hat{\mathcal{K}}$ and $\hat{\boldsymbol{m}}$ since
unlike our definition of the xKS ansatz (3.1), the vector
$\hat{\boldsymbol{m}}$ is not normalized to unity.
In the case $\Lambda=0$, the CCLP metric can be rewritten in terms of
spheroidal coordinates in the form (3.83) with the flat background metric and
the vectors $\boldsymbol{k}$, $\hat{\boldsymbol{m}}$ given by [56]
$\displaystyle\bar{g}_{ab}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}$
$\displaystyle=-\mathrm{d}t^{2}-2\mathrm{d}r\left(\mathrm{d}t-a\sin^{2}\theta\,\mathrm{d}\phi-b\cos^{2}\theta\,\mathrm{d}\psi\right)+\rho^{2}\,\mathrm{d}\theta^{2}\phantom{\frac{1}{2}}$
$\displaystyle\qquad+\left(r^{2}+a^{2}\right)\sin^{2}\theta\,\mathrm{d}\phi^{2}+\left(r^{2}+b^{2}\right)\cos^{2}\theta\,\mathrm{d}\psi^{2},$
(3.84) $\displaystyle k_{a}\,\mathrm{d}x^{a}$
$\displaystyle=-\mathrm{d}t+a\sin^{2}\theta\,\mathrm{d}\phi+b\cos^{2}\theta\,\mathrm{d}\psi,\phantom{\frac{1}{2}}$
(3.85) $\displaystyle\hat{m}_{a}\,\mathrm{d}x^{a}$
$\displaystyle=b\sin^{2}\theta\,\mathrm{d}\phi+a\cos^{2}\theta\,\mathrm{d}\psi.\phantom{\frac{1}{2}}$
(3.86)
The functions $\mathcal{H}$, $\hat{\mathcal{K}}$ and the one-form gauge
potential proportional to the Kerr–Schild vector $\boldsymbol{k}$ then read
$\mathcal{H}=-\frac{M}{\rho^{2}}+\frac{Q^{2}}{2\rho^{4}},\qquad\hat{\mathcal{K}}=-\frac{Q}{\rho^{2}},\qquad
A=-\frac{\sqrt{3}Q}{2\rho}\boldsymbol{k},$ (3.87)
where $r$ is the spheroidal radial coordinate, the angular coordinates have
usual ranges $\phi\in\langle 0,2\pi)$, $\psi\in\langle 0,2\pi)$,
$\theta\in\langle 0,\pi\rangle$ and
$\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta$. One may show that
the vectors $\boldsymbol{k}$ (3.85) and $\hat{\boldsymbol{m}}$ (3.86) in the
background spacetime satisfy [61]
$(\hat{m}_{a\bar{;}b}-\hat{m}_{b\bar{;}a})k^{b}=0,\qquad(k_{a\bar{;}b}-k_{b\bar{;}a})\hat{m}^{b}=0,$
(3.88)
where “$\bar{;}$” denotes the covariant derivative compatible with the
background metric $\bar{g}_{ab}$ and as follows from (3.13) the contractions
can be performed by both the full and background metrics since
$\boldsymbol{k}$ is geodetic. It is also convenient to define
$\nu^{2}\equiv\rho^{2}-r^{2}$.
Since the metric is in the form (3.83) we put it into our definition of the
xKS form (3.1) with a unit vector $\boldsymbol{m}$ by rescaling the vector
$\hat{\boldsymbol{m}}$ and including its norm to the function
$\hat{\mathcal{K}}$. Thus, the norm of the vector $\hat{\boldsymbol{m}}$ is
$|\hat{\boldsymbol{m}}|=\frac{\nu}{r}$, therefore the normalized
$\boldsymbol{m}$ reads
$m_{a}\,\mathrm{d}x^{a}=\frac{br\sin^{2}\theta}{\nu}\mathrm{d}\phi+\frac{ar\cos^{2}\theta}{\nu}\mathrm{d}\psi$
(3.89)
and since the scalar functions are related via
$\mathcal{K}=\hat{\mathcal{K}}|\hat{\boldsymbol{m}}|$ one gets
$\mathcal{K}=-\frac{Q\nu}{r\rho^{2}}.$ (3.90)
Obviously, one may identify $\boldsymbol{k}$ and $\boldsymbol{m}$ with the
null and spacelike frame vectors $\boldsymbol{\ell}$ and
$\boldsymbol{m}^{(2)}$, respectively. The remaining vectors $\boldsymbol{n}$,
$\boldsymbol{m}^{(3)}$ and $\boldsymbol{m}^{(4)}$ can be obtained by solving
the constraints (1.8). Similarly as for the five-dimensional Kerr–(anti-)de
Sitter metric in section 2.6.5, it is easier to find a null frame in the
background spacetime first
$\displaystyle m^{(3)}_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\frac{\rho}{\nu}\cot\theta\left(\mathrm{d}t-b\,\mathrm{d}\psi\right),$
(3.91) $\displaystyle m^{(4)}_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\rho\,\mathrm{d}\theta,\phantom{\frac{1}{2}}$ (3.92)
$\displaystyle\bar{n}_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\frac{1}{2}\frac{\rho^{2}-r^{2}\sin^{2}\theta}{\nu^{2}\sin^{2}\theta}\,\mathrm{d}t+\mathrm{d}r+\frac{1}{2}a\frac{\rho^{2}-r^{2}\sin^{2}\theta}{\nu^{2}}\,\mathrm{d}\phi$
$\displaystyle\qquad-\frac{1}{2}b\cot^{2}\theta\frac{\rho^{2}+r^{2}\sin^{2}\theta}{\nu^{2}}\,\mathrm{d}\psi,$
(3.93)
from which we then construct the frame in the full spacetime using (3.11).
Therefore, the vectors $\boldsymbol{m}^{(3)}$ and $\boldsymbol{m}^{(4)}$ are
same in both frames and $\boldsymbol{n}$ is given by
$\displaystyle n_{a}\,\mathrm{d}x^{a}$
$\displaystyle=\left(\frac{1}{2}\frac{\rho^{2}-r^{2}\sin^{2}\theta}{\nu^{2}\sin^{2}\theta}-\frac{2M\rho^{2}-Q^{2}}{2\rho^{4}}\right)\mathrm{d}t+\mathrm{d}r$
(3.94)
$\displaystyle\qquad+\left(\frac{1}{2}a\frac{\rho^{2}-r^{2}\sin^{2}\theta}{\nu^{2}}+\left(a\frac{2M\rho^{2}-Q^{2}}{2\rho^{4}}+b\frac{Q}{\rho^{2}}\right)\sin^{2}\theta\right)\mathrm{d}\phi$
$\displaystyle\qquad-\left(\frac{1}{2}b\frac{\rho^{2}+r^{2}\sin^{2}\theta}{\nu^{2}\sin^{2}\theta}-\left(b\frac{2M\rho^{2}-Q^{2}}{2\rho^{4}}+a\frac{Q}{\rho^{2}}\right)\right)\cos^{2}\theta\,\mathrm{d}\psi.$
Straightforwardly, one may also obtain the contravariant components of the
frame vectors (3.85), (3.89), (3.91), (3.92) and (3.94)
$\displaystyle k^{a}\,\partial_{a}$
$\displaystyle=\partial_{r},\phantom{\frac{1}{2}}$ (3.95) $\displaystyle
m_{(2)}^{a}\,\partial_{a}$
$\displaystyle=\frac{ab}{r\nu}\,\partial_{t}-\frac{Q\nu^{2}+ab\rho^{2}}{r\rho^{2}\nu}\,\partial_{r}+\frac{b}{r\nu}\,\partial_{\phi}+\frac{a}{r\nu}\,\partial_{\psi},$
(3.96) $\displaystyle m_{(3)}^{a}\,\partial_{a}$
$\displaystyle=\frac{\sin\theta\cos\theta}{\rho\nu}\bigg{[}(a^{2}-b^{2})\,\partial_{t}-\frac{(r^{2}+a^{2})}{\sin^{2}\theta}\,\partial_{r}$
$\displaystyle\qquad+\frac{a}{\sin^{2}\theta}\,\partial_{\phi}-\frac{b}{\cos^{2}\theta}\,\partial_{\psi}\bigg{]},$
(3.97) $\displaystyle m_{(4)}^{a}\,\partial_{a}$
$\displaystyle=\frac{1}{\rho}\,\partial_{\theta},$ (3.98) $\displaystyle
n^{a}\,\partial_{a}$
$\displaystyle=-\frac{b^{2}}{\nu^{2}}\,\partial_{t}-\left(\frac{\rho^{2}-(r^{2}+2b^{2})\sin^{2}\theta}{2\nu^{2}\sin^{2}\theta}+\frac{M}{\rho^{2}}-\frac{Q^{2}}{2\rho^{4}}\right)\partial_{r}$
$\displaystyle\qquad+\frac{a\cot^{2}\theta}{\nu^{2}}\,\partial_{\phi}-\frac{b}{\nu^{2}}\,\partial_{\psi}.$
(3.99)
Having established the frame, we can easily calculate the Ricci rotation
coefficients (1.13). It can be shown that $L_{a0}=0$ and therefore
$\boldsymbol{k}$ is geodetic and affinely parametrized. The frame (3.99) is
not parallelly transported along the geodesics of the Kerr–Schild congruence
$\boldsymbol{k}$ since some of the independent components $N_{i0}$ and
$\stackrel{{\scriptstyle i}}{{M}}_{j0}$ are non-vanishing
$\displaystyle N_{20}$ $\displaystyle=-Q\frac{\nu}{\rho^{4}},\qquad
N_{30}=\frac{r\cot\theta}{\rho\nu},\qquad N_{40}=-\frac{\cot\theta}{\rho},$
(3.100) $\stackrel{{\scriptstyle 2}}{{M}}_{30}$
$\displaystyle=0,\qquad\mbox{$\stackrel{{\scriptstyle
2}}{{M}}_{40}$}=0,\qquad\mbox{$\stackrel{{\scriptstyle
3}}{{M}}_{40}$}=-\frac{\nu}{\rho^{2}}.$ (3.101)
Note that we cannot set this frame to be parallelly transported using
appropriate spins and null rotations with $\boldsymbol{k}$ fixed as in the
case of the Kerr–(anti-)de Sitter black hole in section 2.6.5 since we are not
able to transform away the component $N_{20}$ unless the identification of the
vector $\boldsymbol{m}$ appearing in the xKS metric (3.1) with the frame
vector $\boldsymbol{m}^{(2)}$ is relaxed.
The optical matrix $L_{ij}$ takes the block-diagonal form
$L_{ij}=\begin{pmatrix}\frac{1}{r}&0&0\\\
0&\frac{r}{\rho^{2}}&\frac{\nu}{\rho^{2}}\\\
0&-\frac{\nu}{\rho^{2}}&\frac{r}{\rho^{2}}\end{pmatrix}$ (3.102)
and it immediately follows that $L_{ij}$ is a normal matrix. Moreover,
interestingly, the optical matrix $L_{ij}$ (3.102) of the CCLP black hole
satisfies the optical constraint (2.128). Note that, in section 2.6.1, it has
been pointed out that in a certain sense the optical constraint can be
considered as a possible generalization of the Goldberg–Sachs theorem to
higher dimensions restricted to Einstein GKS spacetimes.
As has been mentioned in section 3.1.2, the vectors $\boldsymbol{k}$ and
$\boldsymbol{m}$ of the CCLP metric satisfy the relation (3.30) and therefore
(3.31) and (3.32) also hold. It can be seen directly from (3.102) that
$L_{22}=\frac{1}{r}$, $L_{2\tilde{\imath}}=0$ and using (3.31) from (3.100) it
follows that $L_{12}=-Q\frac{\nu}{\rho^{4}}$. Substituting these Ricci
rotation coefficients to (3.37), we obtain
$[\boldsymbol{k},\boldsymbol{m}]^{a}=-2Q\frac{\nu}{\rho^{4}}k^{a}+\frac{1}{r}m^{a}$
(3.103)
and therefore the vector fields $\boldsymbol{k}$ and $\boldsymbol{m}$ are
surface-forming. Note also that $L_{22}=\frac{1}{r}$ following from the
optical matrix (3.102) is in accordance with (3.31) since
$\zeta=|\hat{\boldsymbol{m}}|=\frac{\nu}{r}$ implies that
$\frac{\mathrm{D}\zeta}{\zeta}=-\frac{1}{r}$.
Employing the frame (3.99), we can express frame components of the Weyl tensor
and determine the algebraic type. In accordance with proposition 8, the boost
weight 2 components of the Weyl tensor vanish
$C_{0i0j}=0$ (3.104)
and therefore the spacetime is of Weyl type I or more special. Due to the
tracelessness of the Weyl tensor, the boost weight 1 components $C_{010i}$ are
not independent (1.35) and thus vanishing of the components $C_{0ijk}$ is
sufficient condition for a spacetime to be of Weyl type II or more special. In
the case of the CCLP black hole, one gets the non-zero components
$\displaystyle C_{0234}$
$\displaystyle=-C_{0243}=-\frac{4Q\nu^{2}}{\rho^{6}},\qquad$ (3.105)
$\displaystyle C_{0323}$
$\displaystyle=-C_{0332}=C_{0424}=-C_{0442}=-\frac{2Qr\nu}{\rho^{6}},$
$\displaystyle C_{0324}$
$\displaystyle=-C_{0342}=-C_{0423}=C_{0432}=-\frac{2Q\nu^{2}}{\rho^{6}},$
which suggests that the spacetime is of Weyl type I.
In section 2.2.2, we have mentioned the results of [13] that stationary
spacetimes with the metric remaining unchanged under reflection symmetry and
with non-vanishing expansion are of Weyl types G, Ii, D or conformally flat.
The CCLP spacetimes obey these conditions along with the reflection symmetry
$t\rightarrow-t$, $\phi\rightarrow-\phi$, $\psi\rightarrow-\psi$ of the metric
in Boyer–Lindquist-type coordinates given in [57]. Therefore we can conclude
that in general the CCLP solution is of Weyl type Ii with $\boldsymbol{k}$
being the WAND unless a distinct multiple WAND exists. However, the form of
the xKS metric (3.1) suggests that $\boldsymbol{k}$ should have the highest
order of alignment. Then only either in the uncharged case when the metric
corresponds to the five-dimensional Myers–Perry black hole or in the non-
rotating limit $\nu=0$, the spacetime is of more special Weyl type D and the
metric reduces to the GKS form since $\mathcal{K}$ (3.90) vanishes.
## Chapter 4 Quadratic gravity
In this chapter, we mainly present our results published in [65], which will
be occasionally extended. First of all, let us provide a motivation for
considering generalizations of the Einstein theory referred to as quadratic
gravity with a general Lagrangian containing all possible polynomial curvature
invariants up to the second order in the Riemann tensor and summarize some
basic properties of such theories.
As it is known, the Einstein–Hilbert action (1.1) is non-renormalizable [66].
In perturbative quantum gravity, corrections have to be added and, demanding
coordinate invariance, these corrections should consist of various curvature
invariants. If one includes curvature squared terms $\alpha R^{2}+\beta
R_{ab}R^{ab}$ to the four-dimensional Einstein–Hilbert action, the theory
becomes renormalizable [67, 68]. Note that such terms give the most general
action up to the second order in curvature since the term $R_{abcd}R^{abcd}$
can be rewritten using the squared Ricci tensor, squared curvature scalar and
the Gauss–Bonnet term which is topological invariant in four dimensions and
thus does not contribute to the dynamics. Unfortunately, the field equations
then contain the fourth derivatives of the metric and unitarity is lost due to
the introduced massive ghost-like graviton.
In string theory, the ghost freedom of low-energy effective action requires
that the quadratic corrections to the Einstein–Hilbert term, which is the
lowest order term in the Regge slope expansion of strings, are of the
Gauss–Bonnet form [69]. Therefore, the action up to the second order consists
of dimensionally extended Euler densities which correspond to the first three
terms of the Lovelock Lagrangian yielding the field equations of the second
order in derivatives of the metric. However, the forms of the corrections
depend on the type of string theory [70] and in higher orders they are no
longer given only by Euler densities.
Recently, in four dimensions [71] and later in arbitrary dimension [72], it
has been shown that despite non-zero $\alpha$ and $\beta$ producing ghosts,
the massive spin-0 mode can be eliminated and the massive spin-2 mode becomes
massless by an appropriate choice of the parameters of the theory denoted as
the critical point. This fact has led to a current growing interest in such
theories [73, 74, 75, 76, 77].
We will consider a general action of quadratic gravity which can be rearranged
to a more convenient form [78]
$\displaystyle\mathcal{S}$
$\displaystyle=\int\mathrm{d}^{n}x\,\sqrt{-g}\bigg{(}\frac{1}{\kappa}\left(R-2\Lambda_{0}\right)+\alpha
R^{2}+\beta R_{ab}R^{ab}$ (4.1)
$\displaystyle\qquad+\gamma\left(R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}\right)\bigg{)},$
where the term multiplied by $\kappa^{-1}$ is the well-known Einstein–Hilbert
term leading to the Einstein field equations. The last term multiplied by
$\gamma$ is the Gauss–Bonnet term and these both terms appear as the first
three terms in the Lovelock theory [79] with the Lagrangian
$\mathcal{L}=\sqrt{-g}\ \sum^{p}_{k=0}\alpha_{k}\mathcal{L}_{k},$ (4.2)
which consists of a linear combination of dimensionally extended
$2k$-dimensional Euler densities
$\mathcal{L}_{k}=\frac{1}{2^{k}}\delta^{a_{1}b_{1}\dots
a_{k}b_{k}}_{c_{1}d_{1}\dots
c_{k}d_{k}}\prod_{m=1}^{k}R_{a_{m}b_{m}}^{\phantom{a_{m}b_{m}}c_{m}d_{m}},$
(4.3)
where $\delta^{a_{1}b_{1}\dots a_{k}b_{k}}_{c_{1}d_{1}\dots
c_{k}d_{k}}=\frac{1}{k!}\delta_{[c_{1}}^{a_{1}}\delta_{d_{1}}^{b_{1}}\cdots\delta_{c_{k}}^{a_{k}}\delta_{d_{k}]}^{b_{k}}$
is the totally anti-symmetric generalized Kronecker delta. In $n$ dimensions,
$2k$-dimensional Euler densities, where $2k\geq n$, are topological invariants
or vanish identically and thus do not contribute to the field equations. Since
these equations are quasi-linear of the second order in derivatives of the
metric, Lovelock theories are natural generalizations of the Einstein gravity
to higher dimensions, in contrast to general quadratic gravity (4.1) where,
due to non-zero $\alpha$ and $\beta$, the field equations are of the fourth
order.
Varying the action (4.1) with respect to the metric leads to the source-free
field equations of quadratic gravity [80]
$\displaystyle\frac{1}{\kappa}\left(R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda_{0}g_{ab}\right)+2\alpha
R\left(R_{ab}-\frac{1}{4}Rg_{ab}\right)$ (4.4)
$\displaystyle\qquad+\left(2\alpha+\beta\right)\left(g_{ab}\Box-\nabla_{a}\nabla_{b}\right)R+2\gamma\bigg{(}RR_{ab}-2R_{acbd}R^{cd}$
$\displaystyle\qquad+R_{acde}R_{b}^{\phantom{b}cde}-2R_{ac}R_{b}^{\phantom{b}c}-\frac{1}{4}g_{ab}\left(R_{cdef}R^{cdef}-4R_{cd}R^{cd}+R^{2}\right)\bigg{)}$
$\displaystyle\qquad+\beta\Box\left(R_{ab}-\frac{1}{2}Rg_{ab}\right)+2\beta\left(R_{acbd}-\frac{1}{4}g_{ab}R_{cd}\right)R^{cd}=0.$
As in the Gauss–Bonnet theory, quadratic gravity in dimension $n>4$ admits, in
general, two distinct maximally symmetric vacua with the corresponding
cosmological constants $\Lambda$ given by
$\frac{\Lambda-\Lambda_{0}}{2\kappa}+\Lambda^{2}\bigg{(}\frac{(n-4)}{(n-2)^{2}}(n\alpha+\beta)+\frac{(n-3)(n-4)}{(n-2)(n-1)}\gamma\bigg{)}=0.$
(4.5)
In general relativity, Birkhoff’s theorem states that a spherically symmetric
solution of the Einstein field equations in vacuum is locally isometric to the
Schwarzschild solution, consequently, spherically pulsating objects cannot
emit gravitational waves. It has been shown in [81] that Birkhoff’s theorem is
valid also in the Lovelock theories, namely, that solutions of the source-free
Lovelock field equations with spherical, planar or hyperbolic symmetry are
locally isometric to the corresponding static black hole. Although the fact
that the field equations are of the second order is crucial in the proof of
the Birkhoff’s theorem, as has been discussed recently in [82], this theorem
can be extended also to higher derivative theories with the field equations of
the fourth order if the traced field equations are of the second order, i.e.
the massive spin-0 mode is not present in the linearized field equations of
the theory and if the field equations for spherically, plane or hyperbolically
symmetric spacetimes reduce to the second order. However, note that in the
case of quadratic gravity (4.1) the extended Birkhoff’s theorem requires
$\alpha=\beta=0$ and thus the action is given only by the Gauss–Bonnet term.
Let us also point out that at least certain subclasses of quadratic gravity
possess well-posed initial value formulation. For instance, the Cauchy problem
in the case of Einstein–Gauss–Bonnet gravity, i.e. $\alpha=\beta=0$,
$\gamma\neq 0$, was studied in [83]. In four dimensions, effectively with
$\gamma=0$, it has been shown that the Cauchy problem can be solved for
initial data if $\beta\neq 0$ [84].
As has been already mentioned, although quadratic gravity at the linearized
level around any of two vacua describes massless and massive spin-2 and
massive spin-0 modes, by an appropriate choice of the parameters of the
theory, one may eliminate the massive scalar mode and subsequently ensure that
the massive spin-2 mode becomes massless. Following [72], the trace of the
linearized field equations for the metric fluctuations
$h_{ab}=\bar{g}_{ab}+g_{ab}$ around an (A)dS vacuum $\bar{g}_{ab}$ reads
$\left[(4(n-1)\alpha+n\beta)\bar{\Box}-(n-2)\left(\frac{1}{\kappa}+4\epsilon\Lambda\right)\right]R^{L}=0,$
(4.6)
where $R^{L}$ is the linearized Ricci scalar and $\epsilon$ is defined in
(4.15). Setting
$4(n-1)\alpha+n\beta=0,$ (4.7)
we get rid of the massive spin-0 mode provided that
$\kappa\neq-4\epsilon\Lambda$ and then, choosing the gauge
$\bar{\nabla}^{a}h_{ab}=\bar{\nabla}_{b}h$, the linearized field equations
reduce to [72]
$\left(\bar{\Box}-\frac{4\Lambda}{(n-1)(n-2)}-M^{2}\right)\left(\bar{\Box}-\frac{4\Lambda}{(n-1)(n-2)}\right)h_{ab}=0.$
Therefore, vanishing $M^{2}$ defines the critical point only with the massless
spin-2 excitation
$M^{2}\equiv-\frac{4\Lambda}{\beta}\left(\frac{1}{4\Lambda\kappa}+\frac{n\alpha}{n-2}+\frac{\beta}{n-2}+\frac{(n-3)(n-4)}{(n-1)(n-2)}\gamma\right)=0.$
(4.8)
Note that the parameters cannot be tuned so that both distinct (A)dS vacua
become simultaneously critical. However, as has been also shown in [72], one
may still employ the remaining arbitrariness of the parameters to obtain a
theory with only one unique critical (A)dS vacuum. The cosmological constant
of such (A)dS vacuum is then given by
$\Lambda=\Lambda_{0}=\frac{(n-1)(n-2)}{8(n-3)\kappa\gamma}$ (4.9)
and the action reduces to the form of the Einstein–Weyl gravity
$\mathcal{S}=\int\mathrm{d}^{n}x\,\sqrt{-g}\bigg{(}\frac{1}{\kappa}\left(R-2\Lambda_{0}\right)+\gamma
C_{abcd}C^{abcd}\bigg{)}$ (4.10)
with only one additional parameter $\gamma$ besides Einstein’s constant
$\kappa$. Although at first sight it could seem that all type III and N
solutions of the Einstein gravity are also solutions of the Einstein–Weyl
gravity since the last term in the action (4.10) vanishes for these Weyl types
such reasoning is incorrect. In the following sections, we will thus study
type III and N solutions of the field equations of general quadratic gravity
(4.4) in detail. It will be shown, for instance, that apart from a subclass of
type III Einstein spacetimes that are solutions of quadratic gravity with
arbitrary parameters of theory, there exist a subclass of type III Einstein
spacetimes that do not satisfy the field equations if the Gauss–Bonnet term is
present in the action (4.1), i.e. $\gamma\neq 0$.
The field equations of quadratic gravity (4.4) are very complex and a direct
approach to finding exact solutions seems to be hopeless, therefore, so far
the known solutions of quadratic gravity has been obtained mostly by means of
an appropriate ansatz for the metric inspired by the form of known solutions
in the Einstein theory. Apart from the recently found AdS-wave solution of
general quadratic gravity in arbitrary dimension [74] using the Kerr–Schild
ansatz, at least to the author’s knowledge, previously known exact solutions
of quadratic gravity belong only to one of two subclasses with either
$\gamma=0$ or $\alpha=\beta=0$.
In the former case $\gamma=0$, few exact solutions are known. Four-dimensional
plane wave spacetimes have been analyzed in [85]. It has turned out that such
solutions with null radiation in a theory with $\beta=0$ are the same as in
the Einstein gravity, whereas if $\beta\neq 0$, an additional condition is
imposed on the null radiation term. Note that this is in accordance with our
more general results in section 4.2. A perturbative solution of the field
equations has been used in [86] to compare gravitational waves in the
linearized quadratic gravity with those in the linearized Einstein gravity. It
has been shown that the corrections to amplitude depend on the parameter
$\beta$ and the angular frequency of the wave but not on $\alpha$. Charged
black holes were also studied within this subclass of theories. It has been
pointed out in [87] that the four-dimensional Reissner–Nordström solution of
the Einstein field equations is a solution of quadratic gravity if $\beta=0$.
In the case $\alpha=\beta=0$ corresponding to the Einstein–Gauss–Bonnet
gravity, much more exact solutions have been found. The spherically symmetric
solutions in this theory consist of two branches, asymptotically Schwarzschild
black holes with a positive mass parameter and asymptotically
Schwarzschild–AdS spacetimes with a negative one [88]. Both are included in
the more general solution found in [89] using the Kerr–Schild ansatz with an
(anti-)de Sitter background metric in the spheroidal coordinates. However,
this solution does not represent rotating black hole solutions which have been
so far studied only numerically [90, 91, 92] or in the limit of small angular
momentum [93].
In the following section, we employ a different strategy to find a solution.
In section 4.1, we determine under which conditions known solutions of the
source-free Einstein theory solve also the field equations of quadratic
gravity and in section 4.2 we find explicit solutions of reduced field
equations assuming a special form of the Ricci tensor.
### 4.1 Einstein spacetimes
One of the approaches which may lead to the simplification of the very complex
field equations of quadratic gravity (4.4) is to assume a special form of the
Ricci tensor. Let us first study the simplest case of such a form, i.e.
Einstein spacetimes
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab},$ (4.11)
as exact solutions to quadratic gravity.
One may substitute the expression of the Riemann tensor in terms of the Weyl
tensor (1.31)
$R_{abcd}=C_{abcd}+\frac{2}{n-2}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})-\frac{2}{(n-1)(n-2)}Rg_{a[c}g_{d]b},$
(4.12)
along with the Ricci tensor (4.11) and the corresponding scalar curvature
$R=\frac{2n}{n-2}\Lambda$ to the field equations (4.4). Since the Weyl tensor
is traceless its contractions with the metric vanish and (4.4) reduces to the
simple form
$\mathcal{B}g_{ab}-\gamma\left(C_{a}^{\phantom{a}cde}C_{bcde}-\frac{1}{4}g_{ab}C^{cdef}C_{cdef}\right)=0,$
(4.13)
with the constant factor $\mathcal{B}$ given only by the effective
cosmological constant $\Lambda$, parameters of theory $\alpha$, $\beta$,
$\gamma$, $\kappa$, $\Lambda_{0}$ and dimension of spacetime $n$ as
$\mathcal{B}=\frac{\Lambda-\Lambda_{0}}{2\kappa}+\epsilon\Lambda^{2},$ (4.14)
where
$\epsilon=\frac{(n-4)}{(n-2)^{2}}(n\alpha+\beta)+\frac{(n-3)(n-4)}{(n-2)(n-1)}\gamma.$
(4.15)
Obviously, in the special case $\gamma=0$, i.e. when the Gauss–Bonnet term is
not present in the action (4.1), all Einstein spacetimes (4.11) with a
cosmological constant $\Lambda$ solve the field equations of quadratic gravity
(4.4) provided that $\mathcal{B}=0$.
The condition $\mathcal{B}=0$ appears several times in this chapter and, in
fact, it is a quadratic equation determining the effective cosmological
constant $\Lambda$ in terms of the parameters of particular theory $\alpha$,
$\beta$, $\gamma$, $\kappa$, $\Lambda_{0}$. In four dimensions, $\epsilon=0$
and then (4.14) implies that $\mathcal{B}=0$ admits only one root
$\Lambda=\Lambda_{0}$. In dimension $n>4$, there are two possible roots
$\Lambda=-\frac{1}{4\kappa\epsilon}\left(1\pm\sqrt{1+8\kappa\epsilon\Lambda_{0}}\right),$
(4.16)
where the Einstein constant $\kappa$ in our convention is assumed to be
positive. In the case that $1+8\kappa\epsilon\Lambda_{0}=0$, the unique
solution is $\Lambda=2\Lambda_{0}$. If $\mathcal{B}=0$ admits two roots, the
following combinations of their signs depending on the parameters of the
theory are possible:
* •
if $\Lambda_{0}<0$ then either
1. 1.
$0<4\kappa\epsilon<-\frac{1}{2\Lambda_{0}}$ when both roots $\Lambda$ are
negative, or
2. 2.
$4\kappa\epsilon<0$ when one root is positive and the other is negative.
* •
If $\Lambda_{0}=0$ then one of the roots is zero $\Lambda=0$ and the sign of
second root $\Lambda=-\frac{1}{2\kappa\epsilon}$ is either positive or
negative depending on the sign of $\epsilon$.
* •
If $\Lambda_{0}>0$ then either
1. 1.
$0>4\kappa\epsilon>-\frac{1}{2\Lambda_{0}}$ when both roots are positive, or
2. 2.
$4\kappa\epsilon>0$ when one root is positive and the other is negative.
Finally, if $1+8\kappa\epsilon\Lambda_{0}<0$ then $\mathcal{B}=0$ does not
admit any real root. Obviously, the same discussion also applies to the
possible signs of cosmological constants of two conformally flat vacua of
quadratic gravity since in that case the Weyl tensor vanishes.
Note also that in four dimensions the Gauss–Bonnet term is purely topological
and does not contribute to the field equations (4.4). The assumption that the
spacetime is Einstein leads to (4.13) effectively with $\gamma=0$ and thus
$\mathcal{B}=0$. For $n=4$, from (4.15) it follows that $\epsilon=0$ and then
(4.14) implies that the cosmological constants $\Lambda$ and $\Lambda_{0}$
have to be equal. In other words, all four-dimensional Einstein spaces with
$\Lambda=\Lambda_{0}$ are solutions of the field equations of quadratic
gravity (4.4). In fact, it has been already pointed out in [71], where the
four-dimensional action of quadratic gravity has been studied, that the
corresponding field equations reduce for Einstein spaces to the Einstein field
equations. At the level of the Weyl tensor, this can be seen as a consequence
of the identity
$C_{a}^{\phantom{a}cde}C_{bcde}=\frac{1}{4}g_{ab}C^{cdef}C_{cdef}$ which holds
in four dimensions but is not valid without additional restrictions in
dimension $n>4$ [94].
In the rest of this chapter, we study the conditions under which (4.13) can be
satisfied in a general case $\gamma\neq 0$ in arbitrary dimension. Thus we
will look for various classes of spacetimes satisfying
$C_{a}^{\phantom{a}cde}C_{bcde}=\frac{1}{4}g_{ab}C^{cdef}C_{cdef}$.
#### 4.1.1 Type N Einstein spacetimes
Now we show that the relation
$C_{a}^{\phantom{a}cde}C_{bcde}=\frac{1}{4}g_{ab}C^{cdef}C_{cdef}$ holds for
Weyl type N spacetimes since both sides vanish. The Weyl tensor expressed in
the frame (1.8) has only boost weight $-2$ components in the decomposition
(1.34), i.e.
$C_{abcd}=4\Omega^{\prime}_{ij}\ell_{\\{a}{m^{i}}_{b}\ell_{c}{m^{j}}_{d\\}},$
(4.17)
where we adopt more compact notation $\Omega^{\prime}_{ij}\equiv C_{1i1j}$
from [95]. Note that the null vector $\boldsymbol{\ell}$ is a multiple WAND
and $\Omega^{\prime}_{ij}$ is symmetric and traceless. Obviously, the
following contractions of the Weyl tensor appearing in the field equations of
quadratic gravity for Einstein spacetimes (4.13) vanish
$C_{a}^{\phantom{a}cde}C_{bcde}=C^{cdef}C_{cdef}=0$ (4.18)
and thus we are left with the algebraic constraint $\mathcal{B}=0$ which,
similarly as in the case of (A)dS vacua, prescribes two possible effective
cosmological constants $\Lambda$ of the solution for given parameters
$\alpha$, $\beta$, $\gamma$, $\kappa$, $\Lambda_{0}$. Therefore,
###### Proposition 10
All Weyl type N Einstein spacetimes (4.11) in arbitrary dimension with
appropriately chosen effective cosmological constant $\Lambda$ (4.16) are
exact solutions of the vacuum field equations of quadratic gravity (4.4).
Note that in the special case of Ricci-flat spacetimes $\Lambda=0$ in the
Gauss–Bonnet gravity $\alpha=\beta=0$ with vanishing cosmological constant
$\Lambda_{0}=0$ this result has been already pointed out in [96].
We can also relate the result of proposition 10 with chapter 2 using the
statement of proposition 4. It follows that all non-expanding Einstein GKS
spacetimes (2.11) with a cosmological constant (4.16) are consequently
solutions of the field equations of quadratic gravity.
Let us briefly overview known type N Einstein spacetimes in higher dimensions.
The multiple WAND of such spacetimes is always geodetic. This was shown in [2]
for the Ricci-flat case where the Bianchi identity (B.9) in [2] leads to
$C_{1i1[j}L_{k]0}=0$ which then implies that the WAND $\ell$ is geodetic.
Although it is obvious from (2.74), (2.76) and (2.77) that the Riemann tensor
of type N Einstein spacetimes not only has $R_{1i1j}$ components as in the
Ricci-flat case but also $R_{0101}$, $R_{0i1j}$ and $R_{ijkl}$ are non-zero.
Still, the Bianchi identity (B.9) in [2] reduces to the equation
$C_{1i1[j}L_{k]0}=0$ as well and one may follow the same steps as in [2] to
show immediately that the WAND is geodetic. Moreover, without loss of
generality, one may assume the geodetic WAND is affinely parametrized.
In an appropriately chosen frame the optical matrix $L_{ij}$ of type N
Einstein spacetimes consists of just one block $2\times 2$ [46]
$L_{ij}=\left(\begin{array}[]{c|c}\begin{matrix}s&a\\\ -a&s\end{matrix}&\hskip
14.0pt\mathbf{0}\\\
\hline\cr\phantom{\bigg{[}}\mathbf{0}\phantom{\bigg{[}}&\mathbf{0}\end{array}\right).$
(4.19)
From the definition of the optical scalars (1.23) follows that the expansion,
shear and twist are given by
$\theta=\frac{2}{n-2}s,\qquad\sigma^{2}=2\frac{n-4}{n-2}s^{2},\qquad\omega^{2}=2a^{2},$
(4.20)
respectively. Type N Einstein spacetimes can be thus further classified
according to the optical properties of the multiple WAND.
##### Type N Einstein Kundt spacetimes
The Kundt class is defined as spacetimes admitting a non-expanding $\theta=0$,
non-shearing $\sigma^{2}=0$ and non-twisting $\omega^{2}=0$ geodetic null
congruence $\boldsymbol{\ell}$, in other words, the optical matrix vanishes,
$L_{ij}=0$. Spacetimes belonging to this class can be described by a metric of
the form [37, 40]
$\mathrm{d}s^{2}=2\mathrm{d}u\left[\mathrm{d}v+H(u,v,x^{k})\,\mathrm{d}u+W_{i}(u,v,x^{k})\,\mathrm{d}x^{i}\right]+g_{ij}(u,x^{k})\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}.$
(4.21)
In general, higher dimensional Einstein Kundt metrics are of Weyl type II or
more special [3] and the functions $H$, $W_{i}$ and $g_{ij}$ have not been
expressed explicitly in the literature for the Weyl type N yet. However, the
situation differs for Ricci-flat Kundt metrics of types N and III where one
can set [38]
$g_{ij}(u,x^{k})=\delta_{ij}$ (4.22)
and the corresponding functions $W_{i}$ and $H$ for Weyl type N are given in
(2.98) and (2.99). In four dimensions, all type N Kundt metrics are completely
known [43] and given by (2.103), (2.104) and (2.106).
As we have already mentioned, non-expanding Einstein GKS spacetimes (2.11)
discussed in section 2.5 solve the field equations of quadratic gravity since
they belong to this class of spacetimes.
Further details and examples of type N Kundt metrics can be found, for
instance, in section 2.5.1 and in [37, 39]. In section 2.5.2, the Brinkmann
warp product is used to generate new Einstein type N Kundt metrics from the
known ones.
##### Expanding, non-twisting type N Einstein spacetimes
In four dimensions, expanding $\theta\neq 0$ and non-twisting $\omega^{2}=0$
type N Einstein metrics are necessarily shear-free due to the Goldberg–Sachs
theorem. This fact follows also immediately from (4.20). Therefore, such
metrics belong to the Robinson–Trautman class and are completely known [97],
see also [43] and references therein.
In contrast, for type N Einstein spacetimes in dimension $n>4$, it follows
from (4.20) that non-vanishing expansion $\theta\neq 0$ implies non-zero shear
$\sigma^{2}>0$. Higher dimensional metrics belonging to this class of
spacetimes can be constructed by warping four-dimensional type N Einstein
Robinson–Trautman metrics [97]
$\displaystyle\mathrm{d}\tilde{s}^{2}$
$\displaystyle=-2\psi\,\mathrm{d}u\,\mathrm{d}r+2r^{2}(\mathrm{d}x^{2}+\mathrm{d}y^{2})-2r(2rf_{1}+\epsilon
x)\,\mathrm{d}u\,\mathrm{d}x$ (4.23) $\displaystyle\qquad-2r(2rf_{2}+\epsilon
y)\,\mathrm{d}u\,\mathrm{d}y+2(\psi B+A)\,\mathrm{d}u^{2},$
with
$\displaystyle A$
$\displaystyle=\frac{1}{4}\epsilon^{2}(x^{2}+y^{2})+\epsilon(f_{1}x+f_{2}y)r+(f_{1}^{2}+f_{2}^{2})r^{2},$
(4.24) $\displaystyle B$
$\displaystyle=-\frac{1}{2}\epsilon-r\partial_{x}f_{1}+\frac{1}{6}\tilde{\Lambda}r^{2}\psi,$
$\displaystyle\psi$ $\displaystyle=1+\frac{1}{2}\epsilon(x^{2}+y^{2}),$
where $\epsilon=\pm 1$ or $0$, $\tilde{\Lambda}$ is a four-dimensional
cosmological constant and the functions $f_{1}=f_{1}(x,y)$ and
$f_{2}=f_{2}(x,y)$ are subject to
$\partial_{x}f_{1}=\partial_{y}f_{2},\quad\partial_{y}f_{1}=-\partial_{x}f_{2}.$
(4.25)
Then the five-dimensional metric takes one of the forms (1.40), (1.41),
(1.43)–(1.45) depending on the signs of $\Lambda$ and $\tilde{\Lambda}$, where
$\lambda=\frac{\Lambda}{6}$ and the five-dimensional cosmological constant
$\Lambda$ obeys $|\Lambda|=2|\tilde{\Lambda}|$. Whereas, in the case
$\Lambda=0$, $\tilde{\Lambda}>0$, the metric is given by (1.44) with
$\tilde{\Lambda}=3$.
##### Twisting type N Einstein spacetimes
Very few four-dimensional exact solutions of Einstein gravity within this
class are known. This includes the Ricci-flat Hauser metric [98] and the Leroy
metric [99] for a negative cosmological constant, see also [9].
As in the previous case, higher dimensional solutions in this class can be
constructed from four-dimensional twisting solutions using the Brinkmann warp
product. An example of such warped spacetimes can be obtained using the Leroy
metric in the form given in [46] as a seed. Since the cosmological constant of
the seed metric is negative, there is just one possibility of the sign of the
cosmological constant of the warped metric and thus we employ (1.43). After
the substitution $\tilde{z}=\sqrt{-\lambda}z$, the five-dimensional metric
reads
$\begin{split}\mathrm{d}\tilde{s}^{2}&=\frac{1}{-\Lambda
y^{2}\cos^{2}\tilde{z}}\bigg{[}\frac{2}{3}(\mathrm{d}x+y^{3}\,\mathrm{d}u)\big{[}6y\,\mathrm{d}r+y^{3}(1-r^{2})\,\mathrm{d}u\\\
&\qquad+(13-r^{2})\,\mathrm{d}x+12r\,\mathrm{d}y\big{]}+3(r^{2}+1)(\mathrm{d}x^{2}+\mathrm{d}y^{2})+6\mathrm{d}\tilde{z}^{2}\bigg{]},\end{split}$
(4.26)
where $\Lambda=6\lambda<0$ is a five-dimensional cosmological constant.
#### 4.1.2 Type III Einstein spacetimes
In general, for type III Einstein spacetimes the term
$C_{a}^{\phantom{a}cde}C_{bcde}-\frac{1}{4}g_{ab}C^{cdef}C_{cdef}$ in (4.13)
does not vanish as in the case of Weyl type N. Using the compact notation [95]
$\Psi_{i}^{\prime}\equiv C_{101i},\qquad\Psi_{ijk}^{\prime}\equiv
C_{1ijk},\qquad\Omega_{ij}^{\prime}\equiv C_{1i1j},$ (4.27)
where
$\Psi^{\prime}_{ijk}=-\Psi^{\prime}_{ikj},\qquad\Psi^{\prime}_{[ijk]}=0,\qquad\Psi^{\prime}_{i}=\Psi^{\prime}_{kik},$
(4.28)
the Weyl tensor of type III can be expressed from (1.34) as
$C_{abcd}=8\Psi^{\prime}_{i}\ell_{\\{a}n_{b}\ell_{c}m^{i}_{d\\}}+4\Psi^{\prime}_{ijk}\ell_{\\{a}m^{i}_{b}m^{j}_{c}m^{k}_{d\\}}+4\Omega^{\prime}_{ij}\ell_{\\{a}m^{i}_{b}\ell_{c}m^{j}_{d\\}}.$
(4.29)
Obviously, $C^{cdef}C_{cdef}$ vanishes and if we define $\tilde{\Psi}$ as
$\tilde{\Psi}\equiv\frac{1}{2}\Psi^{\prime}_{ijk}\Psi^{\prime}_{ijk}-\Psi^{\prime}_{i}\Psi^{\prime}_{i},$
(4.30)
then
$C_{a}^{\phantom{a}cde}C_{bcde}=\tilde{\Psi}\ell_{a}\ell_{b}.$ (4.31)
Therefore, in the case of Weyl type III, one may further simplify the field
equations of quadratic gravity for Einstein spaces (4.13) to the form
$\mathcal{B}g_{ab}-\gamma\tilde{\Psi}\ell_{a}\ell_{b}=0.$ (4.32)
The trace of (4.32) implies $\mathcal{B}=0$, which again determines two
possible effective cosmological constants $\Lambda$ for the given parameters
of theory, and subsequently it remains to satisfy $\tilde{\Psi}=0$.
###### Proposition 11
Weyl type III Einstein spacetimes with an effective cosmological constant
$\Lambda$ subject to (4.16) are exact solutions of the vacuum field equations
of quadratic gravity (4.4) if and only if $\tilde{\Psi}=0$.
From (4.28) it follows that in four dimensions $\tilde{\Psi}=0$. This is in
agreement with the already mentioned statement that all four-dimensional
Einstein spacetimes with an appropriate cosmological constant are solutions of
quadratic gravity since in this case effectively $\gamma=0$ in (4.13).
A wide class of higher dimensional Einstein spacetimes with $\tilde{\Psi}=0$
can be obtained by using the Brinkmann warp product. In order to show that we
start with the following observation. The components of the Weyl tensor of the
seed $\mathrm{d}\tilde{s}^{2}$ and warped metric $\mathrm{d}s^{2}$ (1.37)
expressed in coordinates $x^{a}=(z,x^{\mu})$ are related by [17]
$C_{\mu\nu\rho\sigma}=f{\tilde{C}}_{\mu\nu\rho\sigma},\qquad
C_{z\mu\nu\rho}=C_{z\mu z\nu}=0.$ (4.33)
Raising the indices by the corresponding metrics, it then follows that the
contractions $C_{a}^{\phantom{a}cde}C_{bcde}$ are given by
$C_{\mu}^{\phantom{\mu}\nu\rho\sigma}C_{\tau\nu\rho\sigma}=\frac{1}{f}{\tilde{C}}_{\mu}^{\phantom{\mu}\nu\rho\sigma}{\tilde{C}}_{\tau\nu\rho\sigma},$
(4.34)
with all $z$-components being zero.
Let us also recall that the Brinkmann warp product preserves the Weyl type of
algebraically special spacetimes. Therefore, if one takes an arbitrary four-
dimensional type III Einstein metric as a seed, for which
${\tilde{C}}_{\mu}^{\phantom{\mu}\nu\rho\sigma}{\tilde{C}}_{\tau\nu\rho\sigma}=0$
holds identically, then the warped metric represents a type III Einstein
spacetime with $\tilde{\Psi}=0$ and thus, by proposition 11, it is also an
exact solution of the field equations of quadratic gravity (4.4) provided that
the effective cosmological constant $\Lambda$ satisfies (4.16). A few examples
of such twisting and non-twisting type III Einstein spacetimes obtained by the
Brinkmann warp product are given in [46].
It should be emphasized that in contrast with the type N case, there exist
type III Einstein spacetimes which are not solutions of quadratic gravity. For
instance, $\tilde{\Psi}$ is clearly non-vanishing for the type III(a) subclass
of type III spacetimes characterized by $\Psi^{\prime}_{i}=0$ [5]. Type III(a)
Kundt spacetimes with null radiation given in [38] contain type III(a) Ricci-
flat metrics (4.21), (4.22) with a covariantly constant null vector, where
$\displaystyle H=H(u,x^{k}),\qquad W_{2}=0,\qquad
W_{\tilde{\imath}}=W_{\tilde{\imath}}(u,x^{k}),\phantom{\frac{1}{2}}$
$\displaystyle
W^{\tilde{\imath}}_{\phantom{\tilde{\imath}},\tilde{\imath}2}=0,\qquad
W^{\tilde{\imath}}_{\phantom{\tilde{\imath}},\tilde{\imath}\tilde{\jmath}}=\Delta
W_{\tilde{\jmath}},\phantom{\frac{1}{2}}$ (4.35) $\displaystyle\Delta
H-\frac{1}{4}(W_{\tilde{\imath},\tilde{\jmath}}-W_{\tilde{\jmath},\tilde{\imath}})(W^{\tilde{\imath},\tilde{\jmath}}-W^{\tilde{\jmath},\tilde{\imath}})-W^{\tilde{\imath}}_{\phantom{\tilde{\imath}},\tilde{\imath}u}=0.$
Here the indices $\tilde{\imath}$, $\tilde{\jmath}$ run from 3 to $n-1$ and
necessarily at least one of the components
$C_{1k\tilde{\jmath}\tilde{\imath}}=\frac{1}{2}(W_{\tilde{\imath},\tilde{\jmath}}-W_{\tilde{\jmath},\tilde{\imath}})_{,k}$
has to be non-vanishing, otherwise the metric reduces to Weyl type N. An
explicit five-dimensional example of such pp-wave metric is given by [27]
$\displaystyle W_{2}=0,\qquad W_{3}=h(u)x^{2}x^{4},\qquad
W_{4}=h(u)x^{2}x^{3},$ (4.36) $\displaystyle
H=H_{0}=h(u)^{2}\left[\frac{1}{24}\left(\left(x^{3}\right)^{4}+\left(x^{4}\right)^{4}\right)+h^{0}(x^{2},x^{3},x^{4})\right],$
(4.37)
where $h^{0}(x^{2},x^{3},x^{4})$ is subject to $\Delta h^{0}=0$. In fact, all
type III Ricci-flat pp-waves belong to the type III(a) subclass since the
existence of the covariantly constant null vector $\boldsymbol{\ell}$ implies
$C_{abcd}\ell^{a}=0$ and thus $\Psi^{\prime}_{i}$ vanishes. Therefore, type
III Ricci-flat pp-waves are not solutions of quadratic gravity.
Motivated by the above results we naturally introduce two new subclasses of
the principal Weyl type III, namely type III(A) characterized by
$\tilde{\Psi}\not=0$ and type III(B) defined by $\tilde{\Psi}=0$. Obviously,
type III(a) is a subclass of type III(A) since, as already mentioned,
$\tilde{\Psi}=\frac{1}{2}\Psi^{\prime}_{ijk}\Psi^{\prime}_{ijk}\neq 0$ in this
case.
#### 4.1.3 Comparison with other classes of spacetimes
It is of interest to compare the set of exact solutions of quadratic gravity
(QG) with other overlapping classes of spacetimes. Namely, spacetimes with
vanishing curvature invariants (VSI) [4], spacetimes with constant curvature
invariants (CSI) [37], Kundt subclass of CSI (KCSI), pp -waves which will be
denoted as ppN, ppIII, etc., depending on a particular Weyl type, and
universal metrics (U) for which quantum corrections, i.e. all symmetric
covariantly conserved tensors of rank 2 constructed from the metric, Riemann
tensor and its covariant derivatives, are a multiple of the metric [100, 101].
Einstein or Ricci-flat subclasses of these sets will be indicated by the
appropriate subscript, for instance, QG${}_{\text{E}}$ and QG${}_{\text{RF}}$.
The class of pp -waves is defined geometrically as spacetimes admitting a
covariantly constant null vector field, say $\boldsymbol{\ell}$, and thus
$\ell_{a;b}=0$. It then follows from the definition of the Riemann tensor that
$R_{abcd}\ell^{a}=0$. The contraction with respect to the second and fourth
index yields $R_{0b}=0$, where we identify $\boldsymbol{\ell}$ with the
corresponding null frame vector (1.8). On the other hand, in the case of
Einstein spaces, the frame component of the Ricci tensor $R_{01}$ is
proportional to the cosmological constant $R_{01}=\frac{2\Lambda}{n-2}$ and
therefore Einstein pp -waves do not admit non-vanishing $\Lambda$, i.e.
pp${}_{\text{E}}$ = pp${}_{\text{RF}}$. In the Ricci-flat case, the Weyl
tensor is given exactly by the Riemann tensor and thus for pp${}_{\text{RF}}$
we obtain $C_{abcd}\ell^{a}=0$. In four dimensions this corresponds to the Bel
criterion ensuring that the spacetime is of type N, whereas, in higher
dimensions, it implies that the spacetime is of Weyl type II or more special.
We will consider $n>4$ since in four dimensions all pp${}_{\text{RF}}$ are of
type N and as discussed above all Einstein spacetimes with
$\Lambda=\Lambda_{0}$ belong to QG which leads to a considerable
simplification.
From the definition of U, it is obvious that U $\subset$ QG since the action
of quadratic gravity (4.1) contains quantum corrections only up to the second
order in curvature. The results of [4] that VSI spacetimes are of Weyl type
III or more special and admit a congruence of non-expanding, non-shearing and
non-twisting null geodesics and thus belong to the Kundt class imply VSI
$\subset$ KCSI. All pp-waves from the set ppN${}_{\text{RF}}$ $\cup$
ppIII${}_{\text{RF}}$ belong to VSI, whereas, as was shown above,
ppIII${}_{\text{RF}}$ $\cap$ QG is $\emptyset$ and therefore
VSI${}_{\text{RF}}$ $\nsubseteq$ QG and pp${}_{\text{RF}}$ $\nsubseteq$ QG. It
also holds that pp${}_{\text{RF}}$ $\nsubseteq$ VSI since ppII${}_{\text{RF}}$
solutions exist in higher dimensions.
Recently, it was conjectured in [101] that U $\subset$ KCSI. Obviously,
ppIII${}_{\text{RF}}$ are examples of spacetimes which are VSI and thus KCSI
but not U. However, note that QG${}_{\text{E}}$ $\nsubseteq$ CSI since
examples of QG${}_{\text{E}}$ metrics with non-vanishing expansion mentioned
in this section have in general non-trivial curvature invariants [46].
### 4.2 Spacetimes with aligned null radiation
One may attempt to find a wider class of solutions of quadratic gravity
considering more general form of the Ricci tensor than for Einstein spacetimes
(4.11) but still sufficiently simple to considerably reduce the field
equations (4.4). Therefore, we will assume that the Ricci tensor contains an
additional aligned null radiation term
$R_{ab}=\frac{2\Lambda}{n-2}g_{ab}+\Phi\ell_{a}\ell_{b}.$ (4.38)
Then the contracted Bianchi identities
$\nabla^{a}R_{ab}=\frac{1}{2}\nabla_{b}R$ imply that the null radiation term
has to be covariantly conserved $(\Phi\ell^{a}\ell^{b})_{;a}=0$, which one may
rewrite using the derivatives (1.17) and the optical scalars (1.25) as
$\left[\mathrm{D}\Phi+\Phi(n-2)\theta\right]\ell_{a}+\Phi\ell_{a;b}\ell^{b}=0.$
(4.39)
We identify $\boldsymbol{\ell}$ with the corresponding null frame vector (1.8)
so that the contraction of (4.39) with the vector $\boldsymbol{m}^{(i)}$
implies $L_{i0}=0$ in terms of the Ricci rotation coefficients (1.13).
Therefore, $\boldsymbol{\ell}$ has to be geodetic and, without loss of
generality, we choose $\boldsymbol{\ell}$ to be affinely parametrized.
Consequently, the contraction of (4.39) with the frame vector $\boldsymbol{n}$
yields
$\mathrm{D}\Phi=-(n-2)\theta\Phi.$ (4.40)
Now, following the same steps as for Einstein spaces in section 4.1, we
express the field equations (4.4) in terms of the Weyl tensor and the Ricci
tensor (4.38) and simplify them using the tracelessness of the Weyl tensor and
the fact that $\boldsymbol{\ell}$ is a null vector
$\displaystyle(\beta\Box+\mathcal{A})(\Phi\ell_{a}\ell_{b})-2\mathcal{B}g_{ab}+2\gamma\left(C_{a}^{\phantom{a}cde}C_{bcde}-\frac{1}{4}g_{ab}C^{cdef}C_{cdef}\right)$
(4.41)
$\displaystyle\qquad+2\Phi\left(\beta-2\frac{n-4}{n-2}\gamma\right)C_{acbd}\ell^{c}\ell^{d}=0,$
where $\mathcal{A}$ is defined as
$\mathcal{A}=\frac{1}{\kappa}+4\Lambda\bigg{(}\frac{n\alpha}{n-2}+\frac{\beta}{n-1}+\frac{(n-3)(n-4)}{(n-2)(n-1)}\gamma\bigg{)}$
(4.42)
and $\mathcal{B}$ is given by (4.14). If the Weyl tensor is of type III or
more special then the last term in (4.41) vanishes and one may rewrite (4.41)
using $\tilde{\Psi}$ (4.31) as
$(\beta\Box+\mathcal{A})(\Phi\ell_{a}\ell_{b})-2\mathcal{B}g_{ab}+2\gamma\tilde{\Psi}\ell_{a}\ell_{b}=0.$
(4.43)
From now on, we restrict ourselves to spacetimes with $\tilde{\Psi}=0$. The
trace of (4.43) yields $\mathcal{B}=0$ which again determines two possible
effective cosmological constants $\Lambda$ via (4.16). The remaining part of
(4.43) reads
$(\beta\Box+\mathcal{A})(\Phi\ell_{a}\ell_{b})=0.$ (4.44)
Using the notion of the subclasses III(A) and III(B) of the Weyl type III
defined at the end of section 4.1.2, we arrive to
###### Proposition 12
All spacetimes of Weyl types III(B), N and O with the Ricci tensor of the form
(4.38) are vacuum solutions of quadratic gravity provided that $\mathcal{B}=0$
and the null radiation term $\Phi\ell_{a}\ell_{b}$ satisfies (4.44).
It should be emphasized that these spacetimes with a null radiation term in
the Ricci tensor, i.e. solutions of the non-vacuum Einstein field equations,
are solutions of the vacuum field equations of quadratic gravity without any
matter field terms in the action.
Let us briefly comment the special case $\beta=0$. Then it follows that both
$\mathcal{A}$ (4.42) and $\mathcal{B}$ (4.14) have to vanish and from these
relations, one may eliminate the parameter $\gamma$ to obtain
$\frac{8n\alpha\kappa}{(n-2)^{2}}\Lambda^{2}-\Lambda+2\Lambda_{0}=0.$ (4.45)
Therefore, the effective cosmological constant $\Lambda$ is determined only by
the parameters $\alpha$, $\kappa$ and $\Lambda_{0}$. If the constraint on
$\Lambda$ (4.45) admits a real solution then the remaining parameter $\gamma$
is subject to $\mathcal{A}=0$ or $\mathcal{B}=0$. In other words, for special
values of the parameters $\alpha$, $\gamma$, $\kappa$ and $\Lambda_{0}$ of a
theory with $\beta=0$, all spacetimes of Weyl types III(B), N and O with the
Ricci tensor of the form (4.38) with an arbitrary $\Phi$ and an effective
cosmological constant $\Lambda$ given by (4.45) are exact solutions of
quadratic gravity. However, none of such spacetimes with null radiation
satisfies the field equations of quadratic gravity if $\mathcal{A}\neq 0$,
which occurs, for instance, in the case $\Lambda=0$. For the pure Gauss–Bonnet
gravity $\alpha=\beta=0$, this implies that if $\Lambda=2\Lambda_{0}$
following from (4.45) and simultaneously
$\gamma=-\frac{(n-2)(n-1)}{(n-3)(n-4)}\frac{1}{8\Lambda_{0}\kappa},$ (4.46)
then both $\mathcal{A}$, $\mathcal{B}$ vanish and $\Phi$ can be arbitrary,
otherwise $\Phi$ has to be zero in order to satisfy the field equations.
Since we are interested in solutions of quadratic gravity with arbitrary
parameters we assume $\beta\neq 0$ in the rest of this chapter. The
contraction of (4.44) with the frame vectors $\boldsymbol{\ell}$ and
$\boldsymbol{n}$ gives
$\Phi L_{ij}L_{ij}=\Phi[(n-2)\theta^{2}+\sigma^{2}+\omega^{2}]=0,$ (4.47)
where the optical matrix $L_{ij}$ is defined in (1.13) and the scalars
$\theta$, $\sigma$ and $\omega$ (1.23) correspond to expansion, shear, and
twist, respectively. This implies that the geodetic vector $\boldsymbol{\ell}$
is non-expanding, $\theta=0$, non-shearing, $\sigma=0$, and non-twisting,
$\omega=0$, and thus the optical matrix vanishes $L_{ij}=0$. Then it
immediately follows from (4.40) that $\mathrm{D}\Phi=0$, i.e. $\Phi$ does not
depend on an affine parameter along the null geodesics $\boldsymbol{\ell}$.
###### Proposition 13
All Weyl type III(B), N or conformally flat solutions of the source-free field
equations of quadratic gravity (4.4) with $\beta\neq 0$ and the Ricci tensor
of the form (4.38) belong to the Kundt class.
Note that in general, not only in quadratic gravity, conformally flat
spacetimes with null radiation belong to the Kundt class as follows directly
from the Bianchi identities.
Contracting (4.44) twice with the frame vector $\boldsymbol{n}$ and
substituting $L_{a0}=0$, i.e. $\boldsymbol{\ell}$ being geodetic and affinely
parametrized, $L_{ij}=0$ and $\mathrm{D}\Phi=0$, we obtain the remaining non-
trivial frame component of (4.44)
$\displaystyle\delta_{i}\delta_{i}\Phi+\left(2L_{1i}+4L_{[1i]}+\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\right)\delta_{i}\Phi+2\Phi\bigg{(}2\delta_{i}L_{[1i]}+L_{i1}L_{i1}$
(4.48)
$\displaystyle\qquad+4L_{1i}L_{[1i]}+2L_{[1i]}\mbox{$\stackrel{{\scriptstyle
i}}{{M}}_{jj}$}\bigg{)}+\frac{4\Lambda\Phi}{n-2}+\mathcal{A}\beta^{-1}\Phi=0,$
where we also employed the Ricci identities [3] and the commutators of the
derivatives along the frame vectors (1.18). The Ricci rotation coefficients
$L_{1i}$, $L_{i1}$ and $\stackrel{{\scriptstyle i}}{{M}}_{jk}$ are defined in
(1.13).
It has been shown in section 2.5 that one may always set the frame in Kundt
spacetimes so that $L_{[1i]}=0$, $L_{12}\neq 0$, $L_{1\tilde{\imath}}=0$. For
Kundt metrics in the canonical form (4.21), the natural frame with the
geodetic WAND $\ell_{a}\mathrm{d}x^{a}=\mathrm{d}u$ as the null frame vector
$\boldsymbol{\ell}$ implies that the antisymmetric part of $L_{1i}$ vanishes
$L_{[1i]}=\ell_{[a;b]}n^{a}m^{b}_{(i)}=\ell_{[a,b]}n^{a}m^{b}_{(i)}-\Gamma^{c}_{[ab]}\ell_{c}n^{a}m^{b}_{(i)}=0,$
(4.49)
since $\ell_{a,b}=0$ in these coordinates and the Christoffel symbols are
symmetric in the lower indices. However, $L_{\tilde{\imath}0}$ are in general
non-vanishing in this frame. Let us emphasize that this result does not depend
on the choice of the frame vectors $\boldsymbol{n}$ and $\boldsymbol{m}^{(i)}$
and it can be used for further simplification of (4.48). Moreover, one may
rewrite $\delta_{i}\delta_{i}\Phi$ in terms of the d’Alembert operator to get
$\Box\Phi+4L_{1i}\delta_{i}\Phi+2L_{1i}L_{1i}\Phi+\frac{4\Lambda\Phi}{n-2}+\mathcal{A}\beta^{-1}\Phi=0.$
(4.50)
Note that, as argued under proposition 4, non-expanding GKS spacetimes (2.11)
with the Ricci tensor of the form (4.38) belong to the Kundt class of Weyl
type N. Therefore, such spacetimes with an appropriate effective cosmological
constant $\Lambda$ solve the field equations of quadratic gravity provided
that the null radiation term $\Phi\ell_{a}\ell_{b}$ satisfies (4.44) or
equivalently (4.50) in the case $L_{[1i]}=0$.
Interestingly, if we assume that the function $\mathcal{H}$ of non-expanding
GKS spacetimes with aligned null radiation is independent on an affine
parameter along null geodesics of the Kerr–Schild congruence
$\boldsymbol{k}\equiv\boldsymbol{\ell}$, i.e. $\mathrm{D}\mathcal{H}=0$, and
set $L_{[1i]}=0$, the corresponding Einstein field equations (2.81), (2.82),
(2.92) reduce to
$\Box\mathcal{H}+4L_{1i}\delta_{i}\mathcal{H}+2L_{1i}L_{1i}\mathcal{H}+\frac{4\Lambda\mathcal{H}}{n-1}=\Phi.$
(4.51)
Let us point out the similarity of (4.50) and (4.51) which effectively
decouples these two equations and thus allows us to solve for $\mathcal{H}$
and $\Phi$ independently.
#### 4.2.1 Explicit solutions of Weyl type N
In this section, we present a few examples of type N solutions of the field
equations of quadratic gravity with the Ricci tensor of the form (4.38). We
solve the cases with a vanishing and non-vanishing effective cosmological
constant $\Lambda$ separately since the form of all higher dimensional type N
Kundt metrics with $\Lambda=0$ and aligned null radiation is explicitly known
[38], whereas, in the case $\Lambda\neq 0$ we employ at least a particular
example of such Kundt metrics known as the Siklos metric.
##### Case $\Lambda=0$
Type N Kundt metrics with aligned null radiation and the vanishing
cosmological constant $\Lambda$ which belong to a subclass of VSI spacetimes,
admit the form (4.21), (4.22). As in the Ricci-flat case discussed in section
2.5.1, these metrics can be split into two subclasses with vanishing
($\epsilon=0$) and non-vanishing ($\epsilon=1$) quantity $L_{1i}L_{1i}$. One
may choose the same null frame as in (2.95) and, consequently, the Ricci
rotation coefficients $L_{1i}$, $L_{i1}$ and $L_{11}$ are given as in (2.96).
The constraints on the undetermined metric functions $W_{i}$ and $H$ imposed
by the form of the Weyl tensor (4.17) and the Ricci tensor (4.38) differ from
the Ricci-flat case since the null radiation term $\Phi$ now occurs [38]
$\begin{split}&W_{2}=0,\qquad
W_{\tilde{\imath}}=x^{2}C_{\tilde{\imath}}(u)+x^{\tilde{\jmath}}B_{\tilde{\jmath}\tilde{\imath}}(u),\phantom{\frac{1}{2}}\\\
&H=H^{0}(u,x^{i}),\qquad\Delta H^{0}-\frac{1}{2}\sum
C^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}+\Phi=0\end{split}$
(4.52)
in the case $\epsilon=0$ and
$\begin{split}&W_{2}=-\frac{{{2}}v}{x^{2}},\qquad
W_{\tilde{\imath}}=C_{\tilde{\imath}}(u)+x^{\tilde{\jmath}}B_{\tilde{\jmath}\tilde{\imath}}(u),\qquad
H=\frac{v^{2}}{2(x^{2})^{2}}+H^{0}(u,x^{i}),\\\
&x^{2}\Delta\left(\frac{H^{0}}{x^{2}}\right)-\frac{1}{(x^{2})^{2}}\sum
W^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}+\Phi=0\end{split}$
(4.53)
in the case $\epsilon=1$, respectively. Where the indices $\tilde{\imath}$,
$\tilde{\jmath}$, …range from 3 to $n-1$ and
$B_{[\tilde{\imath}\tilde{\jmath}]}=0$ in both cases.
Now we determine the form of the function $\Phi$. Obviously,
$\Phi_{,v}=\mathrm{D}\Phi=0$ and thus $\Phi$ does not depend on the coordinate
$v$. In the Einstein gravity, no further conditions are imposed on $\Phi$,
whereas in quadratic gravity theory with $\beta\neq 0$, $\Phi$ still has to
satisfy (4.44) or equivalently (4.50) since $L_{[1i]}=0$ in our chosen frame.
The latter one is simpler to express and leads directly to
$\Phi_{,ii}-\frac{2\epsilon}{x^{2}}\Phi_{,2}+\frac{2\epsilon}{(x^{2})^{2}}\Phi+(\kappa\beta)^{-1}\Phi=0.$
(4.54)
In the case $\epsilon=0$, (4.54) reads
$\Delta\Phi+(\kappa\beta)^{-1}\Phi=0.$ (4.55)
Substituting (4.55) into (4.52), we obtain
$\Delta H^{0}_{\text{vac}}-\frac{1}{2}\sum
C^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}=0,$
(4.56)
where we define $H^{0}_{\text{vac}}=H^{0}-\kappa\beta\Phi$. Thus,
$H^{0}_{\text{vac}}$ corresponds to a vacuum VSI solution of the Einstein
gravity, i.e. it obeys (4.52) with $\Phi=0$.
Similarly, in the case $\epsilon=1$, (4.54) reads
$x^{2}\Delta\left(\frac{\Phi}{x^{2}}\right)+(\kappa\beta)^{-1}\Phi=0.$ (4.57)
Putting (4.57) to (4.53) and denoting
$H^{0}_{\text{vac}}=H^{0}-\kappa\beta\Phi$ gives the Einstein field equations
for Ricci-flat type N VSI metrics, i.e. (4.53) with $\Phi=0$ for
$H^{0}_{\text{vac}}$
$x^{2}\Delta\left(\frac{H^{0}_{\mathrm{vac}}}{x^{2}}\right)-\frac{1}{(x^{2})^{2}}\sum
W^{2}_{\tilde{\imath}}-2\sum_{\tilde{\imath}<\tilde{\jmath}}B^{2}_{\tilde{\imath}\tilde{\jmath}}=0.$
(4.58)
In other words, these steps may be performed backwards. We take an arbitrary
vacuum type N VSI metric (4.21), (4.22) with $H^{0}_{\text{vac}}$, $W_{i}$ and
$\Phi=0$, i.e. a solution of the Einstein field equations (4.56) or (4.58),
and independently find $\Phi$ solving the corresponding equation (4.55) or
(4.57), respectively. Therefore, we finally arrive at a solution of the field
equations of quadratic gravity (4.4) represented by the metric (4.21), (4.22)
with $\Phi\neq 0$, $H^{0}=H^{0}_{\mathrm{vac}}+\kappa\beta\Phi$ and $W_{i}$
unchanged.
Note that in this case where we assume $\Lambda=0$ the condition
$\mathcal{B}=0$ (4.14) determining the effective cosmological constant
$\Lambda$ in terms of the parameters $\alpha$, $\beta$, $\gamma$, $\kappa$ and
$\Lambda_{0}$ implies $\Lambda_{0}=0$ and therefore we are not able to satisfy
the criticality condition (4.7), (4.8) by tuning the remaining parameters.
##### Case $\Lambda\not=0$
One may perform a similar procedure also in the case of a non-vanishing
effective cosmological constant $\Lambda$. Unlike in the previous case
$\Lambda=0$, to our knowledge a general form of the metric describing all type
N Kundt spacetimes with $\Lambda\neq 0$ and aligned null radiation is not
explicitly known. Therefore, we consider the $n$-dimensional Siklos metric
[102]
$\mathrm{d}s^{2}=\frac{1}{-\lambda
z^{2}}\left[2\mathrm{d}u\,\mathrm{d}v+2H(u,v,x^{k})\,\mathrm{d}u^{2}+\delta_{ij}\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}\right],$
(4.59)
where
$\lambda=\frac{2\Lambda}{(n-1)(n-2)},\qquad z=x^{n-1},$ (4.60)
as an example belonging to this class of spacetimes.
The Siklos metric is conformally related to pp -waves. However, note that the
terms inside the parentheses correspond to the metric which describes all pp
-waves only in four dimensions. In higher dimensions, this metric does not
represent the pp -wave class completely and not even the type N pp -wave
subclass.
Obviously, the metric (4.59) takes the GKS form (2.11) since the first and
third terms in the parentheses represent an anti-de Sitter background and
$\mathrm{d}u$ corresponds to the null vector. Then proposition 4 ensures that
this metric is indeed of Weyl type N even if we admit an aligned null
radiation term in the Ricci tensor as commented under this proposition.
We need to split the Kerr–Schild term
$\mathcal{H}k_{a}k_{b}\,\mathrm{d}x^{a}\,\mathrm{d}x^{b}=\frac{H}{\lambda
z^{2}}\,\mathrm{d}u^{2}$ into the vector $\boldsymbol{k}$ and the function
$\mathcal{H}$ so that $\boldsymbol{k}$ is affinely parametrized. The simplest
way how to do that is to employ the coordinate transformation
$v=-\lambda\tilde{v}z^{2}$ to put (4.59) to the canonical Kundt form
$\mathrm{d}s^{2}=2\mathrm{d}u\left[\mathrm{d}\tilde{v}-\frac{H}{\lambda
z^{2}}\,\mathrm{d}u+\frac{2\tilde{v}}{z}\,\mathrm{d}z\right]-\frac{1}{\lambda
z^{2}}\delta_{\tilde{\imath}\tilde{\jmath}}\,\mathrm{d}x^{\tilde{\imath}}\,\mathrm{d}x^{\tilde{\jmath}}.$
(4.61)
In this form, $\mathrm{d}u$ corresponds to the congruence of non-expanding,
non-shearing and non-twisting affinely parametrized null geodesics. Therefore,
$\boldsymbol{k}=k_{a}\mathrm{d}x^{a}=\mathrm{d}u$ and
$\mathcal{H}=\frac{1}{\lambda z^{2}}H$. Furthermore, if we identify the
Kerr–Schild vector $\boldsymbol{k}$ with the frame vector $\boldsymbol{\ell}$,
then from (4.49) follows that $L_{[1i]}=0$. One may complete the frame by a
natural choice of the remaining vectors
$\displaystyle\ell_{a}\,\mathrm{d}x^{a}$ $\displaystyle=\mathrm{d}u,\qquad
n_{a}\,\mathrm{d}x^{a}=\frac{1}{-\lambda z^{2}}\,\mathrm{d}v+\frac{H}{-\lambda
z^{2}}\,\mathrm{d}u,\qquad m^{(i)}_{a}\,\mathrm{d}x^{a}=\mathrm{d}x^{i},$
(4.62) $\displaystyle\ell^{a}\,\partial_{a}$ $\displaystyle=-\lambda
z^{2}\,\partial_{v},\qquad
n^{a}\,\partial_{a}=\partial_{u}-H\,\partial_{v},\qquad
m_{(i)}^{a}\,\partial_{a}=\partial_{i}.$
For the simplicity, we assume that $\mathrm{D}\mathcal{H}=0$, i.e.
$\mathcal{H}$ is independent on an affine parameter along the null geodesics
$\boldsymbol{k}$. On the other hand, using the frame (4.62), it follows that
$\mathrm{D}\mathcal{H}=-\lambda z^{2}\mathcal{H}_{,v}$ and thus $\mathcal{H}$
and consequently $H$ does not depend on the coordinate $v$. Therefore, we can
use the Einstein field equations for non-expanding GKS spacetimes with aligned
null radiation (4.51) which for the Siklos metric (4.59) leads to
$\Delta H-\frac{n-2}{z}H_{,z}=\Phi.$ (4.63)
The condition on $\Phi$ (4.50) for the metric (4.59) reads
$\Delta\\!\left(-\lambda z^{2}\Phi\right)-\frac{n-2}{z}\left(-\lambda
z^{2}\Phi\right)_{,z}-\frac{\mathcal{C}}{z^{2}}(-\lambda z^{2}\Phi)=0,$ (4.64)
where we defined
$\mathcal{C}\equiv
2+\frac{\mathcal{A}}{\beta\lambda}=\frac{2}{\beta}\left(\frac{1}{2\lambda\kappa}+(n-1)(n\alpha+\beta)+(n-3)(n-4)\gamma\right).$
(4.65)
Now we eliminate $\Phi$ from (4.63) by combining (4.63) with (4.64) and
denoting $H^{\text{vac}}=H-\mathcal{C}^{-1}z^{2}\Phi$. This leads to the
vacuum ($\Phi=0$) equation (4.63) for $H^{\text{vac}}$
$\Delta H^{\rm vac}-\frac{n-2}{z}H^{\rm vac}_{,z}=0.$ (4.66)
Therefore, we can take an arbitrary higher dimensional Siklos metric (4.59)
obeying the vacuum Einstein field equations (4.66) and find a solution $\Phi$
of (4.64). Then the metric (4.59) with
$H=H^{\text{vac}}+\mathcal{C}^{-1}z^{2}\Phi$, where the obtained $\Phi$ enters
also the corresponding Ricci tensor (4.38), satisfies the field equations of
quadratic gravity (4.4).
Unfortunately, this procedure cannot be performed for quadratic gravity
theories with the parameters at critical points (4.7), (4.8). Since
$\mathcal{C}=0$ in these cases, we are not able to solve for $\Phi$ and $H$
independently. Instead, one has to find $\Phi$ obeying (4.64) and then solve
(4.63) for $\mathcal{H}$ with the given $\Phi$.
Let us point out how the solution (4.59), (4.63), (4.64) is related to the
AdS-wave solution that has been found in [74] by direct substitution of the
Siklos metric to the field equations of quadratic gravity (4.4). After
necessary long calculations of the Riemann tensor and its various
contractions, the authors of [74] have obtained the constraint for the
function $H$ which can be equivalently expressed from (4.63) and (4.64)
eliminating $\Phi$
$\left(\Delta-\frac{n-2}{z}\,\partial_{z}-\frac{\mathcal{C}}{z^{2}}\right)\left[z^{2}\left(\Delta-\frac{n-2}{z}\,\partial_{z}\right)H\right]=0.$
(4.67)
In [74], it has been further integrated to get a particular solution.
## Chapter 5 Conclusions and outlook
In chapter 2, we have investigated GKS spacetimes (2.11) with an (anti-)de
Sitter background in arbitrary dimension. It has turned out that the
Kerr–Schild vector $\boldsymbol{k}$ is geodetic if and only if the boost
weight zero component $T_{00}$ of the energy–momentum tensor vanishes as
stated in proposition 1. It has been shown that the vector field
$\boldsymbol{k}$ is geodetic in the background spacetime if and only if it is
geodetic in the full spacetime and the same also holds for the affine
parametrization. For GKS spacetimes with a geodetic Kerr–Schild vector
$\boldsymbol{k}$ including Einstein spaces and spacetimes containing matter
fields aligned with $\boldsymbol{k}$ such as aligned Maxwell field or aligned
null radiation, we have given the explicit form of the Ricci tensor and shown
that the optical properties of the Kerr–Schild congruence $\boldsymbol{k}$
encoded in the optical matrix $L_{ij}$ in the full spacetime are same as those
in the background spacetime. If $\boldsymbol{k}$ is geodetic then $T_{0i}$ as
well as the positive boost weight components of the Weyl tensor vanish and
thus such GKS spacetimes are algebraically special, i.e. of Weyl type II or
more special, see proposition 2.
In section 2.5, we have focused on non-expanding Einstein GKS spacetimes. The
Einstein field equations imply that these spacetimes belong to the Kundt class
and are only of Weyl type N, see proposition 4. It has been pointed out that
the same statement also holds if we admit null radiation term in the Ricci
tensor and that the Kerr–Schild function $\mathcal{H}$ of non-expanding
Einstein spacetimes is a linear function of the affine parameter $r$ along the
null geodesics $\boldsymbol{k}$. We have also presented some known examples of
non-expanding Einstein GKS spacetimes in section 2.5.1 and constructed some
new ones in section 2.5.2 using the Brinkmann warp product.
Expanding Einstein GKS spacetimes have been discussed in section 2.6. The
compatible Weyl types are II or D with $\boldsymbol{k}$ being the multiple
WAND as stated in proposition 5 and the corresponding optical matrix $L_{ij}$
satisfies the optical constraint (2.128) implying that $L_{ij}$ is a normal
matrix and in an appropriate frame takes the block diagonal form consisting of
$2\times 2$ and identical $1\times 1$ blocks and zeros. This sparse form has
allowed us to integrate the Sachs equation and explicitly express the
$r$-dependence of the optical matrix $L_{ij}$ and subsequently of the
Kerr–Schild function $\mathcal{H}$ and boost weight zero components of the
Weyl tensor. The $2\times 2$ blocks in the optical matrix correspond to planes
spanned by pairs of the spacelike frame vectors in which the geodetic
congruence $\boldsymbol{k}$ is twisting. Only if $L_{ij}$ is non-degenerate
and does not contain any $2\times 2$ block or if in even dimensions $L_{ij}$
contains only identical $2\times 2$ blocks then the congruence
$\boldsymbol{k}$ is non-shearing. It has been also shown that the rank of
$L_{ij}$ is at least 2. Therefore, in four dimensions, the optical matrix
consists of one $2\times 2$ block or two identical $1\times 1$ blocks which is
in accordance with the Goldberg–Sachs theorem. Expressing the Kretschmann
scalar, we have discussed presence of curvature singularities at the origin
$r=0$ in section 2.6.4. It has turned out that there are three possible cases
depending on the form of the optical matrix $L_{ij}$. Namely, either no
singularity is present or there is a point or Kerr-like singularity. In
section 2.6.5, the analysis of expanding Einstein GKS spacetimes has been
compared with the explicit example of the five-dimensional Kerr–(anti-)de
Sitter black hole. We have established the null frame parallelly transported
along the geodetic Kerr–Schild vector $\boldsymbol{k}$ and expressed the
optical matrix. Using the results of section 2.6.4, we have discussed the
presence of curvature singularities depending on the two rotation parameters
of the black hole.
In future work, the analysis of the GKS spacetimes could be extended to higher
order theories of gravity such as the Gauss–Bonnet or more general Lovelock
theories, some basic analysis in this field has been already done in [103] and
the particular case of the GKS metrics in five-dimensional Gauss–Bonnet
gravity has been studied in [89, 104]. It may be also useful to employ the GKS
ansatz to investigate Weyl type D solutions of quadratic gravity and thus
extend the work started in chapter 4.
Higher dimensional GKS metrics could be also studied in the context of the
Einstein–Maxwell theory. As has been mentioned above, the generalization of
the four-dimensional Kerr–Newman black hole to higher dimensions using its KS
form with the vector potential proportional to the Kerr–Schild vector
$\boldsymbol{k}$ has failed. However, if one assumes the vector potential
given by a linear combination of the Kerr–Schild vector $\boldsymbol{k}$ and a
spacelike vector $\boldsymbol{m}$, i.e. $A_{a}=\alpha k_{a}+\beta m_{a}$,
where $\alpha$ and $\beta$ are scalar functions then it can be shown that this
choice is compatible with $T_{00}=T_{0i}=0$ if the relations (3.31) appearing
in the analysis of xKS spacetimes hold. This suggests that a solution in the
GKS form with this vector potential could be found if $\boldsymbol{m}$
corresponds to a $1\times 1$ block in the optical matrix and thus does not lie
in any plane in which $\boldsymbol{k}$ is twisting.
In chapter 3, we have studied xKS spacetimes (3.1), i.e. an extension of the
GKS ansatz where, in addition to the null Kerr– Schild vector
$\boldsymbol{k}$, a spacelike vector field $\boldsymbol{m}$ appears in the
metric. Unlike for GKS spacetimes, in general we have obtained only the
necessary condition for the component $T_{00}$ under which the Kerr–Schild
vector $\boldsymbol{k}$ is geodetic, see proposition 6. However, if one
appropriately restricts the geometry of the vectors $\boldsymbol{k}$ and
$\boldsymbol{m}$ this condition becomes sufficient, see corollary 7. As for
GKS spacetimes, the Kerr–Schild vector field $\boldsymbol{k}$ is geodetic and
affinely parametrized in the background spacetime if it is geodetic and
affinely parametrized in the full spacetime and vice versa. Similarly, if
$\boldsymbol{k}$ is geodetic the optical matrices in both spacetimes are
identical. In contrast with the GKS case, it has been shown that xKS metrics
with a geodetic Kerr–Schild vector $\boldsymbol{k}$ are of Weyl type I or more
special, as stated in proposition 8, and the components $R_{0i}$ of the Ricci
tensor do not vanish identically.
In section 3.3, we have restricted ourselves to Kundt xKS spacetimes and
determined the $r$-dependence of the scalar function $\mathcal{K}$ in the case
of Weyl type II when the components $T_{0i}$ vanish, see proposition 9. It has
turned out that all VSI spacetimes belong to the xKS class of solutions and
explicit examples with the corresponding forms of the function $\mathcal{K}$
have been given. Higher dimensional pp -waves are of Weyl type II or more
special and type III and N Ricci-flat pp -waves belong to the VSI and
consequently to the xKS class. It has been also shown that type II CSI Ricci-
flat pp -waves admit the xKS form. However, it is not clear whether all Ricci-
flat pp -waves can be cast to the xKS form.
An important example of an expanding xKS spacetime, namely the charged
rotating CCLP black hole in five-dimensional minimal gauged supergravity, has
been studied in section 3.4. In the case with the flat background, we have
established a null frame and expressed the optical matrix which interestingly
satisfies the optical constraint (2.128). Non-vanishing boost weight 1
components of the Weyl tensor suggest that the CCLP black hole is in general
of Weyl type Ii. In the uncharged case which corresponds to the Myers–Perry
black hole and in the non-rotating limit, the metric reduces to the GKS form
and is of Weyl type D.
Although the necessary calculations are more involved than for GKS spacetimes,
it is obvious that the Ricci and Riemann tensors of the xKS metrics
dramatically simplify if one assumes that the relations (3.30) restricting the
geometry of the vectors $\boldsymbol{k}$ and $\boldsymbol{m}$ hold. The
analysis of these simplified expressions is left for future work.
In chapter 4, we have studied exact solutions of the field equations of
quadratic gravity. In fact, we have determined under which conditions certain
exact solutions of the Einstein theory solve also the source-free field
equations of quadratic gravity. In the case of Einstein spacetimes, it has
turned out that only the Gauss–Bonnet term in the action of quadratic gravity
imposes an additional conditions on the Weyl types of a solution. If the
Gauss–Bonnet term is not present, i.e. $\gamma=0$, or in four dimensions when
the Gauss–Bonnet term effectively vanishes then Einstein spaces of all Weyl
types with an appropriate effective cosmological constant $\Lambda$ given by
$\mathcal{B}=0$ (4.14) are solutions of quadratic gravity. Otherwise,
restricting ourselves to types III and more special, Einstein spacetimes only
of Weyl types N or III(B) with an effective cosmological constant $\Lambda$
given by $\mathcal{B}=0$ solve the field equations of quadratic gravity as
stated in propositions 10 and 11. The subclass III(B) of type III is defined
via vanishing quantity $\tilde{\Psi}$ (4.30) constructed from the Weyl tensor.
Note that type III Einstein spaces belonging to the subclass denoted as III(A)
are not solutions of quadratic gravity. Moreover, the class of type III(A)
spacetimes contains the class of III(a) spacetimes and therefore type III(a)
Ricci-flat Kundt metrics including all type III Ricci-flat pp -waves do not
solve the field equations of quadratic gravity. We have also referred to known
examples of higher dimensional type N metrics solving quadratic gravity and
constructed some examples of type III solutions using the fact that the
Brinkmann warp product of four-dimensional type III Einstein spacetimes leads
to the five-dimensional type III(B) Einstein spacetimes.
In section 4.2, we have found a wider class of solutions of quadratic gravity
considering the Ricci tensor with an additional null radiation term aligned
with a WAND. Restricting to the subclasses of type III with vanishing quantity
$\tilde{\Psi}$, i.e. types III(B), N and O, we have obtained the algebraic
condition $\mathcal{B}=0$ determining the effective cosmological constant
$\Lambda$ and also the constraint for the null radiation term which implies
that the only allowed metrics belong to the Kundt class, see propositions 12
and 13.
Recall that GKS spacetimes with null radiation belong to the type N Kundt
class and thus solve the field equations of quadratic gravity with an
appropriate $\Lambda$ given by $\mathcal{B}=0$. In section 4.2.1, we have
given explicit examples of such metrics with $\Lambda=0$ and $\Lambda\neq 0$.
Due to the similarity of the corresponding equations for the null radiation
term $\Phi$ and for the Kerr–Schild function $\mathcal{H}$, we have been able
to express two completely independent equations for $\Phi$ and
$\mathcal{H}_{\text{vac}}$, where $\mathcal{H}_{\text{vac}}$ corresponds to a
Ricci-flat or Einstein part of the solution. However, at critical points of
quadratic gravity the equations for $\Phi$ and $\mathcal{H}$ cannot be
separated in this way.
In future work, type D solutions of quadratic gravity could be also studied,
however, the significant simplification as in the type III cases does not
occur and a non-trivial form of the Ricci tensor could be necessary.
Therefore, it could be convenient to employ, for instance, the GKS ansatz.
Note also that the field equations of quadratic gravity (4.43) for spacetimes
of Weyl type III and the Ricci tensor with aligned null radiation can be also
solved with the assumption $\tilde{\Psi}\neq 0$, i.e. for spacetimes of type
III(A). Therefore, unlike III(A) Einstein spaces which do not solve the field
equations of quadratic gravity, type III(A) solutions of quadratic gravity
with null radiation in the Ricci tensor could exist.
## References
* [1] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Rev. Rel. 11 (2008) 6, arXiv:0801.3471 [hep-th]. http://www.livingreviews.org/lrr-2008-6.
* [2] V. Pravda, A. Pravdová, A. Coley, and R. Milson, “Bianchi identities in higher dimensions,” Class.Quant.Grav. 21 (2004) 2873–2898, arXiv:gr-qc/0401013 [gr-qc].
* [3] M. Ortaggio, V. Pravda, and A. Pravdová, “Ricci identities in higher dimensions,” Class. Quant. Grav. 24 (2007) 1657–1664, arXiv:gr-qc/0701150 [gr-qc].
* [4] A. Coley, R. Milson, V. Pravda, and A. Pravdová, “Vanishing scalar invariant spacetimes in higher dimensions,” Class. Quant. Grav. 21 (2004) 5519–5542, arXiv:gr-qc/0410070 [gr-qc].
* [5] A. Coley, R. Milson, V. Pravda, and A. Pravdová, “Classification of the Weyl tensor in higher dimensions,” Class. Quant. Grav. 21 (2004) L35–L42, arXiv:gr-qc/0401008 [gr-qc].
* [6] R. Milson, A. Coley, V. Pravda, and A. Pravdová, “Alignment and algebraically special tensors in Lorentzian geometry,” Int. J. Geom. Meth. Mod. Phys. 2 (2005) 41–61, arXiv:gr-qc/0401010 [gr-qc].
* [7] A. Coley, “Classification of the Weyl Tensor in Higher Dimensions and Applications,” Class. Quant. Grav. 25 (2008) 033001, arXiv:0710.1598 [gr-qc].
* [8] H. S. Reall, “Algebraically special solutions in higher dimensions,” arXiv:1105.4057 [gr-qc].
* [9] H. Stephani, D. Kramer, M. A. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein’s field equations. Cambridge monographs on mathematical physics. Cambridge Univ. Press, 2003\.
* [10] P.-J. De Smet, “Black holes on cylinders are not algebraically special,” Class. Quant. Grav. 19 (2002) 4877–4896, arXiv:hep-th/0206106 [hep-th].
* [11] M. Godazgar, “Spinor classification of the Weyl tensor in five dimensions,” Class. Quant. Grav. 27 (2010) 245013, arXiv:1008.2955 [gr-qc].
* [12] M. Ortaggio, “Bel–Debever criteria for the classification of the Weyl tensors in higher dimensions,” Class. Quant. Grav. 26 (2009) 195015, arXiv:0906.3818 [gr-qc].
* [13] V. Pravda, A. Pravdová, and M. Ortaggio, “Type D Einstein spacetimes in higher dimensions,” Class. Quant. Grav. 24 (2007) 4407–4428, arXiv:0704.0435 [gr-qc].
* [14] M. Durkee and H. S. Reall, “A Higher dimensional generalization of the geodesic part of the Goldberg–Sachs theorem,” Class. Quant. Grav. 26 (2009) 245005, arXiv:0908.2771 [gr-qc].
* [15] H. W. Brinkmann, “Einstein spaces which are mapped conformally on each other,” Math. Ann. 94 (1925) 119–145.
* [16] J. Podolský and M. Ortaggio, “Robinson–Trautman spacetimes in higher dimensions,” Class. Quant. Grav. 23 (2006) 5785–5797, arXiv:gr-qc/0605136 [gr-qc].
* [17] M. Ortaggio, V. Pravda, and A. Pravdová, “On higher dimensional Einstein spacetimes with a warped extra dimension,” Class. Quant. Grav. 28 (2011) 105006, arXiv:1011.3153 [gr-qc].
* [18] T. Málek and V. Pravda, “Kerr–Schild spacetimes with an (A)dS background,” Class. Quant. Grav. 28 (2011) 125011, arXiv:1009.1727 [gr-qc].
* [19] T. Málek and V. Pravda, “Higher Dimensional Kerr–Schild Spacetimes with (A)dS Background,” J. Phys.: Conf. Ser. 314 (2011) 012111.
* [20] R. P. Kerr and A. Schild, “Some algebraically degenerate solutions of Einstein’s gravitational field equations,” Proc. Symp. Appl. Math. 17 (1965) 199–209.
* [21] R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Phys. Rev. Lett. 11 (1963) 237–238.
* [22] R. C. Myers and M. J. Perry, “Black Holes in Higher Dimensional Space-Times,” Annals Phys. 172 (1986) 304.
* [23] P. C. Vaidya, “The Gravitational Field of a Radiating Star,” Proc. Indian Acad. Sci. A33 (1951) 264.
* [24] B. R. Iyer and C. V. Vishveshwara, “The Vaidya solution in higher dimensions,” Pramana 32 (1989) 749–752.
* [25] W. Kinnersley, “Field of an Arbitrarily Accelerating Point Mass,” Phys. Rev. 186 (1969) 1335–1336.
* [26] M. Gürses and Ö. Sarıoğlu, “Accelerated charge metrics in D dimensions,” Class. Quant. Grav. 19 (2002) 4249–4262, arXiv:gr-qc/0203097 [gr-qc].
* [27] M. Ortaggio, V. Pravda, and A. Pravdová, “Higher dimensional Kerr–Schild spacetimes,” Class. Quant. Grav. 26 (2009) 025008, arXiv:0808.2165 [gr-qc].
* [28] F. Tangherlini, “Schwarzschild field in n dimensions and the dimensionality of space problem,” Nuovo Cim. 27 (1963) 636–651.
* [29] G. Gibbons and S. A. Hartnoll, “A Gravitational instability in higher dimensions,” Phys. Rev. D66 (2002) 064024, arXiv:hep-th/0206202 [hep-th].
* [30] G. Gibbons, H. Lu, D. N. Page, and C. Pope, “The General Kerr–de Sitter metrics in all dimensions,” J. Geom. Phys. 53 (2005) 49–73, arXiv:hep-th/0404008 [hep-th].
* [31] B. Carter, “Hamilton–Jacobi and Schrodinger separable solutions of Einstein’s equations,” Commun. Math. Phys. 10 (1968) 280.
* [32] S. Hawking, C. Hunter, and M. Taylor, “Rotation and the AdS/CFT correspondence,” Phys. Rev. D59 (1999) 064005, arXiv:hep-th/9811056 [hep-th].
* [33] K. Peeters, “A Field-theory motivated approach to symbolic computer algebra,” Comput. Phys. Commun. 176 (2007) 550–558, arXiv:cs/0608005 [cs.SC].
* [34] K. Peeters, “Introducing Cadabra: A Symbolic computer algebra system for field theory problems,” arXiv:hep-th/0701238 [HEP-TH].
* [35] N. Hamamoto, T. Houri, T. Oota, and Y. Yasui, “Kerr–NUT–de Sitter curvature in all dimensions,” J. Phys. A A40 (2007) F177–F184, arXiv:hep-th/0611285 [hep-th].
* [36] A. Coley, R. Milson, N. Pelavas, V. Pravda, A. Pravdova, and R. Zalaletdinov, “Generalizations of pp-wave spacetimes in higher dimensions,” Phys. Rev. D67 (2003) 104020, arXiv:gr-qc/0212063 [gr-qc].
* [37] A. Coley, S. Hervik, and N. Pelavas, “On spacetimes with constant scalar invariants,” Class. Quant. Grav. 23 (2006) 3053–3074, arXiv:gr-qc/0509113 [gr-qc].
* [38] A. Coley, A. Fuster, S. Hervik, and N. Pelavas, “Higher dimensional VSI spacetimes,” Class. Quant. Grav. 23 (2006) 7431–7444, arXiv:gr-qc/0611019 [gr-qc].
* [39] A. Coley, A. Fuster, and S. Hervik, “Supergravity solutions with constant scalar invariants,” Int. J. Mod. Phys. A24 (2009) 1119–1133, arXiv:0707.0957 [hep-th].
* [40] J. Podolský and M. Žofka, “General Kundt spacetimes in higher dimensions,” Class. Quant. Grav. 26 (2009) 105008, arXiv:0812.4928 [gr-qc].
* [41] I. Ozsváth, I. Robinson, and K. Rózga, “Plane fronted gravitational and electromagnetic waves in spaces with cosmological constant,” J. Math. Phys. 26 (1985) 1755–1761.
* [42] J. Bičák and J. Podolský, “Gravitational waves in vacuum space-times with cosmological constant: I. Classification and geometrical properties of nontwisting type N solutions,” J. Math. Phys. 40 (1999) 4495–4505, arXiv:gr-qc/9907048 [gr-qc].
* [43] J. B. Griffiths and J. Podolský, Exact space-times in Einstein’s general relativity. Cambridge monographs on mathematical physics. Cambridge Univ. Press, 2009\.
* [44] M. Ortaggio, V. Pravda, A. Pravdová, and H. Reall, “On a five-dimensional version of the Goldberg–Sachs theorem.” in preparation, 2012.
* [45] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1990.
* [46] M. Ortaggio, V. Pravda, and A. Pravdová, “Type III and N Einstein spacetimes in higher dimensions: General properties,” Phys. Rev. D82 (2010) 064043, arXiv:1005.2377 [gr-qc].
* [47] M. Gürses and Ö. Sarıoğlu, “Some further properties of the accelerated Kerr–Schild metrics,” Gen. Rel. Grav. 36 (2004) 403–410, arXiv:gr-qc/0308020 [gr-qc].
* [48] J. Podolský, “Photon rockets moving arbitrarily in any dimension,” Int. J. Mod. Phys. D20 (2011) 335–360, arXiv:1006.1583 [gr-qc].
* [49] P. C. Vaidya, “The external field of a radiating star in general relativity,” Current Sci. 12 (1943) 183.
* [50] L. K. Patel and L. Desai, “Higher dimensional Vaidya metric in Einstein and de Sitter background,” Pramana 48 (1997) 819–824.
* [51] N. Ibohal and L. Kapil, “Charged black holes in Vaidya backgrounds: Hawking’s Radiation,” Int. J. Mod. Phys. D19 (2010) 437–464, arXiv:1001.2616 [gr-qc].
* [52] P. V. Bhatt and S. K. Vaidya, “Kerr–Schild type solutions of Einstein–Maxwell equations,” Class. Quant. Grav. 8 (1991) 1717.
* [53] L. Randall and R. Sundrum, “A Large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83 (1999) 3370–3373, arXiv:hep-ph/9905221 [hep-ph].
* [54] L. Randall and R. Sundrum, “An Alternative to compactification,” Phys. Rev. Lett. 83 (1999) 4690–4693, arXiv:hep-th/9906064 [hep-th].
* [55] A. Chamblin, S. Hawking, and H. Reall, “Brane world black holes,” Phys. Rev. D61 (2000) 065007, arXiv:hep-th/9909205 [hep-th].
* [56] A. N. Aliev and D. K. Çiftçi, “A Note on Rotating Charged Black Holes in Einstein–Maxwell–Chern–Simons Theory,” Phys. Rev. D79 (2009) 044004, arXiv:0811.3948 [hep-th].
* [57] Z.-W. Chong, M. Cvetič, H. Lü, and C. Pope, “General non-extremal rotating black holes in minimal five-dimensional gauged supergravity,” Phys. Rev. Lett. 95 (2005) 161301, arXiv:hep-th/0506029 [hep-th].
* [58] W. Chen, H. Lü, and C. Pope, “General Kerr–NUT–AdS metrics in all dimensions,” Class. Quant. Grav. 23 (2006) 5323–5340, arXiv:hep-th/0604125 [hep-th].
* [59] W. Chen and H. Lü, “Kerr–Schild structure and harmonic 2-forms on (A)dS–Kerr–NUT metrics,” Phys. Lett. B658 (2008) 158–163, arXiv:0705.4471 [hep-th].
* [60] S.-Q. Wu, “General rotating charged Kaluza–Klein AdS black holes in higher dimensions,” Phys. Rev. D83 (2011) 121502, arXiv:1108.4157 [hep-th].
* [61] B. Ett and D. Kastor, “An Extended Kerr–Schild Ansatz,” Class. Quant. Grav. 27 (2010) 185024, arXiv:1002.4378 [hep-th].
* [62] R. Emparan and H. S. Reall, “A Rotating black ring solution in five dimensions,” Phys. Rev. Lett. 88 (2002) 101101, arXiv:hep-th/0110260 [hep-th].
* [63] V. Pravda and A. Pravdová, “WANDs of the black ring,” Gen. Rel. Grav. 37 (2005) 1277–1287, arXiv:gr-qc/0501003 [gr-qc].
* [64] F. Prüfer, F. Tricerri, and L. Vanhecke, “Curvature invariants, differential operators and local homogeneity,” Trans. Am. Math. Soc. 348 (1996) 4643.
* [65] T. Málek and V. Pravda, “Types III and N solutions to quadratic gravity,” Phys. Rev. D84 (2011) 024047, arXiv:1106.0331 [gr-qc].
* [66] G. ’t Hooft and M. Veltman, “One loop divergencies in the theory of gravitation,” Annales Poincare Phys.Theor. A20 (1974) 69–94.
* [67] K. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D16 (1977) 953–969.
* [68] K. Stelle, “Classical Gravity with Higher Derivatives,” Gen. Rel. Grav. 9 (1978) 353–371.
* [69] B. Zwiebach, “Curvature Squared Terms and String Theories,” Phys. Lett. B156 (1985) 315.
* [70] M. Bento and O. Bertolami, “Maximally Symmetric Cosmological Solutions of higher curvature string effective theories with dilatons,” Phys. Lett. B368 (1996) 198–201, arXiv:gr-qc/9503057 [gr-qc].
* [71] H. Lu and C. Pope, “Critical Gravity in Four Dimensions,” Phys. Rev. Lett. 106 (2011) 181302, arXiv:1101.1971 [hep-th].
* [72] S. Deser, H. Liu, H. Lu, C. Pope, T. C. Sisman, et al., “Critical Points of D-Dimensional Extended Gravities,” Phys. Rev. D83 (2011) 061502, arXiv:1101.4009 [hep-th].
* [73] M. Alishahiha and R. Fareghbal, “D-Dimensional Log Gravity,” Phys. Rev. D83 (2011) 084052, arXiv:1101.5891 [hep-th].
* [74] I. Güllü, M. Gürses, T. Ç. Şişman, and B. Tekin, “AdS Waves as Exact Solutions to Quadratic Gravity,” Phys. Rev. D83 (2011) 084015, arXiv:1102.1921 [hep-th].
* [75] Y. Pang, “Brief Note on AMD Conserved Quantities in Quadratic Curvature Theories,” Phys. Rev. D83 (2011) 087501, arXiv:1101.4267 [hep-th].
* [76] M. Porrati and M. M. Roberts, “Ghosts of Critical Gravity,” Phys. Rev. D84 (2011) 024013, arXiv:1104.0674 [hep-th].
* [77] H. Liu, H. Lu, and M. Luo, “On Black Hole Stability in Critical Gravities,” arXiv:1104.2623 [hep-th].
* [78] S. Deser and B. Tekin, “Energy in generic higher curvature gravity theories,” Phys. Rev. D67 (2003) 084009, arXiv:hep-th/0212292 [hep-th].
* [79] D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12 (1971) 498–501.
* [80] I. Gullu and B. Tekin, “Massive Higher Derivative Gravity in D-dimensional Anti-de Sitter Spacetimes,” Phys. Rev. D80 (2009) 064033, arXiv:0906.0102 [hep-th].
* [81] R. Zegers, “Birkhoff’s theorem in Lovelock gravity,” J. Math. Phys. 46 (2005) 072502, arXiv:gr-qc/0505016 [gr-qc].
* [82] J. Oliva and S. Ray, “Birkhoff’s Theorem in Higher Derivative Theories of Gravity,” Class. Quant. Grav. 28 (2011) 175007, arXiv:1104.1205 [gr-qc].
* [83] Y. Choquet-Bruhat, “The Cauchy problem for stringy gravity,” J. Math. Phys. 29 (1988) 1891–1895.
* [84] P. Teyssandier and P. Tourrenc, “The Cauchy problem for the R+R**2 theories of gravity without torsion,” J. Math. Phys. 24 (1983) 2793.
* [85] M. Madsen, “The plane gravitational wave in quadratic gravity,” Class. Quant. Grav. 7 (1990) 87–96.
* [86] E. C. de Rey Neto, O. D. Aguiar, and J. C. de Araujo, “A Perturbative solution for gravitational waves in quadratic gravity,” Class. Quant. Grav. 20 (2003) 2025–2031, arXiv:gr-qc/0304091 [gr-qc].
* [87] A. Economou and C. Lousto, “Charged black holes in quadratic theories,” Phys. Rev. D49 (1994) 5278–5285, arXiv:gr-qc/9310021 [gr-qc].
* [88] D. G. Boulware and S. Deser, “String Generated Gravity Models,” Phys. Rev. Lett. 55 (1985) 2656.
* [89] A. Anabalon, N. Deruelle, Y. Morisawa, J. Oliva, M. Sasaki, D. Tempo, and R. Troncoso, “Kerr–Schild ansatz in Einstein–Gauss–Bonnet gravity: An exact vacuum solution in five dimensions,” Class. Quant. Grav. 26 (2009) 065002, arXiv:0812.3194 [hep-th].
* [90] Y. Brihaye and E. Radu, “Five-dimensional rotating black holes in Einstein–Gauss–Bonnet theory,” Phys. Lett. B661 (2008) 167–174, arXiv:0801.1021 [hep-th].
* [91] Y. Brihaye, B. Kleihaus, J. Kunz, and E. Radu, “Rotating black holes with equal-magnitude angular momenta in d=5 Einstein–Gauss–Bonnet theory,” JHEP 1011 (2010) 098, arXiv:1010.0860 [hep-th].
* [92] Y. Brihaye, “Charged, rotating black holes in Einstein–Gauss–Bonnet gravity,” arXiv:1108.2779 [gr-qc].
* [93] H. C. Kim and R. G. Cai, “Slowly Rotating Charged Gauss–Bonnet Black holes in AdS Spaces,” Phys. Rev. D77 (2008) 024045, arXiv:0711.0885 [hep-th].
* [94] D. Lovelock, “Dimensionally dependent identities,” Proc. Camb. Phil. Soc. 68 (1970) 345–350.
* [95] M. Durkee, V. Pravda, A. Pravdová, and H. S. Reall, “Generalization of the Geroch–Held–Penrose formalism to higher dimensions,” Class. Quant. Grav. 27 (2010) 215010, arXiv:1002.4826 [gr-qc].
* [96] A. Pravdová and V. Pravda, “Newman–Penrose formalism in higher dimensions: Vacuum spacetimes with a non-twisting multiple WAND,” Class. Quant. Grav. 25 (2008) 235008, arXiv:0806.2423 [gr-qc].
* [97] A. García Díaz and J. F. Plebański, “All nontwisting N’s with cosmological constant,” J. Math. Phys. 22 (1981) 2655.
* [98] I. Hauser, “Type-N gravitational field with twist,” Phys. Rev. Lett. 33 (1974) 1112–1113.
* [99] J. Leroy, “Un espace d’Einstein de type N à rayons non intégrables,” C. R. Acad. Sc. Paris 64 (1970) 1078–1080.
* [100] A. Coley, G. Gibbons, S. Hervik, and C. Pope, “Metrics With Vanishing Quantum Corrections,” Class. Quant. Grav. 25 (2008) 145017, arXiv:0803.2438 [hep-th].
* [101] A. A. Coley and S. Hervik, “Universality and constant scalar curvature invariants,” arXiv:1105.2356 [hep-th].
* [102] A. Chamblin and G. Gibbons, “Nonlinear Supergravity on a Brane without Compactification,” Phys. Rev. Lett. 84 (2000) 1090–1093, arXiv:hep-th/9909130 [hep-th].
* [103] B. Ett and D. Kastor, “Kerr–Schild Ansatz in Lovelock Gravity,” JHEP 1104 (2011) 109, arXiv:1103.3182 [hep-th].
* [104] A. Anabalon, N. Deruelle, D. Tempo, and R. Troncoso, “Remarks on the Myers–Perry and Einstein Gauss–Bonnet Rotating Solutions,” Int. J. Mod. Phys. D20 (2011) 639–647, arXiv:1009.3030 [gr-qc].
###### List of Tables
1. 1.1 The conditions on the frame components of the Weyl tensor determining primary algebraic type of a spacetime.
2. 1.2 Allowed combinations of signs of the Ricci scalar $\tilde{R}$ corresponding to the seed metric $\mathrm{d}\tilde{s}^{2}$ and of the Ricci scalar $R$ corresponding to the warped metric $\mathrm{d}s^{2}$ (1.37).
3. 2.1 Weyl types compatible with Einstein generalized Kerr–Schild spacetimes depending on the values of the expansion scalar $\theta$.
4. 3.1 Properties of higher dimensional Ricci-flat pp -waves.
|
arxiv-papers
| 2012-04-02T01:33:14 |
2024-09-04T02:49:29.282524
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tom\\'a\\v{s} M\\'alek",
"submitter": "Tom\\'a\\v{s} M\\'alek",
"url": "https://arxiv.org/abs/1204.0291"
}
|
1204.0325
|
# Calculation of the Structure Properties of a Strange Quark Star in the
Presence of Strong Magnetic Field Using a Density Dependent Bag Constant
Gholam Hossein Bordbar1,2 111Corresponding author. E-mail:
bordbar@physics.susc.ac.ir, Hajar Bahri1 and Fatemeh Kayanikhoo1 1Department
of Physics, Shiraz University, Shiraz 71454, Iran
and
2Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha, Iran
###### Abstract
In this article we have calculated the structure properties of a strange quark
star in static model in the presence of a strong magnetic field using MIT bag
model with a density dependent bag constant. To parameterize the density
dependence of bag constant, we have used our results for the lowest order
constrained variational calculation of the asymmetric nuclear matter. By
calculating the equation of state of strange quark matter, we have shown that
the pressure of this system increases by increasing both density and magnetic
field. Finally, we have investigated the effect of density dependence of bag
constant on the structure properties of strange quark star.
## I Introduction
The core of a neutron star has been formed from nuclear matter, composed of
the neutrons, protons, electrons (for assurance negation of electric charge)
and other particles like pions, mesons and etc (Lattimer & Prakash rk1
(2004)). It is known that nuclear matter is meta-stable, that after releasing
a lot of energy converts into strange quark matter (SQM) to achieve stability.
This quark matter is the most stable state of matter that has been known until
now. Thus, there is a new class of compact stars that come from the collapse
of neutron stars, and are more stable compared to the neutron stars (Farhi &
Jaffe rk2 (1984)). The best candidates for this conversion are the neutron
stars with masses of $1.5-1.8\ M_{\odot}$ and quick spin (Drake et al. rk3
(2002); Li et al. rk4 (1999); Weber rk5 (2005)).
The collapse of a neutron star may lead to a strange quark star (SQS) or a
hybrid star. Also under special conditions, an SQS may be directly born from
the core collapse of a type II supernova. An SQS, from its center to surface
is made from SQM, and on its surface may exist a nuclear layer (Glendenning &
Weber rk6 (1992)). Hybrid stars are the ones with cores composed of SQM
(Bhattacharyya et al. rk7 (2006)). Here, we just consider the structure
properties of SQS.
The mass and density of an SQS is between the mass and density of a neutron
star and that of a black hole. The mass-radius relation for an SQS is as
$M\propto R^{3}$ which is different from that of a neutron star. This star
does not have the minimum mass. For an SQS with $1M_{\odot}\leq M\leq
2M_{\odot}$, the radius is about $10\ km$ (Farhi & Olinto rk8 (1986); Shapiro
& Tenkolsky rk9 (1983)).
Recent observations indicate that the object SWIFT J1749.4-2807 may be an SQS
(Yu & Xu rk17 (2010)). The given results by Chandra observations also show
that the objects RX J185635-3754 and 3C58 may be bare strange stars (Prakash
et al. rk55 (2003)). It is known that the compact objects such as neutron
star, pulsars, magnetars and strange quark stars, are under the influence of
strong magnetic fields which are typically about $10^{15}-10^{19}\ G$
(Kouveliotou et al. rk10 (1998), rk11 (1999); Haensel et al. rk12 (2007);
Glendenning rk13 (2000); Weber rk14 (2007); Camenzind rk15 (2007)). Therefore,
in astrophysics, it is of special interest to study the effect of a strong
magnetic field on the properties of SQM. We note that in the presence of a
magnetic field, the conversion of neutron stars to bare quark stars cannot
take place unless the value of the magnetic field exceeds $10^{20}\ G$ (Ghosh
& Chakrabarty rk16 (2001)).
In recent years, we have calculated the maximum gravitational mass and other
structure properties of a neutron star with a quark core at zero (Bordbar et
al. rk53 (2006)) and finite temperatures (Yazdizadeh & Bordbar rk54 (2011)).
We have also computed the structure properties of SQS at zero temperature
(Bordbar et al. rk18 (2009)) and finite temperature (Bordbar et al. rk19
(2011)). We have also calculated the structure of a magnetized SQS using MIT
bag model with a fixed bag constant ($90\ \frac{MeV}{fm^{3}}$) (Bordbar &
Peyvand rk20 (2011)). In the present work, we investigate the effect of
density dependence of bag constant on the structure of an SQS in the presence
of strong magnetic field.
## II Computation of strange quark matter equation of state in the presence
of magnetic field
The equation of state (EOS) of strange quark matter (SQM) plays an important
role for determining the structure of stars at high densities. To obtain EOS
of SQM, there are different models based on Quantum Chromodynamics (QCD). At
present, it isn’t possible to achieve an exact EOS of SQM by primary
principles of QCD. Therefore, scientists have tried to find approximate
methods by combining the basic features of QCD, for example, MIT bag model
(Chodos et al. rk21 (1974); Weber rk14 (2007); Peshier et al. rk45 (2000);
Alford et al. rk46 (2005)), NJL model (Rehberg et al. rk47 (1996); Hanauske et
al. rk48 (2001); R¨uster & Rischke rk49 (2004); Menezes et al. rk50 (2006)),
and perturbative QCD model (Baluni rk51 (1978); Fraga et al. rk52 (2001);
Farhi & Jaffe rk2 (1984)).
In MIT bag model, the quarks in the bag are considered as a free Fermi gas,
and the energy per volume for SQM is equal to the kinetic energy of the free
quarks plus a bag constant ($\mathcal{B}$) (Chodos et al. rk21 (1974)). The
bag constant $\mathcal{B}$ can be interpreted as the difference between the
energy densities of the noninteracting quarks and the interacting ones.
Dynamically, its role is as the pressure that keeps the quark gas in constant
density and potential. In the initial MIT bag model, different values such as
$55$ and $90\ \frac{MeV}{fm^{3}}$ are considered for the bag constant. As we
know, the density of SQM increases from surface to the core of SQS, therefore
using a density dependent bag constant instead of a fixed bag constant is more
suitable.
### II.1 Density dependent bag constant
The analysis of the experimental data achieved at CERN shows that the quark-
hadron transition happens at a density about seven times the normal nuclear
matter energy density ($156\ MeVfm^{-3}$) (Heinz rk22 (2001); Heinz & Jacobs
rk44 (2000); Farhi & Jaffe rk2 (1984)). However theoretically, for no density-
independent value of bag constant the hadron to quark matter transition takes
place (Burgio et al. rk26 (2002)). Therefore, it is essential to use a density
dependent bag constant. Recently, a density dependent form has been also
considered for $\mathcal{B}$ (Adami & Brown rk23 (1993); Jin & Jenning rk24
(1997); Blaschke et al. rk25 (1999); Burgio et al. rk26 (2002)). The density
dependence of $\mathcal{B}$ is highly model dependent. According to the
hypothesis of a constant energy density along the transition line, Burgio et
al. tried to determine a range of possible values for B by exploiting the
experimental data obtained at the CERN SPS (Burgio et al. rk26 (2002)). By
assumption that the transition to quark-gluon plasma is determined by the
value of the energy density only, they estimated the value of bag constant and
its possible density dependence. They attempted to provide effective
parameterizations for this density dependence, trying to cover a wide range by
considering some extreme choices in such a way that at asymptotic densities,
the bag constant has some finite value. They employed a Gaussian form as
follows
$\mathcal{B}(\rho)=\mathcal{B}_{\infty}+(\mathcal{B}_{0}-\mathcal{B}_{\infty})e^{-\gamma(\rho/\rho_{0})^{2}}.$
(1)
The parameter $\mathcal{B}_{0}=\mathcal{B}(\rho=0)$ is constant and equal to
$\mathcal{B}_{0}=400\ \frac{MeV}{fm^{3}}$. In the above equation, $\gamma$ is
a numerical parameter which is usually equal to $\rho_{0}\approx 0.17\
fm^{-3}$, the normal nuclear matter density. $\mathcal{B}_{\infty}$ depends
only on the free parameter $\mathcal{B}_{0}$.
The value of the bag constant ($\mathcal{B}$) should be compatible with
experimental data. The experimental results at CERN-SPS confirms a proton
fraction $x_{pt}=0.4$ (Heinz rk22 (2001); Heinz & Jacobs rk44 (2000); Burgio
et al. rk26 (2002)). Therefore, we use the equation of state of asymmetric
nuclear matter to evaluate $\mathcal{B}_{\infty}$. We use the lowest order
constrained variational (LOCV) many-body method based on the cluster expansion
of the energy for calculating the equation of state of asymmetric nuclear
matter as follows (Bodbar & Modarres rk27 (1997), rk28 (1998); Modarres &
Bordbar rk29 (1998); Bordbar & Bigdeli 2007a , 2007b , 2008a , 2008b ; Bigdeli
et al. rk34 (2009); Bigdeli et al. rk35 (2010)).
The asymmetric nuclear matter is defined as a system consisting of $Z$ protons
($pt$) and $N$ neutrons ($nt$) with the total number density
$\rho=\rho_{pt}+\rho_{nt}$ and proton fraction
$x_{pt}=\frac{\rho_{pt}}{\rho}$, where $\rho_{pt}$ and $\rho_{nt}$ are the
number densities of protons and neutrons, respectively. For this system, we
consider a trial wave function of the form,
$\psi=F\phi,$ (2)
where $\phi$ is the slater determinant of the single-particle wave functions,
and $F$ is the A-body correlation operator ($A=Z+N$) which is given by
$F=\mathcal{S}\prod_{i>j}f(ij).$ (3)
In the above equation, $\mathcal{S}$ is a symmetrizing operator.
For the asymmetric nuclear matter, the energy per nucleon up to the two-body
term in the cluster expansion is as follows
$E([f])=\frac{1}{A}\frac{<\psi|H|\psi>}{<\psi|\psi>}=E_{1}+E_{2}.$ (4)
The one-body energy, $E_{1}$, is
$E_{1}=\sum_{i=1}^{2}\sum_{k_{i}}\frac{\hbar^{2}k_{i}^{2}}{2m},$ (5)
where labels 1 and 2 are used for proton and neutron respectively, and $k_{i}$
is the momentum of particle $i$. The two-body energy, $E_{2}$, is given by
$E_{2}=\frac{1}{2A}\sum_{ij}<ij|\mathcal{V}(12)|ij-ji>,$ (6)
where
$\mathcal{V}(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$
(7)
In Eq. (7), $f(12)$ and $V(12)$ are the two-body correlation and nucleon-
nucleon potential, respectively. In our calculations, we use $UV_{14}+TNI$
nucleon-nucleon potential (Lagaris & Pandharipande 1981a , 1981b ). We
minimize the two-body energy with respect to the variations in the correlation
functions subject to the normalization constraint. From the minimization of
the two-body energy, we get a set of differential equations. By numerically
solving these differential equations, we can calculate the correlation
functions. The two-body energy is obtained using these correlation functions
and then we can calculate the energy of asymmetric nuclear matter. The
procedure of these calculations has been fully discussed in reference (Bordbar
& Modarres rk28 (1998)).
The experimental results at CERN-SPS confirm a proton fraction $x_{pt}=0.4$
(Heinz rk22 (2001); Heinz & Jacobs rk44 (2000); Burgio et al. rk26 (2002)).
Therefore, to calculate $\mathcal{B}_{\infty}$, we use our results of the
above formalism for the asymmetric nuclear matter characterized by a proton
fraction $x_{pt}=0.4$. By assuming that the hadron-quark transition takes
place at the energy density equal to $1100\ MeVfm^{-3}$ (Heinz rk22 (2001);
Burgio et al. rk26 (2002)), we find that the baryonic density of nuclear
matter corresponding to this value of the energy density is $\rho_{B}=0.98\
fm^{-3}$ (transition density). At densities lower than this value, the energy
density of SQM is higher than that of the nuclear matter. With increasing the
baryonic density, these two energy densities become equal at the transition
density, and above this value, the nuclear matter energy density remains
always higher. Later, we determine $\mathcal{B}_{\infty}=8.99\
\frac{MeV}{fm^{3}}$ by putting the energy density of SQM and that of the
nuclear matter equal to each other.
### II.2 Energy density calculation of strange quark matter in the presence
of magnetic field
We consider SQM composed of $u$, $d$ and $s$ quarks with spin up (+) and down
(-). We denote the number density of quark $i$ with spin up by
$\rho_{i}^{(+)}$ and spin down by $\rho_{i}^{(-)}$. We introduce the
polarization parameter $\xi_{i}$ by
$\xi_{i}=\frac{\rho_{i}^{(+)}-\rho_{i}^{(-)}}{\rho_{i}},$ (8)
where $0\leq\xi_{i}\leq 1$ and $\rho_{i}=\rho_{i}^{(+)}+\rho_{i}^{(-)}$. Under
the conditions of beta-equilibrium and charge neutrality, we get the following
relation for the number density,
$\rho=\rho_{u}=\rho_{d}=\rho_{s},$ (9)
where $\rho$ is the total baryonic density of the system.
Within the MIT bag model, the total energy of SQM in the presence of magnetic
field ($B$) can be written as
$E_{tot}=E_{K}+\mathcal{B}+E_{M},$ (10)
where $E_{M}$ is the contribution of magnetic energy, $\mathcal{B}$ is the bag
constant (in this article, we use a density dependent bag constant (Eq. (1)),
and $E_{K}$ is the total kinetic energy of SQM. The total kinetic energy of
SQM is as follows,
$E_{K}=\sum_{i=u,d,s}E_{i},$ (11)
where $E_{i}$ is the kinetic energy of quark $i$,
$E_{i}=\sum_{p=\pm}\sum_{k^{(p)}}\sqrt{m_{i}^{2}c^{4}+\hbar^{2}k^{(p)^{2}}c^{2}}$
(12)
we ignore the masses of quark $u$ and $d$, while we assume $m_{s}=150\ MeV$
for quark $s$. After performing some algebra, supposing that
$\xi_{s}=\xi_{u}=\xi_{d}=\xi$, we obtain the following relation for the total
kinetic energy density ($\varepsilon_{K}=\frac{E_{K}}{V}$) of SQM,
$\displaystyle\varepsilon_{K}$ $\displaystyle=$
$\displaystyle\frac{3}{16\pi^{2}\hbar^{3}}\sum_{p=\pm}[\frac{\hbar}{c^{2}}k_{F}^{(p)}E_{F}^{(p)}(2\hbar^{2}k_{F}^{(p)^{2}}c^{2}+m_{s}^{2}c^{4})-m_{s}^{4}c^{5}ln(\frac{\hbar
k_{F}^{(p)}+E_{F}^{(p)}/c}{m_{s}c})]$ (13) $\displaystyle+$
$\displaystyle\frac{3\hbar
c\pi^{2/3}}{4}\rho^{4/3}[(1+\xi)^{4/3}+(1-\xi)^{4/3}]$
where
$k_{F}^{(\pm)}=(\pi^{2}\rho)^{1/3}(1\pm\xi)^{1/3},$ (14)
and
$E_{F}^{(\pm)}=(\hbar^{2}k_{F}^{(\pm)^{2}}c^{2}+m_{s}^{2}c^{4})^{1/2}.$ (15)
For SQM, the contribution of magnetic energy is as $E_{M}=-M.B$. If we assume
the magnetic field is along the $z$ direction, the contribution of the
magnetic energy of SQM is given by
$E_{M}=-\sum_{i=u,d,s}M_{z}^{(i)}B,$ (16)
where $M_{z}^{(i)}$ is the magnetization of the system corresponding to
particle $i$ which is given by
$M_{z}^{(i)}=N_{i}\mu_{i}\xi_{i}.$ (17)
In the above equation, $N_{i}$ and $\mu_{i}$ are the number and magnetic
moment of particle $i$, respectively. By some simplification, the contribution
of the magnetic energy density ($\varepsilon_{M}=\frac{E_{M}}{V}$) of SQM can
be obtained as follows
$\varepsilon_{M}=-\sum_{i=u,d,s}\rho_{i}\mu_{i}\xi_{i}B.$ (18)
Using the above equation and $\rho=\rho_{u}=\rho_{d}=\rho_{s}$ and with the
assumption that $\xi=\xi_{u}=\xi_{d}=\xi_{s}$, we have
$\varepsilon_{M}=-(\rho B\xi\mu_{s}+\rho B\xi\mu_{u}+\rho B\xi\mu_{d}).$ (19)
Now, we take advantage of numerical values of the magnetic moment for quarks (
Wong Samuel rk40 (1990)) :
$\mu_{s}=-0.581\ \mu_{N},\mu_{u}=1.852\ \mu_{N},\mu_{d}=-0.972\ \mu_{N}$.
Using Eq. (19) and above values, we conclude that
$\varepsilon_{M}=-0.299\rho\xi\mu_{N}B,$ (20)
where $\mu_{N}=5.05\times 10^{-27}\ (J/T)$ is nuclear magneton.
### II.3 The results of energy of strange quark matter in the presence of
magnetic field
We have calculated the properties of strange quark matter in the presence of
magnetic field with the density dependent bag constant (Eq. (1)). Our results
are as follows.
Our results for the total energy density
($\varepsilon_{tot}=\frac{E_{tot}}{V}$) of strange quark matter (SQM) in the
presence of magnetic field have been plotted versus the polarization parameter
in Fig. 1 for various densities at $B=5\times 10^{18}\ G$. We see that there
is a minimum point in the energy curve for each density which shows a meta-
stable state for this system. As the density increases, the minimum point of
energy shifts to the lower values of the polarization, and finally it
disappears at high densities in which the system becomes nearly unpolarized.
In Fig. 2, we have plotted the polarization parameter corresponding to the
minimum point of energy versus density for two magnetic fields $B=5\times
10^{18}\ G$ and $B=5\times 10^{19}\ G$. We can see that the value of $\xi$
decreases by increasing the density, and it becomes nearly zero at high
densities. We have also drown the polarization parameter as a function of the
magnetic field at different densities in Fig. 3. As this figure shows, the
polarization parameter increases by increasing the magnetic field for all
densities.
For the strange quark matter, our results for the total energy density at
$B=5\times 10^{18}\ G$, which calculated with the density dependent bag
constant, have been shown as a function of density in Fig. 4. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar &
Peyvand rk20 (2011)) are also given for comparison. It can be observed that
the total energy density has an increasing rate by increasing the density.
Also, it can be found that for $\rho$ greater (lower) than about $0.6\
fm^{-3}$, the energy of SQM with the density dependent bag constant is lower
(greater) than that with the fixed bag constant. From Fig. 4, it is seen that
for $\rho<0.6\ fm^{-3}$, the increasing of energy has a slow slope, whereas
for $\rho>0.6\ fm^{-3}$ this increasing is accomplished with a more quick
slope.
Fig. 5 shows the phase diagram for the strange quark matter. We can see that
as the density increases, the magnetic field grows monotonically. It
explicitly means that at higher densities, the ferromagnetic phase transition
occurs at higher values of the magnetic field.
### II.4 The equation of state for strange quark matter in the presence of
magnetic field
In this section, we calculate the equation of state (EoS) of strange quark
matter (SQM) in the presence of magnetic field with density dependent bag
constant. Generally, we can calculate EoS using the following relation,
$P(\rho)=\rho\frac{\partial\varepsilon_{tot}}{\partial\rho}-\varepsilon_{tot},$
(21)
where $P$ is the pressure and $\varepsilon_{tot}$ is the energy density, which
in the presence of magnetic field, is obtained from Eq. (10). In Fig. 6, we
have compared our results for EoS of SQM at different magnetic fields. This
shows that for all magnetic fields, by increasing the density, pressure has an
increasing rate. Also, we can see that with increasing magnetic field, the
pressure increases. In Fig. 7, we have drawn EOS of SQM by density dependent
bag constant at $B=5\times 10^{18}\ G$. The results for $\mathcal{B}=90\
\frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar & Peyvand rk20 (2011))
are also given for comparison. This figure indicates that for $\rho$ greater
than about $0.52\ fm^{-3}$, when the bag constant is density dependent, the
pressure of SQM is greater than that of the density independent case.
## III Structure of strange quark star
Quark stars are relativistic objects, therefore we used the general relativity
for calculation of their structures. Since most of the massive general
relativistic objects have some forms of rotation (very rapid in the case of
pulsars). In this calculations, we are interested in the investigation of the
strong magnetic field effects on the structure of a static strange quark star.
Using the the equation of state (EoS) of strange quark matter (SQM), we can
obtain the structure of these stars by numerically integrating the general
relativistic equations of hydrostatic equilibrium, the Tolman-Oppenheimer-
Volkoff (TOV) equation (Shapiro & Teukolski rk9 (1983)),
$\displaystyle\frac{dP}{dr}$ $\displaystyle=$
$\displaystyle-\frac{G(\frac{\varepsilon(r)m(r)}{r^{2}})(1+\frac{P(r)}{c^{2}\varepsilon(r)})(1+\frac{4\pi
r^{3}P(r)}{m(r)c^{2}})}{(1-\frac{2Gm(r)}{c^{2}})}$
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi
r^{2}\varepsilon(r).$ (22)
In the above equations, $P$ is pressure and $\varepsilon(r)$ is the energy
density, $G$ is the gravitational constant and $m(r)$ is the mass inside
radius $r$ which is calculated as follows
$\displaystyle m(r)=\int_{0}^{r}4\pi r^{\prime
2}\varepsilon(r^{\prime})dr^{\prime}.$ (23)
Now, by selecting a central energy density $\varepsilon_{c}$, under the
boundary conditions $P(0)=P_{c}$ and $m(0)=0$, we integrate TOV equations
outwards to a radius $r=R$ at which $P$ vanishes (Shapiro & Teukolski rk9
(1983)).
In this section, we calculate the structure of the SQS with density dependent
bag constant in the presence of a magnetic field. We should note that a strong
magnetic field changes the spherical symmetry of the system and for magnetic
fields less than $10^{19}\ G$, this effect is ignorable (Felipe & Martinez
rk42 (2009); Gonzalez Felipe et al. rk43 (2011)). Considering the anisotropy
of the strange quark matter pressures in the presence of magnetic field, it
has been shown that for vanishing AMM (anomalous magnetic moments), the
perpendicular component of the pressure $P_{\perp}$ goes to zero at about
$2\times 10^{19}G$ (Gonzalez Felipe et al. rk56 (2008)). Thus in the case of
SQM, for $B<10^{19}G$ the anisotropy in the pressures is relatively small,
i.e, $P_{\perp}=P_{\parallel}$.
In Fig. 8, we have drawn the gravitational mass versus the central density
($\varepsilon_{c}$) for an SQS in the magnetic fields $B=0$ and $5\times
10^{18}\ G$. We see that as the central density increases, the gravitational
mass of an SQS increases, and finally it reaches a limiting value which is
called maximum gravitational mass. Fig. 8 shows that by presenting the
magnetic field, the gravitational mass decreases. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ at $B=5\times 10^{18}\ G$ (Bordbar &
Peyvand rk20 (2011)) are also given in Fig. 8 for the sake of comparison. This
indicates that the density dependence of bag constant leads to substantially
higher values for the gravitational mass of SQS. With the density dependent
bag constant, we have found that the maximum gravitational mass of SQS is
about $1.62\ M_{\odot}$, while with the fixed bag constant, it is about $1.33\
M_{\odot}$.
We have plotted the gravitational mass of SQS as a function of the radius
(mass-radius relation) for the magnetic fields $B=0$ and $5\times 10^{18}\ G$
in Fig. 9. It is seen that for all cases of SQS, the gravitational mass
increases by increasing the radius. In Fig. 9, we have also compared our
results for the density dependent case of bag constant with those of density
independent case. We can see that for the case of fixed bag constant, the
increasing rate of gravitational mass versus radius is higher than that of
density dependent case. However, it will be more constructive to consider the
effects of rotation on the properties of the star which is beyond our present
investigation. Some authors have shown that considering the rotation of the
star leads to the larger maximum mass for strange quark stars (Shen et al.
rk57 (2005)).
## IV Summary and conclusions
We have investigated a cold static strange quark star in the presence of a
strong magnetic field. For this purpose, some of the bulk properties of the
strange quark matter such as the energy density and equation of state have
been computed using the MIT bag model with the density dependent bag constant.
Calculations of the energy for different magnetic polarization in the presence
of a magnetic field demonstrated that as the density increases, the minimum
point of energy shifts to the lower values of the polarization. We have shown
that the value of the polarization parameter decreases by increasing the
density, and it also increases by increasing the magnetic field. Our results
at $B=5\times 10^{18}\ G$ show that for both cases of the density dependent
bag constant and fixed bag constant, the total energy density have an
increasing rate by increasing the density. We have shown that there is a
ferromagnetic phase transition at high magnetic fields. Our computations
indicate that the pressure increases by increasing the density and magnetic
field. In this work, we have also studied the structure properties of the
strange quark stars. Our results show that the gravitational mass of the
strange quark star increases by increasing the central energy density. It was
shown that this gravitational mass reaches a limiting value at higher values
of the central energy density. We have shown that the maximum mass of the
strange quark star reduces by presenting the magnetic field. Finally, a
comparison has been also made between the results of density dependent bag
constant and those of fixed bag constant. Our calculation with the density
dependent bag constant shows a higher maximum mass with respect to that of
fixed bag constant.
One of the possible astrophysical implications of our results is calculation
of the surface redshift $(z_{s})$ of SQS. This parameter is of special
interest in astrophysics and can be obtained from the mass and radius of the
star using the following relation (Camenzind rk15 (2007)),
$z_{s}=(1-\frac{2GM}{Rc^{2}})^{-\frac{1}{2}}-1.$ (24)
Our results corresponding to the maximum mass and radius of SQS calculated by
density dependent bag constant lead to $z_{s}=0.534\ ms^{-1}$ for the magnetic
field $B=5\times 10^{18}G$.
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University Research Council.
## References
* (1) Adami C., Brown G. E., 1993, Phys. Rep. 234, 1
* (2) Alford M., Braby M., Paris M., Reddy S., 2005, Astrophys. J. 629, 969
* (3) Baluni V., 1978, Phys. Rev. D, 17, 2092
* (4) Bhattacharyya A., Ghosh S. K., Joardar P. S., Mallick R., Raha S., 2006, astro-ph/0606523v1
* (5) Bigdeli M., Bordbar G. H., Rezaei Z., 2009, Phys. Rev. C, 80, 034310
* (6) Bigdeli M., Bordbar G. H., Poostforush A., 2010, Phys. Rev. C, 82, 034309
* (7) Blaschke D., Grigorian H., Poghosyan G., Roberts C. D., Schmidt S., 1999, Phys. Lett. B, 450. 207
* (8) Bordbar G. H., Modarres M., 1997, J. Phys. G: Nucl. Phys. 23, 1631
* (9) Bordbar G. H., Modarres M., 1998, Phys. Rev. C, 57, 714
* (10) Bordbar G. H., Bigdeli M., Yazdizadeh T., 2006, Int. J. of Mod. Phys. A, 21, 5991
* (11) Bordbar G. H., Bigdeli M., 2007a, Phys. Rev. C, 75, 045804
* (12) Bordbar G. H., Bigdeli M., 2007b, Phys. Rev. C, 76, 035803
* (13) Bordbar G. H., Bigdeli M., 2008a, Phys. Rev. C, 77, 015805
* (14) Bordbar G. H., Bigdeli M., 2008b, Phys. Rev. C, 78, 054315
* (15) Bordbar G. H., Nurafshan M., Khosropour B., 2009, Iranian J. Phys. Res. 9, 237
* (16) Bordbar G. H., Peyvand A., 2011, Res. Astron. Astrophysics, 11, 851
* (17) Bordbar G. H., Poostforush A., Zamani A., 2011, Astrophysics, 54, 277
* (18) Burgio G. F., Baldo M., Sahu P. K., Santra B., Schulze H. J., 2002, Phys. Lett. B, 526, 19
* (19) Camenzind M., 2007, Compact objects in astrophysics : White Dwarfs, Neutron Stars, and Black Holes, Springer
* (20) Chodos A., Jaffe R. L., Johnson K., Thorn C. B., 1974, Phys. Rev. D, 9, 3471
* (21) Drake J. J., Herman L. M., Stefan D., Peter E. F., Antonella F., Michael J., Vinay K., Fabrizio N., Deron O. P., Bradford J. W., Klaus W., 2002, Astrophys. J, 572, 996
* (22) Farhi E., Jaffe R. L., 1984, Phys. Rev. D, 30, 2379
* (23) Farhi E., Olinto A., 1986, Astrophys. J. 310, 261
* (24) Felipe R. G., Martinez A. P., 2009, J. Phys. G: Nucl. Phys. 36, 075202
* (25) Fraga E. S., Pisarski R. D., Schaffner-Bielich J., 2001, Phys. Rev.D, 63, 121702
* (26) Ghosh T., Chakrabarty S., 2001, Phys. Rev. D, 63, 043006
* (27) Glendenning N. K., Weber F., 1992, Astrophy. J, 400, 67
* (28) Glendenning N. K., ed., 2000, Compact Stars : Nuclear Physics, Particle Physics, and General Relativity, Springer
* (29) Gonzalez Felipe R., Perez Martinez A., Perez Rojas H., Orsaria M., 2008, Phys. Rev. C, 77, 015807
* (30) Gonzalez Felipe R., Manreza Paret D., Perez Martnez A., 2011, Magnetized color flavor locked state and compact stars, Eur. Phys. J. A, 47, 1
* (31) Hanauske M., Satarov L. M., Mishustin I. N., Stocker H., Greiner W., 2001, Phys. Rev. D, 64, 043005
* (32) Hansel P., Potekhin A. Y., Yakovlev, D. G. eds., 2007, Neutron Stars 1 : Equation of State and Structure, Astrophysics and Space Science Library, vol. 326
* (33) Heinz U., 2001, Nucl. Phys. A, 685, 414
* (34) Heinz U., Jacobs M., 2000, nucl-th/0002042
* (35) Jin Xue-min., Jenning B. K., 1997, Phys. Rev. C, 55, 1567
* (36) Kouveliotou C., Dieters S., Strohmayer T. et al., 1998, Nature, 393, 235
* (37) Kouveliotou C., Strohmayer T., Hurley K. et al., 1999, ApJ, 510, L115
* (38) Lagaris I. E., Pandharipande V. R., 1981a, Nucl. Phys. A, 359, 331
* (39) Lagaris I. E., Pandharipande V. R., 1981b, Nucl. Phys. A, 359, 349
* (40) Lattimer J. M., Prakash M., 2004, astro-ph/0405262v1
* (41) Li X., Bombaci I., Dey M., Dey J., van den Heuvel E. P. J., 1999, Phys. Rev. Lett, 83, 3776
* (42) Menezes D. P., Providencia C., Melrose D. B., 2006, J. Phys. G: Nucl. Part. Phys, 32,1081
* (43) Modarres M., Bordbar G. H., 1998, Phys. Rev. C, 58, 2781
* (44) Peshier A., Kampfer B., Soff G., 2000, Phys. Rev. C, 61, 04520
* (45) Prakash M., Lattimer J. M., Steiner A. W., and Page D., 2003, Nucl. Phys. A, 715, 835
* (46) Rehberg P., Klevansky S. P., Hufner J., 1996, Phys. Rev. C, 53, 410
* (47) Ruster S. B., Rischke D. H., 2004, Phys. Rev. D, 69, 045011
* (48) Shapiro S., Tenkolsky S., 1983, Black Holes, White Dwarfs, and Neutron Stars, Wiley
* (49) Shen J., Zhang Y., Wang B., Su R. K., 2005, Int. J. Mod. Phys. A, 20, 7547
* (50) Weber F. ed., 2007, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics, Iop Publishing
* (51) Weber F., 2005, Prog. Part. Nucl. Phys. 54, 193
* (52) Wong Samuel S. M., 1990, Introductory Nuclear Physics, International Edition, Prentice-Hall
* (53) Yazdizadeh T., Bordbar G. H., 2011, Res. Astron. Astrophys. 11, 471
* (54) Yu H. W., Xu R. X., 2010, Res. Astron. Astrophys. 10, 81
Figure 1: Total energy density ($\varepsilon_{tot}$) as a function of the
polarization parameter ($\xi$) for different densities ($\rho$).
Figure 2: Polarization parameter ($\xi$) versus density ($\rho$) for
$B=5\times 10^{18}$ and $5\times 10^{19}\ G$.
Figure 3: Polarization parameter ($\xi$) corresponding to the minimum points
of energy density versus the magnetic field ($B$) for different values of
density ($\rho$).
Figure 4: Total energy density versus density ($\rho$) calculated by density
dependent bag constant (full curve) at $B=5\times 10^{18}\ G$. The results for
$\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ (dashed curve) at $B=5\times 10^{18}\ G$
have been also given for comparison.
Figure 5: Phase diagram for the strange quark matter in the presence of a
strong magnetic field.
Figure 6: The equation of state of SQM at $B=0$, $\ 5\times 10^{18}\ $ and
$5\times 10^{19}\ G$.
Figure 7: The equation of state of SQM in the case of density dependent bag
constant (full curve) at $B=5\times 10^{18}\ G$. The results for the case of
fixed bag constant ($\mathcal{B}=90\ \frac{MeV}{fm^{3}}$) (dashed curve) at
$B=5\times 10^{18}\ G$ have been also given for comparison.
Figure 8: Gravitational mass versus the central energy density
($\varepsilon_{c}$) at $B=0$ (full curve) and $B=5\times 10^{18}\ G$ (dashed
curve). The results for $\mathcal{B}=90\ \frac{MeV}{fm^{3}}$ (dashed dotted
curve) at $B=5\times 10^{18}\ G$ have been also given for comparison.
Figure 9: The gravitational mass versus radius at $B=0$ (full curve) and
$B=5\times 10^{18}\ G$ (dashed curve). The results for $\mathcal{B}=90\
\frac{MeV}{fm^{3}}$ (dashed dotted curve) at $B=5\times 10^{18}\ G$ have been
also given for comparison.
|
arxiv-papers
| 2012-04-02T06:54:58 |
2024-09-04T02:49:29.302283
|
{
"license": "Public Domain",
"authors": "Gholam Hossein Bordbar, Hajar Bahri and Fatemeh Kayanikhoo",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1204.0325"
}
|
1204.0587
|
# Black-box superconducting circuit quantization
Simon E. Nigg, Hanhee Paik, Brian Vlastakis, Gerhard Kirchmair, Shyam Shankar,
Luigi Frunzio, Michel Devoret, Robert Schoelkopf and Steven Girvin
Departments of Physics and Applied Physics, Yale University, New Haven, CT
06520, USA
###### Abstract
We present a semi-classical method for determining the effective low-energy
quantum Hamiltonian of weakly anharmonic superconducting circuits containing
mesoscopic Josephson junctions coupled to electromagnetic environments made of
an arbitrary combination of distributed and lumped elements. A convenient
basis, capturing the multi-mode physics, is given by the quantized eigenmodes
of the linearized circuit and is fully determined by a classical linear
response function. The method is used to calculate numerically the low-energy
spectrum of a 3D-transmon system, and quantitative agreement with measurements
is found.
###### pacs:
42.50.Ct,85.25.Am,42.50.Pq,03.67.-a
Superconducting electronic circuits containing nonlinear elements such as
Josephson junctions (JJs) are of interest for quantum information processing
Devoret and Martinis (2004); Wallraff _et al._ (2004), due to their
nonlinearity and weak intrinsic dissipation. The discrete low-energy spectrum
of such circuits can now be measured to a precision of better than one part
per million Paik _et al._ (2011). The question thus naturally arises of how
well one can theoretically model such man-made artificial atoms. Indeed,
increasing evidence indicates that due to increased coupling strengths Devoret
_et al._ (2007), current models are reaching their limits Houck _et al._
(2008); Bourassa _et al._ (2009); Niemczyk _et al._ (2010); Filipp _et al._
(2011); Viehmann _et al._ (2011) and in order to further our ability to
design, optimize and manipulate these systems, developing models beyond these
limits becomes necessary. This is the goal of the present work.
An isolated ideal JJ has only one collective degree of freedom: the order
parameter phase difference $\varphi$ across the junction. The zero-
temperature, sub-gap physics of this system, with Josephson energy $E_{J}$ and
charging energy $E_{C}$, is described by the Cooper-pair box Hamiltonian
$H_{\rm CPB}=4E_{C}(\hat{N}-N_{g})^{2}-E_{J}\cos(\hat{\varphi}),$ (1)
where $\hat{N}$ is the Cooper-pair number operator conjugate to
$\hat{\varphi}$ and $N_{g}$ an offset charge. This model is exactly solvable
in terms of Mathieu functions Cottet (2002); Koch _et al._ (2007). The
crucial feature that emerges from this solution is that the charge dispersion,
i.e. the maximal variation of the eigenenergies with $N_{g}$, is exponentially
suppressed with $E_{J}/E_{C}$ while the relative anharmonicity decreases only
algebraically with a slow power-law in $E_{J}/E_{C}$. As a consequence, there
exists a regime with $E_{J}\gg E_{C}$ – the transmon regime – where the
anharmonicity is much larger than the linewidth (e.g. due to fluctuation of
the offset charge $N_{g}$), thus satisfying the operability condition of a
qubit Schreier _et al._ (2008). This is the regime of interest here.
In order to be useful for quantum information processing tasks, several
Josephson qubits must be made to controllably interact with each other and
spurious interactions with uncontrolled (environmental) degrees of freedom
must be minimized. In circuit quantum electrodynamics Blais _et al._ (2004);
Wallraff _et al._ (2004); Koch _et al._ (2007) (cQED), this is achieved by
coupling the JJs to a common microwave environment with a desired discrete
mode structure. So far such systems have mostly been described theoretically
by models well known from quantum optics such as the single-mode Jaynes-
Cummings model and extensions thereof Jaynes and Cummings (1963);
*Tavis-1968a.
When applied to
Figure 1: (Color online) Cartoon of a JJ at the center of a broadband dipole
antenna inside a 3D microwave cavity. The presence of the antenna alters the
geometry of the cavity-mode (full (red) curve) and a precise description
requires the inclusion of many bare modes (dashed curves).
superconducting circuits with multi-level artificial atoms, multi-mode
cavities and increased coupling strengths Devoret _et al._ (2007); Bourassa
_et al._ (2009); Niemczyk _et al._ (2010) however, several technical and
practical difficulties with these approaches arise. For example, capturing
important effects of non-computational qubit states requires going to high
orders in perturbation theory DiCarlo _et al._ (2009). Also, determining the
bare Hamiltonian parameters, in terms of which these models are defined, is
cumbersome and requires iterating between experiment and theory. Perhaps even
more important are the shortcomings of the traditional approaches in dealing
with the multiple modes of the cavity. Indeed high-energy, off-resonant cavity
modes have already been measured to contribute substantially to the inter-
qubit interaction strength DiCarlo _et al._ (2009); Filipp _et al._ (2011)
and, via the multi-mode Purcell effect, also to affect the coherence
properties (relaxation and dephasing) of the qubits Houck _et al._ (2008).
Attempts at including this multi-mode physics in the standard models however,
lead to difficulties with diverging series and QED renormalization issues
Filipp _et al._ (2011); Bourassa and Blais , which to the best of our
knowledge remain unresolved. Fig. 1 illustrates the origin of the problem with
the example of a JJ inside a 3D cavity (3D-transmon) Paik _et al._ (2011).
The presence of a relatively large metallic dipole antenna 111In current
realizations of the 3D-transmon qubits, the length of the antenna is between
$1$ and $10\%$ of the wavelength of the fundamental bare cavity mode. can
strongly alter the geometry of the cavity modes. This essentially classical
effect, can be accounted for precisely only by including a sufficiently large
number of bare modes.
In contrast, we propose to start by considering the coupled but linearized
problem in order to find a basis that incorporates the main effects of the
coupling between multi-level qubits and a multi-mode cavity and then account
for the weak anharmonicity of the Josephson potential perturbatively. The
crucial assumption made here is that charge dispersion effects can be safely
neglected. This is reasonable given that in state-of-the art implementations
of transmon qubits Paik _et al._ (2011); Reed _et al._ (2012), charge
dispersion only contributes a negligible amount to the measured linewidths.
Previous work discussed the nonlinear dynamics of a JJ embedded in an external
circuit classically Manucharyan _et al._ (2007). Here we go one step further
and show how the knowledge of a classical, in principle measurable, linear
response function lets us quantize the circuit, treating qubits and cavity on
equal footing.
Single junction case. We consider a system with a JJ with bare Josephson
energy $E_{J}$ and charging energy $E_{C}$, in parallel with a linear but
otherwise arbitrary electromagnetic environment as depicted in Fig. 2 (a).
Neglecting dissipation, the unbiased junction alone is described by the
Hamiltonian (1). At low energies, when $E_{J}\gg E_{C}$, quantum fluctuations
of the phase $\varphi$ across the junction are small compared with $\pi$ and,
as emphasized in the introduction, the probability of quantum tunneling of the
phase between minima of the cosine potential is negligibly small. It is then
reasonable to expand the latter in powers of $\varphi$, thus obtaining the
approximate circuit representation of Fig. 2 (b), in which the spider symbol
Manucharyan _et al._ (2007) represents the purely nonlinear part and
$L_{J}={\phi_{0}}^{2}/E_{J}$ and $C_{J}=e^{2}/(2E_{C})$ the linear parts of
the Josephson
(a)$E_{C},E_{J}$$\varphi$Cavity
(b)Cavity$L_{J}$$C_{J}$
(c)$Z(\omega)$
(d)$R_{M}$$R_{2}$$R_{1}$$L_{1}$$C_{1}$$L_{2}$$C_{2}$$L_{M}$$C_{M}$
Figure 2: (Color online) (a) Schematics of a JJ ((red) boxed cross) coupled to
an arbitrary linear circuit (striped disk). (b) The Josephson element is
replaced by a parallel combination of: a linear inductance $L_{J}$, a linear
capacitance $C_{J}$ and a purely nonlinear element with energy
$E_{J}(1-\cos(\varphi))-(E_{J}/2)\varphi^{2}$, represented by the spider
symbol. (c) The linear part of the circuit shown in (b) is lumped into an
impedance $Z(\omega)$ seen by the nonlinear element. (d) Foster-equivalent
circuit (pole-decomposition) of the impedance $Z(\omega)$.
element. Here $\phi_{0}=\hbar/(2e)$ is the reduced flux quantum. To leading
order, the energy of the spider element is given by $E_{\rm
nl}=-{\phi_{0}}^{2}\varphi^{4}/(24L_{J})$.
A quantity of central importance in the following is the impedance $Z(\omega)$
of the linear part of the circuit depicted in Fig. 2 (c). The latter is a
complex meromorphic function and by virtue of Foster’s theorem Foster (1924);
Beinger _et al._ (1945) can be synthesized by the equivalent circuit of
parallel LCR oscillators in series shown in Fig. 2 (d). Explicitly
$Z(\omega)=\sum_{p=1}^{M}\left(j\omega C_{p}+\frac{1}{j\omega
L_{p}}+\frac{1}{R_{p}}\right)^{-1},$ (2)
where $M$ is the number of modes 222The case of infinitely many discrete modes
necessitates an extension of Foster’s theorem as discussed in Zinn (1951), but
the results presented here still apply. and we have adopted the electrical
engineering convention of writing the imaginary unit as $j=-i$. This
equivalent circuit mapping corresponds, in electrical engineering language, to
diagonalizing the linearized system of coupled harmonic oscillators. The
resonance frequencies of the linear circuit are determined by the real parts
of the poles of $Z$ or more conveniently by the real parts of the zeros of the
admittance defined as $Y(\omega)=Z(\omega)^{-1}$, and for weak dissipation,
i.e. $R_{p}\gg\sqrt{L_{p}/C_{p}}$, are given by
$\omega_{p}=(L_{p}C_{p})^{-\frac{1}{2}}$. The imaginary parts of the roots
$(2R_{p}C_{p})^{-1}$, give the resonances a finite width. The effective
resistances are given by $R_{p}=1/{\rm Re}Y(\omega_{p})$ and the effective
capacitances are determined by the frequency derivative on resonance of the
admittance as $C_{p}=(1/2){\rm Im}Y^{\prime}(\omega_{p})$. Here and in the
following the prime stands for the derivative with respect to frequency. Note
that ${\rm Im}Y^{\prime}(\omega)>0$ Foster (1924). Together this yields a
compact expression for the quality factor of mode $p$:
$Q_{p}=\frac{\omega_{p}}{2}\frac{{\rm Im}Y^{\prime}(\omega_{p})}{{\rm
Re}Y(\omega_{p})}.$ (3)
When applied to the mode representing the qubit, Eq. (3) gives an estimate for
the Purcell limit on the qubit lifetime $T_{1}=Q_{\rm qb}/\omega_{\rm qb}$ due
to photons leaking out of the cavity.
In order to derive the effective low-energy quantum Hamiltonian of the
circuit, we next neglect dissipation ($R_{p}\rightarrow\infty$) and introduce
the normal (flux) coordinates
$\phi_{p}(t)=f_{p}e^{j\omega_{p}t}+(f_{p})^{*}e^{-j\omega_{p}t}$ associated
with each LC oscillator in the equivalent circuit. We can then immediately
write the classical Hamiltonian function of the equivalent circuit as
$\mathcal{H}_{0}=2\sum_{p=1}^{M}(f_{p})^{*}(L_{p})^{-1}f_{p}$, where the
subscript $0$ indicates that we consider the linear part of the circuit.
Kirchhoff’s voltage law implies that up to an arbitrary constant,
$\phi(t)=\sum_{p=1}^{M}\phi_{p}(t)$, where
$\phi(t)=\int_{-\infty}^{t}V(\tau)d\tau$ is the flux coordinate of the
junction with voltage $V(t)$. Note that by the second Josephson relation, the
order parameter phase difference is related to the latter via
$\varphi(t)=\phi(t)/\phi_{0}$ (modulo $2\pi$).
Quantization is achieved in the canonical way Devoret (1995); Clerk _et al._
(2010) by replacing the flux amplitudes of the equivalent oscillators by
operators as
$f_{p}^{(*)}\rightarrow\sqrt{\frac{\hbar}{2}\mathcal{Z}_{p}^{\rm
eff}}\,a_{p}^{(\dagger)},\quad\mathcal{Z}_{p}^{\rm
eff}=\frac{2}{\omega_{p}{\rm Im}Y^{\prime}(\omega_{p})},$ (4)
with the dimensionless bosonic annihilation (creation) operators $a_{p}$
($a_{p}^{\dagger}$). Direct substitution yields the Hamiltonian
$H_{0}=\sum_{p}\hbar\omega_{p}a_{p}^{\dagger}a_{p}$ of $M$ uncoupled harmonic
oscillators (omitting the zero-point energies) and the Schrödinger operator of
flux across the junction is
$\hat{\phi}=\sum_{p=1}^{M}\sqrt{\frac{\hbar}{2}\mathcal{Z}_{p}^{{\rm
eff}}}\left(a_{p}+a_{p}^{\dagger}\right).$ (5)
We emphasize that the harmonic modes $a_{p}$ represent collective excitations
of the linear circuit and their frequencies $\omega_{p}$ are the equivalent of
dressed oscillator frequencies. The coupling in the linear circuit is treated
exactly and in particular no rotating wave approximation is used.
$\nu_{01}$ (GHz) | $\nu_{c}$ (GHz) | $\nu_{02}$ (GHz) | $\alpha_{\rm qb}$ (MHz) | $\chi$ (MHz) | $L_{J}$ (nH) | $C_{J}$ (ff)
---|---|---|---|---|---|---
$7.77$ | $(7.763)$ | $8.102$ | $(8.105)$ | $15.33$ | $(15.333)$ | -210 | (-193) | $-90$ | $(-80.6)$ | $5.83$ | $7.6$
$7.544$ | $(7.54)$ | $8.126$ | $(8.05)$ | $14.808$ | $(14.830)$ | -280 | (-249) | $-30$ | $(-33.0)$ | $6.12$ | $9.2$
$7.376$ | $(7.376)$ | $7.858$ | $(7.864)$ | $14.489$ | $(14.495)$ | -264 | (-257) | $-37.5$ | $(-38.7)$ | $6.67$ | $4.0$
$7.058$ | $(7.045)$ | $8.005$ | $(8.023)$ | $13.788$ | $(13.794)$ | -328 | (-295) | $-13.2$ | $(-13.3)$ | $7.45$ | $5.2$
$6.808$ | $(6.793)$ | $8.019$ | $(8.017)$ | $13.286$ | $(13.294)$ | -330 | (-293) | $-8$ | $(-8.4)$ | $7.71$ | $7.8$
$6.384$ | $(6.386)$ | $7.832$ | $(7.823)$ | $12.45$ | $(12.449)$ | -318 | (-324) | $-5.4$ | $(-7.6)$ | $9.40$ | $0.34$
Table 1: Low-energy spectrum ($\nu_{01}$, $\nu_{c}$, $\nu_{02}$), qubit
anharmonicity ($\alpha_{\rm qb}$) and state-dependent cavity shift ($\chi$) of
six 3D-transmons. Results are shown in the format: experiment (theory). The
theory values are obtained from a least square fit in $C_{J}$ of the
numerically computed lowest three energy levels of the $\phi^{6}$ model. The
fitted values of $C_{J}$ are given in the last column. Their order of
magnitude (a few femto-farads) agrees with estimates based on the sizes of the
junctions. The Josephson inductances $L_{J}$ are obtained from room-
temperature resistance measurements of the junctions.
The Hamiltonian of the circuit including the JJ is then $H=H_{0}+H_{\rm nl}$,
where $H_{\rm
nl}=-(\hat{\phi})^{4}/(24{\phi_{0}}^{2}L_{J})+\mathcal{O}((\hat{\phi}/\phi_{0})^{6})$.
Physical insight may be gained by treating the nonlinear terms as a
perturbation on top of $H_{0}$ assuming the eigenstates
$\mathinner{|{n_{1},n_{2},\dots,n_{M}}\rangle}$ of the latter with energies
$E_{n_{1},n_{2},\dots,n_{M}}^{(0)}=\sum_{i}n_{i}\hbar\omega_{i}$, to be non-
degenerate. Considering only the leading order $\phi^{4}$ nonlinearity, one
then obtains the reduced Hamiltonian
$H_{4}=H_{0}^{\prime}+\frac{1}{2}\sum_{pp^{\prime}}\chi_{pp^{\prime}}\hat{n}_{p}\hat{n}_{p^{\prime}}.$
(6)
Here $\hat{n}_{p}=a_{p}^{\dagger}a_{p}$ and
$H_{0}^{\prime}=H_{0}+\sum_{p}\Delta_{p}\hat{n}_{p}$ includes a correction to
the Lamb-shift given by
$\Delta_{p}=-\frac{e^{2}}{2L_{J}}\left(\mathcal{Z}^{\rm
eff}_{p}\sum_{q}\mathcal{Z}^{\rm eff}_{q}-(\mathcal{Z}^{\rm
eff}_{p})^{2}/2\right)$. We have further introduced the generalized
$\chi$-shift $\chi_{pp^{\prime}}$ between modes $p$ and $p^{\prime}$. Clearly,
$\alpha_{p}\equiv\chi_{pp}$ is the anharmonicity of the first excited state
(self-Kerr) of mode $p$ while $\chi_{pp^{\prime}}=\chi_{p^{\prime}p}$ with
$p\not=p^{\prime}$ is the state-dependent frequency shift per excitation
(cross-Kerr) of mode $p$ due to the presence of a single excitation in mode
$p^{\prime}$. Explicitly we find
$\chi_{pp}=-\frac{L_{p}}{L_{J}}\frac{C_{J}}{C_{p}}E_{C},\quad\chi_{pp^{\prime}}=-2\sqrt{\chi_{pp}\chi_{p^{\prime}p^{\prime}}}.$
(7)
Note that all modes acquire some anharmonicity due to the presence of the
nonlinear JJ. There is thus no strict separation of qubit and cavity anymore.
Colloquially, a mode with strong (weak) anharmonicity will be called qubit-
like (cavity-like). Interestingly, in this lowest order approximation, the
anharmonicity of mode $p$ is seen to be proportional to the inductive
participation ratios Manucharyan _et al._ (2007) $i_{p}\equiv L_{p}/L_{J}$
and inversely proportional to the capacitive participation ratio $c_{p}\equiv
C_{p}/C_{J}$. In the absence of a galvanic short of the junction in the
resonator circuit, as is the case e.g. for a transmon qubit capacitively
coupled to a cavity, it follows from the sum rule $\lim_{\omega\rightarrow
0}\left[Z(\omega)/(j\omega)\right]=\sum_{p}L_{p}=L_{J}$ that $i_{p}\leq 1$.
Similarly, in the absence of any capacitance in series with $C_{J}$, it
follows that $c_{p}\geq 1$, because $\lim_{\omega\rightarrow 0}\left[j\omega
Z(\omega)\right]=\sum_{p}C_{p}^{-1}=C_{\Sigma}^{-1}$, where
$C_{\Sigma}=C_{J}+C_{\parallel}$ and $C_{\parallel}$ is the total capacitance
in parallel with $C_{J}$. Hence we see that in this experimentally relevant
case, the effective anharmonicity of the qubit-like mode is always reduced as
compared with the anharmonicity of the bare qubit given by $-E_{C}$ Koch _et
al._ (2007). Remarkably, in this approximation we find (see Eq. (7)) that the
cross-Kerr shift between two modes is twice the geometric mean of the
anharmonicities of the two modes.
We emphasize that the above expressions do not however account for higher
order effects in anharmonicity such as the change of sign of the cross-Kerr
shift observed in the straddling regime Koch _et al._ (2007); Boissonneault
_et al._ (2010). Such effects are however fully captured by the full model
$H=H_{0}+H_{\rm nl}$, which can be solved numerically. Remarkably, because the
dressed modes already resum all the bare harmonic modes, typically only a few
dressed modes $M^{*}\ll M$ need to be included for good convergence, thus
considerably reducing the size of the effective Hilbert space, which scales as
$\prod_{p=1}^{M^{*}}(N_{p}+1)$ where $N_{p}$ is the maximal allowed number of
excitations in mode $p$ (e.g. $N_{p}=1$ in a two-level approximation).
Charge dispersion. By assumption charge dispersion effects are neglected in
the above approach. One may however ask how the charge dispersion of an
isolated JJ is affected when the latter is coupled to a cavity. As in the
Caldeira-Leggett model Caldeira and Leggett (1981), the coupling between the
JJ and Harmonic oscillators suppresses the probability of flux tunneling and
hence reduces charge dispersion of the qubit further. A simple estimate of the
suppression factor is provided by the probability $P_{0}$ of leaving the
circuit in the ground state after a flux tunneling event and is found to be
given by the “Lamb-Mössbauer” factor $P_{0}\approx
e^{-\frac{1}{2}\sum_{p\not={\rm qb}}\left(\frac{{\delta
q}^{2}}{2C_{p}}\right)\big{/}(\hbar\omega_{p})}$, where the sum excludes the
qubit mode and $\delta q=C_{J}\phi_{0}/\tau$ is the charge (momentum) kick
generated by a $\phi_{0}$ flux slip through the JJ of duration $\tau$ and
$C_{p}=(1/2){\rm Im}Y^{\prime}(\omega_{p})$. Thus our assumption of neglecting
charge dispersion of the qubit is well justified.
Interestingly though, each eigenmode of the system inherits some charge
dispersion. This effect, essentially a consequence of hybridization, is of
particular importance for applications such as quantum information storage in
high-Q cavities coupled to JJs and is the subject of work in progress.
Generalization to $N$ junctions. The approach can be extended to circuits with
multiple JJs connected in parallel to a common linear circuit. Details about
the derivation are given in the supplementary material sup and we here only
state the results. For $N$ qubits, the resonance frequencies of the linear
part of the circuit are determined by the zeros of the admittance
$Y_{k}(\omega)\equiv Z_{kk}(\omega)^{-1}$ for any choice of reference port
$k=1,\dots,N$, where $\mathbf{Z}$ is the $N\times N$ impedance matrix of the
linear part of the circuit with a port being associated with each junction.
The flux operators of the $N$ junctions, with reference port $k$, are given by
($l=1,\dots,N$)
$\hat{\phi}_{l}^{(k)}=\sum_{p=1}^{M}\frac{Z_{lk}(\omega_{p})}{Z_{kk}(\omega_{p})}\sqrt{\frac{\hbar}{2}\mathcal{Z}_{kp}^{{\rm
eff}}}\left(a_{p}+a_{p}^{\dagger}\right),$ (8)
where $\mathcal{Z}_{kp}^{\rm eff}=2/[\omega_{p}{\rm
Im}Y^{\prime}_{k}(\omega_{p})]$. Note that the resonance frequencies are
independent of the choice of reference port, while the eigenmodes do depend on
it. In lowest order of PT and in the $\phi^{4}$ approximation, we find
$\displaystyle\alpha_{p}$
$\displaystyle=-12\beta_{pppp},\quad\chi_{qp}=-24\beta_{qqpp},\quad q\not=p,$
(9)
as well as the correction to the Lamb-shift
$\Delta_{p}=6\beta_{pppp}-12\sum_{q}\beta_{qqpp}$. Here
$\beta_{qq^{\prime}pp^{\prime}}=\sum_{s=1}^{N}\frac{e^{2}}{24L_{J}^{(s)}}\xi_{sq}\xi_{sq^{\prime}}\xi_{sp}\xi_{sp^{\prime}}$,
and choosing the first port as the reference port ($k=1$),
$\xi_{sp}=\frac{Z_{s1}(\omega_{p})}{Z_{11}(\omega_{p})}\sqrt{\mathcal{Z}_{1p}^{{\rm
eff}}}$. Notice that the Cauchy-Schwarz inequality implies that
$|\chi_{qp}|\leq 2\sqrt{\alpha_{q}\alpha_{p}}$. Also, if $q$ and $q^{\prime}$
refer to two different qubit-like modes, then $\chi_{qq^{\prime}}$ is a
measure for the total interaction strength (cavity mediated and direct dipole-
dipole coupling) between these two qubits.
Comparison with experiment. As a demonstration of this method, we apply it to
the case illustrated in Fig. 1 of a single JJ coupled to a 3D cavity Paik _et
al._ (2011). The admittance at the junction port $Y$ is a parallel combination
of the linearized qubit admittance and the admittance $Y_{c}$ of the cavity-
antenna system, i.e. $Y(\omega)=j\omega C_{J}-j/(\omega L_{J})+Y_{c}(\omega)$.
The junction is assumed to be dissipationless corresponding to a Purcell-
limited qubit and ohmic losses of the cavity are included in $Y_{c}$, which is
complex. The Josephson inductance $L_{J}$ is deduced from the measured
junction resistance at room-temperature $R_{T}$, extrapolating it down to the
operating temperature Gloos _et al._ (2000) of $15\,{\rm mk}$ and using the
Ambegaokar-Baratoff relation, $E_{J}=h\Delta/(8e^{2}R_{T})$. $C_{J}$ – the
only free parameter – is obtained by fitting the lowest three energy levels of
the numerical solution of the $\phi^{6}$ model to the measured spectrum Paik
_et al._ (2011). Although $Y_{c}$ may in principle be obtained from current-
voltage measurements, this is not practical in this system, where the antenna
is hard to access non-invasively, being inside a closed high-Q cavity. Instead
we use a finite element High Frequency Simulation Software (HFSS) and obtain
$Y_{c}(\omega)$ by solving the Maxwell equations numerically. Details on this
simulation step are provided in the supplementary material sup .
From the zeros of the imaginary part of the admittance and their slopes we
build and diagonalize the $\phi^{6}$ Hamiltonian in a truncated Hilbert space,
keeping in total three dressed modes (one qubit and two cavity modes) and
allowing for maximally ten excitations per mode. The results of fitting the
low-energy spectrum of six different samples are presented in Table 1, where
we also compare the predicted and measured qubit anharmonicities and
$\chi$-shifts. We find agreement with the measured spectrum at the sub-per
cent level and to within ten per cent with the measured anharmonicities and
$\chi$-shifts.
Conclusion and outlook. We have presented a simple method to determine the
effective low-energy Hamiltonian of a wide class of superconducting circuits
containing lumped or distributed elements. This method is suitable for weakly
nonlinear circuits, for which the normal modes of the linearized classical
circuit provide a good basis in the quantum case. For an $N$ qubit system it
requires only the knowledge of an $N\times N$ (classical) impedance matrix. By
working in a basis of dressed states, the parameters that appear in the
Hamiltonian incorporate much of the renormalization induced by the coupling
between a multi-level artificial atom and a multi-mode resonator.
Consequently, the number of free parameters is considerably reduced as
compared with standard models based on the Jaynes-Cummings paradigm expressed
in terms of the experimentally inaccessible bare parameters. We have
demonstrated the usefulness of this method in designing superconducting
quantum information processing units by computing the low-energy spectrum of a
3D-transmon. Finally, this model may represent a suitable starting point for
future investigations of the emerging ultra-strong coupling regime of cQED.
Acknowledgments. We thank Claudia De Grandi, Eustace Edwards and Mazyar
Mirrahimi for discussions and Mikhael Guy from the Yale HPC center for support
with numerical simulations. SEN acknowledges financial support from the Swiss
NSF. HP, GK, BV, LF, MD, RS and SG acknowledge financial support from IARPA,
ARO (Contract W911NF-09-1-0514) and the American NSF (Contract DMR-1004406).
All statements of fact, opinion or conclusions, contained herein are those of
the authors and should not be construed as representing the official views or
policies of IARPA, or the U.S. Government.
## References
* Devoret and Martinis (2004) M. H. Devoret and J. M. Martinis, Quantum Information Processing 3, 1 (2004).
* Wallraff _et al._ (2004) A. Wallraff, , D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004).
* Paik _et al._ (2011) H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Phys. Rev. Lett. 107, 240501 (2011).
* Devoret _et al._ (2007) M. Devoret, S. Girvin, and R. Schoelkopf, Ann. Phys 16, 767 (2007).
* Houck _et al._ (2008) A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 101, 080502 (2008).
* Bourassa _et al._ (2009) J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, Phys. Rev. A 80, 032109 (2009).
* Niemczyk _et al._ (2010) T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, T. H. D. Zueco, E. Solano, A. Marx, and R. Gross, Nature Physics 6, 772 (2010).
* Filipp _et al._ (2011) S. Filipp, M. Göppl, J. M. Fink, M. Baur, R. Bianchetti, L. Steffen, and A. Wallraff, Phys. Rev. A 83, 063827 (2011).
* Viehmann _et al._ (2011) O. Viehmann, J. von Delft, and F. Marquardt, Phys. Rev. Lett. 107, 113602 (2011).
* Cottet (2002) A. Cottet, _Implementation of a quantum bit in a superconducting circuit_ , Ph.D. thesis, Université Paris VI (2002).
* Koch _et al._ (2007) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
* Schreier _et al._ (2008) J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B 77, 180502 (2008).
* Blais _et al._ (2004) A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
* Jaynes and Cummings (1963) E. Jaynes and F. Cummings, Proceedings of the IEEE 51, 89 (1963).
* Tavis and Cummings (1968) M. Tavis and F. W. Cummings, Phys. Rev. 170, 379 (1968).
* DiCarlo _et al._ (2009) L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature 460 (2009), doi:10.1038/nature08121.
* (17) J. Bourassa and A. Blais, Private communication.
* Note (1) In current realizations of the 3D-transmon qubits, the length of the antenna is between $1$ and $10\%$ of the wavelength of the fundamental bare cavity mode.
* Reed _et al._ (2012) M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature 482, 382 (2012).
* Manucharyan _et al._ (2007) V. E. Manucharyan, E. Boaknin, M. Metcalfe, R. Vijay, I. Siddiqi, and M. Devoret, Phys. Rev. B 76, 014524 (2007).
* Foster (1924) R. M. Foster, Bell System Technical Journal 3, 260 (1924).
* Beinger _et al._ (1945) E. R. Beinger, R. H. Dicke, N. Marcuvitz, C. G. Montgomery, and E. M. Purcell, _Principles of Microwave Circuits_ , edited by C. G. Montgomery, R. H. Dicke, and E. M. Purcell (MIT Radiation Laboratory, 1945).
* Note (2) The case of infinitely many discrete modes necessitates an extension of Foster’s theorem as discussed in Zinn (1951), but the results presented here still apply.
* Devoret (1995) M. H. Devoret, “Quantum fluctuations in electrical circuits,” (Elsevier Science B. V., 1995) Chap. 10, p. 351, les Houches, Session LXIII.
* Clerk _et al._ (2010) A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
* Boissonneault _et al._ (2010) M. Boissonneault, J. M. Gambetta, and A. Blais, Phys. Rev. Lett. 105, 100504 (2010).
* Caldeira and Leggett (1981) A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981).
* (28) See appended supplementary material.
* Gloos _et al._ (2000) K. Gloos, R. S. Poikolainen, and J. P. Pekola, Applied Physics Letters 77, 2915 (2000).
* Zinn (1951) M. K. Zinn, Bell System Technical Journal 31, 378 (1951).
* Pobel (1937) F. Pobel, _Matter and Methods at Low Temperatures_ , 3rd ed. (Springer, 1937).
* Krupka _et al._ (1999) J. Krupka, K. Derzakowski, M. Tobar, J. Hartnett, and R. G. Geyer, Measurement Science and Technology 10, 387 (1999).
* Note (3) Note that strictly speaking the commutator is rather $[\mathop{exp}\nolimits(i\varphi_{s}),n_{s}]=-\hbar\mathop{exp}\nolimits(i\varphi_{s})$, but as we neglect charge dispersion, it is consistent to neglect the $2\pi$-periodicity of the commutation relation.
Supplementary Material for
“Black-box superconducting circuit quantization”
Simon E. Nigg, Hanhee Paik, Brian Vlastakis, Gerhard Kirchmair, Shyam Shankar
Luigi Frunzio, Michel Devoret, Robert Schoelkopf and Steven Girvin
Departments of Physics and Applied Physics, Yale University, New Haven, CT
06520, USA
(Dated: )
These notes provide further details on the HFSS simulation of the cavity
admittance used to build the effective low-energy Hamiltonian in the black-box
quantization approach to compare with the single junction experiment and on
the black-box quantization method for the multi-qubit case.
## I HFSS modeling of a 3D-transmon
As discussed in the main text, the information about the spectrum of the
quantum circuit, is encoded in the admittance at the port of the Josephson
junction $Y(\omega)=Z(\omega)^{-1}$. More precisely, it is sufficient to know
the real roots and the derivative of $Y$ at these points.
Assuming that the size of the junction is negligibly small compared with the
wavelength of the lower modes of the electromagnetic field in the cavity, it
is appropriate to approximate the admittance of the linear part of the
junction by a simple lumped element parallel LC oscillator with inductance
$L_{J}$ and capacitance $C_{J}$ in parallel with the rest of the linear
resonator. Hence the admittance can be decomposed as
$Y(\omega)=j\omega C_{J}-\frac{j}{\omega L_{J}}+Y_{c}(\omega)\,,$ (10)
where $Y_{c}(\omega)$ is the admittance of the system without the junction.
The latter quantity can in principle be directly measured but in this
particular design a measurement is not practical. Instead we simulate the
classical system without the junction by solving Maxwell’s equations
numerically using HFSS. Fig. 3 shows a graphical representation of the
different meshes used to represent the different elements of the cavity and
antenna system. The smaller the element, the finer the mesh needs to be for
accuracy and convergence. In
Figure 3: (Color online) HFSS model of a 3D-transmon. (a) The 3D resonator
with input and output ports. These are terminated by $50\,{\rm Ohm}$ ports.
(b) Transparent view of the cavity showing the sapphire substrate. Because the
electric field is concentrated in the dielectric, a finer mesh is used. (c)
and (d) Zoom-ins on the antenna placed on top of the substrate. The mesh is
finest around the antenna.
this finite element simulation all metallic parts (Antenna and cavity
boundaries made of pure aluminum), are treated as perfect conductors with zero
resistance. In doing so, we neglect the kinetic inductance of the antenna and
cavity. The finite London penetration depth of roughly $\lambda\approx
15\,{\rm nm}$ would lead to an effective increase of the cavity size and hence
a decrease of the cavity frequency of about $10\,{\rm kHz}$. Furthermore, the
kinetic inductance of the antenna and wire connecting the two antenna pads to
the Josephson junction can be estimated as
$L_{k}=\frac{\lambda\mu_{0}}{2\tanh\left(\frac{d}{2\lambda}\right)}\left[\frac{L}{W}+\frac{l}{w}\right]\,,$
(11)
where $d\approx 100\,{\rm nm}$ is the thickness of the aluminum layer,
$L\approx 1\,{\rm mm}$ is the total length and $W\approx 250\,{\rm\mu m}$ the
width of the antenna and $l\approx 34\,{\rm\mu m}$ is the length and $w\approx
1\,{\rm\mu m}$ the width of the wire. With these numbers we obtain
$L_{k}\approx 1.6\cdot 10^{-3}\,{\rm nH}$, which is about three orders of
magnitude smaller than (the linear part of) the Josephson inductance. A simple
estimate shows that this would lead to a negative shift of the qubit resonance
of only a few hundreds of kHz. These corrections are negligible at the current
level of accuracy but can be easily included in the numerical simulation if
necessary.
The aluminum antenna is evaporated on top of a sapphire substrate, the
thickness of which is $430\,{\rm\mu m}$ for samples $1$, $2$, $4$ and $5$ and
$500\,{\rm\mu m}$ for samples $3$ and $6$. The contraction of aluminum with
decreasing temperature leading to a shrinkage of the cavity of about $0.5\%$
and the reduction of the permittivity of sapphire by less than a per cent are
taken into account Pobel (1937); Krupka _et al._ (1999).
Figure 4: (Color online) Real and imaginary parts of the admittance
$Y(\omega)=j\omega C_{J}-\frac{j}{\omega L_{J}}+Y_{c}(\omega)$ as obtained
from the HFSS simulation.
The imaginary and real parts of the resulting admittance $Y$ are shown in Fig.
4 for $C_{J}=0.34\,{\rm ff}$ over a range of frequencies spanning three modes.
The lowest mode with the largest slope is identified with the qubit mode and
the remaining ones with cavity modes, although it must be kept in mind that
the states corresponding to these modes are all superpositions of the bare
modes. With this input, the corrections due to the nonlinearity of the
junction are computed as explained in the main text. The results of the fit in
$C_{J}$ are given in Table I of the main text and plotted in Fig. 5. Details
on the measurement of the spectrum can be found in Paik _et al._ (2011).
Figure 5: (Color online) Low-energy spectrum of six 3D-transmons. Theory
values (open circles) are obtained by fitting $C_{J}$ for each data set
(stars). The green symbols correspond to the $0\rightarrow 1$ qubit
transition, the red symbols to the lowest cavity resonance and the cyan
symbols to the $0\rightarrow 2$ qubit transition. The left inset show the sub
% level relative errors between theory and experiment and the right inset
shows the fitted values of $C_{J}$.
## II Black-box quantization with multiple Junctions
For simplicity we focus on the dissipationless case. We consider a system with
$N$ Josephson junctions with bare Josephson energies $E_{J}^{(s)}$ and
charging energies $E_{C}^{(s)}$, $s=1,\dots,N$, in parallel with a common
linear dissipationless but otherwise arbitrary electromagnetic resonator as
depicted in Fig. 6 (a).
$1$$2$$s$$N$(a)
$1$$2$$k$$N$(b)$Y_{k}$$s$
(c)$p$$2$$1$$M$
Figure 6: (Color online) (a) Schematics of $N$ JJs (gray (red) boxed crosses)
coupled to an arbitrary linear circuit (striped disk). (b) Corresponding
linearized $N$-port circuit with JJs replaced by parallel LC oscillators. (c)
Foster-equivalent circuit of the impedance $Z_{kk}(\omega)$ of the linearized
circuit shown in (b). The reference port $k\in\\{1,\dots,N\\}$ may be chosen
arbitrarily.
The unbiased isolated junctions alone are described by the Hamiltonian
$H_{J}=\sum_{s=1}^{N}\left(4E_{C}^{(s)}(n_{s})^{2}-E_{J}^{(s)}\cos(\varphi_{s})\right)$,
where $n_{s}$ is the Cooper-pair number operator of the $s$-th junction
conjugate to the phase degree of freedom $\varphi_{s}$, i.e.
$[\varphi_{s},n_{s}]=i\hbar$.333Note that strictly speaking the commutator is
rather $[\exp(i\varphi_{s}),n_{s}]=-\hbar\exp(i\varphi_{s})$, but as we
neglect charge dispersion, it is consistent to neglect the $2\pi$-periodicity
of the commutation relation. A corresponding $N$-port linear circuit, shown in
Fig. 6 (b), is then defined by associating a port with each junction and
replacing the latter with a parallel lumped element LC oscillator with
inductance $L_{J}^{(s)}=(\phi_{0})^{2}/E_{J}^{(s)}$ and capacitance
$C_{J}^{(s)}=e^{2}/(2E_{C}^{(s)})$. Here and in the following
$\phi_{0}=\hbar/(2e)$ is the reduced flux quantum. This corresponds to
expanding the cosines in $H_{J}$ to second order in $\varphi_{s}$. We next
consider this linearized circuit classically.
A quantity of central importance in the following is the $N$-port impedance
matrix $\mathbf{Z}$ with elements
$Z_{ss^{\prime}}(\omega)=V_{s}(\omega)/I_{s^{\prime}}(\omega)\Big{|}_{I_{i}=0,i\not=s^{\prime}}$.
Let us choose arbitrarily one reference port $k$ among the $N$ ports. By
virtue of Foster’s theorem Foster (1924) $Z_{kk}(\omega)$ is a purely
imaginary meromorphic function and can be synthesized by the equivalent
circuit of parallel LC oscillators in series shown in Fig. 6 (c). Explicitly
$Z_{kk}(\omega)=\sum_{p=1}^{M}\left(j\omega C_{p}^{(k)}+\frac{1}{j\omega
L_{p}^{(k)}}\right)^{-1}\,,$ (12)
where $M$ is the number of modes and we have adopted the electrical
engineering convention of writing the imaginary unit as $j=-i$. This
equivalent circuit mapping corresponds, in electrical engineering language, to
diagonalizing the linearized system of coupled harmonic oscillators.
Accordingly, the eigen-frequencies
$\omega_{p}=(L_{p}^{(k)}C_{p}^{(k)})^{-\frac{1}{2}}$ are determined by the
poles of $Z_{kk}$ or more conveniently by the real roots of the admittance
defined as $Y_{k}=Z_{kk}^{-1}$ and the effective capacitances are determined
by the frequency derivative on resonance of the latter as
$C_{p}^{(k)}=(1/2){\rm Im}Y_{k}^{\prime}(\omega_{p})$. Note that Foster (1924)
${\rm Im}Y_{k}^{\prime}(\omega)>0$. The Lagrangian of the system can be
written as
$\mathcal{L}=\frac{1}{2}\sum_{p=1}^{M}\left(C_{p}^{(k)}(\dot{\phi}_{p}^{(k)}(t))^{2}+\frac{(\phi_{p}^{(k)}(t))^{2}}{L_{p}^{(k)}}\right)\,,$
in terms of the normal (flux) coordinates
$\phi_{p}^{(k)}(t)=f_{p}^{k}e^{j\omega_{p}t}+(f_{p}^{k})^{*}e^{-j\omega_{p}t}$,
associated with each of the equivalent LC oscillators. From this, we can
immediately write the Hamiltonian function of the equivalent circuit as
$\mathcal{H}_{0}=2\sum_{p=1}^{M}(f_{p}^{k})^{*}(L_{p}^{(k)})^{-1}f_{p}^{k}$,
where the subscript $0$ indicates that we consider the linear circuit (Fig. 6
(b)). Note that the eigen-frequencies do not depend on the choice of port,
while the eigenmodes do. Kirchhoff’s voltage law implies that up to an
arbitrary constant,
$\varphi_{k}(t)=\phi_{0}^{-1}\sum_{p=1}^{M}\phi_{p}^{(k)}(t)$, where according
to Josephson’s second relation,
$\varphi_{k}(t)=\phi_{0}^{-1}\int_{-\infty}^{t}V_{k}(\tau)d\tau$ is the phase
variable of the $k$-th (reference) junction with voltage $V_{k}$. Importantly
this simple relation holds only for the junction at the reference port $k$. In
order to find the corresponding expressions for the other junctions
($s\not=k$), we notice that the AC voltage amplitude
$V_{s}(\omega)=j\omega\phi_{s}^{(k)}(\omega)$ at frequency $\omega$ generated
across port $s$ in response to a current with amplitude
$I_{s^{\prime}}(\omega)$ applied at port $s^{\prime}$ is given by
$V_{s}(\omega)=Z_{ss^{\prime}}(\omega)I_{s^{\prime}}(\omega)$. Hence we have
$\varphi_{s}^{(k)}(\omega)=(Z_{sk}(\omega)/Z_{kk}(\omega))\varphi_{k}(\omega)$.
Combining this with the above we find that
$\varphi_{s}^{(k)}(t)=\phi_{0}^{-1}\sum_{p=1}^{M}\frac{Z_{sk}(\omega_{p})}{Z_{kk}(\omega_{p})}\left(f_{p}^{k}e^{j\omega_{p}t}+(f_{p}^{k})^{*}e^{-j\omega_{p}t}\right)\,.$
(13)
Quantization is achieve in the canonical way Devoret (1995); Clerk _et al._
(2010) by replacing the flux amplitudes of the equivalent oscillators by
operators as
$f_{p}^{k(*)}\rightarrow\sqrt{\frac{\hbar}{2}\mathcal{Z}_{kp}^{\rm
eff}}\,a_{p}^{(\dagger)}\,,\quad\mathcal{Z}_{kp}^{\rm
eff}=\frac{2}{\omega_{p}{\rm Im}Y_{k}^{\prime}(\omega_{p})}\,,$ (14)
with the dimensionless bosonic annihilation (creation) operators $a_{p}$
($a_{p}^{\dagger}$). Direct substitution yields the Hamiltonian
$H_{0}=\sum_{l}\hbar\omega_{l}a_{l}^{\dagger}a_{l}$ of $M$ uncoupled harmonic
oscillators (omitting the zero point energies) and the Schrödinger operator of
phase of the $l$-th junction is
$\hat{\varphi}_{s}^{(k)}=\phi_{0}^{-1}\sum_{p=1}^{M}\frac{Z_{sk}(\omega_{p})}{Z_{kk}(\omega_{p})}\sqrt{\frac{\hbar}{2}\mathcal{Z}_{kp}^{{\rm
eff}}}\left(a_{p}+a_{p}^{\dagger}\right)\,.$ (15)
This is Eq. (7) of the main text using that
$\hat{\phi}_{s}^{(k)}=\phi_{0}\hat{\varphi}_{s}$. The superscript makes
explicit the dependence on the reference port. Accordingly the root mean
square fluctuation of the flux of junction $s$ in the multi-mode Fock state
$\mathinner{|{n_{1},n_{2},\dots,n_{M}}\rangle}$ is given by
$\sqrt{\mathinner{\langle{(\hat{\phi}_{s}^{(k)})^{2}}\rangle}}=\frac{\hbar}{2}\sum_{p=1}^{M}\left(\frac{Z_{sk}(\omega_{p})}{Z_{kk}(\omega_{p})}\right)^{2}\mathcal{Z}_{kp}^{{\rm
eff}}\left(1+2n_{p}\right)$.
The anharmonic terms generated by the non-linearity of the Josephson
inductance, necessary to build a qubit, are included by expressing the higher
order terms in the expansion of the cosine in the harmonic basis. Including up
to the quartic terms we obtain explicitly after normal ordering
$\displaystyle H$
$\displaystyle=H_{0}-\sum_{pp^{\prime}}\gamma_{pp^{\prime}}\left(2a_{p}^{\dagger}a_{p^{\prime}}+a_{p}^{\dagger}a_{p^{\prime}}^{\dagger}+a_{p}a_{p^{\prime}}\right)$
(16)
$\displaystyle\quad-\sum_{pp^{\prime}qq^{\prime}}\beta_{pp^{\prime}qq^{\prime}}\left(6a_{p}^{\dagger}a_{p^{\prime}}^{\dagger}a_{q}a_{q^{\prime}}+4a_{p}^{\dagger}a_{p^{\prime}}^{\dagger}a_{q}^{\dagger}a_{q^{\prime}}+4a_{p}^{\dagger}a_{p^{\prime}}a_{q}a_{q^{\prime}}+a_{p}a_{p^{\prime}}a_{q}a_{q^{\prime}}+a_{p}^{\dagger}a_{p^{\prime}}^{\dagger}a_{q}^{\dagger}a_{q^{\prime}}^{\dagger}\right)+\sum_{s=1}^{N}\mathcal{O}({\hat{\varphi}_{s}}^{6})\,,$
with coefficients
$\beta_{pp^{\prime}qq^{\prime}}=\sum_{s=1}^{N}\frac{e^{2}}{24L_{J}^{(s)}}\xi_{sp}\xi_{sp^{\prime}}\xi_{sq}\xi_{sq^{\prime}}$
and $\gamma_{pp^{\prime}}=6\sum_{q=1}^{M}\beta_{qqpp^{\prime}}$ where,
choosing the first port as the reference port,
$\xi_{sp}=\frac{Z_{s1}(\omega_{p})}{Z_{11}(\omega_{p})}\sqrt{\mathcal{Z}_{1p}^{{\rm
eff}}}$. Treating the $\varphi^{4}$ nonlinearity in first order perturbation
theory, one obtains the expressions for the energy, generalized chi-shift and
generalized anharmonicity given by Eq. (9)) of the main text.
|
arxiv-papers
| 2012-04-03T03:47:37 |
2024-09-04T02:49:29.311263
|
{
"license": "Public Domain",
"authors": "Simon E. Nigg, Hanhee Paik, Brian Vlastakis, Gerhard Kirchmair, Shyam\n Shankar, Luigi Frunzio, Michel Devoret, Robert Schoelkopf, Steven Girvin",
"submitter": "Simon Nigg",
"url": "https://arxiv.org/abs/1204.0587"
}
|
1204.0650
|
# Variability of Contact Process in Complex Networks
Kai Gong Web Sciences Center, University of Electronic Science and Technology
of China, Chengdu 610054, People’s Republic China Ming Tang
tangminghuang521@hotmail.com Web Sciences Center, University of Electronic
Science and Technology of China, Chengdu 610054, People’s Republic China
Computer Experimental Teaching Center, University of Electronic Science and
Technology of China, Chengdu 610054, People’s Republic of China Hui Yang Web
Sciences Center, University of Electronic Science and Technology of China,
Chengdu 610054, People’s Republic China Mingsheng Shang Web Sciences Center,
University of Electronic Science and Technology of China, Chengdu 610054,
People’s Republic China
###### Abstract
We study numerically how the structures of distinct networks influence the
epidemic dynamics in contact process. We first find that the variability
difference between homogeneous and heterogeneous networks is very narrow,
although the heterogeneous structures can induce the lighter prevalence.
Contrary to non-community networks, strong community structures can cause the
secondary outbreak of prevalence and two peaks of variability appeared.
Especially in the local community, the extraordinarily large variability in
early stage of the outbreak makes the prediction of epidemic spreading hard.
Importantly, the bridgeness plays a significant role in the predictability,
meaning the further distance of the initial seed to the bridgeness, the less
accurate the predictability is. Also, we investigate the effect of different
disease reaction mechanisms on variability, and find that the different
reaction mechanisms will result in the distinct variabilities at the end of
epidemic spreading.
###### pacs:
05.40.Fb,05.60Cd,89.75.Hc
The variability of outbreaks is defined as the relative variation of the
prevalence. In order to assess the accuracy and the forecasting capabilities
of numerical models, the variability of outbreaks has been investigated in
many studies. In numerical models, many factors such as network structures,
travel flows, and initial conditions can affect the reliability of the
epidemic spreading forecast. Recently, a contact process model with identical
infectivity is proposed to study both dynamical processes and phase
transitions of epidemic spreading in complex networks, but the predictability
of the model is totally overlooked. In this paper, by investigating the
variabilities of contact process in distinct networks, we show numerically
that the bridgeness plays a significant role on the predictability of the
epidemic pattern in community network, meaning the further distance of the
initial seed to the bridgeness, the less accurate the predictability is.
Hopefully, this work will provide us further understanding and new perspective
in the variability of contact process in complex networks.
## I Introduction
The great threat of epidemic spreading to human society has been strongly
catching scientists’ eyes Bailey:1975 ; Anderson:1992 . In order to realize
the impact of diseases and develop effective strategies for their control and
containment, the accurate mathematical models of epidemic spreading are the
basic conceptual tools Bailey:1975 ; Anderson:1992 ; Diekmann:2000 ;
Dailey:2001 . In mathematical models, the dynamical patterns of epidemic
spreading will be influenced by many different factors such as the age and
social structure of the population, the contact network among individuals, and
the meta-population characteristics Ferguson:2003 . Especially the
heterogeneity of the population network Albert:2000 can result in the absence
of endemic threshold when the population size is infinite and the exponent of
degree distribution $\gamma\leq 3$ Cohen:2000 ; Satorras:2001a ;
Satorras:2001b ; May:2001 ; Lloyd:2001 . With the further study, the local
structures of complex networks (such as degree correlation, clustering
coefficient, community structure and so on) bring quantitative influences on
epidemic spreading Eguiluz:2002 ; Boguna:2002 ; Boguna:2003 . Considering the
complicated local structures in real networks, the forecasting capabilities
(i.e. variability) of current numerical models have been investigated
Barthelemy:2005 . In addition, both the stochastic nature of travel flows
Tang:2006 ; Kishore:2011 and initial conditions can affect the reliability of
the epidemic spreading forecast Colizza:2006 ; Crepey:2006 ; Gautreau:2007 ;
Gautreau:2008 ; Tang:2009a ; Tang:2009b ; Barthelemy:2010 .
In view of this point, Colizza _et al._ have studied the effect of the airline
transportation network on the predictability of the epidemic pattern by means
of the normalized entropy function Colizza:2006 , and found that the
heterogeneous distribution of this network contributes to enhancing the
predictability. In complex networks, many factors can decrease the forecasting
accuracy of epidemic spreading. Crépey _et al._ have found that initial
conditions such as the degree heterogeneity of the seed show a large
variability on the prediction of the epidemic prevalence, and the infection
time of nodes have non-negligible fluctuations caused by the further distance
and the multiplicity of paths to the seed Crepey:2006 . Comparing the scale-
free network (SFN) with community structure Liu:2005 ; Huang:2006 ; Zhou:2007
; Liu:2008 ; Chu:2009 ; Chen:2009 with the random SFN, the predictability of
the prevalence can be found to be better Huang:2007 .
The common assumption in all the aforementioned works is that each node’s
potential infection-activity (infectivity), measured by its possibly maximal
contribution to the propagation process within one time step, is strictly
equal to its degree. However, there are still many real spreading processes
which can not be described well by this assumption Zhou:2006 . Therefore, a
contact process (CP) model with identical infectivity is proposed to study the
epidemic spreading in complex networks Castellano:2006 . Almost all studies in
CP are focused on dynamical processes and phase transitions Zhou:2006 ;
Castellano:2006 ; Ha:2007 ; Castellano:2007 ; Hong:2007 ; Yang:2007 ;
Yang:2008a ; Yang:2008b ; Noh:2009 ; Lee:2009 ; Munoz:2010 , but the
predictability of the model is totally overlooked. To this end, we study how
the structures of distinct networks (i.e. homogeneous, heterogeneous and
community networks) influence the variabilities of epidemic patterns in CP.
Through numerical experiments, we find that the community structures can
remarkably influence the prevalence and its variability, contrary to non-
community networks (i.e. homogeneous and heterogeneous networks). It is worth
noting that it’s hard for the extraordinarily large variability in a local
community to predict the epidemic prevalence.
This paper is organized as follows. In Sec. II, we briefly describe disease
models in CP in complex networks and provide quantitative measurements of the
predictability of epidemic spreading. In Sec. III, we investigate the
prevalence variabilities in both random graph (RG) Erdos:1960 and SFN
Barabasi:1999 . In Sec. IV, we discuss the essential differences of the
prevalence variabilities both in the global network and the local community.
Finally, we draw conclusions in Sec. V.
## II CP model in complex networks
In our model, three distinct networks, i.e. the homogeneous, heterogeneous and
community networks are adopted to investigate the predictability of epidemic
spreading therein. Firstly, as the mother of all network models, the random
graph of Erdős and Rényi Erdos:1960 is regularly used in the study of complex
networks because networks with a complex topology and unknown organizing
principles often appear randomly Albert:2002 . Random graph is defined as a
graph with $N$ nodes and connection probability $p$, which has a Poisson
distribution. Secondly, since scale-free property is observed in many real
complex systems, dynamics study on scale-free networks have been holding
everyone’s concern. In 1999, Barabási and Albert (BA) put forward the most
classical SFN model which is rooted in two generic mechanisms: growth and
preferential attachment Barabasi:1999 . As there are community structures in
social networks, the last studied structure substrate is community network
Newman:2002 . Here we will adopt a simplified community model proposed by Liu
and Hu, which emphasizes on the community feature in social networks Liu:2005
. For simplicity, two independent random graphs are first produced, and then
two RGs are connected randomly by only one link.
In general, the standard disease models conclude susceptible-infected (SI),
susceptible-infected-susceptible (SIS), and susceptible-infected-refractory
(SIR) epidemiological model. Each node of the network represents an individual
and each link plays as one connection which transmits disease to other node.
In SI (SIS or SIR) model, ’S’, ’I’ and ’R’ represents respectively the
susceptible (healthy), the infected, and the refractory (recovered) state. At
each time step of contact process, each infected node randomly contacts one of
its neighbors, and then the contacted neighboring node will be infected with
probability $\lambda$ if it is in the healthy state, or else its state will
stay the same. At the same time, each infected nodes is cured and becomes
susceptible (refractory) with rate $\mu$ in SIS (SIR) model. To eliminate the
stochastic effect of the disease transmission, we can set $\lambda=1$ and
$\mu=0.2$.
In order to analyze the effect of the underlying network topology on the
predictability of epidemic spreading, the variability of outbreaks is defined
as the relative variation of the prevalence [density of infected individuals
$i(t)$] given by Crepey:2006
$\bigtriangleup[i(t)]=\frac{\sqrt{\langle i(t)^{2}\rangle-{\langle
i(t)\rangle}^{2}}}{\langle i(t)\rangle}.$ (1)
$\bigtriangleup[i(t)]=0$ denotes all independent dynamics realizations are
essentially the same, and the prevalence in the network is deterministic.
Larger $\bigtriangleup[i(t)]$ means worse predictability that a particular
realization is far from average over independent realizations.
## III Predictability in homogeneous and heterogeneous networks
The first issue of our study is how the heterogeneity of network structures
influences the variability of the prevalence in CP. By using a numerical
approach in this section, we analyze the variabilities of outbreaks generated
by different sets of initial nodes, both for random graphs and scale-free
networks with the same network size and average degree. Considering the fact
that the results of a particular network can be generalized to any instances
of network model Crepey:2006 , the numerical simulations we studied here are
run in one network. In Fig. 1, we show the curves $i(t)$ and
$\bigtriangleup[i(t)]$ computed for the different disease models in both RG
and SFN. For SI model in Fig. 1 (a), the density of infected $i(t)$ in RG
reachs its stationary state faster than that in SFN; for SIS model in Fig. 1
(b), the stationary $i(t)$ in RG is greater than that in SFN; and for SIR
model in Fig. 1 (c), RG has the higher peak prevalence. Contrary to the
results for the case of contacting all neighbors, it is first discovered that
the heterogeneous structure can slow down the prevalence of outbreaks in CP.
Because hubs may be contacted many times by their neighboring nodes at each
time step, the total contact ability (i.e. the actual number of contacting
nodes at one time step) of SFN is reduced further accordingly, as a result,
the hub effect holds back the prevalence of diseases. Meaningfully, owing to
the limited contact ability of CP, the infected densities starting from the
initial infected nodes (seeds) with different degrees are almost the same in
SFN, which is distinct from the results for the case of contacting all
neighbors Crepey:2006 .
As shown in Fig. 1 (d), (e), and (f), there is slightly different between
variabilities in RG and SFN when $t<20$, which implies heterogeneous structure
does not visibly alter the predictability of CP before the outbreak of
disease. An important contribution of this study is to analyze the differences
among the variabilities of three kinds of disease models. From the comparison
among them, we find that different recovery mechanisms can result in distinct
variabilities at the end of epidemic spreading. For SI model in Fig. 1 (d),
the time arriving at $i(t)=1$ varies in a mass of realizations, which can
induce an exponential decay of the variability when $t>30$. For SIS model in
Fig. 1 (e), the variability of the prevalence will keep on a steady value in
stationary state. For SIR model in Fig. 1 (f), due to the different lifetimes
of the epidemics in a mass of independent realizations Colizza:2006 , the
greater and greater variabilities are observed by approaching the end of the
epidemics.
Figure 1: (color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$
for the different disease models where the ”triangles” and ”circles” denote
the cases of SF and RG networks with the random initial seeds. $i(t)$ versus
$t$ for SI model (a), SIS model (b), and SIR model (c), and
$\bigtriangleup[i(t)]$ versus $t$ for SI model (d), SIS model (e), and SIR
model (f). The parameters are chosen as $N=0.5\times 10^{4},\langle
k\rangle=10,\lambda=1$, and $\mu=0.2$. The results are averaged over $2\times
10^{4}$ independent realizations in one network.
## IV predictability in the global network and the local community
### IV.1 Global network
Figure 2: (color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$
in community networks where the ”squares”, ”circles”, ”triangleups”,
”triangledowns”, and ”trianglelefts” denote the cases of the bridgeness,
$d=1,2,3$, and $4$, respectively. $i(t)$ versus $t$ for SI model (a), SIS
model (b), and SIR model (c), and $\bigtriangleup[i(t)]$ versus $t$ for SI
model (d), SIS model (e), and SIR model (f). The parameters are chosen as
$N=10^{4},\langle k\rangle=10,\lambda=1$, and $\mu=0.2$. The results are
averaged over $2\times 10^{4}$ independent realizations.
As many social networks combined by several communities, such as Facebook
Facebook , YouTube Youtube , and Xiaonei Xiaonei , information propagation
taking place in community networks Liu:2005 ; Huang:2006 ; Zhou:2007 ;
Liu:2008 ; Chu:2009 is one of the most important subjects studying in complex
networks, but in CP, the related research has been ignored for a long time.
Therefore, in this section, we study the variability of CP in a very simple
community network, where two RGs are connected randomly by only one link.
Obviously, this network has a strong strength of community structure. In order
to normalize the terms of community network, we define the link as weak tie
Onnela:2007 , and call two nodes connected by this link ”bridgeness”
Cheng:2010 . We first investigate the time evolution of epidemics generated by
different seeds staying away from the bridgeness, and it is noted that there
is only one initial seed in each realization. From Fig. 2 (a), (b), and (c),
we can get that the closer the seed to the bridgeness,the epidemic spreads
much faster in the global network, and among all cases, the epidemic starting
at the bridgeness spreads fastest. For SI model, the further distance to the
bridgeness such as $d=3,4$ induces two periods of the quickly rising trend at
the beginning time $t=10,30$, respectively. If the initial seed is far away
from the bridgeness, the disease will be restricted in the first community for
a long time till the bridgeness infected, in which almost all nodes is
infected. As a result, the outbreak in the second community just starts at
that moment the prevalence in the first community get towards the end, which
causes the second outbreak. In the case of SIS and SIR model, the recovery
mechanism reduces this phenomenon occurred, for instance there is the tiny
second peak of the prevalence for $d=3,4$ in Fig. 2 (c). From the above, it is
found that the bridgeness plays a distinctly important role in the rapid
transmission of information in CP.
The variability of prevalence in community network is distinct from that in
the network which has no community structure. As shown in Fig. 2(d), (e), and
(f), the curves $\bigtriangleup[i(t)]$ display two peaks because of the time
delay between two outbreaks occurred in different communities. In addition,
the further distance to the bridgeness makes the second peak occur much later.
In Fig. 2(d), for SI model, the first peak corresponds to the prevalence in
the community with the initial seed, so the variability is almost the same as
that in RG before $t=10$. With the outbreak in the second community, the
second peak occurs. Owing to the greater randomness of the time that disease
first occurs in the second community (see Fig. 3), the second peak of the
variabilities is slightly greater than the first peak. After the infection
density is close to saturated at $t\approx 40$ (see Fig. 2(a)), the
variability will be on exponential decay. As all nodes of community network
are infected in more and more realizations, the variability
$\bigtriangleup[i(t)]$ will rapidly decay to zero. In contrast with the case
of SI model, the second peak in SIS model is less than the first peak, because
the recovery mechanism slows down the propagation velocity of diseases, which
reduces the variability of the prevalence. That is to say, the recovery
mechanism reduces the variability of epidemic spreading. Another extremely
obvious difference is the variability decreases to a steady value at the
stationary state. For SIR model in Fig. 2(f), as time goes by, the epidemics
has the greater and greater variability, which is caused by the different
lifetimes of the epidemics in $2\times 10^{4}$ independent realizations.
### IV.2 The local community
Figure 3: (color online). The distribution of arrival time of disease in the
second community for SI model (a), SIS model (b), and SIR model (c), where the
”squares”, ”circles”, ”triangleups”, ”triangledowns”, and ”trianglelefts”
denote the cases of the bridgeness, $d=1,2,3$, and $4$, respectively. The
results are averaged over $2\times 10^{4}$ independent realizations. Figure 4:
(color online). Evolution of both $i(t)$ and $\bigtriangleup[i(t)]$ in the
second community where the ”squares”, ”circles”, ”triangleups”,
”triangledowns”, and ”trianglelefts” denote the cases of the bridgeness,
$d=1,2,3$, and $4$, respectively. $i(t)$ versus $t$ for SI model (a), SIS
model (b), and SIR model (c), and $\bigtriangleup[i(t)]$ versus $t$ for SI
model (d), SIS model (e), and SIR model (f). The results are averaged over
$2\times 10^{4}$ independent realizations.
Considering the relative independence of a local community, we should take the
prevalence and variability into account. On the other hand, since disease must
be transmitted through bridgenesses from the first community to the second
community, this study contributes to understand the effect of them on epidemic
spreading Zhao:2010 . In this section, we will specifically analyze the effect
of different distances of seeds (to the bridgeness in the first community) on
epidemic spreading in the second community.
At first, the arrival time of disease is defined as the moment that infectious
individual first occurs in the second community in each realization, thus the
distribution of arrival time is obtained through massive realizations. In Fig.
3, the distribution of arrival time for the different initial seeds is showen.
For SI model, the arrival distribution of the bridgeness as seed $d=0$
strictly obey the distribution $P(t)=\lambda(1-\lambda)^{t-1}$. When disease
seed is the node with one step to bridgeness, the arrival time increases
generally, and the distribution becomes much wider and flatter. With the
further increasing of distance of seed to the bridgeness (such as $d=3,4$),
the distributions are nearly the same. It is understood that due to the finite
size effect of network, the disease is transmitted through weak tie to the
second community till overall outbreak happened in the first community. For
SIS model, as a result of the recovery mechanism, the distributions of arrival
time are much more evenly and smoothly than that for SI model, given the
various initial seeds. Compared with Fig. 3 (a), we can find that there are
two peaks for SIS and SIR model with $d=1$. This is because the bridgeness may
be infected through two basic pathways: the bridgeness may be infected
directly by the initial seed (i.e. its neighboring node) in $t\leq 1/\mu$; the
other route is the transmission of infection from the other neighboring nodes
when the disease outbreak in the first community, thus the second peak occurs
at $t\approx 20$ (see Fig. 2(b)). For SIR model, owing to the recovery
mechanism, the prevalence might well disappear in the first community before
arriving at the weak tie. Consequently, arrival rate (i.e. the area of the
distribution) is less than one, and the peak value of the corresponding
distribution is less than that for both SI and SIS model.
Fig. 4 shows the prevalence and variability in the second community which are
generated from the different initial seeds in the first community. From Fig. 4
(a), (b), and (c),we can easily know that the prevalence of the bridgeness
acting as seed increases much faster than that in the other cases generated by
the further seeds. In particular, for SIR model in Fig. 4 (c), the peak of the
prevalence (for the case of the bridgeness) occurs first, and has the maximum
value. In addition, there are two peaks of the prevalence for the case of
$d=1$, which is attributed to the propagation delay between two communities.
Since there is only one interconnection between two communities, the chance of
infecting one from the other is low. In other words, infection within intra-
community will be much faster than inter-community infection. Therefore, the
first peak reflects the outbreak and extinction of disease inside with
infectious seed, and the second peak emerges after the other community is
infected. Note that the two peaks can be only observed in SIR model because of
the fact there will be no peak if virus does not ”die”. With the increase of
distances $d$, the peak value is greater than that for $d=1$, although the
outbreaks occur later.
In Fig. 4 (d), (e), and (f), it is a surprise that the variabilities in the
second community are distinct from that in the global network. Firstly, the
variabilities in the second community are very large and also much greater
than that in the global network in Fig. 2, which implies the huge
unpredictability of prevalence produced in the local community. In particular,
at the beginning of outbreaks, the variability for the case of $d=4$ reaches
about $50$, which is 100 times the maximum value $0.5$ in the global network.
Secondly, the closer distance of the seed to bridgeness, the lower level of
variability it has in the local community. In particular, the maximum
variability value for the case of the bridgeness is only about $1$, which is
much less than $50$ for the case of $d=4$. Thus, the bridgeness plays a
significant role in enhancing the predictability, that the closer initial seed
to the bridgeness, the more accurate the predictability is. Thirdly, each
curve of variabilities can be divided into four parts: the sudden drop stage,
the relatively stable stage, the slowing-down stage, and the final stage of
outbreaks (i.e. the exponential decay stage for SI model, the steady state
stage for SIS model, and the sharp increase stage for SIR model,
respectively). The first stage is originated from the uncertain arrival time
of disease, with the increase of the arrival rate, the variability decreases.
The second stage ($10<t<20$ in most cases) is induced by the interplay between
the outbreak and the arrival of diseases in massive realizations: on the one
hand, the outbreaks in some realizations upgrade the variability; on the other
hand, the increase of the arrival rate counteracts this effect. As the
infection density is close to saturated, the variability will enter the
slowing-down stage. What is noteworthy is that the minimum variability value
just corresponds to the peak value of prevalence for SIR model. In the end of
epidemic spreading, the variabilities $\bigtriangleup[i(t)]$ display the
distinct phenomena for the different disease models, for instance, SIR model
shows the higher and higher variabilities.
## V conclusions and discussions
In conclusions, we have studied the variability of CP in complex networks, and
get the clear understanding that the different network structures can
remarkably influence the prevalence and its variability. Firstly, we find that
the variability difference between homogeneous and heterogeneous networks is
very narrow, although the heterogeneous structure induces a lighter
prevalence. Secondly, two peaks of both the prevalence and variability are
shown in the community network. It’s noted that in the local community, the
extraordinarily large variability in early stage of the outbreak makes the
prediction of disease spreading hard. This result is in accordance with Ref.
Yan:2007 in which the networks with strong community structures are of weak
synchronizability, and the amplitudes of the time series in the local
communities are much larger than that in the global networks. Fortunately, the
bridgeness plays a significant role in enhancing the predictability, the
closer initial seed to the bridgeness, the more accurate the predictability
is. This result suggests that bridgenesses may be the ideal detection stations
in community networks. Moreover, the different reaction mechanisms of disease
models can result in the distinct variabilities. Especially for the case of
SIR model, the greater and greater variabilities are observed at the end of
the epidemics for the different lifetimes of the epidemics in various
realizations.
The community network employed in this study is much more simple, but the
actual community networks have complex structures, such as multifarious
communities, many bridgenesses, and the heterogeneous degree distribution in a
local community. Therefore, the further investigation should be focused on the
more complex community networks.
###### Acknowledgements.
This work is supported by the NNSF of China under Grant Nos. 90924011, and the
Sichuan Provincial Science and Technology Department (Grant No. 2010HH0002).
## References
* (1) N. T. J. Bailey, _The Mathematical Theory of Infectious Diseases_ , second ed.,(Griffin, London, 1975).
* (2) R. M. Anderson and R. M. May, _Infectious Disease of Humans_ ,(Oxford University Press, Oxford, 1992).
* (3) O. Diekmann and J. A. P. Heesterbeek, _Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation_ ,(Wiley, New York, 2000).
* (4) D. J. Dailey and J. Gani, _Epidemic Modelling: An Introduction_ ,(Cambridge University Press, Cambridge, 2001).
* (5) N. M. Ferguson, M. J. Keeling, W. J. Edmunds, R. Gani, B. T. Greenfell, and R. M. Anderson, Nature 425, 681 (2003).
* (6) R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2000).
* (7) R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000).
* (8) R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001).
* (9) R. Pastor-Satorras and A. Vespignani, Phys. Rev. E 63, 066117 (2001).
* (10) R. M. May and A. L. Lloyd, Phys. Rev. E 64, 066112 (2001).
* (11) A. L. Lloyd and R. M. May, Science 292, 1316 (2001).
* (12) V. M. Eguíluz and K. Klemm, Phys. Rev. Lett. 89, 108701 (2002).
* (13) M. Boguñá and R. Pastor-Satorras, Phys. Rev. E 66, 047104 (2002).
* (14) M. Boguñá, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. Lett. 90, 028701 (2003).
* (15) M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, J. Theor. Biol. 235, 275 (2005).
* (16) M. Tang, Z. Liu, and J. Zhou, Phys. Rev. E 74, 036101 (2006); M. Tang and Z. Liu, Physica A 387, 1361 (2008); Commun. Theo. Phys. 49, 252 (2008); Inter. J. Mod. Phys. C 19, 927 (2008).
* (17) V. Kishore, M. S. Santhanam, and R. E. Amritkar, Phys. Rev. Lett. 106, 188701 (2011).
* (18) V. Colizza, A. Barrat, M. Barth lemy, and A. Vespignani, Proc. Natl. Acad. Sci. U.S.A. 103, 2015 (2006).
* (19) P. Crépey, F. P. Alvarez, and M. Barthélemy, Phys. Rev. E 73, 046131 (2006).
* (20) A. Gautreau, A. Barrat, and M. Barthélemy, J. Stat. Mech. L09001 (2007).
* (21) A. Gautreau, A. Barrat, and M. Barthélemy, J. Theor. Biol. 251, 509 (2008).
* (22) M. Tang, L. Liu, and Z. Liu, Phys. Rev. E 79, 016108 (2009).
* (23) M. Tang, L. Liu, and B. Li, Europhys Lett. 87, 18005 (2009).
* (24) M. Barthélemy, C. Godrèche, and J. -M. Luck, J. Theor. Biol. 267, 554 (2010).
* (25) Z. Liu and B. Hu, Europhys. Lett. 72(2), 315 (2005).
* (26) L. Huang, K. Park, and Y. -C. Lai, Phys. Rev. E 73, 035103(R) (2006).
* (27) Y. Zhou, Z. Liu, and J. Zhou, Chin. Phys. Lett. 24, 581 (2007).
* (28) X. Wu and Z. Liu, Physica A 387, 623 (2008).
* (29) X. Chu, J. Guan, Z. Zhang, and S. Zhou, J. Stat. Mech. P07043 (2009).
* (30) D. Chen, Y. Fu, M. Shang, Physica A 388, 2741 (2009).
* (31) W. Huang and C. Li, J. Stat. Mech. P01014 (2007).
* (32) T. Zhou, J. -G. Liu, W. -J. Bai, G. -R. Chen, and B. -H. Wang, Phys. Rev. E 74, 056109 (2006).
* (33) C. Castellano and R. Pastor-Satorras, Phys. Rev. Lett. 96, 038701 (2006).
* (34) M. Ha, H. Hong, and H. Park, Phys. Rev. Lett. 98, 029801 (2007).
* (35) C. Castellano and R. Pastor-Satorras, Phys. Rev. Lett. 98, 029802 (2007).
* (36) H. Hong, M. Ha, and H. Park, Phys. Rev. Lett. 98, 258701 (2007).
* (37) R. Yang, B. -H. Wang, J. Ren, W. -J. Bai, Z. -W. Shi, W. -X. Wang, T. Zhou, Phys. Lett. A 364, 189 (2007).
* (38) R. Yang, T. Zhou, Y. -B. Xie, Y. -C. Lai, and B. -H. Wang, Phys. Rev. E 78, 066109 (2008).
* (39) R. Yang, L. Huang, and Y. -C. Lai, Phys. Rev. E 78, 026111 (2008).
* (40) J. D. Noh and H. Park, Phys. Rev. E 79, 056115 (2009).
* (41) S. H. Lee, M. Ha, H. Jeong, J. D. Noh, and H. Park, Phys. Rev. E 80, 051127 (2009).
* (42) M. A. Muñoz, R. Juhász, C. Castellano, and G. Ódor, Phys. Rev. Lett. 105, 128701 (2010).
* (43) P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).
* (44) A.-L. Barabási and R. Albert, Science 286, 509 (1999).
* (45) R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74,47 (2002).
* (46) M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002); Phys. Rev. E 67, 026126 (2003); M. E. J. Newman and J. Park, Phys. Rev. E 68, 036122 (2003); M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 (2004).
* (47) Facebook, http://www.facebook.com
* (48) Youtube, http://www.youtube.com
* (49) Xiaonei, http://www.xiaonei.com
* (50) J. -P. Onnela, J. Saramäki, J. Hyvönen, G. Szabó, D. Lazer, K. Kaski, J. Kertész, and A. -L. Barabási, Proc. Nat. Acad. Sci. 104, 7332 (2007).
* (51) X. -Q. Cheng, F. -X. Ren, H. -W. Shen, Z. -K. Zhang, and T. Zhou, J. Stat. Mech. P10011 (2010).
* (52) J. Zhao, J. Wu, and K. Xu, Phys. Rev. E 82, 016105 (2010).
* (53) G. Yan, Z.-Q. Fu, J. Ren, and W.-X. Wang, Phys. Rev. E 75, 016108 (2007).
|
arxiv-papers
| 2012-04-03T10:38:57 |
2024-09-04T02:49:29.318311
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kai Gong, Ming Tang, Hui Yang, and Mingsheng Shang",
"submitter": "Ming Tang",
"url": "https://arxiv.org/abs/1204.0650"
}
|
1204.0661
|
11institutetext: Puya Sharif 22institutetext: Department of Physics, Stockholm
University, 10691 Stockholm, Sweden.
22email: ps@puyasharif.net 33institutetext: Hoshang Heydari 44institutetext:
Department of Physics, Stockholm University, 10691 Stockholm, Sweden.
44email: hoshang@fysik.su.se
# An introduction to multi-player, multi-choice quantum games
Puya Sharif and Hoshang Heydari
###### Abstract
We give a self contained introduction to a few quantum game protocols,
starting with the quantum version of the two-player two-choice game of
Prisoners dilemma, followed by a n-player generalization trough the quantum
minority games, and finishing with a contribution towards a n-player m-choice
generalization with a quantum version of a three-player Kolkata restaurant
problem. We have omitted some technical details accompanying these protocols,
and instead laid the focus on presenting some general aspects of the field as
a whole. This review contains an introduction to the formalism of quantum
information theory, as well as to important game theoretical concepts, and is
aimed to work as an introduction suiting economists and game theorists with
limited knowledge of quantum physics as well as to physicists with limited
knowledge of game theory.
## 1 Introduction
Quantum game theory is the natural intersection between three fields. Quantum
mechanics, information theory and game theory. At the center of this
intersection stands one of the most brilliant minds of the 20:th century, John
von Neumann. As one of the early pioneers of quantum theory, he made major
contributions to the mathematical foundation of the field, many of them later
becoming core concepts in the merger between quantum theory and information
theory, giving birth to quantum computing and quantum information theory
Nielsen, today being two of the most active fields of research in both
theoretic and experimental physics. Among economists may he be mostly known as
the father of modern game theory GT-Critical; GT-fudenberg; CourseinGT, the
study of rational interactions in strategic situations. A field well rooted in
the influential book _Theory of Games and Economic Behavior_ (1944), by Von
Neumann and Oscar Morgenstern. The book offered great advances in the analysis
of strategic games and in the axiomatization of measurable utility theory, and
drew the attention of economists and other social scientists to these
subjects. For the last decade or so there has been an active interdisciplinary
approach aiming to extend game theoretical analysis into the framework of
quantum information theory, through the study of quantum games flitney;
pitrowski1; NEinQ; landsburg; Bleiler; MW; offering a variety of protocols
where use of quantum peculiarities like entanglement in quantum
superpositions, and interference effects due to quantum operations has shown
to lead to advantages compared to strategies in a classical framework. The
first papers appeared in 1999. Meyer showed with a model of a penny-flip game
that a player making a _quantum move_ always comes out as a winner against a
player making a _classical_ move regardless of the classical players choice
meyer. The same year Eisert et al. published a quantum protocol in which they
overcame the dilemma in Prisoners dilemma eisert. In 2003 Benjamin and Hayden
generalized Eisert’s protocol to handle multi-player quantum games and
introduced the quantum minority game together with a solution for the four
player case which outperformed the classical randomization strategy benjamin.
These results were later generalized to the $n$-players by Chen et al. in 2004
chen. Multi-player minority games has since then been extensively investigated
by Flitney et al. flitney1; flitney2; schmid. An extension to multi-choice
games, as the Kolkata resturant problem was offered by the authors of this
review, in 2011 puya.
### 1.1 Games as information processing
Information theory is largely formulated independent of the physical systems
that contains and processes the information. We say that the theory is
substrate independent. If you read this text on a computer screen, those bits
of information now represented by pixels on your screen has traveled through
the web encoded in electronic pulses through copper wires, as burst of photons
trough fiber-optic cables and for all its worth maybe on a piece of paper
attached to the leg of a highly motivated raven. What matters from an
information theoretical perspective is the existence of a differentiation
between some states of affairs. The general convention has been to keep things
simple and the smallest piece of information is as we all know a _bit_
$b\in\\{0,1\\}$, corresponding to a binary choice: _true_ or _false_ , _on_ or
_off_ , or simply _zero_ or _one_. Any chunk of information can then be
encoded in strings of bits: $\textbf{b}=b_{n-1}b_{n-2}\cdots
b_{0}\in\\{0,1\\}^{n}$. We can further define functions on strings of bits,
$f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}^{k}$ and call these functions
computations or actions of information processing.
In a similar sense games are in their most general form independent of a
physical realization. We can build up a formal structure for some strategic
situation and model cooperative and competitive behavior within some
constrained domain without regards to who or what these game playing agents
are or what their actions actually is. No matter if we consider people,
animals, cells, multinational companies or nations, simplified models of their
interactions and the accompanied consequences can be formulated in a general
form, within the framework of game theory.
Lets connect these two concepts with an example. We can create a one to one
correspondence with between the conceptual framework of game theory and the
formal structure of information processing. Let there be $n$ agents faced with
a binary choice of joining one of two teams. Each choice is represented by a
binary bit $b_{i}\in\\{0,1\\}$. The final outcome of these individual choices
is then given by a $n$-bit output string $\textbf{b}\in\\{0,1\\}^{n}$. We have
$2^{n}$ possible outcomes, and for each agent we have some preference relation
over these outcomes $\textbf{b}_{j}$. For instance, agent $1$ may prefer to
have agent 3 in her team over agent 4, and may prefer any configuration where
agent 5 is on the other team over any where they are on the same and so on.
For each agent $i$, we’ll have a preference relation of the following form,
fully determining their objectives in the given situation:
$\textbf{b}_{x_{1}}\succeq\textbf{b}_{x_{2}}\succeq\cdots\succeq\textbf{b}_{x_{m}},\;\;\;m=2^{n},$
(1)
where $\textbf{b}_{x_{i}}\succeq\textbf{b}_{x_{j}}$ means that the agent in
question prefers $\textbf{b}_{x_{i}}$ to $\textbf{b}_{x_{j}}$, or is at least
indifferent between the choices. To formalize things further we assign a
numerical value to each outcome $\textbf{b}_{x_{j}}$ for _each_ agent, calling
it the _payoff_ $\$_{i}(\textbf{b}_{x_{j}})$ to agent $i$ due to outcome
$\textbf{b}_{x_{j}}$. This allows us to move from the preference relations in
(1) to a sequence of inequalities.
$\textbf{b}_{x_{i}}\succeq\textbf{b}_{x_{j}}\Longleftrightarrow\$(\textbf{b}_{x_{i}})\geq\$(\textbf{b}_{x_{j}})$.
The aforementioned binary choice situation can now be formulated in terms of
functions $\$_{i}(\textbf{b}_{x_{j}})$ of the output strings
$\textbf{b}_{x_{j}}$, where each entry $b_{i}$ in the strings corresponds to
the choice of an agent $i$. So far has the discussion only regarded the output
string without mentioning any input. We could without loss of generality
define an input as string where all the entries are initialized as 0’s, and
the individual choices being encoded by letting each participant either leave
their bit unchanged or performing a NOT-operation, where $\textrm{NOT}(0)=1$.
More complicated situations with multiple choices could be modeled by letting
each player control more than one bit or letting them manipulate strings of
information bearing units with more states than two; of which we will se an
example of later.
### 1.2 Quantization of information
Before moving on to the quantum formalism of operators and quantum states,
there is one intermediate step worth mentioning, the _probabilistic_ bit,
which has a certain probability $p$ of being in one state and a probability of
$1-p$ of being in the other. If we represent the two states ’0’ and ’1’ of the
ordinary bit by the two-dimensional vectors $(1,0)^{T}$ and $(0,1)^{T}$, then
a probabilistic bit is given by a linear combination of those basis vectors,
with real positive coefficients $p_{0}$ and $p_{1}$, where $p_{0}+p_{1}=1$. In
this formulation, randomization between two different choices in a strategic
situation would translate to manipulating an appropriate probabilistic bit.
##### The quantum bit
Taking things a step further, we introduce the quantum bit or the _qubit_ ,
which is a representation of a two level quantum state, such as the spin state
of an electron or the polarization of a photon. A qubit lives in a two
dimensional complex space spanned by two basis states denoted
$\left|0\right\rangle$ and $\left|1\right\rangle$, corresponding to the two
states of the classical bit.
$\left|0\right\rangle=\left(\begin{array}[]{c}1\\\
0\end{array}\right),\;\left|1\right\rangle=\left(\begin{array}[]{c}0\\\
1\end{array}\right).$ (2)
Unlike the classical bit, the qubit can be in any superposition of
$\left|0\right\rangle$ and $\left|1\right\rangle$:
$\left|\psi\right\rangle=a_{0}\left|0\right\rangle+a_{1}\left|1\right\rangle,$
(3)
where $a_{0}$ and $a_{1}$ are complex numbers obeying
$|a_{0}|^{2}+|a_{1}|^{2}=1$. $|a_{i}^{2}|$ is simply the probability to find
the system in the state $\left|\,i\,\right\rangle,\;i\in\\{0,1\\}$. Note the
difference between this and the case of the probabilistic bit! We are now
dealing with complex coefficients, which means that if we superpose two
qubits, then some coefficients might be eliminated. This interference is one
of many effects without counterpart in the classical case. The state of an
arbitrary qubit can be written in the _computational basis_ as:
$\left|\psi\right\rangle=\left(\begin{array}[]{c}a_{0}\\\
a_{1}\end{array}\right)$ (4)
The state of a general qubit can be parameterized as:
$\left|\psi\right\rangle=\cos\left(\frac{\vartheta}{2}\right)\left|0\right\rangle+e^{i\varphi}\sin\left(\frac{\vartheta}{2}\right)\left|1\right\rangle,$
(5)
where we have factored out and omitted a global phase due to the physical
equivalence between the states $e^{i\phi}\left|\psi\right\rangle$ and
$\left|\psi\right\rangle$. This so called _state vector_ describes a point on
a spherical surface with $\left|0\right\rangle$ and $\left|1\right\rangle$ at
its poles, called the Bloch-sphere, parameterized by two real numbers $\theta$
and $\varphi$, depicted in figure 1.
Figure 1: Bloch sphere
#### Hilbert spaces and composite systems
The state vector of a quantum system is defined in a complex vector space
called Hilbert space $\mathcal{H}$. Quantum states are represented in common
Dirac notation as “ket’s”, written as the right part $\left|\psi\right\rangle$
of a bracket (“bra-ket”). Algebraically a “ket” is column vector in our state
space. This leaves us to define the set of “bra’s” $\langle\phi|$ on the dual
space of $\mathcal{H}$, $\mathcal{H^{\star}}$. The dual Hilbert space
$\mathcal{H^{\star}}$ is defined as the set of linear maps
$\mathcal{H}\rightarrow\textbf{C}$, given by
$\langle\phi|\>:\,\left|\psi\right\rangle\mapsto\langle\phi|\psi\rangle\in\textbf{C},$
(6)
where $\langle\phi|\psi\rangle$ is the inner product of the vectors
$\left|\psi\right\rangle,\left|\phi\right\rangle\in\mathcal{H}$. We can now
write down a more formal definition of a Hilbert space: It is a complex inner
product space with the following properties:
1. 1.
$\langle\phi|\psi\rangle=\langle\psi|\phi\rangle^{\dagger}$, where
$\langle\psi|\phi\rangle^{\dagger}$ is the complex conjugate of
$\langle\psi|\phi\rangle$.
2. 2.
The inner product$\langle\phi|\psi\rangle$ is linear in the first argument:
$\langle
a\phi_{1}+b\phi_{2}|\psi\rangle=a^{\dagger}\langle\phi_{1}|\psi\rangle+b^{\dagger}\langle\phi_{2}|\psi\rangle$.
3. 3.
$\langle\psi|\psi\rangle\geq 0$.
The space of a $n$ qubit system is spanned by a basis of $2^{n}$ orthogonal
vectors $\left|e_{i}\right\rangle$; one for each possible combination of the
basis-states of the individual qubits, obeying the orthogonality condition:
$\langle e_{i}|e_{j}\rangle=\delta_{ij},$ (7)
where $\delta_{ij}=1$ for $i=j$ and $\delta_{ij}=0$ for $i\neq j$. We say that
the Hilbert space of a composite system is the tensor products of the Hilbert
spaces of its parts. So the space of a $n$ qubit system is simply the tensor
product of the spaces of the $n$ qubits.
$\mathcal{H}_{\mathcal{Q}}=\mathcal{H}_{\mathcal{Q}_{n}}\otimes\mathcal{H}_{\mathcal{Q}_{n}-1}\otimes\mathcal{H}_{\mathcal{Q}_{n-2}}...\otimes\mathcal{H}_{\mathcal{Q}_{1}},$
(8)
where $\mathcal{Q}_{i}$ the quantum system $i$ is a vector in
$\textbf{C}^{2}$. A general $n$ qubit system can therefore be written
$\left|\psi\right\rangle=\sum_{x_{n},..,x_{1}=0}^{1}a_{x_{n}...x_{1}}\left|x_{n}\cdots
x_{1}\right\rangle,$ (9)
where
$\left|x_{n}\cdots
x_{1}\right\rangle=\left|x_{n}\right\rangle\otimes\left|x_{n-1}\right\rangle\otimes\cdots\otimes\left|x_{1}\right\rangle\in\mathcal{H}_{\mathcal{Q}}$
(10)
with $x_{i}\in\\{0,1\\}$ and complex coefficients $a_{x_{i}}$. For a two qubit
system,
$\left|x_{2}\right\rangle\otimes\left|x_{1}\right\rangle=\left|x_{2}\right\rangle\left|x_{1}\right\rangle=\left|x_{2}x_{1}\right\rangle$,
we have
$\left|\psi\right\rangle=\sum_{x_{2},x_{1}=0}^{1}a_{x_{2}x_{1}}\left|x_{2}x_{1}\right\rangle=a_{00}\left|00\right\rangle+a_{01}\left|01\right\rangle+a_{10}\left|10\right\rangle+a_{11}\left|11\right\rangle$
(11)
This state space is therefore spanned by four basis vectors:
$\left|00\right\rangle,\left|01\right\rangle,\left|10\right\rangle,\left|11\right\rangle,$
(12)
which are represented by the following 4-dimensional column vectors
respectively:
$\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
0\end{array}\right),\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
0\end{array}\right),\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\
0\end{array}\right),\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\
1\end{array}\right).$ (13)
Figure 2: The classical bit has only two distinct states, the probabilistic
bit can be in any normalized convex combination of those states, whereas the
quantum bit has a much richer state space.
#### Operators
A linear operator on a vector space $\mathcal{H}$ is a linear transformation
$\mathrm{T\,}:\mathcal{\>H}\rightarrow\mathcal{H}$, that maps vectors in
$\mathcal{H}$ to vectors in the same space $\mathcal{H}$. Quantum states are
normalized, and we wish to keep the normalization; we are therefore interested
in transformations that can be regarded as rotations in $\mathcal{H}$. Such
transformations are given by unitary operators $U$. An operator $U$ is called
unitary if $U^{-1}=U^{\dagger}$. They preserve inner products between vectors,
and thereby their norm. A _projection operator_ $P$ is Hermitian i.e.
$P=P^{\dagger}$ and satisfies $P^{2}=P$. We can create a projector $P$, by
taking the outer product of a vector with itself:
$P=\left|\phi\right\rangle\langle\phi|.$ (14)
$P$ is a matrix with every element $P_{ij}$ being the product of the elements
$i,j$ of the vectors in the outer product. This operator projects any vector
$\left|\gamma\right\rangle$ onto the 1-dimensional subspace of $\mathcal{H}$,
spanned by $\left|\phi\right\rangle$:
$P\left|\gamma\right\rangle=\left|\phi\right\rangle\langle\phi|\left|\gamma\right\rangle=\langle\phi|\gamma\rangle\left|\phi\right\rangle.$
(15)
It simply gives the portion of $\left|\gamma\right\rangle$ along
$\left|\phi\right\rangle$.
We will often deal with unitary operators $U\in\mathrm{SU(2)}$, i.e operators
from the special unitary group of dimension 2. The group consists of $2\times
2$ unitary matrices with determinant 1. These matrices will be operating on
single qubits (often in systems of 2 or more qubits). The generators of the
group are the Pauli spin matrices $\sigma_{x},\,\sigma_{y},\,\sigma_{z}$,
shown together with the identity matrix $I$:
$I=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\;\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\;\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\;\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (16)
Note that $\sigma_{x}$ is identical to a classical (bit-flip) ’NOT’-operation.
General $2\times 2$ unitary operators can be parameterized with three
parameters $\theta,\alpha,\beta$, as follows:
$U\left(\theta,\alpha,\beta\right)=\left(\begin{array}[]{cc}e^{i\alpha}\cos\left(\theta/2\right)&ie^{i\beta}\sin\left(\theta/2\right)\\\
ie^{-i\beta}\sin\left(\theta/2\right)&e^{-i\alpha}\cos\left(\theta/2\right)\end{array}\right).$
(17)
An operation is said to be local if it only affects a part of a composite
(multi-qubit) system. Connecting this to the concept of the bit-strings in the
previous section; a local operation translates to just controlling one such
bit. This is a crucial point in the case of modeling the effect of individual
actions, since each agent in a strategic situation is naturally constrained to
decisions regarding their own choices. The action of a set of local operations
on a composite system is given by the tensor product of the local operators.
For a general n-qubit $\left|\psi\right\rangle$ as given in (9) and (10) we
get:
$U_{n}\otimes U_{n-1}\otimes\cdots\otimes
U_{1}\left|\psi\right\rangle=\sum_{x_{n},..,x_{1}=0}^{1}a_{x_{n}...x_{1}}U_{n}\left|x_{n}\right\rangle\otimes
U_{n-1}\left|x_{n-1}\right\rangle\otimes\cdots\otimes
U_{1}\left|x_{1}\right\rangle.$ (18)
#### Mixed states and the density operator
We have so far only discussed _pure states_ , but sometimes we encounter
quantum states without a definite state vector $\left|\psi\right\rangle$,
these are called _mixed states_ and consists of a states that has certain
probabilities of being in some number of different pure states. So for example
a state that is in
$\left|\psi_{1}\right\rangle=a_{0}^{1}\left|0\right\rangle+a_{1}^{1}\left|1\right\rangle$
with probability $p_{1}$ and in
$\left|\psi_{2}\right\rangle=a_{0}^{2}\left|0\right\rangle+a_{1}^{2}\left|1\right\rangle$
with probability $p_{2}$ is mixed. We handle mixed states by defining a
density operator $\rho$, which is a hermitian matrix with unit trace:
$\rho=\sum_{i}p_{i}\left|\psi_{i}\right\rangle\langle\psi_{i}|,$ (19)
where $\sum_{i}p_{i}=1$. A pure state in this representation is simply a state
for which all probabilities, except one is zero. If we apply a unitary
operator $U$ on a pure state, we end up with $U\left|\psi\right\rangle$ which
has the density operator $U\rho
U^{\dagger}=U\left|\psi\right\rangle\langle\psi|U^{\dagger}$. Regardless if we
are dealing with pure or mixed states, we take the expectation value of upon
measurement ending up in a $\left|\phi\right\rangle$ by calculating
$\mathrm{Tr}\left(\left|\phi\right\rangle\langle\phi|\rho\right)$, where
$|\phi\rangle\langle\phi|$ is a so called projector. For calculating the
expectation values of a state to be in _any_ of a number of states
$|\phi_{i}\rangle$, we construct a projection operator
$P=\sum_{i}|\phi_{i}\rangle\langle\phi_{i}|$ and take the trace over $P$
multiplied by $\rho$.
#### Entanglement
Entanglement is the resource our game-playing agents will make use of in the
quantum game protocols to achieve better than classical performance. Non-
classical correlations are thus introduced, by which the players can
synchronize their behavior without any additional communication. An entangled
state is basically a quantum system that _cannot_ be written as a tensor
product of its subsystems, we’ll thus define two classes of quantum states.
Examples below refers to two-qubit states.
Product states:
$|\Psi_{\mathcal{Q}}\rangle=|\Psi_{\mathcal{Q}_{2}}\rangle\otimes|\Psi_{\mathcal{Q}_{1}}\rangle,~{}~{}\textrm{or
using density
matrix}~{}~{}\rho_{\mathcal{Q}}=\rho_{\mathcal{Q}_{2}}\otimes\rho_{\mathcal{Q}_{1}},$
(20)
and entangled states
$|\Psi_{\mathcal{Q}}\rangle\neq|\Psi_{\mathcal{Q}_{2}}\rangle\otimes|\Psi_{\mathcal{Q}_{1}}\rangle,~{}~{}\textrm{or
using density matrix
}~{}~{}\rho_{\mathcal{Q}}\neq\rho_{\mathcal{Q}_{2}}\otimes\rho_{\mathcal{Q}_{1}}.$
(21)
For a mixed state, the density matrix is defined as mentioned by
$\rho_{\mathcal{Q}}=\sum^{N}_{i=1}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ and
it is said to be separable, which we will denote by
$\rho^{sep}_{\mathcal{Q}}$, if it can be written as
$\rho^{sep}_{\mathcal{Q}}=\sum_{i}p_{i}(\rho^{i}_{\mathcal{Q}_{2}}\otimes\rho^{i}_{\mathcal{Q}_{1}}),~{}\sum_{i}p_{i}=1.$
(22)
A set of very important two-qubit entangled states are the Bell states
$|\Phi^{\pm}_{\mathcal{Q}}\rangle=\frac{1}{\sqrt{2}}(|{00}\rangle\pm|11\rangle),~{}~{}~{}~{}|\Psi^{\pm}_{\mathcal{Q}}\rangle=\frac{1}{\sqrt{2}}(|{01}\rangle\pm|10\rangle).$
(23)
The GHZ-type-states
$|\textrm{GHZ}_{n}\rangle=\frac{1}{\sqrt{2}}\left(|00\cdots
0\rangle+e^{i\phi}|11\cdots 1\rangle\right)$ (24)
could be seen as a $n$-qubit generalization of
$|\Phi^{\pm}_{\mathcal{Q}}\rangle$-states.
### 1.3 Classical Games
It is instructive to review the theory of classical games and some major
solution concepts before moving on to examples of quantum games. We’ll start
by defining classical pure and mixed strategy games, and then move on to
introducing some relevant solution concepts and finish off with a definition
of quantum games.
A game is a formal model over the interactions between a number of agents
(_agents, players, participants_ , and _decision makers_ may be used
interchangeably) under some specified sets of choices (_choices, strategies,
actions_ and _moves_ , may be used interchangeably). Each combination of
choices made, or strategies chosen by the different players leads to an
outcome with some certain level of desirability for each of them. The level of
desirability is measured by assigning a real number, a so called _payoff_ $\$$
for each game outcome for each player. Assuming rational players, each will
choose actions that maximizes their expected payoff $E(\$)$, i.e. in an
deterministic as well as in an probabilistic setting acting in a way that,
based on the known information about the situation, maximizes the expectation
value of their payoff. The structure of the game is fully specified by the
relations between the different combinations of strategies and the payoffs
received by the players. A key point is the interdependence of the payoffs
with the strategies chosen by the other players. A situation where the payoff
of one player is independent of the strategies of the others would be of
little interest from a game theoretical point of view. It is natural to extend
the notion of payoffs to _payoff functions_ whose arguments are the chosen
strategies of all players and ranges are the real valued outputs that assigns
a level of desirability for each player to each outcome.
##### Pure strategy classical game
We have a set of $n$ players $\\{1,2,...,n\\}$, $n$ strategy sets $S_{i}$, one
for each player $i$, with $s_{i}^{j}\in S_{i}$, where $s_{i}^{j}$ is the
$j$:th strategy of player $i$. The strategy space $S=S_{1}\times
S_{2}\times\cdots\times S_{n}$ contains all $n$-tuples pure strategies, one
from each set. The elements $\sigma\in S$ are called strategy profiles, some
of which will earn them the status of being a _solution_ with regards to some
solution concept.
We define a game by its payoff-functions $\$_{i}$, where each is a mapping
from the strategy space $S$ to a real number, the payoff or utility of player
$i$. We have:
$\$_{i}:S_{1}\times S_{2}\times\cdots\times S_{n}\rightarrow\textbf{R}.$ (25)
##### Mixed strategy classical game
Let $\Delta(S_{i})$ be the set of convex linear combinations of the elements
$s_{i}^{j}\in S_{i}$. A mixed strategy $s^{mix}_{i}\in\Delta(S_{i})$ is then
given by:
$\sum_{s^{j}_{i}\in
S_{i}}p_{i}^{j}s^{j}_{i}\;\;\;\textrm{with}\;\;\;\sum_{j}p_{i}^{j}=1,$ (26)
where $p^{j}_{i}$ is the probability player $i$ assigns to the choice
$s^{j}_{i}$. The space of mixed strategies
$\Delta(S)=\Delta(S_{1})\times\Delta(S_{2})\times\cdots\times\Delta(S_{n})$
contains all possible mixed strategy profiles $\sigma_{mix}$. We now have:
$\$_{i}:\Delta(S_{1})\times\Delta(S_{2})\times\cdots\times\Delta(S_{n})\rightarrow\textbf{R}.$
(27)
Note that the pure strategy games are fully confined within the definition of
mixed strategy games and can be accessed by assigning all strategies except
one, the probability $p^{j}=0$. This class of games could be formalized in a
framework using probabilistic information units, such as the probabilistic
bit.
### 1.4 Solution concepts
We will introduce two of many game theoretical solution concepts. A solution
concept is a strategy profile $\sigma^{*}\in S$, that has some particular
properties of strategic interest. It could be a strategy profile that one
would expect a group of rational self-maximizing agents to arrive at in their
attempt to maximize their minimum expected payoff. Strategy profiles of this
form i.e. those that leads to a combination of choices where each choice is
the best possible response to any possible choice made by other players tend
to lead to an equilibrium, and are good predictors of game outcomes in
strategic situations. To see how such equilibria can occur we’ll need to
develop the concept of _dominant strategies_.
###### Definition 1
(Strategic dominance): A strategy $s^{dom}\in S_{i}$ is said to be dominant
for player $i$, if for any strategy profile $\sigma_{-i}\in S/S_{i}$, and any
other strategy $s^{j}\neq s^{dom}\in S_{i}$:
$\$_{i}(s^{dom},\sigma_{-i})\geq\$_{i}(s^{j},\sigma_{-i})\;\;\textrm{for
all}\;\;i=1,2,\cdots,n.$ (28)
Lets look at a simple example. Say that we have two players, Alice with legal
strategies $s^{1}_{Alice},s^{2}_{Alice}\in S_{Alice}$ and Bob with
$s^{1}_{Bob},s^{2}_{Bob}\in S_{Bob}$. Now, if the payoff Alice receives when
playing $s^{1}_{Alice}$ against any of Bob’s two strategies is higher than (or
at least as high as) what she receives by playing $s^{2}_{Alice}$, then
$s^{1}_{Alice}$ is her dominant strategy. Her payoff can of course vary
depending on Bob’s move but regardless what Bob does, her dominant strategy is
the _best response_. Now there is no guarantee that such dominant strategy
exists in a pure strategy game, and often must the strategy space be expanded
to accommodate for mixed strategies for them to exist.
If both Alice and Bob has a dominant strategy, then this strategy profile
becomes a _Nash Equilibrium_ , i.e. a combination of strategies for which none
of them can gain by unilaterally deviating from. The Nash equilibrium profile
acts as an attractor in the strategy space and forces the players into it,
even though it is not always an optimal solution. Combinations can exist that
can lead to better outcomes for both (all) players.
|
arxiv-papers
| 2012-04-03T11:32:28 |
2024-09-04T02:49:29.324662
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Puya Sharif and Hoshang Heydari",
"submitter": "Puya Sharif",
"url": "https://arxiv.org/abs/1204.0661"
}
|
1204.0743
|
# Resistive and magnetized accretion flows with convection
Kazem Faghei and Mobina Omidvand
###### Abstract
We considered the effects of convection on the radiatively inefficient
accretion flows (RIAF) in the presence of resistivity and toroidal magnetic
field. We discussed the effects of convection on transports of angular
momentum and energy. We established two cases for the resistive and magnetized
RIAFs with convection: assuming the convection parameter as a free parameter
and using mixing-length theory to calculate convection parameter. A self-
similar method was used to solve the integrated equations that govern the
behavior of the presented model. The solutions showed that the accretion and
rotational velocities decrease by adding the convection parameter, while the
sound speed increases. Moreover, by using mixing-length theory to calculate
convection parameter, we found that the convection can be important in RIAFs
with magnetic field and resistivity.
00footnotetext: School of Physics, Damghan University, Damghan, Iran
e-mail: kfaghei@du.ac.ir
Keywords accretion, accretion discs, convection, magnetohydrodynamics: MHD
## 1 Introduction
The existence of radiatively inefficient accretion flows (RIAFs) have been
confirmed in low-luminosity state of X-ray binaries and nuclei of galaxies
(Narayan et al. 1996; Esin et al. 1997; Di Matteo et al. 2003; Yuan et al.
2003). It was understood that RIAFs are likely to be convectively unstable in
the radial direction due to the inward increase of the entropy of accreting
gas (Narayan & Yi 1994). Moreover, hydrodynamical and magetohydrodynamical
simulations of low-viscosity RIAFs have confirmed these flows are convectively
unstable ( e. g. Igumenshchev et al. 1996; Stone et al. 1999; Machida et al.
2001; Hawley & Balbus 2002; McKinney & Gammie 2002; Igumenshchev et al. 2003).
Self-similar or global solutions for convection-dominated accretion flows
(CDAFs) were presented by several authors (e. g. Narayan et al. 2000; Quataert
& Gruzinov 2000; Abramowicz et al. 2002; Lu et al. 2004; Zhang & Dai 2008).
Igumenshchev et al. (2003) studied the resistive MHD simulations of RIAFs onto
black holes. They assumed two cases for the geometry of the injected magnetic
field: pure toroidal field and pure poloidal field. They found that in the
case of pure toroidal magnetic field, the accreting gas forms a nearly
axisymmetric, geometrically thick, turbulent accretion disc. Moreover, their
solutions represented that the flow resembles in many respects CDAFs found in
previous numerical and analytical investigations of viscous hydrodynamic
flows. Zhang & Dai (2008) investigated the effect of magnetic field on RIAFs
with convection by a semi-analytically method. By exploit of
$\alpha$-prescription for viscosity and convection, they used two methods to
study of magnetized flows with convection, i.e. they take the convective
coefficient $\alpha_{c}$ as a free parameter to discuss the effects of
convection for simplicity. They also established the $\alpha_{c}$-$\alpha$
relation for magnetized flows using the mixing-length theory and compare this
relation with the non-magnetized case. They found that the magnetic field
makes the $\alpha_{c}$-$\alpha$ relation be distinct from that of non-
magnetized flows.
Since the importance of toroidal magnetic field and resistivity in accretion
flows have been confirmed observationally (see Faghei 2011 and references
therein), Faghei (2011) considered the steady, radially self-similar solutions
of accretion flows in the presence of the toroidal magnetic field and the
resistivity. However, he ignored the effects of convection in his model.
Generally semi-analytical studies of magnetized CDAFs are related to non-
resistive magnetized CDAFs (e. g. Zhang & Dai 2008) and the resistive and
magnetized CDAF was studied in MHD simulations (e. g. Igumenshchev & Narayan
2002; Hawley & Balbus 2002; Igumenshchev et al. 2003). Thus, it will be
interesting to study the effects of resistivity on RIAFs with convection.
Here, we adopt the presented solutions by Narayan et al. (2000) and Faghei
(2011). Similar to Narayan et al. (2000), we will discuss the effects of
convection on angular momentum and energy equations. The paper is organized as
follow. In section 2, the basic equations of constructing a model for quasi-
spherical magnetized RIAFs with convection will be defined. In section 3, a
self-similar method for solving equations which govern the behavior of the
accreting gas was utilized. The summary of the model will appear in section 4.
## 2 Basic Equations
Analytical theory of CDAF is based on a self-similar solution of a simplified
set of equations describing RIAFs. We adopted the presented solutions by
Narayan et al. (2000) and Faghei (2011). By using spherical coordinate ($r$,
$\theta$, $\varphi$) centered on a accreting object, let us consider
stationary, axisymmetric, quasi-spherical equations describing an accretion
flow onto the black hole of mass $M$. For the sake of simplicity, the general-
relativistic effect has been neglected and the gravitational force on a fluid
is characterized by Newtonian potential of a point mass, $\psi=-GM/r$. As
magnetic fields, we consider only toroidal fields, $B_{\varphi}$.
Under these assumptions, the continuity equation with mass loss is
$\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\rho v_{r})=\dot{\rho},$ (1)
where $\rho$, $v_{r}$ and $\dot{\rho}$ are the density, the accretion velocity
($v_{r}<0$), and the mass-loss per unit volume, respectively.
The radial momentum equation is
$v_{r}\frac{dv_{r}}{dr}=r\left(\Omega^{2}-\Omega^{2}_{K}\right)-\frac{1}{\rho}\frac{d}{dr}(\rho
c^{2}_{s})-\frac{c^{2}_{A}}{r}-\frac{1}{2\rho}\frac{d}{dr}(\rho c^{2}_{A}),$
(2)
where $c_{s}$ is sound speed, which is defined as $c_{s}^{2}\equiv
p_{gas}/\rho$, with being $p_{gas}$ as the gas pressure, $\Omega$ is the
angular velocity, $\Omega_{K}[=\left(GM/r^{3}\right)^{1/2}]$ is the Keplerian
angular velocity, and $c_{A}$ is the alfven speed, which is defined as
$c_{A}^{2}\equiv B_{\varphi}^{2}/4\pi\rho=2p_{mag}/\rho$, with being $p_{mag}$
as the magnetic pressure. The ram-pressure term $v_{r}dv_{r}/dr$ and last two
terms due to the magnetic field in this equation were ignored in the self-
similar CDAF model of Narayan et al. (2000), while we include them here in
order to consider their effects.
The angular momentum equations can be written in the form of the balance of
advection and diffusion transport terms (Narayan et al. 2000),
$\displaystyle\rho
v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r^{2}}\frac{d}{dr}\left[\nu\rho
r^{4}\frac{d\Omega}{\partial r}\right]+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}\left[\nu_{c}\rho
r^{(5+3g)/2}\frac{d}{dr}\left(\Omega r^{3(1-g)/2}\right)\right],$ (3)
where the two terms of right hand side represent the angular momentum
transport by viscosity and convection. Here, $\nu$ is the kinematic viscosity
coefficient, $\nu_{c}$ is the convective diffusion coefficient, and $g$ is the
parameter to determine the condition of convective angular momentum transport.
When $g=1$, the flux of angular momentum due to convection is
$\dot{J}_{c}=-\nu_{c}\,\rho\,r^{4}\,\frac{d\Omega}{dr}.$ (4)
The above equation implies that the convective angular momentum flux is
oriented down the angular velocity gradient. For a quasi-Keplerian angular
velocity, $\Omega\propto r^{-3/2}$, angular momentum is transported outward.
When $g=-1/3$, the convective angular momentum flux can be written as
$\dot{J}_{c}=-\nu_{c}\,\rho\,r^{2}\,\frac{d\left(\Omega r^{2}\right)}{dr}.$
(5)
This equation represents that the convective angular momentum flux is oriented
down the specific angular momentum gradient. For a quasi-Keplerian angular
velocity, $\Omega\propto r^{-3/2}$, angular momentum is transported inward.
Generally, convection transports angular momentum inward (or outward) for
$g<0$ (or $>0$), and the specific case $g=0$ corresponds to zero angular
momentum transport (Narayan et al. 2000).
In this paper, we assume the kinematic coefficient of viscosity and the
magnetic diffusivity due to turbulence in the accretion flow. So, we use these
parameters in analogy to the $\alpha$-prescription of Shakura & Sunyaev (1973)
for the turbulent,
$\nu=P_{m}\eta=\alpha\frac{c_{s}^{2}}{\Omega_{K}},$ (6)
where $P_{m}$ is the magnetic Prandtl number of the turbulence, which assumed
to be a constant less than unity, $\eta$ is the magnetic diffusivity, and
$\alpha$ is a free parameter less than unity. For the convective diffusion
coefficient, $\nu_{c}$, we adopt the assumptions of Narayan et al. (2000) and
Lu et al. (2004) that all transport phenomena due to convection have the same
diffusion coefficient, which is defined as
$\nu_{c}=\left(\frac{L_{M}^{2}}{4}\right)\sqrt{-N_{eff}^{2}},$ (7)
where $L_{M}$ is the characteristic mixing length and $N_{eff}$ is the
effective frequency of convective blobs. The characteristic mixing length
$L_{M}$ in terms of the pressure scale height, $H_{p}$, can be written as
$L_{M}=2^{-1/4}l_{M}H_{p},~{}~{}~{}~{}H_{p}=-\frac{dr}{d\ln p_{gas}},$ (8)
where $l_{M}$ is the dimensionless mixing-length parameter and its amount is
estimated to be equal to $\sqrt{2}$ in ADAFs (Narayan et al. 2000; Lu et al.
2004). the effective frequency of convective blobs, $N_{eff}$, is given by
$N_{eff}^{2}=N^{2}+\kappa^{2},$ (9)
where $N$ is Brunt-Väisälä frequency, which is defined as
$N^{2}=-{1\over\rho}{dp_{gas}\over dr}{d\over
dr}\ln\left({p_{gas}^{1/\gamma}\over\rho}\right),$ (10)
and $\kappa$ is epicyclic frequency, which is defined as
$\kappa^{2}=2\Omega^{2}\frac{d\ln(\Omega r^{2})}{d\ln r}.$ (11)
For a non-Keplerian flows $\kappa\neq\Omega$, while for a quasi-Keplerian
($\Omega\propto r^{-3/2}$), $\kappa=\Omega$ (Narayan et al. 2000; Lu et al.
2004). Convection appears in flows with $N_{eff}^{2}<0$. We also write the
convective diffusion coefficient in the form similar to usual viscosity of
Shakura & Sunyaev (1973),
$\nu_{c}=\alpha_{c}\frac{c_{s}^{2}}{\Omega_{K}}$ (12)
where $\alpha_{c}$ is a dimensionless coefficient that describes the strength
of convective diffusion. The $\alpha_{c}$ coefficient can be obtained by
equations (8) and (13)
$\alpha_{c}=\frac{\Omega_{K}}{c_{s}^{2}}\left(\frac{L_{M}^{2}}{4}\right)\sqrt{-N_{eff}^{2}}.$
(13)
The energy equation is
$\displaystyle\rho v_{r}T\frac{ds}{dr}\equiv\rho
v_{r}\left[\frac{1}{\gamma-1}\frac{dc_{s}^{2}}{dr}-\frac{c_{s}^{2}}{\rho}\frac{d\rho}{dr}\right]=$
$\displaystyle Q_{diss}+Q_{conv}-Q_{rad},$ (14)
where $T$ is the temperature, $s$ is the specific entropy, $\gamma$ is the
ratio of specific heats, $Q_{diss}$ is dissipative heating rate, $Q_{rad}$ is
the radiative cooling rate, and
$Q_{conv}=-\mathbf{\nabla}\cdot\mathbf{F}_{conv}$, with being
$F_{conv}[=-\rho\nu_{c}Tds/dr]$ as the outward energy flux due to convection.
For the right hand side of the energy equation, we can write
$\displaystyle
Q_{adv}=fQ_{diss}-\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}F_{conv}\right),$ (15)
where $Q_{adv}$ is the advective transport of energy, and
$f[=1-Q_{rad}/Q_{diss}]$ is the advection parameter. The parameter $f$
measures the degree to which the flow is advection-dominated (Narayan & Yi
1994). The dissipative heating rate can be written as
$Q_{diss}=(\nu+g\nu_{c})\rho r^{2}\left(\frac{\partial\Omega}{\partial
r}\right)^{2}+\frac{\eta}{4\pi}{\bf J}^{2},$ (16)
where the right-hand side terms are heating rate due to viscosity, convection,
and resistivity, respectively. In above equation,
${\mathbf{J}}[=\nabla\times{\mathbf{B}}]$ is the current density, with being
${\mathbf{B}}$ as the magnetic field.
Finally, the induction equation with creation/escape of magnetic field can be
written as
$\frac{1}{r}\frac{d}{dr}\left[rv_{r}B_{\varphi}-\eta\frac{d}{dr}(rB_{\varphi})\right]=\dot{B}_{\varphi}.$
(17)
where $B_{\varphi}$ is the toroidal component of magnetic field and
$\dot{B}_{\varphi}$ is the field escaping/creating rate due to a magnetic
instability or dynamo effect. This induction equation is rewritten as
$\displaystyle\dot{B}_{\varphi}=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{1}{r}\frac{d}{dr}\left[\sqrt{4\pi\rho
c^{2}_{A}}\left(rv_{r}-\frac{\alpha}{4\chi
P_{m}}\frac{1}{r\rho\Omega_{K}}\frac{d}{dr}(r^{2}\rho
c^{2}_{A})\right)\right],$ (18)
where $\chi$ is the ratio of the magnetic pressure to the gas pressure, which
is defined by
$\chi=\frac{p_{mag}}{p_{gas}}=\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2}.$
(19)
## 3 Self-Similar Solutions
We seek self-similar solutions in the following form (e.g. Narayan & Yi 1994;
Akizuki & Fukue 2006)
$v_{r}(r)=-c_{1}\alpha\sqrt{\frac{GM_{*}}{r}}$ (20)
$\Omega(r)=c_{2}\sqrt{\frac{GM_{*}}{r^{3}}}$ (21)
$c^{2}_{s}(r)=c_{3}\frac{GM_{*}}{r}$ (22)
$c^{2}_{A}(r)=\frac{B^{2}_{\varphi}}{4\pi\rho}=2\chi c_{3}\frac{GM_{*}}{r}$
(23)
where $c_{1}$, $c_{2}$, and $c_{3}$ are dimensionless constant to be
determined. We use a power-law relation for density
$\rho(r)=\rho_{0}r^{\lambda},$ (24)
where $\rho_{0}$ and $\lambda$ are constant. Using equations (20)-(24), the
mass-loss rate and the magnetic field escaping/creating rate can be written as
$\dot{\rho}(r)=\dot{\rho}_{0}r^{\lambda-3/2},$ (25)
$\dot{B}_{\varphi}(r)=\dot{B}_{0}r^{\frac{\lambda-4}{2}},$ (26)
where $\dot{\rho}_{0}$ and $\dot{B}_{0}$ are constant. Since we have not
applied the effects of wind in the momentum and energy equations, we will
assume a no wind case, $\dot{\rho}=0$ and $\lambda=-3/2$. In this case,
$\dot{B}_{\varphi}\propto r^{-11/4}$, which implies that creation/escape of
magnetic field increases with approaching to central object. This property is
qualitatively consistent with previous studies of accretion flows (Machida et
al. 2006; Oda et al. 2007; Faghei & Mollatayefeh 2012).
Using the self-similar solutions in the continuity, radial momentum, angular
momentum, convection parameter, energy, and induction equations [(1)-(3),
(13), (14), and (18)], we can obtain the following relations:
$\dot{\rho}_{0}=-\left(\lambda+\frac{3}{2}\right)\alpha\rho_{0}c_{1}\sqrt{GM_{*}},$
(27)
$-\frac{1}{2}c^{2}_{1}\alpha^{2}+1-c^{2}_{2}+c_{3}\left[\lambda-1+\chi(1+\lambda)\right]=0,$
(28) $\alpha c_{1}=3(\alpha+g\alpha_{c})(\lambda+2)c_{3},$ (29)
$\displaystyle\alpha
c_{1}\left[\frac{1}{\gamma-1}+\lambda\right]=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{9}{4}\alpha
f\left[(1+\frac{\alpha_{c}}{\alpha}g)c_{2}^{2}+\frac{2\chi}{9P_{m}}c_{3}(1+\lambda)^{2}\right]$
$\displaystyle-\alpha_{c}c_{3}(\lambda+\frac{1}{2})\left[\frac{1}{\gamma-1}+\lambda\right]$
(30)
$\alpha_{c}=\frac{l_{M}^{2}}{4\sqrt{2}c_{3}(\lambda-1)^{2}}\sqrt{\frac{c_{3}(\lambda-1)}{\gamma}[\lambda(1-\gamma)-1]-c_{2}^{2}},$
(31) $\dot{B}_{0}=-\frac{\alpha\lambda}{2}GM_{*}\sqrt{2\pi\rho_{0}\chi
c_{3}}\left[2c_{1}+\frac{c_{3}}{P_{m}}(1+\lambda)\right].$ (32)
Above equations express for $\lambda=-3/2$, there is no mass loss, while for
$\lambda>-3/2$ mass loss (wind) exists.
## 4 Results
Here, similar to Zhang & Dai (2008), we will study the presence of convection
in two cases: $\alpha_{c}$ as a free parameter and $\alpha_{c}$ as a variable.
Fig. 1 : Physical variables as functions of $\chi$ for several values of
convective viscosity. The input parameters are set to $\alpha=0.2$,
$\gamma=1.5$, $P_{m}=1/2$, $f=1$, $l=\sqrt{2}$, $g=-1/3$, and $\lambda=-3/2$.
The solid, dashed, and dotted lines represent $\alpha_{c}=0$, $0.05$, and
$0.1$, respectively.
Fig. 2 : Same as Figure 1, but $\alpha_{c}=0.1$, and the solid, dashed, and
dotted lines represent $P_{m}=\infty$, $1.0$, and $0.5$, respectively.
Fig. 3 : Physical variables as functions of $\chi$ for several values of
magnetic Prandtl number. The input parameters are set to $\alpha=0.5$,
$\gamma=1.5$, $f=1$, $l=\sqrt{2}$, $g=-1/3$, and $\lambda=-3/2$. The solid,
dashed, and dotted lines represent $P_{m}=\infty$, $1.0$, and $0.5$,
respectively.
### 4.1 Case 1: $\alpha_{c}$ as a free parameter
In this case, we take the convective coefficient $\alpha_{c}$ as a free
parameter to discuss the effects of convection for simplicity. Examples of
such solutions are presented in Figures 1 and 2.
In Figure 1, the self-similar coefficients $c_{1}$, $c_{2}$, and $c_{3}$ are
shown as functions of the parameter $\chi$. By adding the parameter $\chi$
which indicates the role of magnetic filed on the dynamics of accretion discs,
we see the coefficients of radial and rotational velocities and sound speed
decrease. This properties are qualitatively consistent with results of Faghei
(2011). In Figure 1, we also studied the effect of convection parameter
$\alpha_{c}$ on the physical variables. The value of $\alpha_{c}$ measures the
strength of convective viscosity and a larger $\alpha_{c}$ denotes a stronger
turbulence due to convection. Figure 1 implies that for non-zero $\alpha_{c}$,
the radial infall velocity is lower than the standard ADAF solution and for
larger $\alpha_{c}$ this reduction of radial infall velocity is more evident.
It can be due to decrease of efficiency of angular momentum transport by
adding the convection parameter $\alpha_{c}$ (see equation 29). The profiles
of angular velocity show that it decreases with the magnitude of $\alpha_{c}$,
while the sound speed increases. These properties are in accord with results
of Zhang & Dai (2008).
In Figure 2, the physical variables are shown as functions of parameter $\chi$
for several values of magnetic Prandtl number. Since inverse of magnetic
Prandtl number is proportional to magnetic diffusivity,
$P_{m}\propto\eta^{-1}$. Thus, reduce of magnetic Prandtl number denotes to
increase of resistivity of the fluid. The solutions in Figure 2 imply that the
accretion velocity and the sound speed both increase with the magnitude of
resistivity, while the rotational velocity decreases. These properties
qualitatively confirm the results of Faghei (2011).
### 4.2 Case 2: $\alpha_{c}$ as a variable
Here, we calculate the dimensionless coefficient $\alpha_{c}$ by using the
mixing-length theory. Because we used a steady self-similar method to derive
$\alpha_{c}$, it becomes a constant throughout of the accreting gas. However,
it is a function of position and time (e. g. Lu et al. 2004). The amount of
convection parameter $\alpha_{c}$ is calculated by equation (31). Using this
equation and equation (28)-(30), we can obtain the behavior of physical
quantities in the presence of convection. Such solutions are shown in Figure
3.
In Figure 3, the coefficients $c_{1}$, $c_{2}$, $c_{3}$, and convection
parameter $\alpha_{c}$ are shown as functions of the degree of magnetic
pressure. Similar to case 1, the accretion and rotational velocities, and
sound speed decrease by adding the parameter $\chi$. While, the convection
parameter $\alpha_{c}$ increases for stronger toroidal magnetic field. This
property is qualitatively consistent with result of Zhang & Dai (2008). In
Figure 3, the physical variables are also studied for several values of
magnetic Prandtl number. The profiles of convection parameter $\alpha_{c}$
imply that it increases by adding the magnetic diffusivity. As for non-zero
magnetic diffusivity, $\alpha_{c}$ is larger than the standard CDAF solution
and for larger magnetic diffusivity this increase of convection parameter
$\alpha_{c}$ is more evident.
## 5 Summary and Discussion
The observational features of low-luminosity state of X-ray binaries and
nuclei of galaxies can be successfully explained by the models of radiatively
inefficient accretion flow (RIAF). The importance of convection in RIAFs was
realized by semi-analytical and direct numerical simulation (e. g. Narayan et
al. 2000; Igumenshchev et al. 2003).
In this research, we considered the effects of convection on the presented
model of Faghei (2011). Similar to Narayan et al. (2000), we assumed the
convection affects on transports of angular momentum and energy. Using a
radially self-similar approach, we studied the effects of convection on the
model for several values of magnetic field and resistivity. The solutions
showed that the accretion and rotational velocities, and sound speed decrease
for stronger magnetic filed. Moreover, we found that the accretion velocity
and sound speed increase with the magnitude of the resistivity, while the
rotational velocity decreased. These properties are qualitatively consistent
with results of Faghei (2011). We studied the effects of convection on a
resistive and magnetized RIAF in two cases: assuming the convection parameter
as a free parameter and using mixing length theory to calculate the convection
parameter. In the first case, we found that by adding the convection
parameter, the radial and rotational velocities decrease and the sound speed
increases. In the second case, we found that the convection parameter
increases by adding the magnetic filed and resistivity. These properties are
in many aspects in accord with results of Zhang & Dai (2008).
The present model have some limitations that can be modified in the future
works. For example, the latitudinal dependence of physical variables have been
ignored in this paper. While, two-dimensional and three-dimensional MHD
simulations of RIAFs show that the disc geometry strongly depends on magnetic
field configuration (e. g. Igumenshchev et al. 2003). Thus, the study of
present model in two/three dimensions can be an interesting subject for future
research. Moreover, it has been understood the magnetic field can change the
criterion for convective instability (e. g. Balbus & Hawley 2002). While, we
igonred the effects of magnetic field on the instability criterion. Thus, the
presented criterion in this paper can be modified in the future research.
## Acknowledgements
I wish to thank the anonymous referee for very useful comments that helped us
to improve the initial version of the paper.
## References
* (1) Abramowicz M. A., Igumenshchev I. V., Quataert E., Narayan R., 2002, ApJ, 565, 1101
* (2)
* (3) Akizuki C., Fukue J., 2006, PASJ, 58, 469
* (4)
* (5) Balbus S. A., Hawley J. F., 2002, ApJ, 573, 749
* (6)
* (7) Di Matteo T., Allen S. W., Fabian A. C., Wilson A. S., Young A. J., 2003, ApJ, 582, 133
* (8)
* (9) Esin A. A., McClintock J. E., Narayan R., 1997, ApJ, 489, 867
* (10)
* (11) Faghei K., 2011, JA&A, arXiv:1111.7302
* (12)
* (13) Faghei K., Mollatayefeh A., 2012, MNRAS,
* (14) doi:10.1111/j.1365-2966.2012.20645.x
* (15)
* (16) Hawley J. F., Balbus S. A., 2002, ApJ, 573, 738
* (17)
* (18) Igumenshchev I. V., Chen X., Abramowicz M. A., 1996, MNRAS, 278, 236
* (19)
* (20) Igumenshchev I. V., Narayan R., 2002, ApJ, 566, 137
* (21)
* (22) Igumenshchev I. V., Narayan R., Abramowicz M. A., 2003, ApJ, 592, 1042
* (23)
* (24) Lu J.-F., Li S.-L., Gu W.-M., 2004, MNRAS, 352, 147
* (25)
* (26) Machida M., Matsumoto R., Mineshige S., 2001, PASJ, 53, L1
* (27)
* (28) Machida M., Nakamura K. E., Matsumoto R., 2006, PASJ, 58, 193
* (29)
* (30) McKinney J. C., Gammie C. F., 2002, ApJ, 573, 728
* (31)
* (32) Narayan R., Igumenshchev I. V., Abramowicz M. A., 2000, ApJ, 539, 798
* (33)
* (34) Narayan R., McClintock J. E., Yi I., 1996, ApJ, 457, 821
* (35)
* (36) Narayan R., Yi I., 1994, ApJ, 428, L13
* (37)
* (38) Oda H., Machida M., Nakamura K. E., Matsumoto R., 2007, PASJ, 59, 457
* (39)
* (40) Quataert E., Gruzinov A., 2000, ApJ, 539, 809
* (41)
* (42) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
* (43)
* (44) Stone J. M., Pringle J. E., Begelmen M. C., 1999, MNRAS, 310, 1002
* (45)
* (46) Yuan F., Quataert E., Narayan R., 2003, ApJ, 598, 301
* (47)
* (48) Zhang D., Dai Z. G., 2008, MNRAS, 388, 1409
* (49)
|
arxiv-papers
| 2012-04-03T17:32:22 |
2024-09-04T02:49:29.331289
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kazem Faghei, Mobina Omidvand",
"submitter": "Kazem Faghei",
"url": "https://arxiv.org/abs/1204.0743"
}
|
1204.0774
|
# A Dark Energy Model in Lyra Manifold
Hoavo Hova111hovhoav@mail.ustc.edu.cn Interdisciplinary Center for Theoretical
Study, University of Science and Technology of China, Hefei, 230026, P. R.
China
(August 27, 2024)
###### Abstract
We consider, in normal-gauge Lyra’s geometry, evolution of a homogeneous
isotropic universe in a gravitational model involving only the standard matter
in interaction with a displacement vector field $\phi_{\mu}$. Considering both
constant and time-dependent displacement vector field we show that the
observed cosmic acceleration could be explained without considering an alien
energy component with a negative pressure.
Lyra’s Geometry, Cosmology, Dark Energy, Interaction, Cosmological Constant.
## I Introduction
There is increasing evidence from recent cosmic observations of Type Ia
SupernovaeRiess ; Perlmutter , Large Scale Structure (LSS)Tegmark1 ; Tegmark2
; Tegmark3 ; Seljak ; Adelman ; Abazajian and Cosmic Microwave
Background(CMB)Spergel1 ; Page ; Hinshaw ; Jarosik that the universe is
undergoing an accelerated expansion at the present stage. This phenomenon
indicates that the universe at present is dominated by a smooth energy
component, dubbed “dark energy”, with a negative pressure that counteracts the
gravitational forces produced by ordinary matter species, such as baryons and
radiation, leading then to an accelerated expansion of the universe. Despite
many years of research and much progress, the nature and the origin of dark
energy have not been confirmed yet.
Obviously, the best and simplest candidate for such dark energy is the so-
called cosmological constant (CC) $\Lambda$ which was introduced by Einstein
into his gravitational field equations in an _ad hoc_ fashion. However, CC
explanation for dark energy usually faces some fundamental problems in
physics, namely the fine-tuning problem and notably the cosmic coincidence
puzzles Weinberg ; Copeland ; Amendola1 . In particle physics, the CC is often
interpreted as the energy, $\rho_{vac}$, of the quantum vacuum, which is close
to the Planck density $M_{P}^{4}$ ($M_{P}=1/\sqrt{8\pi G}$ is the reduced
Planck mass) in magnitude. The observed value of the dark energy density is
much less than that of the quantum vacuum, $\rho_{obs}\approx
10^{-123}\rho_{vac}$. Suppressing this great difference of about 123 orders of
magnitude between the observed value of dark energy and that estimated from
quantum field theory requires some severe fine-tuning mechanisms to
workHawking ; Kachru ; Tye ; Yokoyama ; Mukohyama ; Kane ; Dolgov . On the
other hand, even if this fine-tuning problem could be evaded, the non-
dynamical behaviour of the quantum vacuum energy renders the coincidence
problem unsolved. Attempting to alleviate these two fundamental problems and
to explain the late-time cosmic acceleration, many plausible dynamical models,
such as quintessence Ratra , phantom Onemli ; Caldwell , k-essence Armendariz
, tachyon Padmanabhan , holographic Li ; Limiao , agegraphic Cai , hessence
Wei , Chaplygin gas Kamenshchik , Yang-Mills condensate Zhang , etc., have
been proposed (see also Review article Copeland and references therein).
A different approach to explain the observed accelerating universe with models
involving only the standard matter is a plausible modification of the Einstein
gravitational field equations . Such a modification can arise, either by
extending the Einstein-Hilbert action to a more fundamental theory ($f(R)$
theories of gravity Kerner ; Capozziello ; Nojiri ; Capozziello1 ; Nojiri1 )
or by modifying the Riemannian geometry. In the latter case, a Lyra’s geometry
Lyra ; Sen1 ; Sen2 , which bears a close similarity to Weyl’s geometry Weyl
and is propounded in order to unify gravitation and electromagnetism into a
single space-time geometry, got lots of interest. Indeed, in contrast to
Weyl’s geometry, the connection in Lyra’s geometry is metric preserving, as in
Riemannian geometry, and length transfers are also integrable. In addition,
theories of gravitation, that have been constructed in the framework of Lyra’s
geometry with both a constant and a time-dependent displacement vector field,
involve scalar fields and tensors that are all intrinsic to the geometry Sen1
; Sen2 ; Sen3 ; Sen4 ; Scheibe ; Halford ; Halford1 ; Soleng ; Soleng1 ;
Hudgin ; Beesham ; Beesham1 ; Manoukian ; Matyjasek ; Anirudh ; Mohanty ;
Rahaman ; Katore ; Gad ; Shchigolev . On the other hand, as shown in Sen1 ;
Halford1 these theories predict the same effects within observations limits,
as far as the classical Solar System, as well as tests based on the linearised
form of the field equations, and are free of the Big-Bang singularity and
solve the entropy and horizon problems, which beset the standard models based
on Riemannian geometry.
In the present work, we consider a pressureless matter in interaction with the
displacement vector field. As pointed out in Shchigolev , we will see that, in
the absence of a pressureless matter the displacement vector field alone could
not be considered as a cosmological constant (term) but rather a stiff fluid,
because the associated equation of state is $\omega_{\phi}=+1$ and not
$\omega_{\textsc{cc}}=-1$. Meanwhile, interacting with the pressureless
matter, the displacement vector can play the same role as a cosmological
constant (term), establishing therefore the intrinsic geometrical origin of
the cosmological term. Subsequently, it is shown that the observed
accelerating universe can occur without considering an alien energy component
with a negative pressure.
The outline of the paper is as follows. In Sect. II, firstly we derive the
$\Lambda$CDM model from a model containing the standard matter and a constant
displacement vector field, and secondly we focus on a time-dependent vector
field that yields a variable cosmological term and drives then an accelerating
universe. A summary of the results and the conclusions are presented in Sect.
III. Throughout the paper we adopt the Planck units $c=\hbar=\kappa^{2}=1$ and
use the space-like metric signature $(-,+,+,+)$.
## II Cosmic Acceleration in Normal-Gauge Lyra manifold
The Einstein gravitational field equations in normal gauge for a four-
dimensional Lyra manifold, as obtained by Sen Sen1 are
$G_{\mu\nu}=T_{\mu\nu}+{\cal T}_{\mu\nu},$ (1)
where
${\cal
T}_{\mu\nu}=-\frac{3}{2}\left(\phi_{\mu}\phi_{\nu}-\dfrac{1}{2}g_{\mu\nu}\phi_{\lambda}\phi^{\lambda}\right),$
(2)
is the stress-energy tensor associated with the displacement vector field
$\phi_{\mu}$, i.e., arising as an intrinsic geometrical energy-momentum
tensor, whereas $T_{\mu\nu}$ represents a perfect fluid energy-momentum tensor
defined by
$T_{\mu\nu}=\left(p+\rho\right)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (3)
$u_{\mu}=(1,0,0,0)$ is the 4-velocity of the comoving observer, satisfying
$u^{\mu}u_{\mu}=-1$, $\rho$ and $p=\omega_{b}\rho$ with $0\leq\omega_{b}<1$
are the background energy density and pressure, respectively. In this paper we
shall consider a homogeneous time-depending time-like displacement vector
$\phi_{\mu}=\left(\phi(t),0,0,0\right)$ (4)
With these assumptions the $(0,0)$ and $(i,j)$-components of Eq. (1), in the
flat Friedmann-Lemaître-Robertson-Walker background
$ds^{2}=-dt^{2}+a^{2}d\vec{x}^{2}$, where $a=a(t)$ is the scale factor of an
expanding universe, may be written
$\displaystyle 3H^{2}=\rho-\frac{3}{4}\phi^{2}(t)=\rho_{eff},$ (5)
$\displaystyle-\left(2\dot{H}+3H^{2}\right)=\omega_{b}\rho-\frac{3}{4}\phi^{2}(t)=p_{eff},$
(6)
where an overdot denotes differentiation with respect to the time coordinate
$t$ and $H=\dot{a}/a$ is the Hubble parameter. In the above equations we can
recast the pressure and energy density of the displacement vector in the forms
$p_{\phi}=\rho_{\phi}=-\frac{3}{4}\phi^{2}(t)$, giving a constant equation of
state, $\omega_{\phi}=+1$, of a stiff matter. This proves that the
displacement vector alone cannot be considered as a cosmological constant
which, by contrast, has an equation of state $\omega_{\textsc{cc}}=-1$. In the
following, we set $\theta\equiv\phi^{2}$ and refer to $\theta$ as displacement
field. Thus, arguing the displacement field is a geometrical energy component
contributing to the total energy and interacting with the standard matter, we
will show that the displacement vector can play the role of a cosmological
constant (term). In this context we consider both a constant and time-
dependent displacement vector field.
### II.1 Constant Vector Field and Cosmological Constant
Considering a constant vector field $\phi^{2}=\phi^{2}_{0}=\theta_{0}$ and
starting with a background fluid with a constant equation of state
$\omega_{b}$ the Friedmann equations translate into
$\displaystyle 3H^{2}=\rho-\frac{3}{4}\theta_{0}=\rho_{eff},$ (7)
$\displaystyle
2\dot{H}+3H^{2}=-\omega_{b}\rho+\frac{3}{4}\theta_{0}=-p_{eff}.$ (8)
Resolving the continuity equation of the effective energy density,
$d\rho_{eff}/dx+3(\rho_{eff}+p_{eff})=0$ where $x=\ln a$ is the e-folding
number, one obtains
$\displaystyle\rho=ke^{-3(1+\omega_{b})x}+\frac{3}{2(1+\omega_{b})}\theta_{0},$
(9) $\displaystyle
3H^{2}=\rho_{eff}=ke^{-3(1+\omega_{b})x}+\frac{3}{4}\dfrac{1-\omega_{b}}{1+\omega_{b}}\theta_{0},$
(10) $\displaystyle
p_{eff}=\omega_{b}ke^{-3(1+\omega_{b})x}-\frac{3}{4}\dfrac{1-\omega_{b}}{1+\omega_{b}}\theta_{0},$
(11)
whence we can compute the effective equation of state as follows
$\omega_{eff}=-1+\dfrac{1+\omega_{b}}{1+n_{0}e^{3(1+\omega_{b})x}}$ (12)
with
$n_{0}=\frac{3}{4}\dfrac{1-\omega_{b}}{1+\omega_{b}}\dfrac{\theta_{0}}{k}.$
(13)
Hence the effective equation of state varies from $\omega_{b}$ at
$x\to-\infty$ to $\omega_{\textsc{cc}}=-1$ at $x\to+\infty$.
Considering now a pressureless matter $\omega_{b}=0$, Eqs. (9)-(11) reduce to
the $\Lambda$CDM-like model
$\displaystyle\rho=ke^{-3x}+\frac{3}{2}\theta_{0},$ (14) $\displaystyle
3H^{2}=\rho_{eff}=ke^{-3x}+\frac{3}{4}\theta_{0},$ (15) $\displaystyle
p_{eff}=-\frac{3}{4}\theta_{0}.$ (16)
From Eq. (14) we say that the pressureless background fluid $\rho$ is the
contribution of two terms: the cold dark matter (CDM) and a gain of energy
$\varepsilon=\frac{3}{2}\theta_{0}$ from a modification of the Riemannian
manifold by the presence of a vector field in the geometrically structureless
manifold. On the other hand, Eq. (15) shows that the effective energy density
is also a sum of two terms, namely the CDM and a contribution from the
displacement field, $\tilde{\epsilon}=\frac{3}{4}\theta_{0}$. Since the
effective pressure (16) is the contribution of the displacement field only,
one finds that through the conservation of the total energy density the
equation of state for the displacement field becomes
$\omega_{\phi_{int}}=p_{eff}/\tilde{\epsilon}=-1$. In this case the
displacement field (vector) plays the same role as the cosmological constant
$\Lambda$.
The effective equation of state and the fractional Hubble parameter $E\equiv
H/H_{0}$ are therefore given by
$\omega_{eff}=-\frac{1}{1+\lambda(1+z)^{3}},$ (17)
$E^{2}(z)=\Omega_{m}(1+z)^{3}+(1-\Omega_{m}),$ (18)
where $z=a^{-1}-1$ is the redshift,
$\lambda=\frac{4k}{3\theta_{0}}=\dfrac{\Omega_{m}}{1-\Omega_{m}}$,
$\Omega_{m}\equiv\dfrac{k}{3H_{0}^{2}}$ is the matter density parameter. Thus,
at present time ($x=0$) acceleration occurs for $\lambda<2$ or
$\Omega_{m}<2/3$. A model involving a pressureless background fluid in normal
gauge for Lyra manifold is therefore equivalent to the $\Lambda$CDM model if
one sets $\Lambda=\frac{3}{4}\theta_{0}$. We conclude that this model shows
the intrinsic geometrical origin of the cosmological constant with
$\omega_{\textsc{cc}}=\omega_{\phi_{int}}=-1$, and the constant displacement
vector field arises therefore as the origin of the late time accelerated
expansion of the universe.
### II.2 Time-Dependent Vector Field
The vector field $\phi_{\mu}$ now depending on time interacts mutually with
the background fluid $\rho$. And the effective energy conservation equation
can be written as
$\frac{d\rho}{dx}+3(1+\omega_{b})\rho-\frac{3}{4}\left(\frac{d\theta}{dx}+6\theta\right)=0,$
(19)
Eq. (19) involves two unknown functions $\rho$ and $\theta$, that means in
addition to the Hubble parameter we have three unknown functions to be
determined with only two independent equations. In what follows we construct
the cosmological consequences of these equations, under some conditions that
significantly simplify the search for solutions, but nevertheless show the
richness of the cosmological dynamics of the present model. We can thus encode
the interaction between $\rho$ and $\theta$ into the conservation equations
Amendola1 ; Limiao
$\displaystyle\frac{d\rho}{dx}+3(1+\omega_{b})\rho=\gamma(x),$ (20)
$\displaystyle\frac{3}{4}\left(\frac{d\theta}{dx}+6\theta\right)=\gamma(x),$
(21)
where $\gamma$ denotes the phenomenological interaction term. Owing to the
lack of the knowledge of micro-origin of the interaction and following other
works Limiao we simply parametrize the interaction term in the form:
$\gamma=3b\rho+\frac{9}{2}\tilde{b}\theta\leavevmode\nobreak\ ,$ (22)
where both $b$ and $\tilde{b}$ are dimensionless coupling constants, and the
factors $3$ and $9/2$ before $b$ and $\tilde{b}$ are for convenience in the
following calculations. We emphasize here that $\gamma$ does not contain
explicitly the Hubble parameter $H$, because we are using the e-folding number
$x=\ln a$. However, once we are dealing with the cosmological time $t$, we
instead use $\tilde{\gamma}=H\gamma=\Gamma\rho+\tilde{\Gamma}\theta$, where
$\Gamma$ and $\tilde{\Gamma}$ characterize the strength of the coupling.
Without loss of generality and in order to reduce, furthermore, the number of
parameters we shall consider in this section a pressureless matter and then
set $\omega_{b}=0$. In the following we shall study the cases: $i)$ $b\neq 0$
and $\tilde{b}=0$, $ii)$ $b=0$ and $\tilde{b}\neq 0$ and $iii)$ $b\neq 0$ and
$\tilde{b}\neq 0$, respectively.
#### Case $i$: $b\neq 0$ and $\tilde{b}=0$
Solving (20) and (21) we find
$\displaystyle\theta=-M_{1}e^{-6x}+\frac{4bC_{1}}{3(1+b)}e^{-3(1-b)x},$ (23)
$\displaystyle\rho=C_{1}e^{-3(1-b)x},$ (24) $\displaystyle
3H^{2}=\rho_{eff}=\frac{C_{1}}{1+b}e^{-3(1-b)x}+\frac{3M_{1}}{4}e^{-6x},$ (25)
$\displaystyle
p_{eff}=\frac{3M_{1}}{4}e^{-6x}-\frac{bC_{1}}{1+b}e^{-3(1-b)x},$ (26)
where $C_{1}$ and $M_{1}$ are integration constants. We realize that, with
this choice of $\gamma$ and for $M_{1}>0$, the pressureless fluid energy and
the effective energy density are always positive and monotonically decreasing
for $C_{1}>0$ and $-1<b<1$ during the evolution of the universe. Meanwhile the
effective pressure could be either positive in early times (the pressureless
matter dominates over the displacement field) or negative at the present stage
(vector field dominance epoch). We can now evaluate the dynamically varying
effective equation
$\omega_{eff}=\frac{1-b\xi e^{3(1+b)x}}{1+\xi e^{3(1+b)x}}\leavevmode\nobreak\
,$ (27)
where $\xi=\frac{4C_{1}}{3M_{1}(1+b)}$ is a positive dimensionless constant
for $b>-1$. Assuming a monotonically decreasing effective energy density and
positive $\xi$, the coupling constant $b$ needs to range in the region
$-1<b<1$. The effective equation of state then lies in the range
$\omega_{eff}^{+\infty}=-b<\omega_{eff}<\omega_{eff}^{-\infty}=1$. At the
present stage we have
$\omega^{0}_{eff}=-b+\dfrac{1+b}{1+\xi},$ (28)
which is less than $\omega_{acc}=-1/3$ for $\xi>\dfrac{4}{3b-1}$. For example,
$\omega^{0}_{eff}=-0.9$ if $\xi=\dfrac{1.9}{b-0.9}$.
#### Case $ii$: $b=0$ and $\tilde{b}\neq 0$
In this case the different quantities, previously considered, become
$\displaystyle\theta=M_{2}e^{-6(1-\tilde{b})x},$ (29)
$\displaystyle\rho=C_{2}e^{-3x}+\frac{3\tilde{b}M_{2}}{2(2\tilde{b}-1)}e^{-6(1-\tilde{b})x},$
(30) $\displaystyle
3H^{2}=\rho_{eff}=C_{2}e^{-3x}+\frac{3M_{2}}{4(2\tilde{b}-1)}e^{-6(1-\tilde{b})x},$
(31) $\displaystyle p_{eff}=-\frac{3M_{2}}{4}e^{-6(1-\tilde{b})x},$ (32)
where $C_{2}$ and $M_{2}$ are integration constants. For $\tilde{b}>1/2$ both
effective energy and pressureless background energy densities are positive-
defined. The former is the sum of two terms: CDM and a dynamically variable
quantity, $\Lambda(x)$, that could be thought of as a variable cosmological-
type term associated with the vector field:
$\Lambda(x)=\frac{3\varpi_{1}H_{0}^{2}}{2\tilde{b}-1}e^{-6(1-\tilde{b})x}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{with}\leavevmode\nobreak\ \leavevmode\nobreak\
\varpi_{1}=\dfrac{M_{2}}{4H_{0}^{2}}>0.$ (33)
For $\tilde{b}<1$ the cosmological term is large during the early stages of
the universe and has decayed to its low value at present, explaining readily
the fine-tuning and coincidence puzzles (see fig.(1) where we plot both the
pressureless energy density $\rho$ and the cosmological term $\Lambda$). On
the other hand, the pressureless background fluid is also a sum of two terms,
corresponding to CDM energy and an energy, $\rho_{geo-gained}$, geometrically
gained as the universe evolves, that is,
$\rho=\rho_{\textsc{cdm}}+\rho_{geo-gained},$ (34)
where
$\rho_{geo-
gained}=\frac{6\varpi_{1}H_{0}^{2}}{2\tilde{b}-1}e^{-6(1-\tilde{b})x}$ (35)
is proportional to the cosmological term $\Lambda(x)$.
Figure 1: Evolution of $\rho/3H^{2}_{0}$ (dash-dotted curve) and
$\Lambda/3H^{2}_{0}$ (solid curve) versus the e-folding number $x$ for
$\varpi_{1}=0.3$ and $\tilde{b}=0.7$. In early times the cosmological term was
negligible, compared to the pressureless energy density, while at the present
stage they are of the same order. Hence, this model seems to solve both the
fine-tuning and the coincidence problems.
Since the effective pressure is the sum of the CDM pressure,
$p_{\textsc{cdm}}=0$ and that of the variable cosmological term $\Lambda(x)$,
from (32) and (33) one can determine the equation of state for the variable
cosmological term:
$\omega_{\Lambda}\equiv\frac{p_{eff}}{\Lambda},$
or
$\omega_{\Lambda}=1-2\tilde{b},$ (36)
which, for $\tilde{b}\neq 1$, is different from the equation of state of the
usual cosmological constant $\omega_{\textsc{cc}}=-1$. For example, for
$\tilde{b}=0.9$, $\omega_{\Lambda}=-0.8$, and for $\tilde{b}=0.99$,
$\omega_{\Lambda}=-0.98$. However, the choice $\tilde{b}>1$ will lead to dark
energy behaving as phantom, i.e., $\omega_{\Lambda}$ being less than $-1$
Onemli ; Caldwell ; Kunz ; Alam , and the variable cosmological term will
increase exponentially from $0$ at $x\to-\infty$ to $+\infty$ at $x+\infty$.
The ratio of the effective pressure over the effective energy density
furnishes the effective equation of state in the form
$\omega_{eff}=\frac{\omega_{\Lambda}}{1-\omega_{\Lambda}f_{0}e^{3\omega_{\Lambda}x}},$
(37)
where
$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle\frac{4C_{2}}{3M_{2}}$
(38) $\displaystyle=$
$\displaystyle\dfrac{1}{\varpi_{1}}+\dfrac{1}{\omega_{\Lambda}}$
is supposed to be a positive constant, unlike $\omega_{\Lambda}$ which is
always assumed negative. One thus has
$\omega^{-\infty}_{eff}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\omega^{0}_{eff}=-\varpi_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \omega^{+\infty}_{eff}=\omega_{\Lambda}.$ (39)
As we have pointed out, a positive $f_{0}$ implies
$\omega^{+\infty}_{eff}<\omega^{0}_{eff}$. Hence, $\varpi_{1}>1/3$ can drive
an accelerated expansion. For example, taking $\varpi_{1}$ as the present
fractional density of dark energy, $\omega^{0}_{eff}\approx-0.73$.
#### Case $iii$: $b\neq 0$ and $\tilde{b}\neq 0$
Combining (20) and (21) and after some algebra it is shown that the effective
energy density $\rho_{eff}$, the background fluid energy $\rho$ and the
displacement field $\theta$ satisfy the same ordinary differential equation of
second order under the form
$\frac{d^{2}Q}{dx^{2}}+3(3-b-2\tilde{b})\frac{dQ}{dx}+18(1-b-\tilde{b})Q=0,$
(40)
where $Q\in\left\\{\rho;\theta;\rho_{eff}\right\\}$. We will now consider some
special cases where $b$ is explicitly related to $\tilde{b}$.
A/
$\tilde{b}=1-b$ ($b\neq 1$ or $\tilde{b}\neq 0$ ), one then obtains
$\rho_{eff}(x)=N_{1}+\dfrac{N_{2}}{1+b}e^{-3(1+b)x},$ (41)
and the effective equation of state in this case takes for $b>-1$ the values
$\omega_{eff}^{-\infty}=b$, $\omega_{eff}^{0}=-1+\varpi$ and
$\omega_{eff}^{+\infty}=\omega_{\textsc{cc}}=-1$, where
$\varpi=N_{2}/3H_{0}^{2}$. Acceleration thus occurs if $\varpi<2/3$.
B/
$\tilde{b}=\dfrac{3-b}{2}$ ($b\neq 3$ or $\tilde{b}\neq 0$ ), $\rho_{eff}(x)$
takes the form
$\rho_{eff}(x)=N_{3}e^{3\sqrt{1+b}\leavevmode\nobreak\
x}+N_{4}e^{-3\sqrt{1+b}\leavevmode\nobreak\ x},$ (42)
$N_{3}$ and $N_{4}$ being constants and $b>-1$. For monotonically decreasing
$\rho_{eff}(x)$ as suggested observations we will set $N_{3}=0$, hence the
effective equation of state is a constant, given by
$\omega_{eff}=-1+\sqrt{1+b}$, which is less than $\omega_{acc}=-1/3$ and
greater than $\omega_{\textsc{cc}}=-1$ for $-1<b<-5/9$.
On the other hand, considering a nonzero $N_{3}$ but setting instead
$N_{3}=N_{4}=r/2$, the effective energy density transforms according to
$\rho_{eff}(x)=r\cosh\left(3\sqrt{1+b}\leavevmode\nobreak\ x\right)$ (43)
and
$\omega_{eff}=-1-\sqrt{1+b}\tanh\left(3\sqrt{1+b}\leavevmode\nobreak\
x\right).$ (44)
The effective equation of state $\omega_{eff}$ (44) varies from
$\omega^{-\infty}_{eff}=-1+\sqrt{1+b}$, crosses $\omega_{\textsc{cc}}=-1$ at
$x=0$ and tends to $\omega^{+\infty}_{eff}=-1-\sqrt{1+b}$. That thus leads to
_quintom_ dark energy with the equation of state crossing the cosmological
constant boundary $\omega_{\textsc{cc}}=-1$ Guo .
C/
Now, considering the cases where $\tilde{b}\neq 1-b$ and
$\tilde{b}\neq\dfrac{3-b}{2}$ the general solution is given by
$Q(x)=Q_{1}\exp\left[-\frac{3}{2}\left(3-b-2\tilde{b}+\sqrt{\zeta}\right)x\right]+\tilde{Q}_{1}\exp\left[-\frac{3}{2}\left(3-b-2\tilde{b}-\sqrt{\zeta}\right)x\right],$
(45)
where $Q_{1}\in\left\\{\rho_{1};\theta_{1};\rho^{1}_{eff}\right\\}$ and
$\tilde{Q}_{1}\in\left\\{\tilde{\rho}_{1};\tilde{\theta}_{1};\tilde{\rho}^{1}_{eff}\right\\}$
are integration constants, and
$\zeta=(3-b-2\tilde{b})^{2}-8(1-b-\tilde{b}).$ (46)
The effective energy density reads then:
$\displaystyle 3H^{2}=\rho_{eff}$ $\displaystyle=$ $\displaystyle
Q_{2}\exp\left[-\frac{3}{2}\left(3-b-2\tilde{b}+\sqrt{\zeta}\right)x\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (47a)
$\displaystyle+\tilde{Q}_{2}\exp\left[-\frac{3}{2}\left(3-b-2\tilde{b}-\sqrt{\zeta}\right)x\right],$
where
$Q_{2}=\rho^{1}_{eff}=\rho_{1}-\dfrac{3}{4}\theta_{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{Q}_{2}=\tilde{\rho}^{1}_{eff}=\tilde{\rho}_{1}-\dfrac{3}{4}\tilde{\theta}_{1},$
(47c)
When $\zeta=0$ or $b_{\pm}=-1\pm 2\sqrt{2\tilde{b}}-2\tilde{b}$, and looking
only for solutions leading to an accelerated expansion (which corresponds here
to the value $b_{+}=-1+2\sqrt{2\tilde{b}}-2\tilde{b}$ ), $\rho_{eff}$ and
$\omega_{eff}$ become
$\rho_{eff}=\left(Q_{2}+\tilde{Q}_{2}\right)e^{-3\left(2-\sqrt{2\tilde{b}}\right)x}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\omega_{eff}=1-\sqrt{2\tilde{b}}.$ (48)
$\rho_{eff}$ decreases for $x\leq 0$ and acceleration then occurs if
$8/9<\tilde{b}<2$. On the other hand, $\rho_{eff}$ reduces to the cosmological
constant with $\omega_{eff}=\omega_{\textsc{cc}}=-1$ for $\tilde{b}=2$.
Now, taking $\zeta\neq 0$ and assuming furthermore a monotonically decreasing
effective density during the evolution of the universe the coupling $b$ and
$\tilde{b}$ have to satisfy one of the following constraints:
$\begin{array}[]{rlll}\alpha_{1})&&\tilde{b}<0\Longrightarrow
b<1-\tilde{b},\\\ \alpha_{2})&&0<\tilde{b}\leq 2\Longrightarrow
b\leq-1-2\sqrt{2\tilde{b}}-2\tilde{b}\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\\\
&&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{or}\leavevmode\nobreak\
-1+2\sqrt{2\tilde{b}}-2\tilde{b}\leq b<1-\tilde{b},\\\
\alpha_{3})&&\tilde{b}>2\Longrightarrow
b\leq-1-2\sqrt{2\tilde{b}}-2\tilde{b},\end{array}$ (49)
where we considered both positive and negative values for $b$ and $\tilde{b}$,
and assumed that the integrations constants $Q_{1}$ and $\tilde{Q}_{1}$ are
all nonzero and positive. The effective equation of state may be written as
$\omega_{eff}(x)=\dfrac{1}{2}\left[1-b-2\tilde{b}+\sqrt{\zeta}\leavevmode\nobreak\
\dfrac{1-\chi e^{3\sqrt{\zeta}x}}{1+\chi e^{3\sqrt{\zeta}x}}\right],$ (50)
with $\chi=\dfrac{\varpi_{2}}{1-\varpi_{2}}$ and
$\varpi_{2}=\dfrac{\tilde{Q}_{2}}{3H_{0}^{2}}$. Taking $b=\tilde{b}=1/3$ we
obtain $\omega_{eff}^{-\infty}=\sqrt{3}/3$,
$\omega_{eff}^{+\infty}=-\sqrt{3}/3<-1/3$ and
$\omega_{eff}^{0}=(1-2\varpi_{2})\sqrt{3}/3$ which is less than $-1/3$ if
$\varpi_{2}>1/2+\sqrt{2}/6\approx 0.78$. The Figure (2) shows the plot of
$\omega_{eff}$ in (50) versus $x$ for different values of
$(b,\tilde{b},\chi)$: $b=1/3,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=9$ (dashed curve,
$\omega_{eff}\approx-0.46$); $b=0.66,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=4$ (dash-dotted curve,
$\omega_{eff}\approx-0.66$) and $b=0.66,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=9$ (solid curve,
$\omega_{eff}\approx-0.83$), and the dotted line represents
$\omega_{acc}=-1/3$.
Figure 2: $\omega_{eff}$ (50) versus $x$ for $b=1/3,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=9$ (dashed curve,
$\omega_{eff}\approx-0.46$); $b=0.66,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=4$ (dash-dotted curve,
$\omega_{eff}\approx-0.66$) and $b=0.66,\leavevmode\nobreak\
\tilde{b}=1/3,\leavevmode\nobreak\ \chi=9$ (solid curve,
$\omega_{eff}\approx-0.83$), the dotted line represents $\omega_{acc}=-1/3$.
Notice that in the special case where $Q_{2}=\tilde{Q}_{2}=\dfrac{q}{2}$, Eq.
(47a) becomes
$\rho_{eff}=q\cosh\left(\dfrac{3}{2}\sqrt{\zeta}x\right)\exp\left[-\frac{3}{2}\left(3-b-2\tilde{b}\right)x\right]$
(51)
which decreases monotonically when $x\leq 0$.
### II.3 Concluding Remarks
We have constructed, in normal gauge for a four-dimensional Lyra manifold, a
dark energy model containing only the standard matter that interacts with the
vector field $\phi_{\mu}$. Assuming a constant displacement vector field, the
model mimics the $\Lambda$CDM model, indicating that the constant displacement
field, considered as an energy component of the total energy, plays the role
of a cosmological constant with $\omega_{\textsc{cc}}=\omega_{\phi_{int}}=-1$.
Introduced in general relativity in an ad hoc fashion, the cosmological
constant arises in Lyra’s geometry as a result of the presence of a vector
field in the affine structure and has then an intrinsic geometrical
significance. On the other hand, considering a time-dependent displacement
vector field mutually interacting with the pressureless matter and without an
alien energy component with a negative pressure, we have alleviated the fine-
tuning and coincidence puzzles and shown that the universe could recently
enter an accelerating phase, with even an equation of state crossing the
cosmological constant boundary $\omega_{\textsc{cc}}=-1$. In fact, the
geometrical contribution to the effective energy density, resulting from the
modification of the Riemann manifold is endowed with a negative pressure that
counteracts the gravitational forces produced by the ordinary matter and
therefore favours the acceleration of the universe. In contrast to several
models that consider an alien energy component in the universe, this work
ensures that dark energy responsible for the late time acceleration of the
universe has an intrinsic geometrical origin.
## Acknowledgements
I would like to thank Dr. K. Ayenagbo for valuable discussions
## References
## References
* (1) A. G. Riess et al., Astron. J. 116, (1998) 1009.
* (2) S. Perlmutter et al., Astrophys. J. 517, (1999) 565.
* (3) M. Tegmark et al., Phys. Rev. D 69 (2004) 103501.
* (4) M. Tegmark et al., Astrophys. J. 606 (2004) 702.
* (5) M. Tegmark et al., Phys. Rev. D 74 (2006) 123507.
* (6) U. Seljak et al., Phys. Rev. D 71 (2005) 103515.
* (7) J. K. Adelman-McCarthy et al., Astrophys. J. Suppl. 162 (2006) 38.
* (8) K. Abazajian et al., Astron. J. 126 (2003) 2081; Astron. J. 128 (2004) 502 and Astron. J. 129 (2005) 1755.
* (9) D. N. Spergel et al., Astrophys. J. Suppl. 170 (2007) 377.
* (10) L. Page et al., Astrophys. J. Suppl. 170 (2007) 335.
* (11) G. Hinshaw et al., Astrophys. J. Suppl. 170 (2007) 288.
* (12) N. Jarosik et al., Astrophys. J. Suppl. 170 (2007) 263.
* (13) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989); V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000); S.M. Carroll, Living Rev.Rel. 4, 1 (2001); P.J.E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003); T. Padmanabhan, Phys. Rept. 380, 235 (2003).
* (14) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod.Phys. D 15, 1753 (2006).
* (15) L. Amendola and S. Tsujikawa, Dark energy, CUP, 2010.
* (16) S. W. Hawking, Phys. Lett. B 134(1984) 403.
* (17) S. Kachru, M. B. Schulz and E. Silverstein, Phys. Rev. D 62 (2000) 045021.
* (18) S.-H. H. Tye and I.Wasserman, Phys. Rev. Lett. 86(2001) 1682.
* (19) J. Yokoyama, Phys. Rev. Lett. 88 (2002) 151302.
* (20) S. Mukohyama and L. Randall, Phys. Rev. Lett. 92 (2004) 211302.
* (21) G. L. Kane, M. J. Perry and A. N. Zytkow, Phys. Lett. B 609 (2005) 7.
* (22) A. D. Dolgov and F. R. Urban, Phys. Rev. D 77 (2008) 083503.
* (23) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988); P. J. E. Peebles and B.Ratra, ApJL 325, L17 (1988); C. Wetterich, Nucl. Phys. B 302, 668 (1988); I. Zlatev, L. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999).
* (24) V. K. Onemli and R. P. Woodard, Class. Quant. Grav. 19, 4607 (2002) [arXiv:gr-qc/0204065v2]; V. K. Onemli and R. P. Woodard, Phys. Rev. D 70, 107301 (2004) [arXiv:gr-qc/0406098v2]; T. Brunier, V. K. Onemli and R. P. Woodard, Class. Quant. Grav. 22, 59-84 (2005) [arXiv:gr-qc/0408080v2]; E. O. Kahya and V. K. Onemli, Phys. Rev. D 76, 043512 (2007) [arXiv:gr-qc/0612026v3]; E. O. Kahya, V. K. Onemli and R. P. Woodard, Phys. Rev. D 81, 023508 (2010) [arXiv:0904.4811v1].
* (25) R. R. Caldwell, Phys. Lett. B 545, 23 (2002); S.M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. D 68, 023509 (2003).
* (26) C. Armendáriz-Picón, T. Damour and V. Mukhanov, Phys. Lett. B 458, 209 (1999); C. Armendáriz-Picón, V. Mukhanov and P.J. Steinhardt, Phys. Rev. D 63, 103510 (2001); T. Chiba, T. Okabe and M. Yamaguchi, Phys. Rev. D 62, 023511 (2000).
* (27) T. Padmanabhan, Phys. Rev. D 66, 021301(R) (2002); J. S. Bagla, H. K. Jassal, and T. Padmanabhan, Phys. Rev. D 67, 063504 (2003).
* (28) M. Li, Phys. Lett. B 603, 1 (2004); Q. G. Huang and M. Li, JCAP 0408, 013 (2004); Q. G. Huang and M. Li, JCAP 0503, 001 (2005); Q. G. Huang and Y. G. Gong, JCAP 0408, 006 (2004); X. Zhang and F. Q. Wu, Phys. Rev. D 72, 043524 (2005); M. Li, X. D. Li, S. Wang and X. Zhang, JCAP 0906, 036 (2009); Y. T. Wang and L. X. Xu, Phys. Rev. D 81, 083523 (2010).
* (29) M. Li et al., JCAP 0912, 014, 2009. arXiv:0910.3855v2.
* (30) R. G. Cai, Phys. Lett. B 657, 228 (2007); H. Wei and R. G. Cai, Phys. Lett. B 660, 113 (2008).
* (31) H. Wei, R. G. Cai, and D. F. Zeng, Class. Quant. Grav. 22, 3189 (2005); H. Wei, and R.G. Cai, Phys. Rev. D 72, 123507 (2005).
* (32) A. Y. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, 265 (2001); M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev. D 66, 043507 (2002); X. Zhang, F. Q. Wu and J. Zhang, JCAP 0601, 003 (2006).
* (33) W. Zhao and Y. Zhang, Phys. Lett. B 640, 69-73 (2006); W. Zhao and Y. Zhang, Class. Quant. Grav. 23, 3405-3418 (2006); Y. Zhang, T. Y. Xia, and W. Zhao, Class. Quant. Grav. 24, 3309 (2007); T. Y. Xia and Y. Zhang, Phys. Lett. B 656, 19 (2007); S. Wang, Y. Zhang and T. Y. Xia, JCAP 10, 037 (2008); S. Wang and Y. Zhang, Phys. Lett. B 669, 201 (2008).
* (34) R. Kerner, Gen. Rel. Grav. 14, No. 5, 453-469 (1982).
* (35) S. Capozziello, Int. J. Mod. Phys. D 11, 483-492 (2002) [arXiv:gr-qc/0201033v1].
* (36) S. Nojiri, S.D. Odintsov, Phys. Lett. B 576, 5 (2003); S. Nojiri, S. D. Odintsov, Mod. Phys. Lett. A 19, 627 (2003); S. Nojiri, S. D. Odintsov, Phys. Rev. D, 68, 12352 (2003); S. M. Carroll, V. Duvvuri, M. Trodden, M. Turner, Phys. Rev. D 70, 043528 (2004).
* (37) S. Capozziello, S. Carloni, A. Troisi, Rec. Res. Dev. Astron. Astrophys. 1, 625 (2003) [astro-ph/0303041].
* (38) S. Nojiri, S. D. Odintsov, Gen. Rel. Grav. 36, 1765 (2004); X. H. Meng, P. Wang, Phys. Lett. B 584, 1 (2004).
* (39) G. Lyra, Math. Z. 54, 52 (1951).
* (40) D. K. Sen, Z. Physik, 149, 311 (1957).
* (41) D. K. Sen, Can. Math. Bull. 3, 255 (1960).
* (42) H. Weyl, Sitzber. Preuss. Akad. Wiss., 465-480 (1918).
* (43) D. K. Sen and K. A. Dunn, J. Math. Phys. 12, 578 (1971)
* (44) D. K. Sen and J. R. Vanstone, J. Math. Phys. 13, 990 (1972).
* (45) E. Scheibe, Math. Z. 57, No. 1, (1952) 65-74.
* (46) W. D. Halford, Aust. J. Phys. 23, 863 (1970).
* (47) W. D. Halford, J. Math. Phys. 13, 1699 (1972).
* (48) H. H. Soleng, Class. Quant. Grav. 5, (1988) 1489-1500 .
* (49) H. H. Soleng, Gen. Rel. Grav. 19, No. 12, 1213 (1987).
* (50) R. H. Hudgin, J. Math. Phys. 14, No. 12, (1973) 1794-1799.
* (51) A. Beesham, Aust. J. Phys. 41, 833 (1988).
* (52) A. Beesham, Astrophys. Space Sci. 127, 189 (1986)
* (53) E. B. Manoukian, Phys. Rev. D 5, 2915-2922 (1972).
* (54) J. Matyjasek, Astrophys. Space Sci., 207, 313 (1993).
* (55) A. Pradhan and A. K. Vishwakarma, J. Geom. Phys., 49, (2004) 332-342.
* (56) G. Mohanty, G. C.Samanta, K. L. Mahanta, Theoret. Appl. Mech., 36, No. 2, (2009) 157-166; G. Mohanty, K. L. Mahanta, R. R. Sahoo, Astrophys. Space Sci. 306, (2006) 269; G. Mohanty, K. L. Mahanta, B. K. Bishi, Astrophys. Space Sci. 310, (2007) 273; G. Mohanty, K. L. Mahanta, R. R. Sahoo: Communications in Physics, 17, No. 3 (2007).
* (57) F. Rahaman, Fizika B 11, (2002) 223; F. Rahaman, S. Chakraborty, S. Das, N. Begum, M. Hossain, J. Bera, Pramana-J. Phys. 60, (2003) 453; F. Rahaman, S. Das, N. Begum, M. Hossain, Pramana-J Phys. 61, (2003) 153.
* (58) S. D. Katore, S. V. Thakare and K. S. Adhao, Pramana-J. Phys. 71, No. 1, (2008) 15-22.
* (59) R. M. Gad, Can. J. Phys. 89, 773–778 (2011).
* (60) V. K. Shchigolev and E. A. Semenova, [arXiv:1203.0917v1].
* (61) P. S. Corasaniti, M. Kunz, D. Parkinson, E. J. Copeland and B. A. Bassett, Phys. Rev. D70 (2004) 083006.
* (62) U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004).
* (63) Z. K. Guo et al., Phys. Lett. B 608, 177 (2005); J.-Q. Xia, B. Feng, and X. Zhang, Mod. Phys. Lett. A 20, 2409 (2005); M.-Z Li, B. Feng, and X.-M Zhang, J. Cosmol. Astropart. Phys. 12 (2005) 002; B. Feng, M. Li, Y.-S. Piao and X. Zhang, Phys. Lett. B 634, 101 (2006); M. R. Setare, Phys. Lett. B 641, 130 (2006); M. R. Setare and E. N. Saridakis, Phys. Rev. D 79 043005 (2009); W. Zhao and Y. Zhang, Phys. Rev. D 73, 123509 (2006); G.-B. Zhao, J.-Q. Xia, B. Feng, and X. Zhang, Int. J. Mod. Phys. D 16, 1229 (2007); M. R. Setare, J. Sadeghi, and A. R. Amani, Phys. Lett. B 660, 299 (2008); J. Sadeghi, M. R. Setare, A. Banijamali and F. Milani, Phys. Lett. B 662, 92 (2008); M. R. Setare and E. N. Saridakis, Phys. Lett. B 668, 177 (2008); M. R. Setare and E. N. Saridakis, Int. J. Mod. Phys. D 18, 549-557 (2009) [arXiv:0807.3807v1]; M. R. Setare and E. N. Saridakis, J. Cosmol. Astropart. Phys. 09 (2008) 026.
|
arxiv-papers
| 2012-04-01T05:52:23 |
2024-09-04T02:49:29.337394
|
{
"license": "Public Domain",
"authors": "Hoavo Hova",
"submitter": "Hoavo Hova",
"url": "https://arxiv.org/abs/1204.0774"
}
|
1204.0935
|
arxiv-papers
| 2012-04-04T12:39:34 |
2024-09-04T02:49:29.349628
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaowei Chen",
"submitter": "Shaowei Chen",
"url": "https://arxiv.org/abs/1204.0935"
}
|
|
1204.0951
|
# Quantum entanglement and phase transition in a two-dimensional photon-photon
pair model
Jian-Jun Zhang ruoshui789@gmail.com Jian-Hui Yuan Jun-Pei zhang Ze Cheng
School of Physics, Huazhong University of Science and Technology, Wuhan
430074, China
###### Abstract
We propose a two-dimensional model consisting of photons and photon pairs. In
the model, the mixed gas of photons and photon pairs is formally equivalent to
a two-dimensional system of massive bosons with non-vanishing chemical
potential, which implies the existence of two possible condensate phases.
Using the variational method, we discuss the quantum phase transition of the
mixed gas and obtain the critical coupling line analytically. Moreover, we
also find that the phase transition of the photon gas can be interpreted as
second harmonic generation. We then discuss the entanglement between photons
and photon pairs. Additionally, we also illustrate how the entanglement
between photons and photon pairs can be associated with the phase transition
of the system.
###### keywords:
Bose-Einstein condensate, Quantum phase transition, Entanglement
## 1 Introduction
Bose-Einstein condensate (BEC) is the remarkable state of matter that
spontaneously emerges when a system of bosons becomes cold enough that a
significant fraction of them condenses into a single quantum state to minimize
the system’s free energy. Particles in that state then act collectively as a
coherent wave. The phase transition for an atomic gas was first predicted by
Einstein in 1924 and experimentally confirmed with the discovery of superfluid
helium-4 in 1938. Obviously, atoms aren’t the only option for a BEC. In recent
years, with the development of techniques, the phenomenon of BEC was observed
in several physical system [1-9], including exciton polaritions, solid-state
quasiparticles and so on. We know that photons are the simplest of bosons, so
that it would seem that they could in principle undergo this kind of
condensation. The difficulty is that in the usual blackbody configuration,
which consists of an empty three-dimensional (3D) cavity, the photon is
massless and its chemical potential is zero, so that the BEC of photons under
these circumstances would seem to be impossible. However, very recently, J.
Klaers, etc. have overcome both obstacles using a simple approach [10,11]: By
confining laser light within a two-dimensional (2D) cavity bounded by two
concave mirrors, they create the conditions required for light to thermally
equilibrate as a gas of conserved particles rather than as ordinary blackbody
radiation.
What is more, it is well known that there are many fascinated optical effects
in the nonlinear medium, for instance, reduced fluctuation in one quadrature
(squeezing) [12], sub-Poissonian statistics of the radiation field [13], or
the collapse-revivals phenomenon [14]. Especially, in the nonlinear medium, a
photon from the laser beam can couple with other photons to form a photon-pair
(PP) [15-18]. The essence of PP has been investigated by many authors [19-21].
However, inspired by the experimental discovery of BEC of photons, in this
letter we construct another interesting 2D model consisting of photons and
PPs. In this model, the mixed system of photons and PPs is formally equivalent
to a 2D gas of massive bosons with non-vanishing chemical potential, which
implies the existence of two possible condensate phases, the mixed photon-PP
condensate phase and the pure PP condensate phase. By means of a variational
method we investigate the quantum phase transition of the mixed photon gas.
Especially, we find that the quantum phase transition of the photon gas can be
interpreted as second harmonic generation. We then discuss the entanglement
between photons and PPs. By investigating the entanglement in the ground state
and the dynamics of entanglement, we also illustrate how the entanglement
between photons and PPs can be associated with the phase transition of the
system. The investigation of these questions is important both for its
connection with quantum optics and for its practical applications to harmonic
generation and quantum information.
The remainder of this paper is organized as follows: In Sec. II, we
theoretically investigate the phenomenon of BEC of photons and PPs in a 2D
optical microcavity. The entanglement between photons and PPs is investigated
in Sec. III. Finally, we make a simple conclusion.
## 2 Bose-Einstein condensation of photons and photon pairs
Figure 1: Scheme of the optical microcavity: The microcavity consists of two
curved-mirrors with high reflectivity. A filter filled with a dye solution is
inserted into it, in which photons are repeatedly absorbed and re-emitted by
the dye molecules. The other part of the cavity is filled with a Kerr-like
nonlinear medium.
### 2.1 Description of the photon pair
We start with the description of the PP. History speaking, the essence of the
PP is presently still under discussion [19-21], and there exist many different
ways to obtain it. However, here we will use the standard procedure [22] in
the construction of harmonic generation to derive the PP. We know that the
presence of an electromagnetic field in the nonlinear material causes a
polarization of the medium and the polarization can be expanded in powers of
the instantaneous electric field:
$\displaystyle{\bf{P}}({\bf{r}},t)=\chi^{(1)}{\bf{E}}({\bf{r}},t)+\chi^{(2)}{\bf{E}}^{2}({\bf{r}},t)+....$
(1)
Here, the first term defines the usual linear susceptibility, and the second
term defines the lowest order nonlinear susceptibility. Ignoring the high
order parts (i.e. only expend the polarization to second order in electric
field $E$), we find that the Hamiltonian describing the interaction of the
radiation field with the dielectric medium is decomposed into two terms:
$\displaystyle\begin{array}[]{l}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}H_{{\mathop{\rm
int}}}=H_{line}+H_{nonline}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}H_{line}=-\int{\chi^{(1)}{\bf{E}}^{2}({\bf{r}},t)}d{\bf{r}}\\\
H_{nonline}=-\int{\chi^{(2)}{\bf{E}}^{3}({\bf{r}},t)}d{\bf{r}}\\\
.\end{array}$ (6)
where $H_{line}$ represents the energy of the linear interaction and
$H_{nonline}$ the nonlinear interaction.
It is well known that the electric field operator in a microcavity can be
expanded in terms of normal modes [23] as
$\displaystyle{\bf{E}}(r,t)=i\sum\limits_{\bf{k}}{{\bf{e}}_{\bf{k}}}\left({\frac{{\hbar\omega_{\bf{k}}}}{{2V\varepsilon}}}\right)^{1/2}\left({\hat{a}_{\bf{k}}e^{-i\omega_{\bf{k}}t+i{\bf{k}}\cdot{\bf{r}}}+\hat{a}_{\bf{k}}^{+}e^{i\omega_{\bf{k}}t-i{\bf{k}}\cdot{\bf{r}}}}\right),$
(7)
where $\hat{a}_{\bf{k}}$ and $\hat{a}_{\bf{k}}^{+}$ are the annihilation and
creation operators of photons with frequency $\omega_{\bf{k}}$, and they all
obey the usual boson commutation rules. $V$ is the normalization volume,
$\varepsilon$ is the dielectric constant of the medium and ${\bf{e}}_{\bf{k}}$
is the unit polarization vector with the usual polarization indices omitted
for simplicity. Substituting (3) into (2), for the linear interaction part, we
find that it consists of two processes, dissipation and two-photon
absorption(or emission). Here, dissipation is essentially also a two-photon
process, in which one photon is absorbed by the medium, meanwhile another one
is emitted. The linear interaction can be ignored, if the incident photon
field frequency $\omega_{0}$ is well below the electronic transition
frequencies of the medium. In that case, we need only consider the nonlinear
interaction, which has the simple form
$\displaystyle
H_{nonline}=\frac{\hbar}{{\sqrt{V}}}\sum\limits_{{\bf{k,k^{\prime}}}}{\chi_{{\bf{k}},{\bf{k^{\prime}}}}\left({\hat{b}_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}\hat{a}_{\bf{k}}\hat{a}_{{\bf{k^{\prime}}}}+H.c.}\right)},$
(8)
under the requirements of phase matching. Above, the operator $\hat{a}$
represents the normal photons, $\hat{b}$ represents the coupling PP, and where
$\chi_{{\bf{k}},{\bf{k^{\prime}}}}$ is the coupling matrix element. The
interaction energy in (4) consists of two terms. The first term
$b_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}a_{\bf{k}}a_{{\bf{k^{\prime}}}}$ describes
the process in which two normal photon with wave-vector ${\bf{k}}$ and
${\bf{k^{\prime}}}$ couple into a PP with wave-vector
${\bf{K}}={\bf{k}}+{\bf{k^{\prime}}}$, and the second term describe the
opposite process. The energy is conserved in both the processes.
### 2.2 Free-photon dispersion relation inside the optical microcavity
In this letter, we restrict out investigation inside a 2D optical microcavity.
The microcavity, as shown in Fig. 1, consists of two curved dielectric mirrors
with high reflectivity (about 99.9), which ensure prefect reflection of the
longitudinal component of the electromagnetic field within the cavity. In
addition, the transverse size of the cavity is much larger than its
longitudinal one.
We know that for a free photon, its frequency as a function of transversal
($k_{r}$) and longitudinal ($k_{z}$) wave number is
$\omega=c\left[{k_{z}^{2}+k_{r}^{2}}\right]^{1/2}$. However, in the case of
photons confined inside the microcavity, the vanishing of the electric field
at the reflecting surfaces of the curved-mirrors imposes a quantization
condition on the longitudinal mode number $k_{z}$, $k_{z}=n\pi/D(r)$, where
$n$ is an integer and where $D(r)=D_{0}-2(R-\sqrt{R^{2}-r^{2}})$ is the
separation of two curved-mirrors at distance $r$ from the optical axis, with
$D_{0}$ the mirror separation at distance $r=0$ and $R$ the radius of
curvature.
In the present work, we consider to fix the longitudinal mode number of
photons by inserting a circular filter into the cavity. The filter is filled
with a dye solution, in which photons are repeatedly absorbed and re-emitted
by the dye molecules. Thus, it also plays the role of photon reservoir. We
know that the longitudinal size of the cavity (i.e., the distance between the
mirrors) is very small. The small distance $D(r)$ between the mirrors causes a
large frequency spacing between adjacent longitudinal modes, comparable with
the spectral width of the dye. Modify spontaneous emission such that the
emission of photons with a given longitudinal mode number, $n=q$ in our case,
dominates over other emission processes. In this way, the longitudinal mode
number is frozen out. For fixed longitudinal mode number $q$ and in paraxial
approximation ($r\ll R$, $k_{r}\ll k_{z}$), we also find that the dispersion
relation of photons approximatively becomes $\omega\approx q\pi
c/D_{0}+ck_{r}^{2}D_{0}/2q\pi$. The above frequency-wavevector relation, upon
multiplication by $\hbar$, becomes the energy-momentum relation for the photon
$\displaystyle E\approx
m_{{\rm{ph}}}c^{2}+\frac{{(p_{r})^{2}}}{{2m_{{\rm{ph}}}}},$ (9)
where $m_{{\rm{ph}}}=\hbar q\pi/D_{0}c=\hbar\omega_{eff}/c^{2}$ is the
effective mass of the confined photons. At low temperatures, it is convenient
to redefine the zero of energy, so that only the effective kinetic energy,
$\displaystyle E\approx\frac{{(p_{r})^{2}}}{{2m_{{\rm{ph}}}}},$ (10)
remains. The above analysis shows that for the photon confined inside the 2D
microcavity, it is formally equivalent to a general boson having an effective
mass $m_{{\rm{ph}}}=\hbar\omega_{eff}/c^{2}$, that is moving in the transverse
resonator plane.
Furthermore, we here consider the case that the microcavity (except the filter
part) is filled with a Kerr nonlinear medium exhibiting significant third-
order optical nonlinearity. Due to the nonlinear effect, photons can couple
into PPs. If we connect the non-vanishing effective photon mass to the
previous analysis of the PPs, in this case we then can rewrite the nonlinear
interaction $H_{nonline}$ as
$\displaystyle
H_{nonline}=\frac{\hbar}{{\sqrt{S}}}\sum\limits_{{\bf{k}}_{r}{\bf{,k^{\prime}}}_{r}}{\chi_{{\bf{k}}_{r},{\bf{k^{\prime}}}_{r}}\left({b_{{\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}}^{+}a_{{\bf{k}}_{r}}a_{{\bf{k^{\prime}}}_{r}}+H.c.}\right)}$
(11)
where $S$ is the surface area of the 2D cavity, and where $a_{{\bf{k}}_{r}}$
and $a_{{\bf{k^{\prime}}}_{r}}$ are the annihilation operators of massive
photons with transverse wavevectors ${\bf{k}}_{r}$ and
${\bf{k}}_{r^{\prime}}$, respectively, and
$b_{{\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}}^{+}$ are the creation operator of the
massive PPs with transverse wave-vector
$K_{r}={\bf{k}}_{r}+{\bf{k^{\prime}}}_{r}$. Here, it should be remarked that
the existence of effective photon mass makes the thermodynamics of this 2D
mixed gas of photons and PPs different from the usual 3D photon gas. For the
2D system, thermalization is achieved in a photon-number-conserving way
($N=N_{a}+2N_{b}$) with nonvanishing chemical potential $\mu$, by multiple
scattering with the dye molecules, which acts as heat bath and equilibrates
the transverse modal degrees of freedom of the photon gas to the temperature
of dye molecules.
### 2.3 BEC of photons and photon pairs
In virtue of the above analysis, we consider the following basic Hamiltonian,
to give a simple model of PP formation (with $\hbar=1$ throughout this letter)
$\displaystyle H_{\mu}=H_{a}+H_{b}+H_{ab},$ (12)
with
$\displaystyle\begin{array}[]{l}H_{a}=\sum\limits_{\bf{k}}{\frac{{{\bf{k}}^{2}}}{{2m_{ph}}}a_{\bf{k}}^{+}a_{\bf{k}}}+\frac{{u_{aa}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}},{\bf{k^{\prime\prime}}}}{a_{{\bf{k}}+{\bf{k^{\prime}}}-{\bf{k^{\prime\prime}}}}^{+}a_{{\bf{k^{\prime\prime}}}}^{+}}a_{{\bf{k^{\prime}}}}a_{\bf{k}}\\\
H_{b}=\sum\limits_{\bf{k}}{\left({\frac{{{\bf{k}}^{2}}}{{4m_{ph}}}-2\mu}\right)b_{\bf{k}}^{+}b_{\bf{k}}}+\frac{{u_{bb}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}},{\bf{k^{\prime\prime}}}}{b_{{\bf{k}}+{\bf{k^{\prime}}}-{\bf{k^{\prime\prime}}}}^{+}b_{{\bf{k^{\prime\prime}}}}^{+}}b_{{\bf{k^{\prime}}}}b_{\bf{k}}\\\
H_{ab}=\frac{{u_{a,b}}}{S}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}}}{a_{\bf{k}}^{+}b_{{\bf{k^{\prime}}}}^{+}}b_{{\bf{k^{\prime}}}}a_{\bf{k}}-\frac{1}{{\sqrt{S}}}\sum\limits_{{\bf{k}},{\bf{k^{\prime}}}}{\chi_{{\bf{k}},{\bf{k^{\prime}}}}(b_{{\bf{k}}+{\bf{k^{\prime}}}}^{+}}a_{{\bf{k^{\prime}}}}a_{\bf{k}}+H.c.)\\\
\end{array}$ (16)
Above, $H_{a}$ and $H_{b}$ denote the pure photon and PP contributions, and
$H_{ab}$ refers to the interaction between them. In the dilute gas limit
$u_{a,a}$, $u_{b,b}$, $u_{a,b}$ are proportional to the two-body s-wave
photon-photon, photon-PP, and PP-PP scattering lengths [23], respectively, and
$\chi_{{\bf{k}},{\bf{k^{\prime}}}}$ characterizes the coupling strength,
encoding that PPs are composed of two massive photons. Note that in (9) we
ignore the chemical potential term $\mu N_{a}$ from the Hamiltonian [24].
Additionally, in (9), we also drop the subscript of the transverse wave-
vectors of photons and PPs, i.e. we take ${\bf{k}}_{r}={\bf{k}}$.
It is well known that for a general massive Boson-Boson pairs gas, at absolute
zero temperature, there exists a BEC consisting of two possible condensate
phases [25-27]: (i) Both the single boson and the pair of bosons are
condensed. (ii) The pair of bosons are condensed but the single boson is not.
Now we know that in the case of photons (or PPs) confined inside the
microcavity, the subsystem of photons (or PPs) is formally equivalent to a 2D
gas of massive bosons with non-vanishing chemical potential. Thus, the feature
should also survive for the 2D mixed gas of photons and PPs. In the condensate
phase, a macroscopic number of particles occupy the zero-momentum state, and
it is useful to separate out the condensate modes from the Hamiltonian. Follow
the process, we find that at the BEC state, the grand canonical Hamiltonian of
the mixed system has the form
$\displaystyle H_{\mu}=H_{0}+\delta H,$ (17)
where
$\displaystyle\begin{array}[]{l}H_{0}=-\mu_{b}b^{+}b+\frac{{u_{aa}}}{S}\left({a^{+}a}\right)^{2}+\frac{{u_{bb}}}{S}\left({b^{+}b}\right)^{2}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\frac{{u_{ab}}}{S}a^{+}ab^{+}b-\frac{\chi}{{\sqrt{S}}}\left({a^{+}a^{+}b+b^{+}aa}\right)\\\
\end{array}$ (20)
is the condensate part of the Hamiltonian with $\mu_{b}=2\mu+u_{bb}/S$ the
modified chemical potential of PPs, and where $\delta
H=H\left({a_{\bf{k}},b_{\bf{k}}}\right)$ is the perturbation part and it is a
complex function of the non-condensate modes. Here, we mention that at the
condensate state we only need consider the case of single mode coupling, thus
we can treat the coupling matrix element $\chi$ as an adjustable constant.
Note that in (11) we also drop the zero-momentum subscript of the creation and
annihilation operators of photons and PPs. In the present work, we mainly aim
to investigate the phenomenon of BEC of photons and PPs, thus, hereafter we
will ignore the perturbation part and approximately write the Hamiltonian as
the form $H_{\mu}\approx H_{0}$. The Hamiltonian commutes with the total
photon number $N=a^{+}a+2b^{+}b$ and $n=N/S$ is the particle density.
Up to now we have not made a careful distinction between the two possible
condensate phases. However, for the mixed system, working out the ground-state
phase diagram is very important. Here, we intend to employ the variational
principle method for finding the ground-state configurations of the present
system and examining their dependence from the microscopic parameters. In
other word, we aim to work out the ground-state phase diagram of the mixed
system starting from the study the semiclassical equation. Clearly, we know
that the chemical potential $\mu$ of the mixed system allows the total photon
number (whether free or bound into PPs) to fluctuate around some constant
average value $N$, then the total number of photons need only be conserved on
the average value. For convenience, hereafter we assume that $N$ is an even
number and thus $M=N/2$ denotes the maximum number of the PPs. Furthermore, in
this case we also introduce a new operation, namely the double photon creation
operation with the relation
$\displaystyle\begin{array}[]{l}\left({c^{+}}\right)^{m}\left|0\right\rangle\equiv\sqrt{\frac{{m!}}{{\left({2m}\right)!}}}\left({a^{+}}\right)^{2m}\left|0\right\rangle\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}c^{+}c\left|0\right\rangle\equiv 2a^{+}a\left|0\right\rangle\\\
\end{array},$ (23)
where $\left|0\right\rangle$ is the vacuum state. Using the new operator, we
construct the Gross-Pitaevskii (GP) states [28]
$\displaystyle\left|{\psi_{{\rm{GP}}}}\right\rangle=\frac{1}{{\sqrt{M!}}}\left[{\alpha
c^{+}+\beta b^{+}}\right]^{M}\left|0\right\rangle$ (24)
as the trial macroscopic state. Here,
$\alpha=\left|\alpha\right|e^{i\theta_{a}}$ and
$\beta=\left|\beta\right|e^{i\theta_{b}}$ are complex amplitudes with
$\left|\alpha\right|^{2}=N_{a}/N$ and $\left|\beta\right|^{2}=2N_{b}/N$ the
photon and PP densities, respectively. $\theta_{a}$ and $\theta_{b}$ (real
valued) denoted the phases of each species. Obviously, the parameters $\alpha$
and $\beta$ satisfy the normalized condition
$\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$. With the help of the GP
state, the semiclassical model Hamiltonian $\bar{H}(\alpha,\beta)$ is given by
$\displaystyle\begin{array}[]{l}\bar{H}=\mathop{\lim}\limits_{N\to\infty}\frac{{\left\langle{\psi_{GP}}\right|H_{\mu}\left|{\psi_{GP}}\right\rangle}}{{M{\chi}\sqrt{2n}}}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\left[{-\mu_{b}\left|\beta\right|^{2}+2u_{aa}n\left|\alpha\right|^{4}+u_{bb}n\left|\beta\right|^{4}}\right./2\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{{\left.{+u_{ab}n\left|\alpha\right|^{2}\left|\beta\right|^{2}}\right]}\mathord{\left/{\vphantom{{\left.{+u_{ab}n\left|\alpha\right|^{2}\left|\beta\right|^{2}}\right]}{{\chi}\sqrt{2n}}}}\right.\kern-1.2pt}{{\chi}\sqrt{2n}}}-2\left|\alpha\right|^{2}\sqrt{\left|\beta\right|^{2}}\cos\theta\\\
\end{array},$ (28)
where $\theta=\theta_{b}-2\theta_{a}$ is the phase difference. Considering the
conserved condition $\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1$, we
next introduce a new variables
$s=\left|\alpha\right|^{2}-\left|\beta\right|^{2}$. Using the new notation, we
rewrite the model Hamiltonian as
$\displaystyle\bar{H}=-\lambda s^{2}-2\gamma
s+\xi-\sqrt{2\left({1-s}\right)}(1+s)\cos\theta,$ (29)
with
$\begin{array}[]{l}\lambda=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{ab}}}{4}-\frac{{u_{aa}}}{2}-\frac{{u_{bb}}}{8}}\right)\\\
\gamma=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{bb}}}{8}-\frac{{u_{aa}}}{2}-\frac{{\mu_{b}}}{{4n}}}\right)\\\
\xi=\frac{{\sqrt{2n}}}{{\chi}}\left({\frac{{u_{aa}}}{2}+\frac{{u_{ab}}}{4}+\frac{{u_{bb}}}{8}-\frac{{\mu_{b}}}{n}}\right)\\\
\end{array}$.
According to the variational principle, we minimize the energy
$\bar{H}(\alpha,\beta)$ with $s$ and $\theta$ as variational parameters. We
then obtain the optimum values [i.e.($\bar{s}$, $\bar{\theta}$)] of parameters
for the ground state as follows:
$\displaystyle\left({\bar{s},\bar{\theta}}\right)=\left\\{\begin{array}[]{l}(-1,{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\theta),{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\gamma-\lambda+1<0\\\ (s,{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}0{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}or{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}\pi),{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\gamma-\lambda+1>0\\\ \end{array}\right.,$
(32)
where $-1<s<1$ is the solution of the equation $\lambda
s+\gamma=\left({3s-1}\right)/2\sqrt{2(1-s)}$ (the explicit value can be obtain
by graphical solution method, it is generally too messy to be shown here). The
result, together with the fact that the parameters $\left|\alpha\right|^{2}$
and $\left|\beta\right|^{2}$ denote the photon and PP densities, indicates
that when $\gamma-\lambda+1<0$ the system converts from the mixed photon-PP
phase to the pure PP phase. We therefore can interpret this line
$\gamma-\lambda+1=0$ as the threshold coupling for the formation of a
predominantly PP state. Here, it is to be mentioned that for the pure PP
phase, $\bar{s}=-1$ thus the relative phase $\theta$ cannot be defined.
## 3 Entanglement between photons and photon pairs
### 3.1 Entanglement of the ground state
At the BEC state, one may consider the mixed gas of photons and PPs as a
bipartite system of two modes. For the present system, the entanglement of two
modes is always closely associated with the phase transition of the system.
Moreover, the two modes, be they spatially separated, and differing in some
internal quantum number, are clearly distinguishable subsystems. Thus, the
state of each mode can be characterized by its occupation number. By using the
fact that the total number of photons $N$ is constant, a general state of the
system (in the Heisenberg picture) can be written for even $N$ in terms of the
Fock states by
$\displaystyle\left|\psi\right\rangle=\sum\limits_{m=0}^{M}{c_{m}}\left|{2m,M-m}\right\rangle,$
(33)
where $m$ is the half population of particles in photon mode $a$, and $c_{m}$
is the coefficients of the state. In the Fock representation, the GP state
also can be reexpressed as
$\left|{\psi_{GP}}\right\rangle=\sum\limits_{m=0}^{M}{g_{m}}\left|{2m,M-m}\right\rangle$,
with coefficients
$g_{m}=\sqrt{\frac{{M!}}{{m!\left({M-m}\right)!}}}\alpha^{m}\beta^{M-m}$. The
standard measure of entanglement of the bipartite system is the entropy of
entanglement $S(\rho)$
$\displaystyle
S(\rho)=-\sum\limits_{m=0}^{M}{\left|{c_{m}}\right|^{2}}\log_{2}\left({\left|{c_{m}}\right|^{2}}\right),$
(34)
which is the von Neumann entropy of the reduced density operator of either of
the subsystems[29]. In the present system, the maximal entanglement also can
be obtain by optimizing the expression (18) with respect to
$\left|{c_{m}}\right|^{2}$. By imposing the normalization condition
$\sum\limits_{m=0}^{M}{\left|{c_{m}}\right|^{2}}=1$, we finally get
$S_{\max}=\log_{2}\left({M+1}\right)$, which is related to the dimension $M+1$
of the Hilbert space of the individual modes.
Using expression (18) and the coefficients $c_{m}$ obtained through exact
diagonalization of the Hamiltonian (11) as done in the atom-molecule model
[30], we plot in Fig. 2 the entropy of entanglement of the ground state as a
function of the parameters $\lambda$ and $\gamma$. We note that in this letter
we restrict our attentions to the repulsive case, i.e., we restrict
$\lambda\leq 0$ throughout the letter. From Fig. 2, we observe that the
entanglement entropy exhibits a sudden decrease close $\gamma-\lambda+1=0$.
This is indicative of the fact that across the line $\gamma-\lambda+1=0$ a
quantum phase transition occurs.
Figure 2: Variation in the entropy of entanglement of the ground state with
respect to the parameters $\lambda$ and $\gamma$. Here, we set $N=200$ and
$u_{aa}=u_{bb}/4=0.25$.
To gain more information associated with the quantum phase transition of the
present system, we also depict in Fig. 3 the entropy of entanglement (solid
line) and the expectation value (dashed line) of the scaled PP number operator
of the ground state as a function $\gamma$ for fixed parameter value
$\lambda$. From Fig. 3, we see that the average value of the number of PPs
increases as $\gamma$ increases. Especially, when $\gamma-\lambda+1<0$, the
average number of PPs is maximal. The result confirms that there indeed exists
a phase transition for the present system in the ground state. Furthermore, we
also find that the ground-state entanglement entropy is not maximal at the
critical line, i.e. in the region $\gamma-\lambda+1>0$, the system is always
strongly entangled. Ref. (28) gives the property responsible for the long-
range correlation. Additionally, we also consider that the trait is associated
with the symmetry-broking of the coupling term of the system. Due to the
asymmetric form of the coupling term, the pure photon condensation will be
forbidden. As a result, in the mixed condensate phase the imbalance
$\left({2N_{b}-N_{a}}\right)/N$ between the two modes is always very small,
which is responsible for the strongly entanglement.
Figure 3: The average photon pair occupation number and the entanglement
entropy for the ground state as a function of the $\gamma$ for fixed parameter
value $\lambda=0$. Here, we set $N=200$ and $u_{aa}=u_{bb}/4=0.25$.
In addition, if we connect the non-vanishing photon mass $m_{ph}$ to the
longitudinal wave number $k_{z}$ by the relation $m_{{\rm{ph}}}=\hbar k_{z}/c$
with $k_{z}=q\pi/D_{0}$, then we find that the quantum phase transition of the
photon system can be interpreted as second harmonic generation. When
$\gamma-\lambda+1<0$, almost all photons with frequency $\omega=ck_{z}$ couple
into PPs with frequency $\omega=2ck_{z}$. In this case, the entanglement
between the photons and PPs is very small, and the entropy of entanglement is
close to zero.
### 3.2 Dynamics of entanglement
In the above analysis, we have investigated the entanglement of the ground
state. We found that in the ground state, across the phase transition line the
entanglement entropy exhibits a sudden change. To gain a better understanding
of the influence of ground-state phase transition to entanglement, in this
subsection we investigate the dynamics of entanglement. In studying the
dynamics of the system, we first need express a general state in the form of
temporal evolution (i.e., need change the expression of a general state from
Heisenberg picture to Schrodinger picture). Following the standard procedure,
we can obtain
$\displaystyle\begin{array}[]{l}\left|{\psi\left(t\right)}\right\rangle=U\left(t\right)\left|{\psi\left(0\right)}\right\rangle\\\
=\sum\limits_{m=0}^{M}{c_{m}}\left(t\right)\left|{2m,M-m}\right\rangle\\\
\end{array},$ (37)
where,
$U\left(t\right)=\sum\limits_{n=0}^{M}{\left|{\psi_{n}}\right\rangle\left\langle{\psi_{n}}\right|}\exp\left({-iE_{n}t}\right)$
is the temporal operator with $\left|{\psi_{n}}\right\rangle$ the eigenstates
of the system having energy $E_{n}$, and
$\left|{\psi\left(0\right)}\right\rangle$ is the initial state. Here, the time
dependence of coefficients $c_{m}\left(t\right)$ are given by
$c_{m}\left(t\right)=\left\langle{2m,M-m}\right|U\left(t\right)\left|{\psi\left(0\right)}\right\rangle$.
Subsequently, the entanglement entropy given in (18) can be rewritten as
$S(\rho)=-\sum\limits_{m=0}^{M}{\left|{c_{m}\left(t\right)}\right|^{2}}\log_{2}\left({\left|{c_{m}\left(t\right)}\right|^{2}}\right)$.
In this case, the entanglement entropy depends on both the choice of initial
states and the value of microscopic parameters
$\left\\{{\lambda,\gamma}\right\\}$. At the present work, we consider that the
mixed system is in the BEC state, thus here choosing GP state as the initial
state is suitable. By adjusting the GP coefficients
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}$ and the
microscopic parameters $\left\\{{\lambda,\gamma}\right\\}$, in this subsection
we also want to know if the ground-state phase transition also characterizes
different dynamics.
The time evolution of the entanglement entropy for different initial state and
interaction parameters is shown in Fig. 4. From Fig.4 we observe the features
of quantum dynamics, such as the collapse and revival of oscillations and non-
periodic oscillations. Additionally, we also find that the amplitude of the
entanglement entropy is smaller in the region $\gamma-\lambda+1<0$ contrast
with in the region $\gamma-\lambda+1>0$. Especially, we note that the greater
the imbalance $\left|\beta\right|^{2}-\left|\alpha\right|^{2}$ between the two
modes in the initial state, the clearer the difference can be observed.
Figure 4: Time evolution of the entanglement entropy for different initial
states
$\left|{\psi\left({\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right)}\right\rangle$
and microscopic parameters $\left\\{{\lambda,\gamma}\right\\}$. Form top to
bottom the GP coefficients used are
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0.5,0.5}\right\\}$,
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0.25,0.75}\right\\}$
and
$\left\\{{\left|\alpha\right|^{2},\left|\beta\right|^{2}}\right\\}=\left\\{{0,1}\right\\}$.
From left to right the microscopic parameters used are
$\left\\{{\gamma,\lambda}\right\\}=\left\\{{-2,0}\right\\}$ and
$\left\\{{\gamma,\lambda}\right\\}=\left\\{{-0.5,0}\right\\}$, respectively.
Here, we set $N=20$ and $u_{aa}=u_{bb}/4=0.25$.
To understand the physical reason for the above phenomenon, we also need
rewrite the general state given in (19) in terms of the eigenstates of the
system
$\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{n=0}^{M}{c(n,t)}\left|{\psi_{n}}\right\rangle$,
where
$\left|{c(n,t)}\right|^{2}=\left|{c(n,0)}\right|^{2}=\left|{\left\langle{\psi_{n}}\right|\exp(-iE_{n}t)\left|{\psi\left(0\right)}\right\rangle}\right|^{2}$
can be explained as the transition probability of the system from the initial
state $\left|{\psi\left(0\right)}\right\rangle$ to the corresponding energy
eigenstates $\left|{\psi_{n}}\right\rangle$ at any time $t$. We have already
known that for the ground state across the phase transition line
$\gamma-\lambda+1=0$ the entanglement entropy exhibits a sudden decrease. This
is why in the region $\gamma-\lambda+1<0$ the amplitude of the entanglement
entropy becomes smaller. In addition, in this phase transition region we also
investigative the dependence relation between the ground-state transition
probability $\left|{c\left({0,t}\right)}\right|^{2}$ and the imbalance
$\left|\beta\right|^{2}-\left|\alpha\right|^{2}$ of the initial state. The
result is shown in Fig. 5. From Fig. 5, it is obvious that with the increasing
of the initial-state imbalance the ground-state transition probability becomes
greater. Thus the greater imbalance between the two modes in the initial state
can lead the clearer difference of the amplitude for different region.
Figure 5: Variation of the ground-state transition probability
$\left|{c\left({0,t}\right)}\right|^{2}$ with respect to the initial-state
imbalance $\left|\beta\right|^{2}-\left|\alpha\right|^{2}$. Here, we set
$N=20$ and $u_{aa}=u_{bb}/4=0.25$.
## 4 Conclusion
In this work, we have proposed a 2D model consisting of photons and PPs. In
the model, the mixed gas of photons and PPs is formally equivalent to a 2D
system of massive bosons with non-vanishing chemical potential, which implies
the existence of two possible condensate phase. Based on the GP state and
using the variational method, we have also discussed the quantum phase
transition of the mixed gas and have obtained the critical coupling line
analytically. Especially, we have found that the phase transition of the
photon gas can be interpreted as second harmonic generation. Moreover, by
investigating the entanglement entropy in the ground state and general state,
we have illustrated how the entanglement between photons and PPs can be
associated with the phase transition of the system.
## References
* [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269 (1995) 198.
* [2] K. B. Davis, M-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75 (1995) 3969.
* [3] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78 (1997) 985.
* [4] S. Jochim, _et al_ , Science 302 (2003) 2101.
* [5] H. Deng, G. Weihs, J. Bloch, and Y. Yamamoto, Science 298 (2002) 199.
* [6] J. Kasprzak, _et al_ , Nature 443 (2006) 409.
* [7] R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West, Science 316 (2007) 1007.
* [8] S.O. Demokritov, _et al_ , Nature 443 (2006) 430.
* [9] G. Ortiz and J. Dukelsky, Phys. Rev. A 72 (2005) 043611.
* [10] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, Nature 468 (2010) 545.
* [11] J. Klaers, F. Vewinger, and M. Weitz, Nature Phys. 6 (2010) 512.
* [12] D. F. Walls, and P. Zoller Phys. Rev. Lett. 47 (1981) 709.
* [13] G. Rempe, F. Schmidt-Kaler, and H. Walther, Phys. Rev. Lett 64 (1990) 2783.
* [14] O. V. Kibis, G. Ya. Slepyan, S. A. Maksimenko, and A. Hoffmann, Phys. Rev. Lett 102 (2009) 023601.
* [15] D. F. Walls, and R. Barakat, Phys. Rev. A 1 (1970) 446.
* [16] Z. Cheng, Phys. Rev. Lett. 67 (1991) 2788.
* [17] T. Yamamoto, M. Koashi, S. K. Özdemir, and N. Imoto, Nature 421 (2003) 343.
* [18] Z. D. Walton, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 70 (2004) 052317.
* [19] W. Denk, J. H. Strickler, and W. W. Webb, Science 6 (1990) 73.
* [20] T. Binoth, J.Ph. Guillet, E. Pilon, and M. Werlen, Eur. Phys. J. C 16 (2000) 311.
* [21] Z. Cheng, J. Opt. Soc. Am. B 19 (2002) 1962.
* [22] N. Bloembergen, Non-Linear Optics (W. A. Benjamin, Inc., New York, 1965)
* [23] J. Klaers, J. Schmitt, T. Damm, F. Vewinger, and M. Weitz Appl. Phys. B 105 (2011) 17.
* [24] It may be asked why this term can be ignored. The answer is connected with the fact that in this letter we only consider the infinite particle number case, i.e., we consider the total photon number $N$ as a constant. In the case, the chemical potential term $\mu N_{a}$ cannot give new restriction to the Hamiltonian. Especially, in the following analysis we will point out that at the BEC state the mixed gas of photons and PPs can convert from the mixed photon-PP condensate phase to the pure PP condensate phase. In the pure PP condensate phase, the term $\mu N_{a}$ is meaningless. We therefore ignore the restriction term $\mu N_{a}$ in the model Hamiltonian.
* [25] L. Radzihovsky, J. Park, and P. B. Weichman, Phys. Rev. Lett. 92 (2004) 160402.
* [26] M. W. J. Romans, R. A. Duine, S. Sachdev, and H. T. C. Stoof, Phys. Rev. Lett. 93 (2004) 020405;
* [27] G. Santos, A. Tonel, A. Foerster, and J. Links, Phys. Rev. A 73 (2006) 023609; M. Duncan, A. Foerster, J. Links, E. Mattei, N. Oelkers, and A. P. Tonel, Nucl. Phys. B 767 (2007) 227; G. Santos, A. Tonel, A. Foerster, and J. Links, Phys. Rev. A 73 (2006) 023609.
* [28] S. C. Li, J. Liu, and L. B. Fu, Phys. Rev. A 83, (2011) 042107; S. C. Li, and L. B. Fu, Phys. Rev. A 84, (2011) 023605.
* [29] A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 67, (2003) 013609.
* [30] A. P. Tonel, J. Links, and A. Foerster, J. Phys. A:Math. Gen. 38, (2005) 1235; J. Links, H. Q. Zhou, R. H. McKenzie, and M. D. Gould, J. Phys. A 36 (2003) R63.
|
arxiv-papers
| 2012-04-04T13:58:52 |
2024-09-04T02:49:29.354502
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianjun Zhang, Jianhui Yuan",
"submitter": "Jianjun Zhang",
"url": "https://arxiv.org/abs/1204.0951"
}
|
1204.0956
|
# Electron transport in a ferromagnetic/normal/ferromagnetic tunnel junction
based on the surface of a topological insulator
Jian-Hui Yuan The corresponding author, E-mail: jianhui831110@163.com Yan
Zhang The department of Physics, Guangxi medical university, Nanning,
Guangxi, 530021, China Jian-Jun Zhang Ze Cheng School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China
###### Abstract
We theoretically study the electron transport properties in a
ferromagnetic/normal/ferromagnetic tunnel junction, which is deposited on the
top of a topological surface. The conductance at the parallel (P)
configuration can be much bigger than that at the antiparallel (AP)
configuration. Compared P with AP configuration, there exists a shift of
phase which can be tuned by gate voltage. We find that the exchange field
weakly affects the conductance of carriers for P configuration but can
dramatically suppress the conductance of carriers for AP configuration. This
controllable electron transport implies anomalous magnetoresistance in this
topological spin valve, which may contribute to the development of spintronics
. In addition, we find that there is a Fabry-Perot-like electron interference.
Topological insulator, Electronic transport, Ferrimagnetic
###### pacs:
72.80.Sk, 73.40.-c, 75.50.Gg
## I Introduction
The concept of a topological insulator (TI) dates back to the work of Kane and
Mele, who focused on two-dimensional (2D) systems 1. There has been much
recent interest in TIs, three-dimensional insulators with metallic surface
states protected by time reversal invariance ${[1-25]}$. Its theoretical
${[2]}$ and experimental ${[3]}$ discovery has accordingly generated a great
deal of excitement in the condensed matter physics community. In particular,
the surface of a three-dimensional (3D) TI, such as Bi2Se3 or Bi2Te3 ${[4]}$,
is a 2D metal, whose band structure consists of an odd number of Dirac cones,
centered at time reversal invariant momenta in the surface Brillouin zone
${[5]}$. This corresponds to the infinite mass Rashba model ${[6]}$, where
only one of the spin-split bands exists. This has been beautifully
demonstrated by the spin- and angle-resolved photoemission spectroscopy
${[7,8]}$. Surface sensitive experiments such as angle-resolved photoemission
spectroscopy (ARPES) and scanning tunneling microscopy (STM) ${[9,10]}$ have
confirmed the existence of this exotic surface metal, in its simplest form,
which takes a single Dirac dispersion. Recent theoretical and experimental
discovery of the two dimensional (2D) quantum spin Hall system ${[11-18]}$ and
its generalization to the TI in three dimensions ${[19-21]}$ have established
the state of matter in the time-reversal symmetric systems.
The time-reversal invariant TI is a new state of matter, distinguished from a
regular band insulator by a nontrivial topological invariant, which
characterizes its band structure ${[11]}$. Currently, most works focus on
searching for TI materials and novel transport properties. To my knowledge,
the fabrication of such TI-based nanostructure is still a challenging task.
Usually such structures are fabricated by utilizing the split gate and etching
technique ${[22]}$. On the other hand, the 3D TIs are expected to show several
unique properties when the time reversal symmetry is broken ${[23-25]}$. This
can be realized directly by a ferromagnetic insulating (FI) layer attached to
the 3D TI surface. One remarkable feature of the Dirac fermions is that the
Zeeman field acts like a vector potential: the Dirac Hamiltonian is
transformed as $\sigma\cdot\textbf{k}\longrightarrow\sigma\cdot(\textbf{k+H})$
by the Zeeman field H ${[26]}$. This is in contrast to the
Schr$\ddot{o}$dinger electrons in conventional semiconductor heterostructures
modulated by nanomagnets ${[27-29]}$.
In this work, we study the electron transport properties in a ferromagnetic
(F)/normal/N)/ ferromagnetic (F) tunnel junction, which is deposited on the
top of a topological surface. Ferromagnetic Permalloy electrodes are formed by
electron-beam lithography (EBL) followed by thermal evaporation; a second EBL
step establishes contact to the Permalloy via Cr /Au electrodes ${[30]}$. As
shown in Fig.1, the FI is put on the top of the TI to induce an exchange field
via the magnetic proximity effect. The easy axis of a FI stripe is usually
along its length direction and thus either in parallel (P) or antiparallel
(AP) with the $+y$ axis. We find that the conductance at the P configuration
can be much bigger than that at the AP configuration. Compared P with AP
configuration, there exists a shift of phase which can be tuned by gate
voltage. We find that the exchange field weakly affects the conductance of
carriers for P configuration but can dramatically suppress the conductance of
carriers for AP configuration. This controllable electron transport implies
anomalous magnetoresistance in this topological spin valve, which may
contribute to the development of spintronics. Compared with the conventional
F/N/F tunneling based on two dimensional electron gas (2DEG), the result
implies the existence of Fabry-Perot-like electron interference in F/N/F based
on the TI. In Sec. II , we introduce the model and method for our calculation.
In Sec. III, the numerical analysis to our important issues is reported.
Finally, a brief summary is given in sec. IV.
## II model and method
Figure 1: Schematic illustration of the device. Top: Schematic diagram of two
ferromagnetic barriers on the topological surface divided by a gate electrode
at a distance $L$. Bottom: The magnetization directions of adjacent FI stripes
are parallel (P) in the configuration and antiparallel (AP) in the
configuration.
Now, let us consider a F/N/ F tunnel junction which is deposited on the top of
a topological surface where a gate electrode is attached to the ferromagnetic
material. The ferromagnetism is induced due to the proximity effect by the
ferromagnetic insulators deposited on the top as shown in Fig. 1. We assume
that the initial magnetization of FI stripes in the region I is aligned with
the +y axis. In an actual experiment, one can use a magnet with very strong
(soft) easy axis anisotropy to control the ferromagnetic material. Thus we
focus on charge transport at the Fermi level of the surface of TIs, which is
described by the 2D Dirac Hamiltonian
$\displaystyle
H=\upsilon_{F}\sigma\cdot\textbf{p}+\sigma\cdot\textbf{M}+V{(x)},$ (1)
where $\sigma$ is Pauli matrices ,
$\textbf{M}=M_{y}(x)=M_{0}(\Theta(-x)+\gamma\Theta(x-L))$ is the effective
exchange field and $V(x)=U_{g}\Theta(x)\Theta(L-x)+V_{g}\Theta(x-L)$ is the
gate voltage, where $\gamma=+1$ ($-1$) corresponds to the P (AP)
configurations of magnetization and $\Theta(x)$ is the Heaviside step
function.
Because of the translational invariance of the system along $y$ direction, the
equation $H\Psi(x,y)=E\Psi(x,y)$ admits solutions of the form
$\Psi(x,y)=(\Psi_{1}(x),\Psi_{2}(x))^{T}\exp(ik_{y}y)$. We set
$\hbar=\upsilon_{F}=1$ in the following. Then, with the above Hamiltonian, the
wave function in the whole system is given by
$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lllll
}\exp(ik_{x_{1}}x)+r\exp(-ik_{x_{1}}x),&\quad x<0,\\\ \\\
a\exp(iq_{x}x)+b\exp(-iq_{x}x),&\quad 0<x<L,\\\ \\\
t\exp(ik_{x_{2}}(x-L)),&\quad x>L,\end{array}\right.$ (7)
$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lllll
}\alpha^{+}\exp(ik_{x_{1}}x)+r\alpha^{-}\exp(-ik_{x_{1}}x),&\quad x<0,\\\ \\\
a\beta^{+}\exp(iq_{x}x)+b\beta^{-}\exp(-iq_{x}x),&\quad 0<x<L,\\\ \\\
t\alpha\exp(ik_{x_{2}}(x-L)),&\quad x>L,\end{array}\right.$ (13)
where $k_{x_{1}}=E\cos\theta_{F_{1}}$, $q_{x}=(E-U_{g})\cos\theta$ and
$k_{x_{2}}=(E-V_{g})\cos\theta_{F_{2}}$ are wave vectors in region I, region
II and region III, $\alpha^{\pm}=\pm\exp(\pm i\theta_{F_{1}})$,
$\beta^{\pm}=\pm\exp(\pm i\theta)$ and $\alpha=\exp(i\theta_{F_{2}})$. The
momentum $k_{y}$ conservation should be satisfied everywhere such as
$k_{y}=E\sin\theta_{F_{1}}-M_{0}=(E-U_{g})\sin\theta=(E-V_{g})\sin\theta_{F_{2}}-\gamma
M_{0}$. Also, $r$ and $t$ are reflection and transmission coefficients,
respectively. Continuities of the wave function $\Psi$ at $x=0$ and $x=L$ are
$\Psi(0^{-})=\Psi(0^{+})$ and $\Psi(L^{-})=\Psi(L^{+})$, respectively. We find
that the transmitted electron coefficient $t_{\gamma}$ is given by
$\displaystyle
t_{\gamma}=\frac{2\cos\theta_{F_{1}}\cos\theta\exp(-ik_{x_{2}}L)}{s_{1,\gamma}\cos(q_{x}L)+is_{2,\gamma}\sin(q_{x}L)},$
(14)
with $s_{1,\gamma}=\cos\theta(\exp(i\theta_{F_{2}})+\exp(-i\theta_{F_{1}}))$
and
$s_{2,\gamma}=i\sin\theta(\exp(-i\theta_{F_{1}})-\exp(i\theta_{F_{2}}))-\exp(i(\theta_{F_{2}}-\theta_{F_{1}}))-1$.
Then
$\displaystyle
T_{\gamma}=|t_{\gamma}|^{2}\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}}),$ (15)
where the factor $\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$ is due to
current conservation. In the linear transport regime and for low temperature,
we can obtain the conductance $G$ by introducing it as the electron flow
averaged over half the Fermi surface from the well-known Landauer-Buttiker
formula ${[25,31,32]}$
$\displaystyle G_{\gamma}\sim
1/2\int_{-\pi/2}^{\pi/2}T_{\gamma}(E_{F},E_{F}\cos\theta_{F_{1}})\cos\theta_{F_{1}}d\theta_{F_{1}}.$
(16)
## III Results and Discussions
Figure 2: Gate voltage dependence of the conductances with a P ($\gamma=1$)
and AP ($\gamma=-1$) configuration in the two cases: (a) $V_{g}/E_{F}=0$ and
(b) $V_{g}/E_{F}=2$. The values of the other parameters are $E_{F}=0.1$ and
$\eta=0.5$
For convenience we express all quantities in dimensionless units by means of
the length of the basic unit $L$ and the energy $E_{0}=\hbar v_{F}/L$. For a
typical value of $L=50$ nm and the Bi2Se3 material $v_{F}=5\times 10^{5}$ m/s,
one has $E_{0}=6.6$ meV. We set the energy of electron $E=E_{F}$ and also
define the value $\eta$ with the form $\eta=M_{0}/E_{F}$ in our calculation.
In Fig.2, we show gate voltage dependence of the conductances with a P
($\gamma=1$) and AP ($\gamma=-1$) configuration in the two cases: (a)
$V_{g}/E_{F}=0$ and (b) $V_{g}/E_{F}=2$. The value of the other parameter is
$E_{F}=0.1$ and $\eta=0.5$. The presence of quantum modulation are seen in
these two figures. We can see an oscillation of the electrical conductance
with a period of $\pi$ when the voltage $U_{g}$ is larger than $E_{F}$. The
conductance at the P configuration can be much bigger than that at the AP
configuration. We find that a minimum of conductance at the P configuration
corresponds to a maximum of conductance at the AP configuration [see in fig.2
(a)] when the voltage $U_{g}$ is larger than $E_{F}$. In Fig. 2(b), a similar
tendency to Fig. 2(a) is seen. In distinct contrast to Fig.2(a), a minimum of
conductance at the P configuration here corresponds to a maximum of
conductance at the AP configuration [see in fig.2 (b)]. That is to say, there
exists a shift of $\pi$-phase. To understand these results intuitively, we
consider that the gate voltage $U_{g}$ is larger than the Fermi energy
$E_{F}$. For the given Fermi energy $E_{F}=0.1$, the condition $U_{g}\gg
E_{F}$ is easily satisfied. In this limit we have $\theta\rightarrow 0$ and
hence the transmission probability
$T_{\gamma}\sim(2\cos^{2}\theta_{F_{1}}/(1+\cos\theta_{F_{1}}\cos\theta_{F_{2}}-\cos(2U_{g}L)\sin\theta_{F_{1}}\sin\theta_{F_{2}})\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$.
For $\gamma=1$ and $V_{g}/E_{F}=0$ (or $2$), we find the
$\theta_{F_{1}}\equiv\theta_{F_{2}}$ (or $-\theta_{F_{2}}$), and thus
$T_{\gamma}\sim\cos^{2}\theta_{F_{1}}/(1-\cos^{2}(U_{g}L+\delta)\sin^{2}\theta_{F_{1}})$
where $\delta=0$ (or $\pi/2$) corresponds to $V_{g}/E_{F}=0$ (or $2$). Thus
the phase difference between $V_{g}/E_{F}=0$ and $V_{g}/E_{F}=2$ is given by
$U_{g}L$. We find $G_{\gamma}\propto\cos^{2}(U_{g}L)$ for $V_{g}/E_{F}=0$ but
$G_{\gamma}\propto\sin^{2}(U_{g}L)$ for $V_{g}/E_{F}=2$. When $U_{g}L$ is
equal to the half period of $\pi$, a minimum of conductance will appear for
$V_{g}/E_{F}=0$ but a maximum of conductance will appear for $V_{g}/E_{F}=2$.
When $U_{g}L$ is equal to the period of $\pi$, a maximum of conductance will
appear for $V_{g}/E_{F}=0$ but a minimum of conductance will appear for
$V_{g}/E_{F}=2$. Furthermore, we find that $G_{\gamma}$ oscillates between
$2/3$ and $1$ for $\gamma=1$. For $\gamma=-1$ and $V_{g}/E_{F}=0$ (or $2$),
there is a similar tendency to the case of $\gamma=1$. We can see that
$G_{\gamma}$ is suppressed obviously by the strength of the effective exchange
field . Nevertheless, there exists a shift of $\pi$-phase because of the
factor $\cos(2U_{g}L)$.
Figure 3: Gate voltage dependence of the conductances with a P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for four different values $\eta=0,0.2,0.5$,
and $0.8$. The solid lines are for $V_{g}/E_{F}=0$ while the dashed lines are
for $V_{g}/E_{F}=2$. The value of the other parameter is $E_{F}=0.1$.
In order to observe the effect of the exchange field $\eta$ on the
conductance, in Fig.3 we show the gate voltage dependence of the conductances
with a P ($\gamma=1$) and AP ($\gamma=-1$) configuration for four different
values $\eta=0,0.2,0.5$, and $0.8$. The solid lines are for $V_{g}/E_{F}=0$
while the dashed lines are for $V_{g}/E_{F}=2$. The value of the other
parameter is $E_{F}=0.1$. A similar tendency to Fig. 2 is seen in Fig. 3. It
is easily seen that the exchange field $\eta$ weakly affects the conductance
of carriers for $\gamma=1$ but profoundly influences the conductance of
carriers for $\gamma=-1$. For $\gamma=-1$, $G_{\gamma}$ is suppressed
obviously by increasing the value $\eta$. Due to current conservation, the
factor $\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$ must be real and then we
have $\sin\theta_{F_{1}}=\pm\sin\theta_{F_{2}}+2\eta$ where sign + (or -)
corresponds to $V_{g}/E_{F}=0$ (or 2). We can see
$2\eta-1\leq\sin\theta_{F_{1}}\leq 1$ and
$2\eta-1\leq\sin(\mp\theta_{F_{2}})\leq 1$ where sign - (or +) corresponds to
$V_{g}/E_{F}=0$ (or 2). Thus we find the ranges of the angle-allowable
$\theta_{F_{1}}$ and $\theta_{F_{2}}$ depend on $\eta$. The transmission is
nonzero only for $\theta_{F_{1}}$ and $\theta_{F_{2}}$ in these ranges and
vanishes for $\eta\geq 1$. The number of channels decreases with increasing of
$\eta$, so we can see that $G_{\gamma}$ dramatically decreases with the
increase of $\eta$ for $\gamma=-1$. Noting that the $\eta\geq 1$ for
$\gamma=-1$, the conductance of carriers is forbidden, which implies anomalous
magnetoresistance in this topological spin valve.
Figure 4: Gate voltage dependence of the conductances with a P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for three different values $E_{F}=0.1,1.0$,
and $5.0$. In (a) and (b), the $V_{g}$ is set as $V_{g}/E_{F}=0$ while in (c)
and (d) the $V_{g}$ is set as $V_{g}/E_{F}=2$. The value of the other
parameter is $\eta=0.5$.
In Fig. 4, we show the gate voltage dependence of the conductances with a P
($\gamma=1$) and AP ($\gamma=-1$) configuration for three different values
$E_{F}=0.1,1.0$, and $5.0$. In (a) and (b), the $V_{g}$ is set as
$V_{g}/E_{F}=0$ while in (c) and (d) the $V_{g}$ is set as $V_{g}/E_{F}=2$.
The value of the other parameter is $\eta=0.5$. For $E_{F}=0.1$, we can see
that the $\pi$ periodicity appears. However, the $\pi$ periodicity is broken
for $E_{F}=1$ (or $5$) because the condition $U_{g}\gg E_{F}$ is not satisfied
for the smaller $U_{g}$. Nevertheless, we get the $\pi$ periodicity of
conductance again by choosing a bigger $U_{g}$ for the bigger $E_{F}$.
Furthermore, we find that the minimum of the conductance will appear when the
gate voltage arrives at a certain value. It is easily seen that the minimum of
the conductance shifts to the right with increasing of the Fermi energy. The
larger the Fermi energy is, the smaller the minimum of the conductance is.
This phenomena is very obvious for the P ($\gamma=1$) configuration [see in
figs.4 (a) and (d)]. From Figs.4 and 5, we find that the conductance at the
parallel (P) configuration can be much bigger than that at the antiparallel
(AP) configuration. However it may be not satisfied for the larger Fermi
energy when the gate voltage is not bigger enough. We find that there is a
Fabry-Perot-like electron interference in the F/N/F tunnel junction, which is
deposited on the top of a topological surface. The two ferromagnetic
electrodes and the barrier can compose a Fabry-Perot resonator ${[33,34]}$.
The transmitted electron waves in this resonator can be reflected by the two
ferromagnetic electrodes. The electron waves undergo multiple reflections back
and forth along the resonator between the two ferromagnetic electrodes. The
conductance oscillations are caused by the interference of electron waves
among the modes of the channel-allowable. When the gate voltage $U_{g}$ is
larger than the Fermi energy $E_{F}$, the round trip between the two
ferromagnetic electrodes adds a further phase change $\delta\sim 4\pi/\lambda$
where the Fermi wavelength $\lambda\sim 2\pi/U_{g}$ because of the value
$\theta\sim 0$. When the round trip between the two ferromagnetic electrodes
is equal to the a multiple of wavelength, the quantum interference happens.
This implies that the oscillation period is equal to $\triangle U_{g}=\pi$.
Figure 5: Gate voltage dependence of the conductances with a P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for three different values $E_{F}=1.0,5.0$,
and $10.0$. In (a) , the $V_{g}$ is set as $V_{g}/E_{F}=0$ while in (b)
$V_{g}$ is set as $V_{g}/E_{F}=2$. The value of the other parameter is
$\eta=0.5$.
In order to investigate the solution of the standard electron described by the
Schr$\ddot{o}$dinger equation with a parabolic band structure, we consider a
2DEG in (x,y) plane with a magnetic field B in the z direction as described in
Refs.[26-28]. Thus we can fabricate a F/N/F tunneling based on the 2DEG. We
can apply all the relevant quantities in dimensionless units, which are the
same with the Ref. 28. So we can define the value $\Delta=M_{0}=B$ where
$M_{0}$ is the the effective exchange field corresponding to a F/N/F tunneling
based on the TI and $B$ is the magnetic field corresponding to a F/N/F
tunneling based on 2DEG. Nevertheless, we ignore the splitting of energy
induced by the spin of electron. As described in Fig.1, we set the left
electrode potential $V_{1}=0$. For the electron with parabolic spectrum,
${E_{F}}\sin{\theta_{{F_{1}}}}$ in Eq.(6) should be replaced by
$\sqrt{2{E_{F}}}\sin{\theta_{{F_{1}}}}$. Then the continuity of the wave
function gives the transmission coefficient
$\displaystyle{t_{\gamma}}=\frac{{2{\kern 1.0pt}{k_{1}}{\kern
1.0pt}{q_{x}}}}{{-{q_{x}}{\kern 1.0pt}({k_{1}}+{k_{3}})\cos{\kern 1.0pt}{\kern
1.0pt}({q_{x}}{\kern 1.0pt}L){\kern 1.0pt}+i{\kern 1.0pt}{\kern
1.0pt}({k_{1}}{k_{3}}+q_{x}^{2})s{\kern 1.0pt}in{\kern 1.0pt}({q_{x}}{\kern
1.0pt}L){\kern 1.0pt}{\kern 1.0pt}}},$
and transmission probability
${T_{\gamma}}={\left|{{t_{\gamma}}}\right|^{2}}{\kern 1.0pt}{\kern
1.0pt}{\mathop{\rm\Re}\nolimits}({k_{3}}/{k_{1}})$ where
${k_{1}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{1}}){\kern
1.0pt}{\kern 1.0pt}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\cos{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\theta_{{F_{1}}}}$,
${q_{x}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{2}}){\kern
1.0pt}{\kern 1.0pt}}\cos{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\theta$ and
${k_{3}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{3}}){\kern
1.0pt}{\kern 1.0pt}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\cos{\kern
1.0pt}{\kern 1.0pt}{\theta_{{F_{3}}}}$. We can easily see that
${T_{\gamma}}\equiv 0$ when the Fermi energy is smaller than the right
electrode voltage ${V_{3}}$ , which is different from the case of Dirac band
structure. In Fig.6, we show the wave vector $k_{y}$ dependence of the
transmission probability with a P ($\gamma=1$) and AP ($\gamma=-1$)
configuration for Dirac electrons shown in (a) and the standard electrons
shown in (b). The values of the other parameter are $\Delta=0.5$,
$V_{2}=U_{g}=0$, $V_{3}=V_{g}$ and $k_{x}=k_{x_{1}}=k_{1}=2$. We can find that
transmission is significantly more pronounced for Dirac electrons than for the
usual electrons. Compared with the P ($\gamma=1$) configuration , it can be
seen that the channel of electron transporting from the left electrode to the
right electron is suppressed especially for the standard electron. It is
easily seen that the variety of the tunneling conductance of F/N/F with the
change of the gate voltage is obviously different between the Dirac electron
and standard electron [see in Fig.7]. For the standard electron, the
conductance will decrease monotonously with the increase of the gate voltage
because of evanescent wave modes. However, we can see that the conductance
with a $\pi$ periodicity appears as the gate voltage is large enough for the
Dirac electron. On the one hand, Dirac confined electron exhibits a jittering
motion called Zitterbewegung , originating from the interference of states
with positive and negative energy. On the other hand, the transmitted Dirac
electron waves in this resonator can be reflected by the two ferromagnetic
electrodes one after another, so the phase interference will appear. As a
result, this implies the existence of Fabry-Perot-like electron interference
in a F/N/F tunneling based on the TI.
Figure 6: Wave vector $k_{y}$ dependence of the transmission probability with
a P ($\gamma=1$) and AP ($\gamma=-1$) configuration for the Dirac electrons
shown in (a) and the standard electrons shown in (b). The values of the other
parameter are $\Delta=0.5$, $V_{2}=U_{g}=0$, $V_{3}=V_{g}$ and
$k_{x}=k_{x_{1}}=k_{1}=2$.
Figure 7: Gate voltage $U_{g}$ dependence of the conductance with a P
($\gamma=1$) and AP ($\gamma=-1$) configuration for the Dirac electrons and
the standard electrons. The values of the other parameter are $\eta=0.5$,
$V_{3}=V_{g}=0$ and $E_{F}=1.0$.
## IV Conclusion
In summary, we have theoretically investigated transport features of Dirac
electrons on the surface of a three-dimensional TI under the modulation of a
exchange field provided by an FI stripes. We find that the conductance at the
P configuration can be much bigger than that at the AP configuration. Compared
P with AP configuration, there exists a shift of phase which can be tuned by
gate voltage. We find that the exchange field weakly affects the conductance
of carriers for P configuration but can dramatically suppress the conductance
of carriers for AP configuration. This controllable electron transport implies
anomalous magnetoresistance in this topological spin valve, which may
contribute to the development of spintronics .
Acknowledgments: This work was supported by the National Natural Science
Foundation of China under Grants No. 10174024 and No. 10474025.
## References
* (1) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 2005 ; C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 2005
* (2) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007).
* (3) D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. Hor, R. J. Cava, and M. Z. Hasan, Nature (London) 452, 970 (2008).
* (4) Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansi, D. Grauer, Y. S. Hor, R. J. Cava , M. Z. Hasan, Nature Phys. 5, 398 (2009); H. J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.-C. Zhang, Nature Phys. 5, 438 (2009)
* (5) T. Eguchi, P. Gilkey, and A. Hansen, Phys. Rep. 66, 213 (1980)
* (6) E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984)
* (7) D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Science 323, 919 (2009)
* (8) A. Nishide, A. Taskin, Y. Takeichi, T. Okuda, A. A. Kakizaki, T. Hirahara, K. Nakatsuji, F. Komori, Y. Ando, and I. Matsuda, Phys. Rev. B 81, 041309 (2010)
* (9) X.-L. Qi and S.-C. Zhang, Phys. Today 63, 33 (2010); J. E. Moore, Nature (London) 464, 194 (2010)
* (10) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 ( 2010).
* (11) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 ( 2010)
* (12) B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. 96, 106802 2006
* (13) C. Wu, B. A. Bernevig, and S. C. Zhang, Phys. Rev. Lett. 96, 106401 2006
* (14) C. Xu and J. E. Moore, Phys. Rev. B 73, 045322 2006
* (15) L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 (2006); L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007)
* (16) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008)
* (17) B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757 (2006)
* (18) M. K$\ddot{o}$nig, S. Wiedmann, C. Br$\ddot{u}$ne, A. Roth, H. Buhmann, L. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007)
* (19) J. E. Moore and L. Balents, Phys. Rev. B 75, 121306(R) (2007)
* (20) L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007)
* (21) J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 045426 (2008)
* (22) Chang K and Lou W K, Phys. Rev. Lett. 106 206802 ( 2011)
* (23) Q. Liu, C. X. Liu, C. Xu, X. L. Qi, and S. C. Zhang, Phys. Rev. Lett. 102, 156603 (2009); A. A. Burkov and D. G. Hawthorn, Phys. Rev. Lett. 105, 066802 (2010)
* (24) X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, Science 323, 1184 (2009)
* (25) X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 78, 195424 (2008)
* (26) T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B 81, 121401(R) (2010);
* (27) S. J. Lee, S. Souma, G. Ihm, and K. J. Chang, Phys. Rep. 394, 1 (2004)
* (28) M. Cerchez, S. Hugger, T. Heinzel, and N. Schulz, Phys. Rev. B 75, 035341 (2007); G. Papp and F. M. Peeters, Appl. Phys. Lett 78, 2184 (2001)
* (29) A. Matulis, F. M. Peeters, and P. Vasilopoulos, Phys. Rev. Lett. 72, 1518 (1994)
* (30) Sungjae Cho, Yung-Fu Chen and Michael S. Fuhre, Appl. Phys. Lett. 91, 123105 (2007)
* (31) I. Snyman and C. W. J. Beenakker, Phys. Rev. B 75, 045322 (2007)
* (32) Ya Zhang and Feng Zhai, Appl. Phys. Lett. 96, 172109 (2010)
* (33) A. R. Akhmerov, Johan Nilsson, and C.W. J. Beenakker, Phys. Rev. Lett. 102, 216404 (2009)
* (34) P. Adroguer, C. Grenier, D. Carpentier, J. Cayssol, P. Degiovanni, and E. Orignac, Phys. Rev. B 82, 081303(R) (2010)
|
arxiv-papers
| 2012-04-04T14:13:41 |
2024-09-04T02:49:29.362632
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-Hui Yuan, Yan Zhang, Jian-Jun Zhang and Ze Cheng",
"submitter": "Jianhui Yuan",
"url": "https://arxiv.org/abs/1204.0956"
}
|
1204.0967
|
# Generalization of the Correspondence about
$\operatorname{DTr}$-selfinjective algebras
Fan Kong
Abstract: We give a correspondence between
$(n-1)$-$\operatorname{DTr}$-selfinjective algebras and algebras with dominant
dimension and selinjective dimension being both n for any $n\geq 2$.
Furthermore, we show the relation between the module categories of the two
kinds of algebras.
Key words: n-$\operatorname{DTr}$-selfinjective algebra, dominant dimension,
Gorenstein projective module categories, orthogonal.
## 1 Introduction
The relation between the dominant dimension and the representation property of
an algebra is a very hot topic since $60^{th}$ in the last century. The papers
about this topic in that time are [Mo], [Mu], [T] and so on. The main interest
on this topic is this fact. For any artin algebra, we always can construct
algebras with dominant dimension more than or equal to 2 by their generator-
cogenerators, and these algebras constructed are invariants of the original
algebras to some extent [R]. So those particular algebras with dominant
dimension more than or equal to 2 will reflect the properties of all algebras,
for example, [A], [I1]. On the other hand, the dominant dimension is also
associated with the injective resolution of the regular module. So it has some
relation with Gorenstein(or Cohen-Maucauley) theory, for example, [AS1].
In [AS1], Auslander and Solberg found a correspondence between those algebras
with dominant dimension and selinjective dimension being both 2 and
$\operatorname{DTr}$-selfinjective algebras. What is surprising is that the
Gorenstein projective module categories of those algebras with dominant
dimension and selinjective dimension being both 2 are always module categories
of algebras whose $\operatorname{DTr}$-orbits has some periodic property. This
implies there is a close relation between Auslander-Reiten theory and Cohen-
Maucauley thory.
In 2007, Iyama developed Auslander-Reiten theory. He demonstrated higher
dimensional Auslander-Reiten theory as a generalization of classic Auslander-
Reiten theory in [I2]. In that article, he developed a lot of useful tools
such as higher dimensional Auslander-Reiten translation, maximal orthogonal
subcategories and so on. As an application, in [I1] he showed the higher
dimensional Auslander correspondence which is a generalization of theories in
[A].
If we associate [AS1] with [I1] and [I2], we can find that the higher
dimensional Auslander-Reiten theory should be useful to characterize the
Goreinsten projective module category at least in some particular algebras. In
this article, we will show it. We will show the generalization of the
correspondence in [AS1]. And we will find that the periodic property of higher
dimensional $\operatorname{DTr}$-orbits appears again in our background.
We always assume $R$ is a commutative artin ring, $\operatorname{D}$ is the
duality functor, all agebras are artin $R$-algebras. If there is no special
instruction, we always assume all modules are left finitely generated modules.
## 2 Main theory
Before describing our main theory, we need the following definitions and
notations.
###### Definition 2.1.
Let $\Lambda$ be a basic artin algebra with $dom.\Lambda\geq 1$. Then there
exists a uniquely basic $\Lambda$ module $I$ such that
$\operatorname{add}I=\\{M\mid M\text{ is a projective-injective $\Lambda$
mod-}$ule}. We denote $I$ by $I^{\Lambda}$ . And it is called the minimal
faithful $\Lambda$-module just as in [R].
We follow the notations in [I1] and [I2]. Suppose $\Gamma$ is an artin
algebra. Let $\tau$ be the Auslander Reinten translation of $\Gamma$-mod,
$\tau^{-}$ be the quasi-inverse Auslander Reinten translation of $\Gamma$-mod,
$\Omega$ be the syzygy functor, $\Omega^{-1}$ be the cosyzygy functor. Then
just as in [I1] and [I2], for any $m\geq 1$, let
$\tau^{m}=\tau\cdot\Omega^{m-1}$ and $\tau^{-m}=\tau^{-}\cdot\Omega^{-(m-1)}$.
Also as in [I1] and [I2], suppose $n\geq 1$ and $\mathcal{D}$ is a full
subcategory of $\Gamma$-mod. Then
$\leftidx{{}^{\bot_{n}}}{\mathcal{D}}=\\{M\mid\operatorname{Ext}^{i}(M,X)=0,\forall
X\in\mathcal{D}\text{ and }1\leq i\leq
n\\},{\mathcal{D}}^{\bot_{n}}=\\{M\mid\operatorname{Ext}^{i}(X,M)=0,\forall
X\in\mathcal{D}\text{ and }1\leq i\leq n\\}$. Especially, for a module
$M,\leftidx{{}^{\bot_{n}}}M=\leftidx{{}^{\bot_{n}}}(\operatorname{add}M),M^{\bot_{n}}=(\operatorname{add}M)^{\bot_{n}}$.
For modules M and N, we say $M\bot_{n}N$ if $\operatorname{Ext}(M,N)=0,\forall
1\leq i\leq n$. We say M is n-self-orthogonal if $M\bot_{n}M$.
###### Definition 2.2.
Let $\Gamma$ be a basic artin algebra and $n\geq 2$. If there exists a basic
$\Gamma$ module $\leftidx{{}_{\Gamma}}Q$ which satisfies the following
conditions: $(1)$ it is a generator-cogenerator of $\Gamma$-mod; $(2)$ it is
$(n-2)$-self-orthogonal; $(3)$
$\tau^{n-1}Q\oplus\tau^{-(n-1)}Q\in\operatorname{add}Q$, then we call $\Gamma$
is a $(n-1)$-$\operatorname{DTr}$-selfinjective algebra, Q is a $(n-2)$-self-
orthogonal $(n-1)$-$\operatorname{DTr}$-closed generator-cogenerator.
$1$-$\operatorname{DTr}$-selfinjective algebra is also called
$\operatorname{DTr}$-selfinjective algebra as in [AS1]
Suppose $n\geq 2$, $\Gamma_{1}$, $\Gamma_{2}$ are two
n-$\operatorname{DTr}$-selfinjective modules ,
$\leftidx{{}_{\Gamma_{1}}}Q_{1}$ and $\leftidx{{}_{\Gamma_{2}}}Q_{2}$ are
respectively (n-2)-self-orthogonal $(n-1)$-$\operatorname{DTr}$-closed
generator-cogenerator of $\Gamma_{1}$ and $\Gamma_{2}$. Then we say that the
pair $(\Gamma_{1},\leftidx{{}_{\Gamma_{1}}}Q_{1})$ is equivalent to
$(\Gamma_{2},\leftidx{{}_{\Gamma_{2}}}Q_{2})$ if
$\operatorname{End}\leftidx{{}_{\Gamma_{1}}}Q_{1}$ is Morita equivalent to
$\operatorname{End}\leftidx{{}_{\Gamma_{2}}}Q_{2}$ (or equivalently,
$\operatorname{End}\leftidx{{}_{\Gamma_{1}}}Q_{1}\cong\operatorname{End}\leftidx{{}_{\Gamma_{2}}}Q_{2}$
since both are basic modules). We denote the equivalent class by
$[\Gamma_{1},\leftidx{{}_{\Gamma_{1}}}Q_{1}]$. For a basic artin algebra
$\Lambda$, we denote the equivalent class of $\Lambda$ under algebraic
isomorphism by $[\Lambda]$ (we don’t use Morita equivalent class in order to
ensure all algebras are basic). Then we have the following notations:
$\mathfrak{U}_{n}=\\{[\Lambda]\mid dom.dim\Lambda=inj.dim\Lambda=n\\}$;
$\mathfrak{B}_{n}=\\{[\Gamma,Q]\mid\text{$\Gamma$ is an (n -
1)-$\operatorname{DTr}$-selfinjective algbra, $Q$ is an $(n-2)$-self-
orthogonal}$ $(n-1)$-$\operatorname{DTr}$-closed generator-cogenerator}. Now
we can describing the main theorem.
###### Theorem 2.3.
Suppose $n\geq 2$. Then there is a one to one correspondence
$\mathfrak{U}_{n}\autorightleftharpoons{F}{G}\mathfrak{B}_{n}$
such that
$\forall[\Lambda]\in\mathfrak{U}_{n},F([\Lambda])=[\operatorname{End}^{op}I^{\Lambda},\leftidx{{}_{(\operatorname{End}I^{\Lambda})^{op}}}(\operatorname{D}(I^{\Lambda}))];\forall[\Gamma,Q]\in\mathfrak{B}_{n},G([\Gamma,Q])=[\operatorname{End}^{op}Q]$.
Now suppose
$[\Lambda]\in\mathfrak{U}_{n},\Gamma=\operatorname{End}^{op}I^{\Lambda},\leftidx{{}_{\Gamma}}Q_{\Lambda}=\leftidx{{}_{\Gamma}}(\operatorname{D}(I^{\Lambda}))_{\Lambda}$.
We denote $\\{\leftidx{{}_{\Gamma}}X\mid\text{there is an exact sequence
$0\rightarrow X\rightarrow I_{0}\rightarrow I_{1}\rightarrow\dots I_{m-1}$
such that $I_{0},I_{1},\dots I_{m-1}\in$ }$ $\operatorname{add}I^{\Lambda}\\}$
by $\mathcal{C}^{m}_{\Lambda}$ for any $m\geq 1$. We have the following lemma.
###### Lemma 2.4.
The exact functor
$\operatorname{D}\operatorname{Hom}_{\Lambda}(-,I^{\Lambda})=Q\otimes_{\Lambda}-:\mathcal{C}^{2}_{\Lambda}\rightarrow\Gamma$-mod
is an equivalence between categories.
Proof. For any $\Lambda$-module M,
$\operatorname{D}\operatorname{Hom}_{\Lambda}(M,I^{\Lambda})\cong\operatorname{D}\operatorname{Hom}_{\Lambda}(M,DD(I^{\Lambda}))\cong\operatorname{D}\operatorname{D}(\operatorname{D}(I^{\Lambda})\linebreak\otimes_{\Lambda}M)\cong\operatorname{D}(I^{\Lambda})\otimes_{\Lambda}M$.
So $\operatorname{D}\operatorname{Hom}_{\Lambda}(-,I^{\Lambda})$ and
$Q\otimes_{\Lambda}-$ are naturally isomorphic.
The equivalence between categories is proved for the similar reason as
Proposition 2.5 in chapter 2 of [ARS] .
Using the above lemma we prove the following two corollaries which is also
proved in [R] in a different way.
###### Corollary 2.5.
$Q$ is a generator-cogenerator of $\Gamma$-mod.
Proof.
$\leftidx{{}_{\Gamma}}Q=\leftidx{{}_{\Gamma}}(\operatorname{D}(I^{\Lambda}))=\operatorname{D}\operatorname{Hom}_{\Lambda}(\Lambda,I^{\Lambda})=\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{D}\Lambda)$.
Since
$I^{\Lambda}\in\operatorname{add}\Lambda\bigcap\linebreak\operatorname{add}\operatorname{D}\Lambda,\operatorname{D}(\Gamma_{\Gamma})\oplus\leftidx{{}_{\Gamma}}\Gamma=\operatorname{D}\operatorname{Hom}_{\Lambda}(I^{\Lambda},I^{\Lambda})\oplus\operatorname{Hom}_{\Lambda}(I^{\Lambda},I^{\Lambda})\in\operatorname{add}\leftidx{{}_{\Gamma}}Q$.
So $Q$ is a generator-cogenerator of $\Gamma$-mod.
###### Corollary 2.6.
$\leftidx{{}_{\Gamma}}Q_{\Lambda}$ is faithful balanced.
Proof. The canonical map
$\Gamma\rightarrow\operatorname{End}(\operatorname{D}(I^{\Lambda}))_{\Lambda}$
is an isomorphism since the canonical map
$\Gamma\rightarrow\operatorname{End}^{op}\leftidx{{}_{\Lambda}}I^{\Lambda}$ is
an isomorphism.
On the other hand since
$\operatorname{D}\operatorname{Hom}_{\Lambda}(-,I^{\Lambda})=Q\otimes_{\Lambda}-:\mathcal{C}^{2}_{\Lambda}\rightarrow\Gamma$-mod
is an equivalence between categories,
$\operatorname{End}^{op}\leftidx{{}_{\Gamma}}(\operatorname{D}(I^{\Lambda}))=\operatorname{End}^{op}\leftidx{{}_{\Gamma}}(\operatorname{D}\operatorname{Hom}_{\Lambda}(\Lambda,I^{\Lambda}))=\operatorname{End}^{op}\leftidx{{}_{\Lambda}}\Lambda=\Lambda$.
Since $\leftidx{{}_{\Lambda}}I^{\Lambda}$ is a faithful $\Lambda$-module, we
know the canonical map
$\Lambda\rightarrow\operatorname{End}^{op}\leftidx{{}_{\Gamma}}(\operatorname{D}(I^{\Lambda}))$
is a monomorphism. So it is an isomorphism.
###### Lemma 2.7.
There is an exact sequence: $0\rightarrow\Lambda\rightarrow I_{0}\rightarrow
I_{1}\rightarrow\dots\rightarrow
I_{n-1}\rightarrow\operatorname{D}\Lambda\rightarrow 0$ such that
$I_{0},I_{1},\dots,I_{n-1}\in\operatorname{add}I^{\Lambda}$. Especially,
$\Lambda$ is an n-Gorenstein algebra.
Proof. Since $dom.dim.\Lambda=inj.dim.\leftidx{{}_{\Lambda}}{\Lambda}$, for
any indecomposable projective module $P$, $inj.dim.P=0$ or $n$. If $inj.dim.P$
= 0, $P$ is a projective-injective module. If not, $P$ has a minimal injective
resolution:
$\begin{CD}0\rightarrow P\rightarrow I_{0}\rightarrow
I_{1}\rightarrow\dots\rightarrow I_{n-1}\rightarrow\Omega^{-n}(P)\rightarrow
0\end{CD}$
such that $I_{0}$, $I_{1}$, $\dots$, $I_{n-1}$ are projective-injective
modules. Since this is also a projective resolution of $\Omega^{-n}(P)$,
$\Omega^{-n}(P)$ is an indecomposable module. If not, the injective resolution
of $P$ is not minimal. So $\Omega^{-n}(P)$ is an indecomposable injective
nonprojective module.
On the other hand, the number of non-injective projective modules is equal to
the number of non-projective injective modules. So $\Omega^{-n}$ constructs a
one to one correspondence between non-injective projective modules and non-
projective injective modules. So $\Omega^{-n}(A)$ is a basic module which is
the direct sum of all mutually nonisomorphic nonprojective injective modules.
So the exact sequence in the lemma exists. By duality $\operatorname{D}$, we
know $\Lambda$ is n-Gorenstein algebra.
###### Proposition 2.8.
$\leftidx{{}_{\Gamma}}Q\ \bot_{n-2}\ \leftidx{{}_{\Gamma}}Q$
Proof. If $n=2$, it is clear. Now suppose $n>2$. There exists an injective
resolution of $\leftidx{{}_{\Lambda}}\Lambda$:
$0\rightarrow\Lambda\rightarrow I_{0}\rightarrow
I_{1}\rightarrow\dots\rightarrow I_{n-2}\rightarrow I_{n-1}$
such that $I_{i}\in\operatorname{add}I^{\Lambda}$ for all $i$.
Applying $Q\otimes_{\Lambda}-$ to the exact sequence, we obtain the following
exact sequence since it is an exact functor:
$0\rightarrow\leftidx{{}_{\Gamma}}Q\rightarrow Q\otimes I_{1}\rightarrow
Q\otimes I_{2}\rightarrow\dots\rightarrow Q\otimes I_{n-2}\rightarrow Q\otimes
I_{n-1}$
Since $Q\otimes
I_{i}=\operatorname{D}\operatorname{Hom}_{\Lambda}(I_{i},I^{\Lambda})\in\operatorname{add}\operatorname{D}(\Gamma_{\Gamma})$,
the above sequence is an injective resolution of $\leftidx{{}_{\Gamma}}Q$.
By Lemma 2.4 we have the following commutative diagram:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Lambda}(\leftidx{{}_{\Lambda}}\Lambda,\leftidx{{}_{\Lambda}}\Lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Lambda}(\leftidx{{}_{\Lambda}}\Lambda,I_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Lambda}(\leftidx{{}_{\Lambda}}\Lambda,I_{n-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Gamma}(Q,Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Gamma}(Q,Q\otimes
I_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Gamma}(Q,Q\otimes
I_{n-1})}$
Since the above is an exact sequence, so is the bellow one. Therefore,
$\leftidx{{}_{\Gamma}}Q\ \bot_{n-2}\ \leftidx{{}_{\Gamma}}Q$.
Now we suppose
$\mathcal{N}=\operatorname{D}\operatorname{Hom}_{\Gamma}(-,\Gamma)$ is the
Nakayama functor and
$\mathcal{N}^{-}=\operatorname{Hom}_{\Gamma}(\operatorname{D}(-),\Gamma)$ is
the quasi-inverse Nakayama functor. We denote the sable module category of
$\Gamma$-mod by $\underline{$\Gamma-\text{mod}$}$. Given $M\in\Gamma$-mod, the
corresponding module in $\underline{$\Gamma-\text{mod}$}$ by
$\underline{$M$}$. Dually, we have the notations
$\overline{\Gamma-\text{mod}}$ and $\overline{M}$.
###### Proposition 2.9.
$\leftidx{{}_{\Gamma}}Q=\operatorname{D}(\Gamma_{\Gamma})\oplus\tau^{n-1}Q=\Gamma\oplus\tau^{-(n-1)}Q$
Proof. Step1.
$\leftidx{{}_{\Gamma}}Q=\operatorname{D}(\Gamma_{\Gamma})\oplus\tau^{n-1}Q$.
By Lemma 2.7, there is an exact sequence:
$\begin{CD}\rightline{\hbox{$0\rightarrow\leftidx{{}_{\Lambda}}\Lambda\xrightarrow{d_{-1}}I_{0}\xrightarrow{d_{0}}I_{1}\xrightarrow{d_{1}}I_{2}\xrightarrow{d_{2}}\dots\xrightarrow{d_{n-2}}I_{n-1}\xrightarrow{d_{n-1}}\operatorname{D}(\Lambda_{\Lambda})\rightarrow
0\hskip 28.45274pt\left(*\right)$}}\end{CD}$
such that $I_{i}\in\operatorname{add}I^{\Lambda}$ for all $i$.
Applying $\operatorname{Hom}_{\Lambda}(I^{\Lambda},-)$ to it, we obtain the
exact sequence since it is an exact functor:
$\begin{CD}0\rightarrow\operatorname{Hom}_{\Lambda}(I^{\Lambda},\Lambda)\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{-1})}\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{0})\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{0})}\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{1})\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{1})}\\\
\rightline{\hbox{
$\dots\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{n-2})}\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{n-1})\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{n-1)}}\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{D}(\Lambda_{\Lambda}))\rightarrow
0\hskip 65.44133pt$}}\end{CD}$
Since $\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{i})$ is a projective
$\Gamma$ module for all i,
$\uline{\Omega^{n-2}\text{$\leftidx{{}_{\Gamma}}Q$}}=\uline{\operatorname{Ker}\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{2})}=\uline{\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{Ker}d_{2})}$.
We have the projective resolution of
$\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{Ker}d_{2})$:
$\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{0})\xrightarrow{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{0})}\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{1})\rightarrow\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{Ker}d_{2})\rightarrow
0$
Since
$\operatorname{Hom}_{\Lambda}(I^{\Lambda},-):\operatorname{add}I^{\Lambda}\rightarrow\operatorname{add}\leftidx{{}_{\Gamma}}\Gamma$
is an equivalence between categories. We have the following commutative
diagram:
$\textstyle{\mathcal{N}(\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{N}\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{0})}$$\textstyle{\mathcal{N}(\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{D}\operatorname{Hom}_{\Lambda}(\Lambda,I^{\Lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{D}\operatorname{Hom}_{\Lambda}(d_{-1},I^{\Lambda})}$$\textstyle{\operatorname{D}\operatorname{Hom}_{\Lambda}(I_{0},I^{\Lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{D}\operatorname{Hom}_{\Lambda}(d_{0},I^{\Lambda})}$$\textstyle{\operatorname{D}\operatorname{Hom}_{\Lambda}(I_{1},I^{\Lambda})}$
The vertical morphisms are morphisms. Since the bellow sequence is exact, we
have
$\overline{\tau^{n-1}Q}=\overline{\tau\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{Ker}d_{2})}=\overline{\operatorname{D}\operatorname{Hom}_{\Lambda}(\Lambda,I^{\Lambda})}=\overline{\text{$\leftidx{{}_{\Gamma}}Q$}}$.
Thus $\tau^{n-1}Q\in\operatorname{add}Q$ since $\leftidx{{}_{\Gamma}}Q$ is a
cogenerator. For the same reason in Lemma 2.7, $\tau^{n-1}Q$ is a basic
module. So we have
$\leftidx{{}_{\Gamma}}Q=\operatorname{D}(\Gamma_{\Gamma})\oplus\tau^{n-1}Q$
since $\tau^{n-1}\operatorname{D}(\Gamma_{\Gamma})=0$ and
$\leftidx{{}_{\Gamma}}Q$ is a cogenerator.
Step 2. $\leftidx{{}_{\Gamma}}Q=\Gamma\oplus\tau^{-(n-1)}Q$.
Applying the exact functor $Q\otimes_{\Lambda}-$ to $(*)$, we obtain the
folllowing exact sequence:
$0\rightarrow Q\otimes\Lambda\xrightarrow{Q\otimes d_{-1}}Q\otimes
I_{0}\xrightarrow{Q\otimes d_{0}}\dots\xrightarrow{Q\otimes d_{n-2}}Q\otimes
I_{n-1}\xrightarrow{Q\otimes
d_{n-1}}Q\otimes\operatorname{D}\Lambda\rightarrow 0$
Since $Q\otimes I_{i}$ is an injective $\Gamma$-module for all $i$,
$\overline{\Omega^{-(n-2)}\text{$\leftidx{{}_{\Gamma}}Q$}}=\overline{\operatorname{Ker}(Q\otimes
d_{n-2})}=\overline{Q\otimes\operatorname{Ker}d_{n-2}}$.
We have injective resolution of $Q\otimes\operatorname{Ker}d_{n-2}$:
$0\rightarrow Q\otimes\operatorname{Ker}d_{n-2}\rightarrow Q\otimes
I_{n-2}\xrightarrow{Q\otimes d_{n-2}}Q\otimes I_{n-1}$
By Lemma 2.4, we have the following commutative diagram:
$\begin{CD}\mathcal{N}^{-}(Q\otimes I_{n-2})@>{\mathcal{N}^{-}(Q\otimes
d_{n-2})}>{}>\mathcal{N}^{-}(Q\otimes I_{n-1})\\\ @A{}A{}A@A{}A{}A\\\
\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{n-2})@>{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{n-2})}>{}>\operatorname{Hom}_{\Lambda}(I^{\Lambda},I_{n-1})@>{\operatorname{Hom}_{\Lambda}(I^{\Lambda},d_{n-1})}>{}>\operatorname{Hom}_{\Lambda}(I^{\Lambda},\operatorname{D}\Lambda)@>{}>{}>0\end{CD}$
The vertical morphisms are morphisms. Since the bellow sequence is exact, we
have
$\uline{\tau^{-(n-1)}\text{$\leftidx{{}_{\Gamma}}Q$}}=\uline{\tau^{-}(Q\otimes\operatorname{Ker}d_{n-2})}=\uline{\operatorname{Hom}_{\Lambda}(I^{\Lambda},D\Lambda)}=\uline{\text{$\leftidx{{}_{\Gamma}}Q$}}$
Thus $\tau^{-(n-1)}Q\in\operatorname{add}Q$ since $\leftidx{{}_{\Gamma}}Q$ is
a generator. For the same reason in Lemma 2.7, $\tau^{-(n-1)}Q$ is a basic
module. So we have $\leftidx{{}_{\Gamma}}Q=\Gamma\oplus\tau^{-(n-1)}Q$ since
$\tau^{-(n-1)}\Gamma=0$ and $\leftidx{{}_{\Gamma}}Q$ is a generator.
The following lemma is from [I2].
###### Lemma 2.10.
Suppose $n\geq 2$, $\Sigma$ is an artin algebra. Let $X,Y\in\Sigma$-mod.
$\left(1\right)$ If $X\in\leftidx{{}^{\bot_{n-2}}}\Sigma$. Then we have the
following isomorphism for any $1\leq i\leq
n-2:\operatorname{Ext}^{i}(X,Y)\cong\operatorname{Ext}^{n-1-i}(Y,\tau^{n-1}X)$.
$\left(2\right)$ If $Y\in D(\Sigma_{\Sigma})^{\bot_{n-2}}$. Then we have the
following isomorphism for any $1\leq i\leq
n-2:\operatorname{Ext}^{i}(X,Y)\cong\operatorname{Ext}^{n-1-i}(\tau^{-(n-1)}Y,X)$.
The following lemma may be well known. But we give a proof there.
###### Lemma 2.11.
Suppose $\Sigma$ is an artin algebra. Let $\leftidx{{}_{\Sigma}}M$ be a
generator of $\Sigma$-mod. Then the functor
$\operatorname{Hom}_{\Sigma}(M,-):\Sigma$-mod
$\rightarrow\operatorname{End}^{op}M$-mod is fully faithful.
Dually, if $\leftidx{{}_{\Sigma}}M$ is a cogenerator of $\Sigma$-mod, then the
functor $\operatorname{Hom}_{\Sigma}(-,M):\Sigma$-mod
$\rightarrow\operatorname{End}M$-mod is fully faithful.
Proof. We just prove the first assertion. Since $M$ is a generator it is
faithful.
Suppose $X,Y\in\Gamma$-Mod,
$f:\operatorname{Hom}(M,X)\rightarrow\operatorname{Hom}(M,Y)$ is a
$\operatorname{End}^{op}M$-morphism. Suppose $\pi:T\rightarrow X\rightarrow 0$
is a right $\operatorname{add}M$ approximation. Then there exists an exact
sequence:
$\begin{CD}0@>{}>{}>\operatorname{Hom}(M,\operatorname{ker}\pi)@>{\operatorname{Hom}(M,i)}>{}>\operatorname{Hom}(M,T)@>{\operatorname{Hom}(M,\pi)}>{}>\operatorname{Hom}(M,X)@>{}>{}>0\end{CD}$
Since
$\operatorname{Hom}(T,Y)\rightarrow\operatorname{Hom}(\operatorname{Hom}(M,T),\operatorname{Hom}(M,Y))$
is an isomorphism, there exists $g:T\rightarrow Y$ such that
$f\cdot\operatorname{Hom}(M,\pi)=\operatorname{Hom}(M,g)$. So
$\operatorname{Hom}(M,g\cdot
i)=f\cdot\operatorname{Hom}(M,\pi)\cdot\operatorname{Hom}(M,i)=0\Rightarrow
g\cdot i=0$ $\Rightarrow\exists f^{\prime}:X\rightarrow Y$ such that
$g=f^{\prime}\cdot\pi$ $\Rightarrow\operatorname{Hom}(M,f^{\prime})=f$.
Now we can give the proof of the main theorem.
Proof of the Theorem 2.3. Give $[\Lambda]\in\mathfrak{U}_{n}$, by Corollary
2.5, Proposition 2.8 and Proposition 2.9, $F([\Lambda])\in\mathfrak{B}_{n}$.
By Corollary 2.6, $GF([\Lambda])=[\Lambda]$.
Now suppose $[\Gamma,\leftidx{{}_{\Gamma}}M]\in\mathfrak{B}_{n}$. Let
$\Sigma=\operatorname{End}^{op}{M}$. We prove the another part of the theorem
by 3 steps.
Step1. M has an injective resolution:
$0\rightarrow\leftidx{{}_{\Gamma}}M\rightarrow J_{0}\rightarrow
J_{1}\rightarrow\dots\rightarrow J_{n-2}\rightarrow J_{n-1}$
Since $M\bot_{n-2}M$, applying $\operatorname{Hom}_{\Gamma}(M,-)$ to it, we
have the following exact sequence:
$0\rightarrow\operatorname{Hom}_{\Gamma}(\leftidx{{}_{\Gamma}}M,\leftidx{{}_{\Gamma}}M)\rightarrow\operatorname{Hom}_{\Gamma}(M,J_{0})\rightarrow\operatorname{Hom}_{\Gamma}(M,J_{1})\rightarrow\dots\rightarrow\operatorname{Hom}_{\Gamma}(M,J_{n-1})$
Since $M$ is a generator-cogenerator, by [R],
$\operatorname{Hom}(M,\operatorname{D}(\Gamma_{\Gamma}))$ is a projective-
injective $\Sigma$-module. So the above exact sequence is the injective
resolution of $\leftidx{{}_{\Sigma}}\Sigma$. And
$\operatorname{Hom}_{\Gamma}(M,J_{i})$ is a projective-injective
$\Sigma$-module. So $dom.\Sigma\geq n$.
Step2. Now suppose $Z\in\Gamma$-mod, and the following is an exact sequence:
$0\rightarrow Y\rightarrow M^{\prime}\xrightarrow{f}Z\rightarrow 0$ such that
$f$ is a right $\operatorname{add}M$-approximation of $X$. Then
$\operatorname{Ext}^{1}(M,Y)=0$. So by Lemma Proposition 2.9 and 2.10 .
$\operatorname{Ext}^{n-2}(Y,M)=\operatorname{Ext}^{n-2}(Y,\tau^{n-2}M)=\operatorname{Ext}^{1}(M,Y)=0$.
Suppose $X\in\Gamma$-mod and the following is an exact sequence:
$0\rightarrow
X_{n-1}\xrightarrow{\text{$h$}}M_{n-1}\xrightarrow{\text{$f_{n-1}$}}\dots\xrightarrow{\text{$f_{3}$}}M_{2}\xrightarrow{\text{$f_{2}$}}M_{1}\xrightarrow{\text{$f_{1}$}}X\rightarrow
0\hskip 28.45274pt(**)$
such that $f_{i}:M_{i}\rightarrow\operatorname{Im}f_{i}$ is a right
$\operatorname{add}M$-approximation for all $i$.
If $n>2$, since $M\bot_{n-2}M$,
$\operatorname{Ext}^{1}(\operatorname{Ker}f_{n-2},M)=\operatorname{Ext}^{n-2}(\operatorname{Ker}f_{1},M)=\operatorname{Ext}^{1}(M,\operatorname{Ker}f_{1})=0$.
Thus, $\operatorname{Hom}(h,M)$ is an epic morphism.
If $n=2$, since $f_{1}$ is a right $\operatorname{add}M$-approximation, $h$ is
a left $\operatorname{add}M$-approximation since $M$ is
$\operatorname{DTr}$-closed and a cogenerator. Thus, $\operatorname{Hom}(h,M)$
is an epic morphism.
Applying $\operatorname{Hom}_{\Gamma}(M,-)$ to $(**)$, we have the following
exact diagram:
$0\rightarrow\operatorname{Hom}(M,X_{n-1})\xrightarrow{\operatorname{Hom}(M,\text{$h$})}\dots\xrightarrow{\operatorname{Hom}(M,\text{$f_{1}$})}\operatorname{Hom}(M,X)\rightarrow
0$
By Lemma 2.11, we know that
$\operatorname{Hom}_{\Sigma}(\operatorname{Hom}_{\Gamma}(M,\text{$h$}),\text{$\leftidx{{}_{\Sigma}}\Sigma$})$
is isomorphic to $\operatorname{Hom}(h,\linebreak M)$. So
$\operatorname{Hom}_{\Sigma}(\operatorname{Hom}_{\Gamma}(M,\text{$h$}),\text{$\leftidx{{}_{\Sigma}}\Sigma$})$
is an epic morphism. Since $\operatorname{Hom}(M,M_{i})$ is a projective
$\Sigma$-module for all $i$,
$\operatorname{Ext}^{n-2}_{\Sigma}(\operatorname{Hom}(M,X),\Sigma)=0$.
Suppose $V\in\Sigma$ module. Then there is a morphism $f:M_{1}\rightarrow
M_{2}$ such that there exists a projective resolution of $V$:
$\operatorname{Hom}(M,M_{1})\xrightarrow{\operatorname{Hom}(M,f)}\operatorname{Hom}(M,M_{2})\rightarrow
V\rightarrow 0$
Therefore,
$\operatorname{Ext}^{n+1}_{\Sigma}(V,\Sigma)=\operatorname{Ext}^{n-2}_{\Sigma}(\operatorname{Ker}(\operatorname{Hom}(M,f)),\Sigma)=\operatorname{Ext}^{n-2}_{\Sigma}(\operatorname{Hom}(M,\operatorname{Ker}f),\linebreak\Sigma)=0$.
So $inj.dim.\leftidx{{}_{\Sigma}}\Sigma\leq n$
Step3. Also we know $\operatorname{Hom}(M,\operatorname{D}(\Gamma_{\Gamma}))$
is the minimal faithful $\Sigma$-module by [A]. So $\Sigma$ is not a
selfinjective algebra. So
$inj.dim.\leftidx{{}_{\Sigma}}\Sigma=n=dom.dim.\Sigma$. Therefore,
$G([\Gamma,Q])\in\mathfrak{U}_{n}$. By Lemma 2.11
$\operatorname{End}^{op}\operatorname{Hom}(M,\operatorname{D}(\Gamma_{\Gamma}))=\operatorname{End}^{op}\operatorname{D}(\Gamma_{\Gamma})=\operatorname{End}(\Gamma_{\Gamma})=\Gamma$.
And
$\leftidx{{}_{\Gamma}}(\operatorname{D}\operatorname{Hom}(M,\operatorname{D}(\Gamma_{\Gamma})))=\leftidx{{}_{\Gamma}}(\operatorname{D}\operatorname{D}(\Gamma\otimes
M))=\leftidx{{}_{\Gamma}}M.$ So $GF([\Gamma,M])=[\Gamma,M].$
For an artin algebra $\Sigma$, we denote its finitely generated Gorenstein
projective module category by $Gproj(\Sigma)$.
###### Lemma 2.12.
Suppose $n\geq 2,[\Lambda]\in\mathfrak{U}_{n}$. Then
$Gproj(\Lambda)=\mathcal{C}_{\Lambda}^{n}$
Proof. Suppose $X\in\mathcal{C}_{\Lambda}^{n}$. Then $X$ has an injective
resolution: $0\rightarrow X\rightarrow I_{0}\rightarrow
I_{1}\rightarrow\dots\rightarrow I_{n-1}$ such that $I_{i}$ is a projective
module for all $i$. So
$\operatorname{Ext}^{i}(X,\Lambda)=\operatorname{Ext}^{n+i}(\Omega^{-n}X,\Lambda)=0,\forall
i>0$. So $X\in Gproj(\Lambda)$.
Suppose $\operatorname{Ext}^{n}(Z,\Lambda)=0$. Applying
$\operatorname{Hom}(Z,-)$ to $(*)$ in Proposition 2.9, we have an epic
morphism
$\operatorname{Hom}(Z,I_{n-1})\rightarrow\operatorname{Hom}(Z,\operatorname{D}\Lambda)$.
So $Z$ is cogenerated by $\operatorname{add}I_{n-1}$. Thus it is cogenerated
by $\operatorname{add}I^{\Lambda}$.
Suppose $Y\in Gproj(\Lambda)$. Then
$\operatorname{Ext}^{i}(Y,\Lambda)=0,\forall i\geq 1$. Using the above
assertion by induction on $i$. We know
$\operatorname{Ext}^{n}(\Omega^{-i}Y,\Lambda)=0,$ and $\Omega^{-i}X$ is
cogenerated by $\operatorname{add}I^{\Lambda}$ for all $i\leq n-1$. Then we
have an injective resolution of $Y:0\rightarrow Y\rightarrow
I^{\prime}_{0}\rightarrow I^{\prime}_{1}\rightarrow\dots\rightarrow
I^{\prime}_{n-1}$ such that $I_{i}\in\operatorname{add}I^{\Lambda}$ for all
$i$. So $Y\in\mathcal{C}_{\Lambda}^{n}$.
###### Theorem 2.13.
Suppose $n\geq 2,[\Gamma,Q]\in\mathfrak{B}_{n}$. Let
$\Sigma=\operatorname{End}^{op}Q$. Then
$\leftidx{{}_{\Gamma}}Q^{\bot_{n-2}}=\leftidx{{}^{\bot_{n-2}}}(\leftidx{{}_{\Gamma}}Q)$,
and the functor $\operatorname{Hom}_{\Gamma}(Q,-)$ gives an equivalence
between $Q^{\bot_{n-2}}$ and $Gproj(\Sigma)$.
Proof. Suppose $X\in\Gamma$-mod. By Lemma 2.10
$\operatorname{Ext}^{i}(Q,X)=\operatorname{Ext}^{n-1-i}(X,\tau^{n-1}Q),\forall
1\leq i\leq n-2$. However, since we have the correspondence as in Theorem 2.3,
$Q=\operatorname{D}(\Gamma_{\Gamma})\oplus\tau^{n-1}Q$. So
$\operatorname{Ext}^{i}(Q,X)=\operatorname{Ext}^{n-1-i}(X,Q)$. So the first
assertion is proved.
Now suppose $X\in Q^{\bot_{n-2}}$ and the following is a injective resolution
of $X:0\rightarrow X\rightarrow J_{0}\rightarrow
J_{1}\rightarrow\dots\rightarrow J_{n-1}$. Applying $\operatorname{Hom}(Q,-)$
to it we get an exact sequence:
$0\rightarrow\operatorname{Hom}(Q,X)\rightarrow\operatorname{Hom}(Q,J_{0})\rightarrow\operatorname{Hom}(Q,J_{1})\rightarrow\dots\rightarrow\operatorname{Hom}(Q,J_{n-1})$
since $X\in Q^{\bot_{n-2}}$. $\operatorname{Hom}(Q,J_{i})$ is a projective-
injective module for all $i$ since
$\operatorname{Hom}(Q,\operatorname{D}(\Gamma_{\Gamma}))$ is the minimal
faithful module of $\Sigma$. So
$\operatorname{Hom}(Q,X)\in\mathcal{C}_{\Sigma}^{n}$. Thus
$\operatorname{Hom}(Q,X)\in Gproj(\Sigma)$ by the above lemma.
Conversely, suppose that $M\in Gproj(\Sigma)$. Then there is an injective
resolution of $M:0\rightarrow M\rightarrow
I_{0}\xrightarrow{f_{0}}I_{1}\xrightarrow{f_{1}}\dots\xrightarrow{f_{n-2}}I_{n-1}$
such that $I_{i}\in I^{\Sigma}$. By Lemma 2.11, we know that there exists
$J_{0},J_{1},\dots,J_{n-1}\in\operatorname{add}\operatorname{D}(\Gamma_{\Gamma})$
and morphisms $d_{i}:J_{i}\rightarrow J_{i+1}$ such that
$\operatorname{Hom}(Q,d_{i})$ is isomorphic to $f_{i}$ for all $i$. So there
is a commutative diagram.
$\setcounter{MaxMatrixCols}{11}\begin{CD}\operatorname{Hom}(Q,J_{0})@>{\operatorname{Hom}(Q,d_{0})}>{}>\dots
@>{\operatorname{Hom}(Q,d_{n-2})}>{}>\operatorname{Hom}(Q,J_{n-1})\\\
@V{}V{}V@V{}V{}V\\\ 0@>{}>{}>M@>{}>{}>I_{0}@>{f_{0}}>{}>\dots
@>{f_{n-2}}>{}>I_{n-1}\end{CD}$
The vertical morphisms are morphisms. Since the bellow sequence is exact, so
is the above and $\operatorname{Ker}\operatorname{Hom}(Q,d_{0})=M$. Therefore,
since $\leftidx{{}_{\Gamma}}Q$ is a generator, the sequence
$I_{0}\xrightarrow{d_{0}}I_{1}\xrightarrow{d_{1}}\dots\xrightarrow{d_{n-2}}I_{n-1}$
is exact and $\operatorname{Ker}d_{0}\in Q^{\bot_{n-2}}$. On the other hand,
$\operatorname{Ker}\operatorname{Hom}(Q,d_{0})=\operatorname{Hom}(Q,\operatorname{Ker}d_{0})$.
So $M=\operatorname{Hom}(Q,\operatorname{Ker}d_{0})$. Thus
$\operatorname{Hom}_{\Gamma}(Q,-):Q^{\bot_{n-2}}\rightarrow Gproj(\Sigma)$ is
dense. It is also faithful by Lemma 2.11. So
$\operatorname{Hom}_{\Gamma}(Q,-)$gives an equivalence between
$Q^{\bot_{n-2}}$ and $Gproj(\Sigma)$.
$Q^{\bot_{n-2}}$ has a very interesting property
###### Corollary 2.14.
Suppose $n\geq 2,[\Gamma,Q]\in\mathfrak{B}_{n}$. Then
$\leftidx{{}_{\Gamma}}Q^{\bot_{n-2}}$ is closed under $\tau^{n-1}$ and
$\tau^{-(n-1)}$
Proof. Since
$\leftidx{{}_{\Gamma}}Q^{\bot_{n-2}}=\leftidx{{}^{\bot_{n-2}}}(\leftidx{{}_{\Gamma}}Q)$,
by Lemma 2.10, it’s obvious.
Now we give a homological characterization for
$(n-1)$-$\operatorname{DTr}$-selfinjective algebras. First, we give a lemma.
###### Lemma 2.15.
Suppose $n\geq 2$. If $\Gamma$ is an
$(n-1)$-$\operatorname{DTr}$-Selfinjective algebra, so is $\Gamma^{op}$
Proof. Suppose $Q$ is an $(n-2)$-self-orthogonal
$(n-1)$-$\operatorname{DTr}$-closed generator-cogenerator. Then
$\operatorname{D}Q$ is an (n - 2)-self-orthogonal $\Gamma^{op}$-module. It is
also a generator-cogenerator of $\Gamma^{op}$-mod.
Given $X\in\Gamma$-mod, then
$\tau^{n-1}(\operatorname{D}X)=\tau\Omega^{n-2}(\operatorname{D}X)=\tau\operatorname{D}\Omega^{-(n-2)}X=\operatorname{D}\tau^{-}\Omega^{-(n-2)}\linebreak
X=\operatorname{D}(\tau^{-(n-1)}X)$. For the same reason,
$\tau^{-(n-1)}(\operatorname{D}X)=\operatorname{D}(\tau^{n-1}X)$.
Thus $\operatorname{D}Q$ is a (n - 2)-self-orthogonal
$(n-1)$-$\operatorname{DTr}$-closed generator-cogenerator of
$\Gamma^{op}$-mod. So $\Gamma^{op}$ is an
$(n-1)$-$\operatorname{DTr}$-Selfinjective algebra.
###### Theorem 2.16.
Suppose $n\geq 2$, and $\Gamma$ is a basic artin algebra such that
$\operatorname{D}\Gamma\ \bot_{n-2}\ \Gamma$. Then the following is
equivalent.
$\left(1\right)$ $\Gamma$ is an $(n-1)$-$\operatorname{DTr}$-selfinjective
algebra.
$\left(2\right)$
$\operatorname{Inf}\\{inj.dim.\leftidx{{}_{\Sigma}}\Sigma\mid\Sigma=\operatorname{End}^{op}M,M\text{
is a basic generator-cogenerator of $\Gamma$-mod}$ such that $M\ \bot_{n-2}\
M\\}=n$.
$\left(3\right)$
$\operatorname{Inf}\\{inj.dim.\Sigma_{\Sigma}\mid\Sigma=\operatorname{End}^{op}M,M\text{
is a basic generator-cogenerator of $\Gamma$-mod}$ such that $M\ \bot_{n-2}\
M\\}=n$.
$\left(4\right)$
$\operatorname{Inf}\\{max(inj.dim.\Sigma_{\Sigma},inj.dim.\leftidx{{}_{\Sigma}}\Sigma)\mid\Sigma=\operatorname{End}^{op}M,M\text{
is a basic generator}$ -cogenerator $\text{of $\Gamma$-mod such that }M\
\bot_{n-2}\ M\\}=n$.
Proof. $(1)\Rightarrow(2),(3),(4)$ is obvious since we can choose $M$ is a
$(n-2)$-self-orthogonal $(n-1)$-$\operatorname{DTr}$-closed generator-
cogenerator.
$(2)\Rightarrow(1)$. Suppose $M\text{ is a basic generator-cogenerator of
$\Gamma$-mod such that }M\linebreak\bot_{n-2}\ M$ and
$inj.dim.\leftidx{{}_{\Sigma}}\Sigma=n$ for $\Sigma=\operatorname{End}^{op}M$.
For the same reason in the proof of Theorem 2.3, $dom.dim.\Sigma=n$. So
$\Sigma\in\mathfrak{U}_{n}$. Since
$\operatorname{Hom}_{\Gamma}(M,-):\Gamma$-mod $\rightarrow\Sigma$-mod is fully
faithful By Lemma 2.11. So
$\operatorname{End}^{op}\leftidx{{}_{\Sigma}}\operatorname{Hom}_{\Gamma}(M,\operatorname{D}(\Gamma_{\Gamma}))=\operatorname{End}^{op}\operatorname{D}(\Gamma_{\Gamma})=\operatorname{End}\Gamma_{\Gamma}=\Gamma$.
On the other hand,
$\operatorname{Hom}_{\Gamma}(M,\operatorname{D}(\Gamma_{\Gamma}))$ is a
minimal faithful $\Sigma$-module (by [A],[R]), So we know $\Gamma$ is an
$(n-1)$-$\operatorname{DTr}$-selfinjective algebra by Theorem 2.3.
$(3)\Rightarrow(1)$. If $(3)$ is true, then there exists a basic generator-
cogenerator of $\Gamma^{op}$-module N such that $N\ \bot_{n-2}\ N$ and
$inj.dim.\Sigma_{\Sigma}=n$ for
$\Sigma=\operatorname{End}^{op}\operatorname{D}N$. However,
$\operatorname{End}^{op}\operatorname{D}N=\operatorname{End}N$. So
$\Gamma^{op}$ satisfies $(2)$. By $(2)\Rightarrow(1)$, $\Gamma^{op}$ is an
$(n-1)$-$\operatorname{DTr}$-selfinjective algebra. By Lemma 2.14, so is
$\Gamma$.
$(4)\Rightarrow(2)$. Obvious.
## 3 The case n = 2
When n = 2, $1$-$\operatorname{DTr}$-selfinjective algebras are called
$\operatorname{DTr}$-selfinjective algebras just as in[AS1]. The
correspondence in Theorem 2.3 about it ($n=2$) is the analogy of
representation-finite algebras which is obtained in [A]. So we think
$\operatorname{DTr}$-selfinjective algebras have some similar properties as
representation-finite algebras. For the same reason, the algebras with
diminant dimension and selfinjective dimension being both $2$ should have some
similar properties of Auslander algebras. The homological characterization of
$\operatorname{DTr}$-selfinjective algebras which is demonstrated in Theorem
2.16 when $n=2$ is the analogy of the representation dimension
characterization of representation-finite algebras. In this section we will
give another two similar properties as representation-finite algebras. Firs ,
we will prove the following theorem. The similar property about Auslander-
algebras is placed in the appendix.
###### Theorem 3.1.
Let $\Gamma$ be an artin algebra. Then the following are equivalent.
$\left(1\right)$ $Gproj(\Gamma)$ is an abelian category $\left(\text{Notice:
not necessary an abelian subcategory}\right)$.
$\left(2\right)$ $\operatorname{dom.dim}\Gamma\geq
2,\operatorname{id}_{\Gamma}\Gamma\leq 2$
As a corollary, we can know the form of the Gorenstein projective module
category of an artin algebra when its Gorenstein projective module category is
an abelian category by Theorem 2.3 and Theorem 2.13. They are precisely the
module category of all DTr-selfinjective algebras. We denote
$\\{M\in\Gamma\text{-mod }\mid\operatorname{Ext}^{i}(M,\Gamma)=0\\},i=0,1,2$
by $\leftidx{{}^{\bot_{i}}}{\Gamma}$, the submodule category of
$\operatorname{add}\Gamma$ by $Sub\Gamma$, the Gorenstein Projective dimension
of $X$ by $\operatorname{Gproj.dim}X$ for every $X\in\Gamma$-mod,
$\bigcap\\{\operatorname{ker}f\mid f\in\operatorname{Hom}(X,Y)\\}$ by
$\operatorname{Rej}_{X}(Y)$ for all $X,Y\in\Lambda$-mod. We have the following
lemma.
###### Lemma 3.2.
Let $\Gamma$ be an artin algebra, and $Gproj(\Gamma)$ be an abelian category.
Then
$\left(1\right)$ $\Gamma$ is $2$-Gorenstein algebra.
$\left(2\right)$ If $\operatorname{Gproj.dim}X\leq 1$, then $X\in Sub\Gamma$
for every $X\in\Gamma$-mod.
$\left(3\right)$
$\leftidx{{}^{\bot_{0}}}{\Gamma}\subseteq\leftidx{{}^{\bot_{1}}}{\Gamma}$.
Proof. For every morphism $f:X_{1}\rightarrow X_{2}$ where $X_{1},X_{2}\in
Gproj(\Gamma)$, we denote the kernel and cokernel of $f$ in $Gproj(\Gamma)$ by
$\operatorname{ker}_{Gproj(\Gamma)}f,\operatorname{cok}_{Gproj(\Gamma)}f$
since $Gproj(\Gamma)$ is an abelian category.
(1) Given a morphism $f:X_{1}\rightarrow X_{2}$ where $X_{1},X_{2}\in
Gproj(\Gamma)$, since $Gproj(\Gamma)$ is an abelian category and
$\operatorname{add}\Gamma\subseteq Gproj(\Gamma)$,
$\operatorname{ker}f=\operatorname{ker}_{Gproj(\Gamma)}f\in Gproj(\Gamma)$.
$\Rightarrow$ For every module $X$, there exists an exact sequence
$\begin{CD}0\rightarrow G\rightarrow P_{1}\xrightarrow{f_{X}}P_{0}\rightarrow
X\rightarrow 0\end{CD}$
such that $P_{1},P_{0}\in\operatorname{add}\Gamma$. Then
$G\cong\operatorname{ker}f_{X}\cong\operatorname{ker}_{Gproj(\Gamma)}f_{X}\in
Gproj(\Gamma)$.
$\Rightarrow\operatorname{Ext}^{i}_{\Gamma}(X,\Gamma)=0$, for $i\geq 3$
$\Rightarrow\operatorname{\operatorname{id}}\leftidx{{}_{\Gamma}}{\Gamma}\leq
2$.
Since the left and right Gorenstein projective category are dual, the right
Gorenstein projective module category is also an abelian category. So
$\operatorname{id}\Gamma_{\Gamma}\leq 2$.
(2)Suppose $X\in\Gamma$-mod such that $\operatorname{Gproj.dim}X\leq 1$. Then
there is an exact sequence: $0\rightarrow X_{1}\xrightarrow{f}X_{2}\rightarrow
X\rightarrow 0$ such that $X_{1},X_{2}\in Gproj(\Gamma)$. Suppose
$g:X_{2}\rightarrow X_{3}$ is the cokernal of $f$ in $Gproj(\Gamma)$. So
$f=\operatorname{ker}_{Gproj(\Gamma)}g$ by abelian categories’s axioms. There
exists a commutative diagram:
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{i}$$\textstyle{X_{3}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$
By (1) $\operatorname{ker}_{Gproj(\Gamma)}g=\operatorname{ker}g$
$\Rightarrow\operatorname{ker}g=f\indent\Rightarrow\pi$ is an injective map.
$\Rightarrow$ $X\in Sub\Gamma$ since $Gproj(\Gamma)\subseteq Sub\Gamma$.
(3)Suppose $X\in\leftidx{{}^{\perp_{0}}}{\Gamma}$. There is an exact sequence:
$\begin{CD}0\rightarrow K\xrightarrow{i}P_{1}\xrightarrow{f}P_{0}\rightarrow
X\rightarrow 0\end{CD}$
such that $P_{1},P_{0}\in\operatorname{add}\Gamma$.
Since $X\in\leftidx{{}^{\perp_{0}}}{\Gamma}$, $f$ is a surjective map in
$Gproj(\Gamma)$. On the other hand,
$i=\operatorname{ker}f=\operatorname{ker}_{\mathcal{G}P}f$ by (1). So $f$ is
the cokernel of i in $Gproj(\Gamma)$ by abelian categories’s axioms.
$\Rightarrow$
$\begin{CD}0\rightarrow\operatorname{Hom}(P_{0},\Gamma)\rightarrow\operatorname{Hom}(P_{1},\Gamma)\rightarrow\operatorname{Hom}(K,\Gamma)\end{CD}$
is an exact sequence.
$\Rightarrow\operatorname{Ext}^{1}_{\Gamma}(X,\Gamma)=0$
$\Rightarrow X\in\leftidx{{}^{\perp_{1}}}{\Gamma}$
From now on we can abandon the abstract abelian category structure to prove
the property of $\Gamma$ . What is surprising is that we didn’t use the whole
abelian categories’axioms in the above lemma.
###### Corollary 3.3.
$Sub(\Gamma)$ is extension closed. Moreover,
$\left(\leftidx{{}^{\perp_{0}}}{\Gamma},Sub(\Gamma)\right)$ is a torsion pair
on $\Gamma$-mod.
Proof. If $X\in\leftidx{{}^{\perp_{2}}}{\Gamma}$, then
$\operatorname{Gproj.dim}X\leq 1$. By Lemma 3.2(2), $X\in Sub\Gamma$. On the
other hand, if $X\in Sub\Gamma$, since
$\operatorname{id}\leftidx{{}_{\Gamma}}{\Gamma}\leq 2$, then
$X\in\leftidx{{}^{\perp_{2}}}{\Gamma}$. So
$\leftidx{{}^{\perp_{2}}}{\Gamma}=Sub\Gamma$. $\Rightarrow Sub\Gamma$ is
extension closed.
It is also closed under submodules. So
($\leftidx{{}^{\perp_{0}}}{\Gamma},Sub\Gamma$) is a torsion pair on
$\Gamma$-mod. $\forall M\in\Gamma$-mod. $0\rightarrow
Rej_{M}(\Gamma)\rightarrow M\rightarrow M/Rej_{M}(\Gamma)\rightarrow 0$ is the
decomposition of $M$ by the torsion pair.
Proof of Theorem 3.1. we just need to prove
$\left(1\right)\Longrightarrow\left(2\right)$
Step1. Suppose $X\in Sub\Gamma$. $f:X\hookrightarrow I$ is the injective
envelope of $X$. Suppose $K=\operatorname{Rej}_{I}(\Gamma)$. By
($\leftidx{{}^{\perp_{0}}}{\Gamma},Sub\Gamma$), there is an exact sequence:
$\begin{CD}0\rightarrow K\xrightarrow{i}I\rightarrow L\rightarrow 0\end{CD}$
where $L\in Sub(\Gamma),K\in\leftidx{{}^{\perp_{0}}}{\Gamma}$. By the pull
back of $i$ and $f$, there exists a commutative diagram:
$\begin{CD}00\\\ @V{}V{}V@V{}V{}V\\\
0@>{}>{}>K^{\prime}@>{}>{}>K@>{}>{}>K/K^{\prime}@>{}>{}>0\\\
@V{}V{}V@V{i}V{}V@V{}V{}V\\\ 0@>{}>{}>X@>{f}>{}>I@>{}>{}>I/X@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\ 0@>{}>{}>L^{\prime}@>{}>{}>L\\\ @V{}V{}V@V{}V{}V\\\
00\end{CD}$
Since
$\leftidx{{}^{\perp_{0}}}{\Gamma}\subseteq\leftidx{{}^{\perp_{1}}}{\Gamma}$
and $K/K^{\prime}\in\leftidx{{}^{\perp_{0}}}{\Gamma}$,
$K/K^{\prime}\in\leftidx{{}^{\perp_{1}}}{\Gamma}$. $\Rightarrow
K^{\prime}\in\leftidx{{}^{\perp_{0}}}{\Gamma}$. $\Rightarrow
K^{\prime}\in\leftidx{{}^{\perp_{0}}}{\Gamma}\bigcap Sub\Gamma.\Rightarrow
K^{\prime}=0.\Rightarrow X\cong L^{\prime}$. So there exists a commutative
diagram:
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{I}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$
Since $g$ is an injective map, there exists $h:L\rightarrow I$ such that
$f=hg$. $f$ is left minimal, so $h\pi$ is an isomorphism. $\Rightarrow\pi$ is
an isomorphism. $\Rightarrow I\in Sub\Gamma.\Rightarrow I$ is a projective
module.
Step2. Suppose
$X\in\leftidx{{}^{\perp_{0}}}{\Gamma}\bigcap\leftidx{{}^{\perp_{2}}}{\Gamma}$.
Then $X\in\leftidx{{}^{\perp_{i}}}{\Gamma}$ for $i=0,1,2$. So $X\in
Gproj(\Gamma)$. $\Rightarrow X\in Sub\Gamma$. However,
$X\in\leftidx{{}^{\perp_{0}}}{\Gamma}.$ So $X=0$.
Suppose $f:X_{1}\rightarrow X_{2}$ is an injective morphism such that
$X_{1},X_{2}\in Gproj(\Gamma),X=\operatorname{cok}f$. Then
$X\in\leftidx{{}^{\perp_{2}}}{\Gamma}$. By
($\leftidx{{}^{\perp_{0}}}{\Gamma},Sub\Gamma$), there is an exact sequence:
$\begin{CD}0\rightarrow K\rightarrow X\rightarrow L\rightarrow 0\end{CD}$
such that $K\in\leftidx{{}^{\perp_{0}}}{\Gamma},L\in Sub\Gamma.$
$\Rightarrow\operatorname{Ext}^{2}_{\Gamma}(K,\Gamma)\neq 0$ if $K\neq 0$.
But $\operatorname{Ext}^{2}_{\Gamma}(X,\Gamma)=0$. So
$\operatorname{Ext}^{2}_{\Gamma}(K,\Gamma)=0$. That is contradictive. So
$K=0$. $\Rightarrow X\in Sub\Gamma$
Step3. By step1, there is an exact sequence:
$\begin{CD}0\rightarrow\leftidx{{}_{\Gamma}}{\Gamma}\rightarrow
I_{0}\rightarrow K\rightarrow 0\end{CD}$
such that $I_{0}$ is a projective-injective module.
By step2, $K\in Sub\Gamma$. So by step1, there exists an exact sequence:
$\begin{CD}0\rightarrow K\rightarrow I_{1}\rightarrow I_{2}\rightarrow
0\end{CD}$
such that $I_{1}$ is a projective-injective module.
So there is an exact sequence:
$\begin{CD}0\rightarrow\leftidx{{}_{\Gamma}}{\Gamma}\rightarrow
I_{0}\rightarrow I_{1}\rightarrow I_{2}\rightarrow 0\end{CD}.$
Since $\operatorname{id}\leftidx{{}_{\Gamma}}{\Gamma}\leq 2$, $I_{2}$ is an
injective module.
Now we suppose $k$ be a field, denote $\bigotimes_{k}$ by $\bigotimes$. We
will prove the following theorem which is similar as representation- finite
algebras. And it is also an example of $\operatorname{DTr}$-selfinjective
algebras.
###### Theorem 3.4.
Suppose Q is a acyclic quiver, $\Lambda$ is a finitely dimensional self-
injective $k$ algebra. Let $\Gamma=kQ\bigotimes\Lambda$. Then $\Gamma$ is a
$\operatorname{DTr}$-selfinjective algebra if and only if Q is a Dykin quiver.
For this, we need some lemmas.
###### Lemma 3.5.
Suppose k is a field, A and B are two finitely dimensional algebra over k. Let
$M_{A}$ a right finitely generated A module and $N_{B}$ a right finitely
generated B module. Then $\operatorname{D}(M\bigotimes
N)=\operatorname{D}M\bigotimes\operatorname{D}N$ as $A\bigotimes B$ modules.
Proof. There is an $A\bigotimes B$ homomorphism
$\sigma:\operatorname{D}M\bigotimes\operatorname{D}N\rightarrow\operatorname{D}(M\bigotimes
N)$ such that $\forall f\in\operatorname{D}M,g\in\operatorname{D}N,m\in M,n\in
N,\sigma(f\bigotimes g)(m\bigotimes n)=f(m)g(n)$. We choose the bases and the
dual bases of M and N as $k$ linear spaces. Then it is easy to check $\sigma$
is an isomorphism.
###### Corollary 3.6.
Suppose k is a field, A and B are two finitely dimensional algebra over k, B
is self-injective. Then $\operatorname{D}(A_{A})\bigotimes\leftidx{{}_{B}}B$
is an injective cogenerator of left $A\bigotimes B$ module category.
Proof. $\operatorname{D}(A_{A}\bigotimes
B_{B})=\operatorname{D}(A_{A})\bigotimes\operatorname{D}(B_{B})$ by the above
lemma. since $\leftidx{{}_{B}}B\in\operatorname{add}\operatorname{D}(B_{B})$,
then
$\operatorname{D}(A_{A})\bigotimes\leftidx{{}_{B}}B\in\operatorname{add}\operatorname{D}(A_{A})\bigotimes\operatorname{D}(B_{B})$.
So $\operatorname{D}(A_{A})\bigotimes\leftidx{{}_{B}}B$ is an injective
module. On the other hand, since
$\operatorname{D}(B_{B})\in\operatorname{add}\leftidx{{}_{B}}B$, then
$\operatorname{D}(A_{A})\bigotimes\operatorname{D}(B_{B})\in\operatorname{add}\operatorname{D}(A_{A})\bigotimes\leftidx{{}_{B}}B$.
So $\operatorname{D}(A_{A})\bigotimes\leftidx{{}_{B}}B$ is a cogenerator.
Now, we give the proof of the theorem.
Proof of Proposition 3.4. Suppose $\\{e_{1},e_{2},\dots,e_{n}\\}$ is the set
of all vertices of Q,
$\\{\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{m}\\}$ is a complete
set of primitive idempotents of $\Lambda,M\in kQ\text{-mod}$. Then there
exists the minimal projective resolution of $M$:
$\begin{CD}\rightline{\hbox{$\bigoplus(kQ)e^{i}\xrightarrow{f}\bigoplus(kQ)e^{j}\rightarrow
M\rightarrow 0\hskip 170.71652pt\left(*\right)$}}\end{CD}$
where $e^{i},e^{j}\in\\{e_{1},\dots,e_{n}\\}$, $f=\\{f_{ij}\mid
f_{ij}\in\operatorname{Hom}_{kQ}((kQ)e^{i},(kQ)e^{j})\\}$. So $f$ can be
represented as a matrix $A=\\{a_{ij}\mid a_{ij}\in e^{i}(kQ)e^{j}\\}$.
Suppose
$\varepsilon\in\\{\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{m}\\}$.
$-\bigotimes\Lambda\varepsilon$ acts to ($\ast$). Then we get the following
exact sequence:
$\begin{CD}\bigoplus(kQ)e^{i}\bigotimes\Lambda\varepsilon\xrightarrow{f\bigotimes\Lambda\varepsilon}\bigoplus(kQ)e^{j}\bigotimes\Lambda\varepsilon\rightarrow
M\bigotimes\Lambda\varepsilon\rightarrow 0\end{CD}$
So the following exact sequence is the projective resolution of
$M\bigotimes\Lambda\varepsilon$:
$\begin{CD}\rightline{\hbox{$\bigoplus\Gamma(e^{i}\bigotimes\varepsilon)\xrightarrow{f\bigotimes\Lambda\varepsilon}\bigoplus\Gamma(e^{j}\bigotimes\varepsilon)\rightarrow
M\bigotimes\Lambda\varepsilon\rightarrow 0\hskip
76.82243pt\left(**\right)$}}\end{CD}$
Where $f\bigotimes\Lambda\varepsilon=\\{f_{ij}\bigotimes\Lambda\varepsilon\mid
f_{ij}\bigotimes\Lambda\varepsilon\in\operatorname{Hom}_{\Gamma}(\Gamma(e^{i}\bigotimes\varepsilon),\Gamma(e^{j}\bigotimes\varepsilon))\\}$.
By ($\ast$), $f\bigotimes\Lambda\varepsilon$ can be represented by the matrix
$B=\\{a_{ij}\bigotimes\varepsilon\\}$.
$\operatorname{Hom}_{\Gamma}(-,\Gamma)$ acts to ($**$). Then we get an exact
sequence:
$\begin{CD}\rightline{\hbox{$\bigoplus(e^{j}\bigotimes\varepsilon)\Gamma\xrightarrow{(f\bigotimes\Lambda\varepsilon)^{*}}\bigoplus(e^{j}\bigotimes\varepsilon)\Gamma\rightarrow
N\rightarrow 0\hskip 71.13188pt\left(***\right)$}}\end{CD}$
Where
$(f\bigotimes\Lambda\varepsilon)^{*}=\\{g_{ji}=(f_{ij}\bigotimes\Lambda\varepsilon)^{*}\mid
g_{ji}\in\operatorname{Hom}_{\Gamma}((e^{j}\bigotimes\varepsilon)\Gamma,(e^{i}\bigotimes\varepsilon)\Gamma)\\}$.
By ($**$), $(f\bigotimes\Lambda\varepsilon)^{*}$ can be represented by the
matrix $C=\\{c_{ji}=a_{ij}\bigotimes\varepsilon\\}$, and
$\uline{N}=\uline{\operatorname{Tr}(M\bigotimes\Lambda\varepsilon)}$.
So we have the following commutative diagram:
$\begin{CD}\bigoplus(e^{j}\bigotimes\varepsilon)\Gamma
@>{(f\bigotimes\Lambda\varepsilon)^{*}}>{}>\bigoplus(e^{j}\bigotimes\varepsilon)\Gamma
@>{}>{}>N@>{}>{}>0\\\
@V{\alpha_{1}}V{}V@V{\alpha_{2}}V{}V@V{\alpha_{3}}V{}V\\\
\bigoplus((kQ)e^{j})^{*}\bigotimes(\Lambda\varepsilon)^{*}@>{f^{*}\bigotimes(\Lambda\varepsilon)^{*}}>{}>\bigoplus((kQ)e^{i})^{*}\bigotimes(\Lambda\varepsilon)^{*}@>{}>{}>\operatorname{Tr}M\bigotimes(\Lambda\varepsilon)^{*}@>{}>{}>0\end{CD}$
such that $\alpha_{1},\alpha_{2}$ are isomorphisms. So $\alpha_{3}$ is an
isomorphism.
$\Rightarrow\uline{\operatorname{Tr}(M\bigotimes\Lambda\varepsilon)}\cong\uline{\operatorname{Tr}M\bigotimes(\Lambda\varepsilon)^{*}}$
$\Rightarrow\overline{\operatorname{DTr}(M\bigotimes\Lambda\varepsilon)}\cong\operatorname{D}\uline{\operatorname{Tr}(M\bigotimes\Lambda\varepsilon)}\cong\operatorname{D}\uline{\operatorname{Tr}M\bigotimes\operatorname{D}(\Lambda\varepsilon)^{*}}\cong\overline{\operatorname{DTr}M\bigotimes\operatorname{D}(\Lambda\varepsilon)^{*}}$
by Lemma 5.6
Now we can start to calculate the $\operatorname{DTr}$-obit of the injective
$\Gamma$ module. Since $\operatorname{D}(kQ)\bigotimes\Lambda$ is an injective
cogenerator of $\Gamma$-mod, and it is a direct sum of the modules with the
form $I\bigotimes\Lambda\varepsilon$ where $I$ is an injective $kQ$ module and
$\varepsilon\in\\{\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{m}\\}$,
then we only have to check the length of $I\bigotimes\Lambda\varepsilon$.
Define
$\mathcal{N}(-)=\operatorname{D}\operatorname{Hom}_{\Lambda}(-,\Lambda)$, and
$\mathcal{N}^{n+1}(-)=\mathcal{N}(\mathcal{N}^{n}(-)),\operatorname{DTr}^{n+1}(-)=\operatorname{DTr}(\operatorname{DTr}^{n}(-))$.
Then
$\exists\varepsilon_{k}\in\\{\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{m}\\}$
such that $\Lambda\varepsilon_{k}=\mathcal{N}^{k}(\Lambda\varepsilon)$. So we
have
$\begin{CD}\overline{\operatorname{DTr}^{n}(I\bigotimes\Lambda\varepsilon)}=\overline{\operatorname{DTr}^{n}I\bigotimes\Lambda\varepsilon^{n}}.\end{CD}$
This is easy to be proved by induction. So the length of
$\operatorname{DTr}$-obit of $I\bigotimes\Lambda\varepsilon$ is equal to that
of $I$. The theorem is proved.
## Appendix A Appendix
In this section we will prove the following theorem where k can be a field or
commutative artin ring. Although it can be proved by the way in section 3, we
decide to introduce a way which is more combinatory.
###### Theorem A.1.
If $\mathcal{A}$ is an hom-finite k abeliean category with a finite number of
nonisomorphic indecomposable objects, then $\mathcal{A}$ is equivalent to the
finitely generated module category of a finite dimensional k algebra of
Representation-finite type .
As a corollary,we have
###### Corollary A.2.
Suppose $\Lambda$ is an artin algebra. If the projective module category is an
abelian category, then it is equivalent to the finitely generated module
category of a representation-finite artin algebra. So $\Lambda$ is a Auslander
algebra.
The corollary is a analogy of Theorem 3.1. The above theorem needs several
lemmas. From now on we, we suppose $\mathcal{A}$ is an home-finite k abeliean
category with a finite number of nonisomorphic indecomposable objects
$A_{1},A_{2},\dots,A_{n}$.
###### Lemma A.3.
If $M\in\mathcal{A}$, then $M$ is of finite length.
Proof. We have to prove M satisfies artin conditions and norther conditions.
Step1 $\forall X\in\mathcal{A}$, if $f:X\rightarrow X$ is an injective
morphism(or epicmorphism), then f is an isomorphism.
Suppose $f:X\in\mathcal{A}$ is an injective morphism but not epic and $\forall
i>0,g_{i}=\operatorname{cok}f^{i}$ where$f^{i}=f\dots f,f^{1}=f$. Then
$\forall j$, $g_{i}f^{j}=0$ if and only if $i\leq j$. Now suppose
$h=k_{1}f_{1}+k_{2}f_{2}+\dots+k_{m}f_{m}=0,m>0$. Then
$g_{2}h=k_{1}(g_{2}f_{1})+k_{2}(g_{2}f_{2})+\dots+k_{m}(g_{2}f_{m})=k_{1}(g_{2}f_{1})=0$.
So $k_{1}=0$. By induction, $k_{1}=k_{2}=\dots=k_{m}=o$. So
{$f,f^{2},f^{3},\dots$} is linear independent in $Hom(X,X)$ which is an
contradiction with the hom-finite property of $\mathcal{A}$.
Step2 M satisfies artin conditions.
Because the object in $\mathcal{A}$ is of a Krull-Schmidt category, for each
$X\in\mathcal{A},\exists x^{1},x^{2}\dots$ $x^{n},X\cong A_{1}^{x^{1}}\oplus
A_{2}^{x^{2}}\dots\oplus A_{n}^{x^{n}}$, we denote $x=(x^{1},x^{2}\dots
x^{n})$ as this decomposition. Suppose $\exists$ an infinite chain:
$\dots\stackrel{{\scriptstyle
f_{3}}}{{\rightarrow}}X_{2}\stackrel{{\scriptstyle
f_{2}}}{{\rightarrow}}X_{1}\stackrel{{\scriptstyle f_{1}}}{{\rightarrow}}X$
such that $f_{i}$ is a injective morphism but not an isomorphism. We denote
$x_{i}=(x_{1}^{1},x_{i}^{2}\dots x_{i}^{n})$ if $X_{i}=A_{1}^{x_{i}^{1}}\oplus
A_{2}^{x_{i}^{2}}\dots\oplus A_{n}^{x_{i}^{n}}$. Then we get a sequence in
$N^{n}$. There exists $i>0$ such that $\forall j>i,1\leq k\leq n,x_{i}^{k}\leq
x_{j}^{k}$. Thus there is an injective morphism: $g:X_{i}\rightarrow X_{i+1}$.
So $f_{i+1}g:X_{i}\rightarrow X_{i}$ is an injective morphism. By (1), it is
an isomorphism. So $f_{i+1}$ is also is an injective morphism. That is
contradictive.
Step3 M satisfies noetherian conditions.
Suppose $\exists$ an infinite subobject chain of $X$:
$X_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}X_{2}\stackrel{{\scriptstyle
f_{2}}}{{\rightarrow}}X_{3}\stackrel{{\scriptstyle
f_{3}}}{{\rightarrow}}\dots$ such that $f_{i}$ is a injective morphism but not
an isomorphism. Then we also get a sequence $\\{x_{1},x_{2}\dots\\}$in
$N^{n}$. Denote $S(x_{i})=\sum_{k=1}^{n}x_{i}^{k}$. By step 1, we know
$sup\\{S(x_{1}),S(x_{2})\dots\\}$ $=\infty$ $\Rightarrow\exists
i,sup\\{x_{1}^{i},x_{2}^{i}\dots\\}=\infty$ $\Rightarrow
sup\\{dim_{k}Hom(A_{i},X_{1}),dim_{k}Hom(A_{i},X_{2})\dots\\}=\infty$. But we
know $dim_{k}Hom(A_{i},X_{1})\leq dim_{k}Hom(A_{i},X)$. So
$dim_{k}Hom(A_{i},X)=\infty$ that’s contradicted with the hom-finite property.
The following lemma can be proved similarly by the way in [1, chapter 6].
###### Lemma A.4.
$\exists m\in N$, for every chain $X_{1}\stackrel{{\scriptstyle
f_{1}}}{{\rightarrow}}X_{2}\stackrel{{\scriptstyle
f_{2}}}{{\rightarrow}}X_{3}\stackrel{{\scriptstyle
f_{3}}}{{\rightarrow}}\dots\stackrel{{\scriptstyle
f_{m}}}{{\rightarrow}}X_{m+1}$ with $X_{i}\in\\{A_{1},A_{2},\dots,A_{n}\\}$,
if $f_{j}$ is not an isomorphism for every $j=1,2,\dots,m+1$, then
$f_{m}f_{m-1}\dots f_{1}=0$.
###### Lemma A.5.
Suppose$X\in\mathcal{A}$. The following are equivalent.
$\left(1\right)$ $X$ is a projective object.
$\left(2\right)$ if $f:Y\rightarrow X$ is a right minimal epic morphism, then
$f$ is an isomorphism.
Proof. $\left(1\right)\Rightarrow\left(2\right)$: clear.
$\left(2\right)\Rightarrow\left(1\right)$: Suppose $X$ has the property in
$\left(2\right)$. And $f:Y\rightarrow X$ is an epic morphism. Then
$f=(f_{1},f_{2})$ where
$f_{1}\in\operatorname{Hom}(Y_{1},X),f_{2}\in\operatorname{Hom}(Y_{2},X)$,
$Y=Y_{1}\bigoplus Y_{2}$ such that $f_{1}$ is right minimal and $f_{2}=0$. So
$f_{1}$ is an isomorphism. $f$ is a split epic morphism. So $X$ is a
projective object.
###### Lemma A.6.
$\mathcal{A}$ has enough projective objects
Proof. Suppose $X\in\mathcal{A}$ such that $X$ has no projective cover and $X$
is an indecomposable object. So there exists a right minimal epic morphism
$f_{1}:Y_{1}\rightarrow X$ such that $f_{1}$ is not an isomorphism by the
above lemma. So there exists $Y_{1}=Q_{1}\bigoplus X_{1}$ such that $Q_{1}$ is
a projective object, $X_{1}$ has no projective direct summand, $X_{1}\neq 0$,
and $f_{1}=(g_{1},h_{1})$ where
$g_{1}\in\operatorname{Hom}(Q_{1},X),h_{1}\in\operatorname{Radical}\operatorname{Hom}(X_{1},X)$.
By the way above, we consider the indecomposable direct summand of $X_{1}$.
Then there exists an epic morphism $f_{2}:Y_{2}\rightarrow X_{1}$ such that
$f_{2}\in\operatorname{Radical}\operatorname{Hom}(Y_{2},X_{1}).$ So there
exists $Y_{2}=Q_{2}\bigoplus X_{2}$ such that $Q_{2}$ is a projective object,
$X_{2}$ has no projective direct summand, $X_{2}\neq 0$ since $X$ has no
projective cover, and$f_{2}=(g_{2},h_{2})$ where
$g_{2}\in\operatorname{Hom}(Q_{2},X_{1}),h_{2}\in\operatorname{Radical}\operatorname{Hom}(X_{2},X_{1})$.
By induction, for $k>0$, we get $Y_{k}=Q_{k}\bigoplus X_{k}$ such that $Q_{k}$
is a projective object, $X_{k}$ has no projective direct summand, $X_{k}\neq
0$ since $X$ has no projective cover, and$f_{k}=(g_{k},h_{k})$ where
$g_{k}\in\operatorname{Hom}(Q_{k},X_{k-1}),h_{k}\in\operatorname{Radical}\operatorname{Hom}(X_{k},X_{k-1})$.
We have the following diagram to explain the operation:
---
$\textstyle{\dots}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{1}}$$\textstyle{Q_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\textstyle{X}$$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$
So there exists an epic morphism $(h_{m}\dots
h_{1},\phi_{m}):X_{m}\bigoplus(Q_{1}\bigoplus\dots\bigoplus Q_{m})\rightarrow
X$. Since $X$ has no projective cover, $h_{m}\dots h_{1}\neq 0$. That is
contradicted with the property of $m$.
Thus the above lemma tells us the abelian category has a projective generator.
So by the following well known lemma. The theorem is proved.
###### Lemma A.7.
If an abelian categoryis a hom-finite k category with a projective generator,
then it is equivalent to the left finitely generated module category of the
opposite endomorphism ring of the projective generator.
Acknowledgement. This article is part of the author’s Ph.D. thesis under the
supervision of Pu Zhang. The author is deeply grateful to him for his guidance
and encouragement. The author also thanks Professor Ringel for providing him
the references [Mo], [Mu], [R], [T] and his excellent lectures in Shanghai
Jiao Tong University in 2011. The author also deeply thanks Baolin Xiong for
his helpful discussions and constant encouragement.
## References
* [AF] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, 2nd Edition, Graduate Texts in Mathematics, Vol. 13, Springer, New York, 1992
* [A] M. Auslander. Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971.
* [ARS] M. Auslander and I. Reiten, and S. O. Smal$\phi$. Representation theory of Artin algebras, volume 36 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.
* [AS1] M. Auslander, $\phi$. Solberg, Gorenstein algebras and algebras with dominant dimension at least 2, Comm. in Alg., 21(11)(1993) 3897 - 3934.
* [AS2] M. Auslander, $\phi$. Solberg, Relative homology and representation theory I, Relative homology and homologically fnite subcategories, Comm. in Alg., 21(9)(1993) 2995- 3031.
* [AS3] M. Auslander, $\phi$. Solberg, Relative homology and representation theory II, Relative colilting theory, Comm. in Alg., 21(9)(1993) 3033-3079.
* [AS4] M. Auslander, $\phi$. Solberg, Relative homology and representation theory 111, Cotilting modules and Wedderburn correspondence. Comm. in Alg., 21 (1993), 3081-3097
* [H] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes 119 (University Press, Cambridge, 1988).
* [I1] O. Iyama, Auslander correspondence. Adv. Math., 210(2007) 51 - 82.
* [I2] O. Iyama, Higher dimensional Auslander - Reiten theory on maximal orthogonal subcategories. Adv. Math., 210(1)(2007) 22 - 50.
* [Mo] K. Morita, Duality for modules and its applications in the theory of rings with minimum condition. Sci. Rep. Tokyo Daigaku A 6(1958), 83-142.
* [Mu] BRUNO J. MUELLER, The classification of algebras by dominant dimension, Canad. J. Math., 20(1968) 398-409
* [R] C.M.Ringel, Artin algebras of dominant dimension at least 2, http://www.mathematik.uni-bielefeld.de/ ringel/opus/domdim.pdf
* [T] H.Tachikawa: On dominant dimension of QF-3 algebras. Trans. Amer. Math. Soc. 112 (1964), 249-266.
* [Z] Pu Zhang. A brief introduction to Gorenstein projective modules, http://www.mathematik.uni-bielefeld.de/ sek/sem/abs/zhangpu4.pdf.
Fan Kong, Department of Mathematics, Shanghai Jiao Tong University, 200240
Shanghai, People’s Republic of China.
Email: Kongfan08@yahoo.com.cn
|
arxiv-papers
| 2012-04-04T15:07:03 |
2024-09-04T02:49:29.368889
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fan Kong",
"submitter": "Fan Kong",
"url": "https://arxiv.org/abs/1204.0967"
}
|
1204.1160
|
# Opinion formation in time-varying social networks:
The case of Naming Game
Suman Kalyan Maity sumankalyan.maity@cse.iitkgp.ernet.in Department of
Computer Science and Engineering,
Indian Institute of Technology, Kharagpur, India – 721302 T.Venkat Manoj
manoj.venkat92@gmail.com Department of Computer Science and Engineering,
Indian Institute of Technology, Kharagpur, India – 721302 Animesh Mukherjee
animeshm@cse.iitkgp.ernet.in Department of Computer Science and Engineering,
Indian Institute of Technology, Kharagpur, India – 721302
###### Abstract
We study the dynamics of the Naming Game as an opinion formation model on
time-varying social networks. This agent-based model captures the essential
features of the agreement dynamics by means of a memory-based negotiation
process. Our study focuses on the impact of time-varying properties of the
social network of the agents on the Naming Game dynamics. We investigate the
outcomes of the dynamics on two different types of time-varying data - (i) the
networks vary across days and (ii) the networks vary within very short
intervals of time (20 seconds). In the first case, we find that networks with
strong community structure hinder the system from reaching global agreement;
the evolution of the Naming Game in these networks maintains clusters of
coexisting opinions indefinitely leading to metastability. In the second case,
we investigate the evolution of the Naming Game in perfect synchronization
with the time evolution of the underlying social network shedding new light on
the traditional emergent properties of the game that differ largely from what
has been reported in the existing literature.
###### pacs:
89.75.-k, 05.65.+b, 89.65.Ef
## I Introduction
Social networks are inherently dynamic. Social interactions and human
activities are intermittent, the neighborhood of individuals moving over a
geographic space evolves over time, links appear and disappear in the World-
Wide-Web. The essence of social network lies in its time-varying nature. Links
may exist for a certain time period and may be recurrent. In summary, as time
progresses, the societal structure keeps changing. Similarly, with the
evolution of time, social conventions, shared cultural and linguistic patterns
reshape themselves. Opinions spread, some gets trapped into communities, some
crosses the barrier of local groups/communities and become accepted globally
among different communities and some die competing with others. Most of these
social phenomena can be modeled and analyzed in a time-varying framework.
Almost all previous work is limited to the analysis of the Naming Game
dynamics on static networks baronchelli:06a ; topo ; qu ; dall ; liu ; chaos ;
luca ; luc ; yan . Therefore, in this paper, we focus on the competing opinion
formation over time-varying real-world social networks. One way of viewing at
time-varying networks is as a series of static graphs accumulated over a fixed
time interval; however this kind of networks do not always perfectly capture
the temporal ordering of the links appearing in the system which may sometimes
lead to over/under-estimation of network topologies. Thus, we plan to
investigate the opinion formation process on both the accumulated static
graphs as well as on its detailed time-resolved counterpart.
In this paper, we focus on the basic Naming Game model (NG) baronchelli:06a
to study how opinions spread with time and how societies move towards
consensus in the adoption of a single opinion through negotiation or agree
upon multiple opinions due to non-uniform interaction pattern among different
communities. The evolution of the system in this model takes place through the
usual local pairwise interactions among artificial agents that necessarily
captures the generic and essential features of an agreement process. This
model was expressly conceived to explore the role of self-organization in the
evolution of languages steels96aSelf ; steels96selfOrganizing and has
acquired a paradigmatic role in semiotic dynamics that studies evolution of
languages through invention of new words, grammatical constructions and more
specifically, through adoption of new meaning for different words. NG finds
wide applications in various fields ranging from artificial sensor network as
a leader election model baronchelli:11 to the social media as an opinion
formation model.
The minimal Naming Game (NG) consists of a population of $N$ agents observing
a single object in the environment (may be a discussion on a particular topic)
and opining for that by means of communication to one another through pairwise
interactions, in order to reach a global agreement. The agents have at their
disposal an internal inventory, in which they can store an unlimited number of
different words or opinions. At the beginning, all the individuals have empty
inventories. At each time step, the dynamics consists of a pairwise
interaction between randomly chosen individuals. The chosen individuals can
take part in the interaction as a “speaker” or as a “hearer.” The speaker
voices to the hearer a possible opinion for the object under consideration; if
the speaker does not have one, i.e., ̵͑his inventory is empty͒, he invents an
opinion͔. In case where he already has many opinions ̵͑stored in his
inventory͒, he chooses one of them randomly. The hearer’s move is
deterministic: if she possesses the opinion pronounced by the speaker, the
interaction is a “success”, and in this case both speaker and hearer retain
that opinion as the right one, removing all other competing opinions/words in
their inventories; otherwise, the new opinion is included in the inventory of
the hearer, without any cancellation of opinions in which case the interaction
is termed as a “failure” (see fig 1). The game is played on a fully connected
network, i.e., each agent can, in principle, communicate with all the other
agents, and makes two basic assumptions. One assumes that the number of
possible opinions is so huge that the probability of a opinion being
reinvented is practically negligible (this means that similar opinions is not
taken into account here, although the extension is trivially possible). As a
consequence, one can reduce, without loss of generality, the environment to be
consisting of only one single object/topic of discussion.
Figure 1: (Color online) Agent’s interaction rules in basic NG. Suppose there
is a topic on which a discussion is going on, say “Who is the greatest tennis
player?”. (Top) The speaker chosen at random, opines for “Federer” (also
chosen randomly from his inventory of opinions). Now, the hearer (again chosen
at random) does not have this opinion in her inventory, and therefore she adds
the opinion “Federer” in her inventory and the interaction is a failure .
(Bottom) The speaker opines for “McEnroe” and in this case the opinion is
present in the hearer’s inventory. So, they delete all other opinions except
“McEnroe”. The interaction this time is “success”.
Although the system reaches a global consensus through the invention and decay
of opinions, it is interesting to note the important differences from other
opinion formation models. In Axelrod’s model Axelrod_1997 , each agent is
endowed with a vector of opinions, and can interact with other agents only if
their opinions are already close enough; in Sznajd’s model sznajdweron and in
the Voter model Krapivsky92 , the opinion can take only two discrete values,
and an agent takes deterministically the opinion of one of its neighbors.
Further in Deffuant01 , the opinion is modeled as a unique variable and the
evolution of two interacting agents is deterministic. In the Naming Game model
on the other hand, each agent can potentially have an unlimited number of
possible discrete states (or opinions) at the same time, accumulating in its
memory different possible opinions; the agents are able to “wait” before
reaching a decision. Moreover, each dynamical step can be seen as a
negotiation between a speaker and a hearer, with a certain degree of
stochasticity.
In this paper, we consider the NG dynamics on two different types of time-
varying data; one varying across days while another varying over very short
intervals of time (20 seconds). In the first case, we observe that networks
with strong community structures delay the convergence due to co-existence of
competing and long-lasting clusters of opinions. In the second case, the games
are played in perfect synchronization with the time-evolution of the network.
In this case, we observe that the global observables are markedly different
from the case where the games are played on the static (and composite) version
of the same network as well as from the traditional results reported in the
literature.
The rest of the paper is organized as follows. Section 2 is devoted for the
discussion of the state of the art. In Section 3, we describe the datasets on
which we investigate the Naming Game dynamics in a time-varying social
scenario. Section 4 provides the elaborate model description. In Section 5, we
present the results and provide explanations for our findings. Finally,
conclusions are drawn in section 6.
## II Related work
Most previous studies of the NG model in semiotic/opinion dynamics have
focused on populations of agents in which all pairwise interactions are
allowed, i.e., the agents are placed on the vertices of a fully connected
graph. In statistical mechanics, this topological structure is commonly
referred to as “mean-field” topology baronchelli:06a ; baronchelli08 . Apart
from mean-field case, the model has also been studied on regular lattices topo
; qu ; small world networks qu ; dall ; liu ; chaos ; random geometric graphs
qu ; lu ; hao ; and static luca ; luc ; yan , dynamic who , and empirical con
complex networks.
Lu et. al. con have studied the Naming Game dynamics on a high-school
friendship network and have shown that the presence of community structures
affect the behavior of the dynamics through the formation of long-living late-
stage meta-stable clusters of opinions. Therefore, they propose injection of
committed agents (agents who never change their opinion) into the population
for fast agreement of the dynamics. Nardini et. al. who studied the dynamics
of NG on adaptive networks where the connections can be rewired with the
evolution of the game.
All these prior works have studied the NG dynamics on essentially static
networks. Therefore, our study reported here is unique and different from the
literature since we consider the evolution of the NG dynamics over time-
varying social structure.
## III Datasets
For the purpose of the investigation of the NG dynamics on time-varying
networks, we consider two specific real-world face-to-face contact datasets
and present our results on each of them. Both the datasets are obtained from
http://www.sociopatterns.org/datasets/. The first dataset we consider is the
face-to-face interaction data of visitors of the Science Gallery in Dublin,
Ireland during the spring of 2009 at the event of art-science exhibition
“INFECTIOUS: STAY AWAY”. The dataset contains the cumulative daily networks of
the visitors for sixty-nine days Isella2011166 . The nodes represent visitors
of the Science Gallery while the edges represent close-range face-to-face
proximity (measured using RFID devices carried by each visitor) between the
concerned persons. The weights associated with the edges are the number of 20
seconds intervals during which close-range face-to-face proximity could be
detected. Thus, these daily networks can be thought of as sixty-nine snapshots
of a time-varying societal structure with a periodicity of 24 hours. We will
refer to this as the SG dataset.
We also consider the time-resolved datasets which are the dynamic counterparts
of the daily cumulated contact networks of the SG dataset. From the 69 daily
instances, we consider time-resolved contact pattern of four instances day 9,
20, 22 and 26, which can be considered as the representatives of all the
instances. These time-resolved data are refered to as SGD dataset.
The last dataset we consider is the face-to-face interaction data of the
conference attendees of the ACM Hypertext 2009 conference held in Institute
for Scientific Interchange Foundation in Turin, Italy, from June 29th to July
1st, 2009, where the SocioPatterns project deployed the Live Social Semantics
application. The dataset contains the dynamical network of face-to-face
proximity of 115 conference attendees over about 2.5 days. In future
reference, we will refer to this as the HT dataset.
## IV The model description
The basic NG Model can be summarized as follows. At each time step ($t$ = 1,
2, ..) two agents are randomly selected to interact: one of them plays the
role of speaker, the other one that of hearer. The interactions obey the
following rules
* •
The speaker voices an opinion from its list of opinions to the hearer. (If the
speaker has more than one opinion on his list, he randomly chooses one; if he
has none, he invents one randomly.)
* •
If the hearer has this opinion, the communication is termed “successful”, and
both players delete all other opinions, i.e., collapse their list of opinions
to this one opinion. Therefore, they meet a local agreement.
* •
If the hearer does not have the opinion transmitted by the speaker (termed
“unsuccessful” communication), she adds the opinion to her list of opinions
without any deletion.
Note that in this model any agent is free to interact with any other agent,
i.e., the underlying social structure is assumed to be fully connected. For
the purpose of our analysis however, we assume that the agents are embedded on
realistic social networks (i.e., SG and HT) that are continuously varying over
time. In this case, although the basic rules of the game remain exactly same,
the only issue is to devise a strategy for the speaker-hearer selection. We
consider two variants of this selection, the first one being suitable for the
SG dataset and the second one for the SGD and HT dataset.
Strategy I: Here we randomly select a speaker and preferentially choose a
hearer among its neighbors. Our intention here is to simulate an important
criterion: we talk most preferably to those with whom we had already met
before. This is implemented as follows:
* •
The speaker $i$ is selected randomly.
* •
The hearer $j$ is selected using the preferential rule, with the probability
$p_{ij}=\frac{w_{ij}}{\sum_{j=1}^{k}w_{ij}}$
where $w_{ij}$ can be thought of as the total number of contact events between
the pair $i$ and $j$ while $k$ is the degree of agent $i$ (i.e, the number of
other agents that $i$ is connected to at a particular instant of time).
Strategy II: This variant is quite straight-forward. We choose a random
speaker and a random hearer among its neighbors to impart the equal importance
of each pair of connections.
The main quantities of interest which describe the emergent properties of the
system are
* •
the total number $N_{w}(t)$ of words/opinions in the system at the time $t$
(i.e., the total size of the memory);
* •
the number of different words/opinions $N_{d}(t)$ in the system at the time
$t$;
* •
the average success rate $S(t)$, i.e., the probability, computed averaging
over many simulation runs, that the chosen agent gets involved in a successful
interaction at a given time $t$.
From a global perspective, the quantities which are of interest are the time
to reach the global consensus ($t_{conv}$), the maximum memory required by the
agents during the process ($N_{w}^{max}$) and the time required to reach the
memory peak ($t_{max}$).
## V Results and discussions
In this section, we present the results of the analysis of the NG dynamics on
the SG, SGD and the HT datasets.
### V.1 Analysis on day-wise SG dataset
We have studied the opinion formation process on the sixty-nine days of close
interactions among the visitors for the SG dataset. The primary focus of this
study is to find the behavior of the global quantities of the NG dynamics with
the evolution of the topology over time. We play the NG on each of these daily
networks of sixty-nine days following Strategy I. The networks are not always
fully connected. In case of disconnected components, $N_{d}$ should never
converge to 1 and consequently, the emergence of multi-opinion state is
observed. Therefore, in this case we redefine $t_{conv}$ as the time to reach
the following state: $N_{w}=N$ and $N_{d}=c$ where $c$ is the number of
disconnected components. The natural question that arises is how the opinion
dynamics gets affected as the underlying network structure varies over the
days. It is interesting to note that the memory peak $N_{w}^{max}$ and the
time to reach the memory peak $t_{max}$ have a strong correlation with the
system size $N$. In fact, they have a linear scaling with the population size
$N$ (See fig 2). This relationship is in agreement with what has been observed
in the literature for small-world, scale-free and random networks luca ; chaos
where both $N_{w}^{max}$ and $t_{max}$ scales as $O(N)$.
Figure 2: (Color online) Scaling relation of $N_{w}^{max}$ and $t_{max}$ with
population size $N$. (a) Temporal behavior of $N_{w}^{max}$ and the population
size $N$ for each of the 69 instances. Data are smoothed by taking 7 point
running average. (b) Scaling of $N_{w}^{max}$ with $N$ which is linearly
fitted. The inset shows the scaling of $N_{w}^{max}$ in the thresholded
networks where the threshold is set to 1, 2 and 5 respectively. All these
curves are linearly fitted. (c) Variation of $t_{max}$ and population size $N$
with time. Data are smoothed by taking 7 point running average. (d) Scaling of
$t_{max}$ with $N$ which is linearly fitted. The inset shows the scaling of
$t_{max}$ in the thresholded networks where the threshold is set to 1, 2 and 5
respectively. All these curves are linearly fitted.
Nevertheless, these cumulative daily networks do not resemble any of the well-
known topologies which will be clear when we dig into the results of
$t_{conv}$. We observe that the behavior of $t_{conv}$ (see fig 3(a), (b)) is
not in lines of the existing literature where it is usually noted that
$t_{conv}\sim N^{1.4}$. Therefore the natural question that needs to be
addressed is that what is (are) the property (s) of the underlying network
that leads to such a non-conforming behavior of $t_{conv}$.
Figure 3: (Color online) (a)Temporal behavior of $t_{conv}$ and the population
size $N$. Data are smoothed by taking 7 point running average. (b) Scatter
plot of $t_{conv}$ and $N$ which could not be fitted with $y=x^{1.4}$
($R^{2}\approx 0.03$). (c), (d) and (e) Correlation of $t_{conv}$ with
variance of community sizes $Var(s)$ detected by various community detection
algorithms. The curves are smoothed by taking 20 point running averages.
In fact, the answer to this question lies in the common behavior of the real-
world social networks. These networks typically consist of a number of
communities; nodes within communities are more densely connected, while links
bridging communities are sparse. The effect of the community structure plays a
dominant role with the emergence of long-lasting multi-opinion states at the
late stage of the dynamics which has also been observed in luca and con . In
fact, each community reaches internal consensus fast but the weak connections
between communities are not sufficient for opinions to propagate from one
community to the other leading to long multi-opinion states which are also
known as “metastable states” in the domain of statistical physics. Formally, a
metastable state is a state of the dynamics where global shifts are always
possible but progressively more unlikely and the response properties depend on
the age of the system Mukherjee2011Aging . Community structures are
essentially authentic signatures of metastability which inhibits the dynamics
leading to very slow convergence.
Presence of community structures slows down the dynamics, however, what
renders the system even slower is the presence of different-sized communities.
The reason for this is quite straight-forward: the agents that are part of a
larger size community have a higher probability of being chosen for a game
than those belonging to a smaller size community. This is a reminiscent of the
fact that the agents are chosen randomly which automatically increases the
chances of landing in a larger size community simply because a larger bulk of
the population is confined within this community. Therefore, even when
consensus is reached very fast in a large community, the system keeps on
choosing agents from this community itself mostly resulting in “success with
no outcome”. Further, since the inter-community links are weak, and agents
from smaller commuinities are hardly chosen the overall state of the system
hardly changes thereby always keeping the agents away from the global
consensus. This is reflected through fig 3(c), (d) and (e) where we report the
correlation of $t_{conv}$ with the variance of the community sizes. The basic
idea is as follows: if a network gets decomposed into m communities each of
size $s_{1},s_{2},...,s_{m}$ then we calculate the statistical variance of
this size distribution and plot it against $t_{conv}$. For the purpose of
community analysis, we use three standard algorithms - Newman and Girvan (NGR)
NewGir04 , Newman, Clauset and Moore (NCM) Clauset04 and community detection
by eigen vector (EV) Newman_2006 and in each case we observe that $t_{conv}$
has a strong positive correlation with the variance of the community sizes
(see fig 3 (c), (d) and (e)).
#### V.1.1 Effect of edge weights
As we have suggested earlier, we consider two variants of pair selection, the
weighted and the unweighted one. In this subsection, we attempt to study the
effect of edge-weights on the dynamics. The edge weights play significant role
in pair-selection and so there is a possibility that this affects the
dynamics. However, what we observe here is in the contrary. The global
quantities of interest in case where all the neighboring agents are given
equal preference remain roughly equivalent to the case where the weights are
considered (see fig 4(a), (b) and (c)). The reason behind this is the skewed
distribution of edge weights.
We find that above 60 % edges on average are low-weight edges which somehow
drives the dynamics of the preferential model towards the behavior close to
the dynamics of the unweighted NG dynamics (see table 1). In addition, we also
observe a strong correlation between the average degree $\langle k\rangle$ and
the average strength $\langle s\rangle$ (see fig 4(d)). The weighted
clustering coefficient $C^{w}$ is also close to the topological clustering
coefficient $C$ (see fig 4(e)). Further, the weighted average nearest neighbor
degree $\langle k_{nn}^{w}\rangle$ and the unweighted average nearest neighbor
degree $\langle k_{nn}\rangle$ are perfectly correlated (see fig 4(f)).
Figure 4: (Color online) Effect of edge weights on the dynamics. (a), (b) and
(c) Temporal evolution of $N_{w}^{max}$, $t_{max}$ and $t_{conv}$ for the
weighted and unweighted NG respectively smoothed over a time sliding window of
size 7 . (d) Variation of $\langle k\rangle$ with $\langle s\rangle$. (e)
Variation of $C^{w}$ with $C$. (f) Variation of $\langle k_{nn}^{w}\rangle$
with $\langle k_{nn}\rangle$. Table 1: Distribution of edge weights averaged
over all 69 instances.
Edge Weights | % of edges having that weight
---|---
1 | 0.41
2 | 0.13
3 | 0.06
4 | 0.04
5 | 0.03
Others | 0.33
#### V.1.2 Examples of individual instances
In this subsection, we dig deeper into the individual snapshots to have a more
clear understanding of the ongoing dynamics. From the sixty-nine instances, we
present four representative cases that roughly capture all the different
characteristics found across the instances. Two among these, consist of
disconnected components while the other two are single connected components.
Further, two of them (one connected and the other disconnected) show fast
convergence while another two (again one connected and the other disconnected)
show slow convergence triggered by the presence of community structures
leading to metastability. Here we propose two metrics to capture the two
distinct behaviors of the convergence time. The first one is the average
unique words per community which is denoted by $U(t)$ and defined as follows:
$U(t)=\frac{\sum_{i=1}^{C}|A_{i}|}{C}$
where $C$ is the number of communities and $A_{i}$ is the list of unique words
in community $i$.
The second metric we propose is the average overlap of unique words across
communities which is denoted by $O_{c}(t)$ and defined as follows:
$O_{c}(t)=\frac{2}{C(C-1)}\sum_{i>j}\frac{2(|A_{i}\bigcap
A_{j}|)}{\sqrt{2(|A_{i}|^{2}+|A_{j}|^{2})}}$
Figure 5: (Color online) (a) and (b) Comparison of the evolution of the total
number of words $N_{w}(t)$ and number of different words $N_{d}(t)$ with time
on four representative networks. (c) Average number of unique words per
community $U(t)$ evolving over time. (d) Temporal evolution of average overlap
of words across communities $O_{c}(t)$. Each point in the above curves
represents the average value obtained over 100 simulation runs.
We consider the daily networks of 9th, 20th, 22nd and 26th day. The 9th day
and 22nd day network structures consist of a single connected component with
200 and 240 nodes respectively while the 20th and 26th daily network consist
of multiple disconnected components with 96 and 156 nodes respectively. The
evolution of $N_{w}(t)$ shows a steady growth signifying inventions of new
opinions coupled with a series of failure interactions until the maxima is
reached (see fig 5(a)). From this point onward, the reorganization phase
commences and the players encounter mostly successful interaction resulting in
the drop of $N_{w}(t)$ (fig 5(a)). While for the 20th (disconnected network)
and 22nd (connected network) day consensus is reached fast, for the 9th
(connected network) and the 26th (disconnected network) day the system gets
arrested in a long plateau indicating the presence of metastability and strong
community structures. The growth of $N_{d}(t)$ also signifies similar pattern,
steady rise followed by steady fall and a plateau (signifying a strong
community structure) in case of the 9th and 26th day (see fig 5(b)). To
explore the flat plateau region further we report $U(t)$ and $O_{c}(t)$ in fig
5(c) and 5(d) respectively. It is interesting to note that both $U(t)$ and
$O_{c}(t)$ show a plateau in case of the 9th and the 26th day which is a
signature of the fact that the games played in the plateau region
predominantly produces success with no deletion of opinions leading to the
emergence of metastability.
### V.2 Analysis on the time-resolved dataset
In this section, we consider the datasets containing dynamic face-to-face
interactions. We play the Naming Game on these time-varying networks in
complete synchronization with the real time, i.e., a single game is played on
a single time-resolved snapshot of the same network. Thus, at each time step
$t$ = 1, 2 . . . , the game is played among those agents that are alive at
that particular instant of time in the network. We consider the Strategy II
where at each time step, we choose a random speaker and a random hearer among
its neighbors.
Figure 6: (Color online) (a) and (b) The temporal evolution of $N_{w}(t)$ and
$N_{d}(t)$ on time-varying day 9 SGD dataset. The data are averaged over 100
simulation runs. (c) The behavior of the number of inventions of opinions
$N_{i}(t)$ over time. (d) Comparison of $\Delta N_{w}(t)$ with success rate
$S(t)$. (e) Comparison of temporal evolution of $\Delta N_{w}(t)$ and number
of new connections smoothed by taking 20 point running average. (f) Comparison
of $\Delta N_{w}(t)$ with the variance of community sizes (found by NGR, NCM
and EV algorithm) evolving over time (the curves are suitably scaled by some
constant for the purpose of better visualization). The data are smoothed by
taking 20 point running average.
#### V.2.1 Results from SGD dataset
In this section, we consider time-resolved dataset of four representatives
from the SG dataset. The networks on 9th and 22nd day consist of a single
components; however, the 20th and 26th day networks show existence of multiple
disconnected components. We analyze each of these time-evolving networks and
report the behavior of the global quantities as well as different network
properties infuencing the game dynamics.
The time evolution of $N_{w}(t)$ and $N_{d}(t)$ on the time-varying graph of
day 9 (see fig 6(a) and (b)) show a drastically different behavior from the
case where these quantities are measured on the static (and composite)
counterpart (see fig 5(a) and (b)). The temporal graph shows a slow growth
regime followed by a sharp transition, whereas the static counterpart shows
steady growth regime followed by a steady fall and finally a long-lasting
metastable state (see fig 5(a)). This difference in behavior is due to the
fact that in the time-varying case inventions of opinions prevail throughout
the dynamics (see fig 6(c)) which prevents the disposal of opinions from the
system and hence the memory sizes do not decrease. Further, in fig 6(d) we
show how the absolute change in $N_{w}$ is driven by the success rate; $\Delta
N_{w}$ increases with a decrease in $S(t)$ while it decreases with an increase
in $S(t)$. Fig 6(e) shows the direct correspondence of the $\Delta N_{w}$ with
the new connections. Another interesting property which has an impact on the
dynamics is the community size. Indeed the variance of the community sizes
relates in a similar way to $\Delta N_{w}$ (see fig 6(f)). Therefore, the
continuous inventions, the influx of new connections (causing more failures)
and the fact that the opinions get trapped within local neighborhoods together
contribute to the steeply rising memory size over the time evolution of the
dynamics.
Figure 7: (Color online) (a) and (b) The temporal evolution of $N_{w}(t)$ and
$N_{d}(t)$ on time-varying day 20 SGD dataset. The data are averaged over 100
simulation runs. (c) The behavior of the number of inventions of opinions
$N_{i}(t)$ over time. (d) Comparison of $\Delta N_{w}(t)$ with success rate
$S(t)$. (e) Comparison of temporal evolution of $\Delta N_{w}(t)$ and number
of new connections smoothed by taking 20 point running average. (f) Comparison
of $\Delta N_{w}(t)$ with the variance of community sizes (found by NGR, NCM
and EV algorithm) evolving over time (the curves are suitably scaled by some
constant for the purpose of better visualization). The data are smoothed by
taking 20 point running average. Figure 8: (Color online) (a) and (b) The
temporal evolution of $N_{w}(t)$ and $N_{d}(t)$ on time-varying day 22 SGD
dataset. The data are averaged over 100 simulation runs. (c) The behavior of
the number of inventions of opinions $N_{i}(t)$ over time. (d) Comparison of
$\Delta N_{w}(t)$ with success rate $S(t)$. (e) Comparison of temporal
evolution of $\Delta N_{w}(t)$ and number of new connections smoothed by
taking 20 point running average. (f) Comparison of $\Delta N_{w}(t)$ with the
variance of community sizes (found by NGR, NCM and EV algorithm) evolving over
time (the curves are suitably scaled by some constant for the purpose of
better visualization). The data are smoothed by taking 20 point running
average. Figure 9: (Color online) (a) and (b) The temporal evolution of
$N_{w}(t)$ and $N_{d}(t)$ on time-varying day 26 SGD dataset. The data are
averaged over 100 simulation runs. (c) The behavior of the number of
inventions of opinions $N_{i}(t)$ over time. (d) Comparison of $\Delta
N_{w}(t)$ with success rate $S(t)$. (e) Comparison of temporal evolution of
$\Delta N_{w}(t)$ and number of new connections smoothed by taking 20 point
running average. (f) Comparison of $\Delta N_{w}(t)$ with the variance of
community sizes (found by NGR, NCM and EV algorithm) evolving over time (the
curves are suitably scaled by some constant for the purpose of better
visualization). The data are smoothed by taking 20 point running average.
The time-varying networks of 20th, 22nd and 26th day also behave more or less
in the same way as for day 9 (see fig 7, fig 8, fig 9). All these time-varying
networks typically show a similar slow growth stage followed by a steep rise
which is in contrast to their static counterparts (see fig 5) where we found
mostly 3 distinct phase of the dynamics - a growth stage, followed by a sharp
fall due to series of successful interactions and a meta-stable state due to
presence of community structures. This discrepancy in the behavior of
$N_{w}(t)$ and $N_{d}(t)$ curves is due to the fact that the time-varying
networks witness a continuous influx of new agents into the system with
inventions happening throughout the evolution and the old agents not playing
enough games with the new ones to negotiate and agree upon an opinion.
In fig 10, we show how the frequency of interaction between a pair of
individuals predicts the similarity of the opinions among individuals over
different instants of time. We measure the similarity of opinions between a
pair of individuals by Jaccard Coefficient (JC) of their inventories. It is
formally defined as the size of the intersection divided by the size of the
union of the inventories i.e., $JC(A_{i},A_{j})=\frac{|A_{i}\cap
A_{j}|}{|A_{i}\cup A_{j}|}$ where $A_{i}$ is $i^{th}$ agent’s inventory. From
all the graphs, it is evident that there is a trend of having higher
similarity in opinions with the higher edge-weight where edge-weight reflects
the frequency of interactions between a pair till that particular instant of
time. Thus, with frequent meetings, individuals tend to share similar opinion.
This usually also happens in real-life scenarios where more we meet more
similar-opinionated people we become.
Figure 10: Comparison of mean similarity with edge weights. The graphs on the
first row show the similarity results on SGD 9th day for four different
instances of time at $t=T/4$, $t=T/2$, $t=3T/4$ and $t=T$ where $T$ is the
total time. Similarly, rows 2, 3 and 4 show similarity results for SGD 20th,
22nd and 26th day respectively at four different time instances - $t=T/4$,
$t=T/2$, $t=3T/4$ and $t=T$. The curves are smoothed by taking 10 point
running average. Figure 11: (Color online) Comparison of the global quantities
in the real and the simulated networks. The first row corresponds to the
temporal evolution of $N_{w}$ and $N_{d}$ for SGD 9th day network. The second,
third and fourth row respectively correspond to the temporal evolution of
$N_{w}$ and $N_{d}$ for SGD 20th, 22nd and 26th day network. The datapoints on
the curve are averaged over 100 simulation runs for each of 100 network
realizations (in case of the simulated network).
Results from the control experiments: For the purpose of control experiment,
we create simulated versions for each of the four time-varying networks by
constructing random edges of same number as in the real network in each 20s
time interval. We play the naming game on these simulated networks. These
types of networks resemble stochastic networks where edges randomly appear or
disappear in each epoch. The behavior of the $N_{w}(t)$ and $N_{d}(t)$ (see
fig 11) are not in the lines of what we observe in the real counterparts. In
all the simulated networks, the $N_{w}(t)$ and $N_{d}(t)$ behave similarly as
in case of static Erdős-Rényi graphs luca .
#### V.2.2 Results from the HT dataset
In this section, we consider the second dataset containing dynamic face-to-
face interactions among 113 conference attendees. We first study the global
behavior of the system through the temporal evolution of three main
quantities: the total number
$N_{w}(t)$͒ofopinionsinthesystem,thenumberofdifferentopinions$N_d(t)$͒, and
the rate of success $S(t)$.
Figure 12: (Color online) (a) and (b) The temporal evolution of $N_{w}(t)$ and
$N_{d}(t)$ on time-varying conference network respectively. The insets show
the evolution of $N_{w}(t)$ and $N_{d}(t)$ on their static counterpart. The
data are averaged over 1000 simulation runs. (c) The behavior of the number of
inventions of opinions $N_{i}(t)$ over time. (d) Comparison of $\Delta
N_{w}(t)$ with success rate $S(t)$.
The curve corresponding to $N_{w}(t)$ shows an initial slow growth followed by
a sharp transition and finally reaching a steady growth regime (see fig
12(a)). Note that this result is markedly in contrast to what would have been
observed if the games were played on the composite network constructed at the
end of the conference (see fig 12(a) inset). In fact, this result is in
contrast to most of the other results that have been reported in the
literature so far indicating that the time-varying nature of the underlying
societal structure with new connections being formed, old connections being
dropped and agents entering, leaving and re-entering the system has a strong
impact on the emergent pattern of opinion formation. Similar trends are also
observed for $N_{d}(t)$ \- initially a slow growth followed by a sharp
transition reaching a peak and finally a drop, however, no way close to 1 (see
fig 12(b)). The inset in fig 12(b) shows the evolution of $N_{d}$ if the games
were played on the composite network finally obtained.
Initially, as time proceeds, new individuals join the network that increases
the number of inventions of new opinions (see fig 12(c)) thus causing a rise
in both $N_{w}(t)$ and $N_{d}(t)$. However, later on new inventions stop (fig
12(c)) as the players joining late are less compared to the number that have
already joined and are therefore rarely chosen as speakers thus inhibiting new
inventions. Hence, $N_{d}(t)$ is found to drop in the later stage of the
dynamics although pointing to a clear existence of multiple opinions. In
contrast, $N_{w}(t)$ doesn’t drop because although new opinions are not
formed, old opinions trapped in different groups do not get disposed off the
system.
Further, in fig 12(d) we show how the absolute change in $N_{w}$ is driven by
the rate of success of agents; $\Delta N_{w}$ increases with a decrease in
$S(t)$ while it decreases with an increase in $S(t)$. Finally, an important
analysis that is required to complete the picture centers around the precise
reason for the steady growth in $N_{w}$ in the final regime of the dynamics.
We attempt to provide a plausible explanation for this through a series of
results reported in fig 13.
Figure 13: (Color online) (a) Evolution of inventory sizes $n$ $(n=0,1...)$.
$f_{n}(t)$ is the fraction of agents whose inventory size is $n$ at time $t$.
(b) Comparison of temporal evolution of $\Delta N_{w}(t)$ and number of new
connections smoothed by 20 point running average. (c) Comparison of $\Delta
N_{w}(t)$ with the variance of community sizes (found by NGR, NCM and EV
algorithm) evolving over time (the curves are suitably scaled by some constant
for the purpose of better visualization). The data are smoothed by taking
20-point running average.
In fig 13(a), we present the fraction of agents having 0, 1, 2 and more
opinions in their inventories. Clearly, with the evolution of system, the
fraction of agents with inventory size 0 diminishes; fraction of agents with
size 1 increase steadily while that with size 2 is roughly stable; even larger
size inventories appear only rarely in the course of the evolution. In
addition, we observe that $\Delta N_{w}$ has a direct correspondence with the
number of new connections acquired by the network at each timestep (see fig
13(b)). These new connections trigger an increase in failure events, thereby
increasing $N_{w}$; at the same time success events cannot reduce $N_{w}$
since in most cases the inventory sizes of the agents are already very low
($\sim 1$) and most of these success events are actually again “success with
no outcome”. This last observation indicates that there should be an inherent
community structure in this time-varying network and this is made apparent
through fig 13(c) where we report the variance of the the size of the
communities (using three different algorithms as in the previous cases) and
show that this is highly correlated to $\Delta N_{w}$. In summary, the
presence of community structure coupled with a continuous influx of new
connections (leading to late-stage failures in the system) together lead to
the steady growth of $N_{w}$ in its final regime of evolution.
In fig 14, we present mean similarity of opinions among individuals with edge
weights in different time instances. In all the instances, there is a positive
correlation of having similar opinions with frequency of interactions i.e.,
higher the frequency of interactions (edge-weight), higher is the similarity
in opinion.
Figure 14: Comparison of mean similarity with edge weights on HT dataset. The
mean similarity of opinions with edge weights are shown at different instances
of time (a) at $t=T/4$, (b) $t=T/2$, (c) $t=3T/4$ and (d) $t=T$ where $T$ is
the total time. The curves are smoothed by taking 40 point running average.
Results from control experiments : For this HT dataset also, we create
simulated networks where at each 20s time interval, we construct $m$ number of
random edges with $m$ being the count of edges that appeared on that time
interval in the real network. We observe the two most important observables
$N_{w}(t)$ and $N_{d}(t)$ by playing naming game on these simulated networks.
Both these quantities show a different behavior from its real counterpart. The
$N_{w}(t)$ and $N_{d}(t)$ in the simulated networks show distinct two regions
- a steady growth and then a fall whereas the $N_{w}(t)$ curve in the real
network show a slow growth zone followed by a sharp transition and finally a
zone of steady growth. The simulated networks tend to behave as standard
Erdős-Rényi graphs.
Figure 15: (Color online) Comparison of the global quantities in the real and
the simulated networks. Temporal evolution of (a) $N_{w}$, (b) $N_{d}$ for HT
real and simulated dataset. The datapoints on the curve are averaged over 100
simulation runs for each of 100 network realizations (in case of simulated
network).
## VI Conclusions and future work
In this paper, we studied the Naming Game as a model of opinion formation on
the time-varying social networks. Some of our key observations are:
(a) While considering composite snapshots accumulated over a certain period
(e.g., 69 instances of SG dataset with each instance being an accumulation of
snapshots ) both the maximum memory and the time to reach this memory peak
scale as population size ($N$); however, the time to reach the consensus
strongly depends on the presence of community structure (rather than a
straight-forward $N^{1.4}$ scaling);
(b) While considering the time-evolution of the network in perfect
synchronization with the steps of the game (e.g., SGD and HT dataset) we
observe that the emergent behavior of the most important observables (i.e.,
$N_{w}(t)$ and $N_{d}(t)$) have a nature that is markedly in contrast to what
has been reported so far in the literature thus indicating the strong
influence of the underlying societal structure on the dynamics of opinion
formation. While in case of SGD, we observe that new inventions along with a
continuous influx of new agents keeps both $N_{w}(t)$ and $N_{d}(t)$ sharply
growing, in case of HT, successful interactions among older agents cause
inventions to stop (hence a fall in $N_{d}(t)$) although late-stage failures
continue to exist due to influx of new agents thus contributing to a steady
growth of $N_{w}(t)$ in the final phase of the dynamics. The fall of
$N_{d}(t)$ curve is not observed in case of SGD possibly because in this case
the games are played over a shorter span of time (1 day) in comparison to HT
where the games are played over 2.5 days so that enough successful
interactions could be realized.
There are quite a few interesting directions that can be explored in the
future. One such direction could be to incorporate the dominance index of the
agents into the model. Not all actors in a society are equally dominant; while
some of the actors are more opinionated and dominant the others might be more
passive. This characteristic property can be incorporated into the model by
ranking those agents that are more successful in their past interactions as
more dominant. In this setting, it would be interesting to investigate the
scaling relations most naturally under the constraints that the dominant
agents are allowed to speak more. Another direction could be to investigate
the effect of the flexibility of the agents in adapting to new opinions
(traditionally modeled by a system parameter $\beta$ that encodes the
probability with which the agents update their inventories in case of
successful interactions Baronchelli07 ) when they are embedded on time-varying
networks. Finally, a thorough analytical estimate of the important dynamical
quantities reported only through empirical evidence here is needed to have a
“clear-cut” understanding of the emergent behavior of the system.
## References
* [1] R. Axelrod. The dissemination of culture: A model with local convergence and global polarization. Journal of Conflict Resolution, 41(2), 1997.
* [2] A. Baronchelli. Role of feedback and broadcasting in the naming game. Physical Review E, 83(4), 2011.
* [3] A. Baronchelli, L. Dall’Asta, A. Barrat, and V. Loreto. Topology-induced coarsening in language games. Physical Review E, 73(1), 2006.
* [4] A. Baronchelli, L. Dall’Asta, A. Barrat, and V. Loreto. Nonequilibrium phase transition in negotiation dynamics. Physical Review E, 76, 2007.
* [5] A. Baronchelli, M. Felici, V. Loreto, E. Caglioti, and L. Steels. Sharp transition towards shared vocabularies in multi-agent systems. Journal of Statistical Mechanics: Theory and Experiment, 2006(06), 2006.
* [6] A. Baronchelli and V. Loreto. In-depth analysis of the naming game dynamics: The homogeneous mixing case. International Journal of Modern Physics C, 19(5), 2008.
* [7] A. Barrat, A. Baronchelli, L. Dall’Asta, and V. Loreto. Agreement dynamics on interaction networks with diverse topologies. Chaos, 17(2), 2007.
* [8] A. Clauset, M. E. J. Newman, and C. Moore. Finding community structure in very large networks. Physical Review E, 70(6), 2004.
* [9] L. Dall’Asta and A. Baronchelli. Microscopic activity patterns in the naming game. Journal of Physics A: Mathematical and General, 39(48), 2006.
* [10] L. Dall’Asta, A. Baronchelli, A. Barrat, and V. Loreto. Agreement dynamics on small-world networks. Europhysics Letters, 73(6), 2006.
* [11] L. Dall’asta, A. Baronchelli, A. Barrat, and V. Loreto. Nonequilibrium dynamics of language games on complex networks. Physical Review E, 74(3), 2006.
* [12] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch. Mixing beliefs among interacting agents. Advances in Complex Systems, 3, 2001.
* [13] L. Isella, J. Stehlé, A. Barrat, C. Cattuto, J.-F. Pinton, and W. V. den Broeck. What’s in a crowd? analysis of face-to-face behavioral networks. Journal of Theoretical Biology, 271(1), 2011.
* [14] H. Jia-Bo, Y. Han-Xin, L. Run-Ran, W. Bing-Hong, and Z. Zhi-Yuan. Effect of geometric distance on agreement dynamics of naming game. Chinese Physics Letters, 27(9), 2010.
* [15] P. L. Krapivsky. Kinetics of monomer-monomer surface catalytic reactions. Physical Review A, 45(2), 1992.
* [16] R.-R. Liu, C.-X. Jia, H.-X. Yang, and B.-H. Wang. Naming game on small-world networks with geographical effects. Physica A: Statistical Mechanics and its Applications, 388(17), 2009\.
* [17] Q. Lu, G. Korniss, and B. K. Szymanski. Naming games in spatially-embedded random networks. CoRR, abs/cs/0604075, 2006.
* [18] Q. Lu, G. Korniss, and B. K. Szymanski. Naming games in two-dimensional and small-world-connected random geometric networks. Physical Review E, 77(1), 2008.
* [19] Q. Lu, G. Korniss, and B. K. Szymanski. The naming game in social networks: community formation and consensus engineering. Journal of Economic Interaction and Coordination, 4(2), 2009.
* [20] A. Mukherjee, F. Tria, A. Baronchelli, A. Puglisi, and V. Loreto. Aging in language dynamics. PLoS ONE, 6(2), 2011.
* [21] C. Nardini, B. Kozma, and A. Barrat. Who’s talking first? consensus or lack thereof in coevolving opinion formation models. Physical Review Letters, 100(15), 2008.
* [22] M. E. J. Newman. Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74(3 Pt 2), 2006.
* [23] M. E. J. Newman and M. Girvan. Finding and evaluating community structure in networks. Physical Review E, 69(2), 2004.
* [24] L. Steels. A self-organizing spatial vocabulary. Artificial Life, 2(3), 1995.
* [25] L. Steels. Self-organizing vocabularies. In C. G. Langton and K. Shimohara, editors, Artificial Life V, Nara, Japan, 1996.
* [26] K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community. International Journal of Modern Physics C, 11, 2000.
* [27] H.-X. Yang, W.-X. Wang, and B.-H. Wang. Asymmetric negotiation in structured language games. Physical Review E, 77(2), 2008.
|
arxiv-papers
| 2012-04-05T09:35:05 |
2024-09-04T02:49:29.382984
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Suman Kalyan Maity, T. Venkat Manoj, Animesh Mukherjee",
"submitter": "Suman Kalyan Maity",
"url": "https://arxiv.org/abs/1204.1160"
}
|
1204.1237
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-091 LHCb-PAPER-2011-022 5 April 2012
Measurements of the branching fractions of the decays $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
LHCb collaboration †††Authors are listed on the following pages.
The decay mode $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ allows for one of
the theoretically cleanest measurements of the CKM angle $\gamma$ through the
study of time-dependent $C\\!P$ violation. This paper reports a measurement of
its branching fraction relative to the Cabibbo-favoured mode
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ based on a data sample of 0.37 fb-1
proton-proton collisions at $\sqrt{s}=7$ TeV collected in 2011 with the LHCb
detector. In addition, the ratio of $B$ meson production fractions
$f_{s}/f_{d}$, determined from semileptonic decays, together with the known
branching fraction of the control channel $B^{0}\\!\rightarrow D^{-}\pi^{+}$,
is used to perform an absolute measurement of the branching fractions:
${\cal B}\left(B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}\right)\;=(2.95\pm
0.05\pm 0.17^{\,+\,0.18}_{\,-\,0.22})\times 10^{-3}\,$,
${\cal B}\left(B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}\right)=(1.90\pm
0.12\pm 0.13^{\,+\,0.12}_{\,-\,0.14})\times 10^{-4}\,$,
where the first uncertainty is statistical, the second the experimental
systematic uncertainty, and the third the uncertainty due to $f_{s}/f_{d}$.
Submitted to JHEP
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, H. Voss10, R. Waldi55, S. Wandernoth11, J. Wang53,
D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, K.
Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Unlike the flavour-specific decay $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$,
the Cabibbo-suppressed decay $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$
proceeds through two different tree-level amplitudes of similar strength: a
$\bar{b}\rightarrow\bar{c}u\bar{s}$ transition leading to
$B^{0}_{s}\rightarrow D^{-}_{s}K^{+}$ and a
$\bar{b}\rightarrow\bar{u}c\bar{s}$ transition leading to
$B^{0}_{s}\rightarrow D^{+}_{s}K^{-}$. These two decay amplitudes can have a
large $C\\!P$-violating interference via $B^{0}_{s}-\bar{B}^{0}_{s}$ mixing,
allowing the determination of the CKM angle $\gamma$ with negligible
theoretical uncertainties through the measurement of tagged and untagged time-
dependent decay rates to both the $D^{-}_{s}K^{+}$ and $D^{+}_{s}K^{-}$ final
states [1]. Although the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ decay
mode has been observed by the CDF [2] and BELLE [3] collaborations, only the
LHCb experiment has both the necessary decay time resolution and access to
large enough signal yields to perform the time-dependent $C\\!P$ measurement.
In this analysis, the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ branching
fraction is determined relative to $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$,
and the absolute $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ branching fraction
is determined using the known branching fraction of $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ and the production fraction ratio $f_{s}/f_{d}$ [4]. The two
measurements are then combined to obtain the absolute branching fraction of
the decay $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$. Charge conjugate modes
are implied throughout. Our notation $B^{0}\\!\rightarrow D^{-}\pi^{+}$, which
matches that of Ref. [5], encompasses both the Cabibbo-favoured
$B^{0}\rightarrow D^{-}\pi^{+}$ mode and the doubly-Cabibbo-suppressed
$B^{0}\rightarrow D^{+}\pi^{-}$ mode.
The LHCb detector [6] is a single-arm forward spectrometer covering the
pseudo-rapidity range $2<\eta<5$, designed for studing particles containing
$b$ or $c$ quarks. The detector includes a high-precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$ interaction
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\rm\,Tm}$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream. The combined
tracking system has a momentum resolution $\Delta p/p$ that varies from 0.4%
at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, an impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum, and a decay time
resolution of 50 fs. Charged hadrons are identified using two ring-imaging
Cherenkov detectors. Photon, electron and hadron candidates are identified by
a calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic calorimeter, and a hadronic calorimeter. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers.
The LHCb trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. Two categories of events are recognised based on
the hardware trigger decision. The first category are events triggered by
tracks from signal decays which have an associated cluster in the
calorimeters, and the second category are events triggered independently of
the signal decay particles. Events which do not fall into either of these two
categories are not used in the subsequent analysis. The second, software,
trigger stage requires a two-, three- or four-track secondary vertex with a
large value of the scalar sum of the transverse momenta ($p_{\rm T}$) of the
tracks, and a significant displacement from the primary interaction. At least
one of the tracks used to form this vertex is required to have $p_{\rm T}>1.7$
GeV$/c$, an impact parameter $\chi^{2}$ $>16$, and a track fit $\chi^{2}$ per
degree of freedom $\chi^{2}/\textrm{ndf}$ $<2$. A multivariate algorithm is
used for the identification of the secondary vertices [7]. Each input variable
is binned to minimise the effect of systematic differences between the trigger
behaviour on data and simulated events.
The samples of simulated events used in this analysis are based on the Pythia
6.4 generator [8], with a choice of parameters specifically configured for
LHCb [9]. The EvtGen package [10] describes the decay of the $B$ mesons, and
the Geant4 package [11] simulates the detector response. QED radiative
corrections are generated with the Photos package [12].
The analysis is based on a sample of $pp$ collisions corresponding to an
integrated luminosity of 0.37 fb-1, collected at the LHC in 2011 at a centre-
of-mass energy $\sqrt{s}=7$ TeV. The decay modes $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ are
topologically identical and are selected using identical geometric and
kinematic criteria, thereby minimising efficiency corrections in the ratio of
branching fractions. The decay mode $B^{0}\\!\rightarrow D^{-}\pi^{+}$ has a
similar topology to the other two, differing only in the Dalitz plot structure
of the $D$ decay and the lifetime of the $D$ meson. These differences are
verified, using simulated events, to alter the selection efficiency at the
level of a few percent, and are taken into account.
$B^{0}_{s}$ ($B^{0}$) candidates are reconstructed from a $D^{-}_{s}$
($D^{-}$) candidate and an additional pion or kaon (the “bachelor” particle),
with the $D^{-}_{s}$ ($D^{-}$) meson decaying in the $K^{+}K^{-}\pi^{-}$
($K^{+}\pi^{-}\pi^{-}$) mode. All selection criteria will now be specified for
the $B^{0}_{s}$ decays, and are implied to be identical for the $B^{0}$ decay
unless explicitly stated otherwise. All final-state particles are required to
satisfy a track fit $\chi^{2}/\textrm{ndf}<4$ and to have a high transverse
momentum and a large impact parameter $\chi^{2}$ with respect to all primary
vertices in the event. In order to remove backgrounds which contain the same
final-state particles as the signal decay, and therefore have the same mass
lineshape, but do not proceed through the decay of a charmed meson, the flight
distance $\chi^{2}$ of the $D^{-}_{s}$ from the $B^{0}_{s}$ is required to be
larger than $2$. Only $D^{-}_{s}$ and bachelor candidates forming a vertex
with a $\chi^{2}/\textrm{ndf}<9$ are considered as $B^{0}_{s}$ candidates. The
same vertex quality criterion is applied to the $D^{-}_{s}$ candidates. The
$B^{0}_{s}$ candidate is further required to point to the primary vertex
imposing $\theta_{\textrm{flight}}<0.8$ degrees, where
$\theta_{\textrm{flight}}$ is the angle between the candidate momentum vector
and the line between the primary vertex and the $B^{0}_{s}$ vertex. The
$B^{0}_{s}$ candidates are also required to have a $\chi^{2}$ of their impact
parameter with respect to the primary vertex less than 16.
Further suppression of combinatorial backgrounds is achieved using a gradient
boosted decision tree technique [13] identical to the decision tree used in
the previously published determination of $f_{s}/f_{d}$ with the hadronic
decays [14]. The optimal working point is evaluated directly from a sub-sample
of $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ events, corresponding to 10$\%$
of the full dataset used, distributed evenly over the data taking period and
selected using particle identification and trigger requirements. The chosen
figure of merit is the significance of the $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ signal, scaled according to the Cabibbo suppression
relative to the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ signal, with
respect to the combinatorial background. The significance exhibits a wide
plateau around its maximum, and the optimal working point is chosen at the
point in the plateau which maximizes the signal yield. Multiple candidates
occur in about $2\%$ of the events and in such cases a single candidate is
selected at random.
## 2 Particle identification
Particle identification (PID) criteria serve two purposes in the selection of
the three signal decays $B^{0}\\!\rightarrow D^{-}\pi^{+}$,
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$. When applied to the decay products of the $D^{-}_{s}$ or
$D^{-}$, they suppress misidentified backgrounds which have the same bachelor
particle as the signal mode under consideration, henceforth the “cross-feed”
backgrounds. When applied to the bachelor particle (pion or kaon) they
separate the Cabibbo-favoured from the Cabibbo-suppressed decay modes. All PID
criteria are based on the differences in log-likelihood (DLL) between the
kaon, proton, or pion hypotheses. Their efficiencies are obtained from
calibration samples of $D^{*+}\rightarrow(D^{0}\rightarrow
K^{-}\pi^{+})\pi^{+}$ and $\mathchar 28931\relax\rightarrow p\pi^{-}$ signals,
which are themselves selected without any PID requirements. These samples are
split according to the magnet polarity, binned in momentum and $p_{\rm T}$,
and then reweighted to have the same momentum and $p_{\rm T}$ distributions as
the signal decays under study.
The selection of a pure $B^{0}\\!\rightarrow D^{-}\pi^{+}$ sample can be
accomplished with minimal PID requirements since all cross-feed backgrounds
are less abundant than the signal. The $\overline{}\mathchar
28931\relax^{0}_{b}\\!\rightarrow\overline{}\mathchar
28931\relax_{c}^{-}\pi^{+}$ background is suppressed by requiring that both
pions produced in the $D^{-}$ decay satisfy $\textrm{DLL}_{\pi-p}>-10$, and
the $B^{0}\\!\rightarrow D^{-}K^{+}$ background is suppressed by requiring
that the bachelor pion satisfies $\textrm{DLL}_{K-\pi}<0$.
The selection of a pure $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ or
$B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ sample requires the suppression
of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $\overline{}\mathchar
28931\relax^{0}_{b}\\!\rightarrow\overline{}\mathchar
28931\relax_{c}^{-}\pi^{+}$ backgrounds, whereas the combinatorial background
contributes to a lesser extent. The $D^{-}$ contamination in the $D^{-}_{s}$
data sample is reduced by requiring that the kaon which has the same charge as
the pion in $D^{-}_{s}\rightarrow K^{+}K^{-}\pi^{-}$ satisfies
$\textrm{DLL}_{K-\pi}>5$. In addition, the other kaon is required to satisfy
$\textrm{DLL}_{K-\pi}>0$. This helps to suppress combinatorial as well as
doubly misidentified backgrounds. For the same reason the pion is required to
have $\textrm{DLL}_{K-\pi}<5$. The contamination of $\overline{}\mathchar
28931\relax^{0}_{b}\\!\rightarrow\overline{}\mathchar
28931\relax_{c}^{-}\pi^{+}$, $\overline{}\mathchar
28931\relax_{c}^{-}\rightarrow\overline{}pK^{+}\pi^{-}$ is reduced by applying
a requirement of $\textrm{DLL}_{K-p}>0$ to the candidates that, when
reconstructed under the $\overline{}\mathchar
28931\relax_{c}^{-}\rightarrow\overline{}pK^{+}\pi^{-}$ mass hypothesis, lie
within $\pm 21$ MeV$/c^{2}$ of the $\overline{}\mathchar 28931\relax_{c}^{-}$
mass.
Because of its larger branching fraction, $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ is a significant background to $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$. It is suppressed by demanding that the bachelor satisfies
the criterion $\textrm{DLL}_{K-\pi}>5$. Conversely, a sample of
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$, free of $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ contamination, is obtained by requiring that the bachelor
satisfies $\textrm{DLL}_{K-\pi}<0$. The efficiency and misidentification
probabilities for the PID criterion used to select the bachelor, $D^{-}$, and
$D^{-}_{s}$ candidates are summarised in Table 1.
Table 1: PID efficiency and misidentification probabilities, separated according to the up (U) and down (D) magnet polarities. The first two lines refer to the bachelor track selection, the third line is the $D^{-}$ efficiency and the fourth the $D^{-}_{s}$ efficiency. Probabilities are obtained from the efficiencies in the $D^{*+}$ calibration sample, binned in momentum and $p_{\rm T}$. Only bachelor tracks with momentum below 100 GeV$/c$ are considered. The uncertainties shown are the statistical uncertainties due to the finite number of signal events in the PID calibration samples. | PID Cut | Efficiency ($\%$) | Misidentification ($\%$)
---|---|---|---
| | U | D | U | D
$K$ | $\textrm{DLL}_{K-\pi}>5$ | $83.3\pm 0.2$ | $83.5\pm 0.2$ | $5.3\pm 0.1$ | $4.5\pm 0.1$
$\pi$ | $\textrm{DLL}_{K-\pi}<0$ | $84.2\pm 0.2$ | $85.8\pm 0.2$ | $5.3\pm 0.1$ | $5.4\pm 0.1$
$D^{-}$ | | $84.1\pm 0.2$ | $85.7\pm 0.2$ | - | -
$D^{-}_{s}$ | | $77.6\pm 0.2$ | $78.4\pm 0.2$ | - | -
## 3 Mass fits
The fits to the invariant mass distributions of the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ candidates
require knowledge of the signal and background shapes. The signal lineshape is
taken from a fit to simulated signal events which had the full trigger,
reconstruction, and selection chain applied to them. Various lineshape
parameterisations have been examined. The best fit to the simulated event
distributions is obtained with the sum of two Crystal Ball functions [15] with
a common peak position and width, and opposite side power-law tails. Mass
shifts in the signal peaks relative to world average values [5], arising from
an imperfect detector alignment [16], are observed in the data and are
accounted for. A constraint on the $D^{-}_{s}$ meson mass is used to improve
the $B^{0}_{s}$ mass resolution. Three kinds of backgrounds need to be
considered: fully reconstructed (misidentified) backgrounds, partially
reconstructed backgrounds with or without misidentification (e.g.
$B^{0}_{s}\\!\rightarrow D^{*-}_{s}K^{+}$ or $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\rho^{+}$), and combinatorial backgrounds.
The three most important fully reconstructed backgrounds are
$B^{0}\\!\rightarrow D^{-}_{s}K^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ for $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$, and
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ for $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$. The mass distribution of the $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ events does not suffer from fully reconstructed backgrounds. In
the case of the $B^{0}\\!\rightarrow D^{-}_{s}K^{+}$ decay, which is fully
reconstructed under its own mass hypothesis, the signal shape is fixed to be
the same as for $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ and the peak
position is varied. The shapes of the misidentified backgrounds
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ are taken from data using a reweighting procedure. First, a
clean signal sample of $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ decays is obtained by applying the
PID selection for the bachelor track given in Sect. 2. The invariant mass of
these decays under the wrong mass hypothesis ($B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ or $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$) depends on
the momentum of the misidentified particle. This momentum distribution must
therefore be reweighted by taking into account the momentum dependence of the
misidentifaction rate. This dependence is obtained using a dedicated
calibration sample of prompt $D^{*+}$ decays. The mass distributions under the
wrong mass hypothesis are then reweighted using this momentum distribution to
obtain the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ mass shapes under the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ mass
hypotheses, respectively.
For partially reconstructed backgrounds, the probability density functions
(PDFs) of the invariant mass distributions are taken from samples of simulated
events generated in specific exclusive modes and are corrected for mass
shifts, momentum spectra, and PID efficiencies in data. The use of simulated
events is justified by the observed good agreement between data and
simulation.
The combinatorial background in the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
and $B^{0}\\!\rightarrow D^{-}\pi^{+}$ fits is modelled by an exponential
function where the exponent is allowed to vary in the fit. The resulting shape
and normalisation of the combinatorial backgrounds are in agreement within one
standard deviation with the distribution of a wrong-sign control sample (where
the $D^{-}_{s}$ and the bachelor track have the same charges). The shape of
the combinatorial background in the $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ fit cannot be left free because of the partially
reconstructed backgrounds which dominate in the mass region below the signal
peak. In this case, therefore, the combinatorial slope is fixed to be flat, as
measured from the wrong sign events.
In the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ fit, an additional
complication arises due to backgrounds from $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{*-}_{s}p$, which fall in the signal
region when misreconstructed. To avoid a loss of $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ signal, no requirement is made on the $\textrm{DLL}_{K-p}$
of the bachelor particle. Instead, the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ mass shape is obtained from
simulated $\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ decays,
which are reweighted in momentum using the efficiency of the
$\textrm{DLL}_{K-\pi}>5$ requirement on protons. The $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{*-}_{s}p$ mass shape is obtained by
shifting the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ mass
shape downwards by 200 MeV$/c^{2}$. The branching fractions of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow D^{*-}_{s}p$ are assumed to be equal,
motivated by the fact that the decays $B^{0}\rightarrow D^{-}D_{s}^{+}$ and
$B^{0}\rightarrow D^{-}D_{s}^{*+}$ (dominated by similar tree topologies) have
almost equal branching fractions. Therefore the overall mass shape is formed
by summing the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ and
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{*-}_{s}p$ shapes with equal
weight.
The signal yields are obtained from unbinned extended maximum likelihood fits
to the data. In order to achieve the highest sensitivity, the sample is
separated according to the two magnet polarities, allowing for possible
differences in PID performance and in running conditions. A simultaneous fit
to the two magnet polarities is performed for each decay, with the peak
position and width of each signal, as well as the combinatorial background
shape, shared between the two.
The fit under the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ hypothesis
requires a description of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ background. A
fit to the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ spectrum is first performed to
determine the yield of signal $B^{0}\\!\rightarrow D^{-}\pi^{+}$ events, shown
in Fig. 1. The expected $B^{0}\\!\rightarrow D^{-}\pi^{+}$ contribution under
the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ hypothesis is subsequently
constrained with a $10\%$ uncertainty to account for uncertainties on the PID
efficiencies. The fits to the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
candidates are shown in Fig. 1 and the fit results for both decay modes are
summarised in Table 2. The peak position of the signal shape is varied, as are
the yields of the different partially reconstructed backgrounds (except
$B^{0}\\!\rightarrow D^{-}\pi^{+}$) and the shape of the combinatorial
background. The width of the signal is fixed to the values found in the
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ fit ($17.2$ MeV$/c^{2}$), scaled by the
ratio of widths observed in simulated events between $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ decays ($0.987$).
The accuracy of these fixed parameters is evaluated using ensembles of
simulated experiments described in Sect. 4. The yield of $B^{0}\\!\rightarrow
D^{-}_{s}\pi^{+}$ is fixed to be 2.9% of the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ signal yield, based on the world average branching fraction
of $B^{0}\\!\rightarrow D^{-}_{s}\pi^{+}$ of $(2.16\pm 0.26)\times 10^{-5}$,
the value of $f_{s}/f_{d}$ given in [4], and the value of the branching
fraction computed in this paper. The shape used to fit this component is the
sum of two Crystal Ball functions obtained from the $B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}$ sample with the peak position fixed to the value obtained
with the fit of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$ data sample and the
width fixed to the width of the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
peak.
Figure 1: Mass distribution of the $B^{0}\\!\rightarrow D^{-}\pi^{+}$
candidates (top) and $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ candidates
(bottom). The stacked background shapes follow the same top-to-bottom order in
the legend and the plot. For illustration purposes the plot includes events
from both magnet polarities, but they are fitted separately as described in
the text.
The $\overline{}\mathchar
28931\relax^{0}_{b}\\!\rightarrow\overline{}\mathchar
28931\relax_{c}^{-}\pi^{+}$ background is negligible in this fit owing to the
effectiveness of the veto procedure described earlier. Nevertheless, a
$\overline{}\mathchar 28931\relax^{0}_{b}\\!\rightarrow\overline{}\mathchar
28931\relax_{c}^{-}\pi^{+}$ component, whose yield is allowed to vary, is
included in the fit (with the mass shape obtained using the reweighting
procedure on simulated events described previously) and results in a
negligible contribution, as expected.
Table 2: Results of the mass fits to the $B^{0}\\!\rightarrow D^{-}\pi^{+}$, $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$, and $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ candidates separated according to the up (U) and down (D) magnet polarities. In the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ case, the number quoted for $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ also includes a small number of $B^{0}\\!\rightarrow D^{-}\pi^{+}$ events which have the same mass shape (20 events from the expected misidentification). See Table 3 for the constrained values used in the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ decay fit for the partially reconstructed backgrounds and the $B^{0}\\!\rightarrow D^{-}K^{+}$ decay channel. Channel | $B^{0}\\!\rightarrow D^{-}\pi^{+}$ | $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ | $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$
---|---|---|---
| U | D | U | D | U | D
$N_{\textrm{Signal}}$ | $16304\pm 137$ | $20150\pm 152$ | $2677\pm 62$ | $3369\pm 69$ | $195\pm 18$ | $209\pm 19$
$N_{\textrm{Comb}}$ | $\;\,1922\pm 123$ | $\;\,2049\pm 118$ | $\;\,869\pm 63$ | $\;\,839\pm 47$ | $149\pm 25$ | $255\pm 30$
$N_{\textrm{Part-Reco}}$ | $10389\pm 407$ | $12938\pm 441$ | $2423\pm 65$ | $3218\pm 69$ | - | -
$N_{B^{0}\\!\rightarrow D^{-}_{s}K^{+}}$ | - | - | - | - | $\;\,87\pm 17$ | $100\pm 18$
$N_{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}$ | - | - | - | - | $154\pm 20$ | $164\pm 22$
The fits for the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ candidates are
shown in Fig. 2 and the fit results are collected in Table 2. There are
numerous reflections which contribute to the mass distribution. The most
important reflection is $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$, whose
shape is taken from the earlier $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$
signal fit, reweighted according to the efficiencies of the applied PID
requirements. Furthermore, the yield of the $B^{0}\\!\rightarrow D^{-}K^{+}$
reflection is constrained to the values in Table 3. In addition, there is
potential cross-feed from partially reconstructed modes with a misidentified
pion such as $B^{0}_{s}\\!\rightarrow D^{-}_{s}\rho^{+}$, as well as several
small contributions from partially reconstructed backgrounds with similar mass
shapes. The yields of these modes, whose branching fractions are known or can
be estimated (e.g. $B^{0}_{s}\\!\rightarrow D^{-}_{s}\rho^{+}$,
$B^{0}_{s}\\!\rightarrow D^{-}_{s}K^{*+}$), are constrained to the values in
Table 3, based on criteria such as relative branching fractions and
reconstruction efficiencies and PID probabilities. An important cross-check is
performed by comparing the fitted value of the yield of misidentified
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ events ($318\pm 30$) to the yield
expected from PID efficiencies ($370\pm 11$) and an agreement is found.
Figure 2: Mass distribution of the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ candidates. The stacked background shapes follow the same top-to-bottom order in the legend and the plot. For illustration purposes the plot includes events from both magnet polarities, but they are fitted separately as described in the text. Table 3: Gaussian constraints on the yields of partially reconstructed and misidentified backgrounds applied in the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ fit, separated according to the up (U) and down (D) magnet polarities. Background type | U | D
---|---|---
$B^{0}\\!\rightarrow D^{-}K^{+}$ | $\,16\pm\,3$ | $\,17\pm\,3$
$B^{0}_{s}\\!\rightarrow D^{*-}_{s}\pi^{+}$ | $\;\,63\pm 21$ | $\;\,70\pm 23$
$B^{0}_{s}\\!\rightarrow D^{*-}_{s}K^{+}$ | $\;\,72\pm 34$ | $\;\,80\pm 27$
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\rho^{+}$ | $135\pm 45$ | $150\pm 50$
$B^{0}_{s}\\!\rightarrow D^{-}_{s}K^{*+}$ | $135\pm 45$ | $150\pm 50$
$B^{0}_{s}\\!\rightarrow D^{*-}_{s}\rho^{+}$ | $\;\,45\pm 15$ | $\;\,50\pm 17$
$B^{0}_{s}\\!\rightarrow D^{*-}_{s}K^{*+}$ | $\;\,45\pm 15$ | $\;\,50\pm 17$
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{-}_{s}p$ \+ $\mathchar 28931\relax^{0}_{b}\\!\rightarrow D^{*-}_{s}p$ | $\;\,72\pm 34$ | $\;\,80\pm 27$
## 4 Systematic uncertainties
The major systematic uncertainities on the measurement of the relative
branching fraction of $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ and
$B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ are related to the fit, PID
calibration, and trigger and offline selection efficiency corrections.
Systematic uncertainties related to the fit are evaluated by generating large
sets of simulated experiments using the nominal fit, and then fitting them
with a model where certain parameters are varied. To give two examples, the
signal width is deliberately fixed to a value different from the width used in
the generation, or the combinatorial background slope in the
$B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ fit is fixed to the combinatorial
background slope found in the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ fit.
The deviations of the peak position of the pull distributions from zero are
then included in the systematic uncertainty.
Table 4: Relative systematic uncertainities on the branching fraction ratios. Source | $\frac{B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}}{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}(\%)$ | $\frac{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}{B^{0}\\!\rightarrow D^{-}\pi^{+}}(\%)$ | $\frac{B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}}{B^{0}\\!\rightarrow D^{-}\pi^{+}}(\%)$
---|---|---|---
All non-PID selection | $2.0$ | $2.0$ | $3.0$
PID selection | $1.8$ | $1.3$ | $2.2$
Fit model | $2.4$ | $1.7$ | $2.2$
Efficiency ratio | $1.5$ | $1.6$ | $1.6$
Total | $3.9$ | $3.4$ | $4.6$
In the case of the $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ fit the
presence of constraints for the partially reconstructed backgrounds must be
considered. The generic extended likelihood function can be written as
$\mathcal{L}=e^{-N}N^{N_{\rm
obs}}\times\prod_{j}G(N^{j};N^{j}_{c},\sigma_{N^{j}_{0}})\times\prod_{i=1}^{N_{\rm
obs}}P(m_{i};\vec{\lambda})\,,$ (1)
where the first factor is the extended Poissonian likelihood in which $N$ is
the total number of fitted events, given by the sum of the fitted component
yields $N=\sum_{k}N_{k}$. The fitted data sample contains $N_{\rm obs}$
events. The second factor is the product of the $j$ external constraints on
the yields, $j<k$, where $G$ stands for a Gaussian PDF, and
$N_{c}\pm\sigma_{N_{0}}$ is the constraint value. The third factor is a
product over all events in the sample, $P$ is the total PDF of the fit,
$P(m_{i};\vec{\lambda})=\sum_{k}N_{k}P_{k}(m_{i};\vec{\lambda}_{k})$, and
$\vec{\lambda}$ is the vector of parameters that define the mass shape and are
not fixed in the fit.
Each simulated dataset is generated by first varing the component yield
$N_{k}$ using a Poissonian PDF, then sampling the resulting number of events
from $P_{k}$, and repeating the procedure for all components. In addition,
constraint values $N_{c}^{j}$ used when fitting the simulated dataset are
generated by drawing from $G(N;N^{j}_{0},\sigma_{N^{j}_{0}})$, where
$N^{j}_{0}$ is the true central value of the constraint, while in the nominal
fit to the data $N_{c}^{j}=N^{j}_{0}$.
The sources of systematic uncertainty considered for the fit are signal
widths, the slope of the combinatorial backgrounds, and constraints placed on
specific backgrounds. The largest deviations are due to the signal widths and
the fixed slope of the combinatorial background in the
$B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ fit.
The systematic uncertainty related to PID enters in two ways: firstly as an
uncertainty on the overall efficiencies and misidentification probabilities,
and secondly from the shape for the misidentified backgrounds which relies on
correct reweighting of PID efficiency versus momentum. The absolute errors on
the individual $K$ and $\pi$ efficiencies, after reweighting of the $D^{*+}$
calibration sample, have been determined for the momentum spectra that are
relevant for this analysis, and are found to be 0.5% for
$\textrm{DLL}_{K-\pi}<0$ and 0.5% for $\textrm{DLL}_{K-\pi}>5$.
The observed signal yields are corrected by the difference observed in the
(non-PID) selection efficiencies of different modes as measured from simulated
events:
$\displaystyle\epsilon(B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+})/\epsilon(B^{0}\\!\rightarrow D^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle 1.015\;,$ $\displaystyle\epsilon(B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+})/\epsilon(B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm})$
$\displaystyle=$ $\displaystyle 1.061\;.$
A systematic uncertainty is assigned on the ratio to account for percent level
differences between the data and the simulation. These are dominated by the
simulation of the hardware trigger. All sources of systematic uncertainty are
summarized in Table 4.
## 5 Determination of the branching fractions
The $B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}$ branching fraction relative
to $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ is obtained by correcting the
raw signal yields for PID and selection efficiency differences
$\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}\right)}{{\cal
B}\left(B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}\right)}=\frac{N_{B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}}}{N_{B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}}}\frac{\epsilon^{\textrm{PID}}_{B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}}}{\epsilon^{\textrm{PID}}_{B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}}}\frac{\epsilon^{\textrm{Sel}}_{B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}}}{\epsilon^{\textrm{Sel}}_{B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}}}\;,$ (2)
where $\epsilon_{X}$ is the efficiency to reconstruct decay mode $X$ and
$N_{X}$ is the number of observed events in this decay mode. The PID
efficiencies are given in Table 1, and the ratio of the two selection
efficiencies is $0.943\pm 0.013$.
The ratio of the branching fractions of $B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm}$ relative to $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ is
determined separately for the down ($0.0601\pm 0.0056$) and up ($0.0694\pm
0.0066$) magnet polarities and the two results are in good agreement. The
quoted errors are purely statistical. The combined result is
$\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow D_{s}^{\mp}K^{\pm}\right)}{{\cal
B}\left(B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}\right)}=0.0646\pm 0.0043\pm
0.0025\;,$
where the first uncertainty is statistical and the second is the total
systematic uncertainty from Table 4.
The relative yields of $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ are used to extract the branching fraction
of $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ from the following relation
${\cal B}(B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+})={\cal
B}\left(B^{0}\\!\rightarrow
D^{-}\pi^{+}\right)\frac{\epsilon_{B^{0}\\!\rightarrow
D^{-}\pi^{+}}}{\epsilon_{B^{0}_{s}\\!\rightarrow
D^{-}_{s}\pi^{+}}}\frac{N_{B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}}{\cal
B}\left(D^{-}\rightarrow
K^{+}\pi^{-}\pi^{-}\right)}{\frac{f_{s}}{f_{d}}N_{B^{0}\\!\rightarrow
D^{-}\pi^{+}}{\cal B}\left(D^{-}_{s}\rightarrow K^{-}K^{+}\pi^{-}\right)}\;,$
(3)
using the recent $f_{s}/f_{d}$ measurement from semileptonic decays [4]
$\frac{f_{s}}{f_{d}}=0.268\pm 0.008^{+0.022}_{-0.020}\;,$
where the first uncertainty is statistical and the second systematic. Only the
semileptonic result is used since the hadronic determination of $f_{s}/f_{d}$
relies on theoretical assumptions about the ratio of the branching fractions
of the $B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+}$ and $B^{0}\\!\rightarrow
D^{-}\pi^{+}$ decays. In addition, the following world average values [5] for
the $B$ and $D$ branching fractions are used
$\displaystyle{\cal B}(B^{0}\\!\rightarrow D^{-}\pi^{+})$ $\displaystyle=$
$\displaystyle\left(2.68\pm 0.13\right)\times 10^{-3}\;,$ $\displaystyle{\cal
B}(D^{-}\rightarrow K^{+}\pi^{-}\pi^{-})$ $\displaystyle=$
$\displaystyle\left(9.13\pm 0.19\right)\times 10^{-2}\;,$ $\displaystyle{\cal
B}(D^{-}_{s}\rightarrow K^{+}K^{-}\pi^{-})$ $\displaystyle=$
$\displaystyle\left(5.49\pm 0.27\right)\times 10^{-2}\;,$
leading to
$\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow D^{-}_{s}\pi^{+})$
$\displaystyle=$ $\displaystyle(2.95\pm 0.05\pm 0.17^{+0.18}_{-0.22})\times
10^{-3}\;,$ $\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow
D_{s}^{\mp}K^{\pm})$ $\displaystyle=$ $\displaystyle(1.90\pm 0.12\pm
0.13^{+0.12}_{-0.14})\times 10^{-4}\;,$
where the first uncertainty is statistical, the second is the experimental
systematics (as listed in Table 4) plus the uncertainty arising from the
$B^{0}\\!\rightarrow D^{-}\pi^{+}$ branching fraction, and the third is the
uncertainty (statistical and systematic) from the semileptonic $f_{s}/f_{d}$
measurement. Both measurements are significantly more precise than the
existing world averages [5].
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] R. Fleischer, New strategies to obtain insights into $CP$ violation through $B_{s}\rightarrow D_{s}^{\pm}K^{\mp},D_{s}^{\ast\pm}K^{\mp},...$ and $B_{d}\rightarrow D^{\pm}\pi^{\mp},D^{\ast\pm}\pi^{\mp},...$ decays, Nucl. Phys. B671 (2003) 459, arXiv:hep-ph/0304027
* [2] CDF collaboration, T. Aaltonen et al., First observation of $\bar{B}^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp}$ and measurement of the ratio of branching fractions B$(\bar{B}^{0}_{s}\rightarrow D_{s}^{\pm}K^{\mp})$ /B$(\bar{B}^{0}_{s}\rightarrow D_{s}^{+}\pi^{-})$, Phys. Rev. Lett. 103 (2009) 191802, arXiv:0809.0080
* [3] Belle collaboration, R. Louvot et al., Measurement of the decay $B_{s}^{0}\rightarrow D_{s}^{-}\pi^{+}$ and evidence for $B_{s}^{0}\rightarrow D_{s}^{\mp}K^{\pm}$ in $e^{+}e^{-}$ annihilation at $\sqrt{s}\sim 10.87$ GeV, Phys. Rev. Lett. 102 (2009) 021801, arXiv:0809.2526
* [4] LHCb collaboration, R. Aaij et al., Measurement of b hadron production fractions in 7 TeV pp collisions, Phys. Rev. D 85 (2012) 032008, arXiv:1111.2357
* [5] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
* [6] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [7] V. V. Gligorov, C. Thomas, and M. Williams, The HLT inclusive $B$ triggers, LHCb-PUB-2011-016
* [8] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [9] M. Clemencic et al., The LHCb Simulation Application, Gauss: Design, Evolution and Experience, Journal of Physics: Conference Series 331 (2011), no. 3 032023
* [10] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [11] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [12] P. Golonka and Z. Was, PHOTOS Monte Carlo: A Precision tool for QED corrections in Z and W decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [13] A. Hoecker et al., TMVA: Toolkit for multivariate data analysis, PoS ACAT (2007) 040, arXiv:physics/0703039
* [14] LHCb collaboration, R. Aaij et al., Determination of ${f}_{s}/{f}_{d}$ for 7 TeV $pp$ collisions and measurement of the ${B}^{0}\rightarrow{D}^{-}{K}^{+}$ branching fraction, Phys. Rev. Lett. 107 (2011) 211801, arXiv:1106.4435
* [15] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances. PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [16] LHCb collaboration, R. Aaij et al., Measurement of $b$-hadron masses, Phys. Lett. B708 (2012) 241, arXiv:1112.4896
|
arxiv-papers
| 2012-04-05T14:18:53 |
2024-09-04T02:49:29.391091
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato,\n Y. Amhis, J. Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I.\n Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R.\n Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S.\n Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier,\n C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M.\n De Cian, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi,\n S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli,\n C. Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew,\n J. Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y.M. Kim, M. Knecht, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy,\n L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J.\n Luisier, A. Mac Raighne, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O.\n Maev, J. Magnin, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, S.\n Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan,\n B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, B.K. Pal, J. Palacios,\n A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes,\n C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N.\n Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S. Redford, M.M. Reid,\n A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N.A. Smith, E. Smith, K. Sobczak, F.J.P.\n Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci,\n M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, N. Tuning,\n M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams,\n F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Barbara Storaci",
"url": "https://arxiv.org/abs/1204.1237"
}
|
1204.1258
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-094 LHCb-PAPER-2011-045
Measurement of $\psi{(2S)}$ meson production in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration 222Authors are listed on the following pages.
The differential cross-section for the inclusive production of $\psi(2S)$
mesons in $pp$ collisions at $\sqrt{s}$=7$\mathrm{\,Te\kern-1.00006ptV}$ has
been measured with the LHCb detector. The data sample corresponds to an
integrated luminosity of $36$$\mbox{\,pb}^{-1}$. The $\psi(2S)$ mesons are
reconstructed in the decay channels $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and
$\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\pi^{+}\pi^{-}$, with the $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$
meson decaying into two muons. Results are presented both for promptly
produced $\psi(2S)$ mesons and for those originating from $b$-hadron decays.
In the kinematic range $p_{\rm T}(\psi(2S))\leq
16$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2<y(\psi(2S))\leq 4.5$ we
measure
$\displaystyle\sigma_{\rm prompt}(\psi(2S))$ $\displaystyle=$ $\displaystyle
1.44\pm 0.01~{}(\text{stat})\pm
0.12~{}(\text{syst})^{+0.20}_{-0.40}~{}(\text{pol})~{}{\rm\upmu b},$
$\displaystyle\sigma_{b}(\psi(2S))$ $\displaystyle=$ $\displaystyle 0.25\pm
0.01~{}(\text{stat})\pm 0.02~{}(\text{syst})~{}{\rm\upmu b},$
where the last uncertainty on the prompt cross-section is due to the unknown
$\psi(2S)$ polarization. Recent QCD calculations are found to be in good
agreement with our measurements. Combining the present result with the LHCb
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ measurements we determine the
inclusive branching fraction
$\mathcal{B}(b\rightarrow\psi(2S)X)=(2.73\pm 0.06~{}(\text{stat})\pm
0.16~{}(\text{syst})\pm 0.24~{}(\text{BF}))\times 10^{-3},$
where the last uncertainty is due to the $\mathcal{B}(b\rightarrow J/\psi X)$,
$\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})$ and
$\mathcal{B}(\psi(2S)\rightarrow e^{+}e^{-})$ branching fraction
uncertainties.
Submitted to Eur. Phys. J. C
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, F. Constantin26, A.
Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R.
Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De
Capua21,k, M. De Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P.
De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C.
Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H.
Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F.
Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R.
Dzhelyadin32, A. Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S.
Eidelman31, D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L.
Eklund48, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, E. Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38,
S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30,
C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2,
M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44,
V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V.
Niess5, N. Nikitin29, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-
Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R.
Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, K. Pal53,
J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M.
Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M.
Patel50, S.K. Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-
Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M.
Schiller39, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O.
Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A.
Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N.
Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P.
Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31, O. Shevchenko40, V.
Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49,
E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Topp-
Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A.
Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, P. Urquijo53, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D.
Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37, D.
Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Since its discovery, heavy quarkonium has been one of the most important test
laboratories for the development of QCD at the border between the perturbative
and non-perturbative regimes, resulting in the formulation of the
nonrelativistic QCD (NRQCD) factorisation formalism [1, 2]. However, prompt
production studies carried out at the Tevatron collider in the early 1990s [3]
made clear that NRQCD calculations, based on the leading-order (LO) colour-
singlet model (CSM), failed to describe the absolute value and the transverse
momentum ($p_{\mathrm{T}}$) dependence of the charmonium production cross-
section and polarization data. Subsequently, the inclusion of colour-octet
amplitudes in the NRQCD model has reduced the discrepancy between theory and
experiment, albeit at the price of tuning $ad~{}hoc$ some matrix elements [2].
On the other hand, recent computations of the next-to-leading-order (NLO) and
next-to-next-to-leading-order (NNLO) terms in the CSM yielded predictions in
better agreement with experimental data, thus resurrecting interest in the
colour-singlet framework. Other models have been proposed and it is important
to test them in the LHC energy regime [4, 5].
Heavy quarkonium is also produced from $b$-hadron decays. It can be
distinguished from promptly produced quarkonium exploiting its finite decay
time. QCD predictions are based on the Fixed-Order-Next-to-Leading-Log (FONLL)
approximation for the $b\bar{b}$ production cross-section. The FONLL approach
improves NLO results by resumming $p_{\text{T}}$ logarithms up to the next-to-
leading order [6, 7].
To allow a comparison with theory, promptly produced quarkonia should be
separated from those coming from $b$-hadron decays and from those cascading
from higher mass states (feed-down). The latter contribution strongly affects
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ production and complicates the
interpretation of prompt $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ data. On
the other hand, $\psi(2S)$ charmonium has no appreciable feed-down from higher
mass states and therefore the results can be directly compared with the
theoretical predictions, making it an ideal laboratory for QCD studies.
This paper presents a measurement of the $\psi{(2S)}$ meson production cross-
section in $pp$ collisions at the centre-of-mass energy $\sqrt{s}$ =
7$\mathrm{\,Te\kern-1.00006ptV}$. The data were collected by the LHCb
experiment in 2010 and correspond to an integrated luminosity of
35.9$\pm$1.3$\mbox{\,pb}^{-1}$. The analysis is similar to that described in
Ref. [8] for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production
studies; in particular, the separation between promptly produced $\psi(2S)$
and those originating from $b$-hadron decays is based on the reconstructed
decay vertex information. Two decay modes of the $\psi{(2S)}$ meson have been
used: $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ followed by
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\rightarrow\mu^{+}\mu^{-}$. The
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode, despite a
larger background and a lower reconstruction efficiency, is used to cross-
check and average the results, and to extend the accessible phase space. The
production of $\psi{(2S)}$ meson at the LHC has also been studied at the CMS
experiment [9].
## 2 The LHCb detector and data sample
The LHCb detector is a forward spectrometer [10], designed for precision
studies of $C\\!P$ violation and rare decays of $b$\- and $c$-hadrons. Its
tracking acceptance covers approximately the pseudorapidity region $2<\eta<5$.
The detector elements are placed along the beam line of the LHC starting with
the vertex detector, a silicon strip device that surrounds the $pp$
interaction region and is positioned at 8$\rm\,mm$ from the beams during
collisions. It provides precise measurements of the positions of the primary
$pp$ interaction vertices and decay vertices of long-lived hadrons, and
contributes to the measurement of particle momenta. Other detectors used for
momentum measurement include a large area silicon strip detector located
before a dipole magnet of approximately 4 Tm, and a combination of silicon
strip detectors and straw drift chambers placed downstream. Two ring imaging
Cherenkov detectors are used to identify charged hadrons. Further downstream
an electromagnetic calorimeter is used for photon and electron detection,
followed by a hadron calorimeter. The muon detection consists of five muon
stations equipped with multi-wire proportional chambers, with the exception of
the centre of the first station using triple-GEM detectors.
The LHCb trigger system consists of a hardware level, based on information
from the calorimeter and the muon systems and designed to reduce the frequency
of accepted events to a maximum of 1${\rm\,MHz}$, followed by a software level
which applies a full event reconstruction. In the first stage of the software
trigger a partial event reconstruction is performed. The second stage performs
a full event reconstruction to further enhance the signal purity.
The analysis uses events selected by single muon or dimuon triggers. The
hardware trigger requires one muon candidate with a $p_{\mathrm{T}}$ larger
than 1.4${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or two muon candidates with a
$p_{\mathrm{T}}$ larger than 560${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
480${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. In the first stage of the software
trigger, either of the two following selections is required. The first
selection confirms the single muon trigger candidate and applies a harder cut
on the muon $p_{\mathrm{T}}$ at 1.8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
second selection confirms the dimuon trigger candidate by requiring the
opposite charge of the two muons and adds a requirement to the dimuon mass to
be greater than 2.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the second
stage of the software trigger, two selections are used for the
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ mode. The first tightens the requirement
on the dimuon mass to be greater than
2.9${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and it applies to the firtst
8$\mbox{\,pb}^{-1}$ of the data sample. Since this selection was subsequently
prescaled by a factor five, for the largest fraction of the remaining data
(28$\mbox{\,pb}^{-1}$) a different selection is used, which in addition
requires a good quality primary vertex and tracks for the dimuon system. For
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mode only one
selection is used which requires the combined dimuon mass to be in a
$\pm$120${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window around the
nominal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. To avoid that a
few events with high occupancy dominate the software trigger CPU time, a set
of global event cuts is applied on the hit multiplicity of each subdetector
used by the pattern recognition algorithms, effectively rejecting events with
a large number of pile-up interactions.
The simulation samples used for this analysis are based on the Pythia 6.4
generator [11] configured with the parameters detailed in Ref. [12]. The
prompt charmonium production processes activated in Pythia are those from the
leading-order colour-singlet and colour-octet mechanisms. Their implementation
and the parameters used are described in detail in Ref. [13]. The EvtGen
package [14] is used to generate hadron decays and the Geant4 package [15] for
the detector simulation. The QED radiative corrections to the decays are
generated using the Photos package [16].
## 3 Signal yield
The two modes, $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$, have different
decay and background characteristics, therefore dedicated selection criteria
have been adopted. The optimisation of the cuts has been performed using the
simulation. A common requirement is that the tracks, reconstructed in the full
tracking system and passing the trigger requirements, must be of good quality
($\chi^{2}/\text{ndf}<4$, where ndf is the number of degrees of freedom) and
share the same vertex with fit probability $P(\chi^{2})>0.5\%$
($\psi(2S)\rightarrow\mu^{+}\mu^{-}$) and $P(\chi^{2})>5\%$
($\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\pi^{+}\pi^{-}$). A cut $p_{\mathrm{T}}>$
1.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is applied for the muons from the
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ decay. For muons from
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ we
require a momentum larger than 8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$p_{\mathrm{T}}>$ 0.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Finally the
rapidity of the reconstructed $\psi{(2S)}$ is required to satisfy the
requirement $2<y\leq 4.5$.
The $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ invariant mass spectrum for all
selected candidates is shown in Fig. 1(a). The fitting function is a Crystal
Ball [17] describing the signal plus an exponential function for the
background. In total 90600$\pm$690 signal candidates are found in the
$p_{\mathrm{T}}$ range 0–12${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The mass
resolution is 16.01$\pm$0.12${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the
Crystal Ball parameters that account for the radiative tail are obtained from
the simulation.
For the $\psi{(2S)}\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ decay, both pions are required to have
$p_{\mathrm{T}}>$ 0.3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the sum of the
two-pion transverse momenta is required to be larger than
0.8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The quantity
$Q=M(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\pi^{+}\pi^{-})-M(\pi^{+}\pi^{-})-M(\mu^{+}\mu^{-})$ is required to be
$\leq$ 200${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and to improve the mass
resolution the dimuon invariant mass $M_{\mu^{+}\mu^{-}}$ is constrained in
the fit to the nominal $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ mass value
[18]. Finally, both $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ and
$\psi{(2S)}$ candidates must have $p_{\mathrm{T}}>$
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The invariant mass spectrum is shown
in Fig. 1(b) for all selected candidates. For this decay mode the peak is
described by the sum of two Crystal Ball functions for the signal plus an
exponential function for the background. The number of signal candidates is
12300$\pm$200, the mass resolution is
2.10$\pm$0.07${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the Crystal Ball
tail parameters are fixed to the values obtained from the simulation.
The fits are repeated in each $\psi{(2S)}$ $p_{\mathrm{T}}$ bin to obtain the
number of signal and background candidates for both decays.
Figure 1: Invariant mass distribution for all $\psi(2S)$ candidates passing
the selection cuts for the $\mu^{+}\mu^{-}$ decay (a) and the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$
decay (b).
## 4 Cross-section measurement
The differential cross-section for the inclusive $\psi{(2S)}$ meson production
is computed from
$\frac{d\sigma}{dp_{\mathrm{T}}}(p_{\mathrm{T}})=\frac{N_{\text{sig}}(p_{\mathrm{T}})}{\mathcal{L}~{}\epsilon_{\rm
tot}(p_{\mathrm{T}})~{}{\cal B}~{}\Delta p_{\mathrm{T}}}$ (1)
where $d\sigma/dp_{\mathrm{T}}$ is the average cross-section in the given
$p_{\mathrm{T}}$ bin, integrated over the rapidity range $2<y\leq 4.5$,
$N_{\text{sig}}(p_{\mathrm{T}})$ is the number of signal candidates determined
from the mass fit for the decay under study, $\epsilon_{\rm
tot}(p_{\mathrm{T}})$ is the total detection efficiency including acceptance
and trigger effects, ${\cal B}$ denotes the relevant branching fraction and
$\Delta p_{\mathrm{T}}$ is the bin size. All branching fractions are taken
from Ref. [18]: $\mathcal{B}(\psi(2S)\rightarrow e^{+}e^{-})$ = $(7.72\pm
0.17)\times 10^{-3}$, $\mathcal{B}(\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-})$ = $(33.6\pm
0.4)\times 10^{-2}$ and $\mathcal{B}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\rightarrow\mu^{+}\mu^{-})$ = $(5.93\pm 0.06)\times 10^{-2}$. Assuming
lepton universality, we use the dielectron branching fraction
$\mathcal{B}(\psi(2S)\rightarrow e^{+}e^{-})$ in Eq. (1), since
$\mathcal{B}(\psi(2S)\rightarrow\mu^{+}\mu^{-})$ is less precisely known.
$\mathcal{L}$ is the integrated luminosity, which is calibrated using both Van
der Meer scans [19, 20] and a beam-profile method [21]. A detailed description
of the two methods is given in Ref. [22]. The knowledge of the absolute
luminosity scale is used to calibrate the number of tracks in the vertex
detector, which is found to be stable throughout the data taking period and
can therefore be used to monitor the instantaneous luminosity of the entire
data sample. The integrated luminosity of the data sample used in this
analysis is determined to be 35.9$\mbox{\,pb}^{-1}$.
The total efficiency, $\epsilon_{\rm tot}(p_{\mathrm{T}})$, is a product of
three contributions: the geometrical acceptance, the combined detection,
reconstruction and selection efficiency, and the trigger efficiency. Each
contribution has been determined using simulated events for the two decay
channels. In order to evaluate the trigger efficiency, the trigger selection
algorithms used during data taking are applied to the simulation.
Figure 2: Total efficiency vs. $p_{\mathrm{T}}$ computed from simulation for
unpolarized $\psi{(2S)}$ mesons for $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ (a)
and $\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ (b).
The total efficiency vs. $p_{\mathrm{T}}$ for the two channels, assuming the
$\psi(2S)$ meson unpolarized, is shown in Fig. 2. Extensive studies on dimuon
decays of prompt $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ [8], $\psi(2S)$
and $\Upsilon$ [23] mesons have shown that the total efficiency in the LHCb
detector depends strongly on the initial polarization state of the vector
meson. This effect is absent for $\psi{(2S)}$ mesons coming from $b$-hadron
decays. In fact for these events the natural polarization axis is the
$\psi{(2S)}$ meson flight direction in the $b$-hadron rest frame, while the
$\psi{(2S)}$ meson appears unpolarized along its flight direction in the
laboratory. Simulations [8] and measurements from CDF [24] confirm this. We do
not measure the $\psi{(2S)}$ meson polarization but we assign a systematic
uncertainty to the unpolarized efficiencies in the case of prompt production.
Events are generated with polarizations corresponding to the two extreme cases
of fully transverse or fully longitudinal polarization and the efficiency is
re-evaluated. The difference between these results and those with the
unpolarized sample is taken as an estimate of the systematic uncertainty.
A similar effect exists for the $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$
meson emitted in the $\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ decay. However, in this case, the
$\psi(2S)$ meson polarization is fully transferred to the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ meson since, as measured by the
BES collaboration [25], the two pions are predominantly in the $S$-wave
configuration111The small fraction of $D$-wave measured in Ref. [25] has a
negligible impact on our conclusion. and the
dipion-$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ system is also in a
$S$-wave configuration. This has been verified with data and is correctly
reproduced by the simulation. Therefore the systematics due to polarization
are fully correlated between the two channels and we use the systematic
uncertainties computed for $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ also for the
$\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$
decay.
In order to separate prompt $\psi(2S)$ mesons from those produced in
$b$-hadron decays, we use the pseudo-decay-time variable defined as
$t=d_{z}(M/p_{z})$, where $d_{z}$ is the separation along the beam axis
between the $\psi(2S)$ decay vertex and the primary vertex, $M$ is the nominal
mass of the $\psi(2S)$ and $p_{z}$ is the component of its momentum along the
beam axis. In case of multiple primary vertices reconstructed in the same
event, that which minimises $|d_{z}|$ has been chosen. The prompt component is
distributed as a Gaussian function around $t=0$, with width corresponding to
the experimental resolution, while for the $\psi(2S)$ from $b$-hadron decays
the $t$ variable is distributed according to an approximately exponential
decay law, smeared in the fit with the experimental resolution. The choice of
taking the primary vertex which minimises $|d_{z}|$ could in principle
introduce a background component in the pseudo-decay-time distribution arising
from the association of the $\psi(2S)$ vertex to a wrong primary vertex. The
effect of such background is found to be of the order of 0.5% in the region
around $t=0$ and has been neglected. The function used to fit the $t$
distribution in each $p_{\mathrm{T}}$ bin is
$F(t;f_{\text{p}},\sigma,\tau_{b})=N_{\text{sig}}\left[f_{\text{p}}\delta(t)+(1-f_{\text{p}})\theta(t)\frac{e^{-\frac{t}{\tau_{b}}}}{\tau_{b}}\right]\otimes\frac{e^{-\frac{1}{2}(\frac{t}{\sigma})^{2}}}{\sqrt{2\pi}\sigma}+N_{\text{bkg}}f_{\text{bkg}}(t;\boldsymbol{\Theta})$
(2)
where $N_{\text{sig}}$ and $N_{\text{bkg}}$ are respectively the numbers of
signal and background candidates obtained from the mass fit. The fit
parameters are the prompt fraction, $f_{\text{p}}$, the standard deviation of
the Gaussian resolution function, $\sigma$, and the lifetime describing the
long-lived component of $\psi(2S)$ mesons coming from $b$-hadron decays,
$\tau_{b}$. In principle, all fit parameters are dependent on
$p_{\mathrm{T}}$. The function $f_{\text{bkg}}(t;\boldsymbol{\Theta})$ models
the background component in the distribution and is defined as the sum of a
$\delta$ function and a Gaussian function for the prompt background, plus two
exponential functions for the positive tail and one exponential function for
the negative tail, all convolved with a Gaussian function to account for the
detector resolution. The array of parameters $\boldsymbol{\Theta}$ is
determined from a fit to the $t$ distribution of the events in the mass
sidebands.
Figure 3: Pseudo-decay-time distribution for
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ (a) and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$(b) in the
$p_{\mathrm{T}}$ range $4<p_{\mathrm{T}}\leq 5$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, showing the background and prompt
contributions.
As an example, the pseudo-decay-time distributions for
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ in the
$p_{\mathrm{T}}$ range $4<p_{\mathrm{T}}\leq
5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are presented in Fig. 3. The
contributions of background and prompt $\psi{(2S)}$ mesons are also shown. The
values of the prompt fraction, $f_{\text{p}}$ vs. $p_{\mathrm{T}}$ in the
rapidity range $2<y\leq 4.5$, obtained for the $\mu^{+}\mu^{-}$ and the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ modes, are in good
agreement as shown in Fig. 4.
Figure 4: Fraction of prompt $\psi(2S)$ as a function of $p_{\mathrm{T}}$ for
the $\mu^{+}\mu^{-}$ mode (solid squares) and the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode (open
squares). Error bars include the statistical uncertainties and the systematic
uncertainties due to the fitting procedure.
## 5 Systematic uncertainties on the cross-section measurement
A variety of sources of systematic uncertainties affecting the cross-section
measurement were taken into account and are summarised in Table 1.
Table 1: Systematic uncertainties included in the measurement of the cross-section. Uncertainties labelled with $a$ are correlated between the $\mu^{+}\mu^{-}$ and $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode, while $b$ indicates a correlation between $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and the $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\rightarrow\mu^{+}\mu^{-}$ uncertainties [8]. Uncertainty source | $\mu^{+}\mu^{-}$(%) | $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$(%)
---|---|---
Luminositya,b | 3.5 | 3.5
Size of simulation sample | 0.4–2.2 | 0.6–1.0
Trigger efficiencya | 1–8 | 1–7
Global event cutsa,b | 2.1 | 2.1
Muon identificationa,b | 1.1 | 1.1
Hadron identification | – | 0.5
Track $\chi^{2}$a,b | 1 | 2
Tracking efficiencya | 3.5 | 7.3
Vertex fitb | 0.8 | 1.3
Unknown polarizationa | 15–26 | 15–26
Mass fit function | 1.1 | 0.5
Pseudo-decay-time fits | 2.7 | 2.7
$\mathcal{B}(\psi(2S)\rightarrow e^{+}e^{-})$ | 2.2 | –
$\mathcal{B}(\psi(2S)\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-})$ | – | 1.2
$\mathcal{B}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\rightarrow\mu^{+}\mu^{-})$ | – | 1.0
A thorough analysis of the luminosity scans yields consistent results for the
absolute luminosity scale with a precision of 3.5% [22], this value being
assigned as a systematic uncertainty. The statistical uncertainties from the
finite number of simulated events on the efficiencies are included as a source
of systematic uncertainty; this uncertainty varies from 0.4 to 2.2% for the
$\mu^{+}\mu^{-}$ mode and from 0.6 to 1% for the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode. In addition,
we assign a systematic uncertainty in order to account for the difference
between the trigger efficiency evaluated on data by means of an unbiased
$\mu^{+}\mu^{-}$ sample, and the trigger efficiency computed from the
simulation. This results in a bin-dependent uncertainty up to 8% for the
$\mu^{+}\mu^{-}$ mode and up to 7% for the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode. This
uncertainty is fully correlated between the two decay modes in the overlapping
$p_{\mathrm{T}}$ region. Finally, the statistical uncertainty on the global
event cuts efficiency (2.1% for both modes) is taken as an additional
systematic uncertainty [8].
To assess possible systematic differences in the acceptance between data and
simulation for the $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$
mode, we have studied the dipion mass distribution. The LHCb simulation is
based on the Voloshin-Zakharov model [26] which uses a single phenomenological
parameter $\lambda$
$\frac{d\sigma}{dm_{\pi\pi}}\propto\Phi(m_{\pi\pi})\left[m_{\pi\pi}^{2}-\lambda
m_{\pi}^{2}\right]^{2},$ (3)
where $\Phi(m_{\pi\pi})$ is a phase space factor (see e.g. Ref. [25]) and in
the simulation $\lambda=4$ is assumed. The dipion mass distribution obtained
from the data is shown in Fig. 5. We obtain $\lambda=4.46\pm 0.07(\rm stat)\pm
0.18(\rm syst)$, from which we estimate a negligible systematic effect on the
acceptance (0.25%). Our result is also in good agreement with the BES value
$\lambda=4.36\pm 0.06(\rm stat)\pm 0.17(\rm syst)$ [25].
Figure 5: Dipion mass spectrum for the $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ decay. The curve
shows the result of the fit with Eq. (3) corrected for the acceptance.
To cross-check and assign a systematic uncertainty to the determination of the
muon identification efficiency from simulation, the single track muon
identification efficiency has been measured on data using a tag-and-probe
method [27]. This gives a correction factor for the dimuon of 1.025$\pm$0.011,
which we apply to the simulation efficiencies. The 1.1% uncertainty on the
correction factor is used as systematic uncertainty. The efficiency of the
selection requirement on the dipion identification has been studied on data
and simulation and a difference of 1% has been measured between the two.
Therefore, the simulation efficiencies are corrected for this difference and
an additional systematic uncertainty of 0.5% is included.
The $\psi{(2S)}$ selection also includes a requirement on the track fit
quality. The relative difference between the efficiency of this requirement in
simulation and data is taken as a systematic uncertainty, resulting in an
uncertainty of 0.5% per track. Tracking studies show that the ratio of the
track-finding efficiencies between data and simulation is 1.09 for the
$\mu^{+}\mu^{-}$ mode and 1.06 for the $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\pi^{+}\pi^{-}$ mode, with an uncertainty of 3.5% and 7.3% respectively;
the simulation efficiencies are corrected accordingly and the corresponding
systematic uncertainties are included.
For the requirement on the secondary vertex fit quality, a relative difference
of 1.6% for the $\mu^{+}\mu^{-}$ mode and 2.6% for the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode has been
measured between data and simulation. The simulation efficiency is therefore
corrected for this difference and a corresponding systematic uncertainty of
0.8% ($\mu^{+}\mu^{-}$) and 1.3% ($J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\pi^{+}\pi^{-}$) is assigned.
The systematic uncertainty due to the unknown polarization is computed as
discussed in Section 4. The study done for the two extreme polarization
hypotheses gives an average systematic uncertainty between 15% and 26% for
both modes, relative to the hypothesis of zero polarization, depending on the
$p_{\mathrm{T}}$ bin. These errors are fully correlated between the two decay
modes and strongly asymmetric since the variations of the efficiency are of
different magnitude for transverse and longitudinal polarizations.
A systematic uncertainty from the fitting procedure has been estimated from
the relative difference between the overall number of signal $\psi{(2S)}$ and
the number of signal candidates obtained by summing the results of the fits in
the individual $p_{\mathrm{T}}$ bins. A total systematic uncertainty of 1.1%
for the $\mu^{+}\mu^{-}$ mode and 0.5% for the
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ mode is assigned.
Finally, to evaluate the systematic uncertainty on the prompt fraction from
the $\psi{(2S)}$ pseudo-decay-time fit we recompute $f_{\text{p}}$ with
$\tau_{b}$ (see Eq. (2)) fixed to the largest and smallest value obtained in
the $p_{\mathrm{T}}$-bin fits. The relative variation is at most 2.7% and this
value is assigned as a systematic uncertainty on $f_{\text{p}}$.
## 6 Cross-section results
The differential cross-sections for prompt $\psi(2S)$ and $\psi(2S)$ mesons
from $b$-hadron decays are shown in Fig. 6, where we compare the results
obtained for the $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ channels separately
for the prompt and $b$-hadron decay components.
Figure 6: Comparison of the differential cross-sections measured for prompt
$\psi(2S)$ (circles) and for $\psi(2S)$ from $b$-hadron decay (squares) in the
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$ (solid symbols) and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ (open symbols)
modes. Only the uncorrelated uncertainties are shown.
The values for the two cross-sections estimated using the different decay
modes are consistent within $0.5~{}\sigma$. A weighted average of the two
measurements is performed to extract the final result listed in Table 2.
Table 2: Cross-section values for prompt $\psi(2S)$ and $\psi(2S)$ from $b$-hadrons in different $p_{\mathrm{T}}$ bins and in the range $2<y\leq 4.5$, evaluated as the weighted average of the $\mu^{+}\mu^{-}$ and $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$ channels. The first error is statistical, the second error is systematic, and the last asymmetric uncertainty is due to the unknown polarization of the prompt $\psi(2S)$ meson. $p_{\mathrm{T}}$ [GeV/$c$] | $\frac{d\sigma_{\rm prompt}}{dp_{\mathrm{T}}}$ [$\frac{\rm nb}{{\rm GeV}/c}$] | $\frac{d\sigma_{\rm b}}{dp_{\mathrm{T}}}$ [$\frac{\rm nb}{{\rm GeV}/c}$]
---|---|---
0–1 | 188 $\pm$ 6 $\pm$ 18${}^{+32}_{-67}$ | 22 $\pm$ 2 $\pm$ 2
1–2 | 387 $\pm$ 8 $\pm$ 37${}^{+60}_{-119}$ | 62 $\pm$ 3 $\pm$ 6
2–3 | 317 $\pm$ 7 $\pm$ 26${}^{+44}_{-88}$ | 53 $\pm$ 2 $\pm$ 4
3–4 | 224 $\pm$ 6 $\pm$ 24${}^{+27}_{-53}$ | 39 $\pm$ 2 $\pm$ 4
4–5 | 135 $\pm$ 4 $\pm$ 13${}^{+16}_{-30}$ | 29 $\pm$ 1 $\pm$ 3
5–6 | 77 $\pm$ 2 $\pm$ 7${}^{+9}_{-18}$ | 18 $\pm$ 1 $\pm$ 2
6–7 | 46 $\pm$ 1 $\pm$ 4${}^{+5}_{-10}$ | 10 $\pm$ 1 $\pm$ 1
7–8 | 25 $\pm$ 1 $\pm$ 2${}^{+3}_{-6}$ | 6.3 $\pm$ 0.4 $\pm$ 0.5
8–9 | 14 $\pm$ 1 $\pm$ 1${}^{+2}_{-3}$ | 3.9 $\pm$ 0.3 $\pm$ 0.3
9–10 | 8.3 $\pm$ 0.4 $\pm$ 0.7${}^{+0.9}_{-1.7}$ | 2.5 $\pm$ 0.2 $\pm$ 0.2
10–12 | 4.3 $\pm$ 0.3 $\pm$ 0.4${}^{+0.5}_{-0.9}$ | 1.4 $\pm$ 0.1 $\pm$ 0.1
12–16 | 1.5 $\pm$ 0.1 $\pm$ 0.2${}^{+0.2}_{-0.3}$ | 0.51 $\pm$ 0.04 $\pm$ 0.06
The differential cross-section for promptly produced $\psi(2S)$ mesons, along
with a comparison with some recent theory predictions [28, 29, 30, 31] tuned
to the LHCb acceptance, is shown in Fig. 7. In Ref. [28] and Ref. [29] the
differential prompt cross-section has been computed up to NLO terms in
nonrelativistic QCD, including colour-singlet and colour-octet contributions.
In Ref. [30, 31] the prompt cross-section has been evaluated in a colour-
singlet framework, including up to the dominant $\alpha_{s}^{5}$ NNLO terms.
Experimentally the large-$p_{\mathrm{T}}$ tail behaves like
$p_{\mathrm{T}}^{-\beta}$ with $\beta=4.2\pm 0.6$ and is rather well
reproduced, especially in the colour-octet models.
The differential cross-section for $\psi{(2S)}$ produced in $b$-hadron decays
and the comparison with a recent theory prediction [32] based on the FONLL
approach [6, 7] are presented in Fig. 8. The theoretical prediction of Ref.
[32] uses as input the $b\rightarrow\psi(2S)X$ branching fraction obtained in
the following section. Experimentally the $\psi{(2S)}$ mesons resulting from
$b$-hadron decay have a slightly harder $p_{\mathrm{T}}$ spectrum than those
produced promptly: $\beta=3.6\pm 0.5$. By integrating the differential cross-
section for prompt $\psi(2S)$ and $\psi(2S)$ from $b$-hadrons in the range
$2<y\leq 4.5$ and
$p_{\mathrm{T}}\leq$16${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, we obtain
$\displaystyle\sigma_{\rm prompt}(\psi(2S))$ $\displaystyle=1.44\pm
0.01~{}(\text{stat})\pm
0.12~{}(\text{syst})^{+0.20}_{-0.40}~{}(\text{pol})~{}{\rm\upmu b},$
$\displaystyle\sigma_{b}(\psi(2S))$ $\displaystyle=0.25\pm
0.01~{}(\text{stat})\pm 0.02~{}(\text{syst})~{}{\rm\upmu b},$
where the systematic uncertainty includes all the sources listed in Table 1,
except for the polarization, while the last asymmetric uncertainty is due to
the effect of the unknown $\psi(2S)$ polarization and applies only to the
prompt component.
Figure 7: Differential production cross-section vs. $p_{\mathrm{T}}$ for
prompt $\psi(2S)$. The predictions of three nonrelativistic QCD models are
also shown for comparison. MWC [28] and KB [29] are NLO calculations including
colour-singlet and colour-octet contributions. AL [30, 31] is a colour-singlet
model including the dominant NNLO terms. Figure 8: Differential production
cross-section vs. $p_{\mathrm{T}}$ for $\psi(2S)$ from $b$-hadrons. The shaded
band is the prediction of a FONLL calculation [6, 7, 32].
## 7 Inclusive $\boldsymbol{b\rightarrow\psi(2S)X}$ branching fraction
measurement
The inclusive branching fraction for a $b$-hadron decaying to $\psi{(2S)}$ is
presently known with 50% precision: $\mathcal{B}(b\rightarrow\psi(2S)X)$ =
(4.8 $\pm$ 2.4) $\times 10^{-3}$ [18]. Combining the present result for
$\sigma_{b}(\psi(2S))$ with the previous measurement of
$\sigma_{b}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu)$ [8] we can obtain an
improved value of the aforementioned branching fraction. To achieve this, it
is necessary to extrapolate the two measurements to the full phase space. The
extrapolation factors for the two decays have been determined using the LHCb
simulation [12] and they have been found to be $\alpha_{4\pi}(J/\psi)$=5.88
[8] and $\alpha_{4\pi}(\psi(2S))$=5.48. Most of the theoretical uncertainties
are expected to cancel in the ratio of the two factors
$\xi=\alpha_{4\pi}(\psi(2S))/\alpha_{4\pi}(J/\psi)=0.932$, which is used in
Eq. (4). A systematic uncertainty of $3.4\%$ is estimated for this correction
and included in the final result below. Therefore
$\frac{\mathcal{B}(b\rightarrow\psi(2S)X)}{\mathcal{B}(b\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0muX)}=\xi\frac{\sigma_{b}(\psi(2S))}{\sigma_{b}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu)}.$ (4)
For $\sigma_{b}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu)$ we rescale the
value in [8] for the new determination of the integrated luminosity
($\mathcal{L}$ = 5.49 $\pm$ 0.19$\mbox{\,pb}^{-1}$). For
$\sigma_{b}(\psi(2S))$ we use only the data from the
$\psi(2S)~{}\rightarrow~{}\mu^{+}\mu^{-}$ mode to cancel most of the
systematic uncertainties in the ratio. Effects due to polarization are
negligible for mesons resulting from $b$-hadron decay. We obtain
$\frac{\mathcal{B}(b\rightarrow\psi(2S)X)}{\mathcal{B}(b\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0muX)}=0.235\pm
0.005~{}(\text{stat})\pm 0.015~{}(\text{syst}),$
where the correlated uncertainties (Table 1) between the two cross-sections
are excluded. By inserting the value
$\mathcal{B}(b~{}\rightarrow~{}J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0muX)=(1.16\pm 0.10)\times 10^{-2}$ [18] we get
$\mathcal{B}(b\rightarrow\psi(2S)X)=(2.73\pm 0.06~{}(\text{stat})\pm
0.16~{}(\text{syst})\pm 0.24~{}(\text{BF}))\times 10^{-3},$
where the last uncertainty originates from the uncertainty of the branching
fractions $\mathcal{B}(b\rightarrow J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0muX)$, $\mathcal{B}(\psi(2S)\rightarrow e^{+}e^{-})$ and
$\mathcal{B}(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu\rightarrow\mu^{+}\mu^{-})$.
The ratio of the $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ to
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\rightarrow\mu^{+}\mu^{-}$
differential cross-sections is shown vs. $p_{\mathrm{T}}$ in Fig. 9 for prompt
production ($R_{\rm p}$, Fig. 9(a)) and when the vector mesons originate from
$b$-hadron decays ($R_{b}$, Fig. 9(b)). Since it is not known if the promptly
produced $\psi{(2S)}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ have
similar polarizations [33], we do not assume any correlation of the
polarization uncertainties when computing the uncertainties on $R_{\rm p}$.
The increase of $R_{{\rm p}(b)}$ with $p_{\mathrm{T}}$ is similar to that
measured in the central rapidity region by the CDF [24] and CMS [9]
collaborations.
Figure 9: Ratio of $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ to
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\rightarrow\mu^{+}\mu^{-}$ cross-
sections for prompt
production (a) and for $b$-hadron decay (b), as a function of
$p_{\mathrm{T}}$.
## 8 Conclusions
We have measured the differential cross-section for the process
$pp\rightarrow\psi(2S)X$ at the centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$, as a function of the transverse momentum in
the range $p_{\mathrm{T}}(\psi(2S))\leq
16$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2<y(\psi(2S))\leq 4.5$, via the
decay modes $\psi(2S)\rightarrow\mu^{+}\mu^{-}$ and $\psi(2S)\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu\pi^{+}\pi^{-}$. The data sample
corresponds to about 36$\mbox{\,pb}^{-1}$ collected by the LHCb experiment at
the LHC. Results from the two decay modes agree. The $\psi(2S)$ prompt cross-
section has been separated from the cross-section of $\psi(2S)$ from
$b$-hadrons through the study of the pseudo-decay-time and the two
measurements have been averaged. In the above kinematic range we measure
$\displaystyle\sigma_{\rm prompt}(\psi(2S))$ $\displaystyle=$ $\displaystyle
1.44\pm 0.01~{}(\text{stat})\pm
0.12~{}(\text{syst})^{+0.20}_{-0.40}~{}(\text{pol})~{}{\rm\upmu b},$
$\displaystyle\sigma_{b}(\psi(2S))$ $\displaystyle=$ $\displaystyle 0.25\pm
0.01~{}(\text{stat})\pm 0.02~{}(\text{syst})~{}{\rm\upmu b}.$
The measured $\psi(2S)$ production cross-sections are in good agreement with
the results of several recent NRQCD calculations. In addition, we obtain an
improved value for the $b\rightarrow\psi(2S)X$ branching fraction by combining
the two LHCb production cross-section measurements of the two vector mesons
$J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$ and $\psi(2S)$ from $b$-hadrons.
The result,
$\mathcal{B}(b\rightarrow\psi(2S)X)=(2.73\pm 0.06~{}(\text{stat})\pm
0.16~{}(\text{syst})\pm 0.24~{}(\text{BF}))\times 10^{-3},$
is in good agreement with recent results from the CMS collaboration [9] and is
a significant improvement over the present PDG average [18].
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
We thank B. Kniehl, M. Butenschön and M. Cacciari for providing theoretical
predictions of $\psi{(2S)}$ cross-sections in the LHCb acceptance range.
## References
* [1] W. E. Caswell and G. P. Lepage, Effective lagrangians for bound state problems in QED, QCD, and other field theories, Phys. Lett. B167 (1986) 437.
* [2] G. T. Bodwin, E. Braaten, and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51 (1995) 1125. [Erratum-ibid. D 55 (1997) 5853], arXiv:hep-ph/9407339.
* [3] CDF collaboration, F. Abe et al., Inclusive $J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu$, $\psi{(2S)}$, and b-quark production in $\bar{p}p$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 69 (1992) 3704.
* [4] N. Brambilla et al., Heavy quarkonium physics, (2005) arXiv:hep-ph/0412158v2.
* [5] N. Brambilla et al., Heavy quarkonium: progress, puzzles and opportunities, (2010) arXiv:hep-ph/1010.5827v1.
* [6] M. Cacciari, M. Greco, P. Nason, The $p_{\text{T}}$ spectrum in heavy-flavour hadroproduction, JHEP 9805 (1998) 007.
* [7] M. Cacciari, S. Frixione, M. Mangano, P. Nason and G. Ridolfi, QCD analysis of first b cross section data at 1.96 TeV, JHEP 0407 (2004) 033.
* [8] LHCb collaboration, R. Aaij et al., Measurement of $J/\psi$ production in $pp$ collisions at $\sqrt{s}$=7 TeV, Eur. Phys. J. C71 (2011) 1645.
* [9] CMS collaboration, S. Chatrchyan et al., $J/\psi$ and $\psi(2S)$ production in pp collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, JHEP 02 (2011) 11.
* [10] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005.
* [11] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155.
* [12] M. Clemencic et al., The LHCb simulation application, Gauss: design, evolution and experience, Journal of Physics: Conference Series 331 (2011), no. 3 032023.
* [13] M. Bargiotti and V. Vagnoni, Heavy quarkonia sector in PYTHIA 6.324 : tuning, validation and perspectives at LHCb, CERN-LHCb-2007-042.
* [14] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152.
* [15] GEANT4 collaboration, S. Agostinelli et al., GEANT4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250.
* [16] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in Z and W decays, Eur. Phys. J. C 45 (2006) 97.
* [17] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances. PhD thesis, Institute of Nuclear Physics, Krakow, 1986. DESY-F31-86-02.
* [18] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 Journal of Physics G37 (2010) 075201.
* [19] S. Van der Meer, Calibration of the effective beam height in the ISR, CERN report, ISR-PO/68-31, (1968).
* [20] H. Burkhardt and P. Grafström, Absolute luminosity from machine parameters, CERN-LHC-PROJECT-Report-1019 (2007).
* [21] M. Ferro-Luzzi, Proposal for an absolute luminosity determination in colliding beam experiments using vertex detection of beam-gas interactions, Nucl. Instrum. Meth. A553 (2005) 388-399.
* [22] LHCb collaboration, R. Aaij et al., Absolute luminosity measurements with the LHCb detector at the LHC, JINST 7 (2012) P01010.
* [23] LHCb collaboration, R. Aaij et al., Measurement of $\Upsilon$ production in pp collisions at $\sqrt{s}$ = 7 TeV, arXiv:hep-ex/1202.6579v1.
* [24] CDF collaboration, T. Aaltonen et al., Production of $\psi{(2S)}$ mesons in $p\bar{p}$ collisions at 1.96 TeV, Phys. Rev. D80 (2009) 031103.
* [25] BES collaboration, J. Bai et al., $\pi^{+}\pi^{-}J/\psi$ decay distributions, Phys. Rev. D62 (2000) 032002.
* [26] M. Voloshin and V. Zakharov, Measuring quantum-chromodynamic anomalies in hadronic transitions between quarkonium states, Phys. Rev. Lett. 45, (1980) 688-691.
* [27] A. Sarti et al., Calibration strategy and efficiency measurement of the muon identification procedure at LHCb, LHCb-PUB-2010-002.
* [28] Y.-Q. Ma, K. Wang and K.-T. Chao, A complete NLO calculation of the J/$\psi$ and $\psi^{\prime}$ production at hadron colliders, arXiv:hep-ph/1012.1030.
* [29] B. Kniehl and M. Butenschön, Reconciling J/$\psi$ production at HERA, RHIC, Tevatron, and LHC with nonrelativistic QCD factorization at next-to-leading order, Phys. Rev. Lett. 106 (2011) 022003, and private communication.
* [30] P. Artoisenet et al., $\Upsilon$ production at Fermilab Tevatron and LHC energies, Phys. Rev. Lett. 101 (2008) 152001.
* [31] J.-P. Lansberg, On the mechanisms of heavy-quarkonium hadroproduction, Eur. Phys. J. C61 (2009) 693.
* [32] M. Cacciari, private communication.
* [33] CDF collaboration, A. Abulencia et al., Polarizations of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ mesons produced in $p\bar{p}$ collisions at $\sqrt{s}=$1.96 TeV, Phys. Rev. Lett. 99 (2007) 132001.
|
arxiv-papers
| 2012-04-05T15:31:53 |
2024-09-04T02:49:29.399332
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abell\\'an Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y.\n Amhis, J. Anderson, F. Andrianala, R.B. Appleby, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S.\n Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I.\n Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R.\n Bernet, M.O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, J. Buytaert, A.\n B\\\"uchler-Germann, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, N. Chiapolini, M. Chrzaszcz, P. Ciambrone, K. Ciba, X. Cid\n Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C.\n Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F. Constantin, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G.A. Cowan, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, O. De Aguiar Francisco, K. De Bruyn, M.\n De Cian, F. De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F.\n Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, E.\n Fanchini, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, C.\n F\\\"arber, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao,\n J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar,\n R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V.\n Gligorov, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, C. Gotti, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T.\n Gys, C. G\\\"obel, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, W. Hulsbergen, P.\n Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten,\n J. Imong, A. Inyakin, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F.\n Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B.\n Jost, S. Kandybei, M. Karacson, T.M. Karbach, J. Keaveney, I.R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, R.F. Koopman, P.\n Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, A. Leflat, J. Lefran\\c{c}ois, R. Lef\\`evre, O.\n Leroy, T. Lesiak, L. Li, P.-R. Li, L. Li Gioi, M. Lieng, M. Liles, R.\n Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H. Lopes, E. Lopez Asamar,\n N. Lopez-March, H. Lu, J. Luisier, X. Lyu, A. Mac Raighne, F. Machefert, F.\n Maciuc, O. Maev, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, J.F. Marchand, U. Marconi, J. Marks, G. Martellotti, A. Martens,\n L. Martin, D. Martinez Santos, A. Mart\\'in S\\'anchez, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, R. Messi, S.\n Miglioranzi, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, R. Muresan, B. Muster,\n M. Musy, J. Mylroie-Smith, R. M\\\"arki, K. M\\\"uller, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, A.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A.\n Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, R.\n Oldeman, J.M. Otalora Goicochea, P. Owen, B.K. Pal, J. Palacios, A. Palano,\n M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N. Patrick,\n C. Patrignani, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, P. Perret, M. Perrin-Terrin, A. Petrella, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, I. Polyakov, E.\n Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V.\n Pugatch, A. Puig Navarro, A. P\\'erez-Calero Yzquierdo, W. Qian, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D.A. Roa Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, G.J. Rogers, S.\n Roiser, V. Romanovskiy, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Saur, D. Savrina, P. Schaack, M.\n Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, Y. Shcheglov, T. Shears, L. Shekhtman, V. Shevchenko, A. Shires, R.\n Silva Coutinho, T. Skwarnicki, E. Smith, K. Sobczak, F.J.P. Soler, A.\n Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, O. Stenyakin, S. Stoica, S. Stone, B.\n Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, B. Viaud, I.\n Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D.\n Volyanskyy, D. Voong, A. Vorobyev, S. Wandernoth, J. Wang, D.R. Ward, N.K.\n Watson, A.D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wishahi, M. Witek, W.\n Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Miroslav Saur",
"url": "https://arxiv.org/abs/1204.1258"
}
|
1204.1462
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-068 LHCb-PAPER-2011-030
Measurement of the ratio of prompt $\chi_{c}$ to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in $pp$ collisions
at $\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-2.07413ptV}}$}$
The LHCb collaboration111Authors are listed on the following pages.
The prompt production of charmonium $\chi_{c}$ and $J/\psi$ states is studied
in proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV at
the Large Hadron Collider. The $\chi_{c}$ and $J/\psi$ mesons are identified
through their decays $\chi_{c}\rightarrow J/\psi\gamma$ and
$J/\psi\rightarrow\mu^{+}\mu^{-}$ using 36 pb-1 of data collected by the LHCb
detector in 2010. The ratio of the prompt production cross-sections for
$\chi_{c}$ and $J/\psi$, $\sigma(\chi_{c}\rightarrow
J/\psi\gamma)/\sigma(J/\psi)$, is determined as a function of the $J/\psi$
transverse momentum in the range $2<p_{\mathrm{T}}^{J/\psi}<15$ GeV/$c$. The
results are in excellent agreement with next-to-leading order non-relativistic
expectations and show a significant discrepancy compared with the colour
singlet model prediction at leading order, especially in the low
$p_{\mathrm{T}}^{J/\psi}$ region.
Submitted to Phys. Lett. B
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, L. Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6,
G. Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, K. de Bruyn38, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, K. Ciba35, X. Cid Vidal34, G. Ciezarek50,
P.E.L. Clarke47,35, M. Clemencic35, H.V. Cliff44, J. Closier35, C. Coca26, V.
Coco38, J. Cogan6, P. Collins35, A. Comerma-Montells33, A. Contu52, A. Cook43,
M. Coombes43, G. Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, S. De Capua21,k, M. De
Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P.
Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil
Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A.
Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, C.
Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez
Albor34, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M. Fontana10, F.
Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C. Frei35, M.
Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M. Gandelman2,
P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra Tico44, L.
Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35,
T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D.
Golubkov28, A. Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa
Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35, E. Graugés33, G.
Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B. Gui53, E. Gushchin30,
Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C.
Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N. Harnew52, J.
Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny11, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, A.
Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K. Paterson50, G.N.
Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A.
Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M.
Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste
Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, S. Schleich9, M. Schlupp9, M.
Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H. Schune7, R.
Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A. Semennikov28, K.
Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32,
I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31,
O. Shevchenko40, V. Shevchenko28, A. Shires50, R. Silva Coutinho45, T.
Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A.
Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9, A. Sparkes47, P.
Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S. Stoica26, S.
Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K. Subbiah35, S.
Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S. T’Jampens4, E.
Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van Tilburg11, V.
Tisserand4, M. Tobin37, S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S.
Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35,
A. Ukleja25, P. Urquijo53, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g,
B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A.
Vollhardt37, D. Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, R.
Waldi56, S. Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D.
Webber51, D. Websdale50, M. Whitehead45, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z.
Xing53, Z. Yang3, R. Young47, O. Yushchenko32, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The study of charmonium production provides an important test of the
underlying mechanisms described by Quantum Chromodynamics (QCD). At the
centre-of-mass energies of proton-proton collisions at the Large Hadron
Collider, $c\overline{}c$ pairs are expected to be produced predominantly via
Leading Order (LO) gluon-gluon interactions, followed by the formation of
bound charmonium states. The former can be calculated using perturbative QCD
and the latter is described by non-perturbative models. Other, more recent,
approaches make use of non-relativistic QCD factorization (NRQCD), which
assumes the $c\overline{}c$ pair to be a combination of colour-singlet and
colour-octet states as it evolves towards the final bound system via the
exchange of soft gluons [1]. The fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ produced through the radiative
decay of $\chi_{c}$ states is an important test of both the colour-singlet and
colour-octet production mechanisms. In addition, knowledge of this fraction is
required for the measurement of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ polarisation, since the predicted polarisation is different for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons coming from the
radiative decay of $\chi_{c}$ state compared to those that are directly
produced.
In this paper, we report the measurement of the ratio of the cross-sections
for the production of $P$-wave charmonia $\chi_{cJ}(1P)$, with $J=$ 0, 1, 2,
to the production of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in
promptly produced charmonium. The ratio is measured as a function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum in the
range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and in the rapidity
range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$.
Throughout the paper we refer to the collection of $\chi_{cJ}(1P)$ states as
$\chi_{c}$. The $\chi_{c}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates are reconstructed through their respective decays
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ using
a data sample corresponding to an integrated luminosity of 36
$\mbox{\,pb}^{-1}$ collected during 2010. Prompt (non-prompt) production
refers to charmonium states produced at the interaction point (in the decay of
$b$-hadrons); direct production refers to prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons that are not decay
products of an intermediate resonant state, such as the $\psi{(2S)}$. The
measurements are complementary to the measurements of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section [2]
and the ratio of the prompt $\chi_{c}$ production cross-sections for the $J=1$
and $J=2$ spin states [3], and extend the
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ coverage with
respect to previous experiments [4, 5].
## 2 LHCb detector and selection requirements
The LHCb detector [6] is a single-arm forward spectrometer with a pseudo-
rapidity range $2\,{<}\,\eta\,{<}\,5$. The detector consists of a silicon
vertex detector, a dipole magnet, a tracking system, two ring-imaging
Cherenkov (RICH) detectors, a calorimeter system and a muon system.
Of particular importance in this measurement are the calorimeter and muon
systems. The calorimeter system consists of a scintillating pad detector (SPD)
and a pre-shower system, followed by electromagnetic (ECAL) and hadron
calorimeters. The SPD and pre-shower are designed to distinguish between
signals from photons and electrons. The ECAL is constructed from scintillating
tiles interleaved with lead tiles. Muons are identified using hits in muon
chambers interleaved with iron filters.
The signal simulation sample used for this analysis was generated using the
Pythia $6.4$ generator [7] configured with the parameters detailed in Ref.
[8]. The EvtGen [9], Photos [10] and Geant4 [11] packages were used to decay
unstable particles, generate QED radiative corrections and simulate
interactions in the detector, respectively. The sample consists of events in
which at least one ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay takes place with no constraint on the
production mechanism.
The trigger consists of a hardware stage followed by a software stage, which
applies a full event reconstruction. For this analysis, events are selected
which have been triggered by a pair of oppositely charged muon candidates,
where either one of the muons has a transverse momentum
$p_{\mathrm{T}}\,{>}\,\mbox{${1.8}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
or one of the pair has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.56}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
and the other has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.48}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$.
The invariant mass of the candidates is required to be greater than
${2.9}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. The photons are not
involved in the trigger decision for this analysis.
Photons are reconstructed using the electromagnetic calorimeter and identified
using a likelihood-based estimator, $\mathrm{CL}_{\gamma}$, constructed from
variables that rely on calorimeter and tracking information. For example, in
order to reduce the electron background, candidate photon clusters are
required not to be matched to the trajectory of a track extrapolated from the
tracking system to the cluster position in the calorimeter. For each photon
candidate a value of $\mathrm{CL}_{\gamma}$, with a range between 0
(background-like) and 1 (signal-like), is calculated based on simulated signal
and background samples.
The photons are classified as one of two types: those that have converted to
electrons in the material after the dipole magnet and those that have not.
Converted photons are identified as clusters in the ECAL with correlated
activity in the SPD. In order to account for the different energy resolutions
of the two types of photons, the analysis is performed separately for
converted and non-converted photons and the results are combined. Photons that
convert before the magnet require a different analysis strategy and are not
considered here. The photons used to reconstruct the $\chi_{c}$ candidates are
required to have a transverse momentum
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$,
a momentum
$p^{\gamma}\,{>}\,\mbox{${5}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$ and
$\mathrm{CL}_{\gamma}\,{>}\,0.5$; the efficiency of the $\mathrm{CL}_{\gamma}$
cut for photons from $\chi_{c}$ decays is $72\%$.
All ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are
reconstructed using the decay ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$. The muon and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ identification criteria are
identical to those used in Ref. [2]: each track must be identified as a muon
with $p_{\mathrm{T}}\,{>}\,700{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and have a
track fit $\chi^{2}/\mathrm{ndf}\,{<}\,4$, where $\mathrm{ndf}$ is the number
of degrees of freedom. The two muons must originate from a vertex with a
probability of the vertex fit greater than $0.005$. In addition, the
$\mu^{+}\mu^{-}$ invariant mass is required to be in the range
${3062}\,{-}\,{\mbox{${3120}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$}}$.
The $\chi_{c}$ candidates are formed from the selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and photons.
The non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution
arising from $b$-hadron decays is taken from Ref. [2]. For the $\chi_{c}$
candidates, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-decay
time, $t_{z}$, is used to reduce the contribution from non-prompt decays, by
requiring $t_{z}\,{=}\,(z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{-}\,z_{PV})M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{/}\,p_{z}<\mbox{${0.1}\>{{\rm\,ps}}$}$, where
$M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the reconstructed dimuon
invariant mass, $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}-z_{\mathrm{PV}}$ is the $z$ separation of the reconstructed production
(primary) and decay vertices of the dimuon, and $p_{z}$ is the $z$-component
of the dimuon momentum. The $z$-axis is parallel to the beam line in the
centre-of-mass frame. Simulation studies show that, with this requirement
applied, the remaining fraction of $\chi_{c}$ from $b$-hadron decays is about
$0.1\%$. This introduces an uncertainty much smaller than any of the other
systematic or statistical uncertainties evaluated in this analysis and is not
considered further.
The distributions of the $\mu^{+}\mu^{-}$ mass of selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates and the mass
difference, $\Delta
M\,{=}\,M\left(\mu^{+}\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\mu^{-}\right)$,
of the selected $\chi_{c}$ candidates for the converted and non-converted
samples are shown in Fig. 1. The total number of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates observed in the data
is $\sim 2.6$ million. The fit procedure to extract the three $\chi_{c}$
signal yields using Gaussian functions and one common function for the
combinatorial background is discussed in Ref. [3]. The total number of
$\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ candidates observed are $823$,
$38\,630$ and $26\,114$ respectively. Since the
$\chi_{c0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
branching fraction is $\sim 30$ (17) times smaller than that of the
$\chi_{c1}$ ($\chi_{c2}$), the yield of $\chi_{c0}$ is small as expected [12].
Figure 1: (a) Invariant mass of the $\mu^{+}\mu^{-}$ pair for selected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates. The solid red curve
corresponds to the signal and the background is shown as a dashed purple
curve. (b) and (c) show the $\Delta
M\,{=}\,M\left(\mu^{+}\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\mu^{-}\right)$
distributions of selected $\chi_{c}$ candidates with (b) converted and (c)
non-converted photons. The upper solid blue curve corresponds to the overall
fit function described in Ref. [3]. The lower solid curves correspond to the
fitted $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ contributions from left to
right, respectively (the $\chi_{c0}$ peak is barely visible). The background
distribution is shown as a dashed purple curve.
## 3 Determination of the cross-section ratio
The main contributions to the production of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ arise from direct production
and from the feed-down processes
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ and
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$ where $X$
refers to any final state. The cross-section ratio for the production of
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decays
compared to all prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ can be
expressed in terms of the three $\chi_{cJ}$ $(J=0,1,2)$ signal yields,
$N_{\chi_{cJ}}$, and the prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
yield, $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, as
$\displaystyle\dfrac{\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})}$
$\displaystyle\,\approx\,\dfrac{\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}{\sigma^{\mathrm{dir}}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})+\sigma(\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)+\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}$
$\displaystyle\,\,{=}\,\,\dfrac{\displaystyle{\sum_{J=0}^{J=2}}\;\dfrac{N_{\chi_{cJ}}}{\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}}\cdot\dfrac{\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}R_{2S}+\displaystyle{\sum_{J=0}^{J=2}}\;\dfrac{N_{\chi_{cJ}}}{\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}}\left[\dfrac{\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}{\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}-R_{2S}\right]}$ (1) with $\displaystyle R_{2S}$
$\displaystyle\,\,{=}\,\,\dfrac{1+f_{2S}}{1+f_{2S}\dfrac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{2S}}{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}^{\mathrm{dir}}}}$ (2) and $\displaystyle f_{2S}$
$\displaystyle\,\,{=}\,\,\dfrac{\sigma(\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}X)}{\sigma^{\mathrm{dir}}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})}.$ (3)
The total prompt $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ cross-section is
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=\sum_{J=0}^{J=2}\;\sigma_{\chi_{cJ}}\cdot{\cal
B}(\chi_{cJ}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)$
where $\sigma_{\chi_{cJ}}$ is the production cross-section for each
$\chi_{cJ}$ state and $\cal
B$($\chi_{cJ}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$)
is the corresponding branching fraction. The cross-section ratio $f_{2S}$ is
used to link the prompt $\psi{(2S)}$ contribution to the direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution and $R_{2S}$ takes
into account their efficiencies. The combination of the trigger,
reconstruction and selection efficiencies for direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $\psi{(2S)}$ decay, and
for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay
are $\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, and
$\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
respectively. The efficiency to reconstruct and select a photon from a
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decay,
once the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is already selected,
is $\epsilon^{\chi_{cJ}}_{\gamma}$ and the efficiency for the subsequent
selection of the $\chi_{cJ}$ is $\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$.
The efficiency terms in Eq. (1) are determined using simulated events and are
partly validated with control channels in the data. The results for the
efficiency ratios $\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, $\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
and the product
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$ are
discussed in Sect. 4.
The prompt $N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
$N_{\chi_{cJ}}$ yields are determined in bins of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range
$2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ using the methods
described in Refs. [2] and [3] respectively. In Ref. [2] a smaller data sample
is used to determine the non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ fractions in bins of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and rapidity.
These results are applied to the present
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample without repeating the
full analysis.
## 4 Efficiencies
The efficiencies to reconstruct and select
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ candidates are
taken from simulation. The efficiency ratio
$\epsilon^{2S}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ is consistent with unity for all
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins; hence,
$R_{2S}$ is set equal to 1 in Eq. (2). The ratio of efficiencies
$\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
and the product of efficiencies
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$ for the
$\chi_{c1}$ and $\chi_{c2}$ states are shown in Fig. 2. In general these
efficiencies are the same for the two states, except at low
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ where the
reconstruction and detection efficiencies for $\chi_{c2}$ are significantly
larger than for $\chi_{c1}$. This difference arises from the effect of the
requirement
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$
which results in more photons surviving from $\chi_{c2}$ decays than from
$\chi_{c1}$ decays.
Figure 2: (a) Ratio of the reconstruction and selection efficiency for direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ compared to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $\chi_{c}$ decays,
$\epsilon^{\mathrm{dir}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}/\epsilon^{\chi_{c}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
and (b) the photon reconstruction and selection efficiency multiplied by the
$\chi_{c}$ selection efficiency,
$\epsilon^{\chi_{cJ}}_{\gamma}\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$, obtained
from simulation. The efficiencies are presented separately for the $\chi_{c1}$
(red triangles) and $\chi_{c2}$ (inverted blue triangles) states, and as a
function of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$.
The photon detection efficiency obtained using simulation is validated using
candidate $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
and $B^{+}\rightarrow\chi_{c}K^{+}$ (including charge conjugate) decays
selected from the same data set as the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ candidates. The
efficiency to reconstruct and select a photon from a $\chi_{c}$ in
$B^{+}\rightarrow\chi_{c}K^{+}$ decays, $\epsilon_{\gamma}$, is evaluated
using
$\displaystyle\epsilon_{\gamma}\,{=}\,\frac{N_{B^{+}\rightarrow\chi_{c}K^{+}}}{N_{B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}}\times\frac{{\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})}{{\cal
B}(B^{+}\rightarrow\chi_{c}K^{+})\cdot{\cal
B}(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)}\times R_{\epsilon}$ (4)
where $N_{B^{+}\rightarrow\chi_{c}K^{+}}$ and
$N_{B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ are
the measured yields of $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and ${\cal
B}$ are the known branching fractions. The factor $R_{\epsilon}=1.04\pm 0.02$
is obtained from simulation and takes into account any differences in the
acceptance, trigger, selection and reconstruction efficiencies of the $K$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\chi_{c}$ (except the photon
detection efficiency) and $B^{+}$ in
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\rightarrow\chi_{c}K^{+}$ decays. All branching fractions are taken from
Ref. [12]. The $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ branching fraction is ${\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=(1.013\pm 0.034)\times 10^{-3}$. The dominant process for
$B^{+}\rightarrow\chi_{c}K^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma K^{+}$ decays is via the $\chi_{c1}$ state, with branching
fractions ${\cal B}(B^{+}\rightarrow\chi_{c1}K^{+})=(4.6\pm 0.4)\times
10^{-4}$ and ${\cal
B}(\chi_{c1}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)=(34.4\pm 1.5)\times 10^{-2}$; the contributions from the
$\chi_{c0}$ and $\chi_{c2}$ modes are neglected.
The $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidates
are selected keeping as many of the selection criteria in common as possible
with the main analysis. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\chi_{c}$ selection criteria are the same as for the prompt analysis, apart
from the pseudo-decay time requirement. The bachelor kaon is required to have
a well measured track ($\chi^{2}/\mathrm{ndf}<5$), a minimum impact parameter
$\chi^{2}$ with respect to all primary vertices of greater than $9$ and a
momentum greater than $5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The bachelor
is identified as a kaon by the RICH detectors by requiring the difference in
log-likelihoods between the kaon and pion hypotheses to be larger than $5$.
The $B$ candidate is formed from the $\chi_{c}$ or
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate and the bachelor
kaon. The $B$ vertex is required to be well measured
($\chi^{2}/\mathrm{ndf}<9$) and separated from the primary vertex (flight
distance $\chi^{2}>50$). The $B$ momentum vector is required to point towards
the primary vertex ($\cos\theta>0.9999$, where $\theta$ is the angle between
the $B$ momentum and the direction between the primary and $B$ vertices) and
have an impact parameter $\chi^{2}$ smaller than $9$. The combinatorial
background under the $\chi_{c}$ peak for the $B^{+}\rightarrow\chi_{c}K^{+}$
candidates is reduced by requiring the mass difference $\Delta
M_{\chi_{c}}=M(\mu^{+}\mu^{-}\gamma)-M(\mu^{+}\mu^{-})<600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
A small number of $B^{+}\rightarrow\chi_{c}K^{+}$ candidates which form a good
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ candidate
are removed by requiring $|M(\mu^{+}\mu^{-}\gamma
K)-M(\mu^{+}\mu^{-}K)|>200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The $\Delta M_{B^{+}}=M(\mu^{+}\mu^{-}\gamma K)-M(\mu^{+}\mu^{-}\gamma)$ mass
distribution for the $B^{+}\rightarrow\chi_{c}K^{+}$ candidates is shown in
Fig. 3(a); $\Delta M_{B^{+}}$ is computed to improve the resolution and hence
the signal-to-background ratio. The $B^{+}\rightarrow\chi_{c}K^{+}$ yield,
$142\pm 15$ candidates, is determined from a fit that uses a Gaussian function
to describe the signal peak and a threshold function,
$\displaystyle
f(x)\,{=}\,x^{a}\\!\left(1-e^{\frac{m_{0}}{c}\left(1-x\right)}\right)+b\left(x-1\right),$
(5)
where $x\,{=}\,\Delta M_{B^{+}}\,{/}\,m_{0}$ and $m_{0}$, $a$, $b$ and $c$ are
free parameters, to model the background. The reconstructed $B^{+}$ mass
distribution for the $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates is shown in Fig. 3(b). The
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield,
$8440\pm 96$ candidates, is determined from a fit that uses a Crystal Ball
function [13] to describe the signal peak and an exponential to model the
background.
The photon efficiency from the observation of $B^{+}\rightarrow\chi_{c}K^{+}$
and $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
is measured to be $\epsilon_{\gamma}=(11.3\pm 1.2\pm 1.2)\%$ where the first
error is statistical and is dominated by the observed yield of
$B^{+}\rightarrow\chi_{c}K^{+}$ candidates, and the second error is systematic
and is given by the uncertainty on the branching fraction $\cal
B$($B^{+}\rightarrow\chi_{c1}K^{+}$). The photon efficiency measured in data
can be compared to the photon efficiency, $(11.7\pm 0.3)\%$, obtained using
the same procedure on simulated events. The measurements are in good agreement
and the uncertainty on the difference between data and simulation is
propagated as a $\pm 14\%$ relative systematic uncertainty on the photon
efficiency in the measurement of
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$.
Figure 3: (a) Reconstructed $\Delta M_{B^{+}}=M(\mu^{+}\mu^{-}\gamma
K)-M(\mu^{+}\mu^{-}\gamma)$ mass distribution for
$B^{+}\rightarrow\chi_{c}K^{+}$ candidates and (b) the reconstructed $B^{+}$
mass distribution for $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates. The LHCb data are shown as solid black points, the
full fit functions with a solid blue (upper) curve, the contribution from
signal candidates with a dashed red (lower curve) and the background with a
dashed purple curve.
## 5 Polarisation
The simulation used to calculate the efficiencies and, hence, extract the
result of Eq. (1) assumes that the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and $\chi_{c}$ are unpolarised. The effect of polarised states is
studied by reweighting the simulated events according to different
polarisation scenarios; the results are shown in Table 1. It is also noted
that, since the $\psi{(2S)}$ decays predominantly to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi\pi$, with the $\pi\pi$ in an
$S$ wave state [14], and the $\psi{(2S)}$ polarisation should not differ
significantly from the polarisation of directly produced
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons, the effect of the
polarisation can be considered independent of the
$\psi{(2S)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$
contribution [15].
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$
angular distributions are calculated in the helicity frame assuming azimuthal
symmetry. This choice of reference frame provides an estimate of the effect of
polarisation on the results, pending the direct measurements of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ system is described by the
angle $\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, which is the
angle between the directions of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the laboratory frame. The
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ distribution depends
on the parameter $\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
which describes the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
polarisation; $\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}=+1,-1,0$
corresponds to pure transverse, pure longitudinal and no polarisation
respectively. The $\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ system is described by three angles:
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\theta_{\chi_{c}}$ and $\phi$, where
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the angle
between the directions of the $\mu^{+}$ in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame,
$\theta_{\chi_{c}}$ is the angle between the directions of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame
and the $\chi_{c}$ in the laboratory frame, and $\phi$ is the angle between
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay plane in the
$\chi_{c}$ rest frame and the plane formed by the $\chi_{c}$ direction in the
laboratory frame and the direction of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame.
The general expressions for the angular distributions are independent of the
choice of polarisation axis (here chosen as the direction of the $\chi_{c}$ in
the laboratory frame) and are detailed in Ref. [4]. The angular distributions
of the $\chi_{c}$ states depend on $m_{\chi_{cJ}}$ which is the azimuthal
angular momentum quantum number of the $\chi_{cJ}$ state.
For each simulated event in the unpolarised sample, a weight is calculated
from the distributions of
$\theta^{\prime}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$,
$\theta_{\chi_{c}}$ and $\phi$ in the various polarisation hypotheses compared
to the unpolarised distributions. The weights shown in Table 1 are then the
average of these per-event weights in the simulated sample. For a given
($|m_{\chi_{c1}}|$, $|m_{\chi_{c2}}|$,
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$) polarisation
combination, the central value of the determined cross-section ratio in each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin should be
multiplied by the number in the table. The maximum effect from the possible
polarisation of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$,
$\chi_{c1}$ and $\chi_{c2}$ mesons is given separately from the systematic
uncertainties in Table 3 and Fig. 4.
Table 1: Polarisation weights in
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins for
different combinations of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$,
$\chi_{c1}$ and $\chi_{c2}$ polarisations.
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation parameter;
$\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}=+1,-1,0$ corresponds
to fully transverse, fully longitudinal and no polarisation respectively.
$m_{\chi_{cJ}}$ is the azimuthal angular momentum quantum number corresponding
to total angular momentum $J$; Unpol means the $\chi_{c}$ is unpolarised.
($|m_{\chi_{c1}}|,|m_{\chi_{c2}}|,\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$) | $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ (${\mathrm{\,Ge\kern-0.50003ptV\\!/}c}$)
---|---
2-3 | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 | 9-10 | 10-11 | 11-12 | 12-13 | 13-15
(Unpol,Unpol,-1) | 1.16 | 1.15 | 1.15 | 1.15 | 1.15 | 1.14 | 1.14 | 1.13 | 1.12 | 1.12 | 1.10 | 1.10
(Unpol,Unpol,1) | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.93 | 0.93 | 0.93 | 0.94 | 0.95 | 0.94
(Unpol,0,-1) | 1.16 | 1.14 | 1.13 | 1.11 | 1.10 | 1.09 | 1.09 | 1.08 | 1.07 | 1.06 | 1.06 | 1.07
(Unpol,0,0) | 1.00 | 0.99 | 0.98 | 0.97 | 0.96 | 0.95 | 0.95 | 0.96 | 0.95 | 0.95 | 0.96 | 0.97
(Unpol,0,1) | 0.91 | 0.91 | 0.90 | 0.89 | 0.88 | 0.87 | 0.88 | 0.89 | 0.89 | 0.88 | 0.91 | 0.92
(Unpol,1,-1) | 1.15 | 1.14 | 1.14 | 1.13 | 1.13 | 1.12 | 1.11 | 1.11 | 1.10 | 1.09 | 1.08 | 1.09
(Unpol,1,0) | 0.99 | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98
(Unpol,1,1) | 0.90 | 0.91 | 0.91 | 0.90 | 0.90 | 0.90 | 0.91 | 0.91 | 0.91 | 0.91 | 0.93 | 0.93
(Unpol,2,-1) | 1.18 | 1.17 | 1.18 | 1.20 | 1.21 | 1.21 | 1.20 | 1.19 | 1.19 | 1.19 | 1.16 | 1.15
(Unpol,2,0) | 1.01 | 1.02 | 1.03 | 1.04 | 1.05 | 1.06 | 1.06 | 1.05 | 1.06 | 1.07 | 1.05 | 1.04
(Unpol,2,1) | 0.93 | 0.94 | 0.94 | 0.96 | 0.97 | 0.98 | 0.98 | 0.98 | 0.99 | 1.00 | 1.00 | 0.99
(0,Unpol,-1) | 1.16 | 1.15 | 1.18 | 1.21 | 1.22 | 1.23 | 1.25 | 1.25 | 1.26 | 1.22 | 1.23 | 1.25
(0,Unpol,0) | 0.99 | 1.00 | 1.02 | 1.05 | 1.07 | 1.08 | 1.10 | 1.11 | 1.12 | 1.10 | 1.12 | 1.14
(0,Unpol,1) | 0.91 | 0.93 | 0.94 | 0.97 | 0.98 | 1.00 | 1.02 | 1.04 | 1.05 | 1.03 | 1.06 | 1.08
(1,Unpol,-1) | 1.17 | 1.15 | 1.14 | 1.13 | 1.12 | 1.11 | 1.09 | 1.08 | 1.07 | 1.08 | 1.05 | 1.05
(1,Unpol,0) | 1.00 | 1.00 | 0.99 | 0.98 | 0.97 | 0.97 | 0.96 | 0.95 | 0.95 | 0.96 | 0.95 | 0.95
(1,Unpol,1) | 0.92 | 0.92 | 0.91 | 0.90 | 0.89 | 0.89 | 0.89 | 0.89 | 0.89 | 0.90 | 0.90 | 0.89
(0,0,-1) | 1.15 | 1.14 | 1.15 | 1.17 | 1.18 | 1.18 | 1.20 | 1.21 | 1.20 | 1.17 | 1.19 | 1.22
(0,0,0) | 0.99 | 0.99 | 1.00 | 1.02 | 1.02 | 1.03 | 1.05 | 1.07 | 1.07 | 1.04 | 1.08 | 1.11
(0,0,1) | 0.91 | 0.91 | 0.92 | 0.93 | 0.94 | 0.95 | 0.98 | 1.00 | 1.00 | 0.98 | 1.02 | 1.05
(0,1,-1) | 1.14 | 1.14 | 1.16 | 1.19 | 1.20 | 1.21 | 1.22 | 1.23 | 1.23 | 1.20 | 1.21 | 1.24
(0,1,0) | 0.98 | 0.99 | 1.01 | 1.03 | 1.05 | 1.06 | 1.08 | 1.09 | 1.10 | 1.07 | 1.10 | 1.12
(0,1,1) | 0.90 | 0.92 | 0.93 | 0.95 | 0.96 | 0.98 | 1.00 | 1.02 | 1.03 | 1.01 | 1.04 | 1.07
(0,2,-1) | 1.17 | 1.17 | 1.21 | 1.25 | 1.29 | 1.30 | 1.31 | 1.31 | 1.32 | 1.30 | 1.28 | 1.30
(0,2,0) | 1.01 | 1.02 | 1.05 | 1.09 | 1.12 | 1.14 | 1.16 | 1.17 | 1.19 | 1.17 | 1.17 | 1.18
(0,2,1) | 0.92 | 0.94 | 0.96 | 1.01 | 1.03 | 1.06 | 1.08 | 1.09 | 1.11 | 1.10 | 1.11 | 1.12
(1,0,-1) | 1.16 | 1.13 | 1.12 | 1.09 | 1.07 | 1.05 | 1.04 | 1.04 | 1.02 | 1.02 | 1.01 | 1.01
(1,0,0) | 1.00 | 0.99 | 0.97 | 0.94 | 0.93 | 0.92 | 0.91 | 0.91 | 0.90 | 0.91 | 0.91 | 0.92
(1,0,1) | 0.92 | 0.91 | 0.89 | 0.87 | 0.85 | 0.84 | 0.85 | 0.85 | 0.84 | 0.85 | 0.86 | 0.86
(1,1,-1) | 1.15 | 1.14 | 1.13 | 1.11 | 1.10 | 1.08 | 1.07 | 1.06 | 1.05 | 1.05 | 1.03 | 1.03
(1,1,0) | 0.99 | 0.99 | 0.98 | 0.96 | 0.95 | 0.94 | 0.94 | 0.94 | 0.93 | 0.94 | 0.94 | 0.93
(1,1,1) | 0.91 | 0.91 | 0.90 | 0.88 | 0.87 | 0.87 | 0.87 | 0.87 | 0.87 | 0.88 | 0.88 | 0.88
(1,2,-1) | 1.18 | 1.17 | 1.17 | 1.17 | 1.18 | 1.18 | 1.16 | 1.14 | 1.14 | 1.15 | 1.11 | 1.09
(1,2,0) | 1.02 | 1.01 | 1.01 | 1.02 | 1.03 | 1.03 | 1.02 | 1.01 | 1.02 | 1.03 | 1.01 | 0.99
(1,2,1) | 0.93 | 0.94 | 0.93 | 0.94 | 0.94 | 0.95 | 0.94 | 0.94 | 0.95 | 0.97 | 0.95 | 0.93
## 6 Systematic uncertainties
Table 2: Summary of the systematic uncertainties on $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.80005ptV\\!/}c}$) | $2-3$ | $3-4$ | $4-5$ | $5-6$ | $6-7$ | $7-8$
---|---|---|---|---|---|---
Size of simulation sample | ${}^{+0.0006}_{-0.0005}$ | ${}^{+0.0006}_{-0.0005}$ | ${}^{+0.0007}_{-0.0006}$ | ${}^{+0.0009}_{-0.0009}$ | ${}^{+0.001}_{-0.001}$ | ${}^{+0.002}_{-0.002}$
Photon efficiency | ${}^{+0.011}_{-0.010}$ | ${}^{+0.013}_{-0.011}$ | ${}^{+0.013}_{-0.012}$ | ${}^{+0.016}_{-0.013}$ | ${}^{+0.016}_{-0.013}$ | ${}^{+0.017}_{-0.015}$
Non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction | ${}^{+0.002}_{-0.005}$ | ${}^{+0.003}_{-0.005}$ | ${}^{+0.003}_{-0.006}$ | ${}^{+0.004}_{-0.008}$ | ${}^{+0.005}_{-0.010}$ | ${}^{+0.006}_{-0.011}$
Fit model | ${}^{+0.003}_{-0.003}$ | ${}^{+0.003}_{-0.003}$ | ${}^{+0.002}_{-0.004}$ | ${}^{+0.003}_{-0.005}$ | ${}^{+0.002}_{-0.005}$ | ${}^{+0.003}_{-0.006}$
Simulation calibration | ${}^{+0.010}_{-0.000}$ | ${}^{+0.010}_{-0.000}$ | ${}^{+0.012}_{-0.000}$ | ${}^{+0.012}_{-0.000}$ | ${}^{+0.015}_{-0.000}$ | ${}^{+0.014}_{-0.000}$
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.80005ptV\\!/}c}$) | $8-9$ | $9-10$ | $10-11$ | $11-12$ | $12-13$ | $13-15$
Size of simulation sample | ${}^{+0.002}_{-0.002}$ | ${}^{+0.003}_{-0.003}$ | ${}^{+0.004}_{-0.004}$ | ${}^{+0.006}_{-0.006}$ | ${}^{+0.008}_{-0.008}$ | ${}^{+0.008}_{-0.008}$
Photon efficiency | ${}^{+0.018}_{-0.016}$ | ${}^{+0.020}_{-0.016}$ | ${}^{+0.019}_{-0.016}$ | ${}^{+0.019}_{-0.018}$ | ${}^{+0.021}_{-0.020}$ | ${}^{+0.023}_{-0.019}$
Non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction | ${}^{+0.009}_{-0.011}$ | ${}^{+0.012}_{-0.013}$ | ${}^{+0.011}_{-0.017}$ | ${}^{+0.019}_{-0.019}$ | ${}^{+0.022}_{-0.018}$ | ${}^{+0.018}_{-0.010}$
Fit model | ${}^{+0.002}_{-0.005}$ | ${}^{+0.002}_{-0.003}$ | ${}^{+0.006}_{-0.002}$ | ${}^{+0.001}_{-0.006}$ | ${}^{+0.003}_{-0.008}$ | ${}^{+0.002}_{-0.004}$
Simulation calibration | ${}^{+0.015}_{-0.000}$ | ${}^{+0.017}_{-0.000}$ | ${}^{+0.018}_{-0.000}$ | ${}^{+0.018}_{-0.000}$ | ${}^{+0.017}_{-0.000}$ | ${}^{+0.022}_{-0.000}$
The systematic uncertainties detailed below are measured by repeatedly
sampling from the distribution of the parameter under consideration. For each
sampled value, the cross-section ratio is calculated and the $68.3\%$
probability interval is determined from the resulting distribution.
The statistical errors from the finite number of simulated events used for the
calculation of the efficiencies are included as a systematic uncertainty in
the final results. The uncertainty is determined by sampling the efficiencies
used in Eq. 1 according to their errors. The relative systematic uncertainty
due to the limited size of the simulation sample is found to be in the range
${(0.3}\,{-}\,{3.2)\%}$ and is given for each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in Table
2.
The efficiency extracted from the simulation sample for reconstructing and
selecting a photon in
$\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma$ decays
has been validated using $B^{+}\rightarrow\chi_{c}K^{+}$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays
observed in the data, as described in Sect. 4. The relative uncertainty
between the photon efficiencies measured in the data and simulation, $\pm
14\%$, arises from the finite size of the observed
$B^{+}\rightarrow\chi_{c}K^{+}$ yield and the uncertainty on the known
$B^{+}\rightarrow\chi_{c1}K^{+}$ branching fraction, and is taken to be the
systematic error assigned to the photon efficiency in the measurement of
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$. The
relative systematic uncertainty on the cross-section ratio used in Eq. 1 is
determined by sampling the photon efficiency according to its systematic
error. It is found to be in the range ${(6.4}\,{-}\,{8.7)\%}$ and is given for
each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in
Table 2.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ yield used in Eq. 1 is
corrected for the fraction of non-prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, taken from Ref. [2]. For those
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and rapidity
bins used in this analysis and not covered by Ref. [2]
($13\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,14$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$3.5\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$;
$11\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,13$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$4\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$; and
$14\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), a linear
extrapolation is performed, allowing for asymmetric errors. The systematic
uncertainty on the cross-section ratio is determined by sampling the non-
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction according to a
bifurcated Gaussian function. The relative systematic uncertainty from the
non-prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ fraction is found to
be in the range ${(1.3}\,{-}\,{10.7)\%}$ and is given for each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin in Table
2.
The method used to determine the systematic uncertainty due to the fit
procedure in the extraction of the $\chi_{c}$ yields is discussed in detail in
Ref. [3]. The uncertainty includes contributions from uncertainties on the
fixed parameters, the fit range and the shape of the overall fit function. The
overall relative systematic uncertainty from the fit is found to be in the
range ${(0.4}\,{-}\,{3.2)\%}$ and is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 2.
The systematic uncertainty related to the calibration of the simulation sample
is evaluated by performing the full analysis using simulated events and
comparing to the expected cross-section ratio from simulated signal events.
The results give an underestimate of $10.9\%$ in the measurement of the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ cross-
section ratio. This deviation is caused by non-Gaussian signal shapes in the
simulation which arise from an untuned calorimeter calibration. These are not
seen in the data, which is well described by Gaussian signal shapes. This
deviation is included as a systematic error by sampling from the negative half
of a Gaussian with zero mean and a width of $10.9\%$. The relative uncertainty
on the cross-section ratio is found to be in the range ${(6.3}\,{-}\,{8.2)\%}$
and is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 2. A
second check of the procedure was performed using simulated events generated
according to the distributions observed in the data, i.e. three overlapping
Gaussians and a background shape similar to that in Fig. 1. In this case no
evidence for a deviation was observed. Other systematic uncertainties due to
the modelling of the detector in the simulation are negligible.
In summary, the overall systematic uncertainty is evaluated by simultaneously
sampling the deviation of the cross-section ratio from the central value,
using the distributions of the cross-section ratios described above. The
systematic uncertainty is then determined from the resulting distribution as
described earlier in this section. The separate systematic uncertainties are
shown in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ in Table 2 and the combined uncertainties are shown in Table 3.
## 7 Results and conclusions
The cross-section ratio,
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$,
measured in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ is given in Table 3 and shown in Fig. 4. The measurements are
consistent with, but suggest a different trend to previous results from CDF
using $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV [5] as shown in Fig. 4(a),
and from HERA-$B$ in $p\mathrm{A}$ collisions at $\sqrt{s}=41.6$ GeV, with
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ below roughly
$5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, which gave
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})=0.188\pm 0.013^{+0.024}_{-0.022}$ [4].
Table 3: Ratio $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and in the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$. The first error is statistical and the second is systematic (apart from the polarisation). Also given is the maximum effect of the unknown polarisations on the results as described in Sect. 5. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | $\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ | Polarisation effects
---|---|---
$2-3$ | ${0.140}^{+0.005\;+0.015}_{-0.005\;-0.011}$ | ${}^{+0.025}_{-0.014}$
$3-4$ | ${0.160}^{+0.003\;+0.017}_{-0.004\;-0.012}$ | ${}^{+0.028}_{-0.015}$
$4-5$ | ${0.168}^{+0.003\;+0.019}_{-0.003\;-0.012}$ | ${}^{+0.035}_{-0.018}$
$5-6$ | ${0.189}^{+0.004\;+0.021}_{-0.004\;-0.015}$ | ${}^{+0.048}_{-0.025}$
$6-7$ | ${0.189}^{+0.005\;+0.022}_{-0.004\;-0.016}$ | ${}^{+0.054}_{-0.028}$
$7-8$ | ${0.211}^{+0.005\;+0.024}_{-0.005\;-0.017}$ | ${}^{+0.064}_{-0.033}$
$8-9$ | ${0.218}^{+0.007\;+0.026}_{-0.007\;-0.019}$ | ${}^{+0.068}_{-0.034}$
$9-10$ | ${0.223}^{+0.009\;+0.030}_{-0.009\;-0.019}$ | ${}^{+0.070}_{-0.034}$
$10-11$ | ${0.226}^{+0.011\;+0.030}_{-0.011\;-0.022}$ | ${}^{+0.073}_{-0.036}$
$11-12$ | ${0.233}^{+0.013\;+0.034}_{-0.013\;-0.026}$ | ${}^{+0.070}_{-0.036}$
$12-13$ | ${0.252}^{+0.018\;+0.037}_{-0.017\;-0.029}$ | ${}^{+0.071}_{-0.035}$
$13-15$ | ${0.268}^{+0.018\;+0.038}_{-0.017\;-0.025}$ | ${}^{+0.080}_{-0.037}$
Figure 4: Ratio
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ in
bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the
range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,15~{}{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$. The LHCb results, in
the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,4.5$ and assuming the production of unpolarised
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ mesons, are
shown with solid black circles and the internal error bars correspond to the
statistical error; the external error bars include the contribution from the
systematic uncertainties (apart from the polarisation). The lines surrounding
the data points show the maximum effect of the unknown
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations on
the result. The upper and lower limits correspond to the spin states as
described in the text. The CDF data points, at
$\sqrt{s}\,{=}\,\mbox{${1.8}\>{\mathrm{\,Te\kern-0.90005ptV}}$}$ in $p\bar{p}$
collisions and in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-
rapidity range $|\eta^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}|<1.0$,
are shown in (a) with open blue circles [5]. The two hatched bands in (b)
correspond to the ChiGen Monte Carlo generator prediction [16] and NLO NRQCD
[17].
Theory predictions, calculated in the LHCb rapidity range
$2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$, from
the ChiGen Monte Carlo generator [16] and from the NLO NRQCD calculations [17]
are shown as hatched bands in Fig. 4(b). The ChiGen Monte Carlo event
generator is an implementation of the leading-order colour-singlet model
described in Ref. [18]. However, since the colour-singlet model implemented in
ChiGen does not reliably predict the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section, the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$
prediction uses the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-
section measurement from Ref. [2] as the denominator in the cross-section
ratio.
Figure 4 also shows the maximum effect of the unknown
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\chi_{c}$ polarisations on
the result, shown as lines surrounding the data points. In the first
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin, the upper
limit corresponds to a spin state combination
$(|m_{\chi_{c1}}|,|m_{\chi_{c2}}|,\lambda_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$ equal to $(1,2,-1)$ and the lower limit to $(0,1,1)$. For all
subsequent bins, the upper and lower limits correspond to the spin state
combinations $(0,2,-1)$ and $(1,0,1)$ respectively.
In summary, the ratio of the
$\sigma(\chi_{c}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma)\,/\,\sigma({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ prompt
production cross-sections is measured using 36$\mbox{\,pb}^{-1}$ of data
collected by LHCb during 2010 at a centre-of-mass energy
$\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-1.00006ptV}}$}$. The results
provide a significant statistical improvement compared to previous
measurements [4, 5]. The results are in agreement with the NLO NRQCD model
[17] over the full range of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. However,
there is a significant discrepancy compared to the leading-order colour-
singlet model described by the ChiGen Monte Carlo generator [16]. At high
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, NLO
corrections fall less slowly with
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and become
important, it is therefore not unexpected that the model lies below the data.
At low $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, the
data appear to put a severe strain on the colour-singlet model.
## Acknowledgments
We would like to thank L. A. Harland-Lang, W. J. Stirling and K.-T. Chao for
supplying the theory predictions for comparison to our data and for many
helpful discussions.
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] G. T. Bodwin, E. Braaten, and G. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D51 (1995) 1125, arXiv:hep-ph/9407339, Erratum-ibid. D55 (1997) 5853
* [2] LHCb collaboration, R. Aaij et al., Measurement of $J/\psi$ production in pp collisions at $\sqrt{s}$=7 TeV, Eur. Phys. J. C71 (2011) 1645, arXiv:1103.0423
* [3] LHCb collaboration, R. Aaij et al., Measurement of the cross-section ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ for prompt $\chi_{c}$ production at $\sqrt{s}=7$ TeV, Phys. Lett. B714 (2012) 215, arXiv:1202.1080
* [4] HERA-$B$ collaboration, I. Abt et al., Production of the charmonium states $\chi_{c1}$ and $\chi_{c2}$ in proton nucleus interactions at $\sqrt{s}$ = 41.6 GeV, Phys. Rev. D79 (2009) 012001, arXiv:0807.2167
* [5] CDF Collaboration, F. Abe et al., Production of $J/\psi$ mesons from $\chi_{c}$ meson decays in $p\bar{p}$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. Lett. 79 (1997) 578
* [6] LHCb collaboration, A. A. Alves Jr et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [7] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [8] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC), IEEE (2010) 1155
* [9] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [10] E. Barberio and Z. Wa̧s, Photos: a universal Monte Carlo for QED radiative corrections: version 2.0, Comput. Phys. Commun. 79 (1994) 291
* [11] S. Agostinelli et al., Geant4: a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [12] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021, includes 2011 partial update for the 2012 edition.
* [13] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances. PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [14] BES collaboration, J. Z. Bai et al., $\psi(2S)\rightarrow\pi^{+}\pi^{-}J/\psi$ decay distributions, Phys. Rev. D62 (2000) 032002, arXiv:hep-ex/9909038
* [15] P. Faccioli and J. Seixas, Observation of $\chi_{c}$ and $\chi_{b}$ nuclear suppression via dilepton polarization measurements, Phys. Rev. D85 (2012) 074005, arXiv:1203.2033
* [16] L. A. Harland-Lang and W. J. Stirling, http://projects.hepforge.org/superchic/chigen.html
* [17] Y.-Q. Ma, K. Wang, and K.-T. Chao, QCD radiative corrections to $\chi_{cJ}$ production at hadron colliders, Phys. Rev. D83 (2011) 111503, arXiv:1002.3987, calculation in the LHCb rapidity range given by private communication
* [18] E. W. N. Glover, A. D. Martin, and W. J. Stirling, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production at large transverse momentum at hadron colliders, Z. Phys. C38 (1988) 473, Erratum-ibid. C49 (1991) 526
|
arxiv-papers
| 2012-04-06T12:08:43 |
2024-09-04T02:49:29.409358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito,\n A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back,\n V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I.\n Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R.\n Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S.\n Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, K. de\n Bruyn, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A.\n Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B. Couturier, G.\n A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, S.\n De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De\n Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D.\n Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista,\n F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J.\n Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N.\n Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T.\n Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, R. F.\n Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R.\n McNulty, M. Meissner, M. Merk, J. Merkel, S. Miglioranzi, D. A. Milanes,\n M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez,\n G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H.\n Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler,\n A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin,\n F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr,\n E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B.\n Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, R. Waldi, S. Wandernoth, J.\n Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Valerie Gibson",
"url": "https://arxiv.org/abs/1204.1462"
}
|
1204.1559
|
# Uma Construção Elementar de Códigos de Goppa Geométricos
N. M. S
Universidade Estadual de Campinas
Instituto de Matemática, Estatística
e Computação Científica
Departamento de Matemática
Códigos Geométricos de Goppa Via Métodos Elementares
Autor: Nolmar Melo de Souza
Orientador: Prof. Dr. Paulo Roberto Brumatti
Co-orientador: Prof. Dr. Fernando Eduardo Torres Orihuela
FICHA CATALOGRÁFICA ELABORADA PELA
---
BIBLIOTECA DO IMECC DA UNICAMP
Bibliotecária: Maria Júlia Milani Rodrigues - CRB8a /2116
| Melo, Nolmar
---|---
M491c | Códigos geométricos de Goppa via métodos elementares / Nolmar Melo de
| Souza – Campinas, [S.P.:s.n.], 2006.
| Orientadores: Paulo Roberto Brumatti; Fernado Eduardo Torres Orihuela
| Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de
| Matemática, Estatística e Computação Científica.
| 1\. Códigos de controle de erros (Teoria da informação). 2. Semigrupos. 3.
| Geometria algébrica. 4. Teoria da codificação. I. Brumatti, Paulo Roberto.
II.
| Torres Orihuela, Fernando Eduardo. III. Universidade Estadual de Campinas.
| Instituto de Matemática, Estatística e Computação Científica. IV. Título
Título em inglês: Goppa goemetry codes via elementary methods |
---|---
Palavras-chave em inglês (Keywords): 1. Error control codes (Information theory). 2. |
Semigroups. 3. Algebraic geometry. 4. Coding theory. |
Área de concentração: Álgebra |
Titulação: Mestre em Matemática |
Banca examinadora: | Prof. Dr. Paulo Roberto Brumatti (IMECC-UNICAMP)
---|---
| Prof. Dr. Ercílio Carvalho da Silva (UFU-MG)
| Prof. Dr. Reginaldo Palazzo Junior (FEEC-UNICAMP)
Data da defesa: 17/02/2006 |
---|---
Resumo
O objetivo central desta dissertação foi o de apresentar os Códigos
Geométricos de Goppa via métodos elementares que foram introduzidos por J. H.
van Lint, R. Pellikaan e T. Høhold por volta de 1998. Numa primeira parte da
dissertação são apresentados os conceitos fundamentais sobre corpos de funções
racionais de uma curva algébrica na direção de se definir os códigos de Goppa
de maneira clássica, neste estudo nos baseamos principalmente no livro
“Algebraic Function Fields and Codes” de H. Stichtenoth. A segunda parte
inicia-se com a introdução dos conceitos de funções peso, grau e ordem que são
fundamentais para o estudo dos Códigos de Goppa via métodos elementares de
álgebra linear e de semigrupos, tal estudo foi baseado em “Algebraic geometry
codes” de J. H. van Lint, R. Pellikaan e T. Høhold.
A dissertação termina com a apresentação de exemplos que ilustram os métodos
elementares que nos referimos acima.
Abstract
The central objective of this dissertation was to present the Goppa Geometry
Codes via elementary methods which were introduced by J.H. van Lint,
R.Pellikaan and T. Høhold about 1998. On the first part of such dissertation
are presented the fundamental concepts about fields of rational functions of
an algebraic curve in the direction as to define the Goppa Codes on a
classical manner. In this study we based ourselves mainly on the book
“Algebraic Function Fields and Codes” of H. Stichtenoth. The second part is
initiated with an introduction about the functions weight, degree and order
which are fundamental for the study of the Goppa Codes through elementary
methods of linear algebra and of semigroups and such study was based on
“Algebraic Geometry Codes” of J.H. van Lint, R.Pellikaan and T. Høhold.
The dissertation ends up with a presentation of examples which illustrate the
elementary methods that we have referred to above.
Dedico esse trabalho a aqueles que me apoiaram incondicionalmente, meus pais.
João Ferreira de Souza e Maria de Fátima Mello de Souza.
Agradecimentos
Agradeço:
* Por sua grande atenção, paciencia e dedicação agradeço em especial ao professor Paulo Roberto Brumatti, que me orientou na condução deste trabalho.
* Ao professor Fernado Torres, o qual contribuiu com valiosas dicas de como prosseguir os estudos que levaram a construção deste.
* Aos meus pais e irmãos que me apoiaram em todos os momentos da vida.
* Ao professor Haroldo Benatti, que com sua dedicação me incentivou a continuar os estudos após a graduação.
* À José Santana, que me apoiou em varios momentos.
* Aos meus amigos, sem os quais seria dificil concluir essa etapa.
* À Capes, pelo apoio financeiro.
lance de dados
---
(gessinger)
daqui não tem mais volta, pra frente é sem saber
pequenos paraísos e riscos a correr
os deuses jogam pôquer
e bebem no saloon doses generosas de br 101
tá escrito há 6.000 anos em parachoques de caminhão
atalhos perigosos feito frases feitas
os deuses dão as cartas... o resto é com você
no fundo tudo é ritmo
a dança foge do salão
invade a autoestrada do átomo ao caminhão
o fim é puro ritmo
o último suspiro é purificação
os deuses dão as costas... agora é só você
os deuses dão as costas... agora é só você... querer
###### Contents
1. 1 Noções Básicas
1. 1.1 Códigos
2. 1.2 Códigos Lineares
3. 1.3 Lugares
4. 1.4 O Corpo de Funções Racionais
5. 1.5 Divisores
6. 1.6 O Teorema de Riemann-Roch
7. 1.7 Curvas Algébricas
1. 1.7.1 Variedades Projetivas
2. 1.7.2 Curvas Não Singulares
2. 2 Códigos Algébricos Geométricos
1. 2.1 Códigos Algébricos Geométricos
2. 2.2 Teorema de Bézout
3. 3 Códigos de Avaliação
1. 3.1 Funções Peso, Grau e Ordem
2. 3.2 Códigos de Avaliação
3. 3.3 A Cota Ordem
4. 3.4 Semigrupos
5. 3.5 Código de Avaliação Via Semigrupos
4. 4 Exemplos
1. 4.1 Um Primeiro Exemplo
5. References
## Introdução
A teoria de códigos corretores de erros teve início com as pesquisas de
matemáticos da Bell Lab. na década de 1940. Apesar da grande utilização na
engenharia, a teoria utiliza de sofisticadas técnicas matemáticas fazendo uso
de várias áreas tais como Geometria Algébrica, Teoria dos Números, Teoria dos
Grupos e Combinatória.
Os códigos corretores de erros estão presentes no nosso cotidiano sempre que
usamos o computador (no uso da internet ou na armazenagem de dados, por
exemplo), transmitimos dados, assistimos um DVD, etc.
Os códigos algébricos geométricos, que são uma sub-classe dos códigos lineares
(subespaços linear de um determinado espaço vetorial munido com uma métrica),
foram inicialmente apresentados por V. D. Goppa num artigo, [3], publicado em
1981. Essa classe de códigos, que é uma das mais estudadas atualmente, utiliza
como ferramenta principal a geometria algébrica.
Nesse texto apresentamos os códigos de avaliação, que foram introduzidos por
Tom Høhold , Jacobus H. van Lint e Ruud Pellikaan através das funções ordem e
peso. Tais códigos têm um tratamento muito mais simples do que os códigos
algébricos, geométricos visto que estes usam como teorias base os semigrupos e
a álgebra linear.
O primeiro capítulo desse texto, capítulo onde a maioria das proposições
tiveram suas demonstrações omitidas, traz uma introdução à teoria de códigos e
também algumas ferramentas da geométria algébrica tendo como um dos principais
resultados o teorema de Riemann-Roch.
Já no segundo capítulo fazemos uma apresentação dos códigos algébricos
geométicos, códigos que também são chamados de códigos geométricos de Goppa.
Tal apresentação é feita de maneira sucinta.
No terceiro capítulo trazemos as funções ordem, peso e grau, que são
fundamentais para calcularmos os parâmetros dos códigos de avaliação, que alí
também são apresentados. Tais funções, unidas à teoria de semigrupo, nos
permite descrever os códigos de avaliação de modo simples, fazendo desses uma
alternativa aos códigos algébricos geométricos.
No quarto e último capítulo fazemos a conexão entre códigos de Goppa pontuais
e códigos de avaliação e apresentamos alguns exemplos. Alí tentamos mostrar de
modo prático a diferença entre ambos os códigos e ao mesmo tempo a proximidade
dos dois.
## Chapter 1 Noções Básicas
Este capítulo está dedicado a introdução da notação que utilizaremos, assim
como expor definições e teoremas clássicos da teoria, os quais, na maioria das
vezes, terão suas demonstrações omitidas, visto que a bibliografia indicada ao
final desse texto as fazem muito bem. Um dos principais teoremas que iremos
encontrar aqui é o Teorema de Riemann-Roch.
### 1.1 Códigos
Nessa seção iremos introduzir um dos principais objetos que serão tratados
nesse texto (código). Usaremos aqui um conjunto $Q$ com $q$ elementos e o
chamaremos de alfabeto.
###### Definição 1.1.1.
A um subconjunto próprio não vazio, $C$, de $Q^{n}$, damos o nome de código.
Chamaremos de palavras código de comprimento $n$ aos seus elementos.
Dizemos que um código $C$ é trivial se $\\#C=1$.
Em seguida apresentamos as definições elementares e o resultado, proposição
1.1.5, que caracteriza os códigos perfeitos.
###### Definição 1.1.2 (Distância de Hamming).
Sendo $x,y\in\mathrm{Q}^{n}$ definimos distância de $x$ à $y$ como:
$d(x,y)=\\#\\{i;1\leqslant i\leqslant n\textrm{ e }x_{i}\neq y_{i}\\}.$
###### Definição 1.1.3.
Quando $Q$ é um corpo e $x\in Q^{n}$ o peso de $x$ é definido como
$w(x)=d(x,0)$.
###### Definição 1.1.4.
Chamamos de distância mínima do código $C$ ao natural
$d=min\\{d(x,y);\,\,x,y\in C\\}$.
###### Proposição 1.1.5.
Dado um código com distância mínima $d=2e+1$ temos que as bolas do conjunto
$B=\\{\mathbf{B}[x,e];x\in C\\}$ são disjuntas, onde $\mathbf{B}[x,e]=\\{y\in
Q^{n};d(x,y)\leqslant e\\}$.
###### Proof.
Segue diretamente da desigualdade triangular visto que a distância de Hamming
é uma métrica. ∎
###### Observação 1.1.6.
Dizemos que um código com distância mínima $d=2e+1$ detecta $2e$ e corrige $e$
erros, ou seja, ocorrendo até $e$ erros é possível decodificar a palavra
código transmitida.
###### Definição 1.1.7.
Um código $C\subset\mathrm{Q}^{n}$ com distância mínima $2e+1$ é dito código
perfeito se $\mathrm{Q}^{n}=\stackrel{{\scriptstyle\bullet}}{{\bigcup}}_{x\in
C}B(x,e)$.
### 1.2 Códigos Lineares
Nessa seção definiremos uma das principais classes de códigos e para tais
códigos o alfabeto, $\mathrm{Q}$, é um corpo finito $\mathbb{F}_{q}$ com $q$
elementos, onde $q=p^{r}$ e $p$ é um número primo.
###### Definição 1.2.1 (Código Linear).
Um código linear $C$ sobre um alfabeto $\mathbb{F}_{q}$ é um
$\mathbb{F}_{q}-$subespaço vetorial de $\mathbb{F}_{q}^{n}$. Se
$dim_{F_{q}}(C)=k$ chamamos $C$ de um $[n,k]$-código.
Usaremos a notação $[n,k,d]$-código, para um código linear com distância
mínima $d$.
###### Teorema 1.2.2.
Num código linear a distância mínima é igual ao peso mínimo.
###### Proof.
Como $C$ é um subespaço vetorial e dados $x,y\in C$ temos que $x-y\in C$.
Sejam $x,y\in C$ tais que $d(x,y)$ seja mínima. Logo $d(x,y)=d(x-y,0)=w(x-y)$
e mais $w(x-y)$ é mínimo. ∎
Na descrição de um código linear um importante ingrediente é o que se chama
matriz geradora.
###### Definição 1.2.3.
Uma matriz $k\times n$, $G$, é dita geradora de um código linear $C$ se os
seus vetores linha formam uma base para $C$.
###### Definição 1.2.4.
Dizemos que uma matriz geradora, $G$, de um código linear, $C$, está na forma
padrão se esta matriz está escrita em blocos da seguinte maneira:
$G=[I_{k}|P]$
onde $I_{k}$ é a matriz identidade $k\times k$ e $P$ é uma $k\times(n-k)$
matriz.
Em [4] vemos que nem todo código tem uma matriz geradora na forma padrão,
contudo vê-se também que pode-se definir códigos equivalentes com os mesmos
parâmetros $n,\,k$ e $d$ de modo que essa possua uma matriz geradora na forma
padrão.
###### Proposição 1.2.5.
O complemento ortogonal de um $[n,k,d]-$código $C$, em relação ao produto
interno canônico111Dados dois vetores em $\mathbb{F}_{q}^{n}$,
$a=(a_{1},\ldots,a_{n})$ e $b=(b_{1},\ldots,b_{n})$, definimos o produto
interno canônico como $\langle a,b\rangle=\sum_{i=1}^{n}a_{i}b_{i}$., também é
um código e esse é chamado código dual e denotado por $C^{\bot}$. Além disso a
dimensão do código dual é $n-k$ e mais, sendo $G=[I_{k}|P]$ a matriz geradora
do código $C$, então $H=[-P^{t}|I_{n-k}]$ é a matriz geradora do código dual
$C^{\bot}$. A matriz $H$ assim definida é dita matriz de teste de paridade
pois $x\in C$ se, e somente se, $Hx^{t}=0$.
A proposição acima nos dá uma informação muito importante sobre a pertinência
de uma dada palavra ao código, o que é fundamental para a sua decodificação.
Assim ficam justificadas a definição e o resultado que apresentaremos a
seguir:
###### Definição 1.2.6.
Sendo $C$ um código linear com matriz de teste de paridade $H$, então para
todo $x\in\mathbb{F}_{q}^{n}$ chamamos $Hx^{t}$ de síndrome de $x$.
###### Teorema 1.2.7.
Dados $x,y\in\mathbb{F}_{q}^{n}$ então $x$ e $y$ tem a mesma síndrome se, e
somente se, $x-y\in C$.
###### Proof.
Seja $H$ a matriz de teste de paridade do código $C$, então temos:
$Hx^{t}=Hy^{t}\Leftrightarrow Hx^{t}-Hy^{t}=0\Leftrightarrow
H(x-y)^{t}=0\Leftrightarrow x-y\in C.$
∎
As definições e resultados que apresentaremos abaixo estão todos relacionados
com a codificação de uma mensagem enviada.
###### Definição 1.2.8.
Seja $c$ uma palavra transmitida e $x$ o vetor recebido, definimos o vetor
erro como sendo $e=x-c$, observamos que a quantidade de erros ocorridos na
transmissão é o peso de $e$.
###### Definição 1.2.9.
Dizemos que dois vetores estão numa mesma classe lateral se eles tem a mesma
síndrome. Um vetor de menor peso na classe é dito líder.
###### Teorema 1.2.10.
Sendo $C$ um $[n,k,d]-$código, se $u\in\mathbb{F}_{q}^{n}$ é tal que
$w(u)\leqslant\left[\frac{d-1}{2}\right]$222A notação
$\left[\frac{a}{b}\right]$ denota o maior inteiro menor ou igual a
$\frac{a}{b}$. então $u$ é o líder de sua classe.
###### Proof.
Sejam $u,v\in\mathbb{F}_{q}^{n}$, com
$w(u)\leqslant\left[\frac{d-1}{2}\right]$ e
$w(v)\leqslant\left[\frac{d-1}{2}\right]$, se $u-v\in C$ temos que
$w(u-v)\leqslant
w(u)+w(v)\leqslant\left[\frac{d-1}{2}\right]+\left[\frac{d-1}{2}\right]\leqslant
d-1$, logo $u-v=0$, assim temos que $u=v.$ ∎
###### Teorema 1.2.11.
Sendo $H$ uma matriz de teste de paridade de $C$ temos que $w(C)\geqslant r$
se, e somente se, quaisquer $r-1$ colunas de $H$ são lineramente independentes
(L.I.).
###### Proof.
$\Leftarrow)$ Seja $c\in C$, $c\neq 0$, $H=[h_{1},h_{2},\cdots,h_{n}]$. Então
$0=Hc^{t}=\sum_{i=1}^{n}c_{i}h_{i}$. Logo $w(c)\geqslant r$, pois, caso
contrário teríamos uma combinação linear de $r-1$ vetores L.I. dando zero.
$\Rightarrow)$ Suponha que exista $r-1$ colunas L.D., sem perda de
generalidade podemos supor que são as $r-1$ primeiras colunas $h_{1},\cdots
h_{r-1}$. Logo existem $c_{1},\cdots c_{r-1}\in\mathbb{F}_{q}$, não todos
nulos, com $\sum_{i=1}^{r-1}c_{i}h_{i}=0$. Assim
$c=(c_{1},\cdots,c_{r-1},0,\cdots,0)\in C$ pois $Hc^{t}=0$ e $w(c)\leqslant
r-1$, absurdo. ∎
Uma cota que relaciona os parâmetros de um código aparecem no teorema:
###### Teorema 1.2.12 (Cota de Singleton).
Seja $C$ um $[n,k,d]-$código linear, então $d\leqslant n-k+1$. Chamamos de
código MDS (Maximum Distance Separable) ao código tal que $d=n-k+1$.
###### Proof.
Seja $H$ a matriz de teste de paridade de $C$. Logo, posto de $H=n-k$. Do
teorema 1.2.11 segue que quaisquer $(d-1)$ colunas de $H$ são L.I.. Assim
$d-1\leqslant$ (posto de $H$), temos que $d\leqslant n-k+1$. ∎
### 1.3 Lugares
Nessa seção apresentaremos as estruturas necessárias para a definição dos
códigos algébricos geométricos que veremos no próximo capítulo.
###### Definição 1.3.1.
Um corpo de funções algébricas $F/K$ de uma variável sobre $K$ é uma extensão
de corpos $F\supset K$ tal que $F$ é uma extensão algébrica finita de $K(x)$
com $x\in F$ e transcendente sobre $K$.
O conjunto $\tilde{K}=\\{a\in F;a$ é algébrico em $K\\}$ é um subcorpo de $F$
e mais $F/\tilde{K}$ é um corpo de funções algébricas sobre $\tilde{K}$. Esse
conjunto é chamado de corpo de constantes de $F/K$. Neste trabalho vamos supor
sempre que $K=\tilde{K}$.
###### Exemplo 1.3.2 (Corpo de Funções racionais).
O corpo de funções $F/K$ é dito racional se $F=K(x)$ para algum $x$ que é
transcendente sobre $K$, onde $K(x)$ é o corpo de frações do anel de
polinômios, $K[x]$, em uma variável sobre o corpo $K$.
Qualquer elemento não nulo, $z\in K(x)$, tem uma única representação
$z=a\prod_{i}P_{i}(x)^{n_{i}}$
com $0\neq a\in K$, $P_{i}(x)\in K[x]$ mônicos irredutíveis distintos e
$n_{i}\in\mathbb{Z}$.
A próxima definição nos traz um tipo de anel fundamental para o
desenvolvimento do nosso trabalho.
###### Definição 1.3.3.
Um anel de valorização de um corpo de funções $F/K$ é um sub-anel
$\mathcal{O}\subset F$ com as seguintes propriedades:
1. 1.
$K\varsubsetneq\mathcal{O}\varsubsetneq F$;
2. 2.
Para qualquer $z\in F$, $z\in\mathcal{O}$ ou $z^{-1}\in\mathcal{O}$.
###### Exemplo 1.3.4.
No corpo de funções racionais $K(x)$ aparecem os primeiros anéis de
valorizações fundamentais, a saber:
Seja $p(x)\in K[x]$, um polinômio irredutível. Definimos o conjunto
$\mathcal{O}_{p(x)}=\left\\{\dfrac{f(x)}{g(x)};f(x),g(x)\in K[x],\,\,p(x)\nmid
g(x)\right\\}.$
Assim definido, $\mathcal{O}_{p(x)}$ é um anel de valorização de $K(x)/K$.
Observe que se $q(x)$ for outro polinômio irredutível, não associado a $p(x)$,
temos que $\mathcal{O}_{p(x)}\neq\mathcal{O}_{q(x)}.$
Os anéis de valorizações são caracterizados, segundo sua estrutura, pelos
resultados:
###### Proposição 1.3.5.
Seja $\mathcal{O}$ um anel de valorização do corpo de funções $F/K$ então:
1. a.
$\mathcal{O}$ é anel local, isto é $\mathcal{O}$ tem um único ideal maximal
$P=\mathcal{O}\setminus\mathcal{O}^{*}$, onde $\mathcal{O}^{*}$ é o conjunto
das unidades de $\mathcal{O}$;
2. b.
Para $0\neq x\in F,\,\,x\in P\Leftrightarrow x^{-1}\notin\mathcal{O}$;
3. c.
Para o corpo de constantes $\tilde{K}$ de $F/K$ temos que
$\tilde{K}\subset\mathcal{O}$ e $\tilde{K}\cap P={0}.$
Do item (b), segue que $\mathcal{O}$ é unicamente determinado por $P$. De
fato, $\mathcal{O}_{P}:=\mathcal{O}=\\{x\in F;x\notin P\\}.$
###### Teorema 1.3.6.
Seja $\mathcal{O}$ um anel de valorização do corpo de funções $F/K$ e $P$ o
seu ideal maximal. Então,
1. a.
$P$ é principal;
2. b.
Se $P=t\mathcal{O}$ então qualquer $0\neq z\in F$ tem uma única representação
da forma $z=t^{n}u$, para algum $n\in\mathbb{Z}$ e $u\in\mathcal{O}^{*}$;
3. c.
$\mathcal{O}$ é um domínio principal.
Os anéis de valorarização nos levam a um outro conceito, a saber, o conceito
de lugar.
###### Definição 1.3.7.
1. 1.
Um lugar $P$ do corpo de funções $F/K$ é um ideal maximal de algum anel de
valorização de $F/K$. Qualquer elemento $t$ que gera $P$ ($P=t\mathcal{O}$) é
dito elemento principal de $P$;
2. 2.
$\mathbb{P}_{F}=\\{P;\,\,P$ é lugar em $F/K\\}$.
Também um lugar dá origem a uma função especial que definiremos a seguir:
###### Definição 1.3.8.
Uma valorização discreta de $F/K$ é uma função
$v:F\rightarrow\mathbb{Z}\cup\\{\infty\\}$ com as seguintes propriedades:
1. 1.
$v(x)=\infty\Leftrightarrow x=0$;
2. 2.
$v(xy)=v(x)+v(y)$, para qualquer $x,y\in F$;
3. 3.
$v(x+y)\geqslant min\\{v(x),v(y)\\}$, para qualquer $x,y,\in F$;
4. 4.
Existe $z\in F$ com $v(z)=1$;
5. 5.
$v(a)=0$, para todo $a\in K$.
###### Definição 1.3.9.
Para $P\in\mathbb{P}_{F}$ associamos uma função de valorização,
$v_{P}:F\rightarrow\mathbb{Z}\cup\\{\infty\\}$, da seguinte maneira: dado um
elemento principal $t\in P$ temos que todo elemento $0\neq z$ em $F$ é escrito
de maneira única na forma $z=t^{n}u$, com $n\in\mathbb{Z}$ e
$u\in\mathcal{O}_{P}^{*}$. Assim fazemos $v_{P}(z)=n$ e $v_{P}(0)=\infty$.
Um primeiro resultado importante a respeito de $v_{P}$ é dado no seguinte
teorema:
###### Teorema 1.3.10.
Seja $F/K$ um corpo de funções então:
1. a.
Para qualquer $P\in\mathbb{P}_{F}$, a função de valorização $v_{P}$ definida
como acima é uma valorização discreta que satisfaz:
1. a.1.
$\mathcal{O}_{P}=\\{z\in F;v_{P}(z)\geqslant 0\\};$
2. a.2.
$\mathcal{O}_{P}^{*}=\\{z\in F;v_{P}(z)=0\\};$
3. a.3.
$P=\\{z\in F;v_{P}(z)>0\\};$
4. a.4.
Um elemento $x\in F$ é principal de $P$ se, e somente se, $v_{P}(x)=1$.
2. b.
Qualquer anel de valorização $\mathcal{O}$ de $F/K$ é um subanel maximal de
$F$.
Definiremos agora alguns elementos da teoria que serão muito utilizados em
nossa explanação.
###### Definição 1.3.11.
Sejam $P\in\mathbb{P}_{F}$ e $F_{P}=\mathcal{O}_{P}/P$ o corpo de resíduos de
$P$. A aplicação $x\mapsto x(P)$ de $F$ em $F_{P}\cup\\{\infty\\}$, onde
$x(P)=x+P$ se $x\in\mathcal{O}_{P}$ e $x(P)=\infty$ se
$x\notin\mathcal{O}_{P}$, é chamada de aplicação de resíduos com respeito a
$P$.
Pode-se provar que (veja em [7]) se $P\in\mathbb{P}_{F}$ então $K\subset
F_{P}$ e $F_{P}/K$ é uma extensão finita e assim definimos:
###### Definição 1.3.12.
Se $P\in\mathbb{P}_{F}$ definimos o grau de $P$ como sendo:
$gr(P)=[F_{P}:K].$
Os conceitos de zeros e pólos são de importância fundamental para a descrição
dos códigos algébricos geométricos.
###### Definição 1.3.13.
Seja $z\in F$ e $P\in\mathbb{P}_{F}$. Dizemos que $P$ é um zero de $z$ se, e
somente se, $v_{P}(z)>0$, e $P$ é um pólo de $z$ se, e somente se,
$v_{P}(z)<0$. Se $v_{P}(z)=m>0$ falamos que $P$ é um zero de ordem $m$ e se
$v_{P}(z)=-m<0$ dizemos que é um pólo de ordem $m$.
O próximo resultado, chamado de teorema da aproximação fraca, vai ser útil na
descrição de um dos códigos que iremos definir.
###### Teorema 1.3.14 (Aproximação Fraca).
Sejam $\,\,F/K\,\,$ um corpo de funções, $P_{1},\ldots,P_{n}\in\mathbb{P}_{F}$
lugares de $F/K$, dois a dois distintos, $x_{1},\ldots,x_{n}\in F$ e
$r_{1},\ldots,r_{n}\in\mathbb{Z}$. Então, existe $x\in F$ tal que:
$v_{P_{i}}(x-x_{i})=r_{i},\textrm{ para }i=1,\ldots,n.$
###### Corolário 1.3.15.
O corpo de funções possui infinitos lugares.
### 1.4 O Corpo de Funções Racionais
Como definimos anteriormente, o corpo de funções racionais é o próprio corpo
de frações $K(x)/K$. Já tínhamos definido o anel de valorização
$\mathcal{O}_{p(x)}=\left\\{\dfrac{f(x)}{g(x)};f(x),g(x)\in K[x],\,\,p(x)\nmid
g(x)\right\\},$
para um polinômio $p(x)\in K[x]$ irredutível. Observamos agora que para esse
anel, o ideal
$P_{p(x)}=\left\\{\dfrac{f(x)}{g(x)};f(x),g(x)\in K[x],\,\,p(x)|f(x),p(x)\nmid
g(x)\right\\},$
é seu único ideal maximal. Consideremos agora um outro anel de valorização
$\mathcal{O}_{\infty}=\left\\{\dfrac{f(x)}{g(x)};f(x),g(x)\in
K[x],\,\,gr(f(x))\leqslant gr(g(x))\right\\},$
que tem como ideal maximal
$P_{\infty}=\left\\{\dfrac{f(x)}{g(x)};f(x),g(x)\in
K[x],\,\,gr(f(x))<gr(g(x))\right\\}.$
Na realidade estes são os únicos anéis de valorizações de $K(x)$ e tal fato é
expresso no resultado:
###### Teorema 1.4.1.
Os únicos lugares de $K(x)/K$ são os da forma $P_{p(x)}$ e $P_{\infty}$ como
acima definidos.
### 1.5 Divisores
Nessa seção estaremos preocupados com a definição dos divisores de um corpo de
funções, os quais serão fundamentais para a construção de alguns dos códigos
que trataremos neste trabalho.
###### Definição 1.5.1.
O grupo aditivo abeliano livre que tem como base livre os lugares de $F/K$ é
denotado por $\mathcal{D}_{F}$ e é chamado de grupo de divisores de $F/K$. Os
elementos de $\mathcal{D}_{F}$ são chamados de divisores e são da forma
$D=\sum_{P\in\mathbb{P}_{F}}n_{P}P,\textrm{ com }n_{P}\in\mathbb{Z}\textrm{,
\'{u}nico e quase sempre nulo}.$
Chamamos de suporte de um divisor $D$ o conjunto dos lugares $P$ tais que
$n_{P}\neq 0$ e denotamos por $supp(D)$. Dados dois divisores $D=\sum n_{P}P$
e $D^{\prime}=\sum n^{\prime}_{P}P$ a soma deles é dada por:
$D+D^{\prime}=\sum_{P\in\mathbb{P}_{F}}(n_{P}+n^{\prime}_{P})P.$
E mais, para cada $P\in\mathbb{P}_{F}$, definimos $v_{P}(D)=n_{P}$, assim
também podemos definir uma ordem parcial de $\mathcal{D}_{F}$ por
$D_{1}\leqslant D_{2}\Leftrightarrow v_{P}(D_{1})\leqslant
v_{P}(D_{2}),\,\,\forall P\in\mathbb{P}_{F}.$
Chamamos de grau de um divisor $D$ ao número
$gr(D)=\sum_{P\in\mathbb{P}_{F}}v_{P}(D)gr(P).$
Definiremos agora os divisores principais, os quais trazem importantes
conseqüências para a teoria. Pode-se provar, veja em [7], que se $x\in F$
então o número de zeros e pólos de $x$ é finito e esse fato nos leva a
seguinte definição:
###### Definição 1.5.2.
Seja $0\neq x\in F$ e denotemos por $Z$ e $N$ o conjunto de zeros e pólos de
$x$ em $\mathbb{P}_{F}$, respectivamente, assim definimos:
1. 1.
$(x)_{0}=\sum_{P\in Z}v_{P}(x)P$, o divisor de zeros de $x$;
2. 2.
$(x)_{\infty}=\sum_{P\in N}(-v_{P}(x))P$, o divisor de pólos de $x$;
3. 3.
$(x)=(x)_{0}-(x)_{\infty}$, o divisor principal de $x$.
###### Definição 1.5.3.
Definimos o grupo de divisores principais de $F/K$ como sendo
$\mathcal{P}_{F}=\\{(x);0\neq x\in F\\}$ (este é um subgrupo de
$\mathcal{D}_{F}$ desde que para $0\neq x,y\in F,\,(xy)=(x)+(y)$). O quociente
$\mathcal{C}_{F}=\mathcal{D}_{F}/\mathcal{P}_{F}$ é chamado grupo de classe de
divisores e dizemos que $D$ é equivalente a $D^{\prime}$, denotando por $D\sim
D^{\prime}$, se $[D]=[D^{\prime}]$, isto é, se $D^{\prime}=(x)+D$ para algum
$x\in\mathbb{F}$.
Agora vamos definir um espaço vetorial associado a um divisor, o qual nos dará
a definição de dimensão desse divisor e também de um invariante muito
importante para a teoria.
###### Definição 1.5.4.
Para um divisor $A\in\mathcal{D}_{F}$ definimos o conjunto
$\mathcal{L}(A)=\\{x\in F;(x)\geqslant-A\\}\cup\\{0\\}.$
###### Lema 1.5.5.
Seja $A\in\mathcal{D}_{F}$ então temos
1. a.
$\mathcal{L}(A)$ é um $K-$espaço vetorial, de dimensão finita;
2. b.
Se $A^{\prime}\sim A$ então $\mathcal{L}(A^{\prime})\simeq\mathcal{L}(A)$.
###### Definição 1.5.6.
Para $A\in\mathcal{D}_{F}$, definimos a dimensão do divisor $A$ como sendo
$dim(A)=dim_{K}(\mathcal{L}(A))$.
###### Teorema 1.5.7.
Qualquer divisor principal tem grau zero. Mais precisamente dado $x\in
F\setminus K$ e $(x)_{0},\,(x)_{\infty}$ denotando os divisores de zeros e
pólos, respectivamente, do divisor de $x$, então:
$gr((x)_{0})=gr((x)_{\infty})=[F:K(x)].$
Como conseqüência imediata deste teorema temos:
###### Corolário 1.5.8.
1. 1.
Seja $A,\,A^{\prime}$ divisores tais que $A\sim A^{\prime}$. Assim temos que
$dim(A)=dim(A^{\prime})$ e que $gr(A)=gr(A^{\prime})$;
2. 2.
Se $gr(A)<0$ então $dim(A)=0$;
3. 3.
Para um divisor $A$ de grau zero , temos que são equivalentes:
1. (a)
$A$ é um divisor principal;
2. (b)
$dim(A)>0;$
3. (c)
$dim(A)=1.$
A partir da proposição que enuciaremos a seguir definimos o gênero de um corpo
de funções algébricas que é um invariante que depende apenas do corpo.
###### Proposição 1.5.9.
Existe uma constante $\gamma\in\mathbb{Z}$ tal que, para todo divisor
$A\in\mathcal{D}_{F}$ temos que:
$gr(A)-dim(A)\leqslant\gamma.$
###### Definição 1.5.10.
Definimos o gênero do corpo de funções $F/K$ como sendo
$g=max\\{gr(A)-dim(A)+1;A\in\mathcal{D}_{F}\\}.$
O primeiro resultado envolvendo o gênero de um corpo de funções é o seguinte:
###### Teorema 1.5.11 (Teorema de Riemann).
Seja $F/K$ uma corpo de funções algébricas com gênero $g$ então:
1. 1.
Para qualquer divisor $A\in\mathcal{D}_{F}$, $dim(A)\geqslant gr(A)+1-g$;
2. 2.
Existe um inteiro $c$, dependendo de $F/K$ tal que $dim(A)=gr(A)+1-g$ sempre
que $gr(A)\geqslant c$.
### 1.6 O Teorema de Riemann-Roch
Aqui $F/K$ denota um corpo de funções algébricas com gênero $g$.
###### Definição 1.6.1.
Para $A\in\mathcal{D}_{F}$ definimos o índice de especialidade de $A$ como
sendo:
$i(A)=dim(A)-gr(A)+g-1.$
Como se vê o Teorema de Riemann garante que o índice de especialidade de um
divisor $A$ é inteiro não negativo e na verdade nós vamos apresentar $i(A)$
como a dimensão de certos espaços vetoriais. Assim para isto começamos com as
definições abaixo:
###### Definição 1.6.2.
Um adele de $F/K$ é uma função
$\begin{array}[]{rccl}\alpha:&\mathbb{P}_{F}&\rightarrow&F\\\
&P&\mapsto&\alpha_{P}\end{array}$
tal que $\alpha_{P}\in\mathcal{O}_{P}$ para quase todos $P\in\mathbb{P}_{F}$.
Podemos ver um adele como um elemento do produto direto
$\prod_{P\in\mathbb{P}_{F}}F$ e usamos a notação
$\alpha=(\alpha_{P})_{P\in\mathbb{P}_{F}}$, e para encurtar,
$\alpha=(\alpha_{P})$.
Chamamos de espaço de adeles de $F/K$ ao conjunto:
$\mathcal{A}_{F}=\\{\alpha;\,\,\alpha\textrm{ \'{e} adele de }F/K\\}.$
###### Definição 1.6.3.
1. 1.
Definimos como adele principal de um elemento $x\in F$ como sendo o adele
cujas as componentes são todas iguais a $x$ ou seja a seqüência constante
$(x)$.
2. 2.
Dado $P\in\mathbb{P}_{F}$ e $x\in F$, o adele cuja as componentes são nulas a
menos da componente $P$, a qual é $x$, será denotado por $\iota_{P}(x)$.
###### Definição 1.6.4.
A um adele $\alpha$, associamos uma função de valorização discreta dada por:
$v_{P}(\alpha):=v_{P}(\alpha_{P})$.
###### Definição 1.6.5.
Para um divisor $A\in\mathcal{D}_{F}$ definimos o conjunto:
$\mathcal{A}_{F}(A)=\\{\alpha\in\mathcal{A}_{F};\,\,v_{P}(\alpha)\geqslant-
v_{P}(A),\,\,\forall P\in\mathbb{P}_{F}\\}.$
Facilmente pode se ver que o conjunto acima definido é um $K-$subespaço
vetorial de $\mathcal{A}_{F}$.
No próximo teorema explicitamos um espaço vetorial cuja a dimensão é o índice
de especialidade do divisor $A$.
###### Teorema 1.6.6.
Dado um divisor $A$, o seu índice de especialidade é dado por:
$i(A)=dim_{K}(\mathcal{A}_{F}/(\mathcal{A}_{F}(A)+F)).$
E como conseqüência imediata temos:
###### Corolário 1.6.7.
$g=dim_{K}(\mathcal{A}_{F}/(\mathcal{A}_{F}(0)+F)).$
Agora veremos o conceito de diferenciais de Weil, o qual nos dará mais
informações sobre o indice de especialidade de um divisor.
###### Definição 1.6.8.
Uma aplicação $K-$linear, $\omega:\mathcal{A}_{F}\rightarrow K$, que se anula
em $\mathcal{A}_{F}(A)+F$ para algum divisor $A\in\mathcal{D}_{F}$ é dita
diferencial de Weil de $F/K$ . Denotamos por $\Omega_{F}$ ao conjunto dos
diferenciais de Weil de $F/K$ e por $\Omega_{F}(A)$ ao conjunto de
diferenciais de Weil de $F/K$ que se anulam em $\mathcal{A}_{F}(A)+F$.
O conceito de divisor canônico é de fundamental importância para o Teorema de
Riemann-Roch e ele é definido a partir da próxima definição e do lema
seguinte.
###### Definição 1.6.9.
Para um diferencial de Weil $\omega\neq 0$ definimos o conjunto de divisores:
$M(\omega)=\\{A\in\mathcal{D}_{F};\,\,\omega\textrm{ se anula em
}\mathcal{A}_{F}(A)+F\\}.$
###### Lema 1.6.10.
Seja $\omega\in\Omega_{F},\,\,\omega\neq 0$, então existe um único divisor
$W\in M(\omega)$ tal que $A\leqslant W$ para todo $A\in M(\omega)$.
###### Definição 1.6.11 (Divisor Canônico).
1. 1.
O divisor $(\omega)$ de um diferencial de Weil $\omega\neq 0$ é o divisor de
$F/K$ unicamente determinado por:
1. (a)
$\omega$ se anula em $\mathcal{A}_{F}((\omega))+F$;
2. (b)
Se $\omega$ se anula em $\mathcal{A}_{F}(A)+F$ então $A\leqslant(\omega).$
2. 2.
Para $0\neq\omega\in\Omega_{F}$ e $P\in\mathbb{P}_{F}$ definimos
$v_{P}(\omega)=v_{P}((\omega))$;
3. 3.
Um lugar $P$ é dito zero (resp. pólo) de $\omega$ se $v_{P}(\omega)>0$ (resp.
$v_{P}(\omega)<0$). $\omega$ é chamado de regular em $P$ se
$v_{P}(\omega)\geqslant 0$, e simplesmente de regular se for regular em todos
os lugares em $\mathbb{P}_{F}$;
4. 4.
Um divisor $W$ é dito divisor canônico de $F/K$ se $W=(\omega)$ para algum
$\omega\in\Omega_{F}$.
###### Proposição 1.6.12.
1. 1.
Para $0\neq x\in F$ e $0\neq\omega\in\Omega_{F}$ temos que
$(x\omega)=(x)+(\omega)$;
2. 2.
Quaisquer dois divisores canônicos são equivalentes.
Esse próximo teorema vai nos permitir calcular o indice de especialidade de um
divisor (mais uma vez como dimensão de um espaço vetorial).
###### Teorema 1.6.13.
Seja $A$ um divisor arbitrário e $W=(\omega)$ um divisor canônico de $F/K$.
Então a aplicação
$\begin{array}[]{rccl}\mu:&\mathcal{L}(W-A)&\rightarrow&\Omega_{F}(A)\\\
&x&\mapsto&x\omega\end{array},$
é um isomorfismo de $K-$espaços Vetoriais, e mais $i(A)=dim(W-A)$.
Em fim chegamos ao principal teorema da teoria até aqui apresentada.
###### Corolário 1.6.14 (Riemann-Roch).
Seja $W$ um divisor canônico de $F/K$. Assim para qualquer
$A\in\mathcal{D}_{F}$ temos que:
$dim(A)=gr(A)+1-g+dim(W-A).$
###### Teorema 1.6.15.
Sendo $A$ um divisor de $F/K$ de grau maior ou igual que $2g-1$, temos:
$dim(A)=gr(A)+1-g.$
A próxima definição nos será bastante útil para o entendimento de um de nossos
códigos.
###### Definição 1.6.16 (Componente Local).
Seja um diferencial de Weil $\omega\in\Omega_{F}$. Definimos a componente
local de $\omega$ como sendo a aplicação $K-linear$
$\begin{array}[]{rccl}\omega_{P}:&F&\rightarrow&K\\\
&x&\mapsto&\omega(\iota_{P}(x))\end{array}.$
Sobre as componentes locais temos os seguintes resultados:
###### Proposição 1.6.17.
Sejam um diferencial de Weil $\omega\in\Omega_{F}$ e um adele
$\alpha=(\alpha_{P})\in\mathcal{A}_{F}$. Então $\omega_{P}(\alpha_{P})\neq 0$
no máximo em finitos lugares $P$, e mais,
$\omega(\alpha)=\sum_{P\in\mathbb{P}_{F}}\omega_{P}(\alpha_{P}).$
Em particular,
$\sum_{P\in\mathbb{P}_{F}}\omega_{P}(1)=0.$
O próximo resultado nos mostra que um diferencial de Weil é unicamente
determinado em relação aos seus componentes locais.
###### Proposição 1.6.18.
1. 1.
Seja $\omega\neq 0$ um diferencial de Weil de $F/K$, $P\in\mathbb{P}_{F}$ e
$W=(\omega)$. Então:
$v_{P}(W)=max\\{r\in\mathbb{Z};\,\omega_{P}(x)=0\textrm{ para todo }x\in
F\textrm{ com }v_{P}(x)\geqslant-r\\}.$
2. 2.
Se $\omega,\,\omega^{\prime}\in\Omega_{F}$ e $\omega_{P}=\omega_{p}^{\prime}$
para algum $P\in\mathbb{P}_{F}$, então $\omega=\omega^{\prime}$.
### 1.7 Curvas Algébricas
Nessa seção apresentaremos as curvas algébricas e sua ligação com os corpos de
funções algébricas.
###### Definição 1.7.1.
Seja $I$ um ideal em $\mathbb{F}[x_{1},\cdots,x_{n}]$ (anel de polinômios em
$n$ indeterminadas com coeficientes sobre um corpo $\mathbb{F}$). Definimos o
conjunto algébrico obtido a partir de $I$ como sendo:
$V(I)=\\{a=(a_{1},\cdots,a_{n})\in\mathbb{F}^{n};\,\,f(a)=0,\,\forall f\in
I\\}.$
Dizemos que um conjunto algébrico $B$ é irredutível se não pode ser escrito
como união de dois outros conjuntos algébricos próprios. Temos que $V(I)$ é
irredutível quando o radical de $I$ é ideal primo. Tal resultado nos leva as
duas definições abaixo:
###### Definição 1.7.2.
1. 1.
Dado um ideal primo $I\,\subset\,\mathbb{F}[x_{1},\,\cdots\,,x_{n}]$ o
conjunto $\mathcal{X}\,=\,V(I)$ é dito variedade afim.
2. 2.
O anel $\mathbb{F}[x_{1},\cdots,x_{n}]/I=\mathbb{F}[\mathcal{X}]$ é dito anel
de coordenadas de $\mathcal{X}$.
###### Definição 1.7.3.
Dada uma variedade algébrica $\mathcal{X}$ definimos o corpo de funções
racionais de $\mathcal{X}$ como sendo o corpo de frações do anel de
coordenadas $\mathbb{F}[\mathcal{X}]$ que é denotado por
$\mathbb{F}(\mathcal{X})$.
Um resultado clássico da álgebra comutativa (Teorema de normalização de
Noether) nos diz que podemos dar a dimensão da variedade algébrica da seguinte
forma:
###### Definição 1.7.4.
Definimos a dimensão da variedade $\mathcal{X}$ como sendo o grau de
transcendência de $\mathbb{F}(\mathcal{X})/\mathbb{F}$.
Para um ponto $P\in\mathcal{X}$, o conjunto
$\mathcal{O}_{P}(\mathcal{X})=\left\\{f\in\mathbb{F}(\mathcal{X});f=\dfrac{g}{h},\,\,g,h\in\mathbb{F}[\mathcal{X}]\textrm{
e }h(P)\neq 0\right\\},$
é um anel local que tem como corpo de frações o próprio
$\mathbb{F}(\mathcal{X})$ e mais o seu ideal maximal é dado por:
$M_{P}(\mathcal{X})=\left\\{f\in\mathbb{F}(\mathcal{X});f=\dfrac{g}{h},\,\,g,h\in\mathbb{F}[\mathcal{X}],g(P)=0\textrm{
e }h(P)\neq 0\right\\}.$
#### 1.7.1 Variedades Projetivas
###### Definição 1.7.5.
Dado um corpo $\mathbb{F}$, no conjunto $\mathbb{F}^{n+1}\setminus\\{0\\}$
definimos a relação de equivalência $\sim$ definida por: dados dois vetores
$v=(v_{0},v_{1},\cdots,v_{n})$ e
$w=(w_{0},w_{1},\cdots,w_{n})\in\mathbb{F}^{n+1}\setminus\\{0\\}$ eles são
equivalentes se forem linearmente dependentes sobre $\mathbb{F}$, ou seja,
$v\sim w$ se, e somente se, existe $\lambda\in\mathbb{F}$ tal que $v=\lambda
w$.
O conjunto quociente $(\mathbb{F}^{n+1}\setminus\\{0\\})/\sim$ das classes de
equivalência segundo a relação $\sim$, é chamado espaço projetivo de dimensão
$n$ e denotado por $\mathbf{P}^{n}(\mathbb{F})$ e seus elementos são denotados
por $(a_{0}:a_{1}:\cdots:a_{n})$.
###### Definição 1.7.6.
Dizemos que um polinômio $F\in\mathbb{F}[x_{1},\cdots,x_{n}]$ é homogêneo se
esse for soma de monômios de mesmo grau. Um ideal gerado por polinômios
homogêneos é chamado de ideal homogêneo.
Observe que dados um ponto
$P=(a_{0}:a_{1}:\cdots:a_{n})=(b_{0}:b_{1}:\cdots:b_{n})\in\mathbf{P}^{n}(\mathbb{F})$
e um polinômio homogêneo $F\in\mathbb{F}[x_{0},\cdots,x_{n}]$ podemos definir
$F(P)=0$ se $F(a_{0},a_{1},\cdots,a_{n})=0$, já que
$F(a_{0},a_{1},\cdots,a_{n})=0$ se, e somente se,
$F(b_{0},b_{1},\cdots,b_{n})=0$. Assim, podemos definir o que seja uma
variedade algébrica projetiva.
###### Definição 1.7.7.
Um subconjunto $\mathcal{X}\subset\mathbf{P}^{n}(\mathbb{F})$ é dito uma
variedade algébrica projetiva se for o conjunto de zeros de um ideal homogêneo
$I\subset\mathbb{F}[x_{0},x_{1},\cdots,x_{n}]$, ou seja:
$\mathcal{X}=V(I)=\\{P\in\mathbf{P}^{n}(\mathbb{F});\,\,F(P)=0,\,\forall F\in
I\\}.$
Uma variedade algébrica projetiva $\mathcal{X}=V(I)$ é irredutível se, e
somente se, o ideal $I$ for um ideal homogêneo e o seu radical for primo.
O anel de coordenadas
$\mathbb{F}_{h}[\mathcal{X}]=\mathbb{F}[x_{0},\cdots,x_{n}]/I$ é dito anel de
coordenadas homogêneas e os elementos que são do formato $f=F+I$ com
$F\in\mathbb{F}[x_{0},\cdots,x_{n}]$ e $F$ homogenea,são chamados de forma de
grau $d$ onde $d=gr(F)$.
Faremos agora a associação entre as curvas algébricas e os corpos de funções.
###### Definição 1.7.8.
Se $\mathcal{X}$ é uma variedade algébrica projetiva definimos o corpo de
funções de $\mathcal{X}$ como sendo:
$\mathbb{F}(\mathcal{X})=\left\\{\frac{g}{h};\,\,g,h\in\mathbb{F}_{h}[\mathcal{X}]\textrm{
formas de mesmo grau e }h\neq 0\right\\}.$
A dimensão da variedade $\mathcal{X}$ é dada pelo grau de transcendência de
$\mathbb{F}(\mathcal{X})/\mathbb{F}$.
###### Definição 1.7.9.
Dado um ponto $P\in\mathcal{X}$ e $f=\frac{g}{l}\in\mathbb{F}(\mathcal{X})$
com $g,l\in\mathbb{F}_{h}[\mathcal{X}]$, dizemos que $f$ está definida em $P$
se $l(P)\neq 0$, $f(P)$ é dito valor de $f$ em $P$.
O anel
$\mathcal{O}_{P}(\mathcal{X})=\\{f\in\mathbb{F}(\mathcal{X});\,\,f\textrm{
\'{e} definida em }P\\}\subset\mathbb{F}(\mathcal{X})$ é um anel local com
ideal maximal
$M_{P}(\mathcal{X})=\\{f\in\mathcal{O}_{P}(\mathcal{X});\,\,f(P)=0\\}$.
###### Definição 1.7.10.
Seja $F\in\mathbb{F}[x_{1},\cdots,x_{n}]$ com grau total de $F$ igual a $l$,
definimos a homogeneização de $F$ como sendo:
$F^{*}=x_{0}^{l}F\left(\frac{x_{1}}{x_{0}},\cdots,\frac{x_{n}}{x_{0}}\right).$
###### Definição 1.7.11.
Uma variedade projetiva (afim) $\mathcal{X}$ de dimensão 1 é chamada de curva
algébrica irredutivel projetiva (afim). Desse modo o corpo de funções
racionais em $\mathcal{X}$, $\mathbb{F}(\mathcal{X})$, é um corpo de funções
algébricas de uma variável como na definição 1.3.1. Mais ainda, dizemos que a
curva algébrica projetiva (afim) é plana se
$\mathcal{X}\subset\mathbf{P}^{2}(\mathbb{F})$ (ou $\mathbb{F}^{2}$).
#### 1.7.2 Curvas Não Singulares
Levando-se em conta que os códigos geométricos de Goppa são gerados pelas
curvas planas não singulares, a seguir apresentaremos a definição de curvas
não singulares.
###### Definição 1.7.12.
Seja $\mathcal{X}$ uma curva algébrica definida pelo polinômio
$F\in\mathbb{F}(x,y)$ e $P$ um ponto em $\mathcal{X}$. Dizemos que o ponto $P$
é um ponto não singular se pelo menos uma das derivadas parciais de $F$
aplicadas nesse ponto é não nula, ou seja, $F_{x}(p)\neq 0$ ou $F_{y}(P)\neq
0$. Se todos os pontos da curva forem não singulares dizemos apenas que a
curva é não singular (ou regular).
Observamos que aqui que a derivada parcial de um polinômio é a sua derivada
formal.
## Chapter 2 Códigos Algébricos Geométricos
Neste capítulo estamos interessados em construir os códigos algébricos
geométricos, conhecidos como Códigos de Goppa Geométricos, e então extrair uma
cota para a sua distância mínima e calcular sua dimensão. Aqui
$\mathbb{F}_{q}$ denotará um corpo com $q$ elementos.
### 2.1 Códigos Algébricos Geométricos
###### Definição 2.1.1.
Seja $\mathbb{F}_{q}=\\{\alpha_{0},\,\cdots,\,\alpha_{q-1}\\}$ e considere o
conjunto, $\mathcal{L}_{k}\subset\mathbb{F}_{q}[x]$, dos polinômios com grau
menor que $k$ e $k\leqslant q$, definimos um código de Reed-Solomon de tamanho
$n=q$ como sendo:
$C_{k}=\\{c(f)=(f(\alpha_{0}),\cdots,f(\alpha_{q-1}));\,f\in\mathcal{L}_{k}\\}.$
###### Proposição 2.1.2.
Como acima definido, $C_{k}$ é um código MDS (Maximum Distance Separable), ou
seja, tem distância mínima $d=n-k+1$.
###### Proof.
Seja $f\in\mathcal{L}_{k}$, temos que $f$ tem grau no máximo $k-1$, assim $f$
admite no máximo $k-1$ raízes distintas em $\mathbb{F}_{q}$, desse modo o peso
de $c(f)$ é no mínimo $n-k+1$, ou seja, $d\geqslant n-k+1$ no entanto temos
que o polinômio
$g(x)=(x-\alpha_{i_{1}})\cdots(x-\alpha_{i_{k-1}})\in\mathcal{L}_{k}$ tem grau
$k-1$ e tem exatamente $k-1$ raízes distintas em $\mathbb{F}_{q}$, desse modo
$d(c(g))=n-k+1$. ∎
A seguir utilizaremos as seguintes notações:
1. $\bigstar$
$F/\mathbb{F}_{q}$, um corpo de funções algébricas de gênero $g$;
2. $\bigstar$
$P_{1},P_{2},\cdots,P_{n}$, lugares de grau 1 dois a dois distintos em
$F/\mathbb{F}_{q}$;
3. $\bigstar$
$D=\sum_{i=1}^{n}P_{i}$, um divisor em $F/\mathbb{F}_{q}$;
4. $\bigstar$
$G$, um divisor em $F/\mathbb{F}_{q}$ tal que $supp(G)\cap supp(D)=\emptyset$.
A partir das notações dadas acima e da definição 1.3.11 estamos em condição de
definir o que vem a ser um código gemétrico de Goppa.
###### Definição 2.1.3 (Códigos Geométricos de Goppa).
Definimos o código algébrico geométrico (código geométrico de Goppa) associado
aos divisores $D$ e $G$ como sendo:
$C(D,G)=\\{c(x)=(x(P_{1}),x(P_{2}),\cdots,x(P_{n}));x\in\mathcal{L}(G)\\}.$
###### Teorema 2.1.4.
O código de Goppa $C(D,G)$ é um $[n,k,d]-$código tal que:
$k=dim(G)-dim(G-D)\textrm{ e }d\geqslant n-gr(G).$
###### Proof.
Seja a aplicação de avaliação
$\begin{array}[]{rccl}ev_{D}:&\mathcal{L}(G)&\rightarrow&C(D,G)\\\
&x&\mapsto&c(x)=((x(P_{1}),\cdots,x(P_{n}))\end{array}.$
Sabemos que $ev_{D}$ é sobrejetiva logo
$\dfrac{\mathcal{L}(G)}{Ker(ev_{D})}\simeq C(D,G)$. Queremos agora mostrar que
$Ker(ev_{D})=\mathcal{L}(G-D)$. Seja $x\in Ker(ev_{D})$, temos que para todo
$Q\in\mathbb{P}_{F}$ $v_{Q}(x)\geqslant-v_{Q}(G)$ e $v_{P_{i}}(x)>0$ para
$i=1,\cdots,n$ e mais, como $supp(G)\cap supp(D)=\emptyset$, temos que,
$v_{Q}(x)\geqslant-v_{Q}(G)+v_{Q}(D)$ assim
$Ker(ev_{D})\subset\mathcal{L}(G-D)$.
Tomemos agora $x\in\mathcal{L}(G-D)$, temos que $v_{P_{i}}(x)\geqslant-
v_{P_{i}}(G)+v_{P_{i}}(D)=1$ para $i=1,\cdots,n$, logo $x\in\mathcal{L}(G)$ e
$v_{P_{i}}(x)>0$ ou seja $x(P_{i})=0$ em $F_{P_{i}}$ o que implica que
$\mathcal{L}(G-D)\subset Ker(ev_{D})$ e pelo teorema dos isomorfismos (álgebra
linear) concluímos que $k=dim(G)-dim(G-D).$
Agora acharemos uma cota para a distância mínima do código. Assumiremos que
$C(D,G)\neq\\{0\\}$, pois caso contrário não teriamos uma distância não nula.
Seja $x\in C(D,G)$ tal que $w(x)=d$, assim temos que existe
$y\in\mathcal{L}(G)$ tal que $w(ev_{D}(y))=d$, mas
$w(ev_{D}(y))=d=\\#\\{i;y(P_{i})\neq 0\\}$. Suponha agora que
$y(P_{1})=y(P_{2})=\cdots=y(P_{n-d})=0$, reordenando os $P_{i}$’s caso
necessário, logo
$0\neq y\in\mathcal{L}\left(G-\sum_{i=1}^{n-d}P_{i}\right)\Rightarrow
dim\left(G-\sum_{i=1}^{n-d}P_{i}\right)\neq 0\Rightarrow$ $0\leqslant
gr\left(G-\sum_{i=1}^{n-d}P_{i}\right)=gr(G)-n+d\Rightarrow$ $d\geqslant
n-gr(G).$
∎
###### Corolário 2.1.5.
Suponha que $gr(G)<n$ então $ev_{D}:\mathcal{L}(G)\rightarrow C(D,G)$ é
injetora e se $2g-2<gr(G)<n$ teremos que $k=gr(G)+1-g$.
###### Proof.
Temos que $gr(G-D)=gr(G)-n<0$, logo (por 1.5.8-2)
$\\{0\\}=\mathcal{L}(G-D)=Ker(ev_{D})$ logo $ev_{D}$ é injetora. Pelo teorema
1.6.15 temos que $k=gr(G)+1-g$. ∎
Esses resultados dão uma motivação para a seguinte definição:
###### Definição 2.1.6.
No código geométrico de Goppa $C(D,G)$ chamamos de distância designada ao
inteiro $d^{*}=n-gr(G)$.
Ao ver essa definição surge uma pergunta: Quando a distância designada é igual
a distância mínima do código? Responderemos essa pergunta na próxima
proposição.
###### Proposição 2.1.7.
Seja $C(D,G)$ um código com distância designada $d^{*}$, suponha que
$dim(G)>0$ e que $d^{*}>0$. então $d=d^{*}$ se, e somente se, existe um
divisor $D^{\prime}$ com $0\leqslant D^{\prime}\leqslant D$,
$gr(D^{\prime})=gr(G)$ e $dim(G-D^{\prime})>0$.
###### Proof.
Suponha que $d=d^{*}$. Seja $0\neq x\in\mathcal{L}(G)$ tal que
$w(ev_{D}(x))=d$. Assim, $ev_{D}(x)=(x(P_{1}),\ldots,x(P_{n}))$, reordenando
os $P_{i}^{\prime}s$ caso necessário, temos $x(P_{1})=\cdots=x(P_{gr(G)})=0.$
Seja $D^{\prime}=\sum_{i=1}^{gr(G)}P_{i}$, logo
$gr(D^{\prime})=\sum_{i=1}^{gr(G)}v_{P_{i}}(D^{\prime})=gr(G).$ Assim temos
que $0\leqslant D^{\prime}\leqslant D.$ Observe que $x\in
Ker(ev_{D^{\prime}})$ logo como na demonstração do teorema 2.1.4, tem-se
$dim(G-D^{\prime})>0.$
Queremos agora demonstrar a segunda parte da proposição. Seja
$D^{\prime}\in\mathcal{D}_{F}$ tal que $0\leqslant D^{\prime}\leqslant D$,
$gr(D^{\prime})=gr(G)$ e $dim(G-D^{\prime})>0$, então temos que existe
$y\in\mathcal{L}(G-D^{\prime})$, assim sendo, temos que o peso da palavra
código $(y(P_{1}),\ldots,y(P_{n}))$ é no máximo $n-gr(G)=d^{*}$, contudo $d$ é
a distância mínima no código e mais, como provado anteriormente
$d^{*}\leqslant d$, logo $d=d^{*}$. ∎
Agora definiremos o código que originalmente foi introduzido por V. D. Goppa
em 1981 no artigo “Codes on Algebraic Curves”.
###### Definição 2.1.8.
Seja $G$ e $D=P_{1}+\cdots+P_{n}$ divisores de $F/\mathbb{F}_{q}$ com os
$P_{i}^{\prime}s$ dois a dois disjuntos e $supp(G)\cap supp(D)=\emptyset$.
Definimos o código $C_{\Omega}(D,G)$ por:
$C_{\Omega}=\\{(\omega_{P_{1}}(1),\ldots,\omega_{P_{n}}(1))\in\mathbb{F}_{q}^{n};\,\,\omega\in\Omega(G-D)\\}.$
O próximo resultado tem como objetivo caracterizar o código $C_{\Omega}(D,G)$.
Mas antes disso apresentaremos um lema técnico.
###### Lema 2.1.9.
Sejam $F/\mathbb{F}_{q}$ um corpo de funções, $P\in\mathbb{P}_{F}$ com
$gr(P)=1$ e $\omega\in\Omega_{F}$ um diferencial de Weil tal que
$v_{P}(\omega)\geqslant-1$. Então $\omega_{P}(1)=0$ se, e somente se,
$v_{P}(\omega)\geqslant 0$.
###### Proof.
$\Leftarrow)$ Primeiro vamos supor que $v_{P}(\omega)\geqslant 0$, logo pela
proposição 1.6.18, temos que $\omega_{P}(x)=0$ para todo $x\in F$ com
$v_{P}(x)\geqslant 0$, temos que $1\in\mathbb{F}_{q}\subset F$ logo
$v_{P}(1)=0$ assim $\omega_{P}(1)=0$.
$\Rightarrow)$ Suponha agora que $\omega_{P}(1)=0$, assim temos que
$\omega_{P}(a)=a\omega_{P}(1)=0$ para todo $a\in\mathbb{F}_{q}$. Pelo teorema
1.3.14, existe $x\in F$ tal que $v_{P}(x)\geqslant 0$, ou seja
$x\in\mathcal{O}_{P}$. Como $gr(P)=1$ temos que
$\mathcal{O}_{P}/P=\mathbb{F}_{q}$, assim existem $y\in P$ e
$a\in\mathbb{F}_{q}$ tais que $x-y=a$ e mais, $v_{P}(y)\geqslant 1$ e
$v_{P}(a)=0$. Por hipótese temos que $v_{P}(\omega)\geqslant-1$ e como
$v_{P}(y)\geqslant 1$, pela proposição 1.6.18, temos que $\omega_{P}(y)=0$.
Assim temos que $\omega_{P}(x)=\omega_{P}(a+y)=\omega_{P}(a)+\omega_{P}(y)=0$,
ou seja, $v_{P}(\omega)\geqslant 0$ (novamente pela proposição 1.6.18). ∎
###### Observação 2.1.10.
A proposição 1.6.18 nos garante que $v_{P}(\omega)\geqslant r$ se, e somente
se, $\omega(x)=0$ para todo $x\in F$ com $v_{P}(x)\geqslant-r$.
Agora enuciaremos o teorema que caracteriza o código $C_{\Omega}(D,G)$.
###### Teorema 2.1.11.
O código $C_{\Omega}(D,G)$ é um $[n,k^{\prime},d^{\prime}]-$código com
parâmetros:
$k^{\prime}=i(G-D)-i(G)\textrm{ e }d^{\prime}\geqslant gr(G)-2g+2.$
E mais, se $gr(G)>2g-2$, temos que $k^{\prime}=i(G-D)\geqslant n+g-1-gr(G)$ e
se $2g-2<gr(G)<n$ então $k^{\prime}=n+g-1-gr(G).$
###### Proof.
Seja $\phi$ a seguinte aplicação:
$\begin{array}[]{rccl}\phi:&\Omega_{F}(G-D)&\rightarrow&C_{\Omega}(D,G)\\\
&\omega&\mapsto&(\omega_{P_{1}}(1),\ldots,\omega_{P_{n}}(1))\end{array},$
obviamente $\phi$ é sobrejetiva, logo $C_{\Omega}(D,G)$ é isomorfo a
$\Omega_{F}(G-D)/Ker(\phi)$.
Seja $\omega\in Ker(\phi)$, temos que $\omega_{P_{i}}(1)=0$ para $i=1\ldots
n$, assim pelo lema 2.1.9, $v_{P_{i}}(\omega)\geqslant 0$. Como
$\omega\in\Omega_{F}(G-D)$, temos que $(\omega)\geqslant G-D$. Observe que se
$P\in\\{P_{1},\ldots,P_{n}\\}$ temos que $v_{P}(G)=0$ e $v_{P}(-d)<0$ e mais,
para $P\notin\\{P_{1},\ldots,P_{n}\\}$ temos que $v_{P}(-D)=0$ assim
$\omega\in\Omega_{F}(G)$.
Seja agora $\omega\in\Omega_{F}(G)$, novamente pelo lema 2.1.9 temos que
$v_{P_{i}}(\omega)\geqslant v_{P_{i}}(G)=0$, logo $\omega\in Ker(\phi).$
Assim pelo isomorfismo visto acima temos que
$k^{\prime}=dim_{\mathbb{F}_{q}}(\Omega_{F}(G-D))-dim_{\mathbb{F}_{q}}(\Omega_{F}(G))=i(G-D)-i(G).$
Seja $\phi(\omega)\in C_{\Omega}(D,G)$, uma palavra com peso $m>0$, sem perda
de generalidade podemos supor que $v_{P_{1}}=\cdots=v_{P_{n-m}}$, desse modo
temos que $\omega\in\Omega_{F}(G-(D-\sum_{i=1}^{n-m}P_{i}))$.
Observe que
$i(G-(D-\sum_{i=1}^{n-m}))=dim_{\mathbb{F}_{q}}(G-(D-\sum_{i=1}^{n-m}))$ e
como $\Omega(G-(D-\sum_{i=1}^{n-m}))\neq\\{0\\}$ temos que
$i(G-(D-\sum_{i=1}^{n-m}))>0$, desse modo
$dim(G-(D-\sum_{i=1}^{n-m}))>gr(G-(D-\sum_{i=1}^{n-m}))-g+1$ o que é a contra
positíva do teorema 1.6.15, assim
$2g-2\geqslant
gr\left(G-\left(D-\sum_{i=1}^{n-m}\right)\right)=gr\left(G\right)-gr\left(D-\sum_{i=1}^{n-m}\right)=gr\left(G\right)-m$
concluimos então que $m\geqslant gr(G)-2g+2$, ou seja, $d^{\prime}\geqslant
gr(G)-2g+2$.
Assuma agora que $gr(G)>2g-2$. Pelo teorema 1.6.15, temos que $i(G)=0$, assim
$k^{\prime}=i(G-D)=dim(G-D)-gr(G-D)+g-1$, como $dim(G-D)\geqslant 0$ temos que
$k^{\prime}\geqslant n+g-1-gr(G)$.
Agora se $gr(G)<n$ temos que $gr(G-D)=gr(G)-n<0$ logo pelo corolário 1.5.8
$dim(G-D)=0$ assim $k^{\prime}=n+g-1-gr(G)$. ∎
O próximo teorema mostra a ligação entre o código geométrico de Goppa
($C(D,G)$) e o código que acabamos de definir $C_{\Omega}(D,G)$. Novamente
antes do teorema apresentaremos mais um lema técnico.
###### Lema 2.1.12.
Sejam $P\in\mathbb{P}_{F}$ tal que $gr(P)=1$, $\omega$ um diferencial de Weil
com $v_{P}(\omega)\geqslant-1$ e $x\in F$ com $v_{P}(x)\geqslant 0$, então
$\omega_{P}(x)=x(P)\omega_{P}(1).$
###### Proof.
Como $gr(P)=1$ temos que $\mathcal{O}_{P}/P=\mathbb{F}_{q}$, o fato de
$v_{P}(x)\geqslant 0$ nos dá que $x\in\mathcal{O}_{P}$, logo existe $y\in P$ e
$a\in\mathbb{F}_{q}$ tais que $x-y=a$, observe que $v_{P}(a)=0$ e
$v_{P}(y)\geqslant 1$. Como $v_{P}(y)\geqslant 1$ temos que $\omega_{P}(y)=0$,
assim
$\omega_{P}(x)=\omega_{P}(a+y)=\omega_{P}(a)+\omega_{P}(y)=a\omega_{P}(1).$
Observe agora que $a$ é a classe de resíduos de $x$ em relação a $P$, logo
$\omega_{P}(x)=x(P)\omega_{P}(1).$ ∎
Em fim o teorema que nos dá a relação entre os códigos.
###### Teorema 2.1.13.
Os códigos $C(D,G)$ e $C_{\Omega}(D,G)$ são duais entre si, ou seja,
$C_{\Omega}(D,G)=C(G,D)^{\bot}.$
###### Proof.
Primeiro vamos mostrar que a dimensão dos códigos ($C(D,G)^{\bot}$ e
$C_{\Omega}(D,G)$) são iguais.
Pelo teorema 2.1.11 temos que $dim(C_{\Omega}(D,G))=i(G-D)-i(G)$, agora pelo
corolário 1.6.14 (Riemann-Rock) temos que
$i(G-D)-i(G)=dim(G-D)+g-1-gr(G-D)-(dim(G)+g-1-gr(G))=dim(G-D)+gr(D)-dim(G)=n+dim(G-D)-dim(G)$,
pelo teorema 2.1.4 temos que $dim(C(D,G))=dim(G)-dim(G-D)$, assim
$dim(C_{\Omega}(D,G))=dim(C(D,G)^{\bot})$.
Agora mostraremos que $C_{\Omega}(D,G)\subset C(D,G)^{\bot}$, o que vai nos
garantir a igualdade desejada.
Seja $\omega\in\Omega_{F}(G-D)$ e $x\in\mathcal{L}(G)$. Identificando $x$ como
um adele principal temos que
$0=\omega(x)=\sum_{P\in\mathbb{P}_{F}}\omega_{P}(x)$ (pela proposição 1.6.17).
Para $P\in\mathbb{P}_{F}\setminus\\{P_{1},\ldots,P_{n}\\}$ temos que
$v_{P}(x)\geqslant-v_{P}(\omega)$, então pela observação 2.1.10 temos que
$v_{P}(x)=0$, assim
$\sum_{P\in\mathbb{P}_{F}}\omega_{P}(x)=\sum_{i=1}^{n}\omega_{P_{i}}(x)$.
Agora pelo lema 2.1.12 temos que
$\sum_{i=1}^{n}\omega_{P_{i}}(x)=\sum_{i=1}^{n}x(P_{i})\omega_{P_{i}}(1)=\langle(\omega_{P_{1}},\ldots,\omega_{P_{n}}),(x(P_{1}),\ldots,x(P_{n}))\rangle$,
concluindo então que $C_{\Omega}(D,D)\subset C(D,G)^{\bot}$. ∎
### 2.2 Teorema de Bézout
Nesta seção ficaremos a par do teorema de Bézout, o qual fala sobre o número
de interseções entre curvas algébricas e também o ligaremos a teoria de
códigos, foco deste trabalho.
###### Teorema 2.2.1 (Teorema de Bézout).
Sejam $\mathcal{X}$ e $\mathcal{Y}$ duas curvas algébricas planas irredutíveis
de grau $l$ e $m$ respectivamente sobre um corpo algebricamente fechado
$\mathbb{F}$ tais que elas não tenham uma componente em comum, então o número
de pontos da interseção entre as curvas é exatamente $lm$ (contanto os pontos
com suas multiplicidades).
###### Proof.
A demonstração desse resultado não está no objetivo desse texto, contudo ela
se encontra em [1]. ∎
###### Proposição 2.2.2.
Consideremos um polinômio $G\in\mathbb{F}_{q}[x,y]$ com grau total $m$ tal que
sua forma homogênea $G^{*}$ define uma curva irredutível não singular
$\mathcal{X}$, então $G$ é irredutível em $\mathbb{F}[x,y]$, onde $\mathbb{F}$
é o fecho algébrico de $\mathbb{F}_{q}$.
###### Proof.
Pela nossa definição $\mathcal{X}$ é uma curva gerada por um ideal primo
$I\subset\mathbb{F}[x,y,z]$ e mais $I=\langle G^{*}\rangle$. Logo $G^{*}$ é
irredutível em $\mathbb{F}[x,y,z]$. Agora sendo $G^{*}$ irredutível e supondo,
por absurdo, que $G$ seja redutível temos que $G=fh$ com $gr(f)<gr(G)$ e
$gr(h)<gr(G)$, logo temos que $G^{*}=(fh)^{*}=f^{*}g^{*}$ que é redutível,
absurdo, assim temos que $G$ é irredutível em $\mathbb{F}[x,y]$. ∎
###### Definição 2.2.3.
Seja $\mathcal{X}$ uma curva definida sobre $\mathbb{F}_{q}$, isto é, as
equações que a define tem seus coeficientes em $\mathbb{F}_{q}$. Os pontos de
$\mathcal{X}$ que tem todas as coordenadas em $\mathbb{F}_{q}$ são ditos
pontos racionais.
Seja $V_{l}$ o espaço vetorial de polinômios de grau total no máximo $l$, em
duas variáveis $x,y$ e com coeficientes em $\mathbb{F}_{q}$. Considere $G$ um
polinômio como o da proposição 2.2.2 (em particular, o grau total de $G$ é
$m$), $P_{1},P_{2},\cdots,P_{n}$ pontos racionais da curva definida por $G$.
Definimos o código $C$ por:
$C=\\{(f(P_{1}),f(P_{2}),\cdots,f(P_{n}));\,\,f\in V_{l}\\}.$
###### Teorema 2.2.4.
Se no código $C$, definido acima tem-se que $n>lm$, então para sua distância
mínima $d$ e sua dimensão $k$ são dadas por:
$d\geqslant n-lm;$ $k=\left\\{\begin{array}[]{ll}{l+2\choose 2},&\textrm{ se
}l<m;\\\ lm+1-{m-1\choose 2},&\textrm{ se }l\geqslant m.\end{array}\right.$
###### Proof.
Primeiro queremos encontrar a dimensão do espaço vetorial $V_{l}$ que tem como
base o conjunto formado pelo monômios de grau menor ou igual a $l$, conjunto
esse que tem cardinalidade igual a
$\sum_{i=0}^{l}(l+1)-i=\dfrac{2(l+1)^{2}-(l+1)l}{2}\\!=\dfrac{(l+1)(l+2)}{2}={l+2\choose
2}$, assim a dimensão de $V_{l}$ é ${l+2\choose 2}$.
Seja $F\in V_{l}$, se $G$ for um fator de $F$ temos que a palavra
correspondente a $F$ no código é zero. Agora dada uma palavra nula no código e
$F\in V_{l}$ o polinômio que a gera temos que a curva $\mathcal{Y}$ definida
por $F=0$ e $G=0$ tem grau $l^{\prime}\leqslant l$ e $l^{\prime}\leqslant m$ e
mais, temos que $P_{1},\cdots,P_{n}$ estão na interseção de $\mathcal{Y}$ com
$\mathcal{X}$. O teorema de Bézout nos garante que se $\mathcal{Y}$ e
$\mathcal{X}$ não tem um fator em comum o número de pontos na intersessão é
menor ou igual a $l^{\prime}m\leqslant lm$, mas por hipótese $n>lm$, logo $F$
tem $G$ como seu fator. Assim temos que as funções em $V_{l}$ que geram a
palavra zero é um subespaço vetorial de dimensão $l-m$ dado por
$GV_{l-m}=\\{GH;\,H\in V_{l-m}\\}$.
Se $l<m$ temos que $V_{l-m}=\emptyset$ logo a dimensão do código é dada por
$k={l+2\choose 2}$. Caso contrário, teremos que $k={l+2\choose
2}-{l-m+2\choose 2}=lm+1-{m-1\choose 2}.$
Agora queremos demonstrar que a distância mínima do código é $d\geqslant
n-lm$. De fato, seja $w\in C$ uma palavra não nula, suponha que $w$ tem mais
que $lm$ coordenadas nulas. Seja $F\in V_{l}$ um polinômio que gere $w$,
tomemos a curva $\mathcal{Y}$ definida por $F=0$, como $gr(F)\leqslant l$
temos que $gr(\mathcal{Y})\leqslant l$ logo
$\\#\mathcal{Y}\cap\mathcal{X}\leqslant lm$, pelo teorema de Bézout. Com um
possível reordenamento das coordenadas de $w$ podemos supor que
$F(P_{1})=F(P_{2})=\cdots=F(P_{lm})=\cdots=F(P_{j})=0$, pela nossa construção
temos que $\\{P_{1},\cdots,P_{j}\\}\subset\mathcal{Y}\cap\mathcal{X}$. Logo
$j\leqslant lm$, assim $d\geqslant n-lm$, como queríamos demonstrar. ∎
## Chapter 3 Códigos de Avaliação
Este capítulo está dedicado a construção dos códigos de avaliação.
### 3.1 Funções Peso, Grau e Ordem
###### Definição 3.1.1.
Seja $R=\mathbb{F}[x_{1},\cdots,x_{m}]$ o anel de polinômios a $m$ variáveis
sobre o corpo $\mathbb{F}$, suponha que exista uma ordem total $\prec$ no
conjunto de monômios de $R$ tal que para quaisquer monômios $M_{1},\,\,M_{2}$
e $M$ temos que:
1. 1.
Se $M\neq 1$, então $1\prec M$;
2. 2.
Se $M_{1}\prec M_{2}$, então $MM_{1}\prec MM_{2}$.
Assim dizemos que $\prec$ é uma ordem de admissão ou ordem de redução em
monômios.
No que segue $R$ é uma $\mathbb{F}-$álgebra, ou seja, um anel comutativo com
unidade tal que $\mathbb{F}\subset R$ como subanel. E mais, o simbolo
$-\infty$ é tal que para todo $n\in\mathbb{N}_{0}\cup\\{-\infty\\}$,
$-\infty+n=-\infty$.
###### Definição 3.1.2.
Uma função $\rho:R\rightarrow\mathbb{N}_{0}\cup\\{-\infty\\}$, que satisfaz as
propriedades abaixo é chamada de função ordem.
1. 1.
$\rho(f)=-\infty$ se, e somente se, $f=0$;
2. 2.
$\rho(\lambda f)=\rho(f)$ para todo $\lambda\in\mathbb{F}\setminus\\{0\\}$;
3. 3.
$\rho(f+g)\leqslant max\\{\rho(f),\rho(g)\\}$ e a igualdade é válida quando
$\rho(f)\neq\rho(g)$;
4. 4.
Se $\rho(f)<\rho(g)$ e $h\neq 0$, então $\rho(fh)<\rho(gh)$;
5. 5.
Se $\rho(f)=\rho(g)\neq 0$, então existe
$\lambda\in\mathbb{F}\setminus\\{0\\}$ tal que $\rho(f-\lambda g)<\rho(g)$.
Se além dessas propriedades $\rho$ também satisfizer a próxima a chamaremos de
função peso.
6. 6.
$\rho(fg)=\rho(f)+\rho(g).$
###### Exemplo 3.1.3.
Um primeiro exemplo de uma função peso é a função grau de polinômios no anel
de polinômios em uma variável $\mathbb{F}[x]$.
###### Definição 3.1.4.
Uma função grau em $R$ é uma função que satisfaz as propriedades 1, 2, 3, 4 e
6 da definição 3.1.2.
Vejamos agora um resultado que nos traz propriedades para as funções ordem.
###### Lema 3.1.5.
Seja $\rho$ uma função ordem em $R$, então temos:
1. 1.
Se $\rho(f)=\rho(g),$ então $\rho(fh)=\rho(gh)$ para todo $h\in R$;
2. 2.
Se $f\in R\setminus\\{0\\},$ então $\rho(1)\leqslant\rho(f)$;
3. 3.
$\mathbb{F}=\\{f\in R;\rho(f)\leqslant\rho(1)\\}$;
4. 4.
Se $\rho(f)=\rho(g)$, então existe um único escalar não nulo
$\lambda\in\mathbb{F}$ tal que $\rho(f-\lambda g)<\rho(g)$.
###### Proof.
A demonstração desse lema sairá diretamente da definição de funções ordem.
(1) Seja $\rho(f)=\rho(g)$, temos que existe $\lambda\in\mathbb{F}$ tal que
$\rho(f-\lambda g)<\rho(g)$, logo $\rho(fh-\lambda gh)<\rho(gh)$. Podemos
escrever $fh=(fh-\lambda gh)+\lambda gh$, assim $\rho(fh)=\rho(\lambda
gh)=\rho(gh).$
(2) Suponha por absurdo que $f\in R$ é um elemento não nulo tal que
$\rho(f)<\rho(1)$, então a cadeia $\rho(1)>\rho(f)>\rho(f^{2})>\cdots$ é
estritamente decrescente, absurdo, pois $\mathbb{N}_{0}\cup\\{-\infty\\}$ é
bem ordenado.
(3) Claramente $\mathbb{F}\subset H=\\{f\in R;\rho(f)\leqslant\rho(1)\\}$,
agora seja $f\neq 0$ tal que $\rho(f)\leqslant\rho(1)$ então
$\rho(f)=\rho(1)$, assim existe $\lambda$ tal que $\rho(f-\lambda)<\rho(1)$,
logo $f-\lambda=0$ ou seja $f\in\mathbb{F}$.
(4) Pela definição de função ordem, temos a existência do $\lambda$, falta
assim mostrar a unicidade. Suponha que $\lambda,\nu\in\mathbb{F}$ são tais que
$\rho(f-\lambda g)<\rho(g)$ e $\rho(f-\nu g)<\rho(g)$. Então temos que
$\rho(f-\lambda g-(f-\nu g))\leqslant max\\{\rho(f-\lambda g),\rho(f-\nu
g)\\}<\rho(g)$. Assim temos que $\rho((\lambda-\nu)g)<\rho(g)$ o que implica
que $\lambda-\nu=0$. Assim $\lambda=\nu$. ∎
Uma primeira conseqüência sobre a estrutura de um anel $R$ com uma função
ordem é dada na seguinte proposição:
###### Proposição 3.1.6.
Se existe uma função ordem, $\rho$, em $R$, então $R$ é um domínio de
integridade.
###### Proof.
Suponha que existam $f,\,g\in R\setminus\\{0\\}$ tais que $fg=0$, sem perda de
generalidade assumiremos que $\rho(f)\leqslant\rho(g)$, assim
$\rho(f^{2})\leqslant\rho(fg)=\rho(0)=-\infty$ logo $\rho(f^{2})=-\infty$,
isto é, $f^{2}=0$. Como $f\neq 0$, temos que
$\rho(1)\leqslant\rho(f)\leqslant\rho(f^{2})$, absurdo. Logo $fg\neq 0$ assim
$R$ é um domínio. ∎
Agora apresentaremos um exemplo com o qual mostraremos que a recíproca da
proposição 3.1.6 é falsa.
###### Exemplo 3.1.7.
A $\mathbb{F}-$álgebra $R=\mathbb{F}[x,y]/\langle xy-1\rangle$ é um domínio,
mas não tem uma função ordem. De fato, denotando por $\overline{x}$ a classe
de equivalência $x+\langle xy-1\rangle$ e por $\overline{y}$ a classe
$y+\langle xy-1\rangle$. Como $R$ é um domínio temos que $\overline{x}\neq 0$
e $\overline{y}\neq 0$. Sendo $\rho$ uma função ordem em $R$ temos que
$\rho(1)\leqslant\rho(\overline{x})$ e assim
$\rho(\overline{y})\leqslant\rho(\overline{xy})=\rho(1)$, ou seja,
$\rho(\overline{y})=\rho(1)$, analogamente achamos que
$\rho(\overline{x})=\rho(1)$. Observe que
$R=\mathbb{F}[\overline{x}]+\mathbb{F}[\overline{y}]$ e assim para todo $f\in
R$ concluimos que $\rho(f)\leqslant\rho(1)$, ou seja, $R=\mathbb{F}$, no
entanto $\overline{x}\notin\mathbb{F}$, absurdo.
A seguir mostraremos que dada uma $\mathbb{F}-$álgebra $R$ com uma função
ordem, essa admite uma $\mathbb{F}-$base com “boas” propriedades.
###### Teorema 3.1.8.
Seja $R$ uma $\mathbb{F}-$álgebra com uma função ordem $\rho$, $\mathbb{F}\neq
R$. Então:
1. 1.
Existe uma $\mathbb{F}-$base, $\\{f_{i},i\in\mathbb{N}\\}$, para $R$ tal que
$\rho(f_{i})<\rho(f_{i+1})$ para todo $i\in\mathbb{N}$;
2. 2.
Se $f=\sum_{i=1}^{m}\lambda_{i}f_{i}$ com $\lambda_{i}\in\mathbb{F}$ e
$\lambda_{m}\neq 0$, temos que $\rho(f)=\rho(f_{m})$;
3. 3.
Seja $l(i,j):=l$ o inteiro tal que $\rho(f_{i}f_{j})=\rho(f_{l})$. Assim,
$l(i,j)<l(i+1,j)$ para todo $i,j$;
4. 4.
Seja $\rho_{i}:=\rho(f_{i})$. Se $\rho$ é uma função peso então
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$.
###### Proof.
(1) Temos que existe $f\in R$ com $f\notin\mathbb{F}$, pois $R\neq\mathbb{F}$,
assim $\rho(1)<\rho(f)$ o que nos dá que $\rho(f^{n})<\rho(f^{n+1})$ para todo
$n\in\mathbb{N}_{0}$. Mais ainda, o conjunto dos valores de $\rho$ é infinito.
Seja $(\rho_{i})_{i\in\mathbb{N}}$ a seqüência crescente de inteiros não
negativos tais que os $\rho_{i}^{\prime}s$ são todos os valores da função
ordem, isto é, $\rho(R\setminus\\{0\\})=\\{\rho_{i};\,i\in\mathbb{N}\\}$. Por
definição, para todo $i\in\mathbb{N}$ existe um $f_{i}\in R$ tal que
$\rho(f_{i})=\rho_{i}$ assim $\rho(f_{i})<\rho(f_{i+1})$. E mais, pela nossa
construção, para todo $f\in R\setminus\\{0\\}$ existe um $f_{i}$ tal que
$\rho(f)=\rho(f_{i})$. Observamos que $\rho_{1}=\rho(1)$. Agora falta mostrar
que $B=\\{f_{i};i\in\mathbb{N}\\}$ é uma base. Claramente $B$ é um conjunto
linearmente independente, mostremos que ele gera $R$.
Seja $f\in R$ temos que existe um $f_{k}\in B$ tal que $\rho(f_{k})=\rho(f)$ o
que nos dá que existe $\lambda_{k}\in\mathbb{F}$ de modo que,
$\rho(f-\lambda_{k}f_{k})<\rho(f_{k})$, novamente temos que existe $f_{h}$ com
$h<k$ tal que $\rho(f-\lambda_{k}f_{k})=\rho(f_{h})$ e conseqüentemente existe
$\lambda_{h}$ tal que $\rho(f-\lambda_{k}f_{k}-\lambda_{h}f_{h})<\rho(f_{h})$,
esse processo deve acabar em no máximo $k$ vezes pois temos apenas $k-1$
$\rho_{i}^{\prime}s$ menores que $\rho_{k}$. Assim chegaremos em
$\rho(f-\sum_{i=0}^{k-1}\lambda_{k-i}f_{k-1})<\rho(1)$, observe que alguns dos
$\lambda_{i}^{\prime}s$ que aparecem aqui podem ser nulos,
$\rho(f-\sum_{i=0}^{k-1}\lambda_{k-i}f_{k-1})=-\infty$ o que nos dá que
$f=\sum_{i=0}^{k-1}\lambda_{k-i}f_{k-1}$, assim $B$ é uma $\mathbb{F}-$base de
$R$.
(2) Pela existência da base acima e o fato de que
$\rho(f+g)=max\\{\rho(f),\rho(g)\\}$ se $\rho(f)\neq\rho(g)$, tem-se (2).
(3) Como $\rho(f_{i})<\rho(f_{i+1})$ temos que
$\rho(f_{i}f_{j})<\rho(f_{i+1}f_{j})$ logo $l(i,j)<l(i+1,j).$
(4) Sendo $\rho$ uma função peso temos que $\rho(fg)=\rho(f)+\rho(g)$. Assim
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$. ∎
### 3.2 Códigos de Avaliação
Nessa seção introduziremos o conceito de código de avaliação. Trabalharemos
aqui com um corpo finito com $q$ elementos, $\mathbb{F}_{q}$. Além disso, aqui
$R$ é uma $\mathbb{F}_{q}-$álgebra com função ordem $\rho$ tal que admite uma
base, $\\{f_{i};i\in\mathbb{N}\\}$, de modo que $\rho(f_{i})<\rho(f_{i+1})$
para todo $i\in\mathbb{N}$.
Queremos transformar o espaço vetorial $\mathbb{F}_{q}^{n}$ em uma álgebra e
para isso precisamos definir uma multiplicação de vetores, assim definiremos a
multiplicação $*$ como sendo a multiplicação usual de coordenadas, ou seja,
dados $a=(a_{1},\cdots,a_{n}),b=(b_{1},\cdots,b_{n})\in\mathbb{F}_{q}^{n}$
temos que $a*b=(a_{1}b_{1},\cdots,a_{n}b_{n})$.
###### Definição 3.2.1.
Definimos por $L_{l}$ o espaço vetorial gerado por $f_{1},\cdots,f_{l}$.
###### Definição 3.2.2.
Chamaremos de morfismo de $\mathbb{F}_{q}-$álgebras, a uma função
$\mathbb{F}_{q}-$linear, $\varphi:{R}\rightarrow{\mathbb{F}_{q}^{n}}$, tal que
$\varphi(fg)=\varphi(f)*\varphi(g)$.
Agora definiremos um código de avaliação e o seu código dual.
###### Definição 3.2.3.
Seja $L_{l}$ o espaço vetorial como acima definido e $\varphi$ um morfismo
entre $\mathbb{F}_{q}-$álgebras, assim definimos o código de avaliação $E_{l}$
determinado por $\varphi$, como sendo a imagem de $L_{l}$ por meio da função
$\varphi$, ou seja,
$E_{l}=\varphi(L_{l})=\langle\varphi(f_{1}),\cdots,\varphi(f_{l})\rangle.$
Denotaremos por $C_{l}$ o código dual de $E_{l}$.
Observe que a seqüência de códigos $(E_{l})_{l\in\mathbb{N}}$ é crescente
segundo a inclusão, e mais, como o espaço de chegada da função $\varphi$ tem
dimensão finita temos que essa seqüência estabiliza num certo $N$.
Trabalharemos aqui somente com os morfismos sobrejetivos.
###### Exemplo 3.2.4.
Sejam $R=\mathbb{F}_{q}[x_{1},\ldots,x_{n}]/I$, onde $I$ é um ideal do anel de
polinômios $\mathbb{F}_{q}[x_{1},\ldots,x_{n}]$,
$\mathcal{P}=\\{P_{1},\ldots,P_{m}\\}$ um subconjunto com $m$ pontos de
$V(I)$, consideremos a função de avaliação:
$\begin{array}[]{rccl}ev_{\mathcal{P}}:&R&\rightarrow&\mathbb{F}_{q}^{m}\\\
&f+I&\mapsto&(f(P_{1}),\ldots,f(P_{m}))\end{array}.$
Observe que $ev_{\mathcal{P}}$ está bem definida, pois para $P\in V(I)$ temos
que se $f+I=g+I$ então $f-g\in I$, logo, $(f-g)(P)=0$, ou seja, $f(P)=g(P)$, e
mais, desse modo definido $ev_{\mathcal{P}}$ é um morfismo de
$\mathbb{F}_{q}$-álgebras, visto que $fg(P)=f(P)g(P)$, para todos $f,g\in R$ e
$P\in\mathbb{F}_{q}^{n}$.
###### Lema 3.2.5.
A função de avaliação acima definida é sobrejetiva.
###### Proof.
Sejam $\,P_{j}\,=\,(a_{j1},\,\ldots,\,a_{jm})$ e os conjuntos auxiliares
$A_{il}=\\{a_{jl};\,j=1,\ldots,m\\}\setminus\\{a_{il}\\}$. Definimos os
polinômios
$G_{i}=\prod_{l=1}^{n}\prod_{a\in A_{il}}(x_{l}-a).$
Temos que $G_{i}(P_{j})=0$ sempre que $i\neq j$, e mais $G_{i}(P_{i})\neq 0$.
Os polinômios $G_{i}/G_{i}(P_{i})$ quando aplicados em $ev_{\mathcal{P}}$
chegam na base canônica de $\mathbb{F}_{q}^{n}$ assim $ev_{\mathcal{P}}$ é
sobrejetiva. ∎
### 3.3 A Cota Ordem
Um dos principais parâmetros de uma código é a sua distância mínima,
contudo,na maioria das vezes é difícil de ser calculada, assim torna-se
importante as cotas para essa distância mínima. Nesta seção estamos
interessados em encontrar uma cota inferior para a distância mínima do código
$C_{l}$.
Continuaremos a usar as notações da seção anterior, lembrando que $N$ é o
menor natural tal que a seqüência de códigos de avaliação $E_{l}$ estabiliza.
Os morfismos utilizados serão sobrejetivos.
Definimos aqui uma $N\times n$ matriz $H$ com a $i-$ésima linha sendo
$h_{i}=\varphi(f_{i})$ onde $f_{i}$ são os elementos da base construída
anteriormente e $\varphi$ é o morfismo de $\mathbb{F}_{q}-$álgebras utilizado
aqui.
###### Definição 3.3.1.
Seja $y\in\mathbb{F}_{q}^{n}$. Consideremos as sindromes $s_{i}(y)=\langle
y,h_{i}\rangle$ e $s_{ij}(y)=\langle y,(h_{i}*h_{j})\rangle$. Então a matriz
$S(y)=(s_{ij}(y);\,\,\,1\leqslant i,j\leqslant N)$ é a matriz síndrome de $y$.
###### Lema 3.3.2.
Seja $y\in\mathbb{F}_{q}^{n}$ e $D(y)$ a matriz diagonal com as coordenadas de
$y$ em sua diagonal, então
$S(y)=HD(y)H^{t},$
e mais
$posto(S(y))=w(y).$
###### Proof.
Temos que $s_{ij}(y)=\langle
y,(h_{i}*h_{j})\rangle=\sum_{l}y_{l}h_{il}h_{jl}$, e mais, denotando $C=S(y)$,
temos que $C_{ij}=H_{ik}y_{k}H^{t}_{kj}=H_{ik}y_{k}H_{jk}$, logo a igualdade é
válida.
Agora o posto de $D(y)$ é justamente o peso de $y$. Como a função $\varphi$ é
sobrejetiva temos que a matriz $H$ tem posto máximo, ou seja, posto de $H$ é
$n$. Assim temos que o posto de $S(y)$ é o mesmo posto de $D(y)$ como
queriamos. ∎
###### Definição 3.3.3.
Seja $l\in\mathbb{N}_{0}$, definimos o conjunto
$N_{l}=\\{(i,j)\in\mathbb{N}^{2};\,\,l(i,j)=l+1\\}$, onde $l(i,j)$ é o número
natural definido no item 3 do teorema 3.1.8. A sua cardinalidade denotaremos
por $\nu_{l}$.
###### Lema 3.3.4.
Se $t=\nu_{l}$ e $(i_{1},j_{1}),\ldots,(i_{t},j_{t})$ é a enumeração dos
elementos de $N_{l}$ em ordem crescente segundo a ordem lexicográfica111Dados
$(a,b),\,(c,d)\in\mathbb{N}^{2}$, $(a,b)<(c,d)$ se $a<c$ ou $a=c$ e $b<d$. em
$\mathbb{N}^{2}$ . Então $i_{1}<i_{2}<\cdots<i_{t}$ e
$j_{t}<j_{t-1}<\cdots<j_{1}$. Além disso, se $y\in C_{l}\setminus C_{l+1}$
temos que $s_{i_{u}j_{v}}(y)=0$ se $u<v$ e $s_{i_{u}j_{v}}(y)\neq 0$ se $u=v$.
###### Proof.
Pela ordem da seqüência temos que $i_{1}\leqslant
i_{2}\leqslant\cdots\leqslant i_{t}$, suponha por absurdo que $i_{k}=i_{k+1}$.
Desse modo temos que $j_{k}<j_{k+1}$ logo
$l+1=l(i_{k},j_{k})<l(i_{k},j_{k+1})=l(i_{k+1},j_{k+1})=l+1$
absurdo, logo a seqüência é estritamente crescente.
Agora suponha que $j_{k}\geqslant j_{k+1}$, novamente temos que
$l+1=l(i_{k},j_{k})\geqslant l(i_{k},j_{k-1})>l(i_{k-1},j_{k-1})=l+1,$
outro absurdo, assim essa seqüência é decrescente.
Seja $y\in C_{l}$, se $u<v$ temos que $l(i_{u},j_{v})<l(i_{v},j_{v})=l+1$,
assim $f_{i_{u}}f_{j_{v}}\in L_{l}$ e desse modo $h_{i_{u}}*h_{j_{v}}\in
E_{l}$, logo $s_{i_{u}j_{v}}(y)=\langle y,h_{i_{u}}*h_{j_{v}}\rangle=0$. Do
mesmo modo se $u=v$ temos que $l(i_{u},j_{v})=l+1$ o que nos dá que
$h_{i_{u}}*h_{j_{v}}\in L_{l+1}\setminus L_{l}$, e mais,
$f_{i_{u}}f_{j_{v}}\equiv\mu f_{l+1}\mathsf{mod}{L}_{l}$ para algum
$0\neq\mu\in\mathbb{F}_{q}$. Assim $h_{i_{u}}*h_{j_{v}}\equiv\mu
h_{l+1}\mathsf{mod}{E}_{l}$, como $y\notin C_{l}$ temos que
$s_{l+1}(y)=\langle y,h_{l+1}\rangle\neq 0$ pois $\langle h_{i},y\rangle=0$
para $1\leqslant i\leqslant l$ e $h_{l+1}\notin C_{l}.$ Assim
$s_{i_{u}j_{v}}(y)\neq 0$. ∎
Observamos que a matriz $m_{uv}=s_{i_{u}j_{v}}(y)$ com $1\leqslant
u,v\leqslant\nu_{l}$ como do lema acima, é uma matriz quadrada de posto
$\nu_{l}$, e mais, $(m_{uv})$ é uma sub-matriz de $S(y)$, logo o posto de
$S(y)$ é maior ou igual a $\nu_{l}$. Isso juntamente como os lemas 3.3.2 e
3.3.4 demonstram a seguinte proposição:
###### Proposição 3.3.5.
Se $y\in C_{l}\setminus C_{l+1}$, então $w(y)\geqslant\nu_{l}$.
Definiremos agora algumas cotas relacionadas aos códigos de avaliação.
###### Definição 3.3.6.
Chamaremos de cota ordem aos números
$d(l)=min\\{\nu_{m};m\geqslant l\\},$ $d_{\varphi}(l)=min\\{\nu_{m};m\geqslant
l,\,\,C_{m}\neq C_{m+1}\\}.$
###### Teorema 3.3.7.
Os números $d(l)$ e $d_{\varphi}(l)$ são cotas inferiores para a distância
mínima de $C_{l}$. Mais ainda, $d(C_{l})\geqslant d_{\varphi}(l)\geqslant
d(l).$
###### Proof.
Pela proposição 3.3.5 temos que $d(C_{l})\geqslant\nu_{l}\geqslant
d_{\varphi}(l).$ ∎
### 3.4 Semigrupos
Nesta seção falaremos de semigrupos e a sua associação aos códigos de
avaliação.
###### Definição 3.4.1.
Um semigrupo numérico é um subconjunto $\Lambda$ de $\mathbb{N}_{0}$ com as
seguintes propriedades:
1. 1.
$\Lambda$ é fechado para adição;
2. 2.
$0\in\Lambda$.
Os elementos de $\mathbb{N}_{0}\setminus\Lambda$ são chamados de lacunas e a
quantidade desses elementos será denotada por $g=g(\Lambda)$ (aqui $g$ pode
ser infinito).
Suponha agora $\rho$ é uma função peso em uma $\mathbb{F}_{q}-$álgebra $R$,
por 3.1.2-6 temos que o conjunto $\Lambda=\\{\rho(f);\,f\in R,\,f\neq 0\\}$ é
um semigrupo numérico chamado de semigrupo de $\rho$ . Se $g<\infty$, então
existe um $n\in\Lambda$ tal que se $x\in\mathbb{N}$ e $n\leqslant x$ então
$x\in\Lambda$. Ao menor $n$ com essa propriedade chamamos de condutor de
$\Lambda$ e denotamos por $c=c(\Lambda)$. Observe que $c-1$ é o maior lacuna
de $\Lambda$ desde que $g>0$.
###### Definição 3.4.2.
Seja $(\rho_{l})_{l\in\mathbb{N}}$ uma enumeração do semigrupo $\Lambda$ tal
que $\rho_{l}<\rho_{l+1}$ para todo $l$. Denotamos por $g(l)$ o número de
lacunas menores que $\rho_{l}$.
###### Lema 3.4.3.
Sejam $\Lambda$ um semigrupo com finitos lacunas e $l\in\mathbb{N}$, então:
1. 1.
$g(l)=\rho_{l}-l+1$;
2. 2.
$\rho_{l}\leqslant l+g-1$, valendo a igualdade se, e somente se,
$\rho_{l}\geqslant c$;
3. 3.
Se $l>c-g$, então $\rho_{l}=l+g-1$;
4. 4.
Se $l\leqslant c-g$, então $\rho_{l}<c-1$.
###### Proof.
(1) O elemento $\rho_{l}\in\Lambda$ é o $(\rho_{l}+1)-$ésimo elemento de
$\mathbb{N}_{0}$ e mais, ele é o $(\rho_{l}+1-g(l))-$ésimo elemento de
$\Lambda$. Assim $l=\rho_{l}+1-g(l)$, ou seja, $g(l)=\rho_{l}+1-l$.
(2) Se $g(l)\leqslant g$ temos que $\rho_{l}+1-l\leqslant g$, logo
$\rho_{l}\leqslant g-1+l$, se $\rho_{l}\geqslant c$ temos que todos os lacunas
são menores que $\rho_{l}$ logo $g(l)=g$ valendo assim a igualdade.
(3) Temos que $c$ é o $(c+1)-$ésimo elemento de $\mathbb{N}_{0}$ e mais, é o
$(c+1-g)-$ésimo termo de $\Lambda$m, assim $c=\rho_{c+1-g}$. Tomando $l>c-g$
teremos $\rho_{l}\geqslant\rho_{c+1-g}=c$ e mais, conseguimos assim
$\rho_{l}=g-1+l$.
(4) Seja $l\leqslant c-g$, assim $\rho_{l}\leqslant l+g-1\leqslant c-1$.
Contudo $c-1$ é um lacuna ou é negativo. Para $c=0$ não tem sentido a
proposição, então temos que $c-1$ é um lacuna, assim $\rho_{l}<c-1$. ∎
A próxima proposição nos traz um resultado sobre o condutor de um semigrupo
numérico que vai influenciar em uma importante definição.
###### Proposição 3.4.4.
Seja $\Lambda$ um semigrupo numérico com número de lacunas $g<\infty$, então
$c\leqslant 2g$ e a igualdade é válida se, e somente se, para todo lacuna $s$
temos que $c-1-s$ não é um lacuna.
###### Proof.
Observemos que os pares $(s,t)\in\mathbb{N}^{2}_{0}$ tais que $s+t=c-1$, pelo
fato de $c-1$ ser um lacuna e de $\Lambda$ ser fechado em relação a soma, tem
pelo menos um dos dois termos de cada par como um lacuna. Como temos $c$ pares
desses, levando em consideração a ordem, temos que existem pelo menos
$\left[\dfrac{c+1}{2}\right]$ lacunas, logo $c\leqslant 2g$.
Agora se a igualdade é válida, temos que $g=\dfrac{c}{2}$ assim dados
$s,t\in\mathbb{N}_{0}$ tais que $s+t=c-1$ temos que apenas um dos dois ($s$ ou
$t$) é um lacuna, logo sendo $s$ um lacuna $c-1-s$ não pode ser lacuna.
Supondo que se $s$ for um lacuna temos que $c-1-s$ não é, isso nos dá que
apenas um dos termos dos pares $(s,t)$ tais que $s+t=c-1$ é um lacuna, assim
temos exatamente $\dfrac{c}{2}$ lacunas. ∎
O resultado anterior justifica a seguinte definição:
###### Definição 3.4.5.
Um semigrupo numérico é chamado simétrico se $c=2g$.
###### Definição 3.4.6.
Dizemos que um semigrupo numérico é finitamente gerado se existe um conjunto
$A=\\{a_{1},\ldots,a_{k}\\}\subset\Lambda$ tal que dado $\lambda\in\Lambda$
temos que existem $x_{1},\ldots,x_{k}\in\mathbb{N}_{0}$, tais que
$\lambda=\sum_{i=1}^{k}x_{i}a_{i}$. Assim falamos que $A$ gera $\Lambda$ e
escrevemos $\Lambda=\langle A\rangle$.
A respeito de subgrupos numéricos finitamente gerados o primeiro resultado que
apresentamos é:
###### Proposição 3.4.7.
Sejam $a,b\in\mathbb{N}$ tais que $mdc(a,b)=1$. O semigrupo gerado por $a$ e
$b$ é simétrico, tem como último lacuna o número $ab-a-b$, como seu condutor o
número $(a-1)(b-1)$ e o número total de lacunas é $(a-1)(b-1)/2$.
###### Proof.
Como $mdc(a,b)=1$, temos que todo inteiro $m$ pode ser escrito como $m=xa+yb$,
de maneira única com $0\leqslant y<b$.
Pelo fato acima observamos que o maior lacuna possível é $(b-1)a-b$ e de fato
esse número é um lacuna, pois não é possível escreve-lo como $(b-1)a-b=xa+yb$
com $x,y\in\mathbb{N}_{0}$, assim temos que o condutor é
$c=(b-1)a-b+1=(a-1)(b-1)$.
Agora mostremos que o semigrupo é simétrico. Suponhamos por absurdo que o
semigrupo não seja simétrico, ou seja, existe $s,t\in\mathbb{N}$ lacunas tais
que $s+t=c-1$ onde $c$ é o condutor. Podemos escrever $s=x_{1}a+y_{1}b$ e
$t=x_{2}a+y_{2}b$, assim temos que $c-1=ab-a-b=(x_{1}+x_{2})a+(y_{1}+y_{2})b$.
Observe que $0\leqslant x_{1}+x_{2}\leqslant 2b-2$ e mais
$y_{1}+y_{2}\leqslant-2$. Disto segue que:
$D=(-y_{1}-y_{2}-1)b=(x_{1}+x_{2}-b+1)a\Rightarrow$
$0<b\leqslant(-y_{1}-y_{2}-1)b=(x_{1}+x_{2}-b+1)a\leqslant(b-1)a<ba$
Como $mdc(a,b)=1$ temos que $a|(-y_{1}-y_{2}-1)$ e que $b|(x_{1}+x_{2}-b+1)$,
assim chegamos que $0<\dfrac{D}{ab}<1$, absurdo, logo $\Lambda$ é simétrico.
Como o semigrupo é simétrico temos que $c=2g$, logo $g=(a-1)(b-1)/2$. ∎
Faremos agora mais um lema técnico sobre semigrupos.
###### Lema 3.4.8.
Sejam $\Lambda$ um semigrupo numérico com finitos lacunas e $s\in\Lambda$.
Então temos que $\\#(\Lambda\setminus\\{s+\lambda;\,\lambda\in\Lambda\\})=s$.
###### Proof.
Seja $c$ o condutor de $\Lambda$, $T=\\{t\in\mathbb{N}_{0};\,\,t\geqslant
s+c\\}$, claramente temos que $T\subset\Lambda$, e mais, $T\subset
s+\Lambda=\\{s+\lambda;\,\lambda\in\Lambda\\}$. Seja
$U=\\{u\in\Lambda;u<s+c\\}$, temos que $\\#U=s+c-g$, além disso $\Lambda=U\cup
T$. Seja $V=\\{v\in s+\Lambda;\,\,s\leqslant v<s+c\\}$, temos que
$\\#V=s+c-g-s=c-g$, e mais, $s+\Lambda=V\cup T$. Observe que as uniões acima
são disjuntas e mais, $V\subset U$. Assim temos que:
$\\#(\Lambda\setminus s+\Lambda)=\\#(U\cup T\setminus V\cup T)=\\#(U\setminus
T)=s+c-g-(c-g)=s.$
Como queríamos demonstrar. ∎
Uma conseqüência quase imediata desse lema é:
###### Proposição 3.4.9.
Seja $f$ um elemento não nulo de uma $\mathbb{F}_{q}-$álgebra $R$ com uma
função peso $\rho$. Então $dim_{\mathbb{F}_{q}}(R/\langle f\rangle)=\rho(f)$.
###### Proof.
Sejam $\Lambda$ o semigrupo da função peso $\rho$ e $s=\rho(f)$. Tomemos a
seqüência $(\rho_{i})_{i\in\mathbb{N}}$ dos elementos de $\Lambda$ em ordem
crescente. Pela propriedade 3.1.2-6 temos que a imagem dos elementos não nulos
do ideal $\langle f\rangle$ segundo a função $\rho$ é o conjunto $s+\Lambda$.
Como feito antes, para todo $\rho_{i}\in\Lambda$, existe um $f_{i}\in R$ tal
que $\rho(f_{i})=\rho_{i}$ e caso $\rho_{i}\in s+\Lambda$ podemos tomar
$f_{i}\in\langle f\rangle$. Os conjuntos $\\{f_{i};\,i\in\mathbb{N}\\}$ e
$\\{f_{i};\,i\in\mathbb{N},\,\rho_{i}\in s+\Lambda\\}$ formam uma base para a
álgebra $R$ e o ideal $\langle f\rangle$ respectivamente, vide demonstração de
3.1.8. Desse modo as classes de equivalência $f_{i}$ módulo $\langle f\rangle$
com $i\in\mathbb{N}$ e $\rho_{i}\in\Lambda\setminus(s+\Lambda)$ formam uma
base para o quociente $R/\langle f\rangle$. Assim a sua dimensão é a
cardinalidade da base que é $s$ pelo lema 3.4.8, ou seja, $\rho(f)$. ∎
###### Definição 3.4.10.
Seja $R=\mathbb{F}_{q}[x_{1},\ldots,x_{n}]/I$, onde $I$ é um ideal do anel de
polinômios $\mathbb{F}_{q}[x_{1},\ldots,x_{n}]$, para $f+I\in R$ dizemos que
$P\in\mathbb{F}^{n}_{q}$ é um zero de $f+I$, se $P\in V(I)$ e $f(P)=0$.
###### Lema 3.4.11.
Seja $R$ uma $\mathbb{F}_{q}-$álgebra finita com uma função peso $\rho$. Seja
$f\in R$ um elemento não nulo. Então o número de zeros de $f$ é no máximo
$\rho(f)$.
###### Proof.
Seja $\mathcal{P}$ o conjunto de zeros de $f$ e $t=\\#\mathcal{P}$. A função
de avaliação, $ev_{\mathcal{P}}:R\rightarrow\mathbb{F}_{q}^{t}$, é uma função
linear e pelo lema 3.2.5 temos que $ev_{\mathcal{P}}$ é sobrejetiva. Isso nos
garante que $R/Ker(ev_{\mathcal{P}})\simeq\mathbb{F}_{q}^{t}$. Observe que
$\langle f\rangle\subset Ker(ev_{\mathcal{P}})$ e olhando ambos como sub-
espaço vetorial de $R$ temos que $dim_{\mathbb{F}_{q}}(\langle
f\rangle)\leqslant dim_{\mathbb{F}_{q}}(Ker(ev_{\mathcal{P}}))$. Assim segue
que
$t=dim_{\mathbb{F}_{q}}(R/Ker(ev_{\mathcal{P}}))=dim_{\mathbb{F}_{q}}(R)-dim_{\mathbb{F}_{q}}(Ker(ev_{\mathcal{P}})\leqslant
dim_{\mathbb{F}_{q}}(R)-dim_{\mathbb{F}_{q}}(\langle
f\rangle)=\dim_{\mathbb{F}_{q}}(R/\langle f\rangle)=\rho(f)$ por 3.4.9. ∎
### 3.5 Código de Avaliação Via Semigrupos
Aqui estamos interessados em encontrar uma cota para a distância mínima dos
códigos de avaliação $E_{l}$.
Nessa seção iremos supor que $\rho$ é uma função peso em
$R=\mathbb{F}_{q}[x_{1},\ldots,x_{m}]/I$, onde $I$ é um ideal de
$\mathbb{F}_{q}[x_{1},\ldots,x_{m}]$ (anel de polinômios em $m$ variáveis).
Seja $(\rho_{i})_{i\in\mathbb{N}}$ a enumeração do semigrupo de $\rho$ em
ordem crescente. Tomemos $\mathcal{P}$ como um conjunto com $n$ pontos do
conjunto $V(I)$ (variedade algébrica gerada por $I$). A função de avaliação
$ev_{\mathcal{P}}:R\rightarrow\mathbb{F}_{q}^{n}$ nos define os códigos de
avaliação
$E_{l}=\\{ev_{\mathcal{P}}(f);\,\,f\in R,\,\rho(f)\leqslant\rho_{l}\\}.$
###### Teorema 3.5.1.
A distância mínima do código $E_{l}$ é maior ou igual a $n-\rho_{l}$. Se
$\rho_{l}<n$, temos que $dim_{\mathbb{F}_{q}}(E_{l})=l.$
###### Proof.
Seja $c$ uma palavra código não nula de $E_{l}$, então existe $f\in
R\setminus\\{0\\}$ tal que $\rho(f)\leqslant\rho_{i}$ e
$c=ev_{\mathcal{P}}(f)$. Temos que $c_{i}=f(P_{i})$ para todo $0<i<n+1$, onde
os $c_{i}$’s são as coordenadas da palavra $c$, pelo lema 3.4.11 temos que o
número de zeros de $f$ é no máximo $\rho(f)\leqslant\rho_{l}$, assim
$w(c)\geqslant n-\rho_{l}$.
Agora suponha que $\rho_{l}<n$. Temos que $E_{l}$ é a imagem do espaço
vetorial $L_{l}$ através da função $ev_{\mathcal{P}}$. Se $f\in L_{l}$ e
$ev_{\mathcal{P}}(f)=0$ então $f$ tem pelo menos $n$ zeros, mas pelo lema
3.4.11 se $f$ é não nulo então $f$ admite no máximo $\rho(f)$ zeros, contudo
$\rho(f)\leqslant\rho_{l}<n$, assim $f$ tem de ser nulo, concluímos assim que
$Ker(ev_{\mathcal{P}}|_{L_{l}})=\\{0\\}$, o que nos dá que
$dim_{\mathbb{F}_{q}}(E_{l})=dim_{\mathbb{F}_{q}}(L_{l})$, lembremos que
$L_{l}$ é o espaço vetorial gerado pelo conjunto $\\{f_{1},\ldots,f_{l}\\}$,
logo sua dimensão é $l$, assim $dim_{\mathbb{F}_{q}}(E_{l})=l$. ∎
###### Corolário 3.5.2.
Seja $\rho$ uma função peso com $g$ lacunas. Se $\rho_{k}<n$, então $E_{k}$ é
um $[n,k,d]-$código tal que $d\geqslant n+1-k-g$.
###### Proof.
Pelo teorema 3.5.1 temos que $d\geqslant n-\rho_{k}$. Agora pelo lema 3.4.3
temos que $\rho_{k}\leqslant k+g-1$, assim segue que $d\geqslant n+1-k-g.$ ∎
## Chapter 4 Exemplos
Este capítulo está dedicado a apresentação de alguns exemplos dos códigos que
nesse texto foram definidos. As notações aqui utilizadas são na maioria as
mesmas do capítulo 3, ou seja, $R$ denotará sempre uma
$\mathbb{F}_{q}-$álgebra, $\rho$ uma função peso, $l(i,j)$, o inteiro tal que
$\rho(f_{i}f_{j})=\rho_{l(i,j)}$, onde $\\{f_{1},f_{2},\ldots\\}$ é uma
$\mathbb{F}_{q}$-base para $R$, etc.
### 4.1 Um Primeiro Exemplo
Esse primeiro teorema do capítulo vai nos permitir garantir a existência de
funções ordem e peso em determinadas condições.
###### Teorema 4.1.1.
Sejam $R$ uma $\mathbb{F}-$álgebra e $\\{f_{1},f_{2},\ldots\\}$ uma
$\mathbb{F}-$base do $\mathbb{F}-$espaço vetorial $R$, com $f_{1}=1$. Sejam,
ainda, $(\rho_{i})_{i\in\mathbb{N}}$ uma seqüência estritamente crescente de
inteiros não negativos e $\rho:R\rightarrow\mathbb{N}_{0}\cup\\{-\infty\\}$ a
função definida por $\rho(0)=-\infty$ e $\rho(f)=\rho_{i}$ se $f\neq 0$ e $i$
for o menor inteiro tal que $f\in L_{i}$, onde $L_{i}$ é o
$\mathbb{F}-$subespaço vetorial gerado por $\\{f_{1},\ldots,f_{i}\\}$. Se para
todo $(i,j)\in\mathbb{N}^{2}$ tem-se que $l(i,j)<l(i+1,j)$, então $\rho$ é uma
função ordem e, mais ainda, se $\rho_{l(i,j)}=\rho_{i}+\rho_{j}$, então $\rho$
é uma função peso.
###### Proof.
Diretamente da definição da função $\rho$, vemos que ela satisfaz as condições
1, 2, 3 e 5 para ser uma função peso. Mostraremos que ela também satisfaz as
outras duas condições (4 e 6).
Para $f\in R\setminus\\{0\\}$, associamos o número $\iota(f)$, o qual é o
menor inteiro positivo tal que $f\in L_{\iota(f)}$. Dados $f,g\in
R\setminus\\{0\\}$ temos que
$f=\sum_{i\leqslant\iota(f)}\lambda_{i}f_{i},\,g=\sum_{i\leqslant\iota(g)}\nu_{i}f_{i}\,fg=\sum_{i\leqslant\iota(fg)}\mu_{i}f_{i}\textrm{
e }f_{i}f_{j}=\sum_{l\leqslant l(i,j)}\eta_{ijl}f_{l}.$
com $\lambda_{\iota(f)}\neq 0,\,\,\nu_{\iota(g)}\neq 0,\,\,\mu_{\iota(fg)}\neq
0$ e $\eta_{ijl(i,j)}\neq 0$.
Observe que
$fg=\left(\sum_{i\leqslant\iota(f)}\lambda_{i}f_{i}\right)\left(\sum_{j\leqslant\iota(g)}\nu_{j}f_{j}\right)=\sum_{i\leqslant\iota(f)}\sum_{j\leqslant\iota(g)}\lambda_{i}\nu_{j}f_{i}f_{j}=$
$\sum_{i\leqslant\iota(f)}\sum_{j\leqslant\iota(g)}\lambda_{i}\nu_{j}\sum_{l\leqslant
l(i,j)}\eta_{ijl}f_{l}=\sum_{i\leqslant\iota(f)}\sum_{j\leqslant\iota(g)}\sum_{l\leqslant
l(i,j)}\lambda_{i}\nu_{j}\eta_{ijl}f_{l},$
assim temos que
$\mu_{l}=\sum_{l(i,j)=l}\lambda_{i}\nu_{j}\eta_{l}f_{l}.$
Por hipótese temos que $l(i,j)<l(i+1,j)$, assim $l(i,j)<l(\iota(f),\iota(g))$
se $i<\iota(f)$ ou $j<\iota(g)$. Supondo $i=\iota(f)$ e $j=\iota(g)$, temos
que $\mu_{\iota{fg}}=\lambda_{i}\nu_{j}\eta_{ijl(i,j)}\neq 0$, o que nos
garante que $\iota(fg)=l(\iota(f),\iota(g))$.
Agora dados $f,g,h\in R\setminus\\{0\\}$, com $\rho(f)<\rho(g)$, temos que
$\rho(fh)=\rho_{\iota(fh)}=\rho_{l(\iota(f),\iota(h))}<\rho_{l(\iota(g),\iota(h))}=\rho_{\iota(gh)}=\rho(gh)$,
logo a condição 4 é verificada e $\rho$ é uma função ordem.
Agora assumindo que $\rho_{l(i,j)}=\rho_{l}+\rho_{j}$, temos que
$\rho(fg)=\rho_{\iota(fg)}=\rho_{l(\iota(f),\iota(g))}=\rho_{\iota(f)}+\rho_{\iota(g)}=\rho(f)+\rho(g).$
Ou seja, a condição 6 também é satisfeita sendo então $\rho$ uma função peso.
∎
###### Exemplo 4.1.2.
Seja $\mathcal{X}$ a curva definida pelo polinômio
$P(x,y)\in\mathbb{F}_{q}[x,y]$, $P(x,y)=x^{m}+y^{m-1}+G(x,y)$, com
$gr_{\mathsf{total}}(G(x,y))<m-1$. Como $P(x,y)$ é um polinômio irredutível,
temos que o anel $R=\mathbb{F}_{q}[x,y]/\langle P(x,y)\rangle$ é um domínio de
integridade e mais, $R$ é uma $\mathbb{F}_{q}-$álgebra. Assim, construído $R$
e denotando por $\overline{x},\,\overline{y}$ as classes de equivalências
$x+\langle P(x,y)\rangle$ e $y+\langle P(x,y)\rangle$, respectivamente, temos
que $R$ admite uma função peso, $\rho$, tal que $\rho(\overline{x})=m-1$ e
$\rho(\overline{y})=m$.
###### Proof.
O conjunto $B=\\{\overline{x}^{\alpha}\overline{y}^{\beta}\in
R;\,\alpha<m\\}$, é uma $\mathbb{F}_{q}-$base para $R$. De fato, primeiro
observemos que $\overline{x}^{m}=-\overline{y}^{m-1}-\overline{G}$, e dado
$\overline{h}\in R$ temos que $\overline{h}=h(x,y)+\langle P(x,y)\rangle$ e
mais, $h(x,y)=\sum_{i=0}^{a}\sum_{j=0}^{b}\lambda_{ij}x^{i}y^{j}$, com
$\lambda_{ab}\neq 0$. Para demonstrar que $B$ gera $R$ vamos supor sem perda
de generalidade que $h(x,y)$ é um monômio, visto que no máximo ele é uma soma
de monômios, assim $h(x,y)=x^{a}y^{b}$, se $a<m$ temos que $h(x,y)\in B$ assim
não temos o que fazer, suponha então $a\geqslant m$.
Podemos escrever $a=km+c$ com $c<a$, assim
$\overline{x}^{a}\overline{y}^{b}=\overline{x}^{c}\overline{y}^{b}(-\overline{y}^{m-1}-\overline{g})^{k}$,
temos que $\overline{x}^{c}\overline{y}^{b+k(m-1)}\in B$ e os demais monômios
de $\overline{x}^{c}\overline{y}^{b}(-\overline{y}^{m-1}-\overline{g})^{k}$
tem grau em $x$ menor que $a$, logo repetindo esse processo recursivamente
para os demais temos que $\overline{h}$ é escrito como combinação linear de
elementos de $B$.
Agora queremos mostrar que $B$ é um conjunto linearmente independente. Seja
$\sum\lambda_{ij}\overline{x}^{i}\overline{y}^{j}=0$ uma soma finita de
elementos de $B$, equivalentemente mostraremos para
$\sum\lambda_{ij}x^{i}y^{y}=f(x,y)P(x,y)$ para algum
$f(x,y)\in\mathbb{F}_{q}[x,y]$, temos que se $f(x,y)\neq 0$,
$gr_{x}(f(x,y)P(x,y))>m$ e mais,
$gr_{x}\left(\sum\lambda_{ij}x^{i}y^{y}\right)<m$ logo a igualdade não é
válida, sendo $f(x,y)=0$ temos que os $\lambda_{ij}^{\prime}$s são nulos, pois
esses monômios são linearmente independentes em $\mathbb{F}_{q}[x,y]$ assim
$B$ é uma $\mathbb{F}_{q}-$base para $R$.
Seja $\\{f_{1},f_{2},f_{3},\ldots\\}$ uma enumeração do conjunto $B$. Para
$f_{i}=\overline{x}^{\alpha}\overline{y}^{\beta}$ definimos
$\rho_{i}=\alpha(m-1)+\beta m$. Sendo
$D=\\{(a,b)\in\mathbb{N}_{0}^{2};a<m\\}$, a função
$\varphi:D\rightarrow\mathbb{N}_{0}$, tal que $\varphi(a,b)=a(m-1)+bm$, é
injetiva, já que $mdc(m,m-1)=1$ e $a<m$, assim se $i\neq j$ temos que
$\rho_{i}\neq\rho_{j}$.
Reordenando se preciso, assumiremos que a seqüência
$(\rho_{i})_{i\in\mathbb{N}}$ é estritamente crescente.
Seja $f_{i}=\overline{x}^{\alpha}\overline{y}^{\beta}$ e
$f_{j}=\overline{x}^{\gamma}\overline{y}^{\delta}$, com $\alpha<m$ e
$\gamma<m$. Seguindo as notações do teorema anterior, queremos mostrar que
$l(i,j)<l(i+1,j)$ e para isso vamos mostrar que
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$. Separaremos em 2 casos, $\alpha+\gamma<m$ e
$m\leqslant\alpha+\gamma<2m$.
1. 1.
Se $\alpha+\gamma<m$, temos que $f_{i}f_{j}\in B$, logo
$f_{i}f_{j}=f_{l(i,j)}$, onde $l(i,j)$ é o menor inteiro $l$, tal que
$f_{i}f_{l}\in L_{l}=\langle f_{1},\ldots,f_{l}\rangle$, assim
$\rho_{l}(i,j)=\rho_{i}+\rho_{j}$.
2. 2.
Agora vamos supor $\alpha+\gamma\geqslant m$, assim $\alpha+\gamma=m+\epsilon$
com $0\leqslant\epsilon<m$. Tomamos $n=\beta+\delta$, temos que
$f_{i}f_{j}=(\overline{x}^{\alpha}\overline{y}^{\beta})(\overline{x}^{\gamma}\overline{y}^{\delta})=\overline{x}^{(m+\epsilon)}\overline{y}^{(\beta+\delta)}=\overline{x}^{\epsilon}\overline{y}^{n}(-\overline{y}^{m-1}-\overline{g})=$
$-\overline{x}^{\epsilon}\overline{y}^{(n+m-1)}-\overline{x}^{\epsilon}\overline{y}^{n}\overline{g}.$
Observe que $\overline{x}^{\epsilon}\overline{y}^{(n+m-1)}\in B$ e mais, se
tomarmos $f_{l}=\overline{x}^{\epsilon}\overline{y}^{(n+m-1)}$ temos que
$\rho_{i}+\rho_{j}=(\alpha+\gamma)(m-1)+(\beta+\delta)m=(m+\epsilon)(m-1)+nm=$
$\epsilon(m-1)+(m-1+n)m=\rho_{l}.$
Um monômio de $G$ com coeficiente não nulo é da forma $x^{\kappa}y^{\lambda}$
com $\kappa\leqslant gr_{x}(G)=d$ e $\kappa+\lambda<m-1$.
Se
$(\epsilon,\eta),(\kappa,\lambda)\in\mathbb{N}_{0}^{2},\,\epsilon<m,\,\kappa\leqslant
d,\,\kappa+\lambda<m-1$ e $\rho_{l}=\epsilon(m-1)+(m-1+n)m$, então
$\overline{x}^{\epsilon+\kappa}\overline{y}^{\eta+\lambda}\in L_{l-1}$. De
fato, mostraremos isso nos dois casos que seguem.
1. (a)
Se $\epsilon+\kappa<m$, então
$\overline{x}^{\epsilon+\kappa}\overline{y}^{\eta+\lambda}\in B$ e mais,
$(\eta+\lambda)m+(\epsilon+\kappa)(m-1)<\epsilon(m-1)+(\eta+m-1)m=\rho_{l}$,
assim, $\overline{x}^{\epsilon+\kappa}\overline{y}^{\eta+\lambda}\in L_{l-1}$.
2. (b)
Agora, se $\epsilon+\kappa\geqslant m$, temos que
$\epsilon+\kappa=m+\epsilon^{\prime}$ com $\epsilon^{\prime}<\epsilon$, pois
$\kappa\leqslant d<m$ e $\epsilon<m$. Do mesmo modo fazemos
$\eta+\lambda=\eta^{\prime}$. Assim
$\overline{x}^{\epsilon+\kappa}\overline{y}^{\eta+\lambda}=\overline{x}^{m+\epsilon^{\prime}}\overline{y}^{\eta^{\prime}}=-\overline{x}^{\epsilon^{\prime}}\overline{y}^{m-1+\eta^{\prime}}-\overline{x}^{\epsilon^{\prime}}\overline{y}^{\eta^{\prime}}\overline{g}.$
Observe que
$\rho_{l^{\prime}}=\epsilon^{\prime}(m-1)+(m-1+\eta^{\prime})m=(m+\epsilon^{\prime})(m-1)+\eta^{\prime}m=(\epsilon+\kappa)(m-1)+(\eta+\lambda)m<\rho_{l}$,
assim $f_{l}\in L_{l-1}$, e mais, recursivamente podemos mostrar que
$\overline{x}^{\epsilon^{\prime}}\overline{y}^{\eta^{\prime}}\overline{g}\in
L_{l-1}$.
Assim mostramos que se $f_{i}f_{j}\in L_{l(i,j)}$ então
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$.
Desse modo pelo teorema 4.1.1, temos que $R$ admite uma função peso e que essa
do modo que foi construida é gerada por $m-1$ e $m$. ∎
###### Exemplo 4.1.3.
Seja $\mathcal{X}$ a curva definida no exemplo 4.1.2. O semigrupo numérico
gerado pela função peso $\rho$, do mesmo exemplo, tem $g={{m-1}\choose{2}}$
lacunas. Sejam $Q$ um conjunto de $n$ pontos racionais de $\mathcal{X}$ e $k$
tal que $\rho_{k}=lm$, com $(l>m,\textrm{ e }lm<n)$. O código de avaliação
$C=E_{k}$, determinado por $ev_{Q}$, é um $[n,k,d]-$código com, $d\geqslant
n-lm$ e $k=lm+1-g=lm+1-{{m-1}\choose{2}}$.
###### Proof.
Temos que o semigrupo determinado $\rho$ é gerado por $m$ e $m-1$. Assim, pela
proposição 3.4.7, temos que $g={m-1\choose 2}$. Pelo teorema 3.5.1, temos que
$d\geqslant n-lm$. Agora, do lema 3.4.3(1) segue que $\rho_{k}=k+g-1$, assim
$k=lm-g+1$. ∎
Note que a dimensão e a cota inferior para a distância mínima do código do
último exemplo e do código do teorema 2.2.4 são iguais.
Em vários artigos encontramos referências aos códigos geométricos de Goppa
pontuais, códigos da forma $C(D,mQ)$, onde $Q$ é um ponto racional e $m$ um
inteiro. Os códigos de avaliação que aqui construimos foram propostos como um
modo de estudo dos códigos de Goppa pontuais de modo simples. A principio
pensava-se que os códigos de avaliação generalisavam os de Goppa pontuais, mas
recentemente fora provado que isso é falso. Nos exemplos que seguem fazemos
uma associação dos códigos geométricos de Goppa pontuais aos de avaliação.
No livro Algebraic Function Fields and Codes de Hennin Stichtenoth, página
113, temos um exemplo de um corpo de funções no qual o gênero é dado por,
$g=(m-1)/2$ caso $m$ seja ímpar e $g=(m-2)/2$ caso contrário.
###### Exemplo 4.1.4.
Sejam $K$ um corpo finito de caracteristica diferente de 2, $\mathcal{X}$ a
curva definida por $y^{2}=f(x)=p_{1}(x)\cdots p_{s}(x)\in K[x]$, onde
$p_{1}(x),\ldots,p_{s}(x)$ são polinômios mônicos irredutiveis distintos entre
si, $s\geqslant 1$ e $F$ o corpo de frações do anel de coordenada de
$\mathcal{X}$. Assim $K$ é o corpo de constantes de $F$ e se $m=gr(f(x))$ for
impar temos que o gênero de $F$ é $(m-1)/2$.
Agora sejam $P,P_{1},\ldots,P_{n}$, places de grau 1 dois a dois disjuntos,
$D=P_{1}+\cdots+P_{n}$, $G=lmP$ um divisor em $F/K$, $m<gr(G)=lm<n$ e
$supp(G)\cap supp(D)=\emptyset$. Então o código geométrico de Goppa $C(D,G)$
(definição 2.1.3) tem parâmetros, $k=lm+1-(m-1)/2$ e $d\geqslant n-lm$.
Faremos agora a construção de um código de avaliação.
###### Exemplo 4.1.5.
Sejam $K$ um corpo finito com caracteristica diferente de 2,
$f(x,y)=y^{2}+g(x)$ um polinômio em $K[x,y]$ tal que $g(x)\in K[x]$,
$\mathcal{X}$ a curva plana gerada por $f(x,y)$ e $m=gr(g(x))$ impar, em
particular $f(x,y)$ é irredutível. Faça $R=K[x,y]/\langle f(x,y)\rangle$.
Assim $R$ é uma $K-$álgebra que admite uma função peso, $\rho$, gerada por 2 e
$m$.
###### Proof.
Denotaremos por $\overline{h}$ à classe $h+\langle f(x,y)\rangle$ de $R$. O
conjunto $B=\\{\overline{x}^{b}\overline{y}^{a};a<2\\}$ é uma $K-$base para
$R$. De fato, observe que dada uma soma finita da forma
$\sum\lambda_{ab}\overline{x}^{b}\overline{y}^{a}=0$, teriamos em $K[x,y]$ a
igualdade $\sum\lambda_{ab}x^{b}y^{a}=h(x,y)f(x,y)$, para algum $h(x,y)\in
K[x,y]$, contudo $gr_{y}(h(x,y)f(x,y))\geqslant 2$ enquanto
$gr_{y}\left(\sum\lambda_{ab}x^{b}y^{a}\right)<2$, assim $B$ é um conjunto
linearmente independente. Mostremos agora que $B$ gera $R$.
É suficiente mostrar que $B$ gera os elementos da forma
$\overline{y}^{c}\overline{x}^{d}$, pois todos elementos em $R$ são somas de
elementos desse tipo. Se $c<2$ temos que $\overline{y}^{c}\overline{x}^{d}\in
B$, ou seja, não tem o que fazer, vamos supor agora que $c\geqslant 2$.
Podemos então escrever $c=a+2k$ com $a<2$, assim
$\overline{y}^{c}\overline{x}^{d}=\overline{y}^{a}\overline{x}^{d}\overline{g(x)}^{k}$,
pois $\overline{y}^{2}=\overline{g(x)}$, logo $B$ é uma $K-$base de $R$.
Enumeramos $B$ como $\\{f_{1},f_{2},\ldots\\}$. Agora sendo
$f_{i}=\overline{y}^{a}\overline{x}^{b}\in B$, definimos $\rho_{i}=2b+am$,
observe que $2b+am=2\tilde{b}+\tilde{a}m$, com $a,\tilde{a}<2$ se, e somente,
$a=\tilde{a}$ e $b=\tilde{b}$, visto que $mdc(2,m)=1$, assim reenumerando caso
necessário, assumiremos que $(\rho_{i})_{i\in\mathbb{N}}$ é uma seqüência
estritamente crescente.
Seja $l(i,j)$ o menor inteiro tal que $f_{i}f_{j}\in L_{l(i,j)}$, queremos
mostrar que $l(i+1,j)>l(i,j)$, ou seja, temos que mostrar que
$\rho_{l(i,j)}<\rho_{l(i+1,j)}$, para isso provaremnos que
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$.
Sejam $f_{i}=\overline{x}^{a}\overline{y}^{b}$ e
$f_{j}=\overline{x}^{c}\overline{y}^{d}$, com $b<2$ e $d<2$. Se $b+d<2$ temos
que $f_{i}f_{j}\in B$ e logo $\rho_{l(i,j)}=\rho_{i}+\rho_{j}$. Suponha agora
$b+d\geqslant 2$, temos que $b+d=2+\lambda$ com $\lambda<2$, assim
$f_{i}f_{j}=\overline{y}^{\lambda}\overline{x}^{a+c}\overline{g(x)}$. Note que
para $f_{h}=\overline{y}^{\lambda}\overline{x}^{a+c+m}$, temos que
$\rho_{h}=2a+2c+2m+\lambda
m=2(a+c)+m(\lambda+2)=2(a+c)+m(b+d)=\rho_{i}+\rho_{j}$, observe também que
outro monômio $f_{t}$ de
$\overline{y}^{\lambda}\overline{x}^{a+c}\overline{g(x)}$ é tal que
$\rho_{t}<\rho_{h}$ assim $l(i,j)=h$, concluimos então que
$\rho_{l(i,j)}=\rho_{i}+\rho_{j}$ e assim
$\rho_{l(i+1,j)}=\rho_{i+1}+\rho_{j}>\rho_{i}+\rho_{j}=\rho_{l(i,j)}$.
Agora pelo teorema 4.1.1, temos que existe uma função peso em $R$. ∎
###### Exemplo 4.1.6.
Com as hipóteses do exemplo 4.1.5, tome um conjunto $P$ com $n$ pontos
racionais distintos de $\mathcal{X}$. Além disso, seja $k$ dado por
$\rho_{k}=lm$. Temos então que o código $E_{k}$ tem parâmetros iguais ao do
código $C(D,G)$ visto em 4.1.4.
## References
* [1] Fulton, W., Algebraic Curves. Benjamin, New York, 1969.
* [2] Garcia, Arnaldo e Lequain, Yves, Elementos de Álgebra. Associação Instituto Nacional de Matemática Pura e Aplicada. Rio de Janeiro, 2003. (Projeto Euclides)
* [3] Goppa, V. D., Codes on Algebraic Curves. Soviet Math. Dokl. vol 24, No.1, pp. 170-172, 1981.
* [4] Hefez, A. e Villela, M. L. T., Códigos Corretores de Erros, (Série Computação e Matemática), IMPA. Rio de Janeiro, 2002.
* [5] Lang, Serge, Algebra, Third Edition. Addison-Wesley Publishing Company. Massachusetts, 1997.
* [6] MacWlliams, F. J. and Sloane, N. J. A., The theory of error-correcting codes. (North-Holland Mathematical Library; 16). North-Holland, Amsterdam, 1977.
* [7] Stichtenoth, Hennin, Algebraic Function Fields and Codes. Springer-Verlag, Berlin, 1993.
* [8] van Lint, J. H., Introduction to Coding Theory, 3rd rev. and expand ed. Springer - Verlag Berlin Heidelberg New York, New York, 1998.
* [9] van Lint, Jacobus H. , Pellikaan, Ruud and Høhold, Tom, Algebraic geometry codes In the Handbook of Coding Theory, vol 1, pp. 871-961, Elsevier, Amsterdam, 1998.
* [10] van Lint, Jacobus H. , Pellikaan, Ruud and Høhold, Tom, An Elementary Approach to Algebraic Geometry Codes, Congressus Numerantium 135, pp. 25-35, 1998.
## Index
* Adele Definição 1.6.2
* Anel
* de coordenadas item 2
* de valorização Definição 1.3.3
* Aplicação
* de Resíduos Definição 1.3.11
* Classe
* Lateral Definição 1.2.9
* Código Definição 1.1.1
* de Avaliação Definição 3.2.3
* Linear Definição 1.2.1
* Componente
* Local Definição 1.6.16
* Condutor §3.4
* Conjunto
* algébrico Definição 1.7.1
* Corpo
* de constantes §1.3
* de funções algébricas Definição 1.3.1
* de funções racionais Exemplo 1.3.2, §1.4
* de funções racionais de $\mathcal{X}$ Definição 1.7.3
* Cota
* de Singleton Teorema 1.2.12
* Ordem Definição 3.3.6
* Curva
* não singular Definição 1.7.12
* Diferencial
* de Weil Definição 1.6.8
* Dimensão
* da variedade Definição 1.7.4
* do divisor Definição 1.5.6
* Distancia
* de Hamming Definição 1.1.2
* Divisor
* canônico item 4
* Divisores Definição 1.5.1
* equivalentes Definição 1.5.3
* Espaço
* de adeles §1.6
* projetivo §1.7.1
* Função
* grau Definição 3.1.4
* ordem Definição 3.1.2
* peso item 5
* Gênero Definição 1.5.10
* Grau
* do divisor §1.5
* Grupo
* de divisores principais Definição 1.5.3
* Ideal
* homogêneo Definição 1.7.6
* Índice
* de especialidade Definição 1.6.1
* Lugar item 1
* Matriz
* geradora Definição 1.2.3
* Síndrome Definição 3.3.1
* Morfismo
* de $\mathbb{F}_{q}-$álgebras Definição 3.2.2
* Peso Definição 1.1.3
* Polinômio
* Homogêneo Definição 1.7.6
* Pólo Definição 1.3.13
* Ponto
* não singular Definição 1.7.12
* Semigrupo
* de $\rho$ §3.4
* numérico Definição 3.4.1
* Simétrico Definição 3.4.5
* Sindrome Definição 1.2.6
* Suporte
* do divisor §1.5
* Teorema
* de Bézout Teorema 2.2.1
* de Riemann Teorema 1.5.11
* Riemann-Roch Corolário 1.6.14
* Valorização
* Discreta Definição 1.3.8
* Variedade
* afim item 1
* algébrica projetiva Definição 1.7.7
* Zero Definição 1.3.13
|
arxiv-papers
| 2012-04-06T20:35:19 |
2024-09-04T02:49:29.421294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nolmar Melo",
"submitter": "Nolmar Melo",
"url": "https://arxiv.org/abs/1204.1559"
}
|
1204.1580
|
# Certifying the Restricted Isometry Property is Hard
Afonso S. Bandeira, Edgar Dobriban, Dustin G. Mixon, William F. Sawin A.S.
Bandeira is with the Program in Applied and Computational Mathematics,
Princeton University, Princeton, New Jersey 08544 USA (e-mail:
ajsb@math.princeton.edu).E. Dobriban is with the Department of Statistics,
Stanford University, Stanford, California 94305 USA (e-mail:
dobriban@stanford.edu).D.G. Mixon is with the Department of Mathematics and
Statistics, Air Force Institute of Technology, Wright-Patterson AFB, Ohio
45433 USA (e-mail: dustin.mixon@afit.edu).W.F. Sawin is with the Department of
Mathematics, Princeton University, Princeton, New Jersey 08544 USA (e-mail:
wsawin@math.princeton.edu).The authors thank Boris Alexeev for reading this
manuscript and providing thoughtful comments and suggestions. Bandeira was
supported by NSF Grant No. DMS-0914892, and Mixon was supported by the A.B.
Krongard Fellowship. The views expressed in this article are those of the
authors and do not reflect the official policy or position of the United
States Air Force, Department of Defense, or the U.S. Government.
###### Abstract
This paper is concerned with an important matrix condition in compressed
sensing known as the restricted isometry property (RIP). We demonstrate that
testing whether a matrix satisfies RIP is ${\mathsf{NP}}$-hard. As a
consequence of our result, it is impossible to efficiently test for RIP
provided ${\mathsf{P}}\neq{\mathsf{NP}}$.
## I Introduction
It is now well known that compressed sensing offers a method of taking few
sensing measurements of high-dimensional sparse vectors, while at the same
time enabling efficient and stable reconstruction [1]. In this field, the
restricted isometry property is arguably the most popular condition to impose
on the sensing matrix in order to acquire state-of-the-art reconstruction
guarantees:
###### Definition 1.
We say a matrix $\Phi$ satisfies the $(K,\delta)$-restricted isometry property
(RIP) if
$(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}\leq(1+\delta)\|x\|^{2}$
for every vector $x$ with at most $K$ nonzero entries.
To date, RIP-based reconstruction guarantees exist for Basis Pursuit [2],
CoSaMP [3] and Iterative Hard Thresholding [4], and the ubiquitous utility of
RIP has made the construction of RIP matrices a subject of active research
[5]–[7]. Here, random matrices have found much more success than deterministic
constructions [5], but this success is with high probability, meaning there is
some (small) chance of failure in the construction. Furthermore, RIP is a
statement about the conditioning of all $\binom{N}{K}$ submatrices of an
$M\times N$ sensing matrix, and so it seems computationally intractable to
check whether a given instance of a random matrix fails to satisfy RIP; it is
widely conjectured that certifying RIP for an arbitrary matrix is
${\mathsf{NP}}$-hard. In the present paper, we prove this conjecture.
###### Problem 2.
Given a matrix $\Phi$, a positive integer $K$, and some $\delta\in(0,1)$, does
$\Phi$ satisfy the $(K,\delta)$-restricted isometry property?
In short, we show that any efficient method of solving Problem 2 can be called
in an algorithm that efficiently solves the ${\mathsf{NP}}$-complete subset
sum problem. As a consequence of our result, there is no method by which one
can efficiently test for RIP provided ${\mathsf{P}}\neq{\mathsf{NP}}$. This
contrasts with previous work [8], in which the reported hardness results are
based on less-established assumptions on the complexity of dense subgraph
problems.
In the next section, we review the basic concepts we will use from
computational complexity, and Section 3 contains our main result.
## II A brief review of computational complexity
In complexity theory, problems are categorized into complexity classes
according to the amount of resources required to solve them. For example, the
complexity class ${\mathsf{P}}$ contains all problems which can be solved in
polynomial time, while problems in ${\mathsf{EXP}}$ may require as much as
exponential time. Problems in ${\mathsf{NP}}$ have the defining quality that
solutions can be verified in polynomial time given a certificate for the
answer. As an example, the graph isomorphism problem is in ${\mathsf{NP}}$
because, given an isomorphism between graphs (a certificate), one can verify
that the isomorphism is legitimate in polynomial time. Clearly,
${\mathsf{P}}\subseteq{\mathsf{NP}}$, since we can ignore the certificate and
still solve the problem in polynomial time.
While problem categories provide one way to describe complexity, another
important tool is the polynomial-time reduction, which allows one to show that
a given problem is “more complex” than another. To be precise, a polynomial-
time reduction from problem $A$ to problem $B$ is a polynomial-time algorithm
that solves problem $A$ by exploiting an oracle which solves problem $B$; the
reduction indicates that solving problem $A$ is no harder than solving problem
$B$ (up to polynomial factors in time), and we say “$A$ reduces to $B$,” or
$A\leq B$. Such reductions lead to some of the most popular definitions in
complexity theory: We say a problem $B$ is called ${\mathsf{NP}}$-hard if
every problem $A$ in ${\mathsf{NP}}$ reduces to $B$, and a problem is called
${\mathsf{NP}}$-complete if it is both ${\mathsf{NP}}$-hard and in
${\mathsf{NP}}$. In plain speak, ${\mathsf{NP}}$-hard problems are harder than
every problem in ${\mathsf{NP}}$, while ${\mathsf{NP}}$-complete problems are
the hardest of problems in ${\mathsf{NP}}$.
Contrary to popular intuition, ${\mathsf{NP}}$-hard problems are not merely
problems that seem to require a lot of computation to solve. Of course,
${\mathsf{NP}}$-hard problems have this quality, as an ${\mathsf{NP}}$-hard
problem can be solved in polynomial time only if ${\mathsf{P}}={\mathsf{NP}}$;
this is an open problem, but it is widely believed that
${\mathsf{P}}\neq{\mathsf{NP}}$ [9]. However, there are other problems which
seem hard but are not known to be ${\mathsf{NP}}$-hard (e.g., the graph
isomorphism problem). As such, while testing for RIP in the general case seems
to be computationally intensive, it is not obvious whether the problem is
actually ${\mathsf{NP}}$-hard. Indeed, by the definition of
${\mathsf{NP}}$-hard, one must compare its complexity to the complexity of
every problem in ${\mathsf{NP}}$. To this end, notice that $A\leq B$ and
$B\leq C$ together imply $A\leq C$, and so to demonstrate that a problem $C$
is ${\mathsf{NP}}$-hard, it suffices to show that $B\leq C$ for some
${\mathsf{NP}}$-hard problem $B$.
In the present paper, we demonstrate the hardness of certifying RIP by
reducing from the following problem:
###### Problem 3.
Given a matrix $\Psi$ and some positive integer $K$, do there exist $K$
columns of $\Psi$ which are linearly dependent?
Problem 3 has a brief history in computational complexity. First, McCormick
[10] demonstrated that the analogous problem of testing the girth of a
transversal matroid is ${\mathsf{NP}}$-complete, and so by invoking the
randomized matroid representation of Marx [11], Problem 3 is hard for
${\mathsf{NP}}$ under randomized reductions [12]. Next, Khachiyan [13] showed
that the problem is ${\mathsf{NP}}$-hard by focusing on the case where $K$
equals the number of rows of $\Psi$; using a particular matrix construction
with Vandermonde components, he reduced this instance of the problem to the
subset sum problem. Recently, Tillmann and Pfetsch [14] used ideas similar to
McCormick’s to strengthen Khachiyan’s result: they prove Problem 3 is
${\mathsf{NP}}$-hard without focusing on such a specific instance of the
problem. Each of these complexity results use $M\times N$ matrices with
integer entries whose binary representations take $\leq p(M,N)$ bits for some
polynomial $p$; we will exploit this feature in our proof.
## III Main result
###### Theorem 4.
Problem 2 is ${\mathsf{NP}}$-hard.
###### Proof.
Reducing from Problem 3, suppose we are given a matrix $\Psi$ with integer
entries. Letting $\mathrm{Spark}(\Psi)$ denote the size of the smallest
collection of linearly dependent columns of $\Psi$, we wish to determine
whether $\mathrm{Spark}(\Psi)\leq K$. To this end, we take $P\leq 2^{p(M,N)}$
to be the size of the largest entry in $\Psi$, and define
$C=2^{\lceil\log_{2}\sqrt{MN}P\rceil}$ and $\Phi=\frac{1}{C}\Psi$; note that
we choose $C$ to be of this form instead of $\sqrt{MN}P$ to ensure that the
entries of $\Phi$ can be expressed in ${\mathsf{poly}}(M,N)$ bits without
truncation. Of course, linear dependence between columns is not affected by
scaling, and so testing $\Phi$ is equivalent to testing $\Psi$. In fact, since
we plan to appeal to an RIP oracle, it is better to test $\Phi$ since the
right-hand inequality of Definition 1 is already satisfied for every
$\delta>0$:
$\displaystyle\|\Phi\|_{2}\leq\sqrt{MN}\|\Phi\|_{\mathrm{max}}=\sqrt{MN}\frac{P}{C}\leq
1\leq\sqrt{1+\delta}.$
We are now ready to state the remainder of our reduction: For some value of
$\delta$ (which we will determine later), ask the oracle if $\Phi$ is
$(K,\delta)$-RIP; then
$\begin{array}[]{rcll}\mbox{(i)}&\mbox{$\Phi$ is
$(K,\delta)$-RIP}&\Longrightarrow&\mathrm{Spark}(\Psi)>K,\\\
\mbox{(ii)}&\mbox{$\Phi$ is not
$(K,\delta)$-RIP}&\Longrightarrow&\mathrm{Spark}(\Psi)\leq K.\end{array}$
The remainder of this proof will demonstrate (i) and (ii).
Note that (i) immediately holds for all choices of $\delta\in(0,1)$ by the
contrapositive. Indeed, $\mathrm{Spark}(\Psi)\leq K$ implies the existence of
a nonzero vector $x$ in the nullspace of $\Phi$ with $\leq K$ nonzero entries,
and $\|\Phi x\|^{2}=0<(1-\delta)\|x\|^{2}$ violates the left-hand inequality
of Definition 1. For (ii), we also consider the contrapositive. When
$\mathrm{Spark}(\Psi)>K$, we have that every size-$K$ subcollection of
$\Psi$’s columns is linearly independent. Letting $\Psi_{\mathcal{K}}$ denote
the submatrix of columns indexed by a size-$K$ subset
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$, this implies that
$\lambda_{\mathrm{min}}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})>0$, and so
$\det(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})>0$. Since the entries of
$\Psi$ lie in $\\{-P,\ldots,P\\}$, we know the entries of
$\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}}$ lie in
$\\{-MP^{2},\ldots,MP^{2}\\}$, and since
$\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}}$ is integral with positive
determinant, we must have $\det(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})\geq
1$. In fact,
$\displaystyle 1$
$\displaystyle\leq\det(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})$
$\displaystyle=\prod_{k=1}^{K}\lambda_{k}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})$
$\displaystyle\leq\lambda_{\mathrm{min}}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})\cdot\lambda_{\mathrm{max}}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})^{K-1}$
$\displaystyle\leq\lambda_{\mathrm{min}}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})\cdot\big{(}K\|\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}}\|_{\mathrm{max}}\big{)}^{K-1},$
and so we can rearrange to get
$\displaystyle\lambda_{\mathrm{min}}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})$
$\displaystyle=\frac{1}{C^{2}}\lambda_{\mathrm{min}}(\Psi_{\mathcal{K}}^{*}\Psi_{\mathcal{K}})$
$\displaystyle\geq\frac{1}{C^{2}(KMP^{2})^{K-1}}\geq 2^{-5MNp(M,N)},$
where the last inequality follows from $K\leq M\leq N$ and other coarse
bounds. Therefore, if we pick $\delta:=1-2^{-5MNp(M,N)}$, then since our
choice for $\mathcal{K}$ was arbitrary, we conclude that $\Phi$ is
$(K,\delta)$-RIP whenever $\mathrm{Spark}(\Psi)>K$, as desired. Moreover,
since $\delta$ can be expressed in the standard representation using
${\mathsf{poly}}(M,N)$ bits, we can ask the oracle our question in polynomial
time. ∎
It is important to note that Theorem 4 is a statement about testing for RIP in
the worst case; this result does not rule out the existence of matrices for
which RIP is easily verified (e.g., using coherence in conjunction with the
Gershgorin circle theorem for small values of $K$ [5]).
## References
* [1] E.J. Candès, T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 51 (2005) 4203–4215.
* [2] E.J. Candès, The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 589–592.
* [3] D. Needell, J.A. Tropp, CoSaMP: Iterative signal recovery from incomplete and inaccurate samples, Appl. Comput. Harmon. Anal. 26 (2009) 301–321.
* [4] T. Blumensath, M.E. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal. 27 (2009) 265–274.
* [5] A.S. Bandeira, M. Fickus, D.G. Mixon, P. Wong, The road to deterministic matrices with the restricted isometry property, Available online: arXiv:1202.1234
* [6] R. DeVore, Deterministic constructions of compressed sensing matrices, J. Complexity 23 (2007) 918–925.
* [7] T. Tao, Open question: Deterministic UUP matrices, Available online: http://terrytao.wordpress.com/2007/07/02/open-question-deterministic-uup-matrices
* [8] P. Koiran, A. Zouzias, On the certification of the restricted isometry property, Available online: arXiv:1103.4984
* [9] S. Cook, The P versus NP problem, Available online: http://www.claymath.org/millennium/P_vs_NP/pvsnp.pdf
* [10] S.T. McCormick, A Combinatorial Approach to Some Sparse Matrix Problems. Ph.D. Thesis, Stanford University (1983)
* [11] D. Marx, A parameterized view on matroid optimization problems, Theor. Comput. Sci. 410 (2009) 4471–4479.
* [12] B. Alexeev, J. Cahill, D.G. Mixon, Full spark frames, Available online: arXiv:1110.3548
* [13] L. Khachiyan, On the complexity of approximating extremal determinants in matrices, J. Complexity 11 (1995) 128–153.
* [14] A.M. Tillmann, M.E. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing, Available online: arXiv:1205.2081
|
arxiv-papers
| 2012-04-06T23:58:53 |
2024-09-04T02:49:29.434372
|
{
"license": "Public Domain",
"authors": "Afonso S. Bandeira, Edgar Dobriban, Dustin G. Mixon, William F. Sawin",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1204.1580"
}
|
1204.1587
|
# Schauder Bases and Operator Theory II: strongly irreducible Schauder
Operators
Yang Cao Yang Cao, Department of Mathematics , Jilin university, 130012,
Changchun, P.R.China caoyang@jlu.edu.cn , Youqing Ji Youqing Ji, Department
of Mathematics , Jilin university, 130012, Changchun, P.R.China
jiyq@jlu.edu.cn and Geng Tian Geng Tian, Department of Mathematics , Jilin
university, 130012, Changchun, P.R.China tiangeng09@mails.jlu.edu.cn
(Date: Feb. 14, 2012)
###### Abstract.
In this paper, we will show that for an operator $T$ which is injective and
has dense range, there exists an invertible operator $X$ (in fact we can find
$U+K$, where $U$ is an unitary operator and $K$ is a compact operator with
norm less than a given positive real number) such that $XT$ is strongly
irreducible. As its application, strongly irreducible operators always exist
in the orbit of Schauder matrices.
###### Key words and phrases:
.
###### 2000 Mathematics Subject Classification:
Primary 47A55, 47A53, 47A16; Secondary 54H20, 37B99.
## 1\. Introduction and preliminaries
From the viewpoint of matrices, Schauder bases and operator theory have
natural relations. For example, the column vector sequence of the matrix of an
invertible operator comprise a Riesz basis under any orthonormal basis (ONB).
In our series paper, we focus on the operator which has a matrix
representation consisting of a Schauder basis under appropriate ONB (We shall
call them Schauder operators and Schauder matrices respectively).
In the matrix theory of finite dimensional space, the famous Jordan canonical
form theorem sufficiently reveals the internal structure of matrices. Works of
Jiang C. L., Ji Y. Q. etc. show that the strongly irreducible operators can be
seen as the generalized Jordan block in the case of separable infinite
dimensional Hilbert space. It will be shown that multiplying the matrix of any
invertible operator on the left of a Schauder matrix gets an equivalent
Schauder matrix (Theorem 3.3). Since Jordan blocks are the objects with
relatively simple properties, a natural question can be raised as follows:
###### Question 1.1.
Given any Schauder matrix $M$, does there exist a matrix $X$ of some
invertible operator such that $XM$ is strongly irreducible?
In this paper, we shall give an affirmative answer to this question. Following
theorem is our main result in operator theory which will be used to solve the
question.
###### Theorem 1.2.
Let $T\in\mathcal{L}(\mathcal{H})$ satisfy $KerT=(0)$ and
$\overline{RanT}=\mathcal{H}$. Then for any $\epsilon>0$, there exist an
unitary operator $U$ and a compact operator $K$ with $||K||<\epsilon$ such
that $(U+K)T$ is strongly irreducible. Moreover, since $\epsilon$ can be
chosen small enough, $U+K$ could be invertible.
###### Remark 1.3.
This theorem does not only motivated by basis theory observations. There exist
some other observations. We summarize them as follows:
1\. Physical backgrounds. In the Mathematics and Physics Interdisciplinary
Seminar at Jilin University, Professor Hai-Jun Wang of Physics Department told
us that multiplying an operator $X$ on the left of a Hamiltonian operator $T$
can be viewed as an evolution (or an evolving step) of a physics system which
is encoded by the Hamiltonian operator $T$. Moreover, the strong
irreducibility of an operator $T$ ensures that there is no subsystem
developing independently to the rest of the system. He also suggested us to
consider the Schauder basis as a sequence of "superposition states". From this
viewpoint, the special case of theorem 1.2 in which $T$ is a self-adjoint
operator may be more interesting. It could be useful in characterizing
evolution of observables.
2\. Compare to the classical results in the theory of strongly irreducible
operators. D.A.Herrero very concerned about compact perturbations of strongly
irreducible operators and asked that for any operator $T$ with connected
spectrum and any $\epsilon>0$, does there exist a compact operator $K$ with
$||K||<\epsilon$ such that $T+K$ is strongly irreducible? This question is the
essential strengthen of the question asked by G.H.Gong. Among many years,
C.L.Jiang, Y.Q.Ji, Z.Y.Wang, S.H.Sun and S.Power obtain the affirmative answer
step by step.
###### Theorem 1.4.
[14, 9, 10, 11, 13, 6, 8] Let $T\in\mathcal{L}(\mathcal{H})$, $\sigma(T)$
connected. Then for any $\epsilon>0$, there exists a compact operator
$K\in\mathcal{L}(\mathcal{H})$ with $||K||<\epsilon$ such that $T+K$ is
strongly irreducible.
Theorem 1.2 is a parallel consideration along the line of above theorem.
Theorem 1.2 focus on the action of multiplying an operator on the left while
theorem 1.4 focus on the action of adding an operator.
3\. Basis theory observations. Operator which is injective and has dense range
has a natural basis theory understanding, that is the main topic of our next
paper. We introduce the main result here:
###### Theorem 1.5.
Let $T\in\mathcal{L}(\mathcal{H})$, then $T$ is a Schauder operator if and
only if $T$ is injective and has dense range.
With this theorem in mind, question 1.1 naturally stimulate us to consider it
towards theorem 1.2.
We organize this paper as follows. The second section contains the proof of
our main theorem 1.2. In the last section we apply the main theorem into basis
theory. To generalize the class of bases which can be studied by operator
theory, we also introduce the blowing up matrix in that section.
### 1.1. Notations
Let $\mathcal{H}_{1},\mathcal{H}_{2},\mathcal{H}$ be complex separable Hilbert
spaces. Denote by $\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{2})$ the set of
all bounded linear operators mapping $\mathcal{H}_{1}$ into $\mathcal{H}_{2}$.
Denote by $\mathcal{K}(\mathcal{H}_{1},\mathcal{H}_{2})$ the subset of
$\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{2})$ of all compact operators. We
simply write $\mathcal{L}(\mathcal{H})$ and $\mathcal{K}(\mathcal{H})$ instead
of $\mathcal{L}(\mathcal{H},\mathcal{H})$ and
$\mathcal{K}(\mathcal{H},\mathcal{H})$ respectively. For
$T\in\mathcal{L}(\mathcal{H}_{1},\mathcal{H}_{2})$, denote the kernel of $T$
and the range of $T$ by Ker$T$ and Ran$T$ respectively. Let
$T\in\mathcal{L}(\mathcal{H})$, denote by $\sigma(T)$, $\sigma_{p}(T)$,
$\sigma_{e}(T)$, $\sigma_{lre}(T)$ and $\sigma_{W}(T)$ the spectrum, the point
spectrum, the essential spectrum, the Wolf spectrum and the Weyl spectrum of
$T$ respectively. Denote by $\sigma_{0}(T)$ the set of isolated points of
$\sigma(T)\backslash\sigma_{e}(T)$. For $\lambda\in\rho_{s-F}(T)$ $(\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}\mathbb{C}\backslash\sigma_{lre}(T))$,
${\rm ind}(\lambda-T)={\rm dimKer}(\lambda-T)-{\rm dimKer}(\lambda-T)^{*}$.
Denote
$\rho_{s-F}^{(n)}(T)=\\{\lambda\in\rho_{s-F}(T);~{}\text{ind}(\lambda-T)=n\\}$,
where $-\infty\leq n\leq\infty$. $T$ is said to be quasi-triangular if there
is a sequence $\\{P_{n}\\}_{n=1}^{\infty}$ of finite rank projections
increasing to the unit operator $I$ with respect to the strong operator
topology such that $\lim\limits_{n\rightarrow\infty}||(I-P_{n})TP_{n}||=0$. It
is well-known that $T$ is quasi-triangular if and only if ind$(T-\lambda)\geq
0$ for all $\lambda\in\rho_{s-F}(T)$. $T$ is said to be strongly irreducible
if there are no nontrivial idempotents commuting with $T$. A Cowen-Douglas
operator is an operator $T$ satisfying the following conditions:
(1) There is a nonempty connected open subset $\Omega$ of
$\rho_{s-F}^{(n)}(T)$ for a natural number n;
(2) $T-\lambda$ is surjective for each $\lambda\in\Omega$;
(3) $\bigvee_{\lambda\in\Omega}{\rm ker}(T-\lambda)=\mathcal{H}$.
If the conditions above are satisfied, we shall write
$T\in\mathcal{B}_{n}(\Omega)$.
## 2\. Main Results
### 2.1.
First, let us introduce some known results. Denote by $r(T)$ the spectral
radius of $T$. It is well-known that
$r(T)=\lim\limits_{n\rightarrow\infty}||T^{n}||^{\frac{1}{n}}.$
A bilateral weighted shift $T\in\mathcal{L}(\mathcal{H})$ is an operator that
maps each vector in some orthonormal basis $\\{e_{n}\\}_{n\in\mathbb{Z}}$ into
a scalar multiple of the next vector, $Te_{n}=\omega_{n}e_{n+1}$, for all
$n\in\mathbb{Z}$.
###### Lemma 2.1.
[18] 1) If $T\in\mathcal{L}(\mathcal{H})$ is an invertible bilateral weighted
shift, then the spectrum of $T$ is the annulus
$\\{\lambda\in\mathbb{C};[r(T^{-1})]^{-1}\leq|\lambda|\leq r(T)\\}$.
2) If $T\in\mathcal{L}(\mathcal{H})$ is a bilateral weighted shift that is not
invertible, then the spectrum of $T$ is the disc
$\\{\lambda\in\mathbb{C};|\lambda|\leq r(T)\\}$.
###### Lemma 2.2.
[12] Let $T$ be a bilateral weighted shift operator with weight sequence
$\\{\omega_{n}\\}_{n\in\mathbb{Z}}$. Then $T$ is a Cowen-Douglas operator if
and only if there exists a $\lambda_{0}\in\rho_{F}(T)$ such that
ind$(T-\lambda_{0})=1$.
###### Remark 2.3.
In the above lemma, if there exists a $\lambda_{0}\in\rho_{F}(T)$ such that
ind$(T-\lambda_{0})=1$, then $T$ must belong to $\mathcal{B}_{1}(\Omega)$ for
some open connected subset of $\mathbb{C}$. One can see [12] for details.
###### Lemma 2.4.
[9] Any operator $T\in\mathcal{B}_{1}(\Omega)$ is strongly irreducible.
###### Lemma 2.5.
Let $T\in\mathcal{L}(\mathcal{H})$ be a bilateral weighted shift with weight
sequence $\\{\omega_{n}\\}_{n\in\mathbb{Z}}$, $\omega_{n}>0$ for
$n\in\mathbb{Z}$ and max$\\{\omega_{n};n\geq 0\\}<$ min$\\{\omega_{n};n<0\\}$.
Then $T\in\mathcal{B}_{1}(\Omega)$ for some open connected subset $\Omega$ of
$\mathbb{C}$ and hence strongly irreducible.
###### Proof.
Let $\mathcal{H}_{1}=\bigvee_{n\leq 0}\\{e_{n}\\}$,
$\mathcal{H}_{2}=\bigvee_{n\geq 1}\\{e_{n}\\}$, then
$T=\begin{bmatrix}\begin{bmatrix}\ddots\\\ \ddots&0\\\
&\omega_{-1}&0\end{bmatrix}&0\\\ \begin{bmatrix}\cdots&0&\omega_{0}\\\ &0&0\\\
{\mathinner{\mkern 2.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise
4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\hbox{.}\mkern 1.0mu}}&&\vdots\\\
\end{bmatrix}&\begin{bmatrix}0\\\ \omega_{1}&0\\\
&\ddots&\ddots\end{bmatrix}\end{bmatrix}\begin{matrix}\vdots\\\ e_{-1}\\\
e_{0}\\\ e_{1}\\\ e_{2}\\\ \vdots\end{matrix}\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}\begin{bmatrix}A\\\
C&B\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\end{matrix}.$
Choose any $\lambda$ such that max$\\{\omega_{n};n\geq 0\\}<\lambda<$
min$\\{\omega_{n};n<0\\}$. We will show that $T-\lambda$ is a Fredholm
operator with index 1.
On the one hand, ${\rm inf}_{x\neq 0}\frac{||A^{*}x||}{||x||}\geq$
min$\\{\omega_{n};n<0\\}>\lambda$,
$||(A^{*}-\lambda)(x)||\geq||A^{*}x||-\lambda||x||\geq({\rm
min}\\{\omega_{n};n<0\\}-\lambda)||x||$ for any $x\in\mathcal{H}_{1}$, hence
$(A-\lambda)^{*}$ is bounded from below and $A-\lambda$ is right invertible.
Let $X_{11}\in\mathcal{L}(\mathcal{H}_{1})$ satisfy $(A-\lambda)X_{11}=I$.
On the other hand, $r(B)=\lim\limits_{k\rightarrow\infty}({\rm sup}_{n\geq
0}\\{\omega_{n+1}\omega_{n+2}\cdots\omega_{n+k}\\})^{\frac{1}{k}}<\lambda$,
hence $B-\lambda$ is invertible.
Let
$X=\begin{bmatrix}X_{11}&0\\\
-(B-\lambda)^{-1}CX_{11}&(B-\lambda)^{-1}\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\
\mathcal{H}_{2}\end{matrix},$
then $(T-\lambda)X=I$. It is easy to compute that Ker$(T-\lambda)=\\{\alpha
x;\alpha\in\mathbb{C}\\}$, where
$\displaystyle
x=\sum_{n\in\mathbb{Z}}x_{n}e_{n},~{}x_{0}=1,~{}x_{n}=\dfrac{\omega_{0}\omega_{1}\cdots\omega_{n-1}}{\lambda^{n}},~{}x_{-n}=\dfrac{\lambda^{n}}{\omega_{-1}\omega_{-2}\cdots\omega_{-n}},~{}\forall~{}n>0.$
Hence dimKer$(T-\lambda)=1$ and $T-\lambda$ is a Fredholm operator with index
1. From lemma 2.2, remark 2.3, lemma 2.4, we know that
$T\in\mathcal{B}_{1}(\Omega)$ for some open connected subset $\Omega$ of
$\mathbb{C}$ and strongly irreducible. ∎
###### Lemma 2.6.
[4] Let $A\in\mathcal{L}(\mathcal{H}_{1}),B\in\mathcal{L}(\mathcal{H}_{2})$,
denote by $\mathcal{T}_{A,B}$ the Rosenblum operator, then
(1) the following three statements equivalent:
i) Ran$\mathcal{T}_{A,B}=\mathcal{L}(\mathcal{H}_{2},\mathcal{H}_{1})$;
ii) $\sigma_{r}(A)\cap\sigma_{l}(B)=\emptyset$, where $\sigma_{r}(A)$ and
$\sigma_{l}(B)$ are right spectrum and left spectrum respectively;
iii) $\mathcal{K}(\mathcal{H}_{2},\mathcal{H}_{1})\subseteq$
Ran$\mathcal{T}_{A,B}$.
(2) If $\sigma_{l}(A)\cap\sigma_{r}(B)=\emptyset$, then
Ker$\mathcal{T}_{A,B}=\\{0\\}$.
###### Lemma 2.7.
[15] Let $0<l_{j}\leq\infty,j=1,2,$
$\mathcal{H}=\oplus_{j=1}^{2}(\oplus_{0\leq i<l_{j}}\mathcal{H}_{i}^{j})$,
then
$T=\begin{bmatrix}A&Q\\\ &B\end{bmatrix}\begin{matrix}\oplus_{0\leq
i<l_{1}}\mathcal{H}_{i}^{1}\\\ \oplus_{0\leq
i<l_{2}}\mathcal{H}_{i}^{2}\end{matrix}\in\mathcal{L}(\mathcal{H})$
is a strongly irreducible operator when the following conditions satisfy:
i) $A=\oplus_{0\leq i<l_{1}}A_{i},B=\oplus_{0\leq
i<l_{2}}B_{i},Q=(Q_{ij})_{ij}$, where
$A_{i}\in\mathcal{L}(\mathcal{H}_{i}^{1}),B_{i}\in\mathcal{L}(\mathcal{H}_{i}^{2})$,
$Q_{ij}\in\mathcal{L}(\mathcal{H}_{j}^{2},\mathcal{H}_{i}^{1})$;
ii) $A_{i},B_{j}$ are all strongly irreducible operators;
iii) Ker$\mathcal{T}_{B_{j},A_{i}}=\\{0\\}$ for all $i,j$,
Ker$\mathcal{T}_{A_{i},A_{j}}=\\{0\\}$, Ker$\mathcal{T}_{B_{i},B_{j}}=\\{0\\}$
for all $i,j,i\neq j$;
iv) If $\sigma_{r}(A_{i})\cap\sigma_{l}(B_{j})\neq\emptyset$, then
$Q_{ij}\overline{\in}Ran\mathcal{T}_{A_{i},B_{j}}$;
v) For any $0\leq m,n<l_{1}$, there exist $k_{0}<\infty$,
$\\{i_{k}\\}_{k=1}^{k_{0}+1}$, $\\{j_{k}\\}_{k=1}^{k_{0}}$ such that
$\sigma_{r}(A_{i_{k}})\cap\sigma_{l}(B_{j_{k}})\neq\emptyset$,
$\sigma_{r}(A_{i_{k+1}})\cap\sigma_{l}(B_{j_{k}})\neq\emptyset$, $i_{1}=m$,
$i_{k_{0}+1}=n$;
vi) For any $0\leq m,n<l_{2}$, there exist $k_{1}<\infty$,
$\\{i_{k}\\}_{k=1}^{k_{1}}$, $\\{j_{k}\\}_{k=1}^{k_{1}+1}$ such that
$\sigma_{r}(A_{i_{k}})\cap\sigma_{l}(B_{j_{k}})\neq\emptyset$,
$\sigma_{r}(A_{i_{k}})\cap\sigma_{l}(B_{j_{k+1}})\neq\emptyset$, $j_{1}=m$,
$j_{k_{1}+1}=n$.
###### Lemma 2.8.
[7, 6] Let $T\in\mathcal{L}(\mathcal{H})$ be quasi-triangular,
$\sigma(T),\sigma_{W}(T)$ are all connected and
$\lambda_{0}\in\partial\sigma_{W}(T)$. Then for any given $\epsilon>0$, there
exists a compact operator $K\in\mathcal{L}(\mathcal{H})$, $||K||<\epsilon$
such that
i) $T+K$ is strongly irreducible;
ii) If $\lambda_{0}\overline{\in}\sigma_{p}(B)$, then
Ker$\mathcal{T}_{B,T+K}=\\{0\\}$.
### 2.2. Proof of Main Theorem
###### Proof.
From the polar decomposition theorem, we have $T=V|T|.$ Since $KerT=(0)$ and
$\overline{RanT}=\mathcal{H}$, $V$ is an unitarily operator. If the theorem is
correct for positive operator $|T|$, i.e. there exist unitarily operator $U$
and compact operator $K$, $||K||<\epsilon$ such that $(U+K)|T|$ is strongly
irreducible, then $(UV^{*}+KV^{*})T=(U+K)|T|$ is strongly irreducible and the
norm of compact operator $KV^{*}$ is less than $\epsilon$. Hence it suffices
to prove the theorem when T is a positive operator. We will break it into four
cases.
Case 1. Let $T$ be an invertible operator. Since $\sigma_{e}(T)\neq\emptyset$,
we denote
$\lambda_{min}=min\\{\lambda;\lambda\in\sigma_{e}(T)\\},\lambda_{min}=min\\{\lambda;\lambda\in\sigma_{e}(T)\\}$.
From the Weyl-Von Neumann theorem, for $\epsilon>0$, there exists a compact
operator $K_{1}$, $||K_{1}||<\frac{\epsilon}{2||T^{-1}||}$ such that,
1) $T+K_{1}$ is a diagonal operator with entries
$\\{\lambda_{1},\lambda_{2},\ldots\\}$ under an ONB
$\\{e_{n}\\}_{n=1}^{\infty}$;
2) there exist two subsequences $\\{\alpha_{n}\\}_{n=1}^{\infty}$ and
$\\{\beta_{n}\\}_{n=1}^{\infty}$ of $\\{\lambda_{n}\\}_{n=1}^{\infty}$ which
satisfy
$\displaystyle\lim\limits_{n\rightarrow\infty}\alpha_{n}=\lambda_{min},~{}\alpha_{n}<\alpha_{n+1},$
$\displaystyle\lim\limits_{n\rightarrow\infty}\beta_{n}=\lambda_{max},~{}\beta_{n}>\beta_{n+1};$
3)
$\\{\lambda_{1},\lambda_{2},\ldots\\}\backslash\\{\\{\alpha_{1},\alpha_{2},\ldots\\}\bigcup\\{\beta_{1},\beta_{2},\ldots\\}\\}\subseteq[\lambda_{min},\lambda_{max}]$.
Denote the elements of set
$\\{\lambda_{1},\lambda_{2},\ldots\\}\backslash\\{\\{\alpha_{1},\alpha_{2},\ldots\\}\bigcup\\{\beta_{1},\beta_{2},\ldots\\}\\}$
by $\\{\gamma_{n}\\}_{n=1}^{\infty}$. Then we have
$T+K_{1}=\begin{bmatrix}\begin{bmatrix}\ddots&&&&&\\\ &\beta_{2}&\\\
&&\beta_{1}&\\\ &&&\alpha_{1}&\\\ &&&&\alpha_{2}&\\\ &&&&&\ddots\\\
\end{bmatrix}&\\\ &\begin{bmatrix}\gamma_{1}\\\ &\gamma_{2}\\\ &&\ddots\\\
\end{bmatrix}\ext@arrow 0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}D\\\
\end{bmatrix}\begin{matrix}\vdots\\\ f_{4}\\\ f_{2}\\\ f_{1}\\\ f_{5}\\\
\vdots\\\ f_{3}\\\ f_{7}\\\ \vdots\\\ \end{matrix},$
where $\\{f_{n}\\}_{n=1}^{\infty}$ is a rearrangement of ONB
$\\{e_{n}\\}_{n=1}^{\infty}$.
Let
$U=\begin{bmatrix}\ddots\\\ \ddots&0&\\\ &1&0&\\\ &&1&0&\\\ &&&1&0&\\\
&&&&\ddots&\ddots&\\\ \end{bmatrix}\begin{matrix}\vdots\\\ f_{4}\\\ f_{2}\\\
f_{1}\\\ f_{5}\\\ \vdots\\\ \end{matrix}~{}{\rm
and}~{}U_{1}=\begin{bmatrix}U&\\\ &I\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\end{matrix},$
where $\mathcal{H}_{1}=\bigvee\\{f_{2n+2},f_{4n+1},n=0,1,2,\ldots\\}$,
$\mathcal{H}_{2}=\bigvee\\{f_{4n+3},n=0,1,2,\ldots\\}$, then we have
$U_{1}(T+K_{1})=\begin{bmatrix}\begin{bmatrix}\ddots\\\ \ddots&0\\\
&\beta_{2}&0&\\\ &&\beta_{1}&0\\\ &&&\alpha_{1}&0\\\ &&&&\ddots&\ddots\\\
\end{bmatrix}\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}B&\\\ &D\\\
\end{bmatrix}\begin{matrix}\vdots\\\ f_{4}\\\ f_{2}\\\ f_{1}\\\ f_{5}\\\
\vdots\\\ \mathcal{H}_{2}\end{matrix}.$
It is easy to compute that
$\displaystyle
r(B)=\lim\limits_{n\rightarrow\infty}(\beta_{1}\beta_{2}\cdots\beta_{n})^{\frac{1}{n}}=\lim\limits_{n\rightarrow\infty}\beta_{n}=\lambda_{max},$
$\displaystyle
r(B^{-1})^{-1}=\lim\limits_{n\rightarrow\infty}((\alpha_{1}^{-1}\alpha_{2}^{-1}\cdots\alpha_{n}^{-1})^{\frac{1}{n}})^{-1}=\lim\limits_{n\rightarrow\infty}\alpha_{n}=\lambda_{min}.$
From lemma 2.1, the spectrum of $B$ is the annulus
$\\{\lambda\in\mathbb{C};\lambda_{min}\leq|\lambda|\leq\lambda_{max}\\}$.
Moreover, the diagonal operator $D$ is contained in this annulus. Hence the
spectrum $\sigma(U_{1}(T+K_{1}))$ is connected. It follows from the theorem
1.4 that there exists a compact operator $K_{2}$,
$||K_{2}||<\frac{\epsilon}{2||T^{-1}||}$ such that $U_{1}(T+K_{1})+K_{2}$ is
strongly irreducible. Notice that
$\\{U_{1}+(U_{1}K_{1}T^{-1}+K_{2}T^{-1})\\}T=U_{1}(T+K_{1})+K_{2}$ and
$U_{1}K_{1}T^{-1}+K_{2}T^{-1}$ is a compact operator with
$||U_{1}K_{1}T^{-1}+K_{2}T^{-1}||<\epsilon$, we complete the proof of this
case.
Case 2. Let $T$ be an operator with $\sigma_{e}(T)=\\{0\\}$. Since $KerT=(0)$
and $\overline{RanT}=\mathcal{H}$, $T$ must have the matrix representation as
follows,
$T=\begin{bmatrix}\lambda_{0}&\\\ &\lambda_{1}\\\ &&\lambda_{-1}\\\
&&&\lambda_{2}\\\ &&&&\lambda_{-2}\\\ &&&&&\ddots\\\
\end{bmatrix}\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\
\vdots\end{matrix},$
where $\lambda_{n}\neq 0$,
$\lim\limits_{n\rightarrow+\infty}\lambda_{n}=\lim\limits_{n\rightarrow-\infty}\lambda_{n}=0.$
As case 1, we can find an unitarily operator $U$ such that
$UT=\begin{bmatrix}\ddots\\\ \ddots&0\\\ &\lambda_{-1}&0\\\ &&\lambda_{0}&0\\\
&&&\lambda_{1}&0\\\ &&&&\ddots&\ddots\\\ \end{bmatrix}\begin{matrix}\vdots\\\
e_{2}\\\ e_{0}\\\ e_{1}\\\ e_{3}\\\ \vdots\end{matrix}.$
We will show that $UT$ is strongly irreducible by directly computation. Let
$P=\begin{bmatrix}\ddots&\vdots&\vdots&\vdots&\vdots&{\mathinner{\mkern
2.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}}\\\
\cdots&a_{-1,-1}&a_{-1,0}&a_{-1,1}&a_{-1,2}&\cdots\\\
\cdots&a_{0,-1}&a_{0,0}&a_{0,1}&a_{0,2}&\cdots\\\
\cdots&a_{1,-1}&a_{1,0}&a_{1,1}&a_{1,2}&\cdots\\\
\cdots&a_{2,-1}&a_{2,0}&a_{2,1}&a_{2,2}&\cdots\\\ {\mathinner{\mkern
2.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}}\par&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{bmatrix}\begin{matrix}\vdots\\\ e_{2}\\\ e_{0}\\\ e_{1}\\\ e_{3}\\\
\vdots\end{matrix}$
be an operator commuting with $UT$. Then we have
(2.1) $\displaystyle\lambda_{i}a_{i,j}=\lambda_{j}a_{i+1,j+1},~{}\forall
i,j\in\mathbb{Z}.$
First, $a_{i,j}=0$ when $i\neq j$. If $k>0$, then
(2.2) $\displaystyle
a_{i,i+k}=\dfrac{\lambda_{k-1}\lambda_{k-2}\cdots\lambda_{0}}{\lambda_{i+k-1}\lambda_{i+k-2}\cdots\lambda_{i}}a_{0,k},~{}\forall~{}i\geq
k,$ (2.3) $\displaystyle
a_{i,i+k}=\dfrac{\lambda_{i+k}\lambda_{i+k+1}\cdots\lambda_{k-1}}{\lambda_{i}\lambda_{i+1}\cdots\lambda_{-1}}a_{0,k},~{}\forall~{}i\leq-1.$
If $k<0$, then
(2.4) $\displaystyle
a_{i,i+k}=\dfrac{\lambda_{k+1}\lambda_{k+2}\cdots\lambda_{0}}{\lambda_{i+k-1}\lambda_{i+k}\cdots\lambda_{i-2}}a_{0,k},~{}\forall~{}i\leq
k+2,$ (2.5) $\displaystyle
a_{i,i+k}=\dfrac{\lambda_{i-1}\lambda_{i-2}\cdots\lambda_{0}}{\lambda_{i+k-1}\lambda_{i+k-2}\cdots\lambda_{k}}a_{0,k},~{}\forall~{}i\geq
1.$
From $(2.2),(2.4)$, we know that $a_{0,k}=0$ for all $k\in\mathbb{Z}$. Hence
$a_{i,j}=0$ when $i\neq j$.
Second, $a_{i,i}=a_{j,j}$ for all $i,j\in\mathbb{Z}$. It is easily seen from
$(2.1)$. In summary, $P=I$ or $P=0$.
Case 3. Let $T$ be an operator which satisfy that $0$ is an isolate point of
$\sigma_{e}(T)$ and $\sigma_{e}(T)\backslash\\{0\\}\neq\emptyset$.
From the spectral decomposition theorem of self-adjoint operators, we have
$T=\begin{bmatrix}T_{1}&\\\ &T_{2}\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\end{matrix},$
where $\sigma_{e}(T_{1})=\\{0\\}$ and $0\overline{\in}\sigma(T_{2})$,
$\mathcal{H}_{1}\bot\mathcal{H}_{2}$ and
$\mathcal{H}_{1}\oplus\mathcal{H}_{2}=\mathcal{H}$.
Notice that dim$\mathcal{H}_{2}=\infty$.
As case 2, since $KerT=(0)$ and $\overline{RanT}=\mathcal{H}$, $T_{1}$ must be
a compact operator and
$T_{1}=\begin{bmatrix}\lambda_{0}\\\ &\lambda_{1}\\\ &&\lambda_{2}\\\
&&&\ddots\end{bmatrix}\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\
\vdots\end{matrix},$
where $\\{e_{n}\\}_{n=0}^{\infty}$ is an ONB of $\mathcal{H}_{1}$,
$\lambda_{n}\neq 0$, $\lim\limits_{n\rightarrow\infty}\lambda_{n}=0.$
Since $T_{2}$ is an invertible self-adjoint operator, there exist a compact
operator $K$ with $||K||<\frac{\epsilon}{||T_{2}^{-1}||}$, such that $T_{2}+K$
is also invertible, and
$T_{2}+K=\begin{bmatrix}\eta_{1}\\\ &\eta_{2}\\\ &&\eta_{3}\\\
&&&\ddots\end{bmatrix}\begin{matrix}f_{1}\\\ f_{2}\\\ f_{3}\\\
\vdots\end{matrix},$
where $\\{f_{n}\\}_{n=1}^{\infty}$ is an ONB of $\mathcal{H}_{2}$.
Let
$K_{1}=\begin{bmatrix}0&\\\ &K\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\end{matrix}.$
Choose $U$ as case 1 such that
$U(T+K_{1})=\begin{bmatrix}\ddots\\\ \ddots&0\\\ &\eta_{1}&0\\\
&&\lambda_{0}&0\\\ &&&\lambda_{1}&0&\\\ &&&&\ddots&\ddots\\\
\end{bmatrix}\begin{matrix}\vdots\\\ f_{1}\\\ e_{0}\\\ e_{1}\\\ e_{2}\\\
\vdots\\\ \end{matrix}.$
From lemma 2.5, we know that $U(T+K_{1})$ is strongly irreducible.
Let $K_{2}=UK_{1}T^{-1},$ then $||K_{2}||<\epsilon$ and
$(U+K_{2})T=U(T+K_{1})$ is strongly irreducible. So we complete the proof of
this case.
Case 4. This case is a little harder. Let $T$ be an operator which satisfy
that $0$ is not an isolate point of $\sigma_{e}(T)$, i.e. there exists a
sequence $\lambda_{n}$ such that
$\\{\lambda_{n}\\}_{n=1}^{\infty}\subseteq\sigma_{e}(T)$,
$\lambda_{n}>\lambda_{n+1}$ and
$\lim\limits_{n\rightarrow\infty}\lambda_{n}=0.$
From the spectral decomposition theorem of self-adjoint operators, we have
$T=\begin{bmatrix}T_{1}\\\ &T_{2}\\\ &&T_{3}\\\ &&&\ddots\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\\\
\mathcal{H}_{3}\\\ \vdots\end{matrix},$
where $\\{\mathcal{H}_{j}\\}_{j\geq 1}$ is a pairwise orthogonal family of
subspaces and $\bigoplus_{j\geq 1}\mathcal{H}_{j}=\mathcal{H}$,
$\sigma(T_{j})\subseteq[\lambda_{j+1},\lambda_{j}]$,
$\lambda_{j+1},\lambda_{j}\in\sigma_{e}(T_{j})$ for all $j\geq 1$.
First step, we will find unitarily operators and compact operators $U_{j}$,
$K_{j}$, $||K_{j}||<\frac{\epsilon}{2j}$ such that each of
$(U_{j}+K_{j})T_{j}$ is strongly irreducible.
1) If there exists a point $\lambda_{j+1}<\alpha_{j}<\lambda_{j}$ such that
$\alpha_{j}-T_{j}$ is invertible, then we deal with it as follows. Let
$[\alpha_{j}-\delta_{j},\alpha_{j}+\delta_{j}]$ be the small interval
contained in $(\lambda_{j+1},\lambda_{j})$ such that
$[\alpha_{j}-\delta_{j},\alpha_{j}+\delta_{j}]\subseteq\rho(T_{j})$. From the
Weyl-Von Neumann theorem, there exists a compact operator $K_{j}$,
$||K_{j}||<\frac{\epsilon}{2j||T_{j}^{-1}||}$ such that
$T_{j}+K_{j}=\begin{bmatrix}\ddots\\\ &&\beta_{2}^{j}\\\ &&&\beta_{0}^{j}\\\
&&&&\beta_{1}^{j}\\\ &&&&&&\ddots\\\ \end{bmatrix}\begin{matrix}\vdots\\\
e_{2}^{j}\\\ e_{0}^{j}\\\ e_{1}^{j}\\\ \vdots\end{matrix},$
where
$\\{\beta_{2n}^{j}\\}\subseteq(\alpha_{j}+\delta_{j},\lambda_{j}],\\{\beta_{2n+1}^{j}\\}\subseteq[\lambda_{j+1},\alpha_{j}-\delta_{j})$
and $\bigvee_{i\geq 0}\\{e_{i}^{j}\\}=\mathcal{H}_{j}$.
Let $U_{j}\in\mathcal{L}(\mathcal{H}_{j})$ as case 1 such that
$U_{j}(T_{j}+K_{j})=\begin{bmatrix}\ddots&\\\ \ddots&0&\\\
&\beta_{2}^{j}&0&\\\ &&\beta_{0}^{j}&0\\\ &&&\beta_{1}^{j}&0\\\
&&&&\ddots&\ddots\\\ \end{bmatrix}\begin{matrix}\vdots\\\ e_{2}^{j}\\\
e_{0}^{j}\\\ e_{1}^{j}\\\ e_{3}^{j}\\\ \vdots\end{matrix}\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}B_{j}.$
Then from lemma 2.5, $B_{j}$ is a bilateral shift operator which is strongly
irreducible. Denote $\widetilde{K_{j}}=U_{j}K_{j}T_{j}^{-1}$, then
$||\widetilde{K_{j}}||<\frac{\epsilon}{2j}$ and
$(U_{j}+\widetilde{K_{j}})T_{j}=B_{j}$ is strongly irreducible.
2) If $[\lambda_{j+1},\lambda_{j}]\subseteq\sigma(T_{j})$, then we deal with
it as follows. Choose an interval
$[\eta_{j}^{2},\eta_{j}^{1}]\subseteq(\lambda_{j+1},\lambda_{j})$ arbitrarily.
Then
$T_{j}=\begin{bmatrix}T_{j1}&\\\ &T_{j2}\\\ &&T_{j3}\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{j}^{1}\\\ \mathcal{H}_{j}^{2}\\\
\mathcal{H}_{j}^{3}\end{matrix},$
where $\\{\mathcal{H}_{j}^{i}\\}_{i=1}^{3}$ is pairwise orthogonal family of
subspaces of $\mathcal{H}_{j}$ and
$\bigoplus_{i=1}^{3}\mathcal{H}_{j}^{i}=\mathcal{H}_{j}$,
$\sigma(T_{j1})=[\eta_{j}^{1},\lambda_{j}]$,
$\sigma(T_{j2})=[\lambda_{j+1},\eta_{j}^{2}]$,
$\sigma(T_{j3})=[\eta_{j}^{2},\eta_{j}^{1}]$. From the Weyl-Von Neumann
theorem, there exist compact operators
$C_{j1}\in\mathcal{L}(\mathcal{H}_{j}^{1})$,
$C_{j2}\in\mathcal{L}(\mathcal{H}_{j}^{2})$,
$||C_{j1}||<\frac{\epsilon}{4j||T_{j}^{-1}||},||C_{j2}||<\frac{\epsilon}{4j||T_{j}^{-1}||}$
such that, $T_{j1}+C_{j1}$ is a diagonal operator with entries
$\\{\gamma_{n}^{j}\\}_{n\leq 0}\subseteq[\eta_{j}^{1},\lambda_{j}]$,
$T_{j2}+C_{j2}$ is a diagonal operator with entries
$\\{\gamma_{n+1}^{j}\\}_{n\geq 0}\subseteq[\lambda_{j+1},\eta_{j}^{2}]$,
Card$\\{n;\gamma_{n+1}^{j}=\eta_{j}^{2}\\}=\infty$ and
(2.6)
$\displaystyle\dfrac{\gamma_{0}^{j}\gamma_{1}^{j}\cdots\gamma_{n-1}^{j}}{(\eta_{j}^{2})^{n}}\geq\dfrac{1}{\sqrt{n}},~{}\forall
n\geq 1.$
As case 1, denote
$V_{j}\in\mathcal{L}(\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2})$ be the
unitarily operator such that
$V_{j}\begin{bmatrix}T_{j1}+C_{j1}&\\\ &T_{j2}+C_{j2}\\\
\end{bmatrix}=\begin{bmatrix}\ddots&\\\ \ddots&0\\\ &\gamma_{-1}^{j}&0\\\
&&\gamma_{0}^{j}&0\\\ &&&\gamma_{1}^{j}&0\\\ &&&&\ddots&\ddots\\\
\end{bmatrix}\begin{matrix}\vdots\\\ e_{-1}^{j}\\\ e_{0}^{j}\\\ e_{1}^{j}\\\
e_{2}^{j}\\\ \vdots\end{matrix}\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}D_{j},$
where
$\bigvee_{i\in\mathbb{Z}}\\{e_{i}^{j}\\}=\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}$.
From lemma 2.5, $D_{j}$ is strongly irreducible and moreover
$\eta_{j+1}\overline{\in}\sigma_{p}(D_{j})$. If $(D_{j}-\eta_{j}^{2})(x)=0$,
$x=\Sigma_{i\in\mathbb{Z}}x_{i}e_{i}^{j}\in\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}$,
then for $n\geq 0$
(2.7) $\displaystyle
x_{n+1}=\dfrac{\gamma_{0}^{j}\gamma_{1}^{j}\cdots\gamma_{n}^{j}}{(\eta_{j}^{2})^{n+1}}x_{0},~{}x_{-n}=\dfrac{(\eta_{j}^{2})^{n}}{\gamma_{-1}^{j}\gamma_{-2}^{j}\cdots\gamma_{-n}^{j}}x_{0}.$
From $(2.6)$ and $(2.7)$, $x_{n}=0,~{}\forall n\in\mathbb{Z}$.
Since $T_{j3}$ is a quasi-triangular operator,
$\sigma(T_{j3})=\sigma_{W}(T_{j3})=[\eta_{j}^{2},\eta_{j}^{1}]$ and
$\eta_{j}^{2}\in\partial\sigma_{W}(T_{j3})$, from lemma 2.8, there exists a
compact operator $C_{j3}$, $||C_{j3}||<\frac{\epsilon}{8j||T_{j}^{-1}||}$ such
that $T_{j3}+C_{j3}$ is strongly irreducible and
Ker$(\mathcal{T}_{D_{j},T_{j3}+C_{j3}})=\\{0\\}$. Moreover, since
$\sigma_{r}(T_{j3}+C_{j3})\cap\sigma_{l}(D_{j})\neq\emptyset$, from lemma 2.6,
there exists a compact operator
$E_{j}:\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}\longrightarrow\mathcal{H}_{j}^{3}$,
$||E_{j}||<\frac{\epsilon}{8j||T_{j}^{-1}||}$ such that
$E_{j}\overline{\in}Ran\mathcal{T}_{T_{j3}+C_{j3},D_{j}}$.
Hence from lemma 2.7, we obtain
$S_{j}\ext@arrow
0359\arrowfill@\Relbar\Relbar\Relbar{}{\mbox{def}}\begin{bmatrix}D_{j}&\\\
E_{j}&T_{j3}+C_{j3}\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}\\\
\mathcal{H}_{j}^{3}\end{matrix}$
is strongly irreducible.
Let
$W_{j}=\begin{bmatrix}V_{j}&\\\ &I\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}\\\
\mathcal{H}_{j}^{3}\end{matrix},$
and
$C_{j}=W_{j}\begin{bmatrix}C_{j1}\\\ &C_{j2}\\\ &&0\\\
\end{bmatrix}T_{j}^{-1}+\begin{bmatrix}0&0\\\ E_{j}&C_{j3}\\\
\end{bmatrix}T_{j}^{-1},$
then $C_{j}$ is a compact operator with $||C_{j}||<\frac{\epsilon}{2j}$ and
$(W_{j}+C_{j})T_{j}=S_{j}$ is strongly irreducible.
In summary, we obtain $(U_{j}+\widetilde{K_{j}})T_{j}=B_{j}$ or
$(W_{j}+C_{j})T_{j}=S_{j}$ for any chosen $T_{j}$. For the sake of brevity, we
denote $U_{j}$ or $W_{j}$ by $\overline{U}_{j}$, denote $\widetilde{K_{j}}$ or
$C_{j}$ by $\overline{K}_{j}$, denote $B_{j}$ or $S_{j}$ by
$\overline{B}_{j}$.
Let $U=\bigoplus_{j\geq 1}\overline{U}_{j}$, $K=\bigoplus_{j\geq
1}\overline{K}_{j}$, then $U$ is an unitarily operator, $K$ is a compact
operator with $||K||<\frac{\epsilon}{2}$ and
$(U+K)T=\begin{bmatrix}\overline{B}_{1}\\\ &\overline{B}_{2}\\\
&&\overline{B}_{3}\\\ &&&\ddots\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{2}\\\
\mathcal{H}_{3}\\\ \vdots\end{matrix}.$
Second step, we will use lemma 2.7 to complete the proof. Since
$\sigma_{r}(\overline{B}_{2n+1})\cap\sigma_{l}(\overline{B}_{2n})\neq\emptyset$,
$\sigma_{r}(\overline{B}_{2n+1})\cap\sigma_{l}(\overline{B}_{2n+2})\neq\emptyset$,
there exist compacts operator $E_{n}$, $F_{n}$,
$||E_{n}||<\frac{\epsilon}{4n||T_{2n}^{-1}||}$,
$||F_{n}||<\frac{\epsilon}{4n||T_{2n}^{-1}||}$ such that $E_{n}\overline{\in}$
Ran$\mathcal{T}_{\overline{B}_{2n-1},\overline{B}_{2n}}$ and
$F_{n}\overline{\in}$
Ran$\mathcal{T}_{\overline{B}_{2n+1},\overline{B}_{2n}}$.
Let
$\displaystyle\overline{K}$ $\displaystyle=$ $\displaystyle
K+\begin{bmatrix}0&\begin{bmatrix}E_{1}T_{2}^{-1}\\\
F_{1}T_{2}^{-1}&E_{2}T_{4}^{-1}\\\ &F_{2}T_{4}^{-1}&E_{3}T_{6}^{-1}\\\
&&\ddots&\ddots\\\ \end{bmatrix}\\\
&0\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{3}\\\
\mathcal{H}_{5}\\\ \vdots\\\ \bigoplus_{n\geq 1}\mathcal{H}_{2n}\\\
\end{matrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\begin{bmatrix}\overline{K}_{1}\\\
&\overline{K}_{3}\\\ &&\overline{K}_{5}\\\ &&&\ddots\\\
\end{bmatrix}&\begin{bmatrix}E_{1}T_{2}^{-1}\\\
F_{1}T_{2}^{-1}&E_{2}T_{4}^{-1}\\\ &F_{2}T_{4}^{-1}&E_{3}T_{6}^{-1}\\\
&&\ddots&\ddots\\\ \end{bmatrix}\\\ &\begin{bmatrix}\overline{K}_{2}\\\
&\overline{K}_{4}\\\ &&\overline{K}_{6}\\\ &&&\ddots\\\
\end{bmatrix}\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{3}\\\
\mathcal{H}_{5}\\\ \vdots\\\ \mathcal{H}_{2}\\\ \mathcal{H}_{4}\\\
\mathcal{H}_{6}\\\ \vdots\end{matrix},$
then $\overline{K}$ is a compact operator with $||\overline{K}||<\epsilon$ and
$\displaystyle(U+\overline{K})T$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\begin{bmatrix}\begin{smallmatrix}\overline{U}_{1}+\overline{K}_{1}\\\
&\overline{U}_{3}+\overline{K}_{3}\\\ &&\overline{U}_{5}+\overline{K}_{5}\\\
&&&\ddots\\\
\end{smallmatrix}\end{bmatrix}&\begin{bmatrix}\begin{smallmatrix}E_{1}T_{2}^{-1}\\\
F_{1}T_{2}^{-1}&E_{2}T_{4}^{-1}\\\ &F_{2}T_{4}^{-1}&E_{3}T_{6}^{-1}\\\
&&\ddots&\ddots\\\ \end{smallmatrix}\end{bmatrix}\\\
&\begin{bmatrix}\begin{smallmatrix}\overline{U}_{2}+\overline{K}_{2}\\\
&\overline{U}_{4}+\overline{K}_{4}\\\ &&\overline{U}_{6}+\overline{K}_{6}\\\
&&&\ddots\\\
\end{smallmatrix}\end{bmatrix}\end{bmatrix}\begin{matrix}\begin{smallmatrix}\mathcal{H}_{1}\\\
\mathcal{H}_{3}\\\ \mathcal{H}_{5}\\\ \vdots\\\ \mathcal{H}_{2}\\\
\mathcal{H}_{4}\\\ \mathcal{H}_{6}\\\ \vdots\end{smallmatrix}\end{matrix}$
$\displaystyle\begin{array}[]{c}\times\end{array}$
$\displaystyle\begin{bmatrix}\begin{bmatrix}T_{1}\\\ &T_{3}\\\ &&T_{5}\\\
&&&\ddots\\\ \end{bmatrix}&0\\\ 0&\begin{bmatrix}T_{2}\\\ &T_{4}\\\ &&T_{6}\\\
&&&\ddots\\\ \end{bmatrix}\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\
\mathcal{H}_{3}\\\ \mathcal{H}_{5}\\\ \vdots\\\ \mathcal{H}_{2}\\\
\mathcal{H}_{4}\\\ \mathcal{H}_{6}\\\ \vdots\end{matrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\begin{bmatrix}\overline{B}_{1}\\\
&\overline{B}_{3}\\\ &&\overline{B}_{5}\\\ &&&\ddots\\\
\end{bmatrix}&\begin{bmatrix}E_{1}\\\ F_{1}&E_{2}\\\ &F_{2}&E_{3}\\\
&&\ddots&\ddots\\\ \end{bmatrix}\\\ &\begin{bmatrix}\overline{B}_{2}\\\
&\overline{B}_{4}\\\ &&\overline{B}_{6}\\\ &&&\ddots\\\
\end{bmatrix}\end{bmatrix}\begin{matrix}\mathcal{H}_{1}\\\ \mathcal{H}_{3}\\\
\mathcal{H}_{5}\\\ \vdots\\\ \mathcal{H}_{2}\\\ \mathcal{H}_{4}\\\
\mathcal{H}_{6}\\\ \vdots\end{matrix}.$
In order to obtain the result, we only need to show the conditions $i)-vi)$ of
lemma 2.7.
From the construction above, conditions $i),ii),iv),v),vi)$ are satisfied.
Since $\sigma(\overline{B}_{i})\cap\sigma(\overline{B}_{j})=\emptyset$, when
$|i-j|\geq 2$, from lemma 2.6,
Ker$\mathcal{T}_{\overline{B}_{i},\overline{B}_{j}}=\\{0\\}$ for $i\neq j$,
$|i-j|\geq 2$.
For $j$ even, $|i-j|=1$, we should consider
Ker$\mathcal{T}_{\overline{B}_{j},\overline{B}_{i}}=\\{0\\}$.
Since each $\overline{B}_{j}$ maybe $B_{j}$ or $S_{j}$, we should consider
four cases for any Ker$\mathcal{T}_{\ast,\ast}$. But the proof are all the
same, we just consider one case that
Ker$\mathcal{T}_{\overline{B}_{j},\overline{B}_{j+1}}=\\{0\\}$ where
$\displaystyle\overline{B}_{j}=S_{j}=\begin{bmatrix}D_{j}&\\\
E_{j}&T_{j3}+C_{j3}\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}\\\
\mathcal{H}_{j}^{3}\end{matrix},$
$\displaystyle\overline{B}_{j+1}=S_{j+1}=\begin{bmatrix}D_{j+1}&\\\
E_{j+1}&T_{j+1,3}+C_{j+1,3}\\\
\end{bmatrix}\begin{matrix}\mathcal{H}_{j+1}^{1}\oplus\mathcal{H}_{j+1}^{2}\\\
\mathcal{H}_{j+1}^{3}\end{matrix}.$
Let
$X=\begin{bmatrix}X_{11}&X_{12}\\\
X_{21}&X_{22}\end{bmatrix}\begin{matrix}\mathcal{H}_{j+1}^{1}\oplus\mathcal{H}_{j+1}^{2}\\\
\oplus\\\
\mathcal{H}_{j+1}^{3}\end{matrix}\mapsto\begin{matrix}\mathcal{H}_{j}^{1}\oplus\mathcal{H}_{j}^{2}\\\
\oplus\\\ \mathcal{H}_{j}^{3}\end{matrix}$
belongs to Ker$\mathcal{T}_{\overline{B}_{j},\overline{B}_{j+1}}$. Then
(2.9) $\displaystyle D_{j}X_{11}=X_{11}D_{j+1}+X_{12}E_{j+1},$ (2.10)
$\displaystyle D_{j}X_{12}=X_{12}(T_{j+1,3}+C_{j+1,3}),$ (2.11) $\displaystyle
E_{j}X_{11}+(T_{j3}+C_{j3})X_{21}=X_{21}D_{j+1}+X_{22}E_{j+1},$ (2.12)
$\displaystyle E_{j}X_{12}+(T_{j3}+C_{j3})X_{22}=X_{22}(T_{j+1,3}+C_{j+1,3}).$
Since $\sigma(D_{j})\cap\sigma(T_{j+1,3}+C_{j+1,3})=\emptyset$, from lemma 2.6
and $(2.9)$, $X_{12}=0$. Since
$\sigma(T_{j3}+C_{j3})\cap\sigma(T_{j+1,3}+C_{j+1,3})=\emptyset$, from lemma
2.6, $X_{12}=0$ and $(2.11)$, $X_{22}=0$.
If $X_{11}=0$, then since
$\sigma(T_{j3}+C_{j3})\cap\sigma(D_{j+1})=\emptyset$, $X_{21}=0$, and hence
$X=0$.
Let
$X_{11}=\begin{bmatrix}\ddots&\vdots&\vdots&\vdots&\vdots&{\mathinner{\mkern
2.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}}\\\
\cdots&a_{-1,-1}&a_{-1,0}&a_{-1,1}&a_{-1,2}&\cdots\\\
\cdots&a_{0,-1}&a_{0,0}&a_{0,1}&a_{0,2}&\cdots\\\
\cdots&a_{1,-1}&a_{1,0}&a_{1,1}&a_{1,2}&\cdots\\\
\cdots&a_{2,-1}&a_{2,0}&a_{2,1}&a_{2,2}&\cdots\\\ {\mathinner{\mkern
2.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise
7.0pt\hbox{.}\mkern 1.0mu}}\par&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{bmatrix}\begin{matrix}\vdots\\\ e_{-1}^{j+1}\\\ e_{0}^{j+1}\\\
e_{1}^{j+1}\\\ e_{2}^{j+1}\\\ \vdots\end{matrix}\mapsto\begin{matrix}\vdots\\\
e_{-1}^{j}\\\ e_{0}^{j}\\\ e_{1}^{j}\\\ e_{2}^{j}\\\ \vdots\end{matrix},$
then from $(2.8)$, we have
(2.13)
$\displaystyle\gamma_{m}^{j}a_{m,n}=\gamma_{n}^{j+1}a_{m+1,n+1},~{}\forall
m,n\in\mathbb{Z}.$
Hence
(2.14) $\displaystyle
a_{m+1,m+k+1}=\dfrac{\gamma_{0}^{j}\gamma_{1}^{j}\cdots\gamma_{m}^{j}}{\gamma_{k}^{j+1}\gamma_{k+1}^{j+1}\cdots\gamma_{m+k}^{j+1}}a_{0,k},~{}\forall
m\geq 0,~{}k\in\mathbb{Z},$ (2.15) $\displaystyle
a_{m,m+k}=\dfrac{\gamma_{m+k}^{j+1}\gamma_{m+k+1}^{j+1}\cdots\gamma_{k-1}^{j+1}}{\gamma_{m}^{j}\gamma_{m+1}^{j}\cdots\gamma_{-1}^{j}}a_{0,k},~{}\forall
m<0,~{}k\in\mathbb{Z}.$
Since
$\dfrac{\gamma_{n}^{j}}{\gamma_{m}^{j+1}}\geq\dfrac{\lambda_{j+1}}{\eta_{j+1}^{2}}>1,~{}\forall
m,n\geq 1,$
$\lim\limits_{m\rightarrow+\infty}\dfrac{\gamma_{0}^{j}\gamma_{1}^{j}\cdots\gamma_{m}^{j}}{\gamma_{k}^{j+1}\gamma_{k+1}^{j+1}\cdots\gamma_{m+k}^{j+1}}=\infty.$
Hence from $(2.13)$ and $(2.14)$, $a_{m,n}=0,~{}\forall m,n\in\mathbb{Z}$. ∎
###### Remark 2.9.
If $T$ is a positive operator which satisfies
$(0,\epsilon)\cap\sigma_{e}(T)=\emptyset$ for some $\epsilon>0$,
Ker$T=\\{0\\}$ and $\sigma_{e}(T)$ is not connected, then there exist an
unitary operator $U$ and a compact operator $K$ with norm less than a given
positive number such that $(U+K)T$ is a strongly irreducible Cowen-douglas
operator. If $T$ is a compact operator which is injective and has dense range,
then for any unitary operator $U$ and any compact operator $K$, $(U+K)T$ can
not be a Cowen-Douglas operator. It is an easy corollary of Case 1,2,3 of the
above proof.
## 3\. Application in Basis Theory
### 3.1.
In this subsection, we discuss existence of operators satisfying both certain
basis theory property and strongly irreducibility.
Recall that a sequence $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is called a Schauder
basis of the Hilbert space $\mathcal{H}$ if and only if for every vector
$x\in\mathcal{H}$ there exists a unique sequence
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ of complex numbers such that the partial sum
sequence $x_{k}=\sum_{n=1}^{k}\alpha_{n}f_{n}$ converges to $x$ in norm.
###### Definition 3.1.
An $\omega\times\omega$ matrix $F$ is said to be a Schauder matrix if and only
if the sequence $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ of its column vectors in
$\mathcal{H}$ comprises a Schauder basis.
Note that from the definition, each column vector $f_{k}$ is automatically a
$l^{2}-$sequence since it represents a vector in $\mathcal{H}$. Given an ONB
$\varphi=\\{e_{n}\\}_{n=1}^{\infty}$ and a basis sequence
$\psi=\\{f_{n}=\sum_{k=1}^{\infty}f_{kn}e_{k}\\}_{n=1}^{\infty}$ of
$\mathcal{H}$, then the matrix $F_{\psi}=(f_{kn})$ is a Schauder matrix by
above definition. It shall be called the Schauder matrix corresponding to the
basis $\psi$.
###### Definition 3.2.
A matrix $F$ is called a unconditional, conditional, Riesz, normalized or
quasinormal matrix respectively if and only if the sequence of its column
vectors comprise a unconditional, conditional, Riesz, normalized or
quasinormal basis. Two Schauder matrices $F_{\psi},F_{\varphi}$ are called
equivalent if and only if the corresponding bases $\psi$ and $\varphi$ are
equivalent.
###### Theorem 3.3.
Assume that $F$ is a Schauder matrix and $G^{*}$ is its inverse matrix. We
have
1\. For each invertible matrix $X$, $XF$ is also a Schauder matrix. Moreover,
$XF$ is unconditional(conditional) if and only if $F$ is
unconditional(conditional);
2\. For each diagonal matrix $D=diag(\alpha_{1},\alpha_{2},\cdots)$ in which
each diagonal element $\alpha_{k}$ is nonzero, $FD$ is also a Schauder matrix.
Moreover, $FD$ is unconditional(conditional) if and only if $F$ is
unconditional(conditional);
3\. For a unconditional matrix $F$, $FU$ is also a unconditional matrix for
$U\in\pi_{\infty}$;
4\. Two Schauder matrix $F$ and $F^{{}^{\prime}}$ are equivalent if and only
if there is a invertible matrix $X$ such that $XF=F^{{}^{\prime}}$.
For a Schauder matrix $M_{\psi}$ and a matrix $X$ of some invertible operator
$T$, $M_{\psi^{{}^{\prime}}}=XM_{\psi}$ is the Schauder matrix of the Schauder
basis $\psi^{{}^{\prime}}=\\{Tf_{n}\\}_{n=1}^{\infty}$. Given an ONB
$\varphi$, the orbit set
$O_{gl}(M_{\psi})=\\{XM_{\psi};\mbox{matrix $X$ represents an invertible
operator $T\in Gl(\mathcal{H})$}\\}$
consists of all equivalent bases of $\psi$. Our main theorem 1.2 and above
theorem 3.3 tell us that we always can pick a strongly irreducible operator as
the representation element in the orbit $O_{gl}(M_{\psi})$.
###### Theorem 3.4.
Suppose that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is a basis sequence and its
corresponding Schauder matrix $M_{\psi}$ under some given ONB represents a
bounded operator $T_{\psi}$. Then there is a matrix $X$ of some invertible
operator such that $XM_{\psi}$ represents an strongly irreducible bounded
compact operator in $L(\mathcal{H})$.
###### Corollary 3.5.
For a Schauder matrix $M_{\psi}$ representing a bounded operator, there always
be matrices of strongly irreducible operators in its orbit $O_{gl}(M_{\psi})$.
By the pole decomposition theorem of operators, there is also a unitary matrix
$U$ such that $UM_{\psi}$ represents a self-adjoint operator. Hence the orbit
$O_{gl}(M_{\psi})$ has both the operators having a large number of strongly
reducible subspaces and the operators having no nontrivial strongly reducible
subspace.
### 3.2. The blowing up matrix
To observe the relations between basis theory and operator theory, is it
enough to just consider bounded operators? Let
$\psi=\\{f_{n}\\}_{n=1}^{\infty}$ be a Schauder basis and $M_{\psi}$ is the
corresponding Schauder matrix under some ONB
$\varphi=\\{e_{n}\\}_{n=1}^{\infty}$. In general, $M_{\psi}$ does not
represents a bounded operator even for a quasinormal basis $\psi$. Following
example show this phenomenon.
###### Example 3.6.
(see [19], Example14.5, p429.) Let $\\{\alpha_{n}\\}_{n=1}^{\infty}$ be a
sequence of positive numbers such that
$\sum_{n=1}^{\infty}n\alpha_{n}^{2}<\infty$ and
$\sum_{n=1}^{\infty}\alpha_{n}=\infty$. Then the sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ defined as
$f_{2n-1}=e_{2n-1}+\sum_{i=n}^{\infty}\alpha_{i-n+1}e_{2i},~{}~{}f_{2n}=e_{2n},~{}~{}n=1,2,\cdots$
is a conditional basis of $\mathcal{H}$. Denote by $F$ the corresponding
Schauder matrix and $T$ the operator it represented. Now we shall show that
$T$ is a unbounded operator indeed. To do this, we rewrite the matrix of $T$
under new ONB. Here we use the fact that $T$ is bounded if and only if
$U^{*}TU$ is bounded for each unitary operator $U$. Denote by
$\mathcal{H}_{1}=span\\{e_{2n-1};n=1,2,\cdots\\}$ and
$\mathcal{H}_{2}=span\\{e_{2n};n=1,2,\cdots\\}$. Now we can rewrite $F$ into
the form
$\begin{array}[]{rll}T=&\left(\begin{array}[]{cc}I&0\\\
T_{1}&I\end{array}\right)&\begin{array}[]{c}\mathcal{H}_{1}\\\
\mathcal{H}_{2}\end{array}\end{array}$
in which the operator $T_{1}$ has a matrix as
$\left(\begin{array}[]{cccc}\alpha_{1}&0&0&0\\\ \alpha_{2}&\alpha_{1}&0&0\\\
\alpha_{3}&\alpha_{2}&\alpha_{1}&0\\\
\vdots&\ddots&\ddots&\ddots\end{array}\right).$
Denote by $S^{(2)}$ the shift operator defined as $S^{(2)}e_{2n}=e_{2n+2}$. It
is trivial to check that we have $S^{(2)}T_{1}=T_{1}S^{(2)}$. Hence we have
$T_{2}$ is in the commutant $\mathcal{A}^{{}^{\prime}}(S^{(2)})$ if $T_{2}$ is
a bounded operator. But it is impossible since the holomorphic function
defined by series $\sum_{n=1}^{\infty}\alpha_{n}z^{n}$ is not in the class
$H^{\infty}$ by the fact $\sum_{n=1}^{\infty}\alpha_{n}=\infty$(cf, [18],
theorem 3, p62).
However, we always can relate a basis to a bounded operator as follows. Assume
that $M=(f_{kn})$ is an $\omega\times\omega$ matrix such that each column
vector $f_{n}=\\{f_{kn}\\}_{k=1}^{\infty}$ is an $l^{2}-sequence$. Then under
a fixed ONB sequence $\\{e_{n}\\}_{n=1}^{\infty}$,
$\mathbf{f}_{n}=\sum_{n=1}^{\infty}f_{kn}e_{k}$ is a vector in $\mathcal{H}$.
Then for a nonzero complex number sequence
$\alpha=\\{\alpha_{k}\\}_{k=1}^{\infty}$, it is clear that
$M_{\alpha}=(\alpha_{1}f_{1},\alpha_{2}f_{2},\alpha_{3}f_{3},\cdots)$ is also
a matrix with $l^{2}-$sequences as its column vectors. Moreover, if $M$
represents a bounded operator and $\alpha$ be a bounded sequence, then
$M_{\alpha}$ also be a matrix of a bounded operator since $M^{{}^{\prime}}=MD$
in which $D$ is the bounded diagonal operator with $\alpha_{k}$ as its
diagonal elements. Inspired by the properties of bases(cf, proposition 4.1.5
and 4.2.12 in the book [16]), we have the following definition.
###### Definition 3.7.
$M_{\alpha}$ is called the blowing up matrix of $M$ with sequence $\alpha$.
###### Theorem 3.8.
Suppose that $\\{f_{n}\\}_{n=1}^{\infty}$ is a basis sequence and
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ is a sequence of complex numbers such that
the sum $\sum_{n=1}^{\infty}||\alpha_{n}f_{n}||$ be finite. Moreover, let
$\psi=\\{\alpha_{n}f_{n}\\}_{n=1}^{\infty}$. Then the matrix $M_{\psi}$
represents a bounded compact operator under any ONB.
###### Proof.
Fixed an ONB $\varphi=\\{e_{n}\\}_{n=1}^{\infty}$. Then each vector $f_{n}$
has a unique $l^{2}-$coodinate $f_{n}=\\{f_{kn}e_{k}\\}_{k=1}^{\infty}$. Let
$g_{n}=\\{g_{kn}\\}_{k=1}^{\infty}=\alpha_{n}f_{n}=\\{\alpha
f_{kn}e_{k}\\}_{k=1}^{\infty}$. Then the matrix $M_{\psi}=(g_{kn})$ under the
ONB $\varphi$. For any vector $x=\sum_{k=1}^{\infty}\xi_{k}e_{k}$, the series
$\sum_{k=1}^{\infty}\xi_{k}g_{k}$ converges since we have
$||\sum_{k=m}^{\infty}\xi_{k}g_{k}||\leq\sum_{k=m}^{\infty}||\xi_{k}g_{k}||\leq\sup_{k}\\{|\xi_{k}|\\}(\sum_{k=m}^{\infty}||g_{k}||)\rightarrow
0$
as $m\rightarrow 0$. Hence $M_{\psi}$ represents an operator $T_{\psi}$ well-
defined everywhere on $\mathcal{H}$. Now by the closed graph theorem, we just
need to show that $M_{\psi}$ also represents a closed operator to finish the
proof. Suppose that $x_{n}=\\{\xi_{k}^{(n)}\\}_{k=1}^{\infty}\rightarrow
x_{0}=\\{\xi_{k}^{(0)}\\}_{k=1}^{\infty}$ in the norm. Now for any
$\epsilon>0$, there is some integer $n_{0}$ such that
$|\xi^{(n)}_{k}-\xi^{(0)}_{k}|<1$ holds for any $n>n_{0}$. Let $k_{1}$ be the
integer satisfying $\sum_{k=k_{1}+1}^{\infty}||g_{k}||<\frac{\epsilon}{2}$.
Then there is an integer $n_{1}$ such that we have
$\sum_{k=1}^{k_{1}}|\xi_{k}^{(n)}-\xi_{k}^{(0)}|||g_{k}||<\frac{\epsilon}{2}$
for all $n>n_{1}$. Let $N=\max\\{n_{0},n_{1}\\}$. Hence for $n>N$ we have
$\begin{array}[]{rl}||T_{\psi}(x_{n}-x_{0})||&=||\sum_{k=1}^{\infty}(\xi_{k}^{(n)}-\xi_{k}^{(0)})g_{k}||\\\
&\leq||\sum_{k=1}^{k_{1}}(\xi_{k}^{(n)}-\xi_{k}^{(0)})g_{k}||+||\sum_{k=k_{1}+1}^{\infty}(\xi_{k}^{(n)}-\xi_{k}^{(0)})g_{k}||\\\
&\leq\sum_{k=1}^{k_{1}}|\xi_{k}^{(n)}-\xi_{k}^{(0)}|||g_{k}||+\sum_{k=k_{1}+1}^{\infty}||g_{k}||<\frac{\epsilon}{2}\\\
&\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}.\end{array}$
Now it is left to show that $M_{\psi}$ is also a compact operator. Let
$K_{n}=\sum_{l=1}^{n}g_{l}\otimes e_{l}$ in which the operator $g_{l}\otimes
e_{l}$ is defined as $(g_{l}\otimes e_{l})x=(x,e_{l})g_{l}$. Then we have
$\begin{array}[]{rl}||(T_{\psi}-K_{n})||&=\sup_{||x||=1}||(T_{\psi}-K_{n})x||\\\
&=\sup_{||x||=1}||(\sum_{l=n+1}^{\infty}g_{l}\otimes e_{l})x||\\\
&\leq\sup_{||x||=1}\sum_{l=n+1}^{\infty}||g_{l}\otimes e_{l})x||\\\
&\leq\sum_{l=n+1}^{\infty}||g_{l}||\rightarrow 0\\\ \end{array}$
as $n\rightarrow\infty$. Hence $T_{\psi}$ is a norm limit of a sequence of
compact operators, that is, $T_{\psi}$ is a compact operator. ∎
Above theorem enlarge the class of bases that can be studied by bounded
operators.
###### Theorem 3.9.
Suppose that $\\{f_{n}\\}_{n=1}^{\infty}$ is a basis sequence and
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ is a sequence of complex numbers such that
the sum $\sum_{n=1}^{\infty}||\alpha_{n}f_{n}||$ be finite. Moreover, let
$\psi=\\{\alpha_{n}f_{n}\\}_{n=1}^{\infty}$. Then there is a matrix $X$ of
some invertible operator such that $XM_{\psi}$ represents an strongly
irreducible bounded compact operator in $L(\mathcal{H})$.
### 3.3.
Now we turn to study the existence of strongly irreducible Schauder operator.
Recall that a Schauder operator $T$ is an operator mapping some ONB into a
Schauder basis. In his paper [17], Olevskii call an operator to be generating
if and only if it maps some ONB into a quasinormal conditional basis. Hence
our definition of Schauder operator is a generalization of Olevskii’s one. A
Schauder operator is said to be conditional if it maps some ONB sequence into
a conditional basis sequence. Our theorem 1.2 ensure the existence of
operators satisfying special properties. For convenience and self-sufficiency,
we list some results on an operator theory description of Schauder basis
appearing in paper [1] without proof.
###### Theorem 3.10.
Following conditions are equivalent:
1\. $T$ is a Schauder operator;
2\. $T$ maps some ONB $\\{e_{n}\\}_{n=1}^{\infty}$ into a basis;
3\. $T$ has a polar decomposition $T=UA$ in which $A$ is a Schauder operator;
4\. Assume that $T$ has a matrix representation $F$ under a fixed ONB
$\\{e_{n}\\}_{n=1}^{\infty}$. There is some unitary matrix $U$ such that $FU$
is a Schauder matrix.
###### Corollary 3.11.
([1]) Compact operator $K=diag\\{1,\frac{1}{2},\frac{1}{3},\cdots\\}$ is a
conditional operator.
###### Corollary 3.12.
There is an operator $T\in L(\mathcal{H})$ satisfying following properties:
1\. $T$ is a strongly irreducible compact operator;
2\. There exists some ONB such that $T$ has a matrix which is a conditional
Schauder matrix.
###### Proof.
Fixed an ONB, suppose that $K=diag\\{1,\frac{1}{2},\frac{1}{3},\cdots\\}$ be
the diagonal operator appearing in corollary 3.11. Then there is some unitary
matrix $U$ such that $KU$ is a conditional Schauder matrix. Then apply theorem
1.2 and theorem 3.10 to the matrix $KU$. ∎
###### Corollary 3.13.
There is an operator $T\in L(\mathcal{H})$ satisfying following properties:
1\. $T$ is a strongly irreducible compact operator;
2\. There exists some ONB such that $T$ has a matrix which is a unconditional
Schauder matrix.
###### Proof.
Fixed an ONB, suppose that $K=diag\\{1,\frac{1}{2},\frac{1}{3},\cdots\\}$ be
the diagonal operator appearing in corollary 3.11. Then apply theorem 1.2 and
theorem 3.10 to the matrix $K$. ∎
By the following main theorem 1 of [17], we can get the same results about the
non-compact case.
###### Theorem 3.14.
([17], p479) A bounded operator $T$ is generating if and only if the following
conditions hold:
a) the operators $T$ and $T^{*}$ do not admit the eigenvalue $\lambda=0$;
b) there exists a number $q,0<q<1$ such that to each segment $[q^{n+1},q^{n}]$
in the spectral decomposition of the positive operator
$(T^{*}T)^{\frac{1}{2}}$, there corresponds an infinite dimensional invariant
subspace.
###### Corollary 3.15.
There is an operator $T\in L(\mathcal{H})$ satisfying following properties:
1\. $T$ is a strongly irreducible non-compact operator;
2\. There exists some ONB such that $T$ has a matrix which is a conditional
Schauder matrix.
###### Proof.
Let $T$ be a self-adjopint operator satisfying the conditional b). Then there
is some unitary matrix $U$ such that $TU$ is a conditional Schauder matrix. It
is clear that $TU$ can not be compact. Then apply theorem 1.2 and theorem 3.10
to the operator $T$. ∎
###### Corollary 3.16.
There is an operator $T\in L(\mathcal{H})$ satisfying following properties:
1\. $T$ is a strongly irreducible non-compact operator;
2\. There exists some ONB such that $T$ has a matrix which is a unconditional
Schauder matrix.
###### Proof.
In fact by proposition 2.20 in paper [1] and theorem 1.2, all invertible and
strongly irreducible operators satisfy the requirements of corollary. For
example, given an irreducible operator $T$ and a complex number
$\lambda\notin\sigma(T)$, then $\lambda-T$ is such one operator. ∎
The first requirement of above lemmas are in the operator theory category
while the others are in basis theory category. The first one ask that the
matrix be good in the sense of Jordan block; the second one ask that the
matrix be nice from the basis viewpoint, that is, the column vectors comprise
a basis.
### 3.4. Small disturbance on the basis const
Now we consider the “small” condition appearing in the theorem 1.2, it ensure
us to get a new basis with a small disturbance on both of the basis const and
the unconditional const.
Recall that a sequence $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is called a Schauder
basis of the Hilbert space $\mathcal{H}$ if and only if for every vector
$x\in\mathcal{H}$ there exists a unique sequence
$\\{\alpha_{n}\\}_{n=1}^{\infty}$ of complex numbers such that the partial sum
sequence $x_{k}=\sum_{n=1}^{k}\alpha_{n}f_{n}$ converges to $x$ in norm.
Denote by $P_{k}$ the the diagonal matrix with the first $k-$th entries on
diagonal line equal to 1 and $0$ for others. Then as an operator $P_{k}$
represents the orthogonal projection from $\mathcal{H}$ to the subspace
$\mathcal{H}^{(k)}=span\\{e_{1},e_{2},\cdots,e_{k}\\}$.
###### Lemma 3.17.
Assume that $\\{e_{k}\\}_{k=1}^{\infty}$ is a fixed ONB of $\mathcal{H}$.
Suppose that an $\omega\times\omega$ matrix $F=(f_{ij})$ satisfies the
following properties:
1\. Each column of the matrix $F$ is a $l^{2}-$sequence;
2\. $F$ has a unique left inverse matrix $G^{*}=(g_{kl})$ such that each row
of $G^{*}$ is also a $l^{2}-$sequence;
3\. Operators $Q_{k}$ defined by the matrix $Q_{k}=FP_{k}G^{*}$ are well-
defined projections on $\mathcal{H}$ and converges to the unit operator $I$ in
the strong operator topology.
Then the sequence
$\\{f_{k}\\}_{k=1}^{\infty},f_{k}=\sum_{j=1}^{\infty}f_{ij}e_{i}$ must be a
Schauder basis.
The projection $FP_{n}G^{*}$ is just the $n-$th “natural projection” so called
in [16](p354). It is also the $n-$th partial sum operator so called in
[19](definition 4.4, p25). Now we can translate theorem 4.1.15 and corollary
4.1.17 in [16] into the following
###### Proposition 3.18.
If $F$ is a Schauder matrix, then $M=\sup_{n}\\{||FP_{n}G^{*}||\\}$ is a
finite const.
The const $M$ is called the basis const for the basis
$\\{f_{n}\\}_{n=1}^{\infty}$.
Assume that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ is a basis. For a subset
$\Delta$ of $\mathbb{N}$, denote by $P_{\Delta}$ the diagonal matrix defined
as $P_{\Delta}(nn)=1$ for $n\in\Delta$ and $P_{\Delta}(nn)=0$ for
$n\notin\Delta$. The projection $Q_{\Delta}=F_{\psi}P_{\Delta}G_{\psi}^{*}$
defined in above lemmas is called a natural projection(see, definition 4.2.24,
[16], p378). In fact for a vector $x=\sum_{n=1}^{\infty}x_{n}f_{n}$, it is
trivial to check $Q_{\Delta}x=\sum_{n\in\Delta}x_{n}f_{n}$. Then we have a
same result for the unconditional basis const(cf, definition4.2.28, [16],
p379):
###### Proposition 3.19.
If $F_{\psi}$ is a Schauder matrix, then the unconditional basis const of the
basis $\psi$ is
$M_{ub}=\sup_{\Delta\subseteq\mathbb{N}}\\{||F_{\psi}P_{\Delta}G_{\psi}^{*}||\\}$.
In virtue of the proposition 4.2.29 and theorem 4.2.32 in the book [16], we
have
###### Proposition 3.20.
For a Schauder basis $\psi$, it is a unconditional basis if and only if
$\sup_{\Delta\subseteq\mathbb{N}}\\{||F_{\psi}P_{\Delta}G_{\psi}^{*}||\\}<\infty$.
Now we can show that we can change a Schauder matrix to an equivalent Schauder
matrix which represents a strongly irreducible operator with small disturbance
on basis const.
###### Theorem 3.21.
Suppose that $\psi=\\{f_{n}\\}_{n=1}^{\infty}$ be a basis and its
corresponding Schauder matrix $F_{\psi}$ under a given ONB
$\varphi=\\{e_{n}\\}_{n=1}^{\infty}$ represents a bounded operator $T_{\psi}$
in $L(\mathcal{H})$. Then for any positive number $\epsilon>0$, there is a
matrix $X$ of some invertible operator $T\in L(\mathcal{H})$ such that
following properties hold:
1\. $XF_{\psi}$ is a strongly irreducible operator;
2\. The column vector sequence $\psi^{{}^{\prime}}$ of $XF_{\psi}$ comprise a
basis sequence equivalent to $\psi$. Moreover, if we denote by
$M^{{}^{\prime}}$ the basis const of $\psi^{{}^{\prime}}$, then we can ask
that the condition $|M-M^{{}^{\prime}}|<\epsilon$ holds; If $\psi$ is also a
unconditional basis, we can also ask that the additional condition
$|M_{ub}^{{}^{\prime}}-M_{ub}|<\epsilon$ holds(Here $M_{ub}^{{}^{\prime}}$
denote the unconditional basis const of the basis $\psi^{{}^{\prime}}$).
###### Proof.
By our main theorem 1.2, we only need to show that the second property holds
when $X=U+K$ been chosen carefully. Clearly for a given $\delta>0$, we can
choose a $X=U+K$ such that both
$1-\frac{\delta}{2}<||U+K||<1+\frac{\delta}{2}$ and
$1-\frac{\delta}{2}<||(U+K)^{-1}||<1+\frac{\delta}{2}$ hold. Then we have
$\begin{array}[]{c}||XF_{\psi}P_{k}G^{*}_{\psi}X^{-1}||\leq||X||\cdot||F_{\psi}P_{k}G^{*}_{\psi}||\cdot||X^{-1}||,\\\
||F_{\psi}P_{k}G^{*}_{\psi}||\leq||X^{-1}||\cdot||XF_{\psi}P_{k}G^{*}_{\psi}X^{-1}||\cdot||X||.\end{array}$
Hence following inequality holds for any $k\in\mathbb{N}$:
$(1+\delta)^{-2}\cdot||F_{\psi}P_{k}G^{*}_{\psi}||\leq||XF_{\psi}P_{k}G^{*}_{\psi}X^{-1}||\leq(1+\delta)^{2}\cdot||F_{\psi}P_{k}G^{*}_{\psi}||.$
Now by proposition 3.18, we have
$M=\sup_{n}\\{||FP_{n}G^{*}||\\}\hbox{ and
}M^{{}^{\prime}}=\sup_{n}\\{||XFP_{n}G^{*}X^{-1}||\\}.$
Therefore by choosing $\delta<\frac{\epsilon}{4M}$ we can prove the first part
of property 2; the second part of property 2 can be proved in just the same
way by lemma 3.20. ∎
Acknowledgements A large part of this article was developed during the seminar
on operator theory held at Jilin University in China.
## References
* [1] Cao Y., Tian G., Hou B. Z., Schauder Bases and Operator Theory: A Criterion for Operators to Be Conditional, preprint.
* [2] C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, Approximation of Hilbert space operators, Volume II, Boston: Pitman Publishing Inc, 1984.
* [3] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Mathematica , Vol. 141(1), 1978, 187-261.
* [4] D. A. Herrero, Approximation of Hilbert space operators, Volume I, Second edition, Boston: Pitman Publishing Inc, 1989\.
* [5] D. A. Herrero, The Diagonal Entries in the Formula ‘Quasitriangular - Compact = Triangular’, and Restrictions of Quasitriangularity, Trans. Amer. Math. Soc. 298, 1986, 1-42.
* [6] Y. Q. Ji, Quasitriangular + small compact = strongly irreducible, Trans.Amer.Math.Soc. 11(1999) 4657-4673.
* [7] Y. Q. Ji,
* [8] Y. Q. Ji and C. L. Jiang, Small compact perturbation of strongly irreducible operators, Integral Equations and Operator Theory, 43(2002)417-449.
* [9] Y. Q. Ji, C. L. Jiang and Z. Y. Wang, Essentially normal + small compact = strongly irreducible, Chinese Math.Ann.Series B, 18(1997) No.4 485-494.
* [10] Y. Q. Ji, C. L. Jiang and Z. Y. Wang, The ($\mathcal{U}+\mathcal{K}$)-orbit of essentially normal operators and compact perturbation of strongly irreducible operators, Functional Analysis in China. MIA 356, 307-314. Kluwer Academic Publishers 1996.
* [11] Y. Q. Ji, C. L. Jiang and Z. Y. Wang, Strongly irreducible operators in nest algebras, Integral Equations and Operator Theory, 28(1997)28-44.
* [12] Y. Q. Ji, J. X. Li and Y. H. Yang, A characterization of bilateral operator weighted shifts being Cowen-Douglas operators, Proc.Amer.Math.Soc., 129(2001),no.11,3025-3210.
* [13] C. L. Jiang, S. Power and Z. Y. Wang, Biquasitriangular operators have strongly irreducible perturbations, Quarterly J. Mathematics, 51(2000), 353-369.
* [14] C. L. Jiang, S. H. Sun and Z. Y. Wang, Essentially normal operator + compact operator = strongly irreducible operator, Trans.Amer.Math.Soc. 349(1997)217-233.
* [15] C. L. Jiang and Z. Y. Wang, The spectral picture and the closure of the similarity orbit of strongly irreducible operators, Integral Equations and Operator Theory 24(1996) 81-105.
* [16] Robert E. Megginson, An introuduction to Banach Space Theory, GTM183, Springe-Verlag, 1998.
* [17] A. M. Olevskii, On operators generating conditional bases in a Hilbert space, Translated from Matematicheskie Zametki, Vol(12), No.1, pp. 73-84, July, 1972.
* [18] A. L. Shields, Weighted shift operators and analytic function theory, Math.Surveys, Vol.13, Amer.Math.Soc.,Providence,RI,1974,49-128.
* [19] I. Singer, Bases in Banach Space I, Springer-verlag, 1970.
|
arxiv-papers
| 2012-04-07T02:08:21 |
2024-09-04T02:49:29.439690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Geng Tian, Youqing Ji, Yang Cao",
"submitter": "Cao Yang",
"url": "https://arxiv.org/abs/1204.1587"
}
|
1204.1620
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-099 LHCb-PAPER-2012-008 May 24, 2012
Inclusive $W$ and $Z$ production in the forward region at
$\sqrt{s}=7$$\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration †††Authors are listed on the following pages.
Measurements of inclusive $W$ and $Z$ boson production cross-sections in $pp$
collisions at $\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ using final states
containing muons are presented. The data sample corresponds to an integrated
luminosity of $37$$\mbox{\,pb}^{-1}$ collected with the LHCb detector. The $W$
and $Z$ bosons are reconstructed from muons with a transverse momentum above
20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and pseudorapidity between $2.0$ and
$4.5$, and, in the case of the $Z$ cross-section, a dimuon invariant mass
between $60$ and $120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The cross-
sections are measured to be $831\pm 9\pm 27\pm 29$$\rm\,pb$for $W^{+}$,
$656\pm 8\pm 19\pm 23$$\rm\,pb$for $W^{-}$ and $76.7\pm 1.7\pm 3.3\pm
2.7$$\rm\,pb$ for $Z$, where the first uncertainty is statistical, the second
is systematic and the third is due to the luminosity. Differential cross-
sections, $W$ and $Z$ cross-section ratios and the lepton charge asymmetry are
also measured in the same kinematic region. The ratios are determined to be
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}=$$1.27\pm
0.02\pm 0.01$ and
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}=$$19.4\pm
0.5\pm 0.9$. The results are in general agreement with theoretical
predictions, performed at next-to-next-to-leading order in QCD using recently
calculated parton distribution functions.
Published in JHEP Vol. 2012, Number 6 (2012), 58, DOI: 10.1007/JHEP06(2012)
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28, K. Belous32, I.
Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J.
Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S.
Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36,
C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N.
Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C.
Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M.
Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I.
Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho
Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X. Cid Vidal34, G.
Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J. Closier35, C.
Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A. Comerma-
Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B. Couturier35,
G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y. David38, I. De
Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De Miranda1, L. De
Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del
Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5, F. Dettori39, J.
Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo Bonal33,n, S.
Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A. Dovbnya40, F.
Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A. Dzyuba27, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C.
Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S.
Hansmann-Menzemer11, N. Harnew52, J. Harrison51, P.F. Harrison45, T.
Hartmann55, J. He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando
Morata34, E. van Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D.
Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11,
M. Jahjah Hussein5, E. Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F.
Jing3, M. John52, D. Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S.
Kandybei40, M. Karacson35, T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U.
Kerzel35, T. Ketel39, A. Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, I.
Komarov29, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A. Kozlinskiy38, L.
Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9,
K. Kruzelecki35, M. Kucharczyk20,23,35,j, V. Kudryavtsev31, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O. Leroy6, T.
Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C.
Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N.
Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B.
Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37,
S. Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37,
V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24,
S. T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, C. Voß55, H. Voss10, R. Waldi55, R. Wallace12, S.
Wandernoth11, J. Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D.
Websdale50, M. Whitehead45, J. Wicht35, D. Wiedner11, L. Wiggers38, G.
Wilkinson52, M.P. Williams45,46, M. Williams50, F.F. Wilson46, J. Wishahi9, M.
Witek23, W. Witzeling35, S.A. Wotton44, S. Wright44, S. Wu3, K. Wyllie35, Y.
Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, X. Yuan3, O. Yushchenko32,
M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
The measurement of the production cross-sections for $W$ and $Z$ bosons
constitutes an important test of the Standard Model and provides valuable
input to constrain the proton parton density functions (PDFs). Theoretical
predictions are known to next-to-next-to-leading-order (NNLO) in perturbative
quantum chromodynamics (pQCD). These calculations are in good agreement with
recent measurements at the LHC from the ATLAS [1, 2], and the CMS [3, 4]
experiments as well as with the results from the $p\bar{p}$ collider
experiments at the $\mathrm{S}\bar{\mathrm{p}\mathrm{pS}}$ [5, 6] and the
Tevatron [7, 8, 9, 10]. The dominant theoretical uncertainty on the cross-
sections arises from the present knowledge of the PDFs and the strong coupling
constant. The accuracy strongly depends on the pseudorapidity111The
pseudorapidity $\eta$ is defined to be $\eta=-\ln\tan(\theta/2)$, where the
polar angle $\theta$ is measured with respect to the beam axis. range;
consequently, measurements by LHCb, which is fully instrumented in the forward
region $2.0<\eta<5.0$, can provide input to constrain the PDFs, both for
pseudorapidities $\eta>2.5$ and in the region which is common to ATLAS and
CMS, $2.0<\eta<2.5$. Besides the determination of the $W$ and $Z$ boson cross-
sections, the measurement of their ratios
$R_{WZ}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
and
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and of the $W$ production charge asymmetry constitute important tests of the
Standard Model, as experimental and theoretical uncertainties partially
cancel. The $W$ charge asymmetry is sensitive to the valence quark
distribution in the proton [11] and provides complementary information to the
results from deep-inelastic scattering cross-sections at HERA [12] as those
data do not strongly constrain the ratio of $u$ over $d$ quarks at low Bjorken
$x$, where $x$ is the proton momentum fraction carried by the quark.
Measurements of $W$ and $Z$ boson production at LHCb have a sensitivity to
values of $x$ as low as $1.7\times 10^{-4}$ and will contribute significantly
to the understanding of PDFs at low $x$ and reasonably large four-momentum
transfer $Q^{2}$, which corresponds to the squared mass of the $W$ or the $Z$
boson.
The measurements of the inclusive $W$ and $Z$ cross-sections222Throughout this
paper $Z$ includes both the $Z$ and the virtual photon ($\gamma^{\star}$)
contribution. in $pp$ collisions at a centre-of-mass energy of
$7$$\mathrm{\,Te\kern-1.00006ptV}$, using final states containing muons, are
presented in this paper. The analysis is based on data taken by the LHCb
experiment in $2010$ with an integrated luminosity of $37$$\mbox{\,pb}^{-1}$.
The cross-sections are measured in a fiducial region corresponding to the
kinematic coverage of the LHCb detector, where the final state muons have a
transverse momentum, $p^{\mu}_{\mathrm{T}}$, exceeding
$20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and lie within the pseudorapidity
range $2.0<\eta^{\mu}<4.5$. This range is smaller than the LHCb acceptance in
order to avoid edge effects for the acceptance. In addition, the invariant
mass of the muons from the $Z$ boson must be in the range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Results are
presented for the total cross-sections and cross-section ratios. Cross-
sections are also measured in bins of muon pseudorapidity for $W$, and in bins
of $Z$ rapidity ($y^{Z}$) for $Z$ production. Because of the presence of the
neutrino, the production asymmetry between $W^{+}$ and $W^{-}$ cannot be
reconstructed as a function of the boson rapidity. Instead it is measured as a
function of the experimentally accessible muon pseudorapidity, $\eta^{\mu}$,
and referred to as the lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$.
To constrain the PDFs, it is useful to measure $A_{\textmu}$ for different
$p^{\mu}_{\mathrm{T}}$ thresholds. The data are compared to NNLO and NLO pQCD
predictions with recent parametrisations for the PDFs. The signal efficiency
and background contribution are mostly derived from data.
The remainder of the paper is organised as follows. Section 2 describes the
LHCb detector and the Monte Carlo samples. Section 3 describes the selection
of the $W$ and $Z$ candidates, the backgrounds, the determination of the
purity and the signal efficiencies. The measurement of the cross-sections as
well as the systematic uncertainties are discussed in Sect. 4. The results are
presented in Sect. 5 and conclusions in Sect. 6.
## 2 LHCb detector and Monte Carlo samples
The LHCb detector [13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector (TT) located
upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and
three stations of silicon-strip detectors (IT) and straw drift-tubes (OT)
placed downstream. The combined tracking system has a momentum resolution
$\Delta p/p$ that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a muon system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. To avoid the possibility that a few events with
high occupancy dominate the CPU time of the software trigger, a set of global
event cuts is applied on the hit multiplicities of most subdetectors used in
the pattern recognition algorithms.
Several Monte Carlo (MC) simulated samples are used to develop the event
selection, estimate the backgrounds, cross-check the efficiencies and to
account for the effect of the underlying event. The Pythia $6.4$ [14]
generator, configured as described in Ref. [15], with the CTEQ6ll [16]
parametrisation for the PDFs is used to simulate the processes
$Z\rightarrow\mu\mu$, $Z\rightarrow\tau\tau$, $W\rightarrow\mu\nu$ and
$W\rightarrow\tau\nu$. The hard partonic interaction is calculated in leading
order pQCD and higher order QCD radiation is modelled using initial and final
state parton showers in the leading log approximation [17]. The fragmentation
into hadrons is simulated in Pythia by the Lund string model [18]. All
generated events are passed through a Geant4 [19] based detector simulation,
the trigger emulation and the event reconstruction chain of the LHCb
experiment.
Samples of $W\rightarrow\mu\nu$ and $Z\rightarrow\mu\mu$ simulated events with
one muon in the LHCb acceptance have been reweighted to reproduce the NNLO
$p^{\mu}_{\mathrm{T}}$ distribution. These samples are referred to as $W$-MC
and $Z$-MC, respectively. In the first step a correction factor is calculated
as a function of the generated muon transverse momentum by determining the
ratio of the generated $p^{\mu}_{\mathrm{T}}$ spectrum, as simulated by the
Powheg [20, *powheg1, *powheg2] generator at NLO, to the generated
$p^{\mu}_{\mathrm{T}}$ spectrum from Pythia. In the second step the events are
reweighted with a factor given by the ratio between the NNLO and NLO
prediction as calculated with Dynnlo [23]. This factor is calculated as a
function of the rapidity of the boson. As an alternative, Pythia samples have
been reweighted to reproduce the $p^{\mu}_{\mathrm{T}}$ distribution as
calculated with Resbos [24, *resbos1, *resbos2]. Resbos includes a NLO
calculation plus next-to-next-to-leading-log resummation of QCD effects at low
transverse momentum.
## 3 Selection of $W$ and $Z$ events
### 3.1 Muon reconstruction and identification
Events with high transverse momentum muons are selected using a single muon
trigger with a threshold of
$p^{\mu}_{\mathrm{T}}>10$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Tracks are
reconstructed starting from the VELO, within which particle trajectories are
approximately straight, since the detector is located upstream of the magnet.
Candidate tracks are extrapolated to the other side of the magnet and a search
is made for compatible hits in the IT and OT sub-detectors. An alternative
strategy searches for track segments in both the VELO and IT/OT detectors and
extrapolates each to the bending plane of the magnet, where they are matched.
Once VELO and IT/OT hits have been combined, an estimate of the track momentum
is available and the full trajectory can be defined. Finally, hits in the TT
sub-detector are added if consistent with the candidate tracks. Thus, the
presence of TT hits can be considered as an independent confirmation of the
validity of the track.
Muons are identified by extrapolating the tracks and searching for compatible
hits in the four most downstream muon stations. For the high momentum muons
that concern this analysis, hits must be found in all four muon stations. In
total, the muon candidate must have passed through over $20$ hadronic
interaction lengths of material.
### 3.2 Selection of $Z\rightarrow\mu\mu$ candidates
Candidate $Z\rightarrow\mu\mu$ events are selected by requiring a pair of well
reconstructed tracks identified as muons; the invariant mass of the two muons
must be in the range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Each muon track
must have $p^{\mu}_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
lie in the range $2.0<\eta^{\mu}<4.5$. The relative uncertainty on the
momentum measurement is required to be less than $10$% and the probability for
the $\chi^{2}/$ndf for the track fit larger than $0.1$%, where ndf is the
number of degrees of freedom. In total, $1966$ $Z$ candidates are selected;
their mass distribution is shown in Fig. 1. The data are not corrected for
initial or final state radiation. A Crystal Ball [27] function for the $Z$
peak, and an exponential distribution for both the off-resonance Drell-Yan
($\gamma^{\star}$) production and the small background contribution are fitted
to the distribution. The fitted mass $90.7\pm
0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and width $3.0\pm
0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, where the uncertainties are
statistical, are consistent with expectation from simulation.
Figure 1: Invariant mass of the selected muon pairs. The fitted distribution
to the data is shown as a solid line and the contribution from background and
off-resonance Drell-Yan production as a dashed line.
### 3.3 $Z\rightarrow\mu\mu$ event yield
The background contribution to the $Z\rightarrow\mu\mu$ analysis is very low.
Five different sources are investigated.
1. 1.
Decays from $Z\rightarrow\tau\tau$ contribute, if both taus decay leptonically
to muons and neutrinos. The tau background is estimated from simulation, with
the $Z$ cross-section fixed to the cross-section measured in this analysis, to
contribute $0.6\pm 0.1$ events to the total sample.
2. 2.
Decays of heavy flavour hadrons contribute to the background if they decay
semileptonically (“heavy flavour” background). The contribution is estimated
from a sample, which is enriched in background. “Non-isolated” muons are
selected with $p^{\mu}_{\mathrm{T}}>15$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $M_{\mu\mu}>40$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and the scalar
sum of the transverse momenta of all tracks in a cone of half angle $0.5$ in
$\eta\mbox{-}\phi$ around the muons larger than
$4$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; here $\phi$ is the azimuthal angle
measured in radians. A fit to the invariant mass distribution at low masses is
then used to estimate the background contribution in the $Z$ mass region. The
heavy flavour contribution is estimated to be $3.5\pm 0.8$ events.
3. 3.
Pions or kaons may be misidentified as muons if they decay in flight (“decay-
in-flight” background) or if they travel through the calorimeters and are
identified by the muon chambers (“punch-through” background). This background
should contribute equally in same-sign and opposite-sign combinations of the
muon pair. No event is found in the $Z$ selection with both tracks having the
same charge. The contribution from muon misidentification is estimated to be
less than one event.
4. 4.
$W$ pair production contributes to the sample if both $W$ bosons decay to a
muon and a neutrino. This contribution corresponds to $0.2\pm 0.1$ events as
estimated with Pythia MC simulation.
5. 5.
Decays of top quark pairs may contribute if both top quarks decay
semileptonically. Pythia MC simulation predicts a contribution of $0.5\pm 0.2$
events.
The total background contribution in the $Z$ sample in the range
$60\mbox{--}120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ amounts to $4.8\pm
1.0$ events. This corresponds to a purity $\rho^{Z}=0.997\pm 0.001$. The
purity is defined as the ratio of signal to candidate events. No significant
dependence on the boson rapidity is observed.
### 3.4 Selection of $W\rightarrow\mu\nu$ candidates
In leading order QCD, $W\rightarrow\mu\nu$ events are characterised by a
single high transverse momentum muon that is not associated with other
activity in the event. As only the muon can be reconstructed in LHCb, the
background contribution is larger for the $W$ than for the $Z$ candidates.
Therefore, more stringent requirements are placed on the track quality of the
muon and additional criteria are imposed in order to select $W$ candidates.
The optimisation of the $W$ selection and the evaluation of the selection
efficiency make use of a “pseudo-$W$” control sample obtained from the
previously described $Z$ selection, where each of the muons is masked in turn,
in order to mimic the presence of a neutrino and fake a $W\rightarrow\mu\nu$
decay. Excellent agreement is observed for all variables of interest between
pseudo-$W$ and $W$ simulated samples with the exception of those that have an
explicit dependence on the transverse momentum of the muon, as the underlying
momentum distribution differs for muons from $Z$ and $W$.
The identification of $W\rightarrow\mu\nu$ candidate events starts by
requiring a well reconstructed track which is identified as a muon. The track
must have a transverse momentum in the range
$20<p^{\mu}_{\mathrm{T}}<70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ within a
pseudorapidity range $2.0<\eta^{\mu}<4.5$. The relative error on the momentum
measurement must be less than $10$%, the probability for the $\chi^{2}/$ndf of
the track fit must be greater than $1$%, and there must be TT hits associated
to the track. The last requirement reduces the number of combinations of VELO
and IT/OT information that have been incorrectly combined to form tracks.
Figure 2: Distributions for $p_{\mathrm{T}}^{\mathrm{cone}}$ (top) and
$E_{\mathrm{T}}^{\mathrm{cone}}$ (bottom). The points are for muons from
pseudo-$W$ data, the yellow (shaded) histograms are for $W$-MC simulation,
while the open histograms are for muons from QCD background with IP
$>80$$\,\upmu\rm m$ from data. All distributions are normalised to unity.
To suppress background from $Z\rightarrow\mu\mu$ decays, it is required that
any other identified muon in the event has a transverse momentum below
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. This removes the events where both
muons have entered the LHCb acceptance.
Identified muons can originate from background processes of heavy flavour
decays, or misidentification of pions and kaons due to decay-in-flight or
punch-through (“QCD background”). In all such cases, the identified muon is
usually produced in the same direction as the other fragmentation products, in
contrast to muons from $W$ decays which tend to be isolated. The isolation of
the muon is described using the charged transverse momentum,
$p_{\mathrm{T}}^{\mathrm{cone}}$, and neutral transverse energy,
$E_{\mathrm{T}}^{\mathrm{cone}}$, in a cone around the candidate muon. The
quantity $p_{\mathrm{T}}^{\mathrm{cone}}$ is defined as the scalar sum of the
transverse momentum of all tracks, excluding the candidate muon, satisfying
$\sqrt{(\Delta\phi)^{2}+(\Delta\eta^{\mu})^{2}}<0.5$, where $\Delta\phi$ and
$\Delta\eta^{\mu}$ are the differences in $\phi$ and $\eta$ between the muon
candidate and the track. The quantity $E_{\mathrm{T}}^{\mathrm{cone}}$ is
defined in a similar way, but summing the transverse energy of all
electromagnetic calorimeter deposits not associated with tracks. The
distributions for $p_{\mathrm{T}}^{\mathrm{cone}}$ and
$E_{\mathrm{T}}^{\mathrm{cone}}$ are shown in Fig. 2 for pseudo-W data, $W$-MC
and muons with $p_{\mathrm{T}}^{\mu}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and an IP larger than $80$$\,\upmu\rm m$. The IP of the muon is defined as the
distance of closest approach to the primary vertex calculated from the other
tracks in the event excluding the muon candidate. The sample with high IP is
enriched with muons from decays of heavy flavour hadrons, showing the typical
shape of QCD background. There is agreement between pseudo-W data and $W$-MC,
while the shape for the heavy flavour events is quite different. To suppress
QCD background, it is required that
$p_{\mathrm{T}}^{\mathrm{cone}}<2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$E_{\mathrm{T}}^{\mathrm{cone}}<2$$\mathrm{\,Ge\kern-1.00006ptV}$.
Muons originating from semi-leptonic decays of heavy flavour hadrons can be
further suppressed by a cut on the IP. Due to the lifetimes of the $B$ and $D$
mesons, these muons do not originate from the primary $pp$ interaction. The IP
distribution is shown in Fig. 4 for pseudo-W events, $W$-MC, and simulated
semi-leptonic decays of hadrons containing a $b$ or $c$ quark. The pseudo-W
events and $W$-MC are in agreement and peak at low values of IP, in contrast
to the heavy flavour background. For the $W$ candidate selection it is
required that IP $<40$$\,\upmu\rm m$. This cut also removes a large fraction
of the background from $W\rightarrow\tau\nu$ and $Z\rightarrow\tau\tau$
decays.
Figure 3: Muon IP distribution for pseudo-W events as points, $W$-MC as a
yellow (shaded) histogram, and muons from simulated semi-leptonic decays of
hadrons containing a $b$ quark in the full open histogram or a $c$ quark in
the dashed open histogram. All distributions are normalised to unity.
Figure 4: $E/pc$ for pseudo-W events as points, $W$-MC as a yellow (shaded)
histogram, and for hadrons from randomly triggered events in the open
histogram. The energy $E$ is the sum of the energies in the electromagnetic
and hadronic calorimeter associated with the particle. All distributions are
normalised to unity.
Pions and kaons that punch-through to the muon chambers can be distinguished
from true muons as they leave substantial energy deposits in the calorimeters.
Figure 4 shows the summed energy, $E$, in the electromagnetic and hadronic
calorimeter associated with the particle, divided by the track momentum, $p$,
for pseudo-$W$ events, $W$-MC, and hadrons with
$p_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in randomly
triggered events. By requiring $E/pc<0.04$ the punch-through contamination can
be reduced to a negligible level. The disagreement between pseudo-$W$ data and
simulated $W$-MC in Fig. 4 is caused by the different underlying momentum
distribution for muons from $W$ and $Z$.
### 3.5 $W\rightarrow\mu\nu$ event yield
After the $W$ selection requirements are imposed $14\,660$ $W^{+}$ and
$11\,618$ $W^{-}$ candidate events are observed. The $W\rightarrow\mu\nu$
signal yield has been determined by fitting the $p^{\mu}_{\mathrm{T}}$ spectra
of positive and negative muons in data, to template shapes for signal and
backgrounds in five bins of $\eta^{\mu}$. The fit is performed with the
following sources for signal and background with the shapes and normalisations
as described below.
1. 1.
The $W\rightarrow\mu\nu$ signal template is obtained using the $W$-MC. The
normalisation is left free to vary in each bin of $\eta^{\mu}$ and for each
charge.
2. 2.
The shape of the template of the largest background, $Z\rightarrow\mu\mu$, is
taken from the $Z$-MC. The normalisation is fixed from data by counting the
number of $Z$ events, scaled by the ratio of events with one muon in the LHCb
acceptance to events with both muons in the acceptance, as determined from
$Z$-MC. The ratio is corrected for the different reconstruction and selection
efficiencies for $W$ and $Z$ as derived from data. This gives an expectation
of $2435\pm 101$ background events (($9.3\pm 0.4$)% of the total sample) in
good agreement with $2335\pm 25$ events found from simulation.
3. 3.
The shape of the $W\rightarrow\tau\nu$ and $Z\rightarrow\tau\tau$ templates
are taken from Pythia. The $Z\rightarrow\tau\tau$ template is scaled according
to the observed number of $Z$ events. These $\tau$ backgrounds constitute
$2.7$% of the total sample.
4. 4.
The heavy flavour template is obtained from data by requiring that the muon is
not consistent with originating from the primary vertex (IP $>80$$\,\upmu\rm
m$). The normalisation is determined from data applying all requirements
except for the impact parameter and fitting the resulting IP distribution to
the two templates shown in Fig. 4: the pseudo-$W$ data to describe the signal,
and the simulated heavy flavour events to describe the background. The heavy
flavour contribution is estimated to be ($0.4\pm 0.2$)% of the total sample.
5. 5.
The punch-through contribution from kaons and pions is largely suppressed by
the requirement on $E/pc$. The $E/pc$ distribution in Fig. 4 is fitted to
pseudo-$W$ data for the signal, and a Gaussian for the punch-through, in order
to estimate the punch-through contribution. This is found to be negligible
($0.02\pm 0.01$)% of the total sample, and also has a shape very similar to
the decay-in-flight component. Hence, this component is not considered when
determining the signal yield.
6. 6.
The decay-in-flight shape is found from data in a two-step procedure using all
events selected throughout $2010$ by any trigger requirement. First, tracks
with a transverse momentum between $20$ and
$70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are taken to describe the
$p_{\mathrm{T}}$ spectrum of hadrons; tracks that fired a muon trigger are
excluded from the sample. Second, this spectrum is weighted according to the
probability for a hadron to decay-in-flight. This probability is defined as
the fraction of tracks identified as muons in randomly triggered events and is
parametrised as a function of the momentum, $p$, by a function of the form
$1-e^{-\alpha/p},$ (1)
as would be expected for a particle whose mean lifetime in the laboratory
frame scales with its boost. Consistent values for $\alpha$ are found in each
pseudorapidity bin and are in agreement with a calculation of the decay
probability based on the mean lifetimes for charged pions and kaons, and the
distance to the electromagnetic calorimeter before which the hadron must have
decayed. The average of the determinations in each pseudorapidity bin defines
the central value for $\alpha$. The relative normalisation of positively to
negatively charged tracks in each bin of pseudorapidity is fixed to that
observed in randomly triggered events, but the overall normalisation in each
bin of pseudorapidity is left free.
Figure 5: Distribution of muon $p_{\mathrm{T}}$ for positively (left panel)
and negatively (right panel) charged muons in $W$ candidate events, for the
total fiducial cross-section (a). The plots (b) to (f) give the same
information for the different $\eta^{\mu}$ bins. The data (points) are
compared to the fitted contributions from $W^{-}$ and $W^{+}$ (light shaded).
The background contributions are, from top to bottom in the legend: decay-in-
flight, $Z\rightarrow\mu\mu$, $\tau$ decays of W and $Z$, and heavy flavour
decays.
The default fit has $15$ free parameters: five parameters for the
normalisation of $W^{+}$ in each of the pseudorapidity bins, five parameters
for $W^{-}$, and five parameters for the contribution coming from the decay-
in-flight. The normalisation of the other sources is fixed. The result of the
fit is shown in Fig. 5. Integrated over both charges and
$p^{\mu}_{\mathrm{T}}$ it is found that ($44.3\pm 1.2$)% of the total sample
is due to $W^{+}$, ($34.9\pm 1.1$)% due to $W^{-}$, ($8.5\pm 0.8$)% due to the
decay-in-flight contribution and the remainder due to the other backgrounds.
The $\chi^{2}/$ndf of the fit is $1.002$. The fit is repeated with the
$Z\rightarrow\mu\mu$ and $W\rightarrow\mu\nu$ template corrected with Resbos
instead of NNLO, yielding ($43.6\pm 1.2$)% for $W^{+}$ and ($34.4\pm 1.1$)%
for $W^{-}$ with $\chi^{2}/\mathrm{ndf}=0.983$. The average of the two fits,
which gives a purity $\rho^{W^{+}}=0.788\pm 0.021$ for $W^{+}$ and
$\rho^{W^{-}}=0.784\pm 0.025$ for $W^{-}$, is taken for the final result; half
of the difference is taken as the systematic uncertainty.
## 4 Cross-section measurement
### 4.1 Cross-section definition
Cross-sections are quoted in the kinematical range defined by the
measurements. The cross-sections are measured in bins of $\eta^{\mu}$ for the
$W$ and in bins of $y^{Z}$ in case of the $Z$. The cross-section in a given
bin of $y^{Z}$ ($\eta^{\mu}$) is defined as
$\sigma_{Z\rightarrow\mu\mu}(y^{Z})=\frac{\rho^{Z}f^{Z}_{\mathrm{FSR}}}{\mathcal{L}\mathcal{A}^{Z}}\sum\limits_{\eta_{i}^{\mu},\eta_{j}^{\mu}}{\frac{N^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})}{\varepsilon^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})}},\mbox{~{}~{}~{}~{}~{}}\sigma_{W\rightarrow\mu\nu}(\eta^{\mu})=\frac{\rho^{W}f^{W}_{\mathrm{FSR}}N^{W}}{\mathcal{L}\mathcal{A}^{W}\varepsilon^{W}},$
(2)
where $N^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})$ is the number of $Z$ candidates
in the respective $y^{Z}$ bin with the two muons in the bins $\eta_{i}^{\mu}$
and $\eta_{j}^{\mu}$ being reconstructed with the efficiency
$\varepsilon^{Z}(\eta_{i}^{\mu},\eta_{j}^{\mu})$. Similarly, $N^{W}$ is the
number of $W$ candidates with the muon in the $\eta^{\mu}$ bin. The purity of
the sample ($\rho^{Z(W)}$), the acceptance ($\mathcal{A}^{Z(W)}$), the
correction factor for final state radiation (FSR) ($f^{Z(W)}_{\mathrm{FSR}}$)
and the efficiency ($\varepsilon^{W}$) are determined per bin; $\mathcal{L}$
is the integrated luminosity. The total cross-section is obtained by summing
the contributions of the five $y^{Z}$ or $\eta^{\mu}$ bins.
### 4.2 Signal efficiencies
The data are corrected for efficiency losses due to track reconstruction, muon
identification, and trigger requirements for both analyses. There is an
additional selection efficiency in the $W$ analysis due to the requirements on
the number of additional muons, IP, $E/pc$, $p_{\mathrm{T}}^{\mathrm{cone}}$,
$E_{\mathrm{T}}^{\mathrm{cone}}$, and on TT hits. All efficiencies are
determined from data.
The efficiencies for track reconstruction and muon identification are obtained
using a tag-and-probe method in the $Z$ sample. One of the muons in the $Z$
sample (tag) satisfies all the track criteria. The other muon (probe) is
selected with looser criteria that depend on the efficiency to be measured.
The invariant mass of the dimuon candidates, reconstructed from the tag and
the probe muons, must lie in the window of
$20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $Z$ mass.
The tracking efficiency, which accounts for track reconstruction and the track
quality requirements, is studied using well reconstructed tracks in the muon
stations which are linked to hits in TT. The average track finding efficiency
is about $90$% in the $Z$ sample and about $86$% for the muon in $W$ events.
The tracking efficiency for $W$ is lower due to the more restrictive cuts on
the track quality. The muon identification efficiency is determined with
tracks without the muon identification requirement for the probe muon. The
average single muon efficiency is above $99$%. Both the tracking and the muon
identification efficiencies agree with simulation within errors.
The trigger efficiency contains two components, the first due to the
efficiency of the single muon trigger and the other due to the global event
cuts (GEC). The single muon trigger efficiency is determined using the
$Z\rightarrow\mu\mu$ sample. One muon is required to fire the single muon
trigger. The trigger response of the other muon then defines the trigger
efficiency. The requirement on the occupancy of the events depends on the
multiplicity of the primary interactions. It was checked with a sample which
did not have the GEC applied, that no events are lost if there is only one
primary vertex reconstructed. The GEC efficiency as a function of the number
of primary vertices is determined by adding randomly triggered events to
events with $Z$ ($W$) candidates with one primary vertex; on average it
amounts to $93$%. The overall trigger efficiency is calculated for each event
depending on the lepton pseudorapidity and the primary vertex multiplicity. It
is found to be about $88$% for the $Z$ and $75$% for the $W$ sample.
The $W$ selection requires that there are no other muons with
$p^{\mu}_{\mathrm{T}}>2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$p_{\mathrm{T}}^{\mathrm{cone}}<2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$E_{\mathrm{T}}^{\mathrm{cone}}<2$$\mathrm{\,Ge\kern-1.00006ptV}$, IP
$<40$$\,\upmu\rm m$, and $E/pc<0.04$. The selection efficiency is determined
from the fraction of pseudo-W events that pass these requirements. A similar
method is used to evaluate the efficiency for the requirement of TT hits
associated to the muon track of the $W$ candidate. Simulation studies show
that with the exception of the $E/pc$ distribution, the pseudo-W data provide
a consistent description of $W\rightarrow\mu\nu$ simulation, as shown in Figs.
2 and 4. However, the harder muon $p_{\mathrm{T}}$ spectrum in pseudo-W data
leads to slightly lower values of $E/pc$ than for muons produced in $W$
decays. The simulation is used to determine this difference, which is only
significant for $\eta^{\mu}$ between $2.0$ and $2.5$, where the efficiency for
$W$ events is estimated to be $2.1$% lower than for pseudo-W data. The
selection efficiency is about $67$% for $2.5<\eta^{\mu}<4.0$ and drops to
about $53$% and $33$% for the two bins at the edge of the acceptance with
$2.0<\eta^{\mu}<2.5$ and $4.0<\eta^{\mu}<4.5$, respectively.
All the efficiencies have been checked for possible dependences on
$p_{\mathrm{T}}^{\mu}$, the azimuthal angle of the muon, magnet polarity, and
$\eta^{\mu}$. Only the latter exhibits a significant dependence, which is
taken into account. Since any charge bias of the efficiencies would directly
influence the measurement of the lepton charge asymmetry, it was checked there
is no significant charge dependence within the uncertainties of the
efficiencies. The efficiency corrections are applied as a function of the
pseudorapidity of the muons except the GEC.
The efficiencies are uncorrelated between pseudorapidity bins but correlated
for $W^{+}$, $W^{-}$ and $Z$. These correlations are taken into account for
the measurement of the lepton charge asymmetry and the cross-section ratios.
### 4.3 Acceptance
The selection criteria for the $W$ and $Z$ define the fiducial region of the
measurement. Simulated events are used to determine the acceptance
$\mathcal{A}$, defined as $\mathcal{A}={N_{\mathrm{rec}}}/{N_{\mathrm{gen}}}$.
Here, $N_{\mathrm{rec}}$ is the number of reconstructed events satisfying the
cuts on the pseudorapidity and the minimal momenta of the reconstructed muons,
as well as on the dimuon mass in the case of the $Z$ analysis. Similarly,
$N_{\mathrm{gen}}$ is the number of generated events with the cuts applied on
the generated muons. The acceptance is estimated with $W$-MC and $Z$-MC. It is
found to be consistent with unity for the $Z$ and above $0.99$ for the $W$
analysis. For the latter, the acceptance corrects for the small loss of events
with $p^{\mu}_{\mathrm{T}}>70$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
### 4.4 Luminosity
The absolute luminosity scale was measured at specific periods during the data
taking, using both Van der Meer scans [28] where colliding beams are moved
transversely across each other to determine the beam profile, and a beam-gas
imaging method [29, 30]. For the latter, reconstructed beam-gas interaction
vertices near the beam crossing point determine the beam profile. Both methods
give similar results and are estimated to have a precision of order $3.5$%.
The knowledge of the absolute luminosity scale is used to calibrate the number
of tracks in the VELO, which is found to be stable throughout the data-taking
period and can therefore be used to monitor the instantaneous luminosity of
the entire data sample. The dataset for this analysis corresponds to an
integrated luminosity of $37.1\pm 1.3$$\mbox{\,pb}^{-1}$.
### 4.5 Corrections to the data
The measured cross-sections are corrected to Born level in quantum
electrodynamics (QED) in order to provide a consistent comparison with NLO and
NNLO QCD predictions, which do not include the effects of FSR. Corrections
have been estimated using Photos [31] interfaced to Pythia. The Pythia
$p_{\mathrm{T}}$ spectrum of the electroweak boson has been reweighted to the
NNLO spectrum as determined with Dynnlo [23]. The correction is taken as the
number of events within the fiducial cuts of the measurements after FSR
divided by the number of events generated within the fiducial cuts.
Pythia simulation is used to study bin-to-bin migrations for $\eta^{\mu}$ and
$y^{Z}$. No significant net migration is observed and no correction is
applied.
### 4.6 Systematic uncertainties
Table 1: Contributions to the systematic uncertainty for the total $Z$ and $W$
cross-sections. The different contributions are discussed in Sect. 4.6
Source | $\Delta\sigma_{Z\rightarrow\mu\mu}$ (%) | $\Delta\sigma_{W^{+}\rightarrow\mu^{+}\nu}$ (%) | $\Delta\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ (%)
---|---|---|---
Signal purity | $\pm 0.1$ | $\pm 1.2$ | $\pm 0.9$
Template shape (fit) | – | $\pm 0.9$ | $\pm 1.0$
Efficiency (trigger, tracking, muon id) | $\pm 4.3$ | $\pm 2.2$ | $\pm 2.0$
Additional selection | – | $\pm 1.8$ | $\pm 1.7$
FSR correction | $\pm 0.02$ | $\pm 0.01$ | $\pm 0.02$
Total | $\pm 4.3$ | $\pm 3.2$ | $\pm 2.9$
Luminosity | $\pm 3.5$ | $\pm 3.5$ | $\pm 3.5$
Aside from the uncertainty on the luminosity measurement, the main sources of
experimental uncertainties come from the efficiency determinations and the
background estimate in the $W$ analysis. The following sources have been
considered:
1. 1.
The relative uncertainties of the tracking, muon identification, trigger and
GEC efficiencies are added in quadrature. They lead to a systematic
uncertainty for the total cross-sections of $4.3$% ($2.2$%, $2.0$%) for the
$Z$ ($W^{+}$, $W^{-}$).
2. 2.
The statistical uncertainty on the efficiency of the additional selection cuts
for the $W$ analysis translates into a $1.8$% ($1.7$%) systematic uncertainty
on the total $W^{+}$ ($W^{-}$) cross-section.
3. 3.
The uncertainty of the background contribution for the $Z$ analysis is small;
the uncertainty in the determination of the sample purity leads to a $0.1$%
uncertainty on the total cross-section.
4. 4.
Both the shape and normalisations of the templates used in the $W$ fit are
considered as an additional source of uncertainty. To determine this
systematic uncertainty each of the following sources is varied in turn, the
data are refitted to determine the fraction of $W^{+}$ and $W^{-}$ events, and
the deviations from the original signal yield are combined in quadrature. The
following variations are made:
* •
the difference of the two fits using different $W$ and $Z$ templates (see
Sect. 3.5) leads to a variation on the $W$ fractions of $0.8$%;
* •
the normalisation of the $Z$ component was changed by the statistical
uncertainty with which it was determined, leading to a variation in the $W$
fractions of $0.3$%;
* •
the normalisation of the $W\rightarrow\tau\nu$ template was changed by the
statistical uncertainty with which it was determined, leading to a negligible
change in the $W$ fractions, since this template shape is very similar to the
decay-in-flight template which is allowed to vary in the fit;
* •
the heavy flavour template has also been changed by the statistical
uncertainty with which it was determined leading to a negligible change in the
$W$ fractions;
* •
instead of leaving the relative normalisation of the decays-in-flight template
between pseudorapidity bins to be free in the fit, this is fixed to the values
observed in randomly triggered events, and the full fit performed with a
single free parameter for the background; the $W$ fractions change by $0.2$%;
* •
the shape of the decay-in-flight template has been changed using different
values for $\alpha$ (see Eq. 1) to describe the decay probability,
corresponding to different regions in which the hadron must have
decayed;333Three different decay regions have been considered: from the
interaction point, from the VELO and from the TT stations up to the
electromagnetic calorimeter. no difference in the $W$ fractions is observed.
5. 5.
The uncertainty on the FSR correction is evaluated for each bin as the maximum
of the statistical uncertainty of the correction factor and the difference
between the weighted and unweighted FSR correction factor.
The sources of systematic uncertainties are summarised in Table 1, together
with the size of the resultant uncertainty on the $W$ and $Z$ total cross-
sections. The total systematic uncertainty is the sum of all contributions
added in quadrature.
## 5 Results
The inclusive cross-sections for $Z\rightarrow\mu\mu$ and $W\rightarrow\mu\nu$
production for muons with
$p^{\mu}_{\mathrm{T}}>20$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the
pseudorapidity region $2.0<\eta^{\mu}<4.5$ and, in the case of $Z$, the
invariant mass range
$60<M_{\mu\mu}<120$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are measured to
be
$\sigma_{Z\rightarrow\mu\mu}$ | = | $76.7\pm 1.7\pm 3.3\pm 2.7$$\rm\,pb$
---|---|---
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}$ | = | $831\pm 9\pm 27\pm 29$$\rm\,pb$
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ | = | $656\pm 8\pm 19\pm 23$$\rm\,pb$ ,
Figure 6: Measurements of the $Z$, $W^{+}$ and $W^{-}$ cross-section and
ratios, data are shown as bands which the statistical (dark shaded/orange) and
total (light hatched/yellow) errors. The measurements are compared to NNLO and
NLO predictions with different PDF sets for the proton, shown as points with
error bars. The PDF uncertainty, evaluated at the $68$% confidence level, and
the theoretical uncertainties are added in quadrature to obtain the
uncertainties of the predictions.
where the first uncertainty is statistical, the second systematic and the
third is due to the luminosity. All the measurements are dominated by the
luminosity and the systematic uncertainty. The latter is dominated by the
limited number of events for the background templates and in the determination
of the efficiencies.
The ratios
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and
$R_{WZ}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
are measured to be
$R_{W}$ | = | $1.27\pm 0.02\pm 0.01$
---|---|---
$R_{WZ}$ | = | $19.4\pm 0.5\pm 0.9$ .
Here, the uncertainty from the luminosity completely cancels. The systematic
uncertainty on the trigger, muon identification, tracking and selection
efficiencies, as well as the uncertainty on the purity are assumed to be fully
correlated between $W^{+}$ and $W^{-}$. No correlation is assumed between the
$\eta^{\mu}$ bins, except for the purity. The uncertainty on the $Z$ cross-
section from the reconstruction efficiency is correlated between boson
rapidity bins. The correlation of the uncertainty on the efficiencies between
$W$ and $Z$ are estimated with MC simulation to be $90$%. The full correlation
matrix is given in the Appendix (Table 2). The ratio of the $W$ to $Z$ cross-
section is measured, for each charge separately, to be
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{Z\rightarrow\mu\mu}$ | = | $10.8\pm 0.3\pm 0.5$
---|---|---
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}/\sigma_{Z\rightarrow\mu\mu}$ | = | $8.5\pm 0.2\pm 0.4$ .
A summary of the measurements of the inclusive cross-sections
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}$,
$\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$ and $\sigma_{Z\rightarrow\mu\mu}$,
and the ratios is shown in Fig. 6. The measurements are shown as a band which
represents the total and statistical uncertainties.
Figure 7: Differential cross-section for $Z\rightarrow\mu\mu$ as a function of
$y^{Z}$. The dark shaded (orange) bands correspond to the statistical
uncertainties, the light hatched (yellow) band to the statistical and
systematic uncertainties added in quadrature. Superimposed are NNLO (NLO)
predictions with different parametrisations for the PDF as points with error
bars; they are displaced horizontally for presentation.
Figure 8: Differential $W$ cross-section in bins of muon pseudorapidity. The
dark shaded (orange) bands correspond to the statistical uncertainties, the
light hatched (yellow) band to the statistical and systematic uncertainties
added in quadrature. Superimposed are NNLO (NLO) predictions as described in
Fig 8. Figure 9: Lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$
in bins of muon pseudorapidity. The dark shaded (orange) bands correspond to
the statistical uncertainties, the light hatched (yellow) band to the
statistical and systematic uncertainties added in quadrature. Superimposed are
NNLO (NLO) predictions as described in Fig 8. The MSTW08 values for
$\eta^{\mu}<2$ represent the central value of the prediction.
The results are compared to theoretical predictions calculated at NNLO with
the program Dynnlo [23] for the NNLO PDF sets of MSTW08 [32], ABKM09 [33],
JR09 [34], HERA15 [12] and NNPDF21 [35] and at NLO for the NLO PDF set CTEQ6m
[16].444Dynnlo sets $\alpha_{s}$ to the value of $\alpha_{s}$ at the mass of
the $Z$ boson as given by the different PDF sets. The scale uncertainties are
estimated by varying the renormalisation and factorisation scales by factors
of two around the nominal value, which is set to the boson mass. The
uncertainties for each set correspond to the PDF uncertainties at 68% and the
scale uncertainties added in quadrature.555The uncertainties for the PDF set
from CTEQ6m which is given at $90$% CL are divided by $1.645$.
While the $W^{-}$ and $Z$ cross-sections are well described by all
predictions, the $W^{+}$ cross-section is slightly overestimated by the ABKM09
and NNPDF21 PDF sets. The ratio of the $W^{-}$ to $Z$ cross-sections agrees
reasonably well with the predictions, but the $W^{+}$ to the $Z$ ratio is
overestimated by most of the predictions. The systematic uncertainties for the
$R_{W}$ almost cancel and also the theoretical uncertainties are much reduced.
The $R_{W}$ measurement tests the Standard Model predictions with a precision
of $1.7\%$ which is comparable to the uncertainty of the theoretical
prediction. The ABKM09 prediction overestimates this ratio while all the other
predictions agree with the measurement.
Figure 10:
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
in bins of muon pseudorapidity. The dark shaded (orange) bands correspond to
the statistical uncertainties, the light hatched (yellow) band to the
statistical and systematic uncertainties added in quadrature. Superimposed are
NNLO (NLO) predictions with different parametrisations as described in Fig 8.
The MSTW08 values for $\eta^{\mu}<2$ represent the central value of the
prediction.
Differential distributions are measured in five bins in $y^{Z}$ for the Z and
of $\eta^{\mu}$ for the W. Figure 8 shows the differential cross-section as a
function of the rapidity of the $Z$ boson together with NNLO (NLO) predictions
with different parametrisation for the PDFs of the proton. The predictions
agree with the measurements within uncertainties though all the predictions
are lower than the measured cross-section for $2.5<\eta^{\mu}<3.0$. The
differential cross-sections are listed in Table 4 in the Appendix.
The differential distribution of the $W^{+}$ and $W^{-}$ cross-section, the
lepton charge asymmetry $A_{\mu}$ and the ratio $R_{W}$ as a function of the
muon pseudorapidity are shown in Figs. 8, 9 and 10 and listed in Tables 4 to 6
as a function of $p^{\mu}_{\mathrm{T}}$. The measurement of the charge
asymmetry and the $W$ ratio provides important additional information on the
PDFs particularly on the valence quark distributions [11].
Since the inclusive cross-section for $W^{+}$ is larger than for $W^{-}$, due
to the excess of $u$ over $d$ quarks in the proton, the overall asymmetry is
positive. The asymmetry and the $W$ cross-sections strongly vary as a function
of the pseudorapidity of the charged lepton, and $A_{\mu}$ even changes sign,
owing to differing helicity dependence of the lepton couplings to the boson.
This behaviour is reflected in the differential $W$ cross-sections, where at
large muon pseudorapidities the $W^{-}$ cross-section is higher than the
$W^{+}$ cross-section, as a consequence of the $V-A$ structure of the $W$ to
lepton coupling. The cross-section and the asymmetry measurements are compared
to the NNLO (NLO) predictions with different parameterisation for the PDFs.
The ABKM09 prediction overestimates the measured asymmetry in three of the
five bins. The other predictions describe the measurement within
uncertainties.
Figure 11: Lepton charge asymmetry
$A_{\mu}=(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$
for muons with $p^{\mu}_{\mathrm{T}}>$25 (top) and
30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (bottom), respectively in bins of
muon pseudorapidity. The dark shaded (orange) bands correspond to the
statistical uncertainties, the light hatched (yellow) band to the statistical
and systematic uncertainties added in quadrature. The statistical uncertainty
is undistinguishable from the total uncertainty. Superimposed are the NNLO
predictions with the MSTW08 parametrisation for the PDF.
The asymmetry is also measured for two higher $p^{\mu}_{\mathrm{T}}$
thresholds for the muons, at 25 and 30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The result is shown in Fig. 11 and listed in Table 6. The NNLO prediction with
MSTW08 parametrisation for the PDF also describes the measured asymmetry with
the higher cuts on the transverse momentum of the muon.
## 6 Conclusions
Measurements of inclusive $W$ and $Z$ boson production in $pp$ collisions at
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ with final states containing muons
have been performed using $37$$\mbox{\,pb}^{-1}$ of data collected with the
LHCb detector. The inclusive cross-sections have been measured separately for
$W^{+}$ and $W^{-}$ production as well as the ratios
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$
and
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/\sigma_{Z\rightarrow\mu\mu}$
and the lepton charge asymmetry
$(\sigma_{W^{+}\rightarrow\mu^{+}\nu}-\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})/(\sigma_{W^{+}\rightarrow\mu^{+}\nu}+\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}})$.
The results have been compared to five next-to-next-to-leading order QCD
predictions with different sets for the parton density functions of the proton
and to one calculation at next-to-leading order. There is general agreement
with the predictions, though some of the PDF sets overestimate the ratios of
the cross-sections. The ratio
$\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}=$$1.27\pm
0.02\pm 0.01$ is measured precisely and allows the Standard Model prediction
to be tested with an accuracy of about $1.7$%, comparable to the uncertainty
on the theory prediction. These represent the first measurements of the $W$
and $Z$ production cross-sections and ratios in the forward region at the LHC,
and will provide valuable input to the knowledge of the parton density
functions of the proton. The uncertainty on the cross-section measurements is
dominated by systematic uncertainties. Since most of these are statistical in
nature, the accuracy on the measurement with further data is expected to
significantly improve.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] ATLAS collaboration, G. Aad et al., Measurement of the W $\rightarrow$ l$\nu$ and Z/$\gamma$* $\rightarrow$ ll production cross sections in proton-proton collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, JHEP 12 (2010) 060, arXiv:1010.2130
* [2] ATLAS collaboration, G. Aad et al., Measurement of the muon charge asymmetry from W bosons produced in pp collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, Phys. Lett. B701 (11) 31, arXiv:1103.2929
* [3] CMS collaboration, S. Chatrchyan et al., Measurement of the inclusive W and Z production cross sections in pp collisions at $\sqrt{s}$ = 7 TeV, JHEP 10 (2011) 132, arXiv:1107.4789
* [4] CMS collaboration, S. Chatrchyan et al., Measurement of the lepton charge asymmetry in inclusive $W$ production in pp collisions at $\sqrt{s}$ = 7 TeV, JHEP 04 (2011) 050, arXiv:1103.3470
* [5] UA1 collaboration, C. Albajar et al., Intermediate vector boson cross-sections at the CERN super proton synchrotron collider and the number of neutrino types, Phys. Lett. B198 (1987) 271
* [6] UA2 collaboration, J. Alitti et al., A measurement of the $W$ and $Z$ production cross-sections and a determination of $\Gamma_{W}$ at the CERN $\bar{p}p$ collider, Phys. Lett. B276 (1992) 365
* [7] CDF collaboration, F. Abe et al., Measurement of $\sigma\cdot{\cal{B}}(W\rightarrow e\nu)$ and $\sigma\cdot{\cal{B}}(Z^{0}\rightarrow e^{+}e^{-})$ in $p\bar{p}$ collisions at $\sqrt{s}=$1.8 TeV, Phys. Rev. Lett. 76 (1996) 3070, arXiv:hep-ex/9509010
* [8] CDF collaboration, F. Abe et al., Measurement of $Z^{0}$ and Drell-Yan production cross section using dimuons in $\bar{p}p$ collisions at $\sqrt{s}=1.8$ TeV, Phys. Rev. D59 (1999) 052002
* [9] CDF collaboration, A. Abulencia et al., Measurements of inclusive W and Z cross sections in p anti-p collisions at $\sqrt{s}$ = 1.96 TeV, J. Phys. G G34 (2007) 2457, arXiv:hep-ex/0508029
* [10] DØ collaboration, B. Abbott et al., Extraction of the width of the $W$ boson from measurements of $\sigma(p\bar{p}\rightarrow W+X)\times{\cal{B}}(W\rightarrow e\nu)$ and $\sigma(p\bar{p}\rightarrow Z+X)\times{\cal{B}}(Z\rightarrow ee)$ and their ratio, Phys. Rev. D61 (2000) 072001, arXiv:hep-ex/9906025
* [11] E. L. Berger, F. Halzen, C. S. Kim, and S. Willenbrock, Weak boson production at Tevatron energies, Phys. Rev. D40 (1989) 83
* [12] H1 and ZEUS collaborations, F. D. Aaron et al., Combined measurement and QCD analysis of the inclusive e±p scattering cross sections at HERA, JHEP 01 (2010) 109, arXiv:0911.0884
* [13] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [14] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [15] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [16] P. M. Nadolsky et al., Implications of CTEQ global analysis for collider observables, Phys. Rev. D78 (2008) 013004, arXiv:0802.0007
* [17] M. Bengtsson and T. Sjöstrand, Parton showers in Leptoproduction Events, Z. Phys. C37 (1988) 465
* [18] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjöstrand, Parton fragmentation and string dynamics, Phys. Rept. 97 (1983) 31
* [19] GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [20] P. Nason, A new method for combining NLO QCD with shower Monte Carlo algorithms, JHEP 11 (2004) 040, arXiv:hep-ph/0409146
* [21] S. Frixione, P. Nason, and C. Oleari, Matching NLO QCD computations with parton shower simulations: the POWHEG method, JHEP 11 (2007) 070, arXiv:0709.2092
* [22] S. Alioli, P. Nason, C. Oleari, and E. Re, A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX, JHEP 06 (2010) 043, arXiv:1002.2581
* [23] S. Catani, L. Cieri, G. Ferrera, D. de Florian, and M. Grazzini, Vector boson production at hadron colliders: A Fully exclusive QCD calculation at NNLO, Phys. Rev. Lett. 103 (2009) 082001, arXiv:0903.2120
* [24] G. A. Ladinsky and C.-P. Yuan, The nonperturbative regime in QCD resummation for gauge boson production at hadron colliders, Phys. Rev. D50 (1994) 4239, arXiv:hep-ph/9311341
* [25] C. Balazs and C.-P. Yuan, Soft gluon effects on lepton pairs at hadron colliders, Phys. Rev. D56 (1997) 5558, arXiv:hep-ph/9704258
* [26] F. Landry, R. Brock, P. M. Nadolsky, and C.-P. Yuan, Tevatron Run-1 $Z$ boson data and Collins-Soper-Sterman resummation formalism, Phys. Rev. D67 (2003) 073016, arXiv:hep-ph/0212159
* [27] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
* [28] S. van der Meer, Calibration of the effective beam height in the ISR, ISR-PO/68-31, 1968
* [29] M. Ferro-Luzzi, Proposal for an absolute luminosity determination in colliding beam experiments using vertex detection of beam-gas interactions, Nucl. Instrum. Meth. A553 (2005) 388
* [30] LHCb collaboration, R. Aaij et al., Absolute luminosity measurements with the LHCb detector at the LHC, JINST 7 (2012) P01010, arXiv:1110.2866
* [31] P. Golonka and Z. Was, PHOTOS Monte Carlo: a precision tool for QED corrections in $Z$ and $W$ decays, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
* [32] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Parton distributions for the LHC, Eur. Phys. J. C63 (2009) 189, arXiv:0901.0002
* [33] S. Alekhin, J. Blumlein, S. Klein, and S. Moch, The 3, 4, and 5-flavor NNLO parton from deep-inelastic-scattering data and at hadron colliders, Phys. Rev. D81 (2010) 014032, arXiv:0908.2766
* [34] P. Jimenez-Delgado and E. Reya, Dynamical NNLO parton distributions, Phys. Rev. D79 (2009) 074023, arXiv:0810.4274
* [35] R. D. Ball et al., A first unbiased global NLO determination of parton distributions and their uncertainties, Nucl. Phys. B838 (2010) 136, arXiv:1002.4407
Appendix
## Appendix A Tables of results
Table 2: Correlation coefficients between $W^{+}$, $W^{-}$ and Z in the five bins considered. The luminosity uncertainty is not included. | $2<\eta^{\mu}\,(y^{Z})<2.5$ | $2.5<\eta^{\mu}\,(y^{Z})<3$ | $3<\eta^{\mu}\,(y^{Z})<3.5$ | $3.5<\eta^{\mu}\,(y^{Z})<4$ | $4<\eta^{\mu}\,(y^{Z})<4.5$ |
---|---|---|---|---|---|---
$W^{+}$ | 1 | | | | | | | | | | | | | | | $2<\eta^{\mu}\,(y^{Z})<2.5$
$W^{-}$ | 0.87 | 1 | | | | | | | | | | | | |
$Z$ | 0.36 | 0.34 | 1 | | | | | | | | | | | |
$W^{+}$ | 0.02 | 0.02 | 0.35 | 1 | | | | | | | | | | | | $2.5<\eta^{\mu}\,(y^{Z})<3$
$W^{-}$ | 0.02 | 0.02 | 0.35 | 0.90 | 1 | | | | | | | | | |
$Z$ | 0.47 | 0.44 | 0.45 | 0.45 | 0.45 | 1 | | | | | | | | |
$W^{+}$ | 0.02 | 0.03 | 0.24 | 0.02 | 0.02 | 0.31 | 1 | | | | | | | | | $3<\eta^{\mu}\,(y^{Z})<3.5$
$W^{-}$ | 0.02 | 0.02 | 0.29 | 0.02 | 0.02 | 0.37 | 0.89 | 1 | | | | | | |
$Z$ | 0.46 | 0.43 | 0.44 | 0.45 | 0.44 | 0.58 | 0.31 | 0.37 | 1 | | | | | |
$W^{+}$ | 0.04 | 0.05 | 0.35 | 0.04 | 0.04 | 0.45 | 0.05 | 0.04 | 0.44 | 1 | | | | | | $3.5<\eta^{\mu}\,(y^{Z})<4$
$W^{-}$ | 0.02 | 0.02 | 0.40 | 0.02 | 0.01 | 0.52 | 0.02 | 0.02 | 0.51 | 0.80 | 1 | | | |
$Z$ | 0.32 | 0.29 | 0.30 | 0.30 | 0.30 | 0.39 | 0.21 | 0.25 | 0.39 | 0.30 | 0.35 | 1 | | |
$W^{+}$ | 0.07 | 0.09 | 0.19 | 0.07 | 0.07 | 0.24 | 0.09 | 0.07 | 0.24 | 0.15 | 0.06 | 0.16 | 1 | | | $4<\eta^{\mu}\,(y^{Z})<4.5$
$W^{-}$ | 0.01 | 0.01 | 0.28 | 0.01 | 0.01 | 0.37 | 0.01 | 0.01 | 0.36 | 0.02 | 0.01 | 0.24 | 0.57 | 1 |
$Z$ | 0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.04 | 0.02 | 0.03 | 0.04 | 0.03 | 0.04 | 0.03 | 0.02 | 0.03 | 1
| $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ | $W^{+}$ | $W^{-}$ | $Z$ |
Table 3: Differential $Z\rightarrow\mu\mu$ cross-section,
$d\sigma_{Z\rightarrow\mu\mu}/dy^{Z}$, in bins of boson rapidity. The first
cross-section uncertainty is statistical, the second systematic, and the third
due to the uncertainty on the luminosity determination. The correction factor
$f^{Z}_{\mathrm{FSR}}$ which is used to correct for FSR is listed separately.
$y^{Z}$ | $d\sigma_{Z\rightarrow\mu\mu}/dy^{Z}$ [pb] | $f^{Z}_{\mathrm{FSR}}$
---|---|---
$2.0-2.5$ | $25.5$ | $\pm 1.4$ | $\pm 1.0$ | $\pm 0.9$ | $1.020\pm 0.001$
$2.5-3.0$ | $66.8$ | $\pm 2.3$ | $\pm 2.7$ | $\pm 2.3$ | $1.018\pm 0.001$
$3.0-3.5$ | $49.8$ | $\pm 2.0$ | $\pm 2.2$ | $\pm 1.7$ | $1.018\pm 0.001$
$3.5-4.0$ | $11.1$ | $\pm 0.9$ | $\pm 0.6$ | $\pm 0.4$ | $1.024\pm 0.001$
$4.0-4.5$ | $0.074$ | $\pm 0.074$ | $\pm 0.004$ | $\pm 0.002$ | $1.027\pm 0.027$
Table 4: Differential $W\rightarrow\mu\nu$ cross-section,
$d\sigma_{W\rightarrow\mu\nu}/\eta^{\mu}$, in bins of lepton pseudorapidity.
The first cross-section uncertainty is statistical, the second systematic, and
the third due to the uncertainty on the luminosity determination. The
correction factor $f^{W}_{\mathrm{FSR}}$ which is used to correct for FSR is
listed separately.
| $\eta^{\mu}$ | $d\sigma_{W\rightarrow\mu\nu}/\eta^{\mu}$ [pb] | $f^{W}_{\mathrm{FSR}}$
---|---|---|---
$W^{+}$ | $2.0-2.5$ | $691$ | $\pm 12$ | $\pm 37$ | $\pm 24$ | $1.0146\pm 0.0004$
| $2.5-3.0$ | $530$ | $\pm 9$ | $\pm 30$ | $\pm 19$ | $1.0086\pm 0.0002$
| $3.0-3.5$ | $296$ | $\pm 7$ | $\pm 23$ | $\pm 10$ | $1.0107\pm 0.0006$
| $3.5-4.0$ | $121$ | $\pm 5$ | $\pm 19$ | $\pm 4$ | $1.0097\pm 0.0005$
| $4.0-4.5$ | $23.1$ | $\pm 3.2$ | $\pm 4.9$ | $\pm 0.8$ | $1.0009\pm 0.0009$
$W^{-}$ | $2.0-2.5$ | $393$ | $\pm 9$ | $\pm 22$ | $\pm 13$ | $1.0147\pm 0.0008$
| $2.5-3.0$ | $370$ | $\pm 8$ | $\pm 20$ | $\pm 13$ | $1.0163\pm 0.0004$
| $3.0-3.5$ | $282$ | $\pm 7$ | $\pm 18$ | $\pm 10$ | $1.0147\pm 0.0004$
| $3.5-4.0$ | $200$ | $\pm 6$ | $\pm 14$ | $\pm 7$ | $1.0173\pm 0.0008$
| $4.0-4.5$ | $68$ | $\pm 5$ | $\pm 10$ | $\pm 2$ | $1.0194\pm 0.0009$
Table 5: Lepton charge asymmetry, $A_{\mu}$, in bins of muon pseudorapidity
for a $p^{\mu}_{\mathrm{T}}$ threshold at 20, 25 and 30 GeV/c. The first
uncertainty is statistical and the second systematic. The effect of FSR is at
the level of $10^{-4}$ and is not listed.
$\eta^{\mu}$ | $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$25${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | $A_{\mu}$ ($p_{\mathrm{T}}^{\mu}>$30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$)
---|---|---|---
$2.0-2.5$ | $0.275$ | $\pm 0.014$ | $\pm 0.003$ | $0.256$ | $\pm 0.015$ | $\pm 0.002$ | $0.238$ | $\pm 0.018$ | $\pm 0.002$
$2.5-3.0$ | $0.178$ | $\pm 0.013$ | $\pm 0.002$ | $0.195$ | $\pm 0.015$ | $\pm 0.001$ | $0.219$ | $\pm 0.017$ | $\pm 0.001$
$3.0-3.5$ | $0.024$ | $\pm 0.016$ | $\pm 0.009$ | $0.054$ | $\pm 0.018$ | $\pm 0.003$ | $0.112$ | $\pm 0.022$ | $\pm 0.002$
$3.5-4.0$ | $-0.247$ | $\pm 0.022$ | $\pm 0.011$ | $-0.203$ | $\pm 0.027$ | $\pm 0.005$ | $-0.124$ | $\pm 0.035$ | $\pm 0.003$
$4.0-4.5$ | $-0.493$ | $\pm 0.058$ | $\pm 0.051$ | $-0.413$ | $\pm 0.081$ | $\pm 0.016$ | $-0.353$ | $\pm 0.122$ | $\pm 0.008$
Table 6: $W$ cross-section ratio,
$R_{W}=\sigma_{W^{+}\rightarrow\mu^{+}\nu}/\sigma_{W^{-}\rightarrow\mu^{-}\bar{\nu}}$,
in bins of muon pseudorapidity. The first error is statistical and the second
systematic. The effect of FSR is at the level of $10^{-4}$ and is not listed.
$\eta^{\mu}$ | $R_{W}$
---|---
$2.0-2.5$ | $1.76$ | $\pm 0.05$ | $\pm 0.01$
$2.5-3.0$ | $1.43$ | $\pm 0.04$ | $\pm 0.01$
$3.0-3.5$ | $1.05$ | $\pm 0.03$ | $\pm 0.02$
$3.5-4.0$ | $0.60$ | $\pm 0.03$ | $\pm 0.01$
$4.0-4.5$ | $0.34$ | $\pm 0.05$ | $\pm 0.05$
|
arxiv-papers
| 2012-04-07T11:46:11 |
2024-09-04T02:49:29.452938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C.\n Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J.\n van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook,\n H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A.\n Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba,\n G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini,\n M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L. Clarke, M.\n Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras,\n P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G. Corti, B.\n Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N. Y. David,\n I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De Miranda, L. De\n Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono,\n C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P.\n Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S.\n Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C.\n Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson,\n V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es,\n G. Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, T. Hampson, S. Hansmann-Menzemer, N. Harnew, J. Harrison, P. F.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A.\n Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E.\n Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J.\n Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim,\n M. Knecht, I. Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N.\n La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner,\n C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N.\n Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I. V.\n Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G.\n Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E.\n Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M. Meissner, M.\n Merk, J. Merkel, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J. Mylroie-Smith, P.\n Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A. D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, B. K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch,\n A. Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S.\n Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S.\n Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues,\n F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M.\n Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C.\n Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M.\n Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A.\n Smith, E. Smith, M. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez\n Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, D.\n Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, R. Wallace, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M.\n Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, A. Zvyagin",
"submitter": "Katharina M\\\"uller",
"url": "https://arxiv.org/abs/1204.1620"
}
|
1204.1664
|
Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than $\mathcal{O}(1/N)$. Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.
The first 8 samples from sequential Bayesian quadrature, versus the first 20 samples from herding. Only 8 weighted samples are needed to give an estimator with the same maximum mean discrepancy as using 20 herding samples with uniform weights. Relative sizes of samples indicate their relative weights.
§ INTRODUCTION
The problem: Integrals A common problem in statistical machine learning is to compute expectations of functions over probability distributions of the form:
\begin{equation}
Z_{f,p} = \int f(x) p(x) dx \label{eqn:integral}
\end{equation}
Examples include computing marginal distributions, making predictions marginalizing over parameters, or computing the Bayes risk in a decision problem. In this paper we assume that the distribution $p(x)$ is known in analytic form, and $f(x)$ can be evaluated at arbitrary locations.
Monte Carlo methods produce random samples from the distribution $p$ and then approximate the integral by taking the empirical mean $\hat{Z} = \frac{1}{N}\sum_{n=1}^{N}f_{x_n}$ of the function evaluated at those points. This non-deterministic estimate converges at a rate $\mathcal{O}(\frac{1}{\sqrt{N}})$. When exact sampling from $p$ is impossible or impractical, Markov chain Monte Carlo (MCMC) methods are often used. MCMC methods can be applied to almost any problem but convergence of the estimate depends on several factors and is hard to estimate [Cowles & Carlin, 1996]. The focus of this paper is on quasi-Monte Carlo methods that – instead of sampling randomly – produce a set of pseudo-samples in a deterministic fashion. These methods operate by directly minimising some sort of discrepancy between the empirical distribution of pseudo-samples and the target distribution. Whenever these methods are applicable, they achieve convergence rates superior to the $\mathcal{O}(\frac{1}{\sqrt{N}})$ rate typical of random sampling.
In this paper we highlight and explore the connections between two deterministic sampling and integration methods: Bayesian quadrature () [O'Hagan, 1991, Rasmussen & Ghahramani, 2003] (also known as Bayesian Monte Carlo) and kernel herding [Chen et al., 2010]. Bayesian quadrature estimates integral (<ref>) by inferring a posterior distribution over $f$ conditioned on the observed evaluations $f_{x_n}$, and then computing the posterior expectation of $Z_{f,p}$. The points where the function should be evaluated can be found via Bayesian experimental design, providing a deterministic procedure for selecting sample locations.
Herding, proposed recently by [Chen et al., 2010], produces pseudosamples by minimising the discrepancy of moments between the sample set and the target distribution. Similarly to traditional Monte Carlo, an estimate is formed by taking the empirical mean over samples $\hat{Z} = \frac{1}{N}\sum_{n=1}^{N}f_{x_n}$. Under certain assumptions, herding has provably fast, $\mathcal{O}(\frac{1}{N})$ convergence rates in the parametric case, and has demonstrated strong empirical performance in a variety of tasks.
Summary of contributions In this paper, we make two main contributions. First, we show that the Maximum Mean Discrepancy (MMD) criterion used to choose samples in kernel herding is identical to the expected error in the estimate of the integral $Z_{f,p}$ under a Gaussian process prior for $f$. This expected error is the criterion being minimized when choosing samples for Bayesian quadrature. Because Bayesian quadrature assigns different weights to each of the observed function values $f(\vx)$, we can view Bayesian quadrature as a weighted version of kernel herding. We show that these weights are optimal in a minimax sense over all functions in the Hilbert space defined by our kernel. This implies that Bayesian quadrature dominates uniformly-weighted kernel herding and other non-optimally weighted herding in rate of convergence.
Second, we show that minimising the MMD, when using weights is closely related to the sparse dictionary selection problem studied in [Krause & Cevher, 2010], and therefore is approximately submodular with respect to the samples chosen. This allows us to reason about the performance of greedy forward selection algorithms for Bayesian Quadrature. We call this greedy method Sequential Bayesian Quadrature ().
We then demonstrate empirically the relative performance of herding, i.i.d random sampling, and , and demonstrate that attains a rate of convergence faster than $\mathcal{O}(1/N)$.
§ HERDING
Herding was introduced by [Welling, 2009] as a method for generating pseudo-samples from a distribution in such a way that certain nonlinear moments of the sample set closely match those of the target distribution. The empirical mean $\frac{1}{N}\sum_{n=1}^{N}f_{x_n}$ over these pseudosamples is then used to estimate integral (<ref>).
§.§ Maximum Mean Discrepancy
For selecting pseudosamples, herding relies on an objective based on the maximum mean discrepancy <cit.>. MMD measures the divergence between two distributions, $p$ and $q$ with respect to a class of integrand functions $\mathcal{F}$ as follows:
\begin{align}
\mmd_{\mathcal{F}}\left(p,q\right) = \sup_{f\in\mathcal{F}}\left\vert\int f_x p(x) dx - \int f_x q(x) dx \right\vert
\end{align}
Intuitively, if two distributions are close in the MMD sense, then no matter which function $f$ we choose from $\mathcal{F}$, the difference in its integral over $p$ or $q$ should be small. A particularly interesting case is when the function class $\mathcal{F}$ is functions of unit norm from a reproducing kernel Hilbert space (RKHS) $\He$. In this case, the MMD between two distributions can be conveniently expressed using expectations of the associated kernel $k(x, x')$ only [Sriperumbudur et al., 2010]:
\begin{align}
MMD^2_{\He}(p,q) =& \sup_{\substack{f\in\He\\\Hnorm{f}=1}}\left\vert\int f_x p(x) dx - \int f_x q(x) dx\right\vert^2\label{eqn:rkhs-mmd}\\
=& \Hnorm{\mu_{p} - \mu_{q}}^2\\
\nonumber =&\iint k(x,y) p(x) p(y) dx dy\\
\nonumber -2 &\iint k(x,y) p(x) q(y) dx dy\\
+ &\iint k(x,y) q(x) q(y) dx dy,
\end{align}
where in the above formula $\mu_{p}=\int \phi(\vx)p(\vx)d\vx\in\He$ denotes the mean element associated with the distribution $p$. For characteristic kernels, such as the Gaussian kernel, the mapping between a distribution and its mean element is bijective. As a consequence $\mmd_{\He}(p,q)=0$ if and only if $p=q$, making it a powerful measure of divergence.
Herding uses maximum mean discrepancy to evaluate of how well the sample set $\{\vx_1,\ldots,\vx_{N}\}$ represents the target distribution $p$:
\begin{align}
\epsilon_{herding}&\left(\{\vx_1,\ldots,\vx_{N}\}\right) = \mmd_{\He}\left(p,\frac{1}{N}\sum_{n=1}^{N}\delta_{x_n}\right)\\
\nonumber =&\iint k(x,y) p(x) p(y) dx dy\\
-2 &\frac{1}{N}\sum_{n=1}^{N}\int k(x,x_n) p(x) dx
+ \frac{1}{N^2}\sum_{n,m=1}^{N} k(x_n,x_m)
\label{eq:mmd_assumption}
\end{align}
The herding procedure greedily minimizes its objective $\epsilon_{herding}\left(\{\vx_1,\ldots,\vx_{N}\}\right)$ , adding pseudosamples $\vx_n$ one at a time. When selecting the $n+1$-st pseudosample:
\begin{align}
\vx_{n+1} &\leftarrow \argmin_{\vx \in \mathcal{X}} \label{eqn:herding_criterion} \epsilon_{herding}\left(\{\vx_1,\ldots,\vx_{n},\vx\}\right)\\
&= \argmax_{\vx \in \mathcal{X}} 2 \expectargs{\vx' \sim p}{k(\vx, \vx')} - \frac{1}{n+1}\sum_{m=1}^{n} k(\vx,\vx_m)\mbox{,}\notag
\end{align}
assuming $k(\vx,\vx) = \mbox{const}$.
The formula (<ref>) admits an intuitive interpretation: the first term encourages sampling in areas with high mass under the target distribution $p(\vx)$. The second term discourages sampling at points close to existing samples.
Evaluating (<ref>) requires us to compute $\expectargs{\vx' \sim p}{k(\vx, \vx')} $, that is to integrate the kernel against the target distribution. Throughout the paper we will assume that these integrals can be computed in closed form. Whilst the integration can indeed be carried out analytically in several cases [Song et al., 2008, Chen et al., 2010], this requirement is the most pertinent limitation on applications of kernel herding, Bayesian quadrature and related algorithms.
§.§ Complexity and Convergence Rates
Criterion (<ref>) can be evaluated in only $\mathcal{O}(n)$ time. Adding these up for all subsequent samples, and assuming that optimisation in each step has $\mathcal{O}(1)$ complexity, producing $N$ pseudosamples via kernel herding costs $\mathcal{O}(N^2)$ operations in total.
In finite dimensional Hilbert spaces, the herding algorithm has been shown to reduce $\mmd$ at a rate $\mathcal{O}(\frac{1}{N})$, which compares favourably with the $\mathcal{O}(\frac{1}{\sqrt{N}})$ rate obtained by non-deterministic Monte Carlo samplers. However, as pointed out by [Bach et al., 2012], this fast convergence is not guaranteed in infinite dimensional Hilbert spaces, such as the RKHS corresponding to the Gaussian kernel.
§ BAYESIAN QUADRATURE
An illustration of Bayesian Quadrature. The function $f(x)$ is sampled at a set of input locations. This induces a Gaussian process posterior distribution on $f$, which is integrated in closed form against the target density, $p(\vx)$. Since the amount of volume under $f$ is uncertain, this gives rise to a (Gaussian) posterior distribution over $Z_{f,p}$.
So far, we have only considered integration methods in which the integral (<ref>) is approximated by the empirical mean of the function evaluated at some set of samples, or pseudo-samples. Equivalently, we can say that Monte Carlo and herding both assign an equal $\frac{1}{N}$ weight to each of the samples.
In [Rasmussen & Ghahramani, 2003], an alternate method is proposed: Bayesian Monte Carlo, or Bayesian quadrature (). puts a prior distribution on $\vf$, then estimates integral (<ref>) by inferring a posterior distribution over the function $\vf$, conditioned on the observations $\vf(\vx_n)$ at some query points $\vx_n$. The posterior distribution over $f$ then implies a distribution over $Z_{f,p}$. This method allows us to choose sample locations $\vx_n$ in any desired manner. See Figure <ref> for an illustration of Bayesian Quadrature.
§.§ BQ Estimator
Here we derive the estimate of (<ref>), after conditioning on function evaluations $\vf(\vx_1) \dots \vf(\vx_N)$, denoted as $f(\vX)$. The Bayesian solution implies a distribution over $Z_{f,p}$. The mean of this distribution, $\expectargs{}{Z}$ is the optimal Bayesian estimator for a squared loss.
For simplicity, $\vf$ is assigned a Gaussian process prior with kernel function $k$ and mean $0$. This assumption is very similar to the one made by kernel herding in Eqn. (<ref>).
After conditioning on $\vf_{\vx}$, we obtain a closed-form posterior over $\vf$:
\begin{align}
p(\vf(\vx\st)|\vf(\vX)) = \N{\vf_{\vx\st}}{\mf(\vx\st)}{\cov(\vx\st,\vx\st')}
\end{align}
\begin{align}
\mf(\vx\st) = & k(\vx\st, \vX) K^{-1} \vf(\vX) \\
\cov(\vx\st, \vx\st') = & k(\vx\st,\vx\st) - k(\vx\st, \vX) K^{-1} k(\vX, \vx\st)
\end{align}
and $K = k(\vX, \vX)$.
Conveniently, the posterior allows us to compute the expectation of (<ref>) in closed form:
\begin{align}
\expectargs{\gp}{Z} & = \expectargs{\gp}{\int f(\vx)p(\vx)d\vx}\\
& = \int\!\!\! \int\!\! f(\vx) p(f(\vx)|\vf(\vX)) p(\vx) d\vx df\\
& = \int\!\!\! \mf(\vx) p(\vx) d\vx \\
& = \left[ \int\!\! k(\vx, \vX) p(\vx) d\vx \right] K^{-1} \vf(\vX) \\
& = \vz^T K^{-1} \vf(\vX)
\label{eq:marg_mean_symbolic}
\end{align}
\begin{align}
z_n & = \int\!\! k(\vx, \vx_n) p(\vx) d\vx = \expectargs{\vx' \sim p}{k(\vx_n, \vx')}.
\end{align}
Conveniently, as in kernel herding, the desired expectation of $Z_{f,p}$ is simply a linear combination of observed function values $\vf(\vx)$:
\begin{align}
\expectargs{\gp}{Z} & = \vz^T K^{-1} \vf(\vX) \\
& = \sum_n w_{\bq}^{(n)} \vf_{\vx_n})
\end{align}
\begin{align}
w_{\bq}^{(n)} & = \sum_m \vz_j^T K^{-1}_{nm}
\label{eq:bq_weights}
\end{align}
Thus, we can view the BQ estimate as a weighted version of the herding estimate. Interestingly, the weights $\vw_{\bq}$ do not need to sum to 1, and are not even necessarily positive.
§.§.§ Non-normalized and Negative Weights
A set of optimal weights given by , after 100 samples were selected on the distribution shown in Figure <ref>. Note that the optimal weights are spread away from the uniform weight ($\frac{1}{N}$), and that some weights are even negative. The sum of these weights is 0.93.
When weighting samples, it is often assumed, or enforced <cit.>, that the weights $\vw$ form a probability distribution. However, there is no technical reason for this requirement, and in fact, the optimal weights do not have this property. Figure <ref> shows a representative set of 100 weights chosen on samples representing the distribution in figure <ref>. There are several negative weights, and the sum of all weights is 0.93.
Figure <ref> demonstrates that, in general, the sum of the Bayesian weights exhibits shrinkage when the number of samples is small.
An example of Bayesian shrinkage in the sample weights. In this example, the kernel width is approximately $\nicefrac{1}{20}$ the width of the distribution being considered. Because the prior over functions is zero mean, in the small sample case the weights are shrunk towards zero. The weights given by simple Monte Carlo and herding do not exhibit shrinkage.
§.§ Optimal sampling for BQ
Bayesian quadrature provides not only a mean estimate of $Z_{f,p}$, but a full Gaussian posterior distribution. The variance of this distribution $\varianceargs{}{Z_{f,p}|f_{x_1}, \dots, f_{x_N}}$ quantifies our uncertainty in the estimate. When selecting locations to evaluate the function $f$, minimising the posterior variance is a sensible strategy. Below, we give a closed form formula for the posterior variance of $Z_{f,p}$, conditioned on the observations $f_{x_1} \dots f_{x_N}$, which we will denote by $\epsilon^2_{\bq{}}$. For a longer derivation, see [Rasmussen & Ghahramani, 2003].
\begin{align}
\epsilon^{2}_{\bq{}}(\vx_1,\ldots,\vx_N) & =
\varianceargs{}{Z_{f,p}|f_{x_1}, \dots, f_{x_N}} \\
% \nonumber & = \expectargs{f \sim \gp, p\sim p(x)}{ \left( f(\vx) - \mf(\vx) \right)\left( f(\vx') - \mf(\vx') \right)} \\
%\nonumber & = \int \Bigg( \!\! \left( \int f(\vx) p(\vx) d\vx - \int \mf(\vx') p(\vx') d\vx' \right) \\
%\nonumber & \quad \times \left( \int f(\vx) p(\vx) d\vx - \int \mf(\vx') p(\vx') d\vx' \right) \!\! \Bigg) p(f) df \\
%\nonumber & = \int\!\!\! \int\!\! \int\!\! \left[ f(\vx) - \mf(\vx) \right] \left[ f((\vx') - \mf(\vx') \right] p(f) df \\
%\nonumber & \qquad \times p(\vx) p(\vx') d\vx d\vx' \\
%\nonumber & = \int\!\! \!\int\!\! \Cov \left[ f((\vx), f((\vx') \right] p(\vx) p(\vx') d\vx d\vx' \\
%\nonumber & = \int\!\!\! \int\!\! \left[ k(\vx, \vx') - k(\vx, \vX) K^{-1} k(\vX, \vx') \right] \\
%\nonumber & \qquad \times p(\vx) p(\vx') d\vx d\vx' \\
%\nonumber & = \int\!\!\! \int\!\! k(\vx, \vx') p(\vx) p(\vx') d\vx d\vx' \\
%\nonumber & \quad - \left[ \int\!\! k(\vx, \vX) p(\vx) d\vx \right] K^{-1} \left[ \int\!\! k(\vX, \vx') p(\vx') d\vx' \right] \\
& = \expectargs{\vx, \vx' \sim p}{k(\vx, \vx')} - \vz^T K^{-1} \vz\mbox{,}
\label{eq:marg_var_symbolic}
\end{align}
where $\vz_n = \expectargs{\vx' \sim p}{k(\vx_n, \vx')}$ as before. Perhaps surprisingly, the posterior variance of $Z_{f,p}$ does not depend on the observed function values, only on the location $x_n$ of samples. A similar independence is observed in other optimal experimental design problems involving Gaussian processes [Krause et al., 2006]. This allows the optimal samples to be computed ahead of time, before observing any values of $f$ at all [Minka, 2000].
We can contrast the objective $\epsilon^{2}_{\bq{}}$ in (<ref>) to the objective being minimized in herding, $\epsilon^{2}_{herding}$ of equation (<ref>). Just like $\epsilon^{2}_{herding}$, $\epsilon^{2}_{\bq{}}$ expresses a trade-off between accuracy and diversity of samples. On the one hand, as samples get close to high density regions under $p$, the values in $\vz$ increase, which results in decreasing variance. On the other hand, as samples get closer to each other, eigenvalues of $K$ increase, resulting in an increase in variance.
In a similar fashion to herding, we may use a greedy method to minimise $\epsilon^{2}_{\bq{}}$, adding one sample at a time. We will call this algorithm Sequential Bayesian Quadrature ():
\begin{align}
\vx_{n+1} &\leftarrow \argmin_{\vx \in \mathcal{X}} \epsilon_{\bq{}}\left(\{\vx_1,\ldots,\vx_{n},\vx\}\right)
\end{align}
Using incremental updates to the Cholesky factor, the criterion can be evaluated in $\mathcal{O}(n^2)$ time. Iteratively selecting $N$ samples thus takes $\mathcal{O}(N^3)$ time, assuming optimisation can be done on $\mathcal{O}(1)$ time.
§ RELATING $\VARIANCEARGS{}{Z_{F,P}}$ TO $\MMD$
The similarity in the behaviour of $\epsilon^{2}_{herding}$ and $\epsilon^{2}_{\bq{}}$ is not a coincidence, the two quantities are closely related to each other, and to .
The expected variance in the Bayesian quadrature $\epsilon^{2}_{\bq{}}$ is the maximum mean discrepancy between the target distribution $p$ and $q_{\bq{}}(x) = \sum_{n=1}^{N}w^{(n)}_{\bq{}}\delta_{x_n}(x)$
The proof involves invoking the representer theorem, using bilinearity of scalar products and the fact that if $f$ is a standard Gaussian process then $\forall g\in\He: \left\langle f,g\right\rangle \sim \mathcal{N}(0,\Hnorm{g})$:
\begin{align}
&\varianceargs{}{Z_{f,p}\vert f_{x_1}, \dots, f_{x_N}}=\\
&= \mathbb{E}_{f\sim GP} \left( \int f(x) p(x) dx - \sum_{n=1}^{N}w^{(n)}_{\bq{}} f(x_n)\right)^2\\
&= \mathbb{E}_{f\sim GP} \left( \int \left\langle f, \phi (x)\right\rangle p(x) dx - \sum_{n=1}^{N}w^{(n)}_{\bq{}} \left\langle f, \phi (x_n)\right\rangle\right)^2\\
&= \mathbb{E}_{f\sim GP} \left\langle f , \int\phi(x) p(x) dx - \sum_{n=1}^{N}w^{(n)}_{\bq{}}\phi(x_n)\right\rangle^2\\
&= \Hnorm{\mu_p - \mu_{q_{\bq{}}}}^2\\
&= \mmd^2(p,q_{\bq{}})
\end{align}
We know that the the posterior mean $\expectargs{\gp}{Z_{f,p}\vert f_1,\ldots,f_N}$ is a Bayes estimator and has therefore the minimal expected squared error amongst all estimators. This allows us to further rewrite $\epsilon^{2}_{\bq{}}$ into the following minimax forms:
\begin{align}
\epsilon^{2}_{\bq{}} &= \sup_{\substack{f\in\He\\\Hnorm{f}{\He}=1}} \left| \int f_x p(x) dx - \sum_{n=1}^{N}w^{(n)}_{\bq{}} f_{x_n}\right|^2\\
&= \inf_{\hat{Z}:\mathcal{X}^N\mapsto\mathbb{R}} \sup_{\substack{f\in\He\\\Hnorm{f}{\He}=1}} \left| Z - \hat{Z}\left(f_{x_1},\ldots,f_{x_N}\right)\right|^2\\
&= \inf_{\bm{w}\in\mathbb{R}^N} \sup_{\substack{f\in\He\\\Hnorm{f}{\He}=1}} \left| \int f_x p(x) dx - \sum_{n=1}^{N}w_n f_{x_n}\right|^2
\end{align}
Looking at $\epsilon^{2}_{\bq{}}$ this way, we may discover the deep similarity to the criterion $\epsilon^2_{herding}$ that kernel herding minimises. Optimal sampling for Bayesian quadrature minimises the same objective as kernel herding, but with the uniform $\frac{1}{N}$ weights replaced by the optimal weights. As a corollary
\begin{align}
\epsilon^{2}_{\bq{}}(x_1,\ldots,x_N) \leq \epsilon^{2}_{KH} (x_1,\ldots,x_N)
\end{align}
It is interesting that $\epsilon^{2}_{\bq{}}$ has both a Bayesian interpretation as posterior variance under a Gaussian process prior, and a frequentist interpretation as a minimax bound on estimation error with respect to an RKHS.
§ SUBMODULARITY
In this section, we use the concept of approximate submodularity [Krause & Cevher, 2010], in order to study convergence properties of .
A set function $s:2^\mathcal{X} \mapsto \mathbb{R}$ is submodular if, for all $A\subseteq B\subseteq \mathcal{X}$ and $\forall x \in \mathcal{X}$
\begin{align}
s(A\cup\{x\})-s(A)\geq s(B\cup\{x\})-s(B)
\end{align}
Intuitively, submodularity is a diminishing returns property: adding an element to a smaller set has larger relative effect than adding it to a larger set. A key result <cit.> is that greedily maximising a submodular function is guaranteed not to differ from the optimal strategy by more than a constant factor of $(1-\frac{1}{e})$.
Herding and are examples of greedy algorithms optimising set functions: they add each pseudosamples in such a way as to minimize the instantaneous reduction in $\mmd$. So it is intuitive to check whether the objective functions these methods minimise are submodular. Unfortunately, neither $\epsilon_{herding}$, nor $\epsilon_{\bq{}}$ satisfies all conditions necessary for submodularity. However, noting that is identical to the sparse dictionary selection problem studied in detail by Krause & Cevher, 2010, we can conclude that satisfies a weaker condition called approximate submodularity.
A set function $s:2^\mathcal{X} \mapsto \mathbb{R}$ is approximately submodular with constant $\epsilon>0$, if for all $A\subseteq B\subseteq \mathcal{X}$ and $\forall x \in \mathcal{X}$
\begin{align}
s(A\cup\{x\})-s(A)\geq s(B\cup\{x\})-s(B) - \epsilon
\end{align}
$\epsilon^{2}_{\bq{}}(\emptyset)-\epsilon^{2}_{\bq{}}(\cdot)$ is weakly a weakly submodular set function with constant $\epsilon<4r$, where $r$ is the incoherency
\begin{equation}
r = \max_{x,x'\in\mathcal{P}\subseteq\mathcal{X}} \frac{k(x,x')}{\sqrt{k(x,x)k(x',x')}}
\end{equation}
By the definition of $\mmd$ we can see that
$-\epsilon^{2}_{\bq{}} = \inf_{w\in\mathbb{R}^N}\Hnorm{\mu_p - \sum_{n=1}^N w^{(n)}_{\bq{}}k(\cdot,\vx_n)}^2$ is the negative squared distance between the mean element $\mu_p$ and its projection onto the subspace spanned by the elements $k(\cdot,\vx_n)$. Substituting $k=1$ into Theorem 1 of Krause & Cevher, 2010 concludes the proof.
Unfortunately, weak submodularity does not provide the strong near-optimality guarantees as submodularity does . If $s:2^\mathcal{X} \mapsto \mathbb{R}$ is a weakly submodular function with constant $\epsilon$, and $\vert\mathcal{A}_n\vert=n$ is the result of greedy optimisation of $s$, then
\begin{equation}
s(\mathcal{A}_n) \geq \left(1-\frac{1}{e}\right)\max_{\vert\mathcal{A}\vert\leq n}s(\mathcal{A}) - n\epsilon
\end{equation}
As pointed out by Krause & Cevher, 2010, this guarantee is very weak, as in our case the objective function $\epsilon^{2}_{\bq{}}(\emptyset)-\epsilon^{2}_{\bq{}}(\cdot)$ is upper bounded by a constant. However, establishing a connection between and sparse dictionary selection problem opens up interesting directions for future research, and it may be possible to apply algorithms and theory developed for sparse dictionary selection to kernel-based quasi-Monte Carlo methods.
§ EXPERIMENTS
In this section, we examine empirically the rates of convergence of sequential Bayesian quadrature and herding. We examine both the expected error rates, and the empirical error rates.
In all experiments, the target distribution $p$ is chosen a 2D mixture of 20 Gaussians, whose equiprobability contours are shown in Figure <ref>. To ensure a comparison fair to herding, the target distribution, and the kernel used by both methods, correspond exactly to the one used in <cit.>.
For experimental simplicity, each of the sequential sampling algorithms minimizes the next sample location from a pool of 10000 locations randomly drawn from the base distribution. In practice, one would run a local optimizer from each of these candidate locations, however in our experiments we found that this did not make a significant difference in the sample locations chosen.
§.§ Matching a distribution
We first extend an experiment from [Chen et al., 2010] designed to illustrate the mode-seeking behavior of herding in comparison to random samples. In that experiment,
it is shown that a small number of i. i. d. samples drawn from a multimodal distribution will tend to, by chance, assign too many samples to some modes, and too few to some other modes. In contrast, herding places `super-samples' in such a way as to avoid regions already well-represented, and seeks modes that are under-represented.
We demonstrate that although herding improves upon i. i. d. sampling, the uniform weighting of super-samples leads to sub-optimal performance. Figure <ref> shows the first 20 samples chosen by kernel herding, in comparison with the first 8 samples chosen by . By weighting the 8 samples by the quadrature weights in (<ref>), we can obtain the same expected loss as by using the 20 uniformly-weighted herding samples.
The maximum mean discrepancy, or expected error of several different quadrature methods. Herding appears to approach a rate close to $\mathcal{O}(1/N)$. appears to attain a faster, but unknown rate.
Figure <ref> shows MMD versus the number of samples added, on the distribution shown in Figure <ref>. We can see that in all cases, dominates herding. It appears that converges at a faster rate than $\mathcal{O}(1/N)$, although the form of this rate is unknown.
There are two differences between herding and : chooses samples according to a different criterion, and also weights those samples differently. We may ask whether the sample locations or the weights are contributing more to the faster convergence of . Indeed, in Figure <ref> we observe that the samples selected by are quite similar to the samples selected by kernel herding. To answer this question, we also plot in Figure <ref> the performance of a fourth method, which selects samples using herding, but later re-weights the herding samples with weights. Initially, this method attains similar performance to , but as the number of samples increases, attains a better rate of convergence. This result indicates that the different sample locations chosen by , and not only the optimal weights, are responsible for the increased convergence rate of .
§.§ Estimating Integrals
We then examined the empirical performance of the different estimators at estimating integrals of real functions. To begin with, we looked at performance on 100 randomly drawn functions, of the form:
\begin{align}
f(\vx) & = \sum_{i=1}^{10} \alpha_i k(\vx, \vc_i)
\end{align}
\begin{align}
\Hnorm{f}^2 = \sum_{i=1}^{10} \sum_{j=1}^{10} \alpha_i \alpha_j k(\vc_i, \vc_j) = 1
\end{align}
That is, these functions belonged exactly to the unit ball of the RKHS defined by the kernel $k(\vx, \vx')$ used to model them.
Within-model error: The empirical error rate in estimating $Z_{f,p}$, for several different sampling methods, averaged over 250 functions randomly drawn from the RKHS corresponding to the kernel used.
Figure <ref> shows the empirical error versus the number of samples, on the distribution shown in Figure <ref>. The empirical rates attained by the method appear to be similar to the MMD rates in Figure <ref>.
By definition, MMD provides a upper bound on the estimation error in the integral of any function in the unit ball of the RKHS (Eqn. (<ref>)), including the Bayesian estimator, . Figure <ref> demonstrates this quickly decreasing bound on the empirical error.
The empirical error rate in estimating $Z_{f,p}$, for the estimator, on 10 random functions drawn from the RKHS corresponding to the kernel used. Also shown is the upper bound on the error rate implied by the $\mmd$.
§.§ Out-of-model performance
A central assumption underlying is that the integrand function belongs to the RKHS specified by the kernel. To see how performance is effected if this assumption is violated, we performed empirical tests with functions chosen from outside the RKHS. We drew 100 functions of the form:
\begin{align}
f(\vx) & = \sum_{i=1}^{10} \alpha_i \exp(-\frac{1}{2} (\vx -\vc_i)^T \Sigma_i^{-1} (\vx -\vc_i)
\end{align}
where each $\alpha_i$ $\vc_i$ $\Sigma_i$ were drawn from broad distributions. This ensured that the drawn functions had features such as narrow bumps and ridges which would not be well modelled by functions belonging to the isotropic kernel defined by $k$.
Out-of-model error: The empirical error rates in estimating $Z_{f,p}$, for several different sampling methods, averaged over 250 functions drawn from outside the RKHS corresponding to the kernels used.
Figure <ref> shows that, on functions drawn from outside the assumed RKHS, relative performance of all methods remains similar.
Code to reproduce all results is available at
§ DISCUSSION
§.§ Choice of Kernel
Using herding techniques, we are able to achieve fast convergence on a Hilbert space of well-behaved functions, but this fast convergence is at the expense of the estimate not necessarily converging for functions outside this space.
If we use a characteristic kernel [Sriperumbudur et al., 2010], such as the exponentiated-quadratic or Laplacian kernels, then convergence in MMD implies weak convergence of $q_N$ to the target distribution.
This means that the estimate converges for any bounded measurable function $f$. The speed of convergence, however, may not be as fast.
Therefore it is crucial that the kernel we choose is representative of the function or functions $f$ we will integrate. For example, in our experiments, the convergence of herding was sensitive to the width of the Gaussian kernel. One of the major weaknesses of kernel methods in general is the difficulty of setting kernel parameters. A key benefit of the Bayesian interpretation of herding and MMD presented in this paper is that it provides a recipe for adapting the Hilbert space to the observations $f(x_n)$. To be precise, we can fit the kernel parameters by maximizing the marginal likelihood of Gaussian process conditioned on the observations. Details can be found in [Rasmussen & Williams, 2006].
§.§ Computational Complexity
While we have shown that Bayesian Quadrature provides the optimal re-weighting of samples, computing the optimal weights comes at an increased computational cost relative to herding.
The computational complexity of computing Bayesian quadrature weights for $N$ samples is $\mathcal{O}(N^3)$, due to the necessity of inverting the Gram matrix $K(\vx, \vx)$. Using the Woodbury identity, the cost of adding a new sample to an existing set is $\mathcal{O}(N^2)$. For herding, the computational complexity of evaluating a new sample is only $\mathcal{O}(N)$, making the cost of choosing $N$ herding samples $\mathcal{O}(N^2)$. For Monte Carlo sampling, the cost of adding an i.i.d. sample from the target distribution is only $\mathcal{O}(1)$.
method complexity rate guarantee
MCMC $\mathcal{O}(N)$ variable ergodic theorem
i.i.d. MC $\mathcal{O}(N)$ $\frac{1}{\sqrt{N}}$ law of large numbers
herding $\mathcal{O}(N^2)$ $\frac{1}{\sqrt{N}} \geq \cdot \geq \frac{1}{N}$ [Chen et al., 2010, Bach et al., 2012]
SBQ $\mathcal{O}(N^3)$ unknown approximate submodularity
A comparison of the rates of convergence and computational complexity of several integration methods.
The relative computational cost of computing samples and weights using , herding, and sampling must be weighed against the cost of evaluating $f$ at the sample locations. Depending on this trade-off, the three sampling methods form a Pareto frontier over computational speed and estimator accuracy. When computing $f$ is cheap, we may wish to use Monte Carlo methods. In cases where $f$ is computationally costly, we would expect to choose the method. When $f$ is relatively expensive, but a very large number of samples are required, we may choose to use kernel herding instead. However, because the rate of convergence of is faster, there may be situations in which the $\mathcal{O}(N^3)$ cost is relatively inexpensive, due to the smaller $N$ required by to achieve the same accuracy as compared to using other methods.
There also exists the possibility to switch to a less costly sampling algorithm as the number of samples increases.
Table <ref> summarizes the rates of convergence of all the methods considered here.
§ CONCLUSIONS
In this paper, we have shown three main results: First, we proved that the loss minimized by kernel herding is closely related to the loss minimized by Bayesian quadrature, when selecting sample locations. This implies that sequential Bayesian quadrature can viewed as an optimally-weighted version of kernel herding.
Second, we showed that the loss minimized by the Bayesian method is approximately submodular with respect to the samples chosen, and established connections to the submodular dictionary selection problem studied in [Krause & Cevher, 2010].
Finally, we empirically demonstrated a superior rate of convergence of over herding, and demonstrated a bound on the empirical error of the Bayesian quadrature estimate.
§.§ Future Work
In section <ref>, we showed that is approximately submodular, which provides only weak sub-optimality guarantees of its performance. It would be of interest to further explore the connection between Bayesian Quadrature and the dictionary selection problem to see if algorithms developed for dictionary selection can provide further practical or theoretical developments. The results in section <ref>, specifically Figure <ref>, suggest that the convergence rate of is faster than $\mathcal{O}(1/N)$. However, we are not aware of any work showing what the theoretically optimal rate is. It would be of great interest to determine this optimal rate of convergence for particular classes of kernels.
§.§ Acknowledgements
The authors would like to thank Carl Rasmussen and Francis Bach for helpful discussions, and Yutian Chen for his help in reproducing experiments.
[Bach et al., 2012]
Bach, F., Lacoste-Julien, S., and Obozinski, G.
On the equivalence between herding and conditional gradient
Arxiv preprint arXiv:1203.4523, 2012.
[Chen et al., 2010]
Chen, Y., Welling, M., and Smola, A.
Super-samples from kernel herding.
UAI, 2010.
[Cowles & Carlin, 1996]
Cowles, Mary K. and Carlin, Bradley P.
Markov chain monte carlo convergence diagnostics: A comparative
Journal of the American Statistical Association, 910
(434):0 883–904, 1996.
ISSN 01621459.
URL <http://dx.doi.org/10.2307/2291683>.
[Krause et al., 2006]
Krause, A., Guestrin, C., Gupta, A., and Kleinberg, J.
Near-optimal sensor placements: Maximizing information while
minimizing communication cost.
In Proceedings of the Fifth International Conference on
Information Processing in Sensor Networks (IPSN '06), pp. 2–10,
Nashville, Tennessee, USA, 2006.
[Krause & Cevher, 2010]
Krause, Andreas and Cevher, Volkan.
Submodular dictionary selection for sparse representation.
In Fürnkranz, Johannes and Joachims, Thorsten (eds.),
ICML, pp. 567–574. Omnipress, 2010.
ISBN 978-1-60558-907-7.
[Minka, 2000]
Minka, T. P.
Deriving quadrature rules from Gaussian processes.
Technical report, Statistics Department, Carnegie Mellon University,
[O'Hagan, 1991]
O'Hagan, A.
Bayes-Hermite quadrature.
Journal of Statistical Planning and Inference, 29:0
245–260, 1991.
[Rasmussen & Ghahramani, 2003]
Rasmussen, C. E. and Ghahramani, Z.
Bayesian monte carlo.
In Becker, S. and Obermayer, K. (eds.), Advances in Neural
Information Processing Systems, volume 15. MIT Press, Cambridge, MA, 2003.
[Rasmussen & Williams, 2006]
Rasmussen, C.E. and Williams, CKI.
Gaussian Processes for Machine Learning.
The MIT Press, Cambridge, MA, USA, 2006.
[Song et al., 2008]
Song, Le, Zhang, Xinhua, Smola, Alex, Gretton, Arthur, and Schölkopf,
Tailoring density estimation via reproducing kernel moment matching.
In Proceedings of the 25th international conference on Machine
learning, ICML '08, pp. 992–999, New York, NY, USA, 2008. ACM.
ISBN 978-1-60558-205-4.
URL <http://doi.acm.org/10.1145/1390156.1390281>.
[Sriperumbudur et al., 2010]
Sriperumbudur, Bharath K., Gretton, Arthur, Fukumizu, Kenji, Schölkopf,
Bernhard, and Lanckriet, Gert R.G.
Hilbert space embeddings and metrics on probability measures.
J. Mach. Learn. Res., 99:0 1517–1561, August 2010.
ISSN 1532-4435.
URL <http://dl.acm.org/citation.cfm?id=1859890.1859901>.
[Welling, 2009]
Welling, M.
Herding dynamical weights to learn.
In Proceedings of the 26th Annual International Conference on
Machine Learning, pp. 1121–1128. ACM, 2009.
|
arxiv-papers
| 2012-04-07T17:54:07 |
2024-09-04T02:49:29.464633
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ferenc Husz\\'ar and David Duvenaud",
"submitter": "David Duvenaud",
"url": "https://arxiv.org/abs/1204.1664"
}
|
1204.1665
|
# Size-dependent Transport Study of In0.53Ga0.47As Gate-all-around Nanowire
MOSFETs: Impact of Quantum Confinement and Volume Inversion
Jiangjiang J. Gu, Heng Wu, Yiqun Liu, Adam T. Neal, Roy G. Gordon, and Peide
D. Ye This work was supported in part by National Science Foundation and in
part by Semiconductor Research Corporation (SRC) Focus Center Research Program
(FCRP) Materials, Structures, and Devices (MSD) Focus Center. The authors
would like to thank J. Shao, D. A. Antoniadis and M. S. Lundstrom for the
technical assistance and valuable discussions.J. J. Gu, H. Wu, A. T. Neal, and
P. D. Ye are with the Department of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN, 47907 USA e-mail: (yep@purdue.edu).Y.
Q. Liu is with the GlobalFoundries, Albany, NY, 12207 USA.R. G. Gordon is with
the Department of Chemistry and Chemical Biology, Harvard University,
Cambridge, MA, 02138 USA.
###### Abstract
InGaAs gate-all-around nanowire MOSFETs with channel length down to 50nm have
been experimentally demonstrated by top-down approach. The nanowire size-
dependent transport properties have been systematically investigated. It is
found that reducing nanowire dimension leads to higher on-current,
transconductance and effective mobility due to stronger quantum confinement
and the volume inversion effect. TCAD quantum mechanical simulation has been
carried out to study the inversion charge distribution inside the nanowires.
Volume inversion effect appears at a larger dimension for InGaAs nanowire
MOSFET than its Si counterpart.
###### Index Terms:
InGaAs, gate-all-around, nanowire.
## I Introduction
InGaAs MOSFETs have recently been considered as one of the promising
candidates for beyond 14nm logic applications [1]. To meet the stringent
demands of electrostatic control, non-planar 3D structures have been
introduced to the fabrication of InGaAs MOSFETs, including InGaAs FinFETs [2],
multi-gate InGaAs quantum-well FETs [3] and most recently, InGaAs gate-all-
around (GAA) nanowire MOSFETs [4].
In particular, the InGaAs GAA nanowire MOSFETs have been shown to offer good
scalability down to channel length ($L_{ch}$) of at least 50nm, thanks to the
best electrostatic control of the GAA structure. High drive current ($I_{on}$)
of 1.17mA/$\mu$m and peak transconductance ($g_{m}$) of 0.7mS/$\mu$m have also
been achieved [4] despite the non-optimized source/drain resistance ($R_{SD}$)
and large equivalent oxide thickness (_EOT_), showing great promise of the
InGaAs GAA technology. Moreover, a detailed scaling metrics study has also
revealed that reducing the nanowire size leads to improvements in subthreshold
swing (_SS_), drain induced barrier lowering (_DIBL_), and threshold voltage
($V_{T}$) roll off, due to the tighter gate control [4]. However, the impact
of nanowire size on the transport properties of InGaAs GAA nanowire MOSFETs
has not been studied and could lead to better understanding and design
guidelines for next-generation InGaAs nanowire devices.
In this letter, we systematically investigate the impact of nanowire size on
the on-state performance of InGaAs GAA nanowire MOSFETs. To our surprise,
higher $I_{on}$ and intrinsic $g_{m}$ has been obtained on devices with
smaller nanowire size. The low field mobility ($\mu_{0}$) is extracted using
the Y-function method to further elucidate the transport performance of the
nanowire devices [5], confirming the enhanced mobility for smaller nanowires.
TCAD quantum mechanical simulation is employed to study the underlying
physical mechanism [6]. It is shown that quantum confinement and volume
inversion effects play an important role in the improved transport properties
for the InGaAs GAA nanowire MOSFETs.
## II Device Fabrication
Figure 1: (a) Key fabrication process steps of InGaAs GAA nanowire MOSFETs by
top-down approach. (b) Schematic diagram of an InGaAs GAA nanowire MOSFET and
InGaAs nanowires with two different sizes under investigation
(30nm$\times$30nm and 50nm$\times$30nm). Surface orientation (100) of the top
surface and ideal side surface with vertical sidewall are illustrated. The
nanowire patterning and current transport direction of [100] is depicted. A
cross-sectional TEM image of an InGaAs GAA nanowire MOSFET with $W_{NW}$ of
30nm is also shown. The actual width and height of the nanowire is measured to
be 26nm and 27nm with a $\sim$25o angle between actual and ideal sidewall.
Note that nanowires with vertical sidewall and designed $W_{NW}$ is assumed in
the normalization and simulation in this letter.
Figure 1 shows the fabrication process flow as well as the schematic diagram
of the InGaAs GAA nanowire MOSFET. Fabrication started with a 30nm
In0.53Ga0.47As channel layer with a p-type doping of $2\times 10^{16}cm^{-3}$
epitaxially grown on a heavily p-doped InP (100) substrate by molecular beam
epitaxy (MBE). After source/drain implantation (Si, $1\times
10^{14}cm^{-2},20keV$), fin patterning was performed using BCl3/Ar inductively
coupled plasma (ICP) etching, followed by hydrogen chloride (HCl) based
nanowire release process. The nanowires were aligned along [100] direction as
required by the anisotropic HCl wet etching. After surface passivation with
ammonia sulfide, 10nm Al2O3 and 20nm WN were grown by atomic layer deposition
(ALD) at temperature of 300oC and 385oC respectively. Due to the excellent
conformal coating ability of ALD, the gate stack forms surrounding all facets
of the nanowires. Gate etch using CF4 based ICP etching was then carried out
to define the gate pattern. Finally, ohmic contacts were formed by electron
beam evaporation of Au/Ge/Ni and liftoff process. Details of the fabrication
process can be found in [4].
The fabricated devices have nominal $L_{ch}$ varying from 120nm down to 50nm.
Two different nanowire width ($W_{NW}$) (50nm and 30nm) were defined by
lithography with a fixed nanowire height ($H_{NW}$) of 30nm defined by the MBE
channel thickness. Since the nanowires were aligned along [100] direction,
both the top and side surfaces are (100) surfaces assuming vertical sidewalls.
Due to the non-optimized fin etching process, the actual sidewall leans 10 to
30 degrees towards (110) surface, confirmed by the SEM images [4].
## III Results and discussion
Figure 2: (a) Output characteristics and (b) transfer characteristics and
intrinsic $g_{m}$ versus $V_{gs}$ of two typical InGaAs GAA nanowire MOSFETs
with $L_{ch}=50nm$, $W_{NW}=30nm$ (red) and $50nm$ (black). $I_{s}$ is used
due to relatively large junction leakage current.
Figure 2(a) shows the output characteristics of two InGaAs GAA nanowire
MOSFETs with $L_{ch}$ of 50nm. Devices with $W_{NW}$ of 30nm and 50nm exhibit
a saturation current of 668$\mu$A/$\mu$m and 482$\mu$A/$\mu$m at
$V_{gs}-V_{T}=2V$ and $V_{ds}=1V$, respectively. The current is normalized by
the total _perimeter_ of the nanowires. The device with smaller nanowire size
has a 38% higher $I_{on}$. Similar enhancement in intrinsic $g_{m}$ is also
observed on the smaller nanowire device as shown in Figure 2(b). The $V_{T}$
of the devices with $W_{NW}$ of 30nm and 50nm are $-0.85V$ and $-0.94V$ from
linear extrapolation at $V_{ds}$=$50mV$, respectively. The negative $V_{T}$ is
due to the low work function of WN ($\sim 4.6eV$). Both devices show good
pinch-off characteristics with a _SS_ of 150mV/dec at a $V_{ds}$ of 50mV. The
upper limit of the interface trap density ($D_{it}$) at midgap is estimated to
be 5.6$\times$1012cm-2$\cdot$$eV^{-1}$. The device with $W_{NW}$=30nm shows
lower _DIBL_ ($\sim$180mV/V) compared to device with $W_{NW}$=50nm
($\sim$250mV/V), indicating improved control of short channel effects by
shrinking the nanowire size. Considering the _EOT_ of $\sim$4.5nm and scaled
$L_{ch}$ of 50nm, the _SS_ and _DIBL_ has been significantly improved compared
to previous FinFET work [2], indicating the suitability of GAA structure for
logic applications.
Figure 3(a) shows the average $I_{on}$ measured at $V_{gs}-V_{T}=2V$ and
$V_{ds}=1V$ as a function of $L_{ch}$. A gradual increase of $I_{on}$ is
observed when scaling down the channel length for both nanowire sizes. An
average of 40% increase in $I_{on}$ has been obtained on devices with $W_{NW}$
of 30nm over the entire $L_{ch}$ range. Devices with different $W_{NW}$ show
similar $R_{SD}$, ranging from 950 to 1150$\Omega\cdot\mu$m. The intrinsic
$g_{m}$ of devices with smaller nanowire size is found to be 34% higher than
those with larger nanowire size (not shown). To further characterize the
transport in the nanowire devices, effective mobility was extracted using the
Y-function method, which agrees reasonably well with the split-CV method and
allows for the suppression of the series resistance effect [5]. Figure 3(b)
shows the average $\mu_{0}$ versus $L_{ch}$, demonstrating over 20% mobility
enhancement for devices with smaller $W_{NW}$. The apparent mobility reduction
at shorter $L_{ch}$ can be explained by Shur’s model using a Mathiessen-like
relation considering the ballistic mobility [7]. It is also shown in Figure
3(b) that the extracted $\mu_{0}$ of the InGaAs GAA nanowire MOSFETs are _2
$\sim$4 times_ higher than those from state-of-the-art Si nanowire devices
[8], owing to the better transport properties of the III-V channel.
Figure 3: The average value of (a) $I_{on}$ measured at $V_{gs}-V_{T}=2V$ and
$V_{ds}$=1V and (b) low field mobility $\mu_{0}$ of InGaAs GAA nanowire
MOSFETs extracted using the Y-function method with $L_{ch}$ varying from 50nm
to 120nm and $W_{NW}$ of 30nm (red circle) and 50nm (black square), compared
to Si GAA nanowire NMOSFETs data [8] (diamond). Over 20 devices were measured
at each data point to obtain the average value.
The increase in $I_{on}$, $g_{m}$ and $\mu_{0}$ has confirmed that improved
transport has been obtained in smaller InGaAs nanowires. Normally for top-down
nanowires, it is expected that reducing the nanowire size will degrade
transport due to the relative increase in surface roughness scattering given
the larger surface to volume ratio of the ultra-small nanowires. However, it
has been reported on Si nanowire MOSFETs that the improved transport from
high-mobility sidewall [9], oxidation induced strain inside the nanowire [10],
and the volume inversion effect in nanowires with small cross sectional area
[11] would result in enhanced transport properties with $W_{NW}$ shrinkage.
The InGaAs (111)A surface has been demonstrated to offer higher mobility than
other crystal orientations due to the trap redistribution [12]. However,
(111)A surface can not be the sidewall facet of InGaAs nanowires in this
study, since the nanowires are aligned along [100] direction. Moreover, the
thermal budget of the fabrication process after nanowire release in this
letter is as low as 385oC, which is much lower than the thermal oxidation
temperature (usually over 1000oC) of the Si nanowire MOSFET [10]. Therefore,
strain-induced mobility enhancement can not play a significant role in the
InGaAs nanowires under investigation. On the other hand, due to the much
smaller effective mass and density of states of InGaAs, the inversion layer
thickness can be 3.5 times larger than that of Si. As a result, inversion
carriers can be pushed further away from the interfaces due to a stronger
quantum confinement leading to the volume inversion effect in InGaAs nanowires
at larger dimensions than Si.
Figure 4: (a) Cross sectional distribution of electron density in the
In0.53Ga0.47As nanowire with $W_{NW}$ of 30nm and 50nm at $V_{gs}-V_{T}=1.2V$.
Dashed line shows the inversion layer centroid. (b) Normalized electron
distribution at the middle of the nanowire ($y=0.5H_{NW}$) for square-shape Si
and In0.53Ga0.47As nanowires. Note that vertical sidewalls are assumed in the
simulation. While the sidewalls in the experiments are not vertical, resulting
in reduced gate control, the general scaling trend comparing nanowires with
different sizes remains unchanged.
To further clarify the underlying mechanism, TCAD simulation using Sentaurus
Device [6] was carried out. The electron distribution inside the nanowire in
the strong inversion regime is obtained using a coupled Poisson and quantum
potential solver based on the density gradient model [6, 13], considering only
$\Gamma$ valley for InGaAs. It is found that both InGaAs GAA nanowire MOSFETs
with $W_{NW}$ of 30nm and 50nm operate in the volume inversion regime, where
the inversion charge density inside the entire nanowire is higher than the
background p-type doping. The $W_{NW}$=30nm device show stronger confinement,
resulting in the inversion layer being pushed 1$\sim$2nm further away from the
surface and a higher inversion charge density at the center of nanowire
compared to the $W_{NW}$=50nm case, as shown in Figure 4(a). This would lead
to suppressed surface roughness scattering for the smaller nanowire.
Furthermore, the volume inversion also results in the inversion layer centroid
of the smaller nanowire being closer to the surface and therefore an increase
in electrostatic capacitance with decreasing $W_{NW}$. It is also found that
the two inversion layers inside InGaAs nanowire would merge into one peak at a
dimension of $\sim$10nm, twice as large as that in the Si case ($\sim$5nm) as
shown in Figure 4(b). The inversion layer distributes further inside the
InGaAs nanowire with a higher density at the center compared to the Si case
with the same nanowire size. Further experimental efforts reducing InGaAs
nanowire size are required to illuminate on the ultimate scaling limit of
InGaAs GAA nanowire MOSFETs, which may require development of new nanowire
thinning techniques. The volume inversion at a larger dimension and a stronger
quantum confinement in the InGaAs GAA nanowire MOSFETs may relax the
fabrication complexity and interface quality requirements for InGaAs nanowire
devices.
## IV Conclusion
In this letter, we have fabricated and characterized InGaAs GAA nanowire
MOSFETs with different nanowire size. Enhanced transport properties have been
confirmed on InGaAs nanowires with a smaller dimension, due to a stronger
quantum confinement and volume inversion effect. It is shown that distribution
of inversion carriers moves further away from the surface and volume inversion
occurs at a larger dimension on InGaAs nanowire than its Si counterpart,
making InGaAs GAA MOSFETs favorable for future logic applications.
## References
* [1] J. A. del Alamo, “Nanometre-scale electronics with III-V compound semiconductors,” _Nature_ , vol. 479, no. 7373, pp. 317–323, Nov. 2011.
* [2] Y. Q. Wu, R. S. Wang, T. Shen, J. J. Gu, and P. D. Ye, “First experimental demonstration of 100 nm inversion-mode InGaAs FinFET through damage-free sidewall etching,” in _IEDM Tech. Dig._ , 2009, pp. 331–334.
* [3] M. Radosavljevic, G. Dewey, J. M. Fastenau, J. Kavalieros, R. Kotlyar, B. Chu-Kung, W. K. Liu, D. Lubyshev, M. Metz, K. Millard, N. Mukherjee, L. Pan, R. Pillarisetty, W. Rachmady, U. Shah, and R. Chau, “Non-planar, multi-gate InGaAs quantum well field effect transistors with high-K gate dielectric and ultra-scaled gate-to-drain/gate-to-source separation for low power logic applications,” in _IEDM Tech. Dig._ , 2010, pp. 611–614.
* [4] J. J. Gu, Y. Q. Liu, Y. Q. Wu, R. Colby, R. G. Gordon, and P. D. Ye, “First experimental demonstration of gate-all-around III-V MOSFETs by top-down approach,” in _IEDM Tech. Dig._ , 2011, pp. 769–772.
* [5] A. Cros, K. Romanjek, D. Fleury, S. Harrison, R. Cerutti, P. Coronel, B. Dumont, A. Pouydebasque, R. Wacquez, B. Duriez, R. Gwoziecki, F. Boeuf, H. Brut, G. Ghibaudo, and T. Skotnicki, “Unexpected mobility degradation for very short devices : A new challenge for CMOS scaling,” in _IEDM Tech. Dig._ , 2006, pp. 663–666.
* [6] _Synopsys Sentaurus Device Manuals_ , see http://www.synopsys.com/tools
/tcad/devicesimulation/pages/sentaurusdevice.aspx.
* [7] M. Shur, “Low ballistic mobility in submicron HEMTs,” _IEEE Electron Device Letters_ , vol. 23, no. 9, pp. 511–513, 2002.
* [8] R. Wang, H. Liu, R. Huang, J. Zhuge, L. Zhang, D.-W. Kim, X. Zhang, D. Park, and Y. Wang, “Experimental investigations on carrier transport in Si nanowire transistors: Ballistic efficiency and apparent mobility,” _IEEE Transactions on Electron Devices_ , vol. 55, no. 11, pp. 2960–2967, Nov. 2008.
* [9] J. Chen, T. Saraya, and T. Hiramoto, “Experimental investigations of electron mobility in Silicon nanowire nMOSFETs on (110) silicon-on-insulator,” _IEEE Electron Device Letters_ , vol. 30, no. 11, pp. 1203–1205, Nov. 2009\.
* [10] K. Moselund, M. Najmzadeh, P. Dobrosz, S. Olsen, D. Bouvet, L. De Michielis, V. Pott, and A. Ionescu, “The high-mobility bended n-channel Silicon nanowire transistor,” _IEEE Transactions on Electron Devices_ , vol. 57, no. 4, pp. 866–876, Apr. 2010.
* [11] S. D. Suk, M. Li, Y. Y. Yeoh, K. H. Yeo, K. H. Cho, I. K. Ku, H. Cho, W. Jang, D.-W. Kim, D. Park, and W.-S. Lee, “Investigation of nanowire size dependency on TSNWFET,” in _IEDM Tech. Dig._ , 2007, pp. 891–894.
* [12] H. Ishii, N. Miyata, Y. Urabe, T. Itatani, T. Yasuda, H. Yamada, N. Fukuhara, M. Hata, M. Deura, M. Sugiyama, M. Takenaka, and S. Takagi, “High electron mobility metal–insulator–semiconductor field-effect transistors fabricated on (111)-oriented InGaAs channels,” _Applied Physics Express_ , vol. 2, no. 12, p. 121101, Nov. 2009.
* [13] M. G. Ancona and H. F. Tiersten, “Macroscopic physics of the silicon inversion layer,” _Phys. Rev. B_ , vol. 35, pp. 7959–7965, May 1987.
|
arxiv-papers
| 2012-04-07T18:00:41 |
2024-09-04T02:49:29.470605
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangjiang J. Gu, Heng Wu, Yiqun Liu, Adam T. Neal, Roy G. Gordon, and\n Peide D. Ye",
"submitter": "Jiangjiang J. Gu",
"url": "https://arxiv.org/abs/1204.1665"
}
|
1204.1720
|
# Metamagnetic transition in Ca1-xSrxCo2As2($x$ = 0 and 0.1) single crystals
J. J. Ying, Y. J. Yan, A. F. Wang, Z. J. Xiang, P. Cheng, G. J. Ye and X. H.
Chen Corresponding author chenxh@ustc.edu.cn Hefei National Laboratory for
Physical Science at Microscale and Department of Physics, University of
Science and Technology of China, Hefei, Anhui 230026, People’s Republic of
China
###### Abstract
We report the magnetism and transport measurements of CaCo2As2 and
Ca0.9Sr0.1Co2As2 single crystals. Antiferromagnetic transition was observed at
about 70 K and 90 K for CaCo2As2 and Ca0.9Sr0.1Co2As2, respectively. Magnetism
and magnetoresistance measurements reveal metamagnetic transition from an
antiferromagnetic state to a ferromagnetic state with the critical field of
3.5 T and 1.5 T respectively along c-axis for these two materials at low
temperature. For the field along ab-plane, spins can also be fully polarized
above the field of 4.5 T for Ca0.9Sr0.1Co2As2. While for CaCo2As2, spins can
not be fully polarized up to 7 T. We proposed the cobalt moments of these two
materials should be ordered ferromagnetically within the ab-plane but
antiferromagnetically along the c-axis(A-type AFM).
###### pacs:
74.25.-q, 74.25.Ha, 75.30.-m
## I INTRODUCTION
The layered ThCr2Si2 structure type is commonly observed for $AT_{2}X_{2}$
compounds, in which $A$ is typically a rare earth, alkaline earth, or alkali
element; $T$ is a transition-metal and $X$ is metalloid element. In this
ThCr2Si2 structure, the $T_{2}X_{2}$ layers are made from edge-sharing
$TX_{4}$ tetrahedra and $A$ ions are intercalated between $T_{2}X_{2}$ layers.
Novel physical properties were observed in such ThCr2Si2 structure compounds
including magnetic ordering and superconductivity. Recent discoveries of high
temperature superconductivity in ThCr2Si2 structure pnictides and selenides
have led to renewed interest in this large class of compoundsRotter ; xlchen .
$A$Fe2As2 ($A$=Ca, Sr, Ba, Eu) with the ThCr2Si2-type structure were widely
investigated because it is easy to grow large-size, high-quality single
crystalswugang ; wangxf . Superconductivity can be achieved through doping or
under high pressure. The maximum $T_{\rm c}$ for the hole-doped samples is
about 38 K and for the Co doped samples the maximum $T_{\rm c}$ can reach 26
KRotter ; Sefat . Superconductivity up to 30 K was observed in $A_{x}$Fe2-ySe2
($A$=K, Rb, Cs and Tl) which also has the ThCr2Si2 structurexlchen ; Mizuguchi
; Wang ; Ying ; Krzton ; Fang . It is very interesting to look for other
materials with related structures and investigate their physical properties to
see if there can be potential parent compounds for new high temperature
superconductors.
$A$Co2As2 ($A$ is rare earth element) has the ThCr2Si2 structure, while their
physical properties haven’t been systematically studied. The magnetic moment
of the Co ion in this type of material would not vanish due to the odd number
of 3d electron, thus we would anticipate an appearance of the magnetic
ordering phase. The magnetic properties of SrCo2As2 show Curie-Weiss-like
behavior and BaCo2As2 was reported as a highly renormalized paramagnet in
proximity to ferromagnetic characterLeithe ; Sefat1 . LaOCoAs with the same
CoAs layers shows itinerant ferromagnetismYanagi . While CaCo2P2 was reported
as having ferromagnetically ordered Co planes, which are stacked
antiferromagneticallyReehuis . The magnetic phase diagram of Sr1-xCaxCo2P2 is
very complicated and closely related to the structural changesJia . The
magnetism behavior in CoAs layered compounds is very interesting and it needs
further investigation.
In this article, we investigated the magnetism and transport properties of
CaCo2As2 and Ca0.9Sr0.1Co2As2 single crystals. Antiferromagnetic (AFM)
ordering was observed below $T_{\rm N}$ $\approx$ 70 K in CaCo2As2. And for
Ca0.9Sr0.1Co2As2, $T_{\rm N}$ increases to about 90 K. We determined the
cobalt moments should be ordered ferromagnetically within the ab-plane but
antiferromagnetically along the c-axis(A-type AFM). Metamagnetic transition
corresponds to a spin-flop transition from an AF to a FM state was observed in
these two samples.
## II EXPERIMENTAL DETAILS
High quality single crystals with nominal composition Ca1-xSrxCo2As2(x=0 and
0.1) were grown by conventional solid-state reaction using CoAs as self-
fluxSefat1 . The CoAs precursor was first synthesized from stoichiometric
amounts of Co and As inside the silica tube at 700
$\mathrm{\SIUnitSymbolCelsius}$ for 24 h. Then, the mixture with ratio of
Ca:Sr:CoAs=1-x:x:4 was placed in an alumina crucible, and sealed in an quartz
tube. The mixture was heated to 1200 $\mathrm{\SIUnitSymbolCelsius}$ in 6
hours and then kept at this temperature for 10 hours, and later slowly cooled
down to 950 $\mathrm{\SIUnitSymbolCelsius}$ at a rate of 3
$\mathrm{\SIUnitSymbolCelsius}$/ hour. After that, the temperature was cooled
down to room temperature by shutting down the furnace. The shining platelike
Ca1-xSrxCo2As2 crystals were mechanically cleaved. The actual composition of
the single crystals were characterized by the Energy-dispersive X-ray
spectroscopy (EDX). The actual doping levels are almost the same as the
nominal values. Resistivity was measured using the Quantum Design PPMS-9 and
Magnetic susceptibility was measured using the Quantum Design SQUID-VSM.
## III RESULTS AND DISCUSSION
### III.1 Crystal structure and resistivity
Figure 1: (color online). (a): The single crystal x-ray diffraction pattern of
CaCo2As2 and Ca0.9Sr0.1Co2As2. (b): Temperature dependence of in-plane
resistivity for CaCo2As2 and Ca0.9Sr0.1Co2As2 single crystals. The inset was
the derivation of resistivity curves around $T_{\rm N}$.
Single crystals of Ca1-xSrxCo2As2 ($x$ = 0 and 0.1) were characterized by
X-ray diffractions (XRD) using Cu $K_{\alpha}$ radiations as shown in
Fig.1(a). Only (00$l$) diffraction peaks were observed, suggesting uniform
crystallographic orientation with the c axis perpendicular to the plane of the
single crystal. The c-axis parameter was about 10.27 Å for CaCo2As2 which is
nearly the same as the previous resultJohnston . For Ca0.9Sr0.1Co2As2, c-axis
parameter increases to 10.41 Å due to the larger ion radius of Sr. The
resistivity of both samples shows metallic behavior as shown in Fig.1(b) which
is similar to SrCo2As2 and BaCo2As2Sefat ; Leithe . The derivation of
resistivity curves as shown in the inset of Fig.1(b) show peaks at about 70
and 90 K for CaCo2As2 and Ca0.9Sr0.1Co2As2, respectively. These temperatures
were consistent with the antiferromagnetic transition temperature ($T_{\rm
N}$) which would be shown later in the magnetic susceptibility measurements.
### III.2 Magnetic susceptibility and magnetoresistance
Figure 2: (color online). Temperature dependence of the susceptibility for
CaCo2As2 with magnetic field along (a) and perpendicular (b) to the ab-plane.
Temperature dependence of the susceptibility for Ca0.9Sr0.1Co2As2 with
magnetic field along (c) and perpendicular (d) to the ab-plane. Figure 3:
(color online). Isothermal magnetization hysteresis of CaCo2As2 with magnetic
field along (a) and perpendicular (b) to the ab-plane. MH curves for
Ca0.9Sr0.1Co2As2 with field along (c) and perpendicular (d) to the ab-plane at
certain temperature. Figure 4: (color online). Magnetoresistance of CaCo2As2
with magnetic field up to 9T along (a) and perpendicular (b) to the ab-plane.
MR curves for Ca0.9Sr0.1Co2As2 with field along (c) and perpendicular (d) to
the ab-plane at certain temperature.
Figure 2 (a) and (b) show the temperature dependence of susceptibility for
CaCo2As2 under various magnetic field up to 7 T applied within ab-plane and
along c-axis, respectively. The susceptibility drops at around 70 K with the
field of 0.1 T applied along and perpendicular to the ab-plane, which
indicates the antiferromagnetic state below 70 K in this material. $T_{\rm N}$
was suppressed by increasing the field and was suppressed to 50 K under the
field of 7 T applied along the ab-plane. However, with the field of 7 T
applied along the c-axis, the susceptibility saturated at low temperature
which indicated the ferromagnetic state at low temperature. For
Ca0.9Sr0.1Co2As2, $T_{\rm N}$ increases to 90 K. Under high magnetic field,
the system changes to ferromagnetic state with field applied both along and
perpendicular to the ab-plane as shown in Fig. 2 (c) and (d). These results
indicate that a metamagnetic transition from antiferromagnetism (AFM) to
ferromagnetism (FM) occurs with increasing magnetic field at low temperature.
In order to further investigate such metamagnetic transition, we performed the
isothermal magnetization hysteresis measurement as shown in Figure 3. The
magnetization (M) almost increases linearly for CaCo2As2 with the field H
applied along the ab-plane at various temperatures. The slope of MH curves is
almost the same below 50 K and decrease with increasing the temperature above
$T_{\rm N}$. While for the field applied along the c-axis, M increases very
sharply when H increases to about 3.5 T at 4 K. With increasing the
temperature, such behavior weakened and vanished above $T_{\rm N}$. This
behavior clearly indicates the spin reorientation in CaCo2As2 at high magnetic
field. While for Ca0.9Sr0.1Co2As2, M increases very steeply with H increasing
to 1.5 T and saturates at high field with the field applied along the c-axis
at 4 K. The transition field gradually decreases with increasing the
temperature and vanishes above $T_{\rm N}$. M increases almost linearly with H
applied along ab-plane at low field at 4 K which is similar to the CaCo2As2,
further increasing the field leads to the saturation of M. The values of the
saturated M are almost the same for the field along and perpendicular to the
ab-plane. This result clearly indicates that almost all the magnetic ions can
be tuned by the magnetic field above a critical field. The critical magnetic
field of Ca0.9Sr0.1Co2As2 is lower than CaCo2As2, which is probably due to the
weakening of inter layer AFM coupling. Spins can be tuned much easier with
magnetic field applied along c-axis than along ab-plane, for CaCo2As2, spins
could not be aligned along ab-plane even under 7 T.
We further measured the magnetoresistance (MR) of CaCo2As2 and
Ca0.9Sr0.1Co2As2 from 4 to 200 K up to 9 T as shown in Figure 4.
Magnetoresistance can be hardly detected at high temperature and gradually
became negative with decreasing the temperature. For CaCo2As2, the magnitude
of magnetoresistance gradually increases above $T_{\rm N}$ and gradually
decreases below $T_{\rm N}$ with decreasing the temperature. This is because
the presence of the FM order tends to suppress the spin scattering. Magnetic
field gradually polarized the spins of Co ions with decreasing the temperature
above $T_{\rm N}$, while below the AFM transition temperature, it became much
more difficult to polarize the spins with the temperature cooling down. The MR
with H perpendicular to the ab-plane was almost the same with H along ab-plane
above $T_{\rm N}$. While the temperature below $T_{\rm N}$, MR became positive
at low magnetic field. When the applied magnetic field surpass the critical
field of metamagnetic transition, MR gradually becomes negative and its
magnitude increases with increasing the magnetic field. Similar MR behavior
was observed above $T_{\rm N}$ with field applied along and perpendicular to
the ab-plane. The MR changes greatly at the critical field of metamagnetic
transition. The metamagnetic transition of Ca0.9Sr0.1Co2As2 is much sharper
comparing with CaCo2As2 with H applied along c-axis. MR became constant when
all the spins were tuned to FM state. The values of MR are almost the same for
the field along and perpendicular to the ab-plane which indicates that all the
spins can be tuned by high magnetic field. Similar MR behaviors across the AF
and the FM phase boundary were also observed in other materials such as
Na0.85CoO2, layered ruthenates and colossal magnetoresistance materialsLuo ;
Nakatsuji . The MR result about the metamagnetic transition is consistent with
the MH measurements in these two materials. The magnetoresistance with field
applied along and perpendicular to the ab-plane are almost the same under high
magnetic field which indicates that magnetoresistance was mainly induced from
magnetic scattering from Co2+ and all the Co2+ spins can be tuned above the
critical field of metamagnetic transition at low temperature.
Figure 5: (color online). Field dependence of Hall resistivity at various
temperatures for CaCo2As2 (a) and Ca0.9Sr0.1Co2As2 (b). Figure 6: (color
online). The temperature dependence of anomalous Hall coefficient $R^{A}_{H}$
and normal Hall coefficient $R^{N}_{H}$ for CaCo2As2 (a) and Ca0.9Sr0.1Co2As2
(b). Figure 7: (color online). (a): Possibly magnetic structure deduced from
magnetism and MR measurement. Magnetic structure above the critical field with
H along (b) and perpendicular (c) to c-axis.
### III.3 Hall coefficient and magnetic structure
We also measured the Hall resistivity $\rho_{xy}$ of CaCo2As2 and
Ca0.9Sr0.1Co2As2 as shown in Figure 5. $\rho_{xy}$ is measured by sweeping
field from -7 T to +7 T at various temperature, thus the accurate Hall
resistivity $\rho_{H}$ is obtained by using $[\rho_{xy}(+H)-\rho_{xy}(-H)]/2$,
where $\rho_{xy}(\pm H)$ is $\rho_{xy}$ under positive or negative magnetic
field. We found $\rho_{xy}$ shows a steep decrease at certain field $H_{C}$
below $T_{\rm N}$ similar to isothermal MR, which arises from the jump
magnitudes of magnetization M due to the spin-flop of Co ions induced by
external field H. It is well known that Hall effect arises from two parts of
normal Hall effect and anomalous Hall effect in ferromagnetic metals, in which
anomalous Hall resistivity is proportional to the magnetization M. Hall
resistivity $\rho_{xy}=R^{N}_{H}H+R^{A}_{H}4\pi M$, where $R^{N}_{H}$ is the
normal Hall coefficient, and $R^{A}_{H}$ is the anomalous Hall
coefficientnagaosa . We extracted the $R^{N}_{H}$ from H-linear term of
$\rho_{xy}$ at low field. Temperature dependence of $R^{N}_{H}$ for CaCo2As2
and Ca0.9Sr0.1Co2As2 are shown in Fig.6(a) and (b), respectively. The value of
$R^{N}_{H}$ is negative at high temperature, indicating the primate carrier in
these two materials is electron. The magnitude of $R^{N}_{H}$ gradually
increases with decreasing the temperature and it reaches its maximum value at
around $T_{\rm N}$. Below $T_{\rm N}$, the magnitude of $R^{N}_{H}$ decreases
very quickly with decreasing the temperature. The anomalous Hall coefficient
$R^{A}_{H}$ below $T_{\rm N}$ is inferred from the ratio of the jump
magnitudes $\Delta M$ and $\Delta\rho_{xy}$ around metamagnetic transition
field. $R^{A}_{H}=\Delta\rho_{xy}/4\pi\Delta M$ is also plotted in Fig.6 (a)
and (b). The magnitude of $R^{A}_{H}$ decreases linearly with decreasing
temperature below $T_{\rm N}$. Such behavior was also observed in TaFe1+yTe3
systemLiu .
Based on the above results of magnetic susceptibility, MR and Hall resistivity
measurement, possible magnetic structures for the spins of Co ions are
proposed as shown in Fig. 7 (a), (b) and (c). Below $T_{\rm N}$, Co spins of
these two materials should be ordered ferromagnetically within the ab-plane
but antiferromagnetically along the c-axis(A-type AFM) under low magnetic
field. When the external field surpass the inner field Hint of Co spins. All
the Co spins aligned along the direction of H. The external field H along ab-
plane is much harder to tune the AFM ordering of Co spins than with H along
c-axis.
The AFM ordering found in CaCo2As2 and Ca0.9Sr0.1Co2As2 is very different from
their isostructure compounds SrCo2As2 and BaCo2As2 in which no magnetic
ordering was observed. Such a difference might due to the much smaller c/a
ratio in CaCo2As2 than in SrCo2As2 and BaCo2As2, similar properties were also
observed in CaCo2P2Jia . High magnetic field can tune almost all the spins,
and metamagnetic transition was observed by increasing the magnetic field at
low temperature which was similar to EuFe2As2wutao . The magnetic property in
this system is strongly correlated to the structure. The critical field of
metamagnetic transition decreased by doping Sr, which was probably due to the
enlarged c-axis parameter and decreased the inter layer AFM coupling.
## IV CONCLUSION
In conclusion, we found AFM ordering in CaCo2As2 and Ca0.9Sr0.1Co2As2 at 70
and 90 K, respectively. A metamagnetic transition from AFM to FM occurs with
increasing magnetic field in these two compounds. Little Sr doping in CaCo2As2
would effectively decrease the critical field of metamagnetic transition. The
magnetic susceptibility and MR measurements all indicate that the cobalt
moments of these two materials should be ordered ferromagnetically within the
ab-plane but antiferromagnetically along the c-axis(A-type AFM), which is the
same with their isostructure compound CaCo2P2Reehuis ; Reehuis2 .
ACKNOWLEDGEMENT This work is supported by the National Basic Research Program
of China (973 Program, Grant No. 2012CB922002 and No. 2011CB00101), National
Natural Science Foundation of China (Grant No. 11190021 and No. 51021091), the
Ministry of Science and Technology of China, and Chinese Academy of Sciences.
## References
* (1) M. Rotter, M. Tegel, D. Johrendt, Phys. Rev. Lett. 101, 107006(2008).
* (2) J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He and X. Chen, Phys. Rev. B 82, 180520(R) (2010).
* (3) G Wu, H Chen, TWu, Y L Xie, Y J Yan, R H Liu, X FWang, J J Ying and X H Chen, J. Phys.: Condens. Matter 20 422201 (2008)
* (4) X. F. Wang, T. Wu, G. Wu, H. Chen, Y. L. Xie, J. J. Ying, Y. J. Yan, R. H. Liu and X. H. Chen, Phys. Rev. Lett. 102, 117005(2009).
* (5) Athena S. Sefat, Rongying Jin, Michael A. McGuire, Brian C. Sales, David J. Singh, and David Mandrus, Phys. Rev. Lett. 101, 117004 (2008).
* (6) Yoshikazu Mizuguchi, Hiroyuki Takeya, Yasuna Kawasaki, Toshinori Ozaki, Shunsuke Tsuda, Takahide Yamaguchi and Yoshihiko Takano, Appl. Phys. Lett. 98, 042511 (2011).
* (7) A. F. Wang, J. J. Ying, Y. J. Yan, R. H. Liu, X. G. Luo, Z. Y. Li, X. F. Wang, M. Zhang, G. J. Ye, P. Cheng, Z. J. Xiang, X. H. Chen, Phys. Rev. B 83, 060512(R) (2011).
* (8) J. J. Ying, X. F. Wang, X. G. Luo, A. F. Wang, M. Zhang, Y. J. Yan, Z. J. Xiang, R. H. Liu, P. Cheng, G. J. Ye, X. H. Chen , Phys. Rev. B 83, 212502 (2011).
* (9) A. Krzton-Maziopa, Z. Shermadini, E. Pomjakushina, V. Pomjakushin, M. Bendele, A. Amato, R. Khasanov, H. Luetkens and K. Conder, J. Phys.: Condens. Matter 23, 052203 (2011).
* (10) Minghu Fang, Hangdong Wang, Chiheng Dong, Zujuan Li, Chunmu Feng, Jian Chen, H.Q. Yuan, EPL, 94, 27009 (2011).
* (11) A. S. Sefat, D. J. Singh, R. Jin, M. A. McGuire, B. C. Sales, and D. Mandrus, Phys. Rev. B 79, 024512 (2009).
* (12) A. Leithe-Jasper, W. Schnelle, C. Geibel, and H. Rosner, Phys. Rev. Lett. 101, 207004 (2008).
* (13) H. Yanagi, R. Kawamura, T. Kamiya, Y. Kamihara, M. Hirano, T. Nakamura, H. Osawa, and H. Hosono, Phys. Rev. B 77, 224431 (2008).
* (14) J. L. Luo, N. L.Wang, G.T. Liu, D. Wu, X. N. Jing, F. Hu, and T. Xiang, Phys. Rev. Lett. 93, 187203 (2004).
* (15) S. Nakatsuji, D. Hall, L. Balicas, Z. Fisk, K. Sugahara, M. Yoshioka, and Y. Maeno, Phys. Rev. Lett. 90, 137202 (2003).
* (16) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539(2010).
* (17) R. H. Liu, M. Zhang, P. Cheng, Y. J. Yan, Z. J. Xiang, J. J. Ying, X. F. Wang, A. F. Wang, G. J. Ye, X. G. Luo, and X. H. Chen, Phys. Rev. B 84, 184432 (2011).
* (18) Shuang Jia, A. J. Williams, P. W. Stephens, and R. J. Cava, Phys. Rev. B 80, 165107 (2009).
* (19) David C. Johnston, Advances in Physics 59, 803 C1061 (2010).
* (20) T. Wu, G. Wu, H. Chen, Y. L. Xie, R. H. Liu, X. F. Wang and X. H. Chen, J. Mag. Mag. Mat. 321, 3870-3874 (2009).
* (21) M. Reehuis and W. Jeitschko, J. Phys. Chem. Solids 51, 961 (1990).
* (22) M. Reehuis, W. Jeitschko, G. Kotzyba, B. Zimmer, and X. Hu, J. Alloys Compd. 266, 54 (1998).
|
arxiv-papers
| 2012-04-08T08:52:18 |
2024-09-04T02:49:29.476796
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. J. Ying, Y. J. Yan, A. F. Wang, Z. J. Xiang, P. Cheng, G. J. Ye and\n X. H. Chen",
"submitter": "X. H. Chen",
"url": "https://arxiv.org/abs/1204.1720"
}
|
1204.1841
|
# Strong skew commutativity preserving maps on von Neumann algebras
Xiaofei Qi Department of Mathematics, Shanxi University, Taiyuan 030006, P.
R. China. qixf1980@126.com and Jinchuan Hou Department of Mathematics,
Taiyuan University of Technology, Taiyuan 030024, P. R. of China
jinchuanhou@yahoo.com.cn
###### Abstract.
Let ${\mathcal{M}}$ be a von Neumann algebra without central summands of type
$I_{1}$. Assume that $\Phi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a
surjective map. It is shown that $\Phi$ is strong skew commutativity
preserving (that is, satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^{*}=AB-BA^{*}$
for all $A,B\in{\mathcal{M}}$) if and only if there exists some self-adjoint
element $Z$ in the center of ${\mathcal{M}}$ with $Z^{2}=I$ such that
$\Phi(A)=ZA$ for all $A\in{\mathcal{M}}$. The strong skew commutativity
preserving maps on prime involution rings and prime involution algebras are
also characterized.
2010 Mathematical Subject Classification. 47B49, 46L10
Key words and phrases. Von Neumann algebras, prime rings, general preserving
maps, skew Lie products.
This work is partially supported by the National Natural Science Foundation of
China (11171249, 11101250).
## 1\. Introduction
Let $\mathcal{R}$ be a *-ring. For $A,B\in{\mathcal{R}}$, denote by
$[A,B]_{*}=AB-BA^{*}$ the skew Lie product. This product $AB-BA^{*}$ is found
playing an important role in some research topics. Let $B$ be a fixed element
in $\mathcal{R}$. The additive map
$\Phi:{\mathcal{R}}\rightarrow{\mathcal{R}}$ defined by $\Phi(A)=AB-BA^{*}$
for all $A\in{\mathcal{R}}$ is a Jordan *-derivation, that is, it satisfies
$\Phi(A^{2})=\Phi(A)A^{*}+A\Phi(A)$. The notion of Jordan *-derivations arose
naturally in $\check{S}$emrl’s work [10, 11] investigating the problem of
representing quadratic functionals with sesquilinear functionals. Motivated by
the theory of rings (and algebras) equipped with a Lie product $[T,S]=TS-ST$
or a Jordan product $T\circ S=TS+ST$, Molnár in [9] studied the skew Lie
product and gave a characterization of ideals in ${\mathcal{B}}(H)$ in terms
of the skew Lie product. It is shown in [9] that, if
${\mathcal{N}}\subseteq{\mathcal{B}}(H)$ is an ideal, then ${\mathcal{N}}={\rm
span}\\{AB-BA^{*}:A\in{\mathcal{N}},B\in{\mathcal{B}}(H)\\}={\rm span}\\{AB-
BA^{*}:A\in{\mathcal{B}}(H),B\in{\mathcal{N}}\\}$. In particular, every
operator in ${\mathcal{B}}(H)$ is a finite sum of $AB-BA^{*}$ type operators.
Later, Bre$\check{s}$ar and Fon$\check{s}$ner [2] generalized the above
results in [9] to rings with involution in different ways.
Recall that a map $\Phi:{\mathcal{R}}\rightarrow{\mathcal{R}}$ is skew
commutativity preserving if, for any $A$, $B\in{\mathcal{R}}$, $[A,B]_{*}=0$
implies $[\Phi(A),\Phi(B)]_{*}=0$. The problem of characterizing linear (or
additive) bijective maps preserving skew commutativity had been studied
intensively on various algebras (see [4, 5] and the references therein). More
specially, we say that a map $\Phi:{\mathcal{R}}\rightarrow{\mathcal{R}}$ is
strong skew commutativity preserving (briefly, SSCP) if
$[\Phi(A),\Phi(B)]_{*}=[A,B]_{*}$ for all $A,B\in{\mathcal{R}}$. SSCP maps are
also called strong skew Lie product preserving maps in [6]. We prefer to the
terminology SSCP because many authors call the maps satisfying
$[\Phi(A),\Phi(B)]=[A,B]$ strong commutativity preserving maps. It is obvious
that a strong skew commutativity preserving map must be skew commutativity
preserving, but the inverse is not true generally. In [6], Cui and Park proved
that, if $\mathcal{R}$ is a factor von Neumann algebra, then every SSCP
surjective map $\Phi$ on ${\mathcal{R}}$ has the form $\Phi(A)=\Psi(A)+h(A)I$
for every $A\in{\mathcal{R}}$, where
$\Psi:{\mathcal{R}}\rightarrow{\mathcal{R}}$ is a linear bijective map
satisfying $\Psi(A)\Psi(B)-\Psi(B)\Psi(A)^{*}=AB-BA^{*}$ for
$A,B\in{\mathcal{R}}$ and $h$ is a real functional on $\mathcal{R}$ with
$h(0)=0$; particularly, if $\mathcal{R}$ is a type $I$ factor, then
$\Phi(A)=cA+h(A)I$ for every $A\in{\mathcal{R}}$, where $c\in\\{-1,1\\}$.
In the present paper, we show further that, in the above result, $h=0$ and
$\Psi(A)=A$ for all $A$ or $\Psi(A)=-A$ for all $A$. In fact, the purpose of
the present paper is to give a characterization of the SSCP maps on prime
*-rings or on general von Neumann algebras without central summands of type
$I_{1}$. And the improvement of the above result in [6] is an immediate
consequence of our results.
Before embarking on the main results, we need some notations. Let
$\mathcal{A}$ be a *-ring. Denote by ${\mathcal{Z}}({\mathcal{A}})$ the center
of $\mathcal{A}$. Observe that $Z\in{\mathcal{Z}}({\mathcal{A}})$ if and only
if $Z^{*}\in{\mathcal{Z}}({\mathcal{A}})$.
${\mathcal{Z}}_{S}({\mathcal{A}})=\\{A\in{\mathcal{Z}}({\mathcal{A}}):A=A^{*}\\}$.
An element $A\in{\mathcal{A}}$ is symmetric (respectively, skew symmetric) if
$A^{*}=A$ (respectively, if $A^{*}=-A$). If $\mathcal{A}$ is a unital
*-algebra with unit $I$ over a field $\mathbb{F}$, let $\mathcal{S}$ (resp.
$\mathcal{K}$) be the set of its symmetric (resp. skew symmetric) elements.
Then every element $A\in{\mathcal{A}}$ can be written as $A=S+K$, where
$S\in{\mathcal{S}}$ and $K\in{\mathcal{K}}$. Moreover, this decomposition is
unique. We say that the involution $*$ is of the first kind if
${\mathbb{F}}I\subseteq{\mathcal{S}}$; equivalently, $*$ is an
$\mathbb{F}$-linear map. Otherwise, $*$ is said to be of the second kind (see
[3]). Recall that a ring $\mathcal{A}$ is prime if, for any $A,\
B\in{\mathcal{A}}$, $A{\mathcal{A}}B=0$ implies $A=0$ or $B=0$. A von Neumann
algebra $\mathcal{M}$ is a subalgebra of some ${\mathcal{B}}(H)$, the algebra
of all bounded linear operators acting on a complex Hilbert space $H$, which
satisfies the double commutant property:
${\mathcal{M}}^{\prime\prime}=\mathcal{M}$, where
${\mathcal{M}}^{\prime}=\\{T:T\in{\mathcal{B}}(H)\ \mbox{and}\ TA=AT\ \forall
A\in{\mathcal{M}}\\}$ and
${\mathcal{M}}^{\prime\prime}=\\{{\mathcal{M}}^{\prime}\\}^{\prime}$. For the
theory of von Neumann algebras, ref. [7].
The rest part of this paper is organized as follows. In Section 2, we discuss
the question in a pure algebraic setting. Let $\mathcal{A}$ be a unital prime
*-ring. Assume that $\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a general
strong skew commutativity preserving surjective map. We show that, if
$\mathcal{A}$ contains a nontrivial symmetric idempotent element, then $\Phi$
has the form $\Phi(A)=A+f(A)$ for all $A\in{\mathcal{A}}$, or
$\Phi(A)=-A+f(A)$ for all $A\in{\mathcal{A}}$, where
$f:{\mathcal{A}}\rightarrow{\mathcal{Z}}_{S}({\mathcal{A}})$ is an arbitrary
map (Theorem 2.1). Particularly, if $\mathcal{A}$ is a *-algebra and if the
involution $*$ is of the second kind, then
$\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is strong skew commutativity
preserving if and only if $\Phi(A)=A$ for all $A\in{\mathcal{A}}$, or
$\Phi(A)=-A$ for all $A\in{\mathcal{A}}$ (Theorem 2.10). As an application, we
obtain a characterization of strong skew commutativity preserving general
surjective maps on factor von Neumann algebras (Theorem 2.11) which improves
the main results in [6]. Section 3 is devoted to characterizing the strong
skew commutativity maps on general von Neumann algebras. We prove that, if
$\mathcal{A}$ is a von Neumann algebra without central summands of type
$I_{1}$, then a map $\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is strong
skew commutativity preserving if and only if $\Phi(A)=ZA$ for all
$A\in{\mathcal{A}}$, where $Z\in{\mathcal{Z}}_{S}({\mathcal{A}})$ with
$Z^{2}=I$ (Theorem 3.1). It is clear that Theorem 2.11 above is also an
immediate consequence of this result.
## 2\. SSCP maps on prime rings with involution
In this section, we discuss the question of characterizing the strong skew
commutativity preserving maps on prime rings with involution $*$. The
following is our main result in this section.
Theorem 2.1. Let ${\mathcal{A}}$ be a unital prime *-ring with the unit $I$.
Assume that $\mathcal{A}$ contains a nontrivial symmetric idempotent $P$ and
$\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a surjective map. If $\Phi$ is
strong skew commutativity preserving, that is,
$\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^{*}=AB-BA^{*}$ for all $A,B\in{\mathcal{A}}$,
then there exists a map
$f:{\mathcal{A}}\rightarrow{\mathcal{Z}}_{S}({\mathcal{A}})$, the set of
central symmetric elements, such that $\Phi(A)=A+f(A)$ for all
$A\in{\mathcal{A}}$, or $\Phi(A)=-A+f(A)$ for all $A\in{\mathcal{A}}$.
To prove Theorem 2.1, we need a lemma.
Let $\mathcal{A}$ be a prime ring. Denote by
${\mathcal{Q}}={\mathcal{Q}}_{ml}({\mathcal{A}})$ the maximal left ring of
quotients of $\mathcal{A}$. Note that the center $\mathcal{C}$ of
$\mathcal{Q}$ is a field which is called the extended centroid of
$\mathcal{A}$ (see [1, 3] for details). Moreover,
${\mathcal{Z}}({\mathcal{A}})\subseteq{\mathcal{C}}$. The following result is
well-known.
Lemma 2.2. ([3, Theorem A.7]) Let $\mathcal{A}$ be a prime ring, and let
$A_{i}$, $B_{i}$, $C_{j}$, $D_{j}\in{\mathcal{Q}}_{ml}({\mathcal{A}})$ be such
that
$\sum_{i=1}^{n}A_{i}XB_{i}=\sum_{j=1}^{m}C_{j}XD_{j}\quad{\rm for\ \ all}\ \
X\in{\mathcal{A}}.$
If $A_{1},\ldots,A_{n}$ are linearly independent over $\mathcal{C}$, then each
$B_{i}$ is a $\mathcal{C}$-linear combination of $D_{1},\ldots,D_{m}$.
Similarly, if $B_{1},\ldots,B_{n}$ are linearly independent over
$\mathcal{C}$, then each $A_{i}$ is a $\mathcal{C}$-linear combination of
$C_{1},\ldots,C_{m}$. In particular, for
$A,B\in{\mathcal{Q}}_{ml}({\mathcal{A}})$, if $AXB=BXA$ for all
$X\in{\mathcal{A}}$, then $A$ and $B$ are $\mathcal{C}$-linear dependent.
We will prove Theorem 2.1 by a series of lemmas. Assume in the sequel that
$\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a SSCP surjective map.
Lemma 2.3.
$\Phi({\mathcal{Z}}_{S}({\mathcal{A}}))={\mathcal{Z}}_{S}({\mathcal{A}})$.
Proof. Take any $Z\in{\mathcal{Z}}_{S}({\mathcal{A}})$. Then for any
$T\in{\mathcal{A}}$, we have $[\Phi(Z),\Phi(T)]_{*}=[Z,T]_{*}=0$. So
$\Phi(Z)\Phi(T)=\Phi(T)\Phi(Z)^{*}.$ By the surjectivity of $\Phi$, we get
$None$ $\Phi(Z)X=X\Phi(Z)^{*}\quad{\rm for\ \ all}\quad X\in{\mathcal{A}}.$
Take $X=I$ in Eq.(2.1), we get $\Phi(Z)=\Phi(Z)^{*}$. This and Eq.(2.1) imply
that $\Phi(Z)\in{\mathcal{Z}}_{S}({\mathcal{A}})$. On the other hand, if there
exists some $A\in{\mathcal{A}}$ such that
$\Phi(A)=Z=Z^{*}\in{\mathcal{Z}}_{S}({\mathcal{A}})$, then we get
$[A,S]_{*}=[\Phi(A),\Phi(S)]_{*}=0$ for all $S\in{\mathcal{A}}$. Hence
$AS=SA^{*}$ holds for all $S$, which means that
$A=A^{*}\in{\mathcal{Z}}_{S}({\mathcal{A}})$, that is,
${\mathcal{Z}}_{S}({\mathcal{A}})\subseteq\Phi({\mathcal{Z}}_{S}({\mathcal{A}}))$,
completing the proof. $\Box$
Lemma 2.4. For any $T,S\in{\mathcal{A}}$, there exists an element
$Z_{T,S}\in{\mathcal{Z}}_{S}({\mathcal{A}})$ depending on $T,S$ such that
$\Phi(T+S)=\Phi(T)+\Phi(S)+Z_{T,S}$.
Proof. For any $T,S,R\in{\mathcal{A}}$, we have
$\begin{array}[]{rl}&[\Phi(T+S)-\Phi(T)-\Phi(S),\Phi(R)]_{*}\\\
=&[\Phi(T+S),\Phi(R)]_{*}-[\Phi(T),\Phi(R)]_{*}-[\Phi(S),\Phi(R)]_{*}\\\
=&[T+S,R]_{*}-[T,R]_{*}-[S,R]_{*}=0.\end{array}$
By the surjectivity of $\Phi$ and the above equation, one sees that
$Z_{T,S}=\Phi(T+S)-\Phi(T)-\Phi(S)\in{\mathcal{Z}}_{S}({\mathcal{A}})$ holds
for all $T,S\in{\mathcal{A}}$. $\Box$
In the following, we will use the technique of Peirce decomposition. By the
assumption, we can take a symmetric nontrivial idempotent element $P$ in
$\mathcal{A}$. Set ${\mathcal{A}}_{11}=P{\mathcal{A}}P$,
${\mathcal{A}}_{12}=P{\mathcal{A}}(I-P)$,
${\mathcal{A}}_{21}=(I-P){\mathcal{A}}P$ and
${\mathcal{A}}_{22}=(I-P){\mathcal{A}}(I-P)$. Then
${\mathcal{A}}={\mathcal{A}}_{11}+{\mathcal{A}}_{12}+{\mathcal{A}}_{21}+{\mathcal{A}}_{22}$.
It is clear that ${\mathcal{A}}_{ij}^{*}={\mathcal{A}}_{ji}$, $i,j=1,2$. For
an element $S_{ij}\in{\mathcal{A}}$, we always mean that
$S_{ij}\in{\mathcal{A}}_{ij}$.
Lemma 2.5. $\Phi(P)^{*}=\Phi(P)$ and $\Phi(I-P)^{*}=\Phi(I-P)$. Moreover,
there exist elements $\alpha$, $\beta$, $\mu\in{\mathcal{C}}$ with
$\alpha\not=0$ such that $\Phi(P)=\alpha P+\mu I$ and
$\Phi(I-P)=\alpha(I-P)+\beta I$.
Proof. For any $A\in{\mathcal{A}}$, it is easy to check that
$[P,[P,[P,A]_{*}]_{*}]_{*}=[P,A]_{*}$. So we have
$[P,[P,[\Phi(P),\Phi(A)]_{*}]_{*}]_{*}=[\Phi(P),\Phi(A)]_{*}$. It follows from
the surjectivity of $\Phi$ that
$[P,[P,[\Phi(P),A]_{*}]_{*}]_{*}=[\Phi(P),A]_{*}\quad{\rm for\ all}\
A\in{\mathcal{A}}.$
Write $\Phi(P)=S_{11}+S_{12}+S_{21}+S_{22}$. Then the above equation becomes
$None$
$\begin{array}[]{rl}&PAS_{11}^{*}+PAS_{12}^{*}-PAS_{21}^{*}-PAS_{22}^{*}\\\
&-S_{11}AP-S_{12}AP+S_{21}AP+S_{22}AP\\\
=&S_{21}A+S_{22}A-AS_{21}^{*}-AS_{22}^{*}\end{array}$
for all $A\in{\mathcal{A}}$.
Taking $A=A_{12}$ in Eq.(2.2), we get $A_{12}S_{12}^{*}=S_{21}A_{12}=0$, that
is,
$PA(I-P)S_{12}^{*}=S_{21}PA(I-P)=0\quad{\rm for\ \ all}\ \ A\in{\mathcal{A}}.$
It follows from the primeness of $\mathcal{A}$ that $S_{12}^{*}=S_{21}=0$, and
so $S_{12}=S_{21}=0$.
Now let $A=A_{11}$ in Eq.(2.2), and we get $A_{11}S_{11}^{*}=S_{11}A_{11}$.
This implies that $S_{11}=S_{11}^{*}$ by taking $A_{11}=P$. So
$PAS_{11}=S_{11}AP$ holds for all $A\in{\mathcal{A}}$. It follows from Lemma
2.2 that $S_{11}=\lambda P$ for some $\lambda\in{\mathcal{C}}$.
Similarly, taking $A=A_{22}$ in Eq.(2.2), and one can obtain
$S_{22}=S_{22}^{*}$ and $S_{22}=\mu(I-P)$ for some $\mu\in{\mathcal{C}}$.
Hence $\Phi(P)=S_{11}+S_{22}=S_{11}^{*}+S_{22}^{*}=\Phi(P)^{*}$ and
$\Phi(P)=S_{11}+S_{22}=\lambda P+\mu(I-P)=\alpha P+\mu I,$
where $\alpha=\lambda-\mu\in{\mathcal{C}}$. It is obvious that $\mu
I\in{\mathcal{C}}$. Note that $\Phi(I)\in{\mathcal{Z}}_{S}({\mathcal{A}})$,
$\Phi(I)-\Phi(P)-\Phi(I-P)\in{\mathcal{Z}}_{S}({\mathcal{A}})$ and
${\mathcal{Z}}_{S}({\mathcal{A}})\subseteq{\mathcal{C}}$. So
$\Phi(I-P)^{*}=\Phi(I-P)$ and there exists an element $\beta\in{\mathcal{C}}$
such that
$\Phi(I-P)=\alpha(I-P)+\beta I.$
Finally, we still need to prove that $\alpha\not=0$. On the contrary, if
$\alpha=0$, then $\Phi(P)=\mu I\in{\mathcal{C}}$. Since
$\Phi(P)\in{\mathcal{A}}$, it follows that
$\Phi(P)\in{\mathcal{Z}}_{S}({\mathcal{A}})$. By Lemma 2.2, we get
$P\in{\mathcal{Z}}_{S}{(\mathcal{A})}$, which is impossible as $\mathcal{A}$
is prime. The proof is finished. $\Box$
Note that $\mathcal{C}$ is a field as $\mathcal{A}$ is prime ([3, Theorem
A.6]). So $\alpha\in{\mathcal{C}}$ is invertible. In the following, let
$\lambda=\alpha^{-1}\in{\mathcal{C}}$. Also note that the unit $1$ of
${\mathcal{C}}$ is the same to the unit $I$ of ${\mathcal{A}}$
Lemma 2.6. For any $A_{ij}\in{\mathcal{A}}_{ij}$, we have
$\Phi(A_{ij})=\lambda A_{ij}$, $1\leq i\not=j\leq 2$.
Proof. Take any $A_{12}\in{\mathcal{A}}_{12}$ and let
$\Phi(A_{12})=S_{11}+S_{12}+S_{21}+S_{22}$. Since
$[\Phi(P),\Phi(A_{12})]_{*}=[P,A_{12}]_{*}=A_{12}$, by Lemma 2.5, we get
$\alpha S_{12}-\alpha S_{21}=A_{12}$, which implies that $S_{21}=0$ and
$S_{12}=\alpha^{-1}A_{12}=\lambda A_{12}$.
For any $B\in{\mathcal{A}}$, by the surjectivity of $\Phi$, there exists an
element $T=T_{11}+T_{12}+T_{21}+T_{22}\in{\mathcal{A}}$ such that $\Phi(T)=B$.
Since $[B,\Phi(A_{12})]_{*}=[\Phi(T),\Phi(A_{12})]_{*}=[T,A_{12}]_{*}$, we
have
$None$ $\begin{array}[]{rl}&BS_{11}+B(\lambda
A_{12})+BS_{22}-S_{11}B^{*}-\lambda A_{12}B^{*}-S_{22}B^{*}\\\
=&T_{11}A_{12}+T_{21}A_{12}-A_{12}T_{12}^{*}-A_{12}T_{22}^{*}.\end{array}$
Multiplying by $P$ from the right in Eq.(2.3), one gets
$BS_{11}-S_{11}B^{*}P-\lambda A_{12}B^{*}P-S_{22}B^{*}P=-A_{12}T_{12}^{*}.$
Replacing $B$ by $(I-P)BP$ in the above equation, we have
$(I-P)BPS_{11}=-A_{12}T_{12}^{*}$, which implies that $(I-P)BPS_{11}=0$ for
all $B\in{\mathcal{A}}$. It follows from the primeness of ${\mathcal{A}}$ that
$S_{11}=0$.
Similarly, replacing $B$ by $PB(I-P)$ and multiplying by $I-P$ from the left
in Eq.(2.3), one can show that $S_{22}=0$. Hence we obtain
$\Phi(A_{12})=S_{12}=\lambda A_{12}$.
The proof of $\Phi(A_{21})=\lambda A_{21}$ is similar, and we omit it here.
$\Box$
Lemma 2.7. For any $A_{ii}\in{\mathcal{A}}_{ii}$, we have $\Phi(A_{ii})=\alpha
A_{ii}$, $i=1,2$.
Proof. Still, we only need to prove that the lemma is true for $A_{11}$.
Take any $A_{11}\in{\mathcal{A}}_{11}$ and let
$\Phi(A_{11})=S_{11}+S_{12}+S_{21}+S_{22}$. Since
$[\Phi(P),\Phi(A_{11})]_{*}=[P,A_{11}]_{*}=0$, by Lemma 2.5, we have
$\alpha S_{11}+\alpha S_{12}-S_{11}(\alpha P)-S_{21}(\alpha P)=0.$
This implies $\alpha S_{12}=0$, an so $S_{12}=0$. On the other hand, since
$[\Phi(I-P),\Phi(A_{11})]_{*}=[I-P,A_{11}]_{*}=0$, by Lemma 2.5 again, one
gets $\alpha S_{21}=0$, which implies that $S_{21}=0$.
For any $B_{12}\in{\mathcal{A}}_{12}$, by Lemma 2.6, we have $[\lambda
B_{12},\Phi(A_{11})]_{*}=[\Phi(B_{12}),\Phi(A_{11})]_{*}=[B_{12},A_{11}]_{*}=0,$
that is,
$\lambda B_{12}S_{22}-S_{11}(\lambda B_{12})^{*}-S_{22}(\lambda
B_{12})^{*}=0.$
Note that $\lambda B_{12}\in{\mathcal{A}}_{12}$. It follows that $\lambda
B_{12}S_{22}=0$, which implies $B_{12}S_{22}=0$. That is, $PB(I-P)S_{22}=0$
for all $B\in{\mathcal{A}}$. As ${\mathcal{A}}$ is prime, we get $S_{22}=0$.
For any $B_{21}\in{\mathcal{A}}_{21}$, by Lemma 2.6, we have $[\lambda
B_{21},\Phi(A_{11})]_{*}=[\Phi(B_{21}),\Phi(A_{11})]_{*}=[B_{21},A_{11}]_{*}$,
that is,
$\lambda B_{21}S_{11}-S_{11}(\lambda
B_{21})_{*}=B_{21}A_{11}-A_{11}B_{21}^{*}.$
This forces $B_{21}(\lambda S_{11}-A_{11})=(\lambda
S_{11}-A_{11})B_{21}^{*}=0$. So we get $(I-P)BP(\lambda S_{11}-A_{11})=0$ for
each $B\in{\mathcal{A}}$. It follows from the primeness of ${\mathcal{A}}$
that $\lambda S_{11}=A_{11}$. Hence
$\Phi(A_{11})=S_{11}=\lambda^{-1}A_{11}=\alpha A_{11}$, completing the proof.
$\Box$
Lemma 2.8. $\alpha=\lambda$, and hence $\alpha^{2}=1$ and $\alpha=\pm 1$.
Proof. For any $A_{12}\in{\mathcal{A}}_{12}$ and
$A_{21}\in{\mathcal{A}}_{21}$, by the definition of $\Phi$ and Lemma 2.6, we
have
$A_{12}A_{21}=[A_{12},A_{21}]_{*}=[\Phi(A_{12}),\Phi(A_{21})]_{*}=[\lambda
A_{12},\lambda A_{21}]_{*}=(\lambda A_{12})(\lambda
A_{21})=\lambda^{2}A_{12}A_{21}.$
It follows that $(\lambda^{2}-1)A_{12}A_{21}=0$. First fix $A_{12}$. Then the
equation becomes $(\lambda^{2}-1)A_{12}AP=0$ for all $A\in{\mathcal{A}}$.
Assume that $\lambda^{2}-1\not=0$. Since $\mathcal{C}$ is a field, we get that
$\lambda^{2}-1$ is invertible. So we have $A_{12}AP=0$ for all
$A\in{\mathcal{A}}$. Since $\mathcal{A}$ is prime, it follows that $A_{12}=0$,
that is, $PA(I-P)=0$ for all $A\in{\mathcal{A}}$. This implies that either
$P=0$ or $P=I$, a contradiction. So $\lambda^{2}=1$ and $\lambda=\alpha$.
Since $\mathcal{C}$ is a field, we see that $\alpha=1$ or $-1$, completing the
proof. $\Box$
Lemma 2.9. There exists a map
$f:{\mathcal{A}}\rightarrow{\mathcal{Z}}({\mathcal{A}})$ such that
$\Phi(A)=A+f(A)$ for all $A\in{\mathcal{A}}$ or $\Phi(A)=-A+f(A)$ for all
$A\in{\mathcal{A}}$.
Proof. By Lemmas 2.6-2.8, we have proved that $\Phi(A_{ij})=A_{ij}$ for all
$A_{ij}\in{\mathcal{A}}_{ij}$ or $\Phi(A_{ij})=-A_{ij}$ for all
$A_{ij}\in{\mathcal{A}}_{ij}$ ($i,j=1,2$). Now, for any
$A=A_{11}+A_{12}+A_{21}+A_{22}\in{\mathcal{A}}$, by Lemma 2.4, there exists
some element $Z_{A}\in{\mathcal{Z}}_{S}({\mathcal{A}})$ such that
$\begin{array}[]{rl}&\Phi(A)-(\Phi(A_{11})+\Phi(A_{12})+\Phi(A_{21})+\Phi(A_{22}))\\\
=&\Phi(A_{11}+A_{12}+A_{21}+A_{22})-\Phi(A_{11})-\Phi(A_{12})-\Phi(A_{21})-\Phi(A_{22})=Z_{A}.\end{array}$
Define a map $f:{\mathcal{A}}\rightarrow{\mathcal{Z}}_{S}({\mathcal{A}})$ by
$f(A)=Z_{A}$ for each $A\in{\mathcal{A}}$. Then $\Phi(A)=A+f(A)$ for each $A$,
or $\Phi(A)=-A+f(A)$ for each $A$. $\Box$
By applying Theorem 2.1, we give a characterization of SSCP maps on prime
*-algebras with the second kind involution.
Theorem 2.10. Let ${\mathcal{A}}$ be a unital prime *-algebra over a field
$\mathbb{F}$ with a nontrivial symmetric idempotent $P$. Assume that the
involution $*$ is of the second kind and
$\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a surjective map. Then $\Phi$
is strong skew commutativity preserving (that is,
$\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^{*}=AB-BA^{*}$ for all $A,B\in{\mathcal{A}}$)
if and only if $\Phi(A)=A$ for all $A\in{\mathcal{A}}$ or $\Phi(A)=-A$ for all
$A\in{\mathcal{A}}$.
Proof. Obviously, the “if” part is true. For the “only if” part, by Theorem
2.1, there exists a map
$f:{\mathcal{A}}\rightarrow{\mathcal{Z}}_{S}({\mathcal{A}})$ such that
$\Phi(A)=A+f(A)$ for all $A\in{\mathcal{A}}$, or $\Phi(A)=-A+f(A)$ for all
$A\in{\mathcal{A}}$. So, to complete the proof of the theorem, we only need to
prove $f\equiv 0$.
In fact, $\Phi$ is SSCP implies that
$None$ $f(B)(A-A^{*})=0\quad{\rm for\ \ all}\quad A,B\in{\mathcal{A}}.$
Since $*$ is of the second kind, there exists a nonzero
$\epsilon\in{\mathbb{F}}$ such that $(\epsilon I)^{*}=-\epsilon I$. Thus, let
$A=\epsilon I$ in Eq.(2.4), we get $2\epsilon f(B)=0$, which implies $f(B)=0$
for each $B\in{\mathcal{A}}$ as $\mathbb{F}$ is a field. The proof is
finished. $\Box$
Recall that a von Neumann algebra ${\mathcal{M}}$ is called a factor if its
center is trivial (i.e., ${\mathcal{Z}}({\mathcal{M}})={\mathbb{C}}I$). Note
that factor von Neumann algebras are prime and $*$ is of the second kind. So,
as an application of Theorem 2.10 to the factor von Neumann algebras case, we
improve the main results of [6] immediately.
Theorem 2.11. Let ${\mathcal{A}}$ be a factor von Neumann algebra. Assume that
$\Phi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a surjective map. Then $\Phi$
is strong skew commutativity preserving if and only if $\Phi(A)=\alpha A$ for
all $A\in{\mathcal{A}}$, where $\alpha\in\\{1,-1\\}$.
## 3\. SSCP maps on von Neumann algebras
In this section, we will discuss the strong skew commutativity preserving maps
on von Neumann algebras. The following is the main result of this section.
Theorem 3.1. Let ${\mathcal{M}}$ be a von Neumann algebra without central
summands of type $I_{1}$. Assume that
$\Phi:{\mathcal{M}}\rightarrow{\mathcal{M}}$ is a surjective map. Then $\Phi$
is strong skew commutativity preserving if and only if there exists an element
$Z\in{\mathcal{Z}}_{S}({\mathcal{M}})$ with $Z^{2}=I$ such that $\Phi(A)=ZA$
for all $A\in{\mathcal{A}}$.
Obviously, Theorem 2.11 is also an immediate consequence of the above result.
We remark that the methods used in the proof of Theorem 2.1 are not valid here
since the von Neumann algebras in Theorem 3.1 may not be prime. In order to
overcome the difficulties caused by the absence of primeness, we need some
deep results coming from the theory of von Neumann algebras.
Let $\mathcal{M}$ be a von Neumann algebra and $A\in{\mathcal{M}}$. Recall
that the central carrier of $A$, denoted by $\overline{A}$, is the
intersection of all central projections $P$ such that $PA=A$. If $A$ is self-
adjoint, then the core of $A$, denoted by $\underline{A}$, is
sup$\\{S\in{\mathcal{Z}}({\mathcal{M}}):S=S^{*},S\leq A\\}$. Particularly, if
$A=P$ is a projection, it is clear that $\underline{P}$ is the largest central
projection $\leq P$. A projection $P$ is said to be core-free if
$\underline{P}=0$ [8]. It is easy to see that $\underline{P}=0$ if and only if
$\overline{I-P}=I$.
The following lemmas are useful for our purpose, where Lemma 3.2 and Lemma 3.3
are proved in [8].
Lemma 3.2. ([8]) Let $\mathcal{M}$ be a von Neumann algebra without central
summands of type $I_{1}$. Then each nonzero central projection
$C\in{\mathcal{M}}$ is the carrier of a core-free projection in $\mathcal{M}$.
Particularly, there exists a nonzero core-free projection $P\in{\mathcal{M}}$
with $\overline{P}=I$.
Lemma 3.3. ([8]) Let $\mathcal{M}$ be a von Neumann algebra. For projections
$P,Q\in{\mathcal{M}}$, if $\overline{P}=\overline{Q}\not=0$ and $P+Q=I$, then
$T\in{\mathcal{M}}$ commutes with $PXQ$ and $QXP$ for all $X\in{\mathcal{M}}$
implies that $T\in{\mathcal{Z}}({\mathcal{M}})$.
Lemma 3.4. Let $\mathcal{M}$ be a von Neumann algebra. Assume that
$P\in\mathcal{M}$ is a projection satisfying $\underline{P}=0$ and
$\overline{P}=I$. Then, for any $Z\in{\mathcal{Z}}({\mathcal{M}})$,
$ZP{\mathcal{M}}(I-P)=\\{0\\}$ implies $Z=0$.
Proof. Assume that ${\mathcal{M}}\subseteq{\mathcal{B}}(H)$, where $H$ is a
Hilbert space, and assume that $Z\in{\mathcal{Z}}({\mathcal{M}})$ with
$Z\not=0$ such that $ZP{\mathcal{M}}(I-P)=\\{0\\}$. Let $Q$ be the projection
onto the closure of the range of $Z$. It is clear that
$Q\in{\mathcal{Z}}({\mathcal{M}})$. So we may write
${\mathcal{M}}=Q{\mathcal{M}}Q\oplus(I-Q){\mathcal{M}}(I-Q)={\mathcal{M}}_{1}\oplus{\mathcal{M}}_{2}$.
Thus, according to the space decomposition $H=QH\oplus(I-Q)H$, we have
$A=\left(\begin{array}[]{rl}A_{1}&0\\\ 0&A_{2}\end{array}\right),\quad
Z=\left(\begin{array}[]{rl}Z_{1}&0\\\ 0&0\end{array}\right),\quad
P=\left(\begin{array}[]{rl}P_{1}&0\\\ 0&P_{2}\end{array}\right),$
where $A\in{\mathcal{M}}$ is arbitrary, $A_{i},P_{i}\in{\mathcal{M}}_{i}$ with
$P_{i}=P_{i}^{*}=P_{i}^{2}$ ($i\in\\{1,2\\}$) and
$Z_{1}\in{\mathcal{Z}}({\mathcal{M}}_{1})$ is injective with dense range. It
follows that
$ZPA(I-P)=\left(\begin{array}[]{cc}Z_{1}P_{1}A_{1}(I_{1}-P_{1})&0\\\
0&0\end{array}\right).$
By the assumption, we get $Z_{1}P_{1}A_{1}(I_{1}-P_{1})=0$, which implies that
$P_{1}A_{1}(I_{1}-P_{1})=0$ for all $A_{1}\in{\mathcal{M}}_{1}$ as $Z_{1}$ is
injective. So $P_{1}A_{1}=P_{1}A_{1}P_{1}$ for each
$A_{1}\in{\mathcal{M}}_{1}$. Thus, by the following claim, we have
$P_{1}\in{\mathcal{Z}}({\mathcal{M}}_{1})$.
Claim. Let $\mathcal{N}$ be a von Neumann algebra. Assume that
$P\in{\mathcal{N}}$ is a projection satisfying $P{\mathcal{N}}(I-P)=\\{0\\}$.
Then $P\in{\mathcal{Z}}({\mathcal{N}})$.
In fact, for such $P$, write $P_{1}=P$ and $P_{2}=I-P$. Denote
${\mathcal{N}}_{ij}=P_{i}{\mathcal{N}}P_{j}$, $i,j\in\\{1,2\\}$. Then
${\mathcal{N}}={\mathcal{N}}_{11}+{\mathcal{N}}_{12}+{\mathcal{N}}_{21}+{\mathcal{N}}_{22}$.
For any $A\in{\mathcal{N}}$, we have $PA=PAP+PA(I-P)$. It follows from the
assumption $P{\mathcal{N}}(I-P)=\\{0\\}$ that $PA(I-P)=0$ for any
$A\in{\mathcal{N}}$. Since $A$ is arbitrary, it is true that $PA^{*}(I-P)=0$
holds for any $A\in{\mathcal{N}}$, which implies that $(I-P)AP=0$ for any
$A\in{\mathcal{N}}$. Thus, for any $A\in{\mathcal{N}}$, we must have
$A=PAP+(I-P)A(I-P)$. Now it is clear that $PA=AP$ for each $A$, that is, the
claim is true.
Let us go back to the proof of the lemma and let
$Q_{0}=\left(\begin{array}[]{rl}P_{1}&0\\\ 0&0\end{array}\right)$. Obviously,
$Q_{0}$ is a central projection with $Q_{0}\leq P$. Since $\underline{P}=0$,
we have $Q_{0}=0$, and so $P_{1}=0$. This yields $QP=0$, which implies
$I-Q\geq\overline{P}=I$, a contradiction. Hence $Z=0$ and the proof is
completed. $\Box$
Now we are in a position to give our proof of Theorem 3.1.
Proof of Theorem 3.1. Still, we only need to prove the “only if” part.
By the same argument as that of Lemmas 2.3-2.4, one can obtain that
$None$
$\Phi({\mathcal{Z}}_{S}({\mathcal{M}}))={\mathcal{Z}}_{S}({\mathcal{M}})$
and
$None$ $\Phi(T+S)=\Phi(T)+\Phi(S)+Z_{T,S}\quad{\rm for\ \ all}\quad
T,S\in{\mathcal{M}},$
where $Z_{T,S}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ depending on $T,S$.
By Lemma 3.2, there is a non-central core-free projection $P$ with central
carrier $I$. For such a $P$, by the definitions of core and central carrier,
$I-P$ is also core-free with $\overline{I-P}=I$. Denote
${\mathcal{M}}_{ij}=P_{i}{\mathcal{M}}P_{j}$ ($i,j\in\\{1,2\\}$), where
$P_{1}=P$ and $P_{2}=I-P$. Then
${\mathcal{M}}={\mathcal{M}}_{11}+{\mathcal{M}}_{12}+{\mathcal{M}}_{21}+{\mathcal{M}}_{22}$.
In all that follows, when writing $S_{ij}$, it always indicates
$S_{ij}\in{\mathcal{M}}_{ij}$.
We will complete the proof by several steps.
Step 1. $\Phi(P)=\Phi(P)^{*}$.
For the identity operator $I$, by Eq.(3.1), there exists some
$Z\in{\mathcal{Z}}_{S}({\mathcal{M}})$ such that $\Phi(Z)=I$. Thus
$0=[P,Z]_{*}=[\Phi(P),\Phi(Z)]_{*}=\Phi(P)-\Phi(P)^{*},$
which implies $\Phi(P)=\Phi(P)^{*}$.
Step 2. There exist elements
$Z_{1},Z_{2},Z_{3}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ with $Z_{1}\not=0$ such
that $\Phi(P)=Z_{1}P+Z_{2}$ and $\Phi(I-P)=Z_{1}(I-P)+Z_{3}$.
Note that $[P,[P,[P,A]_{*}]_{*}]_{*}=[P,A]_{*}$ holds for any
$A\in{\mathcal{M}}$. Thus for every $A$, we have
$[P,[P,[\Phi(P),\Phi(A)]_{*}]_{*}]_{*}=[\Phi(P),\Phi(A)]_{*}.$
It follows from the surjectivity of $\Phi$ that
$None$ $[P,[P,[\Phi(P),A]_{*}]_{*}]_{*}=[\Phi(P),A]_{*}\quad{\rm for\ all}\
A\in{\mathcal{M}}.$
Write $[\Phi(P),A]_{*}=B$. Then Eq.(3.3) becomes
$PB-2PBP+BP=B.$
Multiplying by $P$ and $I-P$ from both sides in the above equation,
respectively, one gets $PBP=0$ and $(I-P)B(I-P)=0$. Therefore
$None$ $P(\Phi(P)A-A\Phi(P)^{*})P=0\quad{\rm
and}\quad(I-P)(\Phi(P)A-A\Phi(P)^{*})(I-P)=0$
hold for all $A\in{\mathcal{M}}$.
Write $\Phi(P)=S_{11}+S_{12}+S_{21}+S_{22}$. Replacing $A$ by $PT(I-P)$ for
any $T\in{\mathcal{M}}$ in Eq.(3.4), we get
$PT(I-P)S_{12}^{*}=0\quad{\rm and}\quad S_{21}PT(I-P)=0,$
which, together with $\Phi(P)=\Phi(P)^{*}$, implies that
$None$ $PT(I-P)S_{21}=S_{21}PT(I-P)=0$
holds for all $T\in{\mathcal{M}}$. It is obvious that
$S_{21}(I-P)TP=(I-P)TPS_{21}=0$. Hence, by Lemma 3.3, we see that
$S_{21}\in{\mathcal{Z}}({\mathcal{M}})$, which forces $S_{21}=0$.
Similarly, replacing $A$ by $(I-P)TP$ for any $T\in{\mathcal{M}}$ in Eq.(3.4)
and applying Step 1, it is easily checked $S_{12}=0$.
Now taking $A=P$ in Eq.(3.3), and by Step 1, one gets
$S_{11}PTP=PTPS_{11}^{*}=PTPS_{11},$
which means $S_{11}\in P{\mathcal{Z}}_{S}({\mathcal{M}})$. Symmetrically, by
taking $A=I-P$ in Eq.(3.3), we obtain
$S_{22}\in(I-P){\mathcal{Z}}_{S}({\mathcal{M}})$. Write $S_{11}=Z_{11}P$ and
$S_{22}=Z_{22}(I-P)$, where
$Z_{11},Z_{22}\in{\mathcal{Z}}_{S}({\mathcal{M}})$. Then
$\Phi(P)=S_{11}+S_{22}=Z_{11}P+Z_{22}(I-P)=(Z_{11}-Z_{22})P+Z_{22}.$
Note that $\Phi(I)-\Phi(I-P)-\Phi(P)\in{\mathcal{Z}}_{S}({\mathcal{M}})$ by
Eqs.(3.1)-(3.2). So there exists some
$Z_{33}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ such that
$\begin{array}[]{rl}\Phi(I-P)=&Z_{33}-\Phi(P)=Z_{33}-Z_{22}-(Z_{11}-Z_{22})P\\\
=&(Z_{11}-Z_{22})(I-P)+(Z_{33}-Z_{11}).\end{array}$
Let $Z_{1}=Z_{11}-Z_{22}$, $Z_{2}=Z_{22}$ and $Z_{3}=Z_{33}-Z_{11}$. Thus
$\Phi(P)=Z_{1}P+Z_{2}$ and $\Phi(I-P)=Z_{1}(I-P)+Z_{3}$.
Finally, we still need to prove that $Z_{1}\not=0$. On the contrary, if
$Z_{1}=0$, then $\Phi(P)=Z_{2}\in{\mathcal{Z}}({\mathcal{M}})$. By Eq.(3.1),
we get $P\in{\mathcal{Z}}_{S}{(\mathcal{M})}$, which is impossible.
Step 3. If $\Phi(T)=P$ and $\Phi(S)=I-P$, where
$T=T_{11}+T_{12}+T_{21}+T_{22}\in{\mathcal{M}}$ and
$S=S_{11}+S_{12}+S_{21}+S_{22}\in{\mathcal{M}}$, then $T_{12}=T_{21}=0$ and
$S_{12}=S_{21}=0$.
In fact, by the equation $[P,T]_{*}=[\Phi(P),\Phi(T)]_{*}=[\Phi(P),P]_{*}$ and
Step 2, one can get $T_{12}=T_{21}=0$; by the equation
$[I-P,S]_{*}=[\Phi(I-P),\Phi(S)]_{*}=[\Phi(I-P),I-P]_{*}$ and Step 2, one can
get $S_{12}=S_{21}=0$.
Step 4. For any $A_{ij}\in{\mathcal{M}}_{ij}$, we have
$\Phi(A_{ij})\in{\mathcal{M}}_{ij}$, $1\leq i\not=j\leq 2$. Moreover,
$Z_{1}P\Phi(A_{12})(I-P)=A_{12}$ holds for all $A_{12}\in{\mathcal{M}}_{12}$
and $Z_{1}(I-P)\Phi(A_{21})P=A_{21}$ holds for all
$A_{21}\in{\mathcal{M}}_{21}$.
We only check the assertion for $A_{12}$, and the case of $A_{21}$ is
similarly dealt with.
For any $A_{12}$, write $\Phi(A_{12})=S_{11}+S_{12}+S_{21}+S_{22}$. By Step 3,
there exists some $T=T_{11}+T_{22}\in{\mathcal{M}}$ such that $\Phi(T)=P$.
Then $[P,\Phi(A_{12})]_{*}=[T,A_{12}]_{*}$, that is,
$S_{12}-S_{21}=T_{11}A_{12}-A_{12}T_{22}^{*}.$
Multiplying by $I-P$ and $P$ from the left side and the right side
respectively in the above equation, one gets
$None$ $S_{21}=(I-P)\Phi(A_{12})P=0.$
On the other hand, since $[\Phi(A_{12}),P]_{*}=[A_{12},T]_{*}$, by Eq.(3.6),
we have $S_{11}-S_{11}^{*}=A_{12}T_{22}-T_{22}A_{12}^{*}.$ Multiplying by $P$
from both sides of the equation, it follows that
$None$ $S_{11}=S_{11}^{*}.$
Similarly, by using the equation
$[\Phi(A_{12}),I-P]_{*}=[\Phi(A_{12}),\Phi(S)]_{*}=[A_{12},S]_{*}$ and Step 3,
it is easily checked that
$None$ $S_{22}=S_{22}^{*}.$
Now, for any $X\in{\mathcal{M}}$, by the surjectivity of $\Phi$, there exists
an element $R=R_{11}+R_{12}+R_{21}+R_{22}\in{\mathcal{M}}$ such that
$\Phi(R)=X$. Since
$[X,\Phi(A_{12})]_{*}=[\Phi(R),\Phi(A_{12})]_{*}=[R,A_{12}]_{*}$, applying
Eq.(3.6), we get
$None$
$\begin{array}[]{rl}&XS_{11}+XS_{12}+XS_{22}-S_{11}X^{*}-S_{12}X^{*}-S_{22}X^{*}\\\
=&R_{11}A_{12}+R_{21}A_{12}-A_{12}R_{12}^{*}-A_{12}R_{22}^{*}.\end{array}$
Replacing $X$ by $PB(I-P)+(I-P)BP$ for any $B\in{\mathcal{M}}$, and
multiplying by $I-P$ and $P$ from the left and the right respectively in
Eq.(3.9), one obtains
$None$ $(I-P)BPS_{11}=S_{22}(I-P)B^{*}P\quad\mbox{\rm holds for all }\quad
B\in{\mathcal{M}};$
Replacing $X$ by $(I-P)BP$ for any $B\in{\mathcal{M}}$, and multiplying by
$I-P$ and $P$ from the left and the right respectively in Eq.(3.9), one gets
$None$ $(I-P)BPS_{11}=0\quad{\rm holds\ for\ all}\quad B\in{\mathcal{M}}.$
Equivalently,
$None$ $(I-P)B^{*}PS_{11}=0\quad{\rm holds\ for\ all}\quad B\in{\mathcal{M}}.$
By Eqs.(3.7) and (3.12), we have
$None$ $S_{11}PB(I-P)=S_{11}^{*}PB(I-P)=((I-P)B^{*}PS_{11})^{*}=0.$
Moreover, it is obvious that $PB(I-P)S_{11}=S_{11}(I-P)BP=0$. This, together
with Eqs.(3.11), (3.13) and Lemma 3.3, yields
$S_{11}\in{\mathcal{Z}}_{S}({\mathcal{M}})$. Hence
$None$ $S_{11}=0.$
Combining Eq.(3.10) with Eq.(3.11), we get
$None$ $S_{22}(I-P)B^{*}P=0\quad{\rm for\ \ all}\quad B\in{\mathcal{M}}.$
Equivalently,
$S_{22}(I-P)BP=0\quad{\rm for\ \ all}\quad B\in{\mathcal{M}}.$
It follows from Eqs.(3.8) and (3.15) that
$PB(I-P)S_{22}=PB(I-P)S_{22}^{*}=(S_{22}(I-P)B^{*}P)^{*}=0.$
Thus we have
$\begin{array}[]{rl}&PB(I-P)S_{22}=S_{22}PB(I-P)\\\
=&S_{22}(I-P)BP=(I-P)BP(I-P)S_{22}=0,\end{array}$
which implies that $S_{22}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ by Lemma 3.3,
and so
$None$ $S_{22}=0.$
Combining Eqs.(3.6), (3.14) and (3.16), we see that
$\Phi(A_{12})\in{\mathcal{M}}_{12}$, as desired.
Finally, since $[\Phi(P),\Phi(A_{12})]_{*}=[P,A_{12}]_{*}$, by Step 2, it is
easy to check that $A_{12}=Z_{1}S_{12}=Z_{1}P\Phi(A_{12})(I-P)$.
Step 5. For any $A_{ii}\in{\mathcal{M}}_{ii}$, we have
$\Phi(A_{ii})=Z_{1}A_{ii}$, $i=1,2$.
Still, we only prove that $\Phi(A_{11})=Z_{1}A_{11}$ holds for all
$A_{11}\in{\mathcal{A}}_{11}$. The case of $i=2$ is checked similarly.
Take any $A_{11}\in{\mathcal{M}}_{11}$ and any $B\in{\mathcal{M}}$. Write
$\Phi(A_{11})=S_{11}+S_{12}+S_{21}+S_{22}$. On the one hand, since
$0=[PB(I-P),A_{11}]_{*}=[\Phi(PB(I-P)),\Phi(A_{11})]_{*}$, by Step 4, we have
$\begin{array}[]{rl}&P\Phi(PB(I-P))(I-P)S_{21}+P\Phi(PB(I-P))(I-P)S_{22}\\\
&-S_{12}(I-P)\Phi(PB(I-P))^{*}P-S_{22}(I-P)\Phi(PB(I-P))^{*}P=0,\end{array}$
which implies that
$P\Phi(PB(I-P))(I-P)S_{21}=S_{12}(I-P)\Phi(PB(I-P))^{*}P,$
$P\Phi(PB(I-P))(I-P)S_{22}=0$
and
$S_{22}(I-P)\Phi(PB(I-P))^{*}P=0.$
Multiplying by $Z_{1}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ in the above three
equations, and applying Step 4, one gets
$None$ $PB(I-P)(I-P)S_{21}=S_{12}(I-P)B^{*}P,$ $None$ $PB(I-P)S_{22}=0$
and
$None$ $S_{22}(I-P)B^{*}P=0$
hold for all $B\in{\mathcal{M}}$. Furthermore, Eq.(3.19) yields
$None$ $S_{22}(I-P)BP=0\quad{\rm for\ \ all}\quad B\in{\mathcal{M}}.$
Note that $(I-P)BP(I-P)S_{22}=0$ and $S_{22}PB(I-P)=0$. These, together with
Lemma 3.3, Eqs.(3.18) and (3.20), imply
$S_{22}\in{\mathcal{Z}}({\mathcal{M}})$. So we must have
$None$ $S_{22}=(I-P)\Phi(A_{11})(I-P)=0.$
On the other hand, by using the equation
$[A_{11},PB(I-P)]_{*}=[\Phi(A_{11}),\Phi(PB(I-P))]_{*},$ we get
$\begin{array}[]{rl}A_{11}PB(I-P)=&S_{11}P\Phi(PB(I-P))(I-P)\\\
&+S_{21}P\Phi(PB(I-P))(I-P)\\\ &-P\Phi(PB(I-P))(I-P)S_{12}^{*}.\end{array}$
It follows that
$A_{11}PB(I-P)=S_{11}P\Phi(PB(I-P))(I-P),$ $S_{21}P\Phi(PB(I-P))(I-P)=0$
and
$P\Phi(PB(I-P))(I-P)S_{12}^{*}=0.$
Multiplying by $Z_{1}$ in the above three equations leads to
$None$ $(Z_{1}A_{11}-S_{11})PB(I-P)=0,$ $None$ $S_{21}PB(I-P)=0$
and
$None$ $PB(I-P)S_{12}^{*}=0$
for all $B\in{\mathcal{M}}$. Eq.(3.24) implies that
$None$ $S_{12}(I-P)B^{*}P=0,$
and so
$None$ $S_{12}(I-P)BP=0\quad{\rm for\ \ all}\quad B\in{\mathcal{M}}.$
Eq.(3.17) and Eq.(3.25) together yield
$None$ $PB(I-P)S_{21}=0\quad{\rm for\ \ all}\quad B\in{\mathcal{M}}.$
It is obvious that $(I-P)BP(I-P)S_{21}=S_{21}(I-P)BP=0$. By Eqs.(3.23), (3.27)
we get $S_{21}\in{\mathcal{Z}}({\mathcal{M}})$ and hence
$None$ $S_{21}=(I-P)\Phi(A_{11})P=0.$
Now consider the equation
$[(I-P)BP,A_{11}]_{*}=[\Phi((I-P)BP),\Phi(A_{11})]_{*}.$ It follows from Step
4, Eqs.(3.21) and (3.28) that
$\begin{array}[]{rl}&(I-P)BPA_{11}-A_{11}PB^{*}(I-P)=(I-P)\Phi((I-P)BP)PS_{11}\\\
&+(I-P)\Phi((I-P)BP)PS_{12}-S_{11}P\Phi(PB(I-P))^{*}(I-P).\end{array}$
This implies that
$(I-P)BPA_{11}=(I-P)\Phi((I-P)BP)PS_{11}$
and
$(I-P)\Phi((I-P)BP)PS_{12}=0.$
Multiplying by $Z_{1}$ in the above two equations, we obtain
$None$ $(I-P)BP(Z_{1}A_{11}-S_{11})=0$
and
$None$ $(I-P)BPS_{12}=0.$
Since $S_{12}PB(I-P)=PB(I-P)PS_{12}=0$, by Eqs.(3.26) and (3.30), we get
$S_{12}\in{\mathcal{Z}}({\mathcal{M}})$, and so
$None$ $S_{12}=P\Phi(A_{11})(I-P)=0.$
By Eqs.(3.22) and (3.29), also noting that
$(Z_{1}A_{11}-S_{11})(I-P)BP=PB(I-P)(Z_{1}A_{11}-S_{11})=0$, we see that
$Z_{1}A_{11}-S_{11}\in{\mathcal{Z}}({\mathcal{M}})$, and hence
$None$ $Z_{1}A_{11}=S_{11}=P\Phi(A_{11})P.$
Now it is clear by Eqs.(3.21), (3.28), (3.31) and (3.32) that
$\Phi(A_{11})=Z_{1}A_{11}$.
Step 6. $Z_{1}^{2}=I$.
For any $A,B\in{\mathcal{M}}$, by the assumptions on $\Phi$ and Step 4, we
have
$\begin{array}[]{rl}PA(I-P)BP=&[PA(I-P),(I-P)BP]_{*}\\\
=&[\Phi(PA(I-P)),\Phi((I-P)BP)]_{*}\\\
=&P\Phi(PA(I-P))(I-P)\Phi((I-P)BP)P.\end{array}$
Multiplying by $Z_{1}^{2}$ in the above equation and applying Step 4 again,
one gets $Z^{2}_{1}PA(I-P)(I-P)BP=PA(I-P)(I-P)BP$. Let us fix $A$. Then the
equation becomes
$None$ $(Z^{2}_{1}-I)PA(I-P)(I-P)BP=0\quad{\rm for\ \ all}\quad
B\in{\mathcal{M}}.$
Similarly, by using the equation
$[(I-P)BP,PA(I-P)]_{*}=[\Phi((I-P)BP),\Phi(PA(I-P))]_{*}$, one obtains
$None$ $(I-P)BP(Z^{2}_{1}-I)PA(I-P)=(Z^{2}_{1}-I)(I-P)BPPA(I-P)=0\quad{\rm
for\ \ all}\quad B\in{\mathcal{M}}.$
By Lemma 3.3, Eqs.(3.33)-(3.34) imply that
$(Z^{2}_{1}-I)PA(I-P)\in{\mathcal{Z}}({\mathcal{M}})$. Hence
$(Z^{2}_{1}-I)PA(I-P)=0$ holds for all $A\in{\mathcal{M}}$. Now by Lemma 3.4,
one gets $Z^{2}_{1}=I$.
Step 7. $\Phi(A)=ZA$ for all $A\in{\mathcal{M}}$, where
$Z\in{\mathcal{Z}}_{S}({\mathcal{M}})$ with $Z^{2}=I$. Therefore, Theorem 3.1
is true.
By Steps 4-6, we have proved that $\Phi(A_{ij})=Z_{1}A_{ij}$ for all
$A_{ij}\in{\mathcal{M}}_{ij}$, where
$Z_{1}\in{\mathcal{Z}}_{S}({\mathcal{M}})$ with $Z_{1}^{2}=I$ ($i,j=1,2$).
Define $Z=Z_{1}$. Now, by a similar argument to that in the proof of Lemma
2.9, one can show that there exists a map
$f:{\mathcal{M}}\rightarrow{\mathcal{Z}}_{S}({\mathcal{M}})$ such that
$\Phi(A)=ZA+f(A)$ for all $A\in{\mathcal{M}}$.
To complete the proof of the theorem, we have to prove $f\equiv 0$. Indeed,
since $[\Phi(A),\Phi(B)]_{*}=[A,B]_{*}$ for every $A,B\in{\mathcal{M}}$, we
have $[ZA+f(A),ZB+f(B)]_{*}=[A,B]_{*}$. It follows that $Zf(B)(A-A^{*})=0$. As
$Z^{2}=I$, we get
$f(B)(A-A^{*})=0\quad{\rm for\ \ all}\quad A,B\in{\mathcal{M}}.$
Take $A=iI$ in the above equation, one gets $2if(B)=0$, and so $f(B)=0$ for
every $B\in{\mathcal{M}}$, completing the proof of Theorem 3.1. $\Box$
## References
* [1] K. I. Beidar, W. S. Martindale 3rd, A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc., New York-Basel-Hong Kong, 1996.
* [2] M. Bre$\check{s}$ar, M. Fso$\check{n}$er, On ring with involution equipped with some new product, Publ. Meth. Debrecen, 57 (2000), 121-134.
* [3] M. Bre$\check{s}$ar, M. A. Chebotar and W. S. Martindale III, Functional identities, Birkh$\ddot{\rm a}$user Basel, 2007.
* [4] M. A. Chebotar, Y. Fong, P.-H. Lee, On maps preserving zeros of the polynomial $xy-yx^{*}$, Lin. Alg. Appl., 408 (2005), 230-243.
* [5] J. Cui, J. Hou, Linear maps preserving elements annihilated by a polymnomial $XY-YX^{\dagger}$, Studia Math. 174(2) (2006), 183-199.
* [6] J. Cui, C. Park, Maps preserving strong skew Lie product on factor von Neumann algebras, Acta Math. Sci., 32B(2) (2012), 531-538.
* [7] R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I, Academic Press, New York, 1983, Vol. II, Academic Press, New York, 1986.
* [8] C. R. Miers, Lie isomorphisms of operator algebras, Pacific J. Math., 38 (1971), 717-735.
* [9] L. Molnár, A condition for a subspace of $B(H)$ to be an ideal, Lin. Alg. Appl. 235 (1996), 229-234.
* [10] P. Šemrl, Quadratic functionals and Jordan *-derivations, Studia Math. 97 (1991), 157-165.
* [11] P. Šemrl, On Jordan *-derivations and an application, Colloq. Math. 59 (1990) 241-251.
|
arxiv-papers
| 2012-04-09T09:49:50 |
2024-09-04T02:49:29.489386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaofei Qi, Jinchuan Hou",
"submitter": "Jinchuan Hou",
"url": "https://arxiv.org/abs/1204.1841"
}
|
1204.1899
|
# A non-overlapping domain decomposition method for incompressible Stokes
equations with continuous pressure
Jing Li Department of Mathematical Sciences, Kent State University, Kent, OH
44242, li@math.kent.edu, http://www.math.kent.edu/$\sim$li/. Xuemin Tu
Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence,
KS 66045-7594, xtu@math.ku.edu, http://www.math.ku.edu/$\sim$xtu/. This
author’s work was supported in part by National Science Foundation contract
DMS-1115759.
###### Abstract
A non-overlapping domain decomposition algorithm is proposed to solve the
linear system arising from mixed finite element approximation of
incompressible Stokes equations. A continuous finite element space for the
pressure is used. In the proposed algorithm, Lagrange multipliers are used to
enforce continuity of the velocity component across the subdomain domain
boundary. The continuity of the pressure component is enforced in the primal
form, i.e., neighboring subdomains share the same pressure degrees of freedom
on the subdomain interface and no Lagrange multipliers are needed. After
eliminating all velocity variables and the independent subdomain interior
parts of the pressures, a symmetric positive semi-definite linear system for
the subdomain boundary pressures and the Lagrange multipliers is formed and
solved by a preconditioned conjugate gradient method. A lumped preconditioner
is studied and the condition number bound of the preconditioned operator is
proved to be independent of the number of subdomains for fixed subdomain
problem size. Numerical experiments demonstrate the convergence rate of the
proposed algorithm.
keywords domain decomposition, incompressible Stokes, FETI-DP, BDDC
AMS 65F10, 65N30, 65N55
## 1 Introduction
Domain decomposition methods have been studied well for solving incompressible
Stokes equations and similar saddle-point problems; see, e.g., [16, 24, 20,
10, 3, 22, 11, 28, 29, 25]. In many of those work, special care need be taken
to deal with the divergence-free constraints across subdomain boundaries,
which often lead to large coarse level problems. The large coarse level
problem will be a bottleneck in large scale parallel computations, and
additional efforts in the algorithm are needed to reduce its impact, cf. [31,
32, 30, 17, 4, 15, 33]. Some recent progress has been made by Dohrmann and
Widlund [5, 6] for the almost incompressible elasticity, where the coarse
level space is built from discrete subdomain saddle-point harmonic extensions
of certain subdomain interface cut-off functions and its dimension is much
smaller than those in the previous studies. Kim and Lee [13, 14, 12, with
Park] studied both the FETI-DP and BDDC algorithms for incompressible Stokes
equations where a lumped preconditioner is used and reduction in the dimension
of the coarse level space is also achieved.
In most above mentioned applications and analysis of domain decomposition
methods for incompressible Stokes equations, the mixed finite element space
contains discontinuous pressures. Application of discontinuous pressures in
domain decomposition methods is natural. The decomposing of the pressure
components to independent subdomains can be handled conveniently and no
continuity of pressures across the subdomain boundary need be enforced.
However, a big class of mixed finite elements used for solving incompressible
Stokes and Navier-Stokes equations have continuous pressures, e.g., the well
known Taylor-Hood type [27]. There have been a variety of approaches using
continuous pressures in domain decomposition methods for solving
incompressible Stokes equations, e.g., by Goldfeld [9], by Šístek et. al.
[26], and by Benhassine and Bendali [1]. In their work, an indefinite system
of linear equations need be solved, either by a generalized minimal residual
method or simply by a conjugate gradient method. To the best of our knowledge,
no scalable convergence rate has been proved analytically for any of those
approaches using continuous pressures.
In this paper, we propose a non-overlapping domain decomposition algorithm for
solving incompressible Stokes equations with continuous pressure finite
element space. The scalability of its convergence rate is proved. In this
algorithm, the subdomain boundary velocities are dealt with in the same way as
in the FETI-DP method: a few for each subdomain are selected as the coarse
level primal variables, which are shared by neighboring subdomains; the others
are subdomain independent and Lagrange multipliers are used to enforce their
continuity. The subdomain boundary pressure degrees of freedom are all in the
primal form. They are shared by neighboring subdomains and no Lagrange
multipliers are needed for their continuity. After eliminating all velocity
variables and the independent subdomain interior parts of the pressures, the
system for the subdomain boundary pressures and the Lagrange multipliers is
shown to be symmetric positive semi-definite. A preconditioned conjugate
gradient method with a lumped preconditioner is studied. As strong condition
number bounds as for the scalar elliptic case are established. In the proposed
algorithm and in the estimate of its condition number bound, no additional
coarse level variables, except those necessary for solving scalar elliptic
problems, are required for incompressible Stokes problems. The resulting
coarse level problem is also symmetric positive definite.
To stay focused on the purpose of this paper, the discussion of the proposed
algorithm and its analysis are based on two-dimensional problems, even though
the same approach can be extended to the three-dimensional case without
substantial obstacles. It is also worth pointing out that the domain
decomposition algorithm and its analysis presented in this paper apply equally
well, with only minor modifications, to the case where discontinuous pressures
are used in the mixed finite element space.
The remainder of this paper is organized as follows. The finite element
discretization of the incompressible Stokes equation is introduced in Section
2. A domain decomposition approach is described in Section 3. The system for
the subdomain boundary pressures and the Lagrange multipliers is derived in
Section 4. Section 5 provides some techniques used in the condition number
bound estimate. In Section 6, a lumped preconditioner is proposed and a
scalable condition number bound of the preconditioned operator is established.
At the end, in Section 7, numerical results for solving a two-dimensional
incompressible Stokes problem are shown to demonstrate the convergence rate of
the proposed algorithm.
## 2 Finite element discretization
We consider solving the following incompressible Stokes problem on a bounded,
two-dimensional polygonal domain $\Omega$ with a Dirichlet boundary condition,
(1) $\left\\{\begin{array}[]{rcll}-\Delta{\bf u}+\nabla p&=&{\bf f},&\mbox{ in
}\Omega\mbox{ , }\\\ -\nabla\cdot{\bf u}&=&0,&\mbox{ in }\Omega\mbox{ , }\\\
{\bf u}&=&{\bf u}_{\partial\Omega},&\mbox{ on }\partial\Omega\mbox{ , }\\\
\end{array}\right.$
where the boundary data ${\bf u}_{\partial\Omega}$ satisfies the compatibility
condition $\int_{\partial\Omega}{\bf u}_{\partial\Omega}\cdot{\bf n}=0$. For
simplicity, we assume that ${\bf u}_{\partial\Omega}={\bf 0}$ without losing
any generality.
The weak solution of (1) is given by: find
${\mathbf{u}}\in\left(H^{1}_{0}(\Omega)\right)^{2}=\\{{\mathbf{v}}\in(H^{1}(\Omega))^{2}~{}\big{|}~{}{\mathbf{v}}={\mathbf{0}}\mbox{
on }\partial\Omega\\}$ and $p\in L^{2}(\Omega)$, such that
(2)
$\left\\{\begin{array}[]{lcll}a({\mathbf{u}},{\mathbf{v}})+b({\mathbf{v}},p)&=&({\mathbf{f}},{\mathbf{v}}),&\forall{\mathbf{v}}\in\left(H^{1}_{0}(\Omega)\right)^{2},\\\\[2.15277pt]
b({\mathbf{u}},q)&=&0,&\forall q\in L^{2}(\Omega)\mbox{ , }\\\
\end{array}\right.$
where
$a({\mathbf{u}},{\mathbf{v}})=\int_{\Omega}\nabla{\bf u}\cdot\nabla{\bf
v},\quad
b({\mathbf{u}},q)=-\int_{\Omega}(\nabla\cdot{\mathbf{u}})q,\quad({\mathbf{f}},{\mathbf{v}})=\int_{\Omega}{\mathbf{f}}\cdot{\mathbf{v}}.$
We note that the solution of (2) is not unique, with the pressure $p$
different up to an additive constant.
A modified Taylor-Hood mixed finite element is used in this paper to solve
(2). The domain $\Omega$ is triangulated into shape-regular elements of
characteristic size $h$. The pressure finite element space, $Q\subset
L^{2}(\Omega)$, is taken as the space of continuous piecewise linear functions
on the triangulation. The velocity finite element space,
${\mathbf{W}}\in\left(H^{1}_{0}(\Omega)\right)^{2}$, is formed by the
continuous piecewise linear functions on the finer triangulation obtained by
dividing each triangle into four subtriangles by connecting the middle points
of its edges. A demonstration of this mixed finite element on a triangulation
of a square domain is shown in Figure 1.
Figure 1: A modified Taylor-Hood mixed finite element
The finite element solution $({\mathbf{u}},p)\in{\mathbf{W}}\bigoplus Q$ of
(2) satisfies
(3) $\left[\begin{array}[]{cccc}A&B^{T}\\\ B&0\\\
\end{array}\right]\left[\begin{array}[]{c}{\bf u}\\\ p\\\
\end{array}\right]=\left[\begin{array}[]{l}{\bf f}\\\ 0\\\
\end{array}\right],$
where $A$, $B$, and ${\mathbf{f}}$ represent respectively the restrictions of
$a(\cdot,\cdot)$, $b(\cdot,\cdot)$ and $({\mathbf{f}},\cdot)$ to the finite-
dimensional spaces ${\mathbf{W}}$ and $Q$. We use the same notation in this
paper to represent both a finite element function and the vector of its nodal
values.
The coefficient matrix in (3) is rank deficient. $A$ is symmetric positive
definite. The kernel of $B^{T}$, denoted by $Ker(B^{T})$, is the space of all
constant pressures in $Q$. The range of $B$, denoted by $Im(B)$, is orthogonal
to $Ker(B^{T})$ and is the subspace of $Q$ consisting of all vectors with zero
average. The solution of (3) always exists and is uniquely determined when the
pressure is considered in the quotient space $Q/Ker(B^{T})$. In this paper,
when $q\in Q/Ker(B^{T})$, $q$ always has zero average. For a more general
right-hand side vector $({\bf f},~{}g)$ given in (3), the existence of its
solution requires that $g\in Im(B)$, i.e., $g$ has zero average.
The modified Taylor-Hood mixed finite element space ${\mathbf{W}}\times Q$, as
shown in Figure 1, is inf-sup stable in the sense that there exists a positive
constant $\beta$, independent of $h$, such that
(4)
$\sup_{{\mathbf{w}}\in{\mathbf{W}}}\frac{b({\mathbf{w}},q)}{|{\mathbf{w}}|_{H^{1}}}\geq\beta\|q\|_{L^{2}},\hskip
14.22636pt\forall q\in Q/Ker(B^{T}),$
cf. [2, Chapter III, §7], or equivalently in matrix/vector form,
(5) $\sup_{{\bf w}\in{\bf
W}}\frac{\left<q,B{\mathbf{w}}\right>^{2}}{\left<{\mathbf{w}},A{\mathbf{w}}\right>}\geq\beta^{2}\left<q,Zq\right>,\hskip
14.22636pt\forall q\in Q/Ker(B^{T}).$
Here, as always in this paper, $\left<\cdot,\cdot\right>$ represents the inner
product of two vectors. The matrix $Z$ represents the mass matrix defined on
the pressure finite element space $Q$, i.e., for any $q\in Q$,
$\|q\|_{L^{2}}^{2}=\left<q,Zq\right>$. It is easy to see, cf. [34, Lemma
B.31], that $Z$ is spectrally equivalent to $h^{2}I$ for two-dimensional
problems, where $I$ represents the identity matrix of the same dimension,
i.e., there exist positive constants $c$ and $C$, such that
(6) $ch^{2}I\leq Z\leq Ch^{2}I.$
Here, as in other places of this paper, $c$ and $C$ represent generic positive
constants which are independent of the mesh size $h$ and the subdomain
diameter $H$ (discussed in the following section).
## 3 A non-overlapping domain decomposition approach
The domain $\Omega$ is decomposed into $N$ non-overlapping polygonal
subdomains $\Omega_{i}$, $i=1,2,...,N$. Each subdomain is the union of a
bounded number of elements, with the diameter of the subdomain in the order of
$H$. The nodes on the interface of neighboring subdomains match across the
subdomain boundaries
$\Gamma={(\cup\partial\Omega_{i})}\backslash\partial\Omega$. $\Gamma$ is
composed of subdomain edges, which are regarded as open subsets of $\Gamma$,
and of the subdomain vertices, which are end points of edges.
The velocity and pressure finite element spaces ${\bf W}$ and $Q$ are
decomposed into
${\bf W}={\bf W}_{I}\bigoplus{\bf W}_{\Gamma},\quad Q=Q_{I}\bigoplus
Q_{\Gamma},$
where ${\bf W}_{I}$ and $Q_{I}$ are direct sums of independent subdomain
interior velocity spaces ${\bf W}^{(i)}_{I}$, and interior pressure spaces
$Q^{(i)}_{I}$, respectively, i.e.,
${\bf W}_{I}=\bigoplus_{i=1}^{N}{\bf W}^{(i)}_{I},\quad
Q_{I}=\bigoplus_{i=1}^{N}Q^{(i)}_{I}.$
${\bf W}_{\Gamma}$ and $Q_{\Gamma}$ are subdomain boundary velocity and
pressure spaces, respectively. All functions in ${\bf W}_{\Gamma}$ and
$Q_{\Gamma}$ are continuous across the subdomain boundaries $\Gamma$; their
degrees of freedom are shared by neighboring subdomains.
To formulate our domain decomposition algorithm, we introduce a partially sub-
assembled subdomain boundary velocity space
${\mathbf{{\widetilde{W}}}}_{\Gamma}$,
${\mathbf{{\widetilde{W}}}}_{\Gamma}={\mathbf{W}}_{\Pi}\bigoplus{\mathbf{W}}_{\Delta}={\mathbf{W}}_{\Pi}\bigoplus\left(\bigoplus_{i=1}^{N}{\mathbf{W}}^{(i)}_{\Delta}\right).$
Here, ${\mathbf{W}}_{\Pi}$ is the continuous, coarse level, primal velocity
space which is typically spanned by subdomain vertex nodal basis functions,
and/or by interface edge basis functions with constant values, or with values
of positive weights on these edges. The primal, coarse level velocity degrees
of freedom are shared by neighboring subdomains. The complimentary space
${\mathbf{W}}_{\Delta}$ is the direct sum of independent subdomain dual
interface velocity spaces ${\mathbf{W}}_{\Delta}^{(i)}$, which correspond to
the remaining subdomain boundary velocity degrees of freedom and are spanned
by basis functions which vanish at the primal degrees of freedom. Thus, an
element in the space ${\mathbf{{\widetilde{W}}}}_{\Gamma}$ typically has a
continuous primal velocity component and a discontinuous dual velocity
component.
The functions ${\bf w}_{\Delta}$ in ${\bf W}_{\Delta}$ are in general not
continuous across $\Gamma$. To enforce their continuity, we define a boolean
matrix $B_{\Delta}$ constructed from $\\{0,1,-1\\}$. On each row of
$B_{\Delta}$, there are only two non-zero entries, $1$ and $-1$, corresponding
to the same velocity degree of freedom on each subdomain boundary node, but
attributed to two neighboring subdomains, such that for any ${\bf w}_{\Delta}$
in ${\bf W}_{\Delta}$, each row of $B_{\Delta}{\bf w}_{\Delta}=0$ implies that
these two degrees of freedom from the two neighboring subdomains be the same.
When non-redundant continuity constraints are enforced, $B_{\Delta}$ has full
row rank. We denote the range of $B_{\Delta}$ applied on ${\bf W}_{\Delta}$ by
$\Lambda$, the vector space of the Lagrange multipliers.
In order to define a certain subdomain boundary scaling operator, we introduce
a positive scaling factor $\delta^{\dagger}(x)$ for each node $x$ on the
subdomain boundary $\Gamma$. Let ${\cal N}_{x}$ be the number of subdomains
sharing $x$, and we simply take $\delta^{\dagger}(x)=1/{\cal N}_{x}$. In
applications, these scaling factors will depend on the heat conduction
coefficient and the first of the Lamé parameters for scalar elliptic problems
and the equations of linear elasticity, respectively; see [19, 18]. Given such
scaling factors at the subdomain boundary nodes, we can define a scaled
operator $B_{\Delta,D}$. We note that each row of $B_{\Delta}$ has only two
nonzero entries, $1$ and $-1$, corresponding to the same subdomain boundary
node $x$. Multiplying each entry by the scaling factor $\delta^{\dagger}(x)$
gives us $B_{\Delta,D}$.
Solving the original fully assembled linear system (3) is then equivalent to:
find $\left({\bf u}_{I},~{}p_{I},~{}{\bf u}_{\Delta},~{}{\bf
u}_{\Pi},~{}p_{\Gamma},~{}\lambda\right)\in{\bf W}_{I}\bigoplus
Q_{I}\bigoplus{\bf W}_{\Delta}\bigoplus{\bf W}_{\Pi}\bigoplus
Q_{\Gamma}\bigoplus\Lambda$, such that
(7)
$\left[\begin{array}[]{cccccc}A_{II}&B_{II}^{T}&A_{I\Delta}&A_{I\Pi}&B_{\Gamma
I}^{T}&0\\\\[3.44444pt] B_{II}&0&B_{I\Delta}&B_{I\Pi}&0&0\\\\[3.44444pt]
A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}&A_{\Delta\Pi}&B_{\Gamma\Delta}^{T}&B_{\Delta}^{T}\\\\[3.44444pt]
A_{\Pi
I}&B_{I\Pi}^{T}&A_{\Pi\Delta}&A_{\Pi\Pi}&B_{\Gamma\Pi}^{T}&0\\\\[3.44444pt]
B_{\Gamma I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}&0&0\\\\[3.44444pt]
0&0&B_{\Delta}&0&0&0\end{array}\right]\left[\begin{array}[]{c}{\bf
u}_{I}\\\\[3.44444pt] p_{I}\\\\[3.44444pt] {\bf u}_{\Delta}\\\\[3.44444pt]
{\bf u}_{\Pi}\\\\[3.44444pt] p_{\Gamma}\\\\[3.44444pt]
\lambda\end{array}\right]=\left[\begin{array}[]{l}{\bf f}_{I}\\\\[3.44444pt]
0\\\\[3.44444pt] {\bf f}_{\Delta}\\\\[3.44444pt] {\bf f}_{\Pi}\\\\[3.44444pt]
0\\\\[3.44444pt] 0\end{array}\right]\mbox{ , }$
where the sub-blocks in the coefficient matrix represent the restrictions of
$A$ and $B$ in (3) to appropriate subspaces. The leading three-by-three block
can be made block diagonal with each diagonal block representing one
independent subdomain problem.
Corresponding to the one-dimensional null space of (3), we consider a vector
of the form $\left({\bf u}_{I},~{}p_{I},~{}{\bf u}_{\Delta},~{}{\bf
u}_{\Pi},~{}p_{\Gamma},~{}\lambda\right)=\left({\bf 0},~{}1_{p_{I}},~{}{\bf
0},~{}{\bf 0},~{}1_{p_{\Gamma}},\lambda\right)$, where $1_{p_{I}}\in Q_{I}$
and $1_{p_{\Gamma}}\in Q_{\Gamma}$ represent vectors with value $1$ on each
entry. Substituting it into (7) gives zero blocks on the right-hand side,
except at the third block
(8) ${\bf
f}_{\Delta}=[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right]+B^{T}_{\Delta}\lambda.$
The first term on the right-hand side represents the line integral of the
normal component of the velocity finite element basis functions across the
subdomain boundary on neighboring subdomains. Corresponding to the same
subdomain boundary velocity degree of freedom, their values on the two
neighboring subdomains are negative of each other. Therefore
$[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right]=B^{T}_{\Delta}B_{\Delta,D}[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right],$
from which we know that ${\mathbf{f}}_{\Delta}={\bf 0}$, for
$\lambda=-B_{\Delta,D}[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right].$
Therefore, a basis of the one-dimensional null space of (7) is
(9)
$\left(\begin{array}[]{cccccc}0,&1_{p_{I}},&0,&0,&1_{p_{\Gamma}},&-B_{\Delta,D}[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right]\end{array}\right).$
## 4 A reduced symmetric positive semi-definite system
The system (7) can be reduced to a Schur complement problem for the variables
$\left(p_{\Gamma},~{}\lambda\right)$. Since the leading four-by-four block of
the coefficient matrix in (7) is invertible, the variables $\left({\bf
u}_{I},~{}p_{I},~{}{\bf u}_{\Delta},~{}{\bf u}_{\Pi}\right)$ can be eliminated
and we obtain
(10) $G\left[\begin{array}[]{c}p_{\Gamma}\\\\[3.44444pt]
\lambda\end{array}\right]~{}=~{}g,$
where
(11) $G=\left[\begin{array}[]{cccc}B_{\Gamma
I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}\\\\[3.44444pt]
0&0&B_{\Delta}&0\end{array}\right]\left[\begin{array}[]{cccc}A_{II}&B_{II}^{T}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
B_{II}&0&B_{I\Delta}&B_{I\Pi}\\\\[3.44444pt] A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&B_{I\Pi}^{T}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right]^{-1}\left[\begin{array}[]{cc}B_{\Gamma
I}^{T}&0\\\\[3.44444pt] 0&0\\\\[3.44444pt]
B_{\Gamma\Delta}^{T}&B_{\Delta}^{T}\\\\[3.44444pt]
B_{\Gamma\Pi}^{T}&0\end{array}\right],$
and
(12) $g=\left[\begin{array}[]{cccc}B_{\Gamma
I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}\\\\[3.44444pt]
0&0&B_{\Delta}&0\end{array}\right]\left[\begin{array}[]{cccc}A_{II}&B_{II}^{T}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
B_{II}&0&B_{I\Delta}&B_{I\Pi}\\\\[3.44444pt] A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&B_{I\Pi}^{T}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right]^{-1}\left[\begin{array}[]{l}{\bf
f}_{I}\\\\[3.44444pt] 0\\\\[3.44444pt] {\bf f}_{\Delta}\\\\[3.44444pt] {\bf
f}_{\Pi}\end{array}\right].$
We denote
(13)
$\widetilde{A}=\left[\begin{array}[]{cccc}A_{II}&B_{II}^{T}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
B_{II}&0&B_{I\Delta}&B_{I\Pi}\\\\[3.44444pt] A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&B_{I\Pi}^{T}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right]\quad\mbox{and}\quad
B_{C}=\left[\begin{array}[]{cccc}B_{\Gamma
I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}\\\\[3.44444pt]
0&0&B_{\Delta}&0\end{array}\right].$
We can see that $-G$ is the Schur complement of the coefficient matrix of (7)
with respect to the last two row blocks, i.e.,
$\left[\begin{array}[]{cc}I&0\\\\[3.44444pt]
-B_{C}\widetilde{A}^{-1}&I\end{array}\right]\left[\begin{array}[]{cc}\widetilde{A}&B_{C}^{T}\\\\[3.44444pt]
B_{C}&0\end{array}\right]\left[\begin{array}[]{cc}I&-\widetilde{A}^{-1}B_{C}^{T}\\\\[3.44444pt]
0&I\end{array}\right]=\left[\begin{array}[]{cc}\widetilde{A}&0\\\\[3.44444pt]
0&-G\end{array}\right].$
From the Sylvester’s law of inertia, namely, the number of positive, negative,
and zero eigenvalues of a symmetric matrix is invariant under a change of
coordinates, we can see that the number of zero eigenvalues of $G$ is the same
as the number of zero eigenvalues (with multiplicity counted) of the original
coefficient matrix of (7), which is one, and all other eigenvalues of $G$ are
positive. Therefore $G$ is symmetric positive semi-definite. The null space of
$G$ is derived from the null space of the original coefficient matrix of (7),
and its basis is given by, cf. (9),
$\left(\begin{array}[]{cc}1_{p_{\Gamma}},&-B_{\Delta,D}[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{p_{I}}\\\
1_{p_{\Gamma}}\end{array}\right]\end{array}\right).$
We denote $X=Q_{\Gamma}\bigoplus\Lambda$. The range of $G$, denoted by
$R_{G}$, is the subspace of $X$ orthogonal to the null space of $G$, and has
the form
(14) $R_{G}=\left\\{\left[\begin{array}[]{c}g_{p_{\Gamma}}\\\\[3.44444pt]
g_{\lambda}\end{array}\right]\in
X~{}{\Big{|}}~{}g_{p_{\Gamma}}^{T}1_{{p_{\Gamma}}}-g_{\lambda}^{T}\left(B_{\Delta,D}[B_{I\Delta}^{T}~{}~{}B_{\Gamma\Delta}^{T}]\left[\begin{array}[]{c}1_{{p_{I}}}\\\
1_{{p_{\Gamma}}}\end{array}\right]\right)=0\right\\}.$
The restriction of $G$ to its range $R_{G}$ is positive definite. The fact
that the solution of (7) always exists for any given $\left({\bf
f}_{I},~{}{\bf f}_{\Delta},~{}{\bf f}_{\Pi}\right)$ on the right-hand side
implies that the solution of (10) exits for any $g$ defined by (12). Therefore
$g\in R_{G}$. When the conjugate gradient method (CG) is applied to solve (10)
with zero initial guess, all the iterates are in the Krylov subspace generated
by $G$ and $g$, which is also a subspace of $R_{G}$, and where the CG cannot
break down. After obtaining $\left(p_{\Gamma},~{}\lambda\right)$ from solving
(10), the other components $\left({\bf u}_{I},~{}p_{I},~{}{\bf
u}_{\Delta},~{}{\bf u}_{\Pi}\right)$ in (7) are obtained by back substitution.
In the rest of this section, we discuss the implementation of multiplying $G$
by a vector. The main operation is the product of ${\widetilde{A}}^{-1}$ with
a vector, cf. (11) and (12). We denote
$A_{rr}=\left[\begin{array}[]{ccc}A_{II}&B_{II}^{T}&A_{I\Delta}\\\\[3.44444pt]
B_{II}&0&B_{I\Delta}\\\\[3.44444pt] A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}\end{array}\right],\quad A_{\Pi
r}=A_{r\Pi}^{T}=\left[A_{\Pi I}\quad B_{I\Pi}^{T}\quad
A_{\Pi\Delta}\right],\quad f_{r}=\left[\begin{array}[]{l}{\bf
f}_{I}\\\\[3.44444pt] 0\\\\[3.44444pt] {\bf f}_{\Delta}\end{array}\right],$
and define the Schur complement
$S_{\Pi}=A_{\Pi\Pi}-A_{\Pi r}A_{rr}^{-1}A_{r\Pi},$
which is symmetric positive definite from the Sylvester’s law of inertia.
$S_{\Pi}$ defines the coarse level problem in the algorithm. The product
$\left[\begin{array}[]{cccc}A_{II}&B_{II}^{T}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
B_{II}&0&B_{I\Delta}&B_{I\Pi}\\\\[3.44444pt] A_{\Delta
I}&B_{I\Delta}^{T}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&B_{I\Pi}^{T}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right]^{-1}\left[\begin{array}[]{l}{\bf
f}_{I}\\\\[3.44444pt] 0\\\\[3.44444pt] {\bf f}_{\Delta}\\\\[3.44444pt] {\bf
f}_{\Pi}\end{array}\right]$
can then be represented by
$\left[\begin{array}[]{c}A_{rr}^{-1}f_{r}\\\\[3.44444pt]
{\mathbf{0}}\end{array}\right]~{}+~{}\left[\begin{array}[]{c}-A_{rr}^{-1}A_{r\Pi}\\\\[3.44444pt]
I_{\Pi}\end{array}\right]~{}S_{\Pi}^{-1}~{}\left({\bf f}_{\Pi}-A_{\Pi
r}A_{rr}^{-1}f_{r}\right),$
which requires solving the coarse level problem once and independent subdomain
Stokes problems with Neumann type boundary conditions twice.
## 5 Some techniques
We first define certain norms for several vector/function spaces. We denote
(15) ${\mathbf{{\widetilde{W}}}}={\bf
W}_{I}\bigoplus{\mathbf{{\widetilde{W}}}}_{\Gamma}.$
For any ${\mathbf{w}}$ in ${\mathbf{{\widetilde{W}}}}$, we denote its
restriction to subdomain $\Omega_{i}$ by ${\bf w}^{(i)}$. A subdomain-wise
$H^{1}$-seminorm can be defined for functions in ${\mathbf{{\widetilde{W}}}}$
by
$|{\mathbf{w}}|^{2}_{H^{1}}=\sum_{i=1}^{N}|{\mathbf{w}}^{(i)}|^{2}_{H^{1}(\Omega_{i})}.$
We also define
${\widetilde{W}}={\bf W}_{I}\bigoplus Q_{I}\bigoplus{\bf
W}_{\Delta}\bigoplus{\bf W}_{\Pi},$
and its subspace
(16) ${\widetilde{W}}_{0}=\left\\{w=\left({\bf w}_{I},~{}p_{I},~{}{\bf
w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}~{}\big{|}~{}B_{II}{\mathbf{w}}_{I}+B_{I\Delta}{\mathbf{w}}_{\Delta}+B_{I\Pi}{\mathbf{w}}_{\Pi}=0\right\\}.$
For any $w=\left({\bf w}_{I},~{}p_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}_{0}$, let ${\mathbf{w}}=\left({\bf
w}_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\mathbf{{\widetilde{W}}}}$. Then
(26) $\displaystyle\left<w,w\right>_{\widetilde{A}}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{l}{\bf w}_{I}\\\\[3.44444pt] {\bf
w}_{\Delta}\\\\[3.44444pt] {\bf
w}_{\Pi}\end{array}\right]^{T}\left[\begin{array}[]{ccc}A_{II}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
A_{\Delta I}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right]\left[\begin{array}[]{l}{\bf
w}_{I}\\\\[3.44444pt] {\bf w}_{\Delta}\\\\[3.44444pt] {\bf
w}_{\Pi}\end{array}\right]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[\begin{array}[]{c}{\bf w}_{I}^{(i)}\\\ {\bf
w}_{\Delta}^{(i)}\\\ {\bf
w}_{\Pi}^{(i)}\end{array}\right]^{T}\left[\begin{array}[]{cccc}A_{II}^{(i)}&A_{I\Delta}^{(i)}&A_{I\Pi}^{(i)}\\\\[3.44444pt]
A_{\Delta I}^{(i)}&A_{\Delta\Delta}^{(i)}&A_{\Delta\Pi}^{(i)}\\\\[3.44444pt]
A_{\Pi
I}^{(i)}&A_{\Pi\Delta}^{(i)}&A_{\Pi\Pi}^{(i)}\end{array}\right]\left[\begin{array}[]{c}{\bf
w}_{I}^{(i)}\\\ {\bf w}_{\Delta}^{(i)}\\\ {\bf
w}_{\Pi}^{(i)}\end{array}\right]=\sum_{i=1}^{N}\left|\left[\begin{array}[]{c}{\bf
w}_{I}^{(i)}\\\ {\bf w}_{\Delta}^{(i)}\\\ {\bf
w}_{\Pi}^{(i)}\end{array}\right]\right|_{H^{1}(\Omega_{i})}^{2}$
$\displaystyle=$ $\displaystyle|{\mathbf{w}}|^{2}_{H^{1}},$
i.e., $\left<\cdot,\cdot\right>_{{\widetilde{A}}}$ defines an inner product on
${\widetilde{W}}_{0}$. In (5), the superscript (i) is used to represent the
restrictions of corresponding vectors and matrices to subdomain $\Omega_{i}$.
Since ${\mathbf{W}}$ is essentially the subspace of
${\mathbf{{\widetilde{W}}}}$ with continuous subdomain boundary velocities,
the inf-sup condition (4) and (5) also holds for the mixed space
${\mathbf{{\widetilde{W}}}}\times Q$. Denote
(40)
$\widetilde{B}=\left[\begin{array}[]{ccc}B_{II}&B_{I\Delta}&B_{I\Pi}\\\\[3.44444pt]
B_{\Gamma
I}&B_{\Gamma\Delta}&B_{\Gamma\Pi}\end{array}\right],\qquad\overline{\widetilde{A}}=\left[\begin{array}[]{ccc}A_{II}&A_{I\Delta}&A_{I\Pi}\\\\[3.44444pt]
A_{\Delta I}&A_{\Delta\Delta}&A_{\Delta\Pi}\\\\[3.44444pt] A_{\Pi
I}&A_{\Pi\Delta}&A_{\Pi\Pi}\end{array}\right],$
as in (7), then
(41) $\sup_{{\bf
w}\in{\mathbf{{\widetilde{W}}}}}\frac{\left<q,\widetilde{B}{\mathbf{w}}\right>^{2}}{\left<{\mathbf{w}},\overline{\widetilde{A}}{\mathbf{w}}\right>}\geq\beta^{2}\left<q,Zq\right>,\hskip
14.22636pt\forall q\in Q/Ker(B^{T}),$
where $\beta$ is the same as in (4) and (5).
We also have the following lemma on the stability of the operator
$\widetilde{B}$.
###### Lemma 1
For any ${\mathbf{w}}\in{\mathbf{{\widetilde{W}}}}$ and $q\in Q$,
$\left<{\widetilde{B}}{\bf
w},q\right>\leq|{\mathbf{w}}|_{H^{1}}\|q\|_{L^{2}}$.
Proof:
$\displaystyle\left<{\widetilde{B}}{\bf w},q\right>^{2}$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{N}\int_{\Omega_{i}}\nabla\cdot{\bf
w}^{(i)}q\right)^{2}\leq\left(\sum_{i=1}^{N}\sqrt{\int_{\Omega_{i}}|\nabla{\bf
w}^{(i)}|^{2}}\sqrt{\int_{\Omega_{i}}q^{2}}\right)^{2}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{i=1}^{N}\int_{\Omega_{i}}|\nabla{\bf
w}^{(i)}|^{2}\right)\left(\sum_{i=1}^{N}\int_{\Omega_{i}}q^{2}\right)=|{\mathbf{w}}|^{2}_{H^{1}}\|q\|^{2}_{L^{2}}.\qquad\Box$
The finite element space for subdomain boundary pressures, $Q_{\Gamma}$, is a
subspace of $L^{2}(\Gamma)$. For each $p_{\Gamma}\in Q_{\Gamma}$, its finite
element extension by zero to the interior of subdomains is denoted by
$p_{\Gamma}^{E}$, which equals $p_{\Gamma}$ on all subdomain boundary nodes
and equals zero on all subdomain interior nodes. We can see that
$p_{\Gamma}^{E}\in Q\subset L^{2}(\Omega)$, and
$\|p_{\Gamma}^{E}\|_{L^{2}(\Omega)}^{2}=\left<p^{E}_{\Gamma},p^{E}_{\Gamma}\right>_{Z}$,
from the definition of $Z$ in Section 2.
From (11) and (13), we can see that
$G=B_{C}{\widetilde{A}}^{-1}B_{C}^{T}.$
In particular, we denote the first row of $B_{C}$ by
$\widetilde{B}_{\Gamma}=\left[B_{\Gamma I}\quad 0\quad B_{\Gamma\Delta}\quad
B_{\Gamma\Pi}\right];$
for the second row, we denote the restriction operator from ${\widetilde{W}}$
onto ${\bf W}_{\Delta}$ by $\widetilde{R}_{\Delta}$, such that for any
$w=\left({\bf w}_{I},~{}p_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}$, $\widetilde{R}_{\Delta}w={\bf
w}_{\Delta}$. Then $G$ can be represented by the following two-by-two block
structure
(42)
$G=\left[\begin{array}[]{cc}G_{p_{\Gamma}p_{\Gamma}}&G_{p_{\Gamma}\lambda}\\\\[3.44444pt]
G_{\lambda p_{\Gamma}}&G_{\lambda\lambda}\end{array}\right],$
where
$\displaystyle
G_{p_{\Gamma}p_{\Gamma}}=\widetilde{B}_{\Gamma}\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T},\qquad
G_{p_{\Gamma}\lambda}=\widetilde{B}_{\Gamma}\widetilde{A}^{-1}\widetilde{R}_{\Delta}^{T}B_{\Delta}^{T},$
$\displaystyle G_{\lambda
p_{\Gamma}}=B_{\Delta}\widetilde{R}_{\Delta}\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T},\qquad
G_{\lambda\lambda}=B_{\Delta}\widetilde{R}_{\Delta}\widetilde{A}^{-1}\widetilde{R}_{\Delta}^{T}B_{\Delta}^{T}.$
The pressure components of all vectors in $R_{G}$ with $g_{\lambda}=0$, cf.
(14), form a subspace of $Q_{\Gamma}$ and we denote this subspace by
$R_{G|Q_{\Gamma}}$. From the definition of $R_{G}$, we can see that for any
vector $p_{\Gamma}\in R_{G|Q_{\Gamma}}$, $p_{\Gamma}^{T}1_{p_{\Gamma}}=0$, and
then its extension by zero to the interior of subdomains, $p_{\Gamma}^{E}$,
also has zero average.
The following lemma follows essentially from [34, Lemma 9.1].
###### Lemma 2
For all $p_{\Gamma}\in R_{G|Q_{\Gamma}}$,
$\beta^{2}\|p_{\Gamma}^{E}\|_{L^{2}(\Omega)}^{2}~{}\leq~{}\left<p_{\Gamma},G_{p_{\Gamma}p_{\Gamma}}p_{\Gamma}\right>~{}\leq~{}\|p_{\Gamma}^{E}\|_{L^{2}(\Omega)}^{2},$
where $p_{\Gamma}^{E}$ represents the extension by zero of $p_{\Gamma}$ to the
interior of subdomains, and $\beta$ is the same as in (4) and (5).
Proof: Note that even though $\widetilde{A}^{-1}$ is indefinite in
${\widetilde{W}}$, it is positive definite when restricted to a subspace of
${\widetilde{W}}$, where the pressure component equals zero, and the norm
$\|\cdot\|_{\widetilde{A}^{-1}}$ is well defined.
To prove the left side inequality, denote for any ${\bf v}=\left({\bf
v}_{I},~{}{\bf v}_{\Delta},~{}{\bf
v}_{\Pi}\right)\in{\mathbf{{\widetilde{W}}}}$, ${\bf v}^{\dagger}=\left({\bf
v}_{I},~{}0,~{}{\bf v}_{\Delta},~{}{\bf v}_{\Pi}\right)\in{\widetilde{W}}$. We
have
$\displaystyle\left<p_{\Gamma},\widetilde{B}_{\Gamma}\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\right>=\|\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\|_{\widetilde{A}^{-1}}^{2}=\sup_{{\bf
v}\in{\mathbf{{\widetilde{W}}}}}\frac{\left<{\bf
v}^{\dagger},\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\right>^{2}_{\widetilde{A}^{-1}}}{\|{\bf
v}^{\dagger}\|^{2}_{\widetilde{A}^{-1}}}=\sup_{{\bf
v}\in{\mathbf{{\widetilde{W}}}}}\frac{\left(p_{\Gamma}^{T}\widetilde{B}_{\Gamma}\widetilde{A}^{-1}{\bf
v}^{\dagger}\right)^{2}}{{\bf v}^{\dagger^{T}}{\widetilde{A}^{-1}}{\bf
v}^{\dagger}}$ $\displaystyle=\sup_{{\bf
w}\in{\mathbf{{\widetilde{W}}}}}\frac{\left(p_{\Gamma}^{T}\widetilde{B}_{\Gamma}{\bf
w}^{\dagger}\right)^{2}}{{\bf w}^{\dagger^{T}}\widetilde{A}{\bf
w}^{\dagger}}=\sup_{{\bf
w}\in{\mathbf{{\widetilde{W}}}}}\frac{\left(p_{\Gamma}^{E^{T}}\widetilde{B}{\bf
w}\right)^{2}}{{\bf w}^{T}\overline{\widetilde{A}}{\bf
w}}\geq\beta^{2}\left<p_{\Gamma}^{E},p_{\Gamma}^{E}\right>_{Z}=\beta^{2}\|p_{\Gamma}^{E}\|_{L^{2}(\Omega)}^{2},$
where we have used the inf-sup condition (41) for the inequality in the
middle.
To prove the right side inequality, for any given $p_{\Gamma}\in
R_{G|Q_{\Gamma}}$, denote ${\bf v}^{\dagger}=\left({\bf
v}_{I},~{}p_{I},~{}{\bf v}_{\Delta},~{}{\bf
v}_{\Pi}\right)=\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma}$, and
the shorter vector ${\bf v}=\left({\bf v}_{I},~{}{\bf v}_{\Delta},~{}{\bf
v}_{\Pi}\right)$. From the continuity of ${\widetilde{B}}$ in Lemma 1 and (5),
we have
$\displaystyle\left<p_{\Gamma},\widetilde{B}_{\Gamma}\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\right>=\left<p_{\Gamma},\widetilde{B}_{\Gamma}{\bf
v}^{\dagger}\right>=\left<p_{\Gamma}^{E},\widetilde{B}{\mathbf{v}}\right>\leq\|p_{\Gamma}^{E}\|_{L^{2}}~{}|{\mathbf{v}}|_{H^{1}}$
$\displaystyle=$
$\displaystyle\|p_{\Gamma}^{E}\|_{L^{2}}~{}\sqrt{\left<\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma},\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\right>_{{\widetilde{A}}}}=\|p_{\Gamma}^{E}\|_{L^{2}}~{}\left<p_{\Gamma},\widetilde{B}_{\Gamma}\widetilde{A}^{-1}\widetilde{B}_{\Gamma}^{T}p_{\Gamma}\right>^{1/2}.\qquad\Box$
The following corollary of Lemma 2 is an immediate result from (6) and the
facts that
$\|p_{\Gamma}^{E}\|_{L^{2}(\Omega)}^{2}=\left<p_{\Gamma}^{E},p_{\Gamma}^{E}\right>_{Z}$,
$\left<p_{\Gamma}^{E},p_{\Gamma}^{E}\right>=\left<p_{\Gamma},p_{\Gamma}\right>$.
###### Corollary 1
There exist positive constants $c$ and $C$, such that
$ch^{2}\beta^{2}I_{p_{\Gamma}}~{}\leq~{}G_{p_{\Gamma}p_{\Gamma}}~{}\leq~{}Ch^{2}I_{p_{\Gamma}}$
where $I_{p_{\Gamma}}$ is the identity matrix of the same dimension as
$G_{p_{\Gamma}p_{\Gamma}}$, and $\beta$ is the same as in (4) and (5).
###### Remark 1
Lemma 2 and Corollary 1 are not used in our proof of the condition number
bound in Section 6. However, it is intuitive to see from Corollary 1 that the
first diagonal block $G_{p_{\Gamma}p_{\Gamma}}$ in matrix $G$ can be
approximated spectrally equivalently by the identity matrix multiplied by
$h^{2}$, which is what is being done in our block diagonal preconditioner
discussed in Section 6.
We also need define a certain jump operator across the subdomain boundaries
$\Gamma$. Let $P_{D}:{\widetilde{W}}\rightarrow{\widetilde{W}}$, be defined
by, cf. [21],
$P_{D}=\widetilde{R}_{\Delta}^{T}B_{\Delta,D}^{T}B_{\Delta}\widetilde{R}_{\Delta}.$
We can see that application of $P_{D}$ to a vector essentially computes the
difference (jump) of the dual velocity component across the subdomain
boundaries and then distributes the jump to neighboring subdomains according
to the scaling factor $\delta^{\dagger}(x)$. In fact, the dual velocity
component is the only part of the vector involved in the application of
$P_{D}$; all other components are kept zero and are added into the definition
to make $P_{D}$ more convenient to use in the presentation of the algorithm.
We also have, for any $w=\left({\bf w}_{I},~{}p_{I},~{}{\bf
w}_{\Delta},~{}{\bf w}_{\Pi}\right)\in{\widetilde{W}}$,
$\left<P_{D}w,P_{D}w\right>_{\widetilde{A}}=\left<B_{\Delta,D}^{T}B_{\Delta}{\bf
w}_{\Delta},B_{\Delta,D}^{T}B_{\Delta}{\bf
w}_{\Delta}\right>_{A_{\Delta\Delta}}.$
The following lemma can be found essentially from [23, Section 6]; see also
(5).
###### Lemma 3
There exists a function $\Phi(H/h)$, such that for all
$w\in{\widetilde{W}}_{0}$,
$\left<P_{D}w,P_{D}w\right>_{\widetilde{A}}\leq\Phi(H/h)\left<w,w\right>_{\widetilde{A}}.$
###### Remark 2
Just as for the positive definite elliptic problems discussed in [23, Section
6], for two-dimensional problems, when only subdomain corner velocities are
chosen as coarse level primal variables, $\Phi(H/h)=C(H/h)(1+\log{(H/h)})$;
when both subdomain corner and edge-average velocity degrees of freedom are
chosen as primal variables, $\Phi(H/h)=CH/h$.
The following lemma is also used and can be found at [10, Lemma 2.3].
###### Lemma 4
Consider the saddle point problem: find
$({\mathbf{u}},p)\in{\mathbf{W}}\bigoplus Q$, such that
(43) $\left[\begin{array}[]{cc}A&B^{T}\\\\[3.44444pt]
B&0\end{array}\right]\left[\begin{array}[]{l}{\mathbf{u}}\\\\[3.44444pt]
p\end{array}\right]=\left[\begin{array}[]{l}{\mathbf{f}}\\\\[3.44444pt]
g\end{array}\right],$
where $A$ and $B$ are as in (3), ${\mathbf{f}}\in{\mathbf{W}}$, and $g\in
Im(B)\subset Q$. Let $\beta$ be the inf-sup constant specified in (5). Then
$\|{\mathbf{u}}\|_{A}\leq\|{\mathbf{f}}\|_{A^{-1}}+\frac{1}{\beta}\|g\|_{Z^{-1}},$
where $Z$ is the mass matrix defined in Section 2.
## 6 A lumped preconditioner
The lumped preconditioner was first used in the FETI algorithm [7] for solving
positive definite elliptic problems. Compared with the Dirichlet
preconditioner, also used for the FETI algorithm [8], the lumped
preconditioner is less effective in the improvement of convergence rate, but
it is also less expensive in the computational costs. The main operation in
the lumped preconditioner is subdomain matrix and vector products, while the
implementation of the Dirichlet preconditioner requires solving subdomain
systems of equations. In this paper, we discuss only the lumped preconditioner
in our algorithm for solving the incompressible Stokes equation; study of the
Dirichlet preconditioner will be addressed in forthcoming work.
We consider a block diagonal preconditioner for (10). From Corollary 1, the
inverse of the first diagonal block $G_{p_{\Gamma}p_{\Gamma}}$ of $G$ can be
effectively approximated by $1/h^{2}$ times the identity matrix. The inverse
of the second diagonal block
$B_{\Delta}\widetilde{R}_{\Delta}\widetilde{A}^{-1}\widetilde{R}_{\Delta}^{T}B_{\Delta}^{T}$,
can be approximated by the following lumped block
$M^{-1}_{\lambda}=B_{\Delta,D}\widetilde{R}_{\Delta}\widetilde{A}\widetilde{R}_{\Delta}^{T}B_{\Delta,D}^{T}.$
This leads to the lumped preconditioner
$M^{-1}=\left[\begin{array}[]{cc}\frac{1}{h^{2}}I_{p_{\Gamma}}&\\\\[3.44444pt]
&M^{-1}_{\lambda}\end{array}\right],$
for solving (10).
###### Remark 3
The mesh size $h$ is used in the above preconditioner. For applications where
the mesh size is not explicitly provided and only the coefficient matrix in
(3) is given, an estimate of $h$ can be obtained by comparing the nonzero
entries in $A$ and $B$ blocks. From the definition of $A$ and $B$ for the
incompressible Stokes problem (2), entries in $A$ and entries in $B$ have a
difference of factor $h$ in general.
$M^{-1}$ is symmetric positive definite. Multiplication of $M^{-1}$ by a
vector requires mainly the product of $\widetilde{A}$ with a vector. When the
CG iteration is applied to solve the preconditioned system
(44) $M^{-1}G\left[\begin{array}[]{c}p_{\Gamma}\\\\[3.44444pt]
\lambda\end{array}\right]~{}=~{}M^{-1}g,$
with zero initial guess, all the iterates belong to the Krylov subspace
generated by the operator $M^{-1}G$ and the vector $M^{-1}g$, which is also a
subspace of the range of $M^{-1}G$. We denote the range of $M^{-1}G$ by
$R_{M^{-1}G}$. The following lemma shows that the CG iteration applied to
solving (44) cannot break down.
###### Lemma 5
Let the preconditioner $M^{-1}$ be symmetric positive definite. The CG
iteration applied to solving (44) with zero initial guess cannot break down.
Proof: We just need to show that for any $0\neq x\in R_{M^{-1}G}$, $Gx\neq 0$.
Let $0\neq x=M^{-1}Gy$, for a certain $y\in X$ and $y\neq 0$. $Gx=GM^{-1}Gy$,
which cannot be zero since $Gy\neq 0$ and $y^{T}GM^{-1}Gy\neq 0$. $\qquad\Box$
###### Lemma 6
Let $M^{-1}$ be symmetric positive definite. For any
$x=(p_{\Gamma},~{}\lambda)\in R_{M^{-1}G}$,
$\left<Mx,x\right>=\max_{y\in R_{G},y\neq
0}\frac{\left<y,x\right>^{2}}{\left<M^{-1}y,y\right>}.$
Proof: Denote the range of $M^{-\frac{1}{2}}G$ by $R_{M^{-1/2}G}$. For any
$x\in R_{M^{-1}G}$,
$\displaystyle\left<Mx,x\right>$ $\displaystyle=$
$\displaystyle\left<M^{\frac{1}{2}}x,M^{\frac{1}{2}}x\right>=\max_{z\in
R_{M^{-1/2}G},z\neq
0}\frac{\left<M^{\frac{1}{2}}x,z\right>^{2}}{\left<z,z\right>}$
$\displaystyle=$ $\displaystyle\max_{y\in R_{G},y\neq
0}\frac{\left<M^{\frac{1}{2}}x,M^{-\frac{1}{2}}y\right>^{2}}{\left<M^{-\frac{1}{2}}y,M^{-\frac{1}{2}}y\right>}=\max_{y\in
R_{G},y\neq
0}\frac{\left<y,x\right>^{2}}{\left<M^{-1}y,y\right>}~{}.\qquad\Box$
In the following, we establish a condition number bound of the preconditioned
operator $M^{-1}G$. We first have the following lemma.
###### Lemma 7
For any $w\in{\widetilde{W}}_{0}$,
$\left<M^{-1}B_{C}w,B_{C}w\right>\leq\Phi(H/h)\left<{\widetilde{A}}w,w\right>,$
where $\Phi(H/h)$ is as defined in Lemma 3.
Proof: Given $w=\left({\bf w}_{I},~{}q_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}_{0}$, let $g_{p_{\Gamma}}=B_{\Gamma I}{\bf
w}_{I}+B_{\Gamma\Delta}{\bf w}_{\Delta}+B_{\Gamma\Pi}{\bf w}_{\Pi}$. We have
(45) $\displaystyle\left<M^{-1}B_{C}w,B_{C}w\right>$ $\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+\left(B_{\Delta}{\widetilde{R}}_{\Delta}w\right)^{T}M^{-1}_{\lambda}B_{\Delta}{\widetilde{R}}_{\Delta}w$
$\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+\left(B_{\Delta}{\widetilde{R}}_{\Delta}w\right)^{T}B_{\Delta,D}{\widetilde{R}}_{\Delta}{\widetilde{A}}{\widetilde{R}}_{\Delta}^{T}B_{\Delta,D}^{T}\left(B_{\Delta}{\widetilde{R}}_{\Delta}w\right)$
$\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+\left<P_{D}w,P_{D}w\right>_{\widetilde{A}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+\Phi(H/h)\left<w,w\right>_{\widetilde{A}},$
where we used Lemma 3 for the last inequality. It is sufficient to bound the
first term of the right-hand side in the above inequality.
We denote ${\mathbf{w}}=\left({\bf w}_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\mathbf{{\widetilde{W}}}}$. Since
$B_{II}{\mathbf{w}}_{I}+B_{I\Delta}{\mathbf{w}}_{\Delta}+B_{I\Pi}{\mathbf{w}}_{\Pi}=0$,
cf. (16), we have
$\displaystyle\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c}B_{II}{\mathbf{w}}_{I}+B_{I\Delta}{\mathbf{w}}_{\Delta}+B_{I\Pi}{\mathbf{w}}_{\Pi}\\\
B_{\Gamma I}{\bf w}_{I}+B_{\Gamma\Delta}{\bf w}_{\Delta}+B_{\Gamma\Pi}{\bf
w}_{\Pi}\end{array}\right]^{T}\left[\begin{array}[]{c}B_{II}{\mathbf{w}}_{I}+B_{I\Delta}{\mathbf{w}}_{\Delta}+B_{I\Pi}{\mathbf{w}}_{\Pi}\\\
B_{\Gamma I}{\bf w}_{I}+B_{\Gamma\Delta}{\bf w}_{\Delta}+B_{\Gamma\Pi}{\bf
w}_{\Pi}\end{array}\right]$ $\displaystyle=$
$\displaystyle\left<{\widetilde{B}}{\bf w},{\widetilde{B}}{\bf w}\right>,$
where ${\widetilde{B}}$ is defined in (40). From (6) and the stability of
${\widetilde{B}}$, cf. Lemma 1, we have
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>$
$\displaystyle=$ $\displaystyle\frac{1}{h^{2}}\left<{\widetilde{B}}{\bf
w},{\widetilde{B}}{\bf w}\right>\leq C\left<{\widetilde{B}}{\bf
w},{\widetilde{B}}{\bf w}\right>_{Z^{-1}}=C\max_{q\in
Q}\frac{\left<{\widetilde{B}}{\bf w},q\right>^{2}}{\left<q,q\right>_{Z}}$
$\displaystyle\leq$ $\displaystyle C\max_{q\in
Q}\frac{|{\mathbf{w}}|^{2}_{H^{1}}\|q\|^{2}_{L^{2}}}{\|q\|^{2}_{L^{2}}}=C|{\mathbf{w}}|^{2}_{H^{1}}=C\left<w,w\right>_{\widetilde{A}},$
where for the last equality, we used the fact that
$B_{II}{\mathbf{w}}_{I}+B_{I\Delta}{\mathbf{w}}_{\Delta}+B_{I\Pi}{\mathbf{w}}_{\Pi}=0$,
and (5). $\quad\Box$
###### Lemma 8
For any given $y=(g_{p_{\Gamma}},g_{\lambda})\in R_{G}$, there exits
$w\in{\widetilde{W}}_{0}$, such that $B_{C}w=y$, and
$\left<{\widetilde{A}}w,w\right>\leq\frac{C}{\beta^{2}}\left<M^{-1}y,y\right>$.
Proof: Given $y=(g_{p_{\Gamma}},g_{\lambda})\in R_{G}$, take ${\bf
w}_{\Delta}^{(I)}=B_{\Delta,D}^{T}g_{\lambda}$. Let ${\bf w}^{(I)}=({\bf
0},~{}{\bf w}_{\Delta}^{(I)},{\bf 0})\in{\bf W}_{I}\bigoplus{\bf
W}_{\Delta}\bigoplus{\bf W}_{\Pi}$ and $w^{(I)}=({\bf 0},~{}0,~{}{\bf
w}_{\Delta}^{(I)},~{}{\bf 0})\in{\bf W}_{I}\bigoplus Q_{I}\bigoplus{\bf
W}_{\Delta}\bigoplus{\bf W}_{\Pi}$. We have
(48) $|{\bf w}^{(I)}|^{2}_{H^{1}}=\left<A_{\Delta\Delta}{\bf
w}^{(I)}_{\Delta},{\bf w}^{(I)}_{\Delta}\right>,$
and
(49) $B_{c}w^{(I)}=\left[\begin{array}[]{cccc}B_{\Gamma
I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}\\\\[3.44444pt]
0&0&B_{\Delta}&0\end{array}\right]\left[\begin{array}[]{c}{\bf
0}\\\\[3.44444pt] 0\\\\[3.44444pt] B_{\Delta,D}^{T}g_{\lambda}\\\\[3.44444pt]
{\bf 0}\end{array}\right]=\left[\begin{array}[]{c}B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\\\\[3.44444pt] g_{\lambda}\end{array}\right],$
where we used the fact that $B_{\Delta}B_{\Delta,D}^{T}=I$.
We consider the solution to the following fully assembled system of linear
equations of the form (3): find $({\bf w}_{I}^{(II)},~{}q_{I}^{(II)},~{}{\bf
w}_{\Gamma}^{(II)},~{}q_{\Gamma}^{(II)})\in{\bf W}_{I}\bigoplus
Q_{I}\bigoplus{\bf W}_{\Gamma}\bigoplus Q_{\Gamma}$, such that
(50) $\left[\begin{array}[]{cccc}A_{II}&B_{II}^{T}&A_{I\Gamma}&B_{\Gamma
I}^{T}\\\\[3.44444pt] B_{II}&0&B_{I\Gamma}&0\\\\[3.44444pt] A_{\Gamma
I}&B_{I\Gamma}^{T}&A_{\Gamma\Gamma}&B_{\Gamma\Gamma}^{T}\\\\[3.44444pt]
B_{\Gamma
I}&0&B_{\Gamma\Gamma}&0\end{array}\right]\left[\begin{array}[]{c}{\bf
w}_{I}^{(II)}\\\\[3.44444pt] q_{I}^{(II)}\\\\[3.44444pt] {\bf
w}_{\Gamma}^{(II)}\\\\[3.44444pt]
q_{\Gamma}^{(II)}\end{array}\right]=\left[\begin{array}[]{l}{\bf
0}\\\\[3.44444pt] -B_{I\Delta}{\bf w}^{(I)}_{\Delta}\\\\[3.44444pt] {\bf
0}\\\\[3.44444pt] g_{p_{\Gamma}}-B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\end{array}\right]\mbox{ , }$
where a particular right-hand side is chosen. We first note that, since
$(g_{p_{\Gamma}},g_{\lambda})\in R_{G}$, the right-hand side vector of the
above system satisfies, cf. (14),
$(-B_{I\Delta}{\bf
w}^{(I)}_{\Delta})^{T}1_{p_{I}}+(g_{p_{\Gamma}}-B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta})^{T}1_{p_{\Gamma}}=g_{p_{\Gamma}}^{T}1_{p_{\Gamma}}-g_{\lambda}^{T}B_{\Delta,D}\left(B_{I\Delta}^{T}1_{p_{I}}+B_{\Gamma\Delta}^{T}1_{p_{\Gamma}}\right)=0,$
i.e., it has zero average, which implies existence of the solution to (50).
Denote ${\bf w}^{(II)}=({\bf w}_{I}^{(II)},~{}{\bf w}_{\Gamma}^{(II)})\in{\bf
W}$. From the inf-sup stability of the original problem (3) and Lemma 4, we
have
(51) $|{\bf
w}^{(II)}|^{2}_{H^{1}}\leq\frac{1}{\beta^{2}}\left\|\left[\begin{array}[]{l}-B_{I\Delta}{\bf
w}^{(I)}_{\Delta}\\\\[3.44444pt] g_{p_{\Gamma}}-B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\end{array}\right]\right\|^{2}_{Z^{-1}}\leq\frac{1}{\beta^{2}}\left\|\left[\begin{array}[]{l}B_{I\Delta}{\bf
w}^{(I)}_{\Delta}\\\\[3.44444pt] B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\end{array}\right]\right\|^{2}_{Z^{-1}}+\frac{1}{\beta^{2}}\left\|\left[\begin{array}[]{l}0\\\\[3.44444pt]
g_{p_{\Gamma}}\end{array}\right]\right\|^{2}_{Z^{-1}}.$
The first term on the right-hand side of (51) can be bounded in the same way
as done in (6), and we have
(52) $\left\|\left[\begin{array}[]{l}B_{I\Delta}{\bf
w}^{(I)}_{\Delta}\\\\[3.44444pt] B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\end{array}\right]\right\|^{2}_{Z^{-1}}\leq
C\left<A_{\Delta\Delta}{\bf w}^{(I)}_{\Delta},{\bf w}^{(I)}_{\Delta}\right>;$
the second term can be bounded by, using (6),
(53) $\left\|\left[\begin{array}[]{l}0\\\\[3.44444pt]
g_{p_{\Gamma}}\end{array}\right]\right\|^{2}_{Z^{-1}}\leq\frac{C}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>.$
Split the continuous subdomain boundary velocity ${\bf w}_{\Gamma}^{(II)}$
into the dual part ${\bf w}_{\Delta}^{(II)}\in{\bf W}_{\Delta}$ and the primal
part ${\bf w}_{\Pi}^{(II)}\in{\bf W}_{\Pi}$, and denote $w^{(II)}=({\bf
w}_{I}^{(II)},~{}q_{I}^{(II)},~{}{\bf w}_{\Delta}^{(II)},~{}{\bf
w}_{\Pi}^{(II)})$. We have, from (50),
(54)
$\left[\begin{array}[]{cccc}B_{II}&0&B_{I\Delta}&B_{I\Pi}\end{array}\right]\left[\begin{array}[]{c}{\bf
w}_{I}^{(II)}\\\\[3.44444pt] q_{I}^{(II)}\\\\[3.44444pt] {\bf
w}_{\Delta}^{(II)}\\\\[3.44444pt] {\bf
w}_{\Pi}^{(II)}\end{array}\right]=-B_{I\Delta}{\bf w}^{(I)}_{\Delta},$
and
(55) $B_{c}w^{(II)}=\left[\begin{array}[]{cccc}B_{\Gamma
I}&0&B_{\Gamma\Delta}&B_{\Gamma\Pi}\\\\[3.44444pt]
0&0&B_{\Delta}&0\end{array}\right]\left[\begin{array}[]{c}{\bf
w}_{I}^{(II)}\\\\[3.44444pt] q_{I}^{(II)}\\\\[3.44444pt] {\bf
w}_{\Delta}^{(II)}\\\\[3.44444pt] {\bf
w}_{\Pi}^{(II)}\end{array}\right]=\left[\begin{array}[]{c}g_{p_{\Gamma}}-B_{\Gamma\Delta}{\bf
w}^{(I)}_{\Delta}\\\\[3.44444pt] 0\end{array}\right].$
Let $w=w^{(I)}+w^{(II)}$. We can see from (54) that $w\in{\widetilde{W}}_{0}$,
cf. (16). We can also see from (49) and (55) that $B_{C}w=y$. Furthermore, by
(5),
$|w|^{2}_{\widetilde{A}}=|{\bf w}^{(I)}+{\bf w}^{(II)}|^{2}_{H^{1}}\leq|{\bf
w}^{(I)}|^{2}_{H^{1}}+|{\bf
w}^{(II)}|^{2}_{H^{1}}\leq\frac{C}{\beta^{2}}\left<A_{\Delta\Delta}{\bf
w}^{(I)}_{\Delta},{\bf
w}^{(I)}_{\Delta}\right>+\frac{C}{\beta^{2}h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>,$
where we used (48), (51), (52), and (53) for the last inequality.
On the other hand, we have
$\displaystyle\left<M^{-1}y,y\right>$ $\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+g_{\lambda}^{T}M^{-1}_{1,\lambda}g_{\lambda}=\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+g_{\lambda}^{T}B_{\Delta,D}{\widetilde{R}}_{\Delta}{\widetilde{A}}{\widetilde{R}}_{\Delta}^{T}B_{\Delta,D}^{T}g_{\lambda}$
$\displaystyle=$
$\displaystyle\frac{1}{h^{2}}\left<g_{p_{\Gamma}},g_{p_{\Gamma}}\right>+\left<A_{\Delta\Delta}{\mathbf{w}}^{(I)}_{\Delta},{\mathbf{w}}^{(I)}_{\Delta}\right>.\qquad\Box$
We also need the following lemma.
###### Lemma 9
For any $w=\left({\bf w}_{I},~{}p_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}_{0}$, $B_{C}w\in R_{G}$.
Proof: We know for any $\left({\bf f}_{I},~{}{\bf f}_{\Delta},~{}{\bf
f}_{\Pi}\right)\in{\mathbf{W}}_{I}\bigoplus{\mathbf{W}}_{\Delta}\bigoplus{\mathbf{W}}_{\Pi}$,
$g$ defined by (12) is in $R_{G}$. For any $w=\left({\bf
w}_{I},~{}p_{I},~{}{\bf w}_{\Delta},~{}{\bf
w}_{\Pi}\right)\in{\widetilde{W}}_{0}$, from the definition of
${\widetilde{A}}$ in (13), there always exists $\left({\bf f}_{I},~{}{\bf
f}_{\Delta},~{}{\bf
f}_{\Pi}\right)\in{\mathbf{W}}_{I}\bigoplus{\mathbf{W}}_{\Delta}\bigoplus{\mathbf{W}}_{\Pi}$,
such that
${\widetilde{A}}w=\left[\begin{array}[]{l}{\bf f}_{I}\\\\[3.44444pt]
0\\\\[3.44444pt] {\bf f}_{\Delta}\\\\[3.44444pt] {\bf
f}_{\Pi}\end{array}\right],\quad\mbox{i.e.,}\quad
w={\widetilde{A}}^{-1}\left[\begin{array}[]{l}{\bf f}_{I}\\\\[3.44444pt]
0\\\\[3.44444pt] {\bf f}_{\Delta}\\\\[3.44444pt] {\bf
f}_{\Pi}\end{array}\right].$
Taking such $\left({\bf f}_{I},~{}{\bf f}_{\Delta},~{}{\bf f}_{\Pi}\right)$,
$g$ defined in (12) is $B_{C}w$. $\qquad\Box$
The following lemma is an immediate result of Lemmas 8 and 9.
###### Lemma 10
The space $R_{G}$ is the same as the range of $B_{C}$ applied on
${\widetilde{W}}_{0}$.
The condition number bound of the preconditioned operator $M^{-1}G$ is given
in the following theorem.
###### Theorem 4
For all $x=(p_{\Gamma},~{}\lambda)\in R_{M^{-1}G}$,
$C\beta^{2}\left<Mx,x\right>\leq\left<Gx,x\right>\leq\Phi(H/h)\left<Mx,x\right>,$
where $\Phi(H/h)$ is as defined in Lemma 3, $\beta$ as in (5).
Proof:
$\left<Gx,x\right>=x^{T}B_{C}{\widetilde{A}}^{-1}B_{C}^{T}x=x^{T}B_{C}{\widetilde{A}}^{-1}{\widetilde{A}}{\widetilde{A}}^{-1}B_{C}^{T}x=\left<{\widetilde{A}}^{-1}B_{C}^{T}x,{\widetilde{A}}^{-1}B_{C}^{T}x\right>_{{\widetilde{A}}}.$
Since ${\widetilde{A}}^{-1}B_{C}^{T}x\in{\widetilde{W}}_{0}$ and
$\left<\cdot,\cdot\right>_{{\widetilde{A}}}$ defines an inner product on
${\widetilde{W}}_{0}$, we have
(56) $\left<Gx,x\right>=\max_{v\in{\widetilde{W}}_{0},v\neq
0}\frac{\left<v,{\widetilde{A}}^{-1}B_{C}^{T}x\right>^{2}_{\widetilde{A}}}{\left<v,v\right>_{\widetilde{A}}}=\max_{v\in{\widetilde{W}}_{0},v\neq
0}\frac{\left<B_{C}v,x\right>^{2}}{\left<{\widetilde{A}}v,v\right>}.$
Lower bound: From Lemma 8, we know that for any given
$y=(g_{p_{\Gamma}},g_{\lambda})\in R_{G}$, there exits
$w\in{\widetilde{W}}_{0}$, such that $B_{C}w=y$ and
$\left<{\widetilde{A}}w,w\right>\leq\frac{C}{\beta^{2}}\left<M^{-1}y,y\right>$.
From (56), we have
$\left<Gx,x\right>\geq\frac{\left<B_{C}w,x\right>^{2}}{\left<{\widetilde{A}}w,w\right>}\geq
C\beta^{2}\frac{\left<y,x\right>^{2}}{\left<M^{-1}y,y\right>}.$
Since $y$ is arbitrary, using Lemma 6, we have
$\left<Gx,x\right>\geq C\beta^{2}\max_{y\in R_{G},y\neq
0}\frac{\left<y,x\right>^{2}}{\left<M^{-1}y,y\right>}=C\beta^{2}\left<Mx,x\right>.$
Upper bound: From (56), Lemmas 7, 10, and 6, we have
$\displaystyle\left<Gx,x\right>$ $\displaystyle\leq$
$\displaystyle\Phi(H/h)\max_{v\in{\widetilde{W}}_{0},v\neq
0}\frac{\left<B_{C}v,x\right>^{2}}{\left<M^{-1}B_{C}v,B_{C}v\right>}$
$\displaystyle=$ $\displaystyle\Phi(H/h)\max_{y\in R_{G},y\neq
0}\frac{\left<y,x\right>^{2}}{\left<M^{-1}y,y\right>}=\Phi(H/h)\left<Mx,x\right>.\qquad\Box$
###### Remark 5
From Theorem 4 and Remark 2, we can see that the condition number bound of the
preconditioned operator $M^{-1}G$ is independent of the number of subdomains
when $H/h$ is fixed. If only subdomain corner velocities are chosen as coarse
level primal variables in the algorithm, the upper eigenvalue bound of the
preconditioned operator depends on $H/h$ in terms of $(H/h)(1+\log{(H/h)})$;
if both subdomain corner and edge-average velocity degrees of freedom are
chosen as primal variables, the upper eigenvalue bound grows as $H/h$.
###### Remark 6
With only minor modifications, the algorithm proposed in this paper and its
analysis apply equally well to the discontinuous pressure case. In that
situation, $p_{\Gamma}$ and the blocks related to it in (7) can simply be
replaced by the vector containing subdomain constant pressures and its
corresponding blocks, respectively. The formulation of the algorithm then
follows the same way as presented in Section 4, and the same condition number
bounds as in Theorem 4 will be obtained. Numerical experiments of our
algorithm for the discontinuous pressure case will also be reported in the
next section.
###### Remark 7
The same condition number bound has been proved by Kim and Lee [14, 12, with
Park] for their FETI-DP algorithms for solving incompressible Stokes
equations. In their algorithms, discontinuous pressure is considered and their
approaches do not apply to the continuous pressure case.
###### Remark 8
We also note that, no additional coarse level degrees of freedom, except those
necessary for solving positive definite elliptic problems, are required in our
algorithm to achieve a scalable convergence rate. For example, for two-
dimensional problems, it is sufficient to include only the subdomain corner
velocity degrees of freedom in the coarse level problem. This represents a
progress compared with earlier work, e.g., [20, 22], where additional
continuity constraints enforcing the divergence-free conditions on subdomain
boundaries are required in the coarse level problem. Reduction in the coarse
level problem size has also been achieved for algorithms discussed in [5, 6,
13, 14, 12], even though discontinuous pressures are considered there.
## 7 Numerical experiments
We consider solving the incompressible Stokes problem (1) in the square domain
$\Omega=[0,1]\times[0,1]$. Zero Dirichlet boundary condition is used. The
right-hand side function ${\mathbf{f}}$ is chosen such that the exact solution
is
${\bf u}=\left[\begin{array}[]{c}\sin^{3}(\pi x)\sin^{2}(\pi y)\cos(\pi
y)\\\\[3.44444pt] -\sin^{2}(\pi x)\sin^{3}(\pi y)\cos(\pi
x)\end{array}\right]\quad\mbox{and}\quad p=x^{2}-y^{2}.$
The modified Taylor-Hood mixed finite element, as shown in Figure 1, is used
for the finite element solution. The preconditioned system (44) is solved by
the CG iteration; the iteration is stopped when the $L^{2}-$norm of the
residual is reduced by a factor of $10^{-6}$.
Table 1 shows the minimum and maximum eigenvalues of the iteration matrix
$M^{-1}G$, and the iteration counts. The coarse level variable space in this
experiment is spanned by the subdomain corner velocities. We can see from
Table 1 that the minimum eigenvalue is independent of the mesh size. The
maximum eigenvalue is independent of the number of subdomains for fixed $H/h$;
for fixed number of subdomains, it depends on $H/h$, presumably in the order
of $(H/h)(1+\log{(H/h)})$ as predicted in Remark 5.
Table 1: Solving (44), with only subdomain corner velocities in coarse space. $H/h$ (fixed) | #sub | $\lambda_{min}$ | $\lambda_{max}$ | iteration
---|---|---|---|---
8 | $4\times 4$ | 0.35 | 8.92 | 21
| $8\times 8$ | 0.35 | 10.07 | 28
| $16\times 16$ | 0.35 | 10.23 | 29
| $24\times 24$ | 0.35 | 10.30 | 29
| $32\times 32$ | 0.35 | 10.33 | 29
#sub (fixed) | $H/h$ | $\lambda_{min}$ | $\lambda_{max}$ | iteration
$8\times 8$ | 4 | 0.30 | 4.22 | 21
| 8 | 0.35 | 10.07 | 28
| 16 | 0.35 | 24.22 | 36
| 24 | 0.35 | 40.12 | 43
| 32 | 0.35 | 57.15 | 50
Table 2: Solving (44), with both subdomain corner and edge-average velocities in coarse space. $H/h$ (fixed) | #sub | $\lambda_{min}$ | $\lambda_{max}$ | iteration
---|---|---|---|---
8 | $4\times 4$ | 0.36 | 4.29 | 17
| $8\times 8$ | 0.36 | 5.29 | 21
| $16\times 16$ | 0.36 | 5.56 | 21
| $24\times 24$ | 0.36 | 5.61 | 21
| $32\times 32$ | 0.36 | 5.64 | 21
#sub (fixed) | $H/h$ | $\lambda_{min}$ | $\lambda_{max}$ | iteration
$8\times 8$ | 4 | 0.33 | 4.00 | 18
| 8 | 0.36 | 5.29 | 21
| 16 | 0.36 | 11.63 | 26
| 24 | 0.36 | 18.67 | 31
| 32 | 0.36 | 26.12 | 36
For the experiment reported in Table 2, the coarse level variable space is
spanned by both the subdomain corner velocities and the subdomain edge-average
velocity components. Even though the edge-average velocity components are not
necessary for the analysis, including them in the coarse level problem
improves the convergence rate, for which the maximum eigenvalue in Table 2
grows in the order of $H/h$, as discussed in Remark 5.
Tables 3 and 4 show the performance of our algorithm for solving the same
problem, but using a mixed finite element with discontinuous pressure. We use
a uniform mesh of triangles, shown on the left in Figure 2; the velocity
finite element space contains the piecewise linear functions on the mesh and
the pressure is a constant on each union of four triangles as shown on the
right in the figure. The same mixed finite element has also been used in [22].
Figure 2: The mesh and the mixed finite element.
Comparing Tables 1 and 2 with Tables 3 and 4, we can see that the convergence
rates of our algorithm, using either continuous or discontinuous pressure, are
quite similar.
Table 3: Solving (44) (using discontinuous pressure), with only corner constraints. $H/h$ (fixed) | #sub | $\lambda_{min}$ | $\lambda_{max}$ | iteration
---|---|---|---|---
8 | $4\times 4$ | 0.48 | 7.93 | 22
| $8\times 8$ | 0.48 | 9.00 | 25
| $16\times 16$ | 0.48 | 9.20 | 25
| $24\times 24$ | 0.48 | 9.20 | 25
| $32\times 32$ | 0.48 | 9.21 | 25
#sub (fixed) | $H/h$ | $\lambda_{min}$ | $\lambda_{max}$ | iteration
$8\times 8$ | 4 | 0.41 | 3.91 | 19
| 8 | 0.48 | 9.00 | 25
| 16 | 0.49 | 21.39 | 36
| 24 | 0.50 | 35.56 | 43
| 32 | 0.50 | 50.87 | 50
Table 4: Solving (44) (using discontinuous pressure), with both corner and edge-average constraints. $H/h$ (fixed) | #sub | $\lambda_{min}$ | $\lambda_{max}$ | iteration
---|---|---|---|---
8 | $4\times 4$ | 0.48 | 3.78 | 17
| $8\times 8$ | 0.49 | 4.47 | 18
| $16\times 16$ | 0.49 | 4.68 | 19
| $24\times 24$ | 0.50 | 4.77 | 19
| $32\times 32$ | 0.50 | 4.80 | 19
#sub (fixed) | $H/h$ | $\lambda_{min}$ | $\lambda_{max}$ | iteration
$8\times 8$ | 4 | 0.43 | 2.80 | 16
| 8 | 0.49 | 4.47 | 18
| 16 | 0.50 | 9.85 | 26
| 24 | 0.50 | 16.05 | 32
| 32 | 0.50 | 22.67 | 37
## Acknowledgment
The authors are very grateful to Olof Widlund and Clark Dohrmann for their
suggestion of this problem.
## References
* [1] H. Benhassine and A. Bendali, A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem, Math. Model. Numer. Anal., 45 (2011), pp. 675–696.
* [2] D. Braess, Finite Elements, 2nd ed., Cambridge, Cambridge, UK, 2001.
* [3] C. R. Dohrmann, Preconditioning of saddle point systems by substructuring and a penalty approach, in Lecture Notes in Computational Science and Engineering, 55, Springer, New York, 2006, pp. 53–64.
* [4] C. R. Dohrmann, An approximate BDDC preconditioner, Numer. Linear Algebra Appl., 14 (2007), pp. 149–168.
* [5] C. R. Dohrmann and O. B. Widlund, An overlapping Schwarz algorithm for almost incompressible elasticity, SIAM J. Numer. Anal., 47 (2009), pp. 2897–2923.
* [6] C. R. Dohrmann and O. B. Widlund, Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity, Internat. J. Numer. Methods Engrg., 82 (2010), pp. 157–183.
* [7] C. Farhat and F. X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Internat. J. Numer. Meths. Engrg., 32 (1991), pp. 1205–1227.
* [8] C. Farhat, J. Mandel, and F. X. Roux, Optimal convergence properties of the FETI domain decomposition method, Comput. Meths. Appl. Mech. Engrg., 115 (1994), pp. 367–388.
* [9] P. Goldfeld, Balancing Neumann-Neumann preconditioners for the mixed formulation of almost-incompressible linear elasticity, Technical Report TR2003-847, Courant Institute of Mathematical Sciences, New York University, September, 2003.
* [10] P. Goldfeld, L. Pavarino, and O. B. Widlund, Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity, Numer. Math., 95 (2003), pp. 283–324.
* [11] H. H. Kim and C.-O. Lee, A Neumann-Dirichlet preconditioner for a FETI–DP formulation of the two-dimensional Stokes problem with mortar methods, SIAM J. Sci. Comput., 28 (2006), pp. 1133–1152.
* [12] H. H. Kim, C.-O. Lee, and E.-H. Park, A FETI-DP formulation for the Stokes problem without primal pressure components, SIAM J. Numer. Anal., 47 (2010), pp. 4142–4162.
* [13] H. H. Kim and C.-O. Lee, A two-level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns, Internat. J. Numer. Methods Engrg., 88 (2011), pp. 1390–1410.
* [14] H. H. Kim and C.-O. Lee, A FETI-DP formulation for the three-dimensional Stokes problem without primal pressure unknowns, SIAM J. Sci. Comput., 28 (2010), pp. 3301–3322.
* [15] H. H. Kim and X. Tu, A three-level BDDC algorithm for mortar discretization, SIAM J. Numer. Anal., 47 (2009), pp. 1576-1600.
* [16] A. Klawonn and L. F. Pavarino, Overlapping Schwarz methods for mixed linear elasticity and Stokes problems, Comput. Methods Appl. Mech. Engrg., 165 (1998), pp. 233–245.
* [17] A. Klawonn and O. Rheinbach, Inexact FETI-DP methods, Internat. J. Numer. Methods Engrg., 69 (2007), pp. 284–307.
* [18] A. Klawonn and O. B. Widlund, Dual-Primal FETI methods for linear elasticity, Comm. Pure Appl. Math., 59 (2006), pp. 1523–1572.
* [19] A. Klawonn, O. B. Widlund, and M. Dryja, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40 (2002), pp. 159–179.
* [20] J. Li, A dual-primal FETI method for incompressible Stokes equations, Numer. Math., 102 (2005), pp. 257–275.
* [21] J. Li and O. B. Widlund, FETI–DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg., 66 (2006), pp. 250–271.
* [22] J. Li and O. B. Widlund, BDDC algorithms for incompressible Stokes equations, SIAM J. Numer. Anal., 44 (2006), pp. 2432–2455.
* [23] J. Li and O. B. Widlund, On the use of inexact subdomain solvers for BDDC algorithms, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 1415–1428.
* [24] L. F. Pavarino and O. B. Widlund, Balancing Neumann-Neumann methods for incompressible Stokes equations, Comm. Pure Appl. Math., 55 (2002), pp. 302–335.
* [25] L. F. Pavarino, O. B. Widlund, and S. Zampini, BDDC preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions, SIAM J. Sci. Comput., 32 (2010), pp. 3604–3626.
* [26] J. Šístek, B. Sousedik, P. Burda, J. Mandel, and J. Novotny, Application of the parallel BDDC preconditioner to the Stokes flow, Comput. Fluids, 46 (2011), pp. 429–435.
* [27] C. Taylor and P. Hood, Numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids, 1 (1973), pp. l–28.
* [28] X. Tu, A BDDC algorithm for a mixed formulation of flows in porous media, Electron. Trans. Numer. Anal., 20 (2005), pp. 164–179.
* [29] X. Tu, A BDDC algorithm for flow in porous media with a hybrid finite element discretization, Electron. Trans. Numer. Anal., 26 (2007), pp. 146–160.
* [30] X. Tu, Three-level BDDC, in Lecture Notes in Computational Science and Engineering, 55, Springer, New York, 2006, pp. 437–444.
* [31] X. Tu, Three-level BDDC in two dimensions, Internat. J. Numer. Methods Engrg., 69 (2007), pp. 33–59.
* [32] X. Tu, Three-level BDDC in three dimensions, SIAM J. Sci. Comput., 29 (2007), pp. 1759–1780.
* [33] X. Tu, A three-level BDDC algorithm for saddle point problems, Numer. Math., 119 (2011), pp. 189–217.
* [34] A. Toselli and O. B. Widlund, Domain Decomposition Methods - Algorithms and Theory, Springer, Berlin-Heidelberg-New York, 2004.
|
arxiv-papers
| 2012-04-09T15:27:17 |
2024-09-04T02:49:29.499797
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jing Li and Xuemin Tu",
"submitter": "Xuemin Tu",
"url": "https://arxiv.org/abs/1204.1899"
}
|
1204.1934
|
# Collective Almost Synchronization in Complex Networks
M. S. Baptista1, Hai-Peng Ren2,1, J. C. M. Swarts3, R. Carareto4,1, H.
Nijmeijer3, C. Grebogi1 1Institute for Complex Systems and Mathematical
Biology, University of Aberdeen, SUPA, AB24 3UE Aberdeen, United Kingdom
2Department of Information and Control Engineering, Xi’an University of
technology, 5 Jinhua South Road, Xi’an, 710048, China 3Department of
Mechanical Engineering, Dynamics and Control Group, Eindhoven University of
Technology, WH 0.144, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
4Escola Politecnica, Universidade de São Paulo, Avenida Prof. Luciano
Gualberto, travessa 3, n. 158, 05508-900 São Paulo, SP, Brazil
###### Abstract
This work introduces the phenomenon of Collective Almost Synchronization
(CAS), which describes a universal way of how patterns can appear in complex
networks even for small coupling strengths. The CAS phenomenon appears due to
the existence of an approximately constant local mean field and is
characterized by having nodes with trajectories evolving around periodic
stable orbits. Common notion based on statistical knowledge would lead one to
interpret the appearance of a local constant mean field as a consequence of
the fact that the behavior of each node is not correlated to the behaviors of
the others. Contrary to this common notion, we show that various well known
weaker forms of synchronization (almost, time-lag, phase synchronization, and
generalized synchronization) appear as a result of the onset of an almost
constant local mean field. If the memory is formed in a brain by minimising
the coupling strength among neurons and maximising the number of possible
patterns, then the CAS phenomenon is a plausible explanation for it.
Spontaneous emergence of collective behavior is common in nature
bouchaud_MD2000 ; couzin_ASB2003 ; helbing_nature2000 . It is a natural
phenomenon characterized by a group of individuals that are connected in a
network by following a dynamical trajectory that is different from the
dynamics of their own. Since the work of Kuramoto kuramoto_LNP1975 , the
spontaneous emergence of collective behavior in networks of phase oscillators
with full connected nodes or with nodes connected by some special topologies
acebron_RMP2005 is analytically well understood. Kuramoto considered a fully
connected network of an infinite number of phase oscillators. If $\theta_{i}$
is the variable describing the phase of an oscillator $i$ in the network, and
$\overline{\theta}$ represents the mean field defined as
$\overline{\theta}=\frac{1}{N}\sum_{i=1}^{N}\theta_{i}$, collective behavior
appears in the network because every node becomes coupled to the mean field.
Peculiar characteristics of this collective behavior is that not only
$\theta_{i}\neq\overline{\theta}$ but also nodes evolve in a way that cannot
be described by the evolution of only one individual node, when isolated from
the network.
In contrast to collective behavior, another widely studied behavior of a
network is when all nodes behave equally, and their evolution can be described
by an individual node when isolated from the network. This state is known as
complete synchronization fujisaka_PTP1983 . If $x_{i}$ represents the state
variables of an arbitrary node $i$ of the network and $x_{j}$ of another node
$j$, and $\overline{x}$ represents the mean field of a network, complete
synchronization appears when $x_{i}=x_{j}=\overline{x}$, for all time. The
main mechanisms responsible for the onset of complete synchronization in
dynamical networks were clarified in pecora_PRL1998 ; nijmeijer_PHYSICAD2009 ;
nijmeijer_IEEE2011 . In networks whose nodes are coupled by non-linear
functions, such as those that depend on time-delays nijmeijer_IEEE2011 or
those that describe how neurons chemically connect baptista_PRE2010 , the
evolution of the synchronous nodes might be different from the evolution of an
individual node, when isolated from the network. However, when complete
synchronization is achieved in such networks, $x_{i}=x_{j}=\overline{x}$.
In natural networks as biological, social, metabolic, neural networks, etc,
barabasi_RMP2002 , the number of nodes is often large but finite; the network
is not fully connected and heterogeneous. The later means that each node has a
different dynamical description or the coupling strengths are not all equal
for every pair of nodes, and one will not find two nodes, say it $x_{i}$ and
$x_{j}$, that have equal trajectories. For such heterogeneous networks, as in
zhou_CHAOS2006 ; gardenes_chaos2011 , found in natural networks and in
experiments juergen_book , one expects to find other weaker forms of
synchronous behavior, such as practical synchronization femat_PLA1999 , phase
synchronization juergen_book , time-lag synchronization rosemblum_PRL1997 ,
and generalized synchronization rulkov_PRE1995 .
We report a phenomenon that may appear in complex networks “far away” from
coupling strengths that typically produce complete synchronization or these
weaker forms of synchronization. However, the reported phenomenon can be
characterized by the same conditions used to verify the existence of these
weaker forms of synchronization. We call it Collective Almost Synchronization
(CAS). It is a consequence of the appearance of an approximately constant
local mean field and is characterized by having nodes with trajectories
evolving around stable periodic orbits, denoted by $\mathbf{\Xi}_{p_{i}}(t)$,
and regarded as a CAS pattern. The appearance of an almost constant mean field
is associated with a regime of weak interaction (weak coupling strength) in
which nodes behave independently jirsa_CN2008 ; batista_PRE2007 . In such
conditions, even weaker forms of synchronization are ruled out to exist. But,
contrary to common notion based on basic statistical arguments, we show that
actually it is the existence of an approximately constant local mean field
that paves the way for weaker forms of synchronization (such as almost, time-
lag, phase, or generalized synchronization) to occur in complex networks.
Denote all the $d$ variables of a node $i$ by ${\mathbf{x}}_{i}$, then we
define that this node presents CAS if the following inequality
$|\mathbf{x}_{i}(t)-\mathbf{\Xi}_{p_{i}}(t-\tau_{i})|<\epsilon_{i}$ (1)
is satisfied for most of the time. The double vertical bar $|\ |$ represents
that we are taking the absolute difference between vector components appearing
inside the bars ($L1$ norm). $\epsilon_{i}$ is a small quantity, not
arbitrarily small, but reasonably smaller than the envelop of the oscillations
of the variables $\mathbf{x}_{i}(t)$. $\mathbf{\Xi}_{p_{i}}(t)$ is the
$d$-dimensional CAS pattern. It is determined by the effective coupling
strength $p_{i}$, a quantity that measures the influence on the node $i$ of
the nodes that are connected to it, and the expected value of the local mean
field at the node $i$, denoted by $\mathbf{C}_{i}$. The local mean field,
denoted by $\overline{\mathbf{x}}_{i}$, is defined only by the nodes that are
connected to the node $i$. The CAS pattern is the solution of a simplified set
of equations describing the network when
$\overline{\mathbf{x}}_{i}=\mathbf{C}_{i}$. According to Eq. (1), if a node in
the network presents the CAS pattern, its trajectory stays intermittently
close to the CAS pattern but with a time-lag between the trajectories of the
node and of the CAS pattern. This property of the CAS phenomenon shares
similarities with the way complete synchronization appears in networks of
nodes coupled under time-delay functions nijmeijer_IEEE2011 . In such
networks, nodes become completely synchronous to a solution of the network
that is different from the solution of an isolated node of the network.
Additionally, the trajectory of the nodes present a time-lag to this solution.
The CAS phenomenon inherits the three main characteristics of a collective
behavior: (a) the variables of a node $i$ ($\mathbf{x}_{i}$) differ from both
the mean field $\overline{\mathbf{x}}$ and the local mean field
$\overline{\mathbf{x}}_{i}$; (b) if the local mean fields of a group of nodes
and their effective coupling are either equal or approximately equal, that
causes all the nodes in this group to follow the same or similar behaviors;
(c) there can exist an infinitely large number of different behaviors (CAS
patterns).
If the CAS phenomenon is present in a network, other weaker forms of
synchronization can be detected. This link is fundamental when making
measurements to detect the CAS phenomenon.
In Ref. femat_PLA1999 , the phenomenon of almost synchronization is
introduced, when a master and a slave in a master-slave system of coupled
oscillators have equal phases but their amplitudes can be different. If a node
$i$ presents the CAS phenomenon [satisfying Eq. (1)] and $\tau_{i}=0$ in Eq.
(1), then the node $i$ is almost synchronous to the pattern
$\mathbf{\Xi}_{p_{i}}$.
Time-lag synchronization rosemblum_PRL1997 is a phenomenon that describes two
identical signals, but whose variables have a time-lag with respect to each
other, i.e. $\mathbf{x}_{i}(t)=\mathbf{x}_{j}(t-\tau)$. In practice, however,
an equality between $\mathbf{x}_{i}(t)$ and $\mathbf{x}_{j}(t-\tau)$ should
not be expected to be typically found, but rather
$\mathbf{x}_{i}(t)\cong\mathbf{x}_{j}(t-\tau),$ (2)
meaning that there is not a constant $\tau$ that can be found such that
$\mathbf{x}_{i}(t)=\mathbf{x}_{j}(t-\tau)$. Another suitable way of writing
Eq. (2) is by $|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau)|\leq\gamma$. If two
nodes $i$ and $j$ that present the CAS phenomenon, have the same CAS pattern,
and $\tau_{i}\neq\tau_{j}\neq 0$, then
$|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau_{ij})|\leq\epsilon_{ij}$ (3)
or alternatively $\mathbf{x}_{i}(t)\cong\mathbf{x}_{j}(t-\tau_{ij})$, for most
of the time, $\tau_{ij}$ representing the time-lag between $\mathbf{x}_{i}$
and $\mathbf{x}_{j}$. This means that almost time-lag synchronization occurs
for two nodes that present the CAS phenomenon and that are almost locked to
the same CAS pattern. Even though nodes that have equal or similar local mean
field (which usually happens for nodes that have equal or similar degrees)
become synchronous with the same CAS pattern (a stable periodic orbit), the
value of their trajectories at a given time might be different, since their
trajectories reach the neighborhood of their CAS patterns in different places
of the orbit. As a consequence, we expect that two nodes that exhibit the same
CAS should present between themselves a time-lag synchronous behavior. For
some small amounts of time, the difference
$|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau_{ij})|$ can be large, since
$\tau_{i}\neq\tau_{j}$ and $\epsilon_{i}\neq\epsilon_{j}$, in Eq. (1). The
closer $\overline{\mathbf{x}}_{i}$ and $\overline{\mathbf{x}}_{j}$ are to
$\mathbf{C}_{i}$, the smaller is $\epsilon_{ij}$ in Eq. (3).
Phase synchronization juergen_book is a phenomenon where the phase
difference, denoted by $\Delta\phi_{ij}$, between the phases of two signals
(or nodes in a network), $\phi_{i}(t)$ and $\phi_{j}(t)$, remains bounded for
all time
$\Delta\phi_{ij}=\left|\phi_{i}(i)-\frac{p}{q}\phi_{j}(t)\right|\leq S.$ (4)
In Ref. juergen_book $S=2\pi$ and $p$ and $q$ are two rational numbers. If
$p$ and $q$ are irrational numbers and $S$ is a reasonably small constant,
then phase synchronization can be referred as to irrational phase
synchronization baptista_PRE2004 . The value of $S$ is calculated in order to
encompass oscillatory systems that possess either a time varying time-scale or
a variable time-lag. Simply make the constant $S$ to represent the growth of
the phase in the faster time scale during one period of the slower time scale.
Phase synchronization between two coupled chaotic oscillators was explained as
being the result of a state where the two oscillators have all their unstable
periodic orbits phase-locked juergen_book . Nodes that present the CAS
phenomenon have unstable periodic orbits that are locked to the stable
periodic orbits described by $\mathbf{\Xi}_{i}(t)$. If $\mathbf{\Xi}_{i}(t)$
has a period $P_{i}$ and the phase of this CAS pattern changes $D\phi_{i}$
within one period, so the angular frequency is $\omega_{i}=D\phi_{i}/P_{i}$.
If $\mathbf{\Xi}_{j}(t)$ has a period $P_{j}$ and the phase of its CAS patter
changes $D\phi_{j}$ within one period, so the angular frequency is
$\omega_{j}=D\phi_{j}/P_{j}$. Then, the CAS patterns of these nodes are phase
synchronous by a ratio of $\frac{p}{q}=\omega_{i}/\omega_{j}$. Since the
trajectories of these nodes are locked to these patterns, the nodes are phase
synchronous by this same ratio, which can be rational or irrational. Assume
additionally that, as one changes the coupling strengths between the nodes,
the expected value $\mathbf{C}_{i}$ of the local mean field of a group of
nodes remains the same. As a consequence, as one changes the coupling
strengths, both the CAS pattern and the ratio
$\frac{p}{q}=\frac{p_{j}D\phi_{i}}{p_{i}D\phi_{j}}$ remain unaltered, and the
observed phase synchronization between nodes in this group is stable under
parameter alterations.
Consider a network of $N$ nodes with nodes connected diffusively (more general
networks are treated in the Supplementary Information) described by
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})+\sigma\sum_{j=1}^{N}{\mathbf{A}_{ij}}{\mathbf{E}}(\mathbf{x}_{j}-\mathbf{x}_{i}),$
(5)
where $\mathbf{x}_{i}\in\Re^{d}$ is a d-dimensional vector describing the
state variables of the node $i$, $\mathbf{F}_{i}$ represents the dynamical
system of the node $i$, and ${\mathbf{A}_{ij}}$ is the adjacent matrix. If
$A_{ij}=1$, then, the node $j$ is connected to the node $i$. ${\mathbf{E}}$ is
the coupling function The degree of a node can be calculated by
$k_{i}=\sum_{j=1}^{N}A_{ij}$.
The CAS phenomenon appears when the local mean field of a node $i$,
$\overline{\mathbf{x}}_{i}(t)=1/k_{i}\sum_{j}A_{ij}\mathbf{x}_{j}$, is
approximately constant and
$\overline{\mathbf{x}}_{i}(t)\approxeq\mathbf{C}_{i}$. Then, the equations for
the network can be described by
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}E(\mathbf{x}_{i})+p_{i}E(\mathbf{C}_{i})+\mathbf{\delta}_{i},$
(6)
where $p_{i}=\sigma k_{i}$ and the residual term is
$\mathbf{\delta}_{i}=p_{i}(\overline{\mathbf{x}}_{i}(t)-\mathbf{C}_{i})$. The
CAS pattern of the node $i$ (a stable periodic orbit) is calculated in the
variables that produce a finite bounded local average field. If all components
of $\mathbf{x}_{i}$ are bounded, then the CAS pattern is given by a solution
of
$\dot{\mathbf{\Xi}}_{p_{i}}=F_{i}(\mathbf{\Xi}_{p_{i}})-p_{i}E(\mathbf{\Xi}_{p_{i}})+p_{i}E(\mathbf{C}_{i}).$
(7)
which is just the same set of equations (6) without the residual term. So, if
$\overline{\mathbf{x}}_{i}(t)=\mathbf{C}_{i}$, the residual term
$\mathbf{\delta}_{i}=0$, and if Eq. (7) has no positive Lyapunov exponents
($\mathbf{\Xi}_{p_{i}}$ is a stable periodic orbit), then the node $x_{i}$
describes a stable periodic orbit. If
$\overline{\mathbf{x}}_{i}(t)-\mathbf{C}_{i}$ is larger than zero but
$\mathbf{\Xi}_{p_{i}}$ is a stable periodic orbit, then the node $x_{i}$
describes a perturbed version of $\mathbf{\Xi}_{p_{i}}$. The closer
$\overline{\mathbf{x}}_{i}$ is to $\mathbf{C}_{i}$, the larger the time that
Eq. (1) is satisfied at a given time. The more stable the periodic orbit is
[the larger the largest negative Lyapunov exponents of Eq. (7)], the longer
Eq. (1) is satisfied at a given time.
If the network has unbounded state variables (as it is the case of Kuramoto
networks kuramoto_LNP1975 ), the CAS pattern is the periodic orbit of period
$T_{i}$ defined in the velocity space such that
$\dot{\mathbf{\Xi}}_{p_{i}}(t)=\dot{\mathbf{\Xi}}_{p_{i}}(t+T_{i})$.
Notice that whereas Eqs. (5) and (6) represent a $Nd$-dimensional system, Eq.
(7) has only dimension $d$.
The existence of this approximately constant local mean field is a consequence
of the Central Limit Theorem, applied to variables with correlation (for more
details, see Supplementary Information). The expected value of the local mean
field can be calculated by
$\mathbf{C}_{i}=_{\lim
t\rightarrow\infty}\frac{1}{t}\int\overline{\mathbf{x}}_{i}(t)dt,$ (8)
where in practice we consider $t$ to be large, but finite. The larger the
degree of a node, the higher is the probability for the local mean field to be
close to an expected value and smaller its variance. If the probability to
find a certain value for the local mean field of the node $i$ does not depend
on the higher order moments of $\overline{\mathbf{x}}_{i}(t)$, then this
probability tends to be Gaussian for sufficiently large $k_{i}$. As a
consequence, the variance $\mu^{2}$ of the local mean field is proportional to
$k_{i}^{-1}$.
There are two criteria for the node $i$ to present the CAS phenomenon:
Criterion 1:
The Central Limit Theorem can be applied, i.e., $\mu^{2}_{i}\propto
k_{i}^{-1}$. Therefore, the larger the degree of a node, the smaller the
variation of the local mean field $\overline{\mathbf{x}}_{i}(t)$ about its
expected value $\mathbf{C}_{i}$.
Criterion 2:
The CAS pattern $\mathbf{\Xi}_{i}(t)$ describes a stable periodic orbit. The
node trajectory can be considered to be a perturbed version of its CAS
pattern. The more stable the faster trajectories of nodes come to the
neighborhood of the periodic orbits (CAS patterns), and the longer they stay
around them.
Whenever the Central Limit Theorem applies, the random variables involved are
independent. But, the Central Limit Theorem can also be applied to variables
with correlation. If nodes that present the CAS phenomenon are locked to the
same CAS pattern, their trajectories still arrive to the CAS pattern at
different “random” times, allowing for the Central Limit Theorem to be
applied. But the time-lag between two nodes ($\tau_{ij}$) is approximately
constant, since the CAS pattern has a well defined period, and the
trajectories of these nodes are locked into it. The local mean field measured
in a node $i$ remains unaltered as one changes the coupling strength either
when the network has an infinite number of nodes (e.g. Kuramoto networks) or
the nodes have a symmetric natural measured (See Secs. C, D, and E of
Supplementary Information). However, as we show in the following example, the
local mean field remains unaltered even when the network has only a finite
number of nodes and it has a natural measure with no special symmetrical
properties.
As an example to illustration how the CAS phenomenon appears in a complex
network, we consider a scaling-free network formed by, say, $N=1000$
Hindmarsh-Rose neurons, with neurons coupled electrically. The network is
described by
$\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
y_{i}+3x_{i}^{2}-x_{i}^{3}-z_{i}+I+\sigma\sum_{j=1}^{N}A_{ij}(x_{j}-x_{i})$
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle 1-5x_{i}^{2}-y_{i}$
(9) $\displaystyle\dot{z}_{i}$ $\displaystyle=$ $\displaystyle-
rz_{i}+4r(x_{i}+1.618)$
where $I$=3.25 and $r$=0.005. The first coordinate of the equations that
describe the CAS pattern is given by
$\dot{\Xi}_{{x}_{i}}=\Xi_{{y}_{i}}+3{\Xi}_{{x}_{i}}^{2}-{\Xi}_{{x}_{i}}^{3}-{\Xi}_{{z}_{i}}+I_{i}-p_{i}{\Xi}_{{x}_{i}}+p_{i}C_{i}.$
(10)
Figure 1: [Color online] (a) Expected value of the local mean field of the
node $i$ against the node degree $k_{i}$. The error bar indicates the variance
($\mu^{2}_{i}$) of $\overline{x}_{i}$. (b) Black points indicate the value of
$C_{i}$ and $p_{i}$ for Eq. (10) to present a stable periodic orbit (no
positive Lyapunov exponents). The maximal values of the periodic orbits
obtained from Eq. (10) is shown in the bifurcation diagram in (c) considering
$C_{i}=-0.82$ and $\sigma=0.001$. (d) The CAS pattern for a neuron $i$ with
degree $k_{i}$=25 (with $\sigma=0.001$ and $C=-0.82$). In the inset, the same
CAS pattern of the neuron $i$ and some sampled points of the trajectory for
the neuron $i$ and another neuron $j$ with degree $k_{j}=25$. (e) The
difference between the first coordinates of the trajectories of neurons $i$
and $j$, with a time-lag of $\tau_{ij}=34.2$. (f) Phase difference between the
phases of the trajectories for neurons $i$ and $j$.
The others are given by
$\dot{\Xi}_{{y}_{i}}=1-\Xi_{{x}_{i}}^{2}-\Xi_{{y}_{i}}$,
$\dot{\Xi}_{{z}_{i}}=-r\Xi_{{z}_{i}}+4r(\Xi_{{x}_{i}}+1.618)$. In this
network, we have numerically verified that criterion 1 is satisfied for
neurons that have degrees $k\geq 10$ if $\sigma\leq\sigma^{*}$, with
$\sigma^{*}\cong 0.001$. In Fig. 1(a), we show the expected value $C_{i}$ of
the local mean field of the first coordinate ${x}_{i}$ of a neuron $i$ with
respect to the neuron degree (indicated in the horizontal axis), for
$\sigma=0.001$. The error bar indicates the variance of $C_{i}$ which fits to
$\propto k_{i}^{-1.0071}$. In (b), we show a parameter space to demonstrate
that the CAS phenomenon is a robust and stable phenomenon. Numerical
integration of Eqs. (9) for $p_{i}\in[0.001,1]$ produces $C_{i}\in[-0.9,0.7]$.
We integrate Eq. (10) by using $C_{i}\in[-0.9,0.7]$ and $p_{i}\in[0,0.2]$, to
show that the CAS pattern is stable for most of the values. So, variations in
$C_{i}$ of a network caused by changes in a parameter do not modify the
stability of the CAS pattern calculated by Eq. (10). For $\sigma=0.001$, Eqs.
(9) yields many nodes for which $\overline{x}_{i}\cong-0.82$. So, to calculate
the CAS pattern for these nodes, we use $C_{i}=-0.82$ and $\sigma=0.001$ in
Eqs. (10). The CAS pattern obtained, as we vary $p_{i}$, is shown in the
bifurcation diagram in (c), by plotting the local maximal points of the CAS
patterns. Criterion 2 is satisfied for most of the range of values of $p_{i}$
that produces a stable periodic CAS pattern. A neuron that has a degree
$k_{i}$ is locked to the CAS pattern calculated by integrating Eqs. (10) using
$k_{i}\sigma=p_{i}$ and the measured expected value for the local mean field,
$C_{i}$. In (d), we show the periodic orbit corresponding to a CAS pattern
associated to a neuron $i$ with degree $k_{i}=25$ (for $\sigma$=0.001) and in
the inset the sampled points of the trajectories of this same neuron $i$ and
of another neuron $j$ that has not only equal degree ($k_{j}$=25), but it
feels also a local mean field of $C_{j}\cong-0.82$. In (e), we show that these
two neurons have a typical time-lag synchronous behavior. In (f), we observe
$p/q=1$ phase synchronization between these two neurons for a long time,
considering that the phase difference remains bounded by $S=6\times 2\pi$ as
defined in Eq. (4), where the number 6 is the number of spikings within one
period of the slower time-scale. In order to verify Eq. (4) for all time, we
need to choose a ratio that is approximately equal to 1 ($p/q\cong 1$), but
not exactly 1 to account for slight differences in the local mean field of
these two neurons. Since $C_{i}$ depends on $\sigma$ for networks that have
neurons possessing a finite degree, we do not expect to observe a stable phase
synchronization in this network. Small changes in $\sigma$ may cause small
changes in the ratio $p/q$. Notice however that Eq. (4) might be satisfied for
a very long time, for $p/q=1$. If neurons are locked to different CAS patterns
(and therefore have different local mean field), Eqs. (1) and (4) are both
satisfied, but phase synchronization will not be 1:1, but with a ratio of
$p/q$ (see Sec. E in Supplementary Information for an example).
If neurons in this scaling-free network become completely synchronous, it is
necessary that $\sigma(N)\geq 2\sigma^{CS}(N=2)/|\lambda_{2}|$ (Ref.
pecora_PRL1998 ). $\sigma^{CS}(N=2)\cong 0.5$ represents the value of the
coupling strength when two bidirectionally coupled neurons become completely
synchronous. $\lambda_{2}=-2.06$ is the largest non-positive eigenvalue of the
Laplacian matrix defined as $A_{ij}-\mbox{diag}{(k_{i})}$. So,
$\sigma^{CS}(N)\geq 1/2.06\cong 0.5$. The CAS phenomenon appears when
$\sigma^{CAS}(N=1000)\leq 0.001$, a coupling strength 500 times smaller than
the one which produces complete synchronization. Similar conclusions would be
obtained when one considers networks of different sizes, with nodes having the
same dynamical descriptions and same connecting topology.
Concluding, in this work we introduce the phenomenon of Collective Almost
Synchronization (CAS), a phenomenon that is characterized by having nodes
possessing approximately constant local mean fields. The appearance of an
approximately constant mean field is a consequence of a regime of weak
interaction between the nodes responsible to place the node trajectory around
stable periodic orbits. A network has the CAS phenomenon if the Central Limit
Theorem can be applied, and it exists an approximately constant mean field. In
other words, the CAS is invariant to changes in the value of the expected
value of the local mean field, that might appear due to parameter alterations
(e.g. coupling strength). If the expected value of the local field changes,
but the Central limit Theorem can still be applied, nodes of the network will
present the CAS phenomenon and the observed weak forms of synchronization
among the nodes might (or not) be preserved. As examples of how common this
phenomenon could be, we have asserted its appearance in a large networks of
chaotic maps (see supplementary information), Hindmarsh-Rose neurons, and
Kuramoto oscillators (see supplementary information). In the Supplementary
Information, we also discuss that the CAS phenomenon is a possible source of
coherent motion in systems that are models for the appearance of collective
motion in social, economical, and animal behaviour.
## I Supplementary Information
### I.1 CAS and generalized synchronization
Generalized synchronization rulkov_PRE1995 ; abarbanel_PRE1996 is a common
behavior in complex networks hung_PRE2008 ; guan_chaos2009 ; hu_chaos2010 ,
and should be expected to be found typically. This phenomenon is defined as
$x_{i}=\Phi(y_{i})$, where $\Phi$ is considered to be a continuous function.
As explained in Refs. rulkov_PRE1995 ; abarbanel_PRE1996 , generalized
synchronization appears due to the existence of a low-dimensional synchronous
manifold, often a very complicated and unknown manifold.
Recent works zhou_CHAOS2006 ; ballerini_PNAS2008 ; pereira_PRE2010 ;
gardenes_chaos2011 have reported that nodes in the network that are highly
connected become synchronous. As shown in ref. guan_chaos2009 , that is a
manifestation of generalized synchronization rulkov_PRE1995 ;
abarbanel_PRE1996 in complex networks. For a fixed coupling strength among
the nodes with heterogeneous degree distributions and for the usual
diffusively coupling configuration one should expect that the set of hub nodes
(highly connected nodes) provides a skeleton about which synchronization is
developed. Reference hramov_PRE2005 demonstrates how ubiquitous generalized
synchronization is in complex networks. It is shown that a necessary condition
for its appearance in oscillators coupled in a driven-response (master-slave)
configuration is that the modified dynamics of the response system presents a
stable periodic behavior. The modified dynamics is a set of equations
constructed by considering only the variables of the response system. In a
complex network, a modified dynamics of a node is just a system of equations
that contains only variables of that node.
An important contribution to understand why generalized synchronization is a
ubiquitous property in complex network is given by the numerical work of Ref.
guan_chaos2009 and the theoretical work of Ref. hu_chaos2010 . In Refs.
guan_chaos2009 ; hu_chaos2010 the ideas of Ref. hramov_PRE2005 are extended
to complex networks. In particular, the work of Ref. hu_chaos2010 shows that
generalized synchronization occurs whenever there is at least one node whose
modified dynamics is periodic. All the nodes that have a stable and periodic
modified dynamics become synchronous in the generalized sense with the nodes
that have a chaotic modified dynamics. The general theorem presented in Ref.
hu_chaos2010 is a powerful tool for the understanding of weak forms of
synchronization or desynchronous behaviors in complex networks. However,
identifying the occurrence of generalized synchronization does not give much
information about the behavior of the network, since the function that relates
the trajectory among the nodes that are generalized synchronous is usually
unknown. The CAS phenomenon allows one to calculate, at least in an
approximate sense, the equations of motion that describes the pattern to which
the nodes are locked to. More specifically, we can derive the set of equations
governing, in an approximate sense, the time evolution of the nodes, not
covered by the theorem in Ref. hu_chaos2010 .
Finally, if there is a node whose modified dynamics describes a stable
periodic behavior and its CAS pattern is also a stable periodic stable
behavior, then the CAS phenomenon appears when the network presents
generalized synchronization.
### I.2 CAS and other synchronous and weak-synchronous phenomena
Consider a network of $N$ nodes described by
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})+\sigma\sum_{j=1}^{N}{\mathbf{A}_{ij}}{\mathbf{E}}[\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})]+\mathbf{\zeta}_{i}(t),$
(11)
where $\mathbf{x}_{i}\in\Re^{d}$ is a d-dimensional vector describing the
state variables of the node $i$, $\mathbf{F}_{i}$ is a $d$-dimensional vector
function representing the dynamical system of the node $i$,
${\mathbf{A}_{ij}}$ is the adjacent connection matrix, ${\mathbf{E}}$ is the
coupling function as defined in pecora_PRL1998 , $\mathcal{H}$ is an arbitrary
differentiable transformation, and $\mathbf{\zeta}_{i}(t)$ is an arbitrary
random fluctuation. Assume in the following that $\mathbf{\zeta}_{i}(t)=0$.
Assume that the nodes in the network (11) have equal dynamical descriptions,
i.e., $\mathbf{F}_{i}=\mathbf{F}$, that the network is fully connected, so
every node has a degree $k_{i}=N-1$, and that
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
We can rewrite it in terms of the average field
$\overline{\mathbf{x}}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}(t)$:
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i}-\overline{\mathbf{x}}),$
(12)
where $p_{i}=\sigma k_{i}$. Therefore every node becomes “decoupled” from the
network in the sense that their interaction is all mediated by the average
field. Collective behavior is dictated by the behavior of the average field
and the individual dynamics of the node. The linear stability of the network
(12) was used in Ref. zhou_CHAOS2006 as an approximation to justify how
desynchronous behavior about the average field can appear in complex networks.
Notice that this assumption can only be rigorously fulfilled if the network is
fully connected and, therefore, it is natural to understand why the
desynchronous phenomena reported in Ref. zhou_CHAOS2006 happens for nodes
that are highly connected. One can interpret the desynchronous behavior
observed in Ref. zhou_CHAOS2006 as an almost synchronization between a node
and the mean field $\overline{\mathbf{x}}$.
The differences between complete synchronization and synchronization in the
collective sense can be explained through the following example. An
interesting solution of Eq. (12) can be obtained when
$\overline{\mathbf{x}}=\mathbf{x}_{i}(t)$, $\mathbf{x}_{i}(t)$ varying in
time. In this case, the average field is along the synchronization manifold.
The network being completely synchronous, all nodes having equal trajectories,
and $\mathbf{F}_{i}(\mathbf{x}_{i}(t))=\mathbf{x}_{i}(t)$. For such a special
network, collective behavior and complete synchronization are the same. On the
other hand, collective behavior typically appears when the coupling term
$\sigma E(\mathbf{x}_{i}-\overline{\mathbf{x}})$ is different from zero for
most of the time and $\mathbf{F}_{i}(\mathbf{x}_{i})\neq\mathbf{x}_{i}$, but
there is a majority of nodes with similar behavior. In this sense, the
desynchronous behaviors reported in Ref. zhou_CHAOS2006 can be considered as
a collective phenomena that happens to parameters close to the ones that
yields complete synchronization.
To understanding when the CAS phenomenon occurs, consider the solution of Eq.
(12) in the thermodynamics limit $N\rightarrow\infty$ when
$\overline{\mathbf{x}}$ is a constant in time, $\overline{\mathbf{x}}=C$. For
such a situation, the evolution of a node can be described by the same
following d-dimensional system of ODEs
$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x}-\mathbf{C}),$
(13)
where $p=\sigma(N-1)$. If complete synchronization takes place, then
$\mathbf{F}_{i}(\mathbf{C})=0$, meaning that there can only exist complete
synchronization if all the nodes lock into the same stable steady state
equilibrium point, likely to happen if $\mathbf{F}_{i}$ is the same for all
the nodes.
Another possible network configuration that leads to
$\overline{\mathbf{x}}=\mathbf{C}$ happens when each node is only weakly
coupled (“independent”) with the others such that the Central Limit Theorem
could be applied. If the network has only a finite number of nodes and
$\overline{\mathbf{x}}(t)$ is not exactly constant in time, but
$\overline{\mathbf{x}}(t)\approxeq\mathbf{C}$, the nodes still behave in the
same predictable way if the dynamics described by
$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x})+p\mathbf{E}(\mathbf{C})$
is a sufficiently stable periodic orbit. This is how the CAS phenomenon
appears in fully connected networks. All nodes become locked to the stable
periodic orbit described by
$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x})+p\mathbf{E}(\mathbf{C})$.
Now, we break the symmetry of the network, allowing the nodes to be connected
arbitrarily to their neighbors. We still consider diffusive linear couplings,
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
The equations of such a network can be written as
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i})+p_{i}\mathbf{E}(\overline{\mathbf{x}}_{i}(t)),$
(14)
where $k_{i}$ is the degree of node $i$ with $k_{l}\leq k_{m}$, if $l<m$, and
$\overline{\mathbf{x}}_{i}(t)$ is the local mean field defined as
$\overline{\mathbf{x}}_{i}(t)=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\mathbf{x}_{j}(t).$
(15)
Our main assumption is that the local mean field of a variable that is
bounded, either $\overline{\mathbf{x}}_{i}(t)$ or
$\overline{\dot{\mathbf{x}}}_{i}(t)$, exhibits small oscillations about an
expected constant value $\mathbf{C}$. In other words, one can define a time
average $\mathbf{C}$ by either
$\mathbf{C}_{i}=\frac{1}{t}\int_{0}^{t}\overline{\mathbf{x}}_{i}(t)dt,$ (16)
or
$\mathbf{C}_{i}=\frac{1}{t}\int_{0}^{t}\overline{\dot{\mathbf{x}}}_{i}(t)dt.$
(17)
Notice that ${\mathbf{x}}_{i}\in\Re^{d}$ (or
$\dot{\mathbf{x}}_{i}\in\Re^{d}$), and so does $\mathbf{C}\in\Re^{d}$. The CAS
phenomenon appears for a node that has at least one component of the local
mean field ($\overline{\mathbf{x}}_{i}$ or $\overline{\dot{\mathbf{x}}}_{i}$)
that is approximately constant. The appearance of this almost constant value
is a consequence of the Central Limit Theorem. For networks whose nodes are
described by only bounded variables, when calculating the local mean field we
only take into consideration the component receiving the couplings from other
nodes. For networks of Kuramoto oscillators that have one variable (the phase
$\theta$) that is not bounded, a constant local mean field appears in the
component that describes the instantaneous frequency ($\dot{\theta}_{i}$).
In Ref. hu_chaos2010 , it was shown that for chaotic networks described by a
system of equations similar to Eq. (14), generalized synchronization can
appear if the modified dynamics described by
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-\sigma
k_{i}\mathbf{E}(\mathbf{x}_{i})$ of a certain number of nodes are either
stable equilibrium points ($\dot{\mathbf{x}}_{i}$=0) or they describe stable
periodic solutions (limit cycle). Generalized synchronization appears between
the nodes that have modified dynamics describing stable periodic states and
the nodes that have modified dynamics describing chaotic states.
To understand the phenomenon of collective almost synchronization (CAS),
introduced in this work, consider that
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
It is a phenomena that appears necessarily when
$\overline{\mathbf{x}}_{i}\approxeq\mathbf{C}_{i}$ or
$\overline{\dot{\mathbf{x}}}_{i}\approxeq\mathbf{C}_{i}$. The equations for
the network can then be described by
$\dot{\mathbf{x}}_{i}=F_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i})+p_{i}\mathbf{E}(\mathbf{C}_{i})+\mathbf{\delta}_{i},$
(18)
where the residual term is
$\delta_{i}=p_{i}(\overline{\mathbf{x}}_{i}-\mathbf{C}_{i})$. This term is
small most of the time but large for some intervals of time;
$\mathbf{\delta}_{i}(t)>0$ for all time, but $\mathbf{\delta}_{i}(t)<\epsilon$
for most of the times. Another requirement for the CAS phenomenon to appear is
that the CAS pattern $\mathbf{\Xi}_{i}(t)$ of a node $i$ that is described by
Eq. (18) ignoring the residual term
$\dot{\mathbf{\Xi}}_{i}=\mathbf{F}_{i}(\mathbf{\Xi}_{i})-p_{i}\mathbf{E}(\mathbf{\Xi}_{i})+p_{i}\mathbf{E}(\mathbf{C}_{i}).$
(19)
must be a stable periodic orbit. We define that a node presents collective
almost synchronization (CAS) if
$|\mathbf{x}_{i}(t)-\mathbf{\Xi}_{i}(t-\tau_{i})|<\epsilon_{i},$ (20)
for most of the time,
Notice from Eq. (19) that for $p_{i}>0$, the CAS pattern will not be described
by $\mathbf{F}(\mathbf{x}_{i})$ and therefore does not belong to the
synchronization manifold. On the other hand, $\mathbf{\Xi}_{i}$ is induced by
the local mean field as typically happens in synchronous phenomenon due to
collective behavior. This property of the CAS phenomenon shares similarities
with the way complete synchronization appears in networks of nodes coupled
under time-delay functions nijmeijer_IEEE2011 . In such networks, nodes become
completely synchronous to a solution of the network that is different from the
solution of an isolated node of the network. Additionally, the trajectory of
the nodes present a time-lag to this solution.
To understand the reason why the CAS phenomenon appears when
$\mathbf{\Xi}_{i}(t)$ is a sufficiently stable periodic orbit, we study the
variational equation of the CAS pattern (19)
$\dot{\mathbf{\xi}}_{i}=[D\mathbf{F}_{i}(\mathbf{\xi}_{i})-p_{i}\mathbf{E}]\mathbf{\xi}_{i}.$
(21)
obtained by linearizing Eq. (19) around $\mathbf{\Xi}_{i}$ by making
$\mathbf{\xi}_{i}=\mathbf{x}_{i}-\mathbf{\Xi}_{i}$. This equation produces no
positive Lyapunov exponents. As a consequence, neglecting the existence of the
time-lag between $\mathbf{x}_{i}(t)$ and $\mathbf{\Xi}(t)_{i}$, the trajectory
of the node $i$ oscillates about $\mathbf{\Xi}_{i}$, and
$\mathbf{x}_{i}-\mathbf{\Xi}_{i}\leq\epsilon_{i}$, for most of the time,
satisfying Eq. (20), where $\epsilon_{i}$ depends on $\mathbf{\delta}_{i}$. If
there are two nodes $i$ and $j$, which feel similar local mean fields,
$\mathbf{\Xi}_{i}\approxeq\mathbf{\Xi}_{j}$, then
$\mathbf{x}_{i}\approxeq\mathbf{x}_{j}$, for most of the time.
To understand why the nodes that present CAS have also between them a time-lag
type of synchronization, integrate Eq. (18), using Eq. (19), to obtain
$\mathbf{x}_{i}(t)=\int_{0}^{t}[\dot{\mathbf{\Xi}}_{i}(t)+\mathbf{\delta}_{i}(t)]dt.$
(22)
This integral is not trivial in the general case. But we have a simple
phenomenological explanation for its solution. When the CAS pattern is
sufficiently stable, the asymptotic time limit state of the variable
$\mathbf{x}_{i}(t)$ is the CAS pattern $\mathbf{\Xi}_{i}(t)$. But due to the
residual term $\mathbf{\delta}_{i}(t)$, the trajectory of $\mathbf{x}_{i}(t)$
arrives in the neighborhood of $\mathbf{\Xi}(t)$ at time $t$ with a time-lag.
As a result, nodes that are collectively almost synchronous obey Eq. (20). In
addition, two nodes that present CAS have also a time-lag between their
trajectories for the same reason. There is an extra contribution to the time-
lag between the trajectories of two nodes if their initial conditions differ.
Phase synchronization juergen_book is a phenomena where the phase difference,
denoted by $\Delta\phi_{ij}$ between the phases of two signals (or nodes in a
network), $\phi_{i}(t)$ and $\phi_{j}(t)$, remains bounded for all time
$\Delta\phi_{ij}=\left|\phi_{i}(i)-\frac{p}{q}\phi_{j}(t)\right|\leq S,$ (23)
where $S=2\pi$, and $p$ and $q$ are two rational numbers juergen_book . For
coupled chaotic oscillators one can also find irrational phase synchronization
baptista_PRE2004 , where Eq. (23) can be satisfied for all time with $p$ and
$q$ irrational. $S$ is a reasonably small constant, that can be larger than
2$\pi$ in order to encompass oscillatory systems that either have a time
varying time-scale or whose time-lag varies in time. This bound can be simply
calculated by making $S$ to represent the growth of the phase in the faster
time scale after one period of the slower time scale.
The link between the CAS phenomenon and phase synchronization can be explained
by thinking that it is a synchronous phenomenon among the nodes that is
mediated by their CAS patterns. The phase of the periodic orbit of the CAS
pattern of the node $i$ grows as
$\tilde{\phi}_{i}(t)=\omega_{i}t+\xi_{i}(t)+\phi_{i}^{0}$ and of the node $j$
grows as $\tilde{\phi}_{j}(t)=\omega_{j}t+\xi_{j}(t)+\phi_{j}^{0}$. The
quantities $\phi_{i}^{0}$ and $\phi_{j}^{0}$ are displacements of the phase
caused by the existence of time-lag, and $\xi_{i}(t)$ and $\xi_{j}(t)$ are
small fluctuations. For $t\rightarrow\infty$ these can be neglected and we
have that
$\frac{\tilde{\phi}_{i}(t)}{\tilde{\phi}_{j}(t)}=\frac{\omega_{i}}{\omega_{j}}=\frac{p}{q},$
(24)
where $\omega_{i}=\lim_{t\rightarrow\infty}\frac{\tilde{\phi}_{i}(t)}{t}$
gives the average frequency of oscillation of the CAS pattern of node $i$, and
$p$ and $q$ are two real numbers.
The phase of the nodes can be written as a function of the phase of the
periodic orbits of the CAS pattern. So,
$\phi(t)_{i}=\tilde{\phi(t)}_{i}+\delta\phi_{i}(t)$ and
$\phi(t)_{j}=\tilde{\phi(t)}_{j}+\delta\phi_{j}(t)$, $\delta_{i}(t)$
represents a variation of the phase of the node $i$ with respect to the phase
of the CAS pattern, and depends on the way the phase is defined
pereira_PLA2007 . The phase difference $\Delta\phi_{ij}(t)$, as written in Eq.
(23), becomes equal to
$|t(q\omega_{i}-p\omega_{j})+q\delta_{i}(t)-p\delta\phi_{j}(t)|$. But, from
Eq. (24), $q\omega_{i}-p\omega_{j}=0$, and therefore,
$\Delta\phi_{ij}(t)\leq\max{(q\delta\phi_{i}(t)-p\delta\phi_{j}(t))}$. But
since the node orbit is locked to the CAS pattern, $\Delta\phi_{ij}(t)$ is
always a small quantity.
In practice, for networks composed by a finite number of nodes, we do not
expect that the quantities $\delta\phi_{i}(t)$ and $\delta\phi_{j}(t)$ to
remain small for all the time. The reason is that the CAS pattern can only be
approximately calculated and in general we do not know the precise real value
of the local mean field. However, our simulations show that these quantities
remain small for time intervals that comprise many periods of oscillations of
the node trajectories. For networks having an expected value of the mean field
$\mathbf{C}_{i}$ that is independent on the coupling strength $\sigma$, the
ratio $p/q$ does not change as one changes the value of $\sigma$, and then
phase synchronization is stable under a parameter variation. For the network
of Kuramoto oscillators, Eq. (23) can be verified for all time with a value of
$p/q$ that remains invariant as one changes $\sigma$.
Assume for now that the nodes have equal dynamics, so
$\mathbf{F}_{i}=\mathbf{F}$. If a node $i$ with degree $k_{i}$ has a periodic
CAS pattern that is sufficiently stable under Eq. (21), all the nodes with
degrees close to $k_{i}$ also have similar CAS patterns that are sufficiently
stable under Eq. (21). Node $i$ is locked to $\mathbf{\Xi}_{i}$ and node $j$
is locked to $\mathbf{\Xi}_{j}$. But since $\mathbf{\Xi}_{i}$ is approximately
equal to $\mathbf{\Xi}_{j}$, thus, $\mathbf{x}_{i}\cong\mathbf{x}_{j}$, for
most of the time. So, if the pattern solution is sufficiently stable, the
external noise $\mathbf{\zeta}_{i}(t)$ can be different from zero, and still
have similar trajectories for that interval of time. The same argument remains
valid if $\mathbf{F}_{i}\neq\mathbf{F}_{j}$, as long as the CAS pattern is
sufficiently stable.
In Ref. pereira_PRE2010 , synchronization was defined in terms of the node
$\mathbf{x}_{N}$ that has the largest number of connections, when
$\mathbf{x}_{i}(t)\cong\mathbf{x}_{N}$ (which is equivalent to stating that
$|\mathbf{x}_{i}(t)-\mathbf{x}_{N}|<\epsilon$), where $\mathbf{x}_{N}$ is
assumed to be very close to the synchronization manifold $\mathbf{s}$ defined
by $\dot{\mathbf{s}}=\mathbf{F}(\mathbf{s})$. This type of synchronous
behavior was shown to exist in scaling free networks whose nodes have equal
dynamics and that are linearly connected. This was called hub synchronization.
The link between the CAS phenomenon with the hub synchronization phenomenon
pereira_PRE2010 , and generalized synchronization can be explained as in the
following. It is not required for nodes that present the CAS phenomenon for
their error dynamics $\mathbf{x}_{j}-\mathbf{x}_{i}$ to be small. But for the
following comparison, assume that
$\mathbf{\vartheta}_{ij}=\mathbf{x}_{j}-\mathbf{x}_{i}$ is small so that we
can linearise Eq. (14) about another node $j$. Assume also that
$\mathbf{F}_{i}=\mathbf{F}$. The variational equations of the error dynamics
between two nodes $i$ and $j$ that have equal degrees are described by
$\dot{\mathbf{\vartheta}}_{ij}=[D\mathbf{F}(\mathbf{x}_{i})-p_{i}E]\mathbf{\vartheta}_{ij}+\mathbf{\eta}_{i}.$
(25)
In Ref. pereira_PRE2010 , hub synchronization exists if Eq. (25), neglecting
the coupling term $\mathbf{\eta}_{i}$, has no positive Lyapunov exponents.
That is another way of stating that hub synchronization between $i$ and $j$
occurs when the variational equations of the modified dynamics
$[\dot{\mathbf{x}}_{i}=\mathbf{F}(\mathbf{x}_{i})-p_{i}E(\mathbf{x}_{i})]$
presents no positive Lyapunov exponent. In other words, in order to have hub
synchronization it is necessary that the modified dynamics of both nodes be
describable by stable periodic oscillations. Hub synchronization is the result
of a weak form of generalized synchronization, defined in terms of the linear
stability of the error dynamics between two highly connected nodes. Unlike
generalized synchronization, hub synchronization offers a way to predict, in
an approximate sense, the trajectory of the synchronous nodes.
In contrast, the CAS phenomenon appears when the CAS pattern, which is
different from the solution of the modified dynamics, becomes periodic.
Another difference between the CAS and the hub synchronization phenomenon is
that whereas $\overline{\mathbf{x}}_{i}\approxeq\mathbf{C}$ in the CAS
phenomenon, $\overline{\mathbf{x}}_{i}\approxeq{\mathbf{x}}_{i}$ in the hub
synchronization, in order for $\mathbf{\eta}_{i}$ to be very small, and
$\mathbf{x}_{i}$ to be close to the synchronization manifold. So, whereas hub
synchronization can be interpreted as being a type of practical
synchronization femat_PLA1999 , CAS is a type of almost synchronization.
In the work of Refs. politi_PRE2006 ; politi_PRL2010 , it was numerically
reported a new desynchronous phenomenon in complex networks. The network has
no positive Lyapunov exponents but it presents a desynchronous non-trivial
collective behavior. A possible situation for the phenomenon to appear is when
$\mathbf{\delta}_{i}$ and $\mathbf{C}_{i}$ in Eq. (18) are either zero or
sufficiently small such that the stability of the network is completely
determined by Eq. (21), and this equation produces no positive Lyapunov
exponent. Assume now that $p_{i}$ in Eq. (19) is appropriately adjusted such
that the CAS pattern for every node $i$ is a stable periodic orbit. The
variational Eqs. (21) for all nodes have no positive Lyapunov exponents. If
additionally, $\overline{\mathbf{x}}_{i}(t)\approxeq\mathbf{C}$, then the
network in Eq. (14) possesses no positive Lyapunov exponent. Therefore,
networks that present the CAS phenomenon for all nodes might present the
desynchronous phenomenon reported in Refs. politi_PRE2006 ; politi_PRL2010 .
The CAS phenomenon becomes different from the phenomenon of
Refs.politi_PRE2006 ; politi_PRL2010 if for at least one node, Eq. (19)
produces a chaotic orbit.
To understand the occurrence of CAS in networks formed by heterogeneous nodes
connected by nonlinear functions such as networks of Kuramoto oscillators, we
rewrite the Kuramoto’s network model in terms of the local mean field,
$\overline{\theta}_{i}=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\theta_{j}$. Using
the coordinate transformation
$\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\exp{{}^{\mathbb{j}(\theta_{j}-\theta_{i})}}=\tilde{r}_{i}\exp{{}^{\mathbb{j}(\overline{\theta}_{i}-\theta_{i})}},$
(26)
the dynamics of the node $i$ is described by
$\dot{\theta_{i}}=\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i}).$
(27)
The phase ${\theta_{i}}$ is not a bounded variable and therefore we expect
that typically $\overline{\theta}_{i}$ has not a well defined average. But,
$\overline{\dot{\theta_{i}}}(t)$ is bounded and has a well defined average
value which is an approximately constant quantity ($C_{i}$) for nodes in
networks with sufficiently large number of connections and with sufficiently
small coupling strengths. When $\overline{\dot{\theta}_{i}}\cong C_{i}$, the
node $i$ has the propensity to exhibit the CAS phenomenon, and the CAS pattern
is calculated by Eq. (27) considering that $\overline{\theta_{i}}=C_{i}t$.
Notice that $\overline{\theta_{i}}=\overline{\dot{\theta}_{i}}t\cong C_{i}t$.
Phase synchronization between two nodes in the networks of Eq. (27) is stable
under parameter variations (coupling strength in this case) if these nodes
present the CAS phenomenon. There is irrational (rational) phase
synchronization if
$\frac{\overline{\dot{\theta_{i}}}}{\overline{\dot{\theta_{j}}}}$ is
irrational (rational). If nodes are sufficiently “decoupled” we expect that
$\frac{\overline{\dot{\theta_{i}}}}{\overline{\dot{\theta_{j}}}}\approxeq\omega_{i}/\omega_{j}$.
Phase synchronization will be rational whenever nodes with different natural
frequencies become locked to Arnold tongues’s, induced by the coupling
$p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i})$.
There is a special solution of Eq. (27) that produces a bounded state in the
variable ${\theta_{i}}$ when the network is complete synchronous to an
equilibrium point. In such case, $\overline{\theta}_{i}$ becomes constant, and
Eq. (27) has one stable equilibrium
$\theta_{i}=\arcsin{\left(\frac{\omega_{i}}{p_{i}}\right)}$, obtained when
$p_{i}>\omega_{i}$. But, the local mean field becomes constant due to complete
synchronization and not due to the fact that the nodes are “decoupled”. These
conditions do not produce the CAS phenomenon.
We take the thermodynamics limit when the network has infinite nodes with
infinite degrees. $C_{i}$ calculated using Eq. (17) does not change as one
change the coupling $\sigma$, since
$\overline{\dot{\theta}_{i}}=\lim_{k_{i},N\rightarrow\infty}\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}(\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta_{i}}-\theta_{i}))]$=$\lim_{k_{i},N\rightarrow\infty}\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]+[\sum_{j=1}^{N}A_{ij}(\sigma\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))]$
=
$\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]+\sigma\sum_{j=1}^{N}A_{ij}(\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))$.
But, if nodes are sufficiently decoupled
$\sum_{j=1}^{N}A_{ij}(\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))$
approaches zero, and therefore, $C_{j}$ only depends on the natural
frequencies:
$\overline{\dot{\theta}_{i}}=C_{i}=\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]$.
Assume that there are two nodes, $i$ and $j$, and that for most of the time
$\Xi_{i}\approxeq\Xi_{j}$. Then, for most of the time it is also true that
$\Xi_{i}-\theta_{i}\approxeq\Xi_{j}-\theta_{j}$, which allow us to write that
$sin(\Psi_{j}-\theta_{j})-sin(\Psi_{i}-\theta_{i})\approxeq\cos{(\Psi_{i}-\theta_{i})}[(\Psi_{j}-\theta_{j})-(\Psi_{j}-\theta_{j})]\approxeq\cos{(\Psi_{i}-\theta_{i})}[\theta_{j}-\theta_{j}]$.
Since $\Psi_{i}\approxeq\theta_{i}$, then
$\cos{(\Psi_{i}-\theta_{i})}\approxeq 1$ and
$sin(\Psi_{j}-\theta_{j})-sin(\Psi_{i}-\theta_{i})\approxeq[\theta_{j}-\theta_{j}]$.
Defining the error dynamics between the two nodes to be
$\xi_{ij}=\theta_{j}-\theta_{i}$, we arrive that
$\dot{\xi}_{ij}\approxeq(\omega_{j}-\omega_{i})-p_{i}\xi_{ij}.$ (28)
Therefore, it implies that we expect to find two nodes having the same similar
CAS behavior when both the local mean field is close and when the difference
between their natural frequencies $(\omega_{j}-\omega_{i})$ is small.
The CAS phenomenon can also appear in a system of driven particles
vicsek_PRL1995 that is a simple but powerful model for the onset of pattern
formation in population dynamics couzin_ASB2003 , economical systems
gregoire_physicaD2003 and social systems helbing_nature2000 . In the work of
Ref. vicsek_PRL1995 , it was assumed that individual particles were moving at
a constant speed but with an orientation that depends on the local mean field
of the orientation of the individual particles within a local neighborhood and
under the effect of additional external noise. Writing an equivalent time-
continuous description of the Vicsek particle model vicsek_PRL1995 , the
equations of motion for the direction of movement of a particle $i$, can be
written as
$\dot{\mathbf{x}}_{i}=-\mathbf{x}_{i}+\overline{\mathbf{x}}_{i}+\Delta\mathbf{\theta}_{i},$
(29)
where $\overline{\mathbf{x}}_{i}$ represents the local mean field of the
orientation of the particle $i$ within a local neighborhood and
$\Delta\mathbf{\theta}_{i}$ represents a small noise term. When
$\overline{\mathbf{x}}_{i}$ is approximately constant, the CAS pattern is
described by a solution of
$\dot{\mathbf{x}}_{i}=-\mathbf{x}_{i}+\overline{\mathbf{x}}_{i}$, which will
be a stable equilibrium point as long as $\Delta\mathbf{\theta}_{i}$ is
sufficiently small. From the Central Limit Theorem,
$\overline{\mathbf{x}}_{i}$ will be approximately constant as long as the
neighborhood considered is sufficiently large or the density of particles is
sufficiently large.
### I.3 About the expected value of the local mean field: the Central Limit
Theorem
The Theorem states that, given a set of $t$ observations, each set of
observation containing $k$ measurements
($x_{1},x_{2},x_{3},x_{4},\ldots,x_{k}$), the sum
$S_{N}=\sum_{i=1}^{k}x_{i}(N)$ (for $N=1,2,\ldots,t$), with the variables
$x_{i}(N)$ drawn from an independent random process that has a distribution
with finite variance $\mu^{2}$ and mean $\overline{x}$, converges to a Normal
distribution for sufficiently large $k$. As a consequence, the expected value
of these $t$ observations is given by the mean $\overline{x}$ (additionally,
$\overline{x}=\frac{1}{t}\sum_{N=1}^{t}S_{N}$), and the variance of the
expected value is given by $\frac{\mu^{2}}{k}$. The larger the number $k$ of
variables being summed, the larger is the probability with which one has a sum
close to the expected value. There are many situations when one can apply this
theorem for variables with some sort of correlation hilhorst_BJP2009 , as it
is the case for variables generated by deterministic chaotic systems with
strong mixing properties, for which the decay of correlation is exponentially
fast. In other words, a deterministic trajectory that is strongly chaotic
behaves as an independent random variable in the long-term. For that reason,
the Central Limit Theorem holds for the time average value $\overline{x}(t)$
produced by summing up chaotic trajectories from nodes belonging to a network
that has nodes weakly connected. Consequently, the distribution of
$\overline{x}_{i}(t)=\frac{1}{N}\sum_{j}A_{ij}x_{j}(t)$ for node $i$ should
converge to a Gaussian distribution centered at
$C_{i}=\frac{1}{t}\int_{0}^{t}\overline{x}_{i}(t)dt$ as the degree of the node
is sufficiently large. In addition, the variance $\mu^{2}_{i}$ of the local
mean field $\overline{x}(t)_{i}$ decreases proportional to $k_{i}^{-1}$, as we
have numerically verified for networks of Hindmarsh-Rose neurons
($\mu^{2}_{i}\propto k_{i}^{-1.0071}$) and networks of Kuramoto oscillators
($\mu^{2}_{i}\propto k_{i}^{-1.055}$).
If the network has no positive Lyapunov exponents, we still expect to find an
approximately constant local mean field at a node $i$, as long as the nodes
are weakly connected and its degree is sufficiently large. To understand why,
imagine that every node in the network stays close to a CAS pattern and one of
its coordinates is described by $sin(\omega_{i}t)$. Without loss of generality
we can make that every node has the same frequency $\omega_{i}=\omega$. The
time-lag property in the node trajectories, when they exhibit the CAS pattern,
results in that every node is close to $sin(\omega_{i}t)$ but they will have a
random time-lag in relation to the CAS pattern (due to the decorrelated
property between the node trajectories). So, the selected coordinate can be
described by $sin(\omega t+\phi^{0}_{i})+\delta_{i}(t)$, where $\phi^{0}_{i}$
is a random initial phase and $\delta_{i}(t)$ is a small random term
describing the distance between the node trajectory and the CAS pattern.
Neglecting the term $\delta_{i}(t)$, the distribution of the sum
$\sum_{i=1}^{k}sin(\omega t+\phi^{0}_{i})$ converges to a normal distribution
with a variance that depends on the variance of $sin(\phi^{0}_{i})$.
From previous considerations, if the degree of some of the nodes tend to
infinite, the variance of the local mean field for those nodes tends to zero
and, in this limit, the residual term $\delta_{i}$ in Eq. (18) is zero and the
local mean field of these nodes is a constant. As a consequence, the node is
perfectly locked with the CAS pattern ($\epsilon=0$ in Eq. (20)).
### I.4 CAS in a network of coupled maps
As another example to illustrate how the CAS phenomenon appears in a complex
network, we consider a network of maps whose node dynamics is described by
$F_{i}(x_{i})=2x_{i}$ mod(1). The network composed, say, by $N=1000$ maps, is
represented by
$x_{i}^{(n+1)}=F_{i}(x_{i}^{(n)})+\sigma\sum_{j=1}^{N}A_{ij}(x_{j}^{(n)}-x_{i}^{(n)})$
mod(1), where the upper index $n$ represents the discrete iteration time, and
$A_{ij}$ is the adjacency matrix of a scaling-free network. The map has a
constant probability density. When such a map is connected in a network, the
density is no longer constant, but still symmetric and having an average value
of 0.5. As a consequence, nodes that have a sufficient amount of connections
($k\geq 10$) feel a local mean field, say, within $[0.475,0.525]$, (deviating
of 5$\%$ about $C_{i}$=0.5) and $\mu^{2}_{i}\propto k_{i}^{-1}$ (criterion 1),
as shown in Fig. 2(a). Therefore, such nodes have propensity to present the
CAS phenomenon. In (b) we show a bifurcation diagram of the CAS pattern,
$\Xi_{i}$, obtained from Eq. (19) by using $C_{i}=C=0.5$, as we vary $p_{i}$.
Nodes in this network that have propensity to present the CAS phenomenon will
present it if additionally $p_{i}\in[1,3]$; the CAS pattern is described by a
period-2 stable orbit (criterion 2). This interval can be calculated by
solving $|2-p_{i}|\leq 1$. In (c) we show the probability density function of
the trajectory of a node that present the CAS phenomenon. The density is
centered at the position of the period-2 orbit of the CAS pattern and for most
of the time Eq. (20) is satisfied. The filled circles are fittings assuming
that the probability density is given by a Gaussian distribution. Therefore,
there is a high probability that $\epsilon_{i}$ in Eq. (20) is small. In (d)
we show a plot of the trajectories of two nodes that have the same degree
which is equal to 80. We chose nodes which present no time-lag between their
trajectories and the trajectory of the pattern. If there was a time-lag, the
points in (d) would not be only aligned along the diagonal (identity) line,
but they would also appear off-diagonal.
Figure 2: (a) Expected value of the local mean field of the node $i$ against
the node degree $k_{i}$. The error bar indicates the variance ($\mu^{2}_{i}$)
of $\overline{x}_{i}$. (b) A bifurcation diagram of the CAS pattern [Eq. (19)]
considering $C_{i}=0.5$. (c) Probability density function of the trajectory of
a node with degree $k_{i}$=80 (therefore, $p_{i}=\sigma k_{i}=1.3$,
$\sigma=1.3/80$). (d) A return plot considering two nodes ($i$ and $j$) with
the same degree $k_{i}=k_{j}=$80.
### I.5 CAS in the Kuramoto network
An illustration of this phenomenon in a network composed by nodes having
heterogeneous dynamical descriptions and a nonlinear coupling function is
presented in a random network of $N$=1000 Kuramoto oscillators. We rewrite the
Kuramoto network model in terms of the local mean field,
$\overline{\theta}_{i}=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\theta_{j}$. Using
the coordinate transformation
$\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\exp{{}^{\mathbb{j}(\theta_{j}-\theta_{i})}}=\tilde{r}_{i}\exp{{}^{\mathbb{j}(\overline{\theta}_{i}-\theta_{i})}}$,
the dynamics of node $i$ is described by
$\dot{\theta_{i}}=\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i}),$
(30)
where $\omega_{i}$ is the natural frequency of the node $i$, taken from a
Gaussian distribution centered at zero and with standard deviation of 4\. If
$\tilde{r}_{i}$=1, all nodes coupled to node $i$ are completely synchronous
with it. If $\tilde{r}_{i}$=0, there is no synchronization between the nodes
that are coupled to the node $i$. Since the phase is an unbounded variable,
the CAS phenomenon should be verified by the existence of an approximate
constant local mean field in the frequency variable $\dot{\theta_{i}}$. If
$\overline{\dot{\theta}_{i}}(t)\cong C_{i}$, which means that
$\overline{\theta_{i}}=\overline{\dot{\theta}_{i}}t\cong C_{i}t$, then Eq.
(30) describes a periodic orbit (the CAS pattern), regardless the values of
$\omega_{i}$, $p_{i}$, and $\tilde{r}_{i}$, since it is an autonomous two-
dimensional system; chaos cannot exist. Therefore, criterion 2 is always
satisfied in a network of Kuramoto oscillators. We have numerically verified
that criterion 1 is satisfied for this network for
$\sigma\leq\sigma^{CAS}(N=1000)$, where $\sigma^{CAS}(N=1000)\cong 0.075$.
Complete synchronization is achieved in this network for
$\sigma\geq\sigma^{CS}=1.25$. So, the CAS phenomenon is observed for a
coupling strength that is 15 times smaller than the one that produces complete
synchronization.
For the following results, we choose $\sigma=0.001$. Since the natural
frequencies have a distribution centered at zero, it is expected that, for
nodes with higher degrees, the local mean field is close to zero (see Fig.
3(a)). In (b), we show the variance of the local mean field of the nodes with
degree $k_{i}$. The fitting produces $\mu^{2}_{i}\propto k_{i}^{-1.055}$
(criterion 1). In (c), we show the relationship between the value of
$p_{i}\tilde{r}_{i}$ and the value of the degree $k_{i}$. In order to
calculate the CAS pattern of a node with degree $k_{i}$, we need to use the
value of $p_{i}\tilde{r}_{i}$ (which is obtained from this figure) and the
measured $C_{i}$ as an input in Eq. (30). We pick two arbitrary nodes, $i$ and
$j$, with degrees $k_{i}=96$ and $k_{j}=56$, respectively, with natural
frequencies $\omega_{i}\approxeq-5.0547$ and $\omega_{j}\approxeq-5.2080$. In
(d), we show that phase synchronization is verified between these two nodes,
with $p/q=\omega_{i}/\omega_{j}$. We also show the phase difference
$\delta\phi_{j}=\theta_{j}-\Xi_{\theta_{j}}$ between the phases of the
trajectory of the node $i$ with degree $k_{j}=96$ and the phase of its CAS
pattern, for a time interval corresponding to approximately 2500/$P$ cycles,
where the period of the cycles in node $i$ is calculated by
$P=\frac{2\pi}{5.0547}$. Phase synchronization between nodes $i$ and $j$ is a
consequence of the fact that the phase difference between the nodes and their
CAS patterns is bounded.
Figure 3: Results for $\sigma=0.001$. (a) Expected value of the local mean
field $\overline{\dot{\theta}_{i}}$ of a node with degree $k_{i}$. (b) The
variance $\mu^{2}_{i}$ of the local mean field. (c) Relationship between the
value of $p_{i}\tilde{r}_{i}$ and $k_{i}$. (d) Phase difference
$\Delta\phi_{ij}=\theta_{i}-p/q\theta_{j}$ between two nodes, one with degree
$k_{i}=96$ and the other with degree $k_{j}=56$; the phase difference
$\delta\phi_{i}=\theta_{i}-\Xi_{\theta_{i}}$ between the phases of the
trajectory of the node $i$ with degree $k_{i}=96$ and the phase of its CAS
pattern.
In the thermodynamic limit, when a fully connected network has an infinite
number of nodes, $C_{i}$ does not change as one changes the coupling $\sigma$,
since it only depends on the mean field of the frequency variable
($\overline{\dot{\theta}}$). As a consequence, if there is the CAS phenomenon
and phase synchronization between two nodes with a ratio of $p/q$ for a given
value of $\sigma$, changing $\sigma$ does not change the ratio $p/q$.
Therefore phase synchronization is stable under alterations in $\sigma$. Phase
synchronization will be rational and stable whenever nodes with different
natural frequencies $\omega_{i}$ become locked to Arnold tongues jensen ;
arnold_tongue induced by the coupling
$p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i})$.
There is a special solution of Eq. (30) that produces a bounded state in the
variable ${\theta_{i}}$ when the network is complete synchronous to an
equilibrium point. In such case, $\overline{\theta}_{i}$ becomes constant, and
Eq. (30) has one stable equilibrium
$\theta_{i}=\arcsin{\left(\frac{\omega_{i}}{p_{i}}\right)}$, obtained when
$p_{i}>\omega_{i}$. But, the local mean field becomes constant due to complete
synchronisation and not due to the fact that the nodes are “decoupled”. These
conditions do not produce the CAS phenomenon.
### I.6 Preserving the CAS pattern in different networks: a way to predict
the onset of the CAS phenomenon in larger networks
Consider two networks, $n_{1}$ and $n_{2}$, whose nodes have equal dynamical
descriptions, the network $n_{1}$ with $N_{1}$ nodes and the network $n_{2}$
with $N_{2}$ nodes ($N_{2}>N_{1}$), and two nodes, $i$ in the network $n_{1}$
and $j$ in the network $n_{2}$. Furthermore, assume that both nodes have
stable periodic CAS patterns (criteria 1 is satisfied), and assume that the
nodes have sufficiently large degrees such that the local mean field of node
$i$ is approximately equal to node $j$. Then the CAS pattern of node $i$ will
be approximately the same as the one of node $j$ if
$\sigma^{CAS}(n_{1})k_{i}(n_{1})=\sigma^{CAS}(n_{2})k_{j}(n_{2}).$ (31)
$\sigma^{CAS}(n_{1})$ and $\sigma^{CAS}(n_{2})$ represent the largest coupling
strengths for which the variance of the local mean field of a node decays with
the inverse of the degree of the node (criterion 2 is satisfied) in the
networks, respectively, and $k_{i}(n_{1})$ and $k_{j}(n_{2})$ are the degrees
of the nodes $i$ and $j$, respectively. In other words, the CAS phenomenon
occur in the network if $\sigma\leq\sigma^{CAS}$.
Therefore, if $\sigma^{CAS}(N_{1})$ is known, $\sigma^{CAS}(N_{2})$ can be
calculated from Eq. (31). In other words, if the CAS phenomenon is observed at
node $i$ for $\sigma\leq\sigma^{CAS}(N_{1})$, the CAS phenomenon will also be
observed at node $j$ for $\sigma(n_{2})\leq\sigma^{CAS}(n_{2})$, where
$\sigma^{CAS}(n_{2})$ satisfies Eq. (31).
Acknowledgment MSB acknowledges the partial financial support of the Northern
Research Partnership. HPR acknowledges the partial financial support of NSFC
Grant 60804040.
## References
* (1) R. Cont and J. P. Bouchaud, Macroecon. Dyn. 4, 170 (2000).
* (2) I. D. Couzin and J. Krause, Adv. Study Behavior 32, 1 (2003).
* (3) D. Helbing, I. Farkas, and T. Vicsek, Nature 407, 487 (2000).
* (4) Y. Kuramoto, in International Symposium on Mathematical Problems in Theoretical Physics, Vol. 39 of Lecture Notes in Physics, edited by H. Araki (Springer Berlin / Heidelberg, ADDRESS, 1975), pp. 420–422.
* (5) J. A. Acebrón et al., Rev. Mod. Phys. 77, 137 (2005).
* (6) H. Fujisaka and T. Yamada, Progress of Theoretical Physics 69, 32 (1983).
* (7) L. M. Pecora and T. Carroll, Phys. Rev. Lett. 80, 2109 (1998).
* (8) E. Steur, I. Tyukin, and H. Nijmeijer, Physica D 238, 2119 (2009).
* (9) E. Steur and H. Nijmeijer, IEEE Trans Circuits I 58, 1358 (2011).
* (10) M. S. Baptista, F. M. M. Kakmeni, and C. Grebogi, Phys. Rev. E 82, 036203 (2010).
* (11) R. Albert and A. L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
* (12) C. Zhou and J. Kurths, Chaos 16, 015104 (2006).
* (13) J. Gomez-Gardenes, Y. Moreno, and A. Arenas, Chaos 21, 016105 (2011).
* (14) A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, ADDRESS, 2001).
* (15) R. Femat and G. Solís-Perales, Phys. Lett. A 262, 50 (1999).
* (16) M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 78, 4193 (1997).
* (17) N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Phys. Rev. E 51, 980 (1995).
* (18) V. Jirsa, Cognitive Neurodynamics 2, 29 (2008), 10.1007/s11571-007-9030-0.
* (19) C. A. S. Batista et al., Phys. Rev. E 76, 016218 (2007).
* (20) M. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
* (21) H. D. I. Abarbanel, N. F. Rulkov, and M. M. Sushchik, Phys. Rev. E 53, 4528 (1996).
* (22) Y.-C. Hung, Y.-T. Huang, M.-C. Ho, and C.-K. Hu, Phys. Rev. E 77, 016202 (2008).
* (23) S. Guan et al., Chaos 19, 013130 (2009).
* (24) A. Hu, Z. Xu, and L. Guo, Chaos 20, 013112 (2010).
* (25) M. Ballerini et al., Proc. of the Nat. Acad. of Sci. 105, 1232 (2008).
* (26) T. Pereira, Phys. Rev. E 82, 1 (2010).
* (27) A. E. Hramov and A. A. Koronovskii, Phys. Rev. E 71, 067201 (2005).
* (28) T. Pereira, M. Baptista, and J. Kurths, Phys. Lett. A 362, 159 (2007).
* (29) R. Zillmer, R. Livi, A. Politi, and A. Torcini, Phys. Rev. E 74, 1 (2006).
* (30) S. Luccioli and A. Politi, Phys. Rev. Lett. 105, 1 (2010).
* (31) T. Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995).
* (32) G. Grégoire, H. Chaté, and Y. Tu, Physica D 181, 157 (2003).
* (33) H. J. Hilhorst, Brazilian J. of Physics 39, 371 (2009).
* (34) M. H. Jensen, P. Bak, and T. Bohr, Phys. Rev. A 30, 1960 (1984).
* (35) V. I. Arnold, AMS Transl. Series 46, 213 (1965).
|
arxiv-papers
| 2012-04-09T17:30:02 |
2024-09-04T02:49:29.508717
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. S. Baptista, Hai-Peng Ren, J. C. M. Swarts, R. Carareto, H.\n Nijmeijer, C. Grebogi",
"submitter": "Murilo Baptista S.",
"url": "https://arxiv.org/abs/1204.1934"
}
|
1204.1995
|
Attribute Exploration
of Gene Regulatory Processes
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
vorgelegt dem Rat der Fakultät für Mathematik und Informatik
der Friedrich-Schiller-Universität Jena
Eingereicht von : Dipl. Theol. / Dipl. Math. Johannes Wollbold Geboren am :
28.10.1958 in Saarbrücken
Gutachter
1. 1.
PD Dr. Peter Dittrich (Universität Jena)
2. 2.
Prof. Dr. Bernhard Ganter (Technische Universität Dresden)
3. 3.
PD Dr. Reinhard Guthke (Universität Jena)
Tag der öffentlichen Verteidigung: 15\. Juli 2011
###### Contents
1. ymbols and abbreviations
1. Abstract
2. 1 Introduction
1. 1.1 Acknowledgements
3. 2 Formal Concept Analysis
1. 2.1 Formal contexts and concept lattices
2. 2.2 Attribute exploration
3. 2.3 Temporal Concept Analysis (TCA)
4. 3 Algebraic and logic process modelling
1. 3.1 An unifying approach: Universal coalgebras
1. 3.1.1 Automata theory
2. 3.1.2 Kripke structures
3. 3.1.3 Labelled Transition Systems with Attributes (LTSA)
4. 3.1.4 TCA – LTSA – automata theory
2. 3.2 Temporal logics
1. 3.2.1 Propositional tense logic
2. 3.2.2 Computation Tree Logic (CTL)
3. 3.2.3 Description logics
3. 3.3 Systems biology
1. 3.3.1 Discrete models
2. 3.3.2 Boolean networks
5. 4 Modelling discrete temporal transitions by FCA
1. 4.1 Example: Installing a wireless device
2. 4.2 The state context $\mathbb{K}_{s}$ and some useful scalings
3. 4.3 The transition context $\mathbb{K}_{t}$
4. 4.4 The transitive context $\mathbb{K}_{tt}$
5. 4.5 The temporal context $\mathbb{K}_{tmp}$
6. 5 Using attribute exploration of the defined formal contexts
1. 5.1 Example
2. 5.2 Integration of knowledge and data
3. 5.3 Transition contexts and automata
7. 6 Inference rules for the integration of background knowledge
1. 6.1 A hierarchy of formal contexts
1. 6.1.1 Excursus: Model theory and Galois connections
2. 6.1.2 Background knowledge for the exploration of $\mathbb{K}_{s},\mathbb{K}_{tt},\mathbb{K}_{t}$ and $\mathbb{K}_{tmp}$
2. 6.2 Transitive and temporal contexts
1. 6.2.1 Incomplete determination of $\mathbb{K}_{tmp}$ by the stem base of $\mathbb{K}_{tt}$
2. 6.2.2 The test context
3. 6.3 Inference rules for a three element attribute set $M$
4. 6.4 Inference rules for Boolean attributes
5. 6.5 Overview of the implemented R scripts
8. 7 Gene regulatory networks I: Analysis of a Boolean network from literature
1. 7.1 Gene regulatory networks
2. 7.2 Sporulation in Bacillus subtilis
3. 7.3 Simulation starting from a state typical for the vegetative growth phase
4. 7.4 Analysis of all possible transitions
9. 8 Gene regulatory networks II: Adapted Boolean network models for extracellular matrix formation
1. 8.1 Biomedical and bioinformatics background
2. 8.2 Methods
1. 8.2.1 Clinical data
2. 8.2.2 Creating network and Boolean functions
3. 8.2.3 Data discretisation
4. 8.2.4 Principles of simulation
5. 8.2.5 Creation of a temporal rule knowledge base
6. 8.2.6 Expert analysis of transition rules
7. 8.2.7 Overview of the implemented R scripts
8. 8.2.8 Additional files
3. 8.3 Results and discussion
1. 8.3.1 Creating a regulatory network from literature information
2. 8.3.2 Boolean functions
3. 8.3.3 Gene expression time courses following TGF$\beta$1 and TNF$\alpha$ stimulation
4. 8.3.4 Data discretisation
5. 8.3.5 Boolean functions adapted to the data
6. 8.3.6 Computing temporal rules by attribute exploration
7. 8.3.7 Results of the attribute exploration
8. 8.3.8 Querying the knowledge base
9. 8.3.9 Expert centered attribute exploration
10. 9 Conclusion and outlook
1. 9.1 Transfer to Description Logics
2. 9.2 Open mathematical and logical questions
3. 9.3 Assessment of the biological applications
11. Attachments
## ymbols and abbreviations
$\bot$ | Complete attribute set, 28, 40
---|---
′ | Derivation operator for object or attribute sets, 12
$\top$ | Empty attribute set, 53
$\bigwedge$ | Infimum of an ordered set, 49
$\models$ | Modelling relation, 16
$\models$ | Semantic inference, 48
$\Box$ | Necessity operator, 25, 27, 37, 93
$\neg$ | Negation, 27
$\overline{x}$ | Negation of the proposition $x$, 64
$\sim$ | Negation, 25
$\neg F$ | Never, 37, 93
$\Diamond$ | Possibility operator, 25, 27, 37, 93
⋈ | Semiproduct of formal contexts, 15
$\bigvee$ | Supremum of an ordered set, 49
$<\,\cdot\,>$ | Support of an implication, 40
$A$ | Set of actions, 24
$\mathcal{A}$ | Coalgebra, 21
$\alpha$ | Variable assignment, 53
$\alpha_{S}$ | $S\rightarrow\Omega(S)$, mapping of a coalgebra, 21
$\operatorname{alw}$ | Always, $\Box\,G$, 38
$C(a)$ | Concept assertion (DL), 28
$D$ | Data (automaton), 21
$\Delta$ | Domain (DL), 29, 92
$\delta$ | Transition function, 21
$E$ | Entities, universe, 32
$\mathcal{EL}$ | Weak DL with tractable subsumption algorithms, 92
$\operatorname{ev}$ | Eventually, $\Diamond F$, 38
$F$ | Eventually, 25, 27, 37
$F$ | Fluents, 32
$F^{E}$ | Set of mappings $\\{E\rightarrow F\\}$, 32
$G$ | Always, 27, 37
$\gamma$ | Output function, 21
$gIm$ | $(g,m)\in I$ for a formal context $(G,M,I)$, 12
$I$ | Relation of a state context, 32
$.^{I}$ | Derivation operator for a formal context $(G,M,I)$, 12
$IF$ | Set of implication forms, 53
$\operatorname{Imp}$ | Implications valid in a formal context (for its set of intents), 47, 48
$J$ | Relation of a many-valued context, 14
$\mathbb{K}_{1}\mid\mathbb{K}_{2}$ | Apposition of formal contexts, 15
$\mathbb{K}_{T}$ | Part of $\mathbb{K}_{tmp}$ with attributes $T$, 37
$\mathbb{K}_{s}$ | State context, 32
$\mathbb{K}_{t}$ | Transition context, 34
$\mathbb{K}_{test}$ | Test context, 53
$\mathbb{K}_{tmp}$ | Temporal context, 37
$\mathbb{K}_{t}^{obs}$ | Observed transition context, 43
$\mathbb{K}_{tt}$ | Transitive context, 35
$M$ | Attribute set of a state context, 32
$\operatorname{Mod}$ | Models of a set of implications, 48
$\mathbb{N}$ | Set of integers, $0\notin\mathbb{N}$, 37
$N_{C}$ | Concept names (DL), 28
$N_{R}$ | Role names (DL), 28
$\nabla$ | Relation of a transition context, 34
$\operatorname{nev}$ | Never, $\Box\,\neg F$, 38
$O$ | Set of objects in TCA, 17
$\Omega$ | Endofunctor of an (universal) coalgebra, 21
$P$ | Pseudo-intent, 16
$\mathfrak{P}(M)$ | Power set of $M$, 17
$\phi$ | State formula (CTL), 27
$\psi$ | Path formula (CTL), 27
$R$ | Transition relation, 33
$r(a,b)$ | Role assertion (DL), 28
$S$ | State set, 21
$[s]R$ | Set of output states for $s$ given $R$, 50
$S^{\Sigma}$ | Set of mappings from an alphabet $\Sigma$ to the state set $S$, 21, 22
$s^{in}$ | Input state, 34
$s^{out}$ | Output state, 34
$Seq_{R}$ | Sequences generated by a relation $R$, 36
$\Sigma$ | Input symbols (automaton), 21
$T$ | Set of temporal attributes, 37
$T$ | Set of time granules in TCA, 17
$\mathcal{TDL}-Lite_{bool}$ | Temporally extended DL, 29
$\mathcal{TL}_{\mathcal{EL}}$ | Temporal extension of the DL $\mathcal{EL}$, 95
$U$ | Until, 27, 38
$X$ | Next, 27, 38
$t(R)$ | Transitive closure of the relation $R$, 35
BF | Boolean function, 82
BN | Boolean network, 30, 35
CTL | Computation tree logic, 27
CTSOT | Conceptual Time System with Actual Objects and a Time Relation, 17
DL | Description logics, 28
ECM | Extracellular matrix, 69
FCA | Formal concept analysis, 12
GCI | General concept inclusion (DL), 28
LTL | Linear temporal logic, 27
LTSA | Labelled Transition System with Attributes, 24
OA | Osteoarthritis, 70
RA | Rheumatoid arthritis, 69
SFB | Synovial fibroblast cell, 69
SM | Synovial membrane, 69
TCA | Temporal Concept Analysis, 17
TF | Transcription factor, 69
TGF$\beta$1 | Transforming growth factor beta I, 70
TNF$\alpha$ | Tumor necrosis factor alpha, 69, 70
### Abstract
The present thesis aims at the logical analysis of discrete processes, in
particular of such generated by gene regulatory networks. States, transitions
and operators from temporal logics are expressed in the language of Formal
Concept Analysis (FCA). This mathematical discipline is a branch of the theory
of ordered sets. It has practical applications in various fields including
data and text mining, knowledge management, semantic web, software
engineering, economics or biology. By the attribute exploration algorithm, an
expert or a computer program is enabled to validate a minimal and complete set
of implications, e.g. by comparison of predictions derived from literature
with observed data. Within gene regulatory networks, the rules of this
knowledge base represent temporal dependencies, e.g. coexpression of genes,
reachability of states, invariants or possible causal relationships.
This new approach is embedded into the theory of universal coalgebras,
particularly automata, Kripke structures and Labelled Transition Systems. A
comparison with the temporal expressivity of Description Logics (DL) is made,
since there are applications of attribute exploration to the construction of
DL knowledge bases. The main theoretical results concern the integration of
background knowledge into the successive exploration of the defined data
structures (formal contexts).
In the practical part of this work, a Boolean network from literature
modelling the initiation of sporulation in Bacillus subtilis is examined.
Coregulation and mutual exclusion of genes were checked systematically, also
dependent from specific initial states. Conditions for sporulation were
clarified by queries to the knowledge base generated by attribute exploration.
Finally, by interdisciplinary collaboration, we extracted literature
information to develop an interaction network containing 18 genes important
for extracellular matrix formation and destruction in the context of
rheumatoid arthritis. Subsequently, we constructed an asynchronous Boolean
network with biologically plausible time intervals for mRNA and protein
production, secretion and inactivation. Experimental gene expression data was
obtained from synovial fibroblast cells stimulated by transforming growth
factor beta I (TGF$\beta$1) or by tumor necrosis factor alpha (TNF$\alpha$)
and discretised thereafter. The Boolean functions of the initial network were
improved iteratively by the comparison of the simulation runs to the observed
time series and by exploitation of expert knowledge. This resulted in adapted
networks for both cytokine stimulation conditions.
The simulations were further analysed by the attribute exploration algorithm
of FCA, integrating the observed time series in a fine-tuned and automated
manner. The resulting temporal rules yielded new contributions to
controversially discussed aspects of fibroblast biology (e.g. considerable
expression of TNF and MMP9 by fibroblasts following TNF$\alpha$ stimulation)
and corroborated previously known facts (e.g. co-expression of collagens and
MMPs after TNF$\alpha$ stimulation), but also generated new hypotheses
regarding literature knowledge (e.g. MMP1 expression in the absence of FOS).
### Zusammenfassung
Ziel dieser Doktorarbeit ist die logische Analyse diskreter Prozesse,
insbesondere von genregulatorischen Netzwerken. Zustände, Transitionen und
Operatoren der temporalen Logik werden in der Sprache der Formalen
Begriffsanalyse (FCA) ausgedrückt. Diese mathematische Disziplin ist ein
Teilgebiet der Ordnungstheorie. Sie findet in vielfältigen Bereichen
praktische Anwendung, wie Data und Text mining, Wissensmanagement, Semantic
Web, Softwareentwicklung, Wirtschaft oder Biologie. Mittels des
Merkmalexplorations-Algorithmus kann ein Experte oder ein Computerprogramm
eine minimale und vollständige Menge von Implikationen validieren,
beispielsweise durch Vergleich von aus der Literatur abgeleiteten Vorhersagen
mit Beobachtungsdaten. Im Rahmen genregulatorischer Netzwerke drücken die
Regeln dieser Wissensbasis zeitliche Abhängigkeiten aus, etwa Koexpression von
Genen, Erreichbarkeit von Zuständen, Invarianten oder mögliche kausale
Zusammenhänge.
Dieser neue Ansatz wird in die Theorie der Universellen Coalgebren
eingebettet, die insbesondere Automatentheorie, Kripkestrukturen und Labelled
Transition Systems einschließt. Außerdem wird ein Vergleich zu temporalen
Aspekten von Beschreibungslogiken gezogen; Anwendungen der Merkmalexploration
auf die Konstruktion beschreibungslogischer Wissensbasen stellen ein neues
Forschungsgebiet dar. Die wichtigsten theoretischen Resultate der vorliegenden
Arbeit betreffen die Integration von Hintergrundwissen in die sukzessive
Exploration der definierten Datenstrukturen (formalen Kontexte).
Im praktischen Teil der Arbeit wird zunächst ein Boolesches Netzwerk aus der
Literatur untersucht, das die Einleitung der Sporenbildung in Bacillus
subtilis modelliert. Koregulation und gegenseitiger Ausschluss von Genen
werden systematisch untersucht, auch in Abhängigkeit von spezifischen
Ausgangszuständen. Bedingungen für die Einleitung der Sporenbildung werden
durch Abfragen der Wissensbasis geklärt, die mittels Merkmalexploration
erzeugt wurde.
Schließlich wurde in interdisziplinärer Zusammenarbeit und nach umfassender
Literaturrecherche ein Netzwerk entwickelt, das 18 Gene enthält, die für die
Bildung und den Abbau extrazellulärer Matrix im Kontext rheumatoider Arthritis
Bedeutung haben. Daraus wurde ein asynchrones Boolesches Netzwerk konstruiert
mit biologisch plausiblen Zeitintervallen für mRNA- und Proteinsynthese,
Sekretion und Inaktivierung. Experimentelle Genexpressionsdaten stammten von
synovialen Fibroblasten, die mit Transforming growth factor beta I
(TGF$\beta$1) beziehungsweise Tumor necrosis factor alpha (TNF$\alpha$)
stimuliert wurden. Die Booleschen Funktionen des anfänglichen Netzwerks wurden
in mehreren Durchläufen optimiert, indem Simulationen mit den beobachteten
Zeitverläufen verglichen wurden. Dabei wurde zusätzliches Expertenwissen auch
zur Signaltransduktion eingebracht. Daraus resultierten zwei Netzwerke, die
jeweils an eine der beiden Bedingungen angepasst waren (Stimulation durch die
beiden Zytokine).
Die endgültigen Simulationen wurden mittels Merkmalexploration untersucht,
wobei die gemessenen Zeitreihen weiter und automatisch integriert wurden. Die
erhaltenen Regeln bringen neue Aspekte in kontrovers diskutierte Fragen der
Biologie von Fibroblasten ein (z.B. beträchtliche Expression von TNF und MMP9
nach Stimulation durch TNF$\alpha$). Sie bestätigen bekannte Tatsachen wie die
Koexpression von Kollagenen und Matrix-Metalloproteasen (MMPs) nach
TNF$\alpha$-Stimulation, erzeugten aber auch bezüglich des Literaturwissens
neue Hypothesen (z.B. Expression von MMP1 in Abwesenheit des
Transkriptionsfaktors FOS).
## Chapter 1 Introduction
During the early 1980s, the mathematical methodology of Formal Concept
Analysis (FCA) emerged within the community of set and order theorists,
algebraists and discrete mathematicians. The aim was to find a new, concrete
and meaningful approach to the understanding of complete lattices (ordered
sets such that for every subset the supremum and infimum exist). The following
discovery proved fruitful: Every complete lattice is representable as a
hierarchy of concepts, which were conceived as sets of objects sharing a
maximal set of attributes. This paved the way for using the field of lattice
theory for a transparent and complete representation of very different types
of knowledge.
Originally FCA was inspired by the educationalist Hartmut von Hentig [99] and
his program of restructuring sciences aiming at interdisciplinary
collaboration and democratic control. The philosophical background traces back
to Charles S. Peirce (1839 - 1914), who condensed some of his main ideas to
the pragmatic maxim:
> _Consider what effects, that might conceivably have practical bearings, we
> conceive the objects of our conception to have. Then, our conception of
> these effects is the whole of our conception of the object._ [78, 5.402]
In that tradition, FCA aims at unfolding the observable, elementary properties
defining the objects subsumed by scientific concepts. If applied to temporal
transitions, effects of specific combinations of state attributes can be
modelled and predicted in a clear and concise manner. Thus, FCA seems to be
appropriate to describe causality – and the limits of its understanding.
At present, FCA is a well developed mathematical theory and there are
practical applications in various fields such as data and text mining,
knowledge management, semantic web, software engineering or economics [36].
The main application of this thesis is related to molecular and systems
biology. Due to the rapid accumulation of data about molecular inter-
relationships, there is an increasing demand for approaches to analyse the
resulting regulatory network models (for a short introduction see Section 7.1,
an example is represented in Figure 8.4). Therefore, we developed a formal
representation of processes, especially biological processes. The purpose was
to construct knowledge bases of rules expressing temporal dependencies within
gene regulatory (or signal transduction and metabolic) networks.
As algorithm, attribute exploration was employed: For a given set of
interesting properties, it builds a sound, complete and non-redundant set of
implications (logically strict rules). During this process, each implication
can be approved or rejected by an expert or a computer program, e.g. by
comparison of knowledge-based predictions with data. Attribute exploration
provides a mathematically strict framework for the validation of rules, and
the resulting implicational base presents the related domain knowledge to the
expert in a compact manner. This stem base is open to intuitive human
discoveries, activating resonance effects with the whole knowledge of a
scientist. Its completeness related to the explored context ensures that the
validity of an arbitrary implication of interest can be decided by logical
derivations from the stem base (automatic reasoning).
Corresponding to the discrete, logical and interactive focus of FCA, we
selected classical Boolean networks [61] for modelling, which are easy to
interpret. They consist of sets of Boolean functions, i.e. the value of one
variable (e.g. gene expressed or not) after one time step depends on the
present values of a subset of the variables. It is also possible to use
mathematical and logical derivations in order to decide many implications
automatically. Furthermore, sets of Boolean rules are applied as knowledge
bases in decision support or expert systems.
FCA was used for the analysis of gene expression data in [75] and [62]. The
present study is the first approach of applying it to the dynamics of (gene)
regulatory network models. With this application domain in mind, we developed
a formal structure as general as possible, since discrete temporal transitions
occur in a variety of domains: control of engineering processes, development
of the values of variables in a computer program, change of interactions in
social networks, a piece of music, etc. Within the domain of discrete and
symbolic process modelling, the present work aims at providing a framework
that may be useful to validate and further analyse models formulated in very
general classes of languages, the most important being the $\mu$-calculus
(comprising as a subset Linear Temporal Logic (LTL) and Computation Tree Logic
(CTL)) on the syntactic, and several types of universal coalgebras on the
semantic level. Simulations and analyses by Petri nets may be integrated as
well; for an example see Chapter 7 and page 7.1.
The thesis is structured as follows: In Chapter 2, the basic data structures
of FCA, the attribute exploration algorithm and Temporal Concept Analysis
(TCA) (applied to the analysis of gene expression data in [112]) are
introduced. In Chapter 3, automata, Kripke structures and Labelled Transition
Systems (LTS) with attributes are presented as universal coalgebras,
furthermore Propositional Tense Logic, CTL and Description Logics (DL). The DL
notion of a role may be interpreted temporally. In addition, an explicit
temporal extension is presented [15].
The starting point of the thesis consisted in further developing an FCA
language for discrete dynamic systems sketched in [40]. Results of this
modelling will be presented in Chapter 4, and the application of attribute
exploration to the defined data structures (formal contexts) in Chapter 5.
They express knowledge concerning states, transitions and attributes from
temporal logics. Chapter 6 presents a method to derive inference rules
integrating already acquired knowledge into the successive exploration of the
four formal contexts. This method uses attribute exploration on a higher
level. Section 6.4 is a revised part of my paper [111]. It investigates
inference rules for Boolean attributes.
The second major part of the present work develops a systems biology method to
analyse the dynamics of gene regulatory networks. The rules of the knowledge
bases generated by attribute exploration represent temporal relationships
within gene regulatory networks, e.g. coexpression of genes. Reachability of
states is mainly expressed by rules with the temporal operators eventually and
never, invariants by always. We focus on the corresponding semantic level,
i.e. implications pertaining to transitions (compare Remark 4.5.3 or
Proposition 6.2.1). Rules pointing at possible causal relations have a
structure like:
> _If gene 1 is expressed and gene 2 is not expressed at some state, then at
> the next state / at all following states / eventually / always gene 3 will
> be expressed._
In Chapters 7 and 8, the background, methods and results of two published
papers of own work are reported:
1. 1.
Sporulation of Bacillus subtilis: A simulation from literature was further
analysed by concept lattices and attribute exploration [111].
2. 2.
In [109], we developed by interdisciplinary collaboration a Boolean network
model for extracellular matrix (ECM) formation and degradation within the
context of rheumatic diseases. It is based on interactions reported in
literature and was adapted to gene expression time series for fibroblast cells
following transforming growth factor beta I (TGF$\beta$1) and tumor necrosis
factor alpha (TNF$\alpha$) stimulation, respectively. The resulting
simulations were further analysed by attribute exploration, integrating the
observed time series in a fine-tuned and automated manner.
In Chapter 9, the results of the biological applications are discussed.
Possibilities of further research aiming at an even better usability of this
approach are sketched. Among other mathematical and logical questions I will
outline how a state, transition or temporal context may be expressed by DL.
This is the basis so that results applying to the computation and extension of
DL knowledge bases by attribute exploration can be used as well as fast DL
reasoners [19] [20]. I will also point at reasons for concentrating on a
classical FCA framework and on parts of temporal logic being particularly
meaningful to human experts in real world applications.
### 1.1 Acknowledgements
First, I am grateful to PD Dr. Reinhard Guthke for posing the biomedical
question and for many critical hints aiming at biological meaning, realistic
applicability and suggestion of hypotheses by modelling. He offered to me
great freedom for developing a method to integrate knowledge and data. Since
Prof. Dr. Bernhard Ganter was equally broad-minded regarding the application
of FCA to biology, a large search space was opened, ranging from pure
mathematics (lattice and FCA theory, categories) over FCA applications,
different logics, gene expression data analysis, systems biology up to
specialised questions of molecular biology and genetics. I am very grateful to
Bernhard Ganter for creative discussions which generated the main ideas of
this thesis to structure the large area, to solve mathematical and logical
questions and to show the applicability on two biologically relevant
questions. PD Dr. Peter Dittrich accepted to resume the final supervision and
made many useful remarks aiming in particular at a better comprehensibility of
the FCA framework.
The collaboration with Ulrike Gausmann, René Huber, Raimund Kinne and Reinhard
Guthke resulting in [109] was very inspiring to me. At all steps of the work a
feedback was given between biological knowledge, data and formal abstractions.
Beyond programming and developing the methods for discretisation, simulation
and attribute exploration, I participated in structuring the comprehensive
literature search and constructed the Boolean network accordingly. After
intense discussions, we adapted the Boolean functions to the data. I analysed
the final temporal rules together with René Huber.
I thank Christian Hummert (HKI Jena), Michael Hecker (HKI Jena and STZ for
Proteome Analysis Rostock), Felix Steinbeck (STZ for Proteome Analysis
Rostock), Mike Behrisch and Daniel Borchmann (Institute of Algebra of Dresden
University) as well as other colleagues for fruitful discussions, for some
programming (see Section 6.5) and for reading parts of the manuscript.
I got encouragement from my supervisors as well as from many other people, to
whom I expressed my gratitude personally.
## Chapter 2 Formal Concept Analysis and the attribute exploration algorithm
This chapter provides formal definitions, a theorem as well as more intuitive
introductions to known FCA notions used from Chapter 4. There, I will give
examples of new formal structures and applications, which mostly should be
sufficient to follow the argumentation of this thesis. Instead of reading this
chapter in advance, it might therefore serve as a reference in order to
clarify notions as needed. For more detailed questions, I refer to the
textbook [41].
### 2.1 Formal contexts and concept lattices
| MMP1 | TIMP1 | MMP9
---|---|---|---
(190,0) | | |
(190,1) | | |
(190,2) | $\times$ | |
(190,4) | $\times$ | |
(190,12) | $\times$ | |
(202,0) | | |
(202,1) | | |
(202,2) | $\times$ | |
(202,4) | $\times$ | |
(202,12) | $\times$ | | $\times$
(205,0) | | |
(205,1) | | |
(205,2) | $\times$ | | $\times$
(205,4) | $\times$ | | $\times$
(205,12) | | |
(220,0) | | $\times$ | $\times$
(220,1) | | | $\times$
(220,2) | | | $\times$
(220,4) | $\times$ | $\times$ | $\times$
(220,12) | $\times$ | $\times$ | $\times$
(221,0) | | $\times$ |
(221,1) | | $\times$ |
(221,2) | $\times$ | $\times$ |
(221,4) | $\times$ | $\times$ |
(221,12) | $\times$ | $\times$ | $\times$
(87,0) | | $\times$ |
(87,1) | $\times$ | |
(87,2) | $\times$ | $\times$ |
(87,4) | $\times$ | $\times$ |
(87,12) | $\times$ | $\times$ | $\times$
Table 2.1: (One-valued) formal context representing a part of a gene
expression data set for cells from osteoarthritis patients 190, 202, 205 and
rheumatoid arthritis patients 220, 221, 87. The objects (rows) designate
measurements for a cell culture at t=0, 1, 2, 4 and 12 hours after stimulation
with TNF$\alpha$. A cross in the column for the attribute MMP1, TIMP1 or MMP9
means: The gene collagenase, tissue inhibitor of metalloproteaseses 1 or
matrix metalloprotease 9 is expressed above a threshold set by the
discretisation method proposed in [112, Section 2.3]. For the biomedical
background see Section 8.1.
One of the classical aims of FCA is the structured, compact but complete
visualisation of a data set by a conceptual hierarchy. The subsequent basic
definitions of formal contexts, scaling and formal concepts are applied in
Sections 4.1, 4.2 and 7.3.
A data table with binary attributes is called a (one-valued) formal context
(Table 2.1):
###### Definition 2.1.1.
[41, Definitions 18 and 19] A Formal Context $(G,M,I)$ defines a relation
$I\subseteq G\times M$ between objects from a set $G$ and attributes from a
set $M$. The set of the attributes common to all objects in $A\subseteq G$ is
denoted by the ${}^{\prime}\,$-operator:
$A^{\prime}:=\\{m\in M\mid(g,m)\in I\text{ for all }g\in A\\}.$
The set of the objects sharing all attributes in $B\subseteq M$ is
$B^{\prime}:=\\{g\in G\mid(g,m)\in I\text{ for all }m\in B\\}.$
If the derivation operators ′ are ambiguous, they will be denoted by the
relation of the respective formal context, e.g. $B^{I}$. Also $gIm$ will be
used instead of $(g,m)\in I$.
A formal context is called clarified, if there are no objects with the same
attribute set (object intent, see Definition 2.1.7) and no attributes with the
same extent, i.e. the context does not contain rows / columns identical except
for object / attribute names. A formal context is row reduced, if all objects
are deleted of which the intent is an intersection of other object intents.
The definition of a column reduced context is analogous. With the exception of
the test context in Section 6.2.2, the formal contexts will not be reduced,
since then the information regarding a part of the objects (for instance gene
expression measurements) or attributes (e.g. genes) is lost.
###### Definition 2.1.2.
[41, Definition 27] A Many-Valued Context $(G,M,W,J)$ consists of sets $G$,
$M$ and $W$ and a ternary relation $J\subseteq G\times M\times W$ for which it
holds that
$(g,m,w)\in J\wedge(g,m,v)\in J\Rightarrow w=v.$
$(g,m,w)\in J$ means “for the object $g$, the attribute $m$ has the value
$w$”. Thus, the many-valued attributes are identifiable with partial maps
$m\colon G\rightarrow W,$ where $m(g)=w\Leftrightarrow(g,m,w)\in I$. A many-
valued context represents a specifically formalised view on an arbitrary table
in a relational database. It is translated into a derived ordinary, one-valued
formal context by a process called conceptual scaling. Scaling makes the
mathematical results of FCA applicable to many-valued contexts and offers
manifold possibilities of data discretisation.
###### Definition 2.1.3.
[41, Definition 28] A Scale for the attribute $m$ of a many-valued context
$(G,M,W,J)$ is a (one-valued) context $\mathbb{S}_{m}:=(G_{m},M_{m},I_{m})$
with $m(G)\subseteq G_{m}$. The objects of a scale are called Scale Values,
the attributes Scale Attributes.
a) $m_{1}$ $m_{2}$ $g_{1}$ 0 1 $g_{2}$ 1 1 b) $\mathbb{N}_{2}$ $0$ $1$ $0$
$\times$ $1$ $\times$ c) $m_{1}.0$ $m_{1}.1$ $m_{2}.0$ $m_{2}.1$ $g_{1}$
$\times$ $\times$ $g_{2}$ $\times$ $\times$
Table 2.2: a) A many-valued context with object set $G:=\\{g_{1},g_{2}\\}$,
attribute set $M:=\\{m_{1},m_{2}\\}$ and values $W:=\\{0,1\\}$, b) the nominal
scale with two values, i.e. the dichotomic scale for each of the two
attributes, c) the scaled context. For instance, in Table 2.1 the attribute
MMP1 could be replaced by MMP1.0 and MMP1.1. This provides explicit
information whether a gene is not expressed and has practical relevance mainly
for attribute exploration (see Table 7.2 and the implications on page 7.2).
By (plain) scaling, an attribute $m$ is replaced by the respective row of the
scale context $\mathbb{S}_{m}$.
###### Definition 2.1.4.
[41, Definition 29] If $(G,M,W,J)$ is a many-valued context and
$\mathbb{S}_{m},\>m\in M$ are scale contexts, then the Derived Context With
Respect To Plain Scaling is the context $(G,N,I)$ with
$N:=\bigcup_{m\in M}\\{m\\}\times M_{m},$
and
$gI(m,n):\Leftrightarrow m(g)=w\text{ and }wI_{m}n.$
The most elementary example is nominal scaling [41, Definition 31]: The scale
context for an attribute $m$ of the many-valued context is a diagonal matrix,
i.e. each attribute value $w\in G_{m}\subseteq W$ is represented by itself.
Then, $m$ is replaced by derived attributes which can be mapped bijectively to
the (possible) attribute values $G_{m}=M_{m}$. Nominal scaling with two scale
attributes is called dichotomic scaling (Table 2.2). In the following, mostly
a variant of a dichotomic scale is applied, where the threshold discretisation
- for each gene separately - presupposed in Table 2.1 is made explicit, e.g.:
MMP1 | $0$ | $1$
---|---|---
$\leq 5710$ | $\times$ |
$>5710$ | | $\times$
MMP9 $0$ $1$ $\leq 144$ $\times$ $>144$ $\times$ TIMP1 $0$ $1$ $\leq 34890$
$\times$ $>34890$ $\times$
Further scaling methods will be introduced in Section 4.2.
In this thesis, two context constructions are needed.
$\overset{.}{M}_{1}\cup\overset{.}{M}_{2}$ denotes the disjoint union of two
attribute sets $M_{1}$ and $M_{2}$, i.e. $\overset{.}{M}_{j}:=\\{j\\}\times
M_{j}$ for $j\in\\{1,2\\}$. Analogously,
$\overset{\>.}{I}_{j}:=\\{(g,(j,m))\mid(g,m)\in I_{j}\\}$ for two relations
$I_{j}\subseteq G\times M_{j}$.
###### Definition 2.1.5.
[41, Definition 30] The Apposition of two formal contexts
$\mathbb{K}_{1}:=(G,M_{1},I_{1})$ and $\mathbb{K}_{2}:=(G,M_{2},I_{2})$ is
defined by
$\mathbb{K}_{1}\mid\mathbb{K}_{2}:=(G,\overset{.}{M}_{1}\cup\overset{.}{M}_{2},\overset{\>.}{I}_{1}\cup\overset{\>.}{I}_{2}).$
###### Definition 2.1.6.
[41, Definition 33] The Semiproduct of two formal contexts
$\mathbb{K}_{1}:=(G_{1},M_{1},I_{1})$ and
$\mathbb{K}_{2}:=(G_{2},M_{2},I_{2})$ is defined by
$\mathbb{K}_{1}\,\begin{sideways}\bowtie\end{sideways}\>\mathbb{K}_{2}:=(G_{1}\times
G_{2},\overset{.}{M}_{1}\cup\overset{.}{M}_{2},\nabla)$
with
$(g_{1},g_{2})\nabla(j,m):\Leftrightarrow g_{j}I_{j}m\qquad\text{for
}j\in\\{1,2\\}.$
Figure 2.1: Concept lattice (Hasse diagram) of the formal context in Table
2.1. Formal concepts are represented by black circles and the order relation
by lines. The extent of a formal concept is the set of all objects
(measurements at specific time points) at or below the circle, following the
lines. Dually, the intent is the set of all attributes above. Thus, the
semantics of the bottom concept is: all three genes are expressed for patients
87 at 12 h, 220 at 4 h and 12 h, and 221 at 12 h after stimulation with
TNF$\alpha$.
The central notion of FCA is a formal concept, a maximal set of objects
$A\subseteq G$ together with all attributes $B\subseteq M$ shared by them, or
a maximal rectangle in a formal context, after row and column permutation.
###### Definition 2.1.7.
[41, Definition 20] A Formal Concept of the context $(G,M,I)$ is a pair
$(A,B)$ with $A\subseteq G,\>B\subseteq M,\>A^{\prime}=B$ and $B^{\prime}=A$.
$A$ is the Extent, $B$ the Intent of the concept $(A,B)$.
For a context $(G,M,I)$,
$\displaystyle\mathcal{C}_{M}\colon\mathfrak{P}(M)$
$\displaystyle\rightarrow\mathfrak{P}(M)$ $\displaystyle B\subseteq M$
$\displaystyle\mapsto B^{\prime\prime}\text{, and}$
$\displaystyle\mathcal{C}_{G}\colon\mathfrak{P}(G)$
$\displaystyle\rightarrow\mathfrak{P}(G)$ $\displaystyle A\subseteq G$
$\displaystyle\mapsto A^{\prime\prime}$
are closure operators, i.e. operators with the properties monotony, extension
and idempotency [41, Definition 14]. From [41, Theorem 1] it follows that the
set of all extents and intents, respectively, of a formal context is a closure
system, i.e. it is closed under arbitrary intersections. Hence, an intent is
sometimes called a closed set [80], [53].
Formal concepts can be ordered by set inclusion of the extents or – dually and
with the inverse order relation – of the intents. With this order, the set of
all concepts of a given formal context is a complete lattice, i.e. a partially
ordered set, where supremum (join) and infimum (meet) exist for any subset. It
is visualised by a Hasse diagram (Figure 2.1).
### 2.2 Attribute exploration
Compared to a concept lattice, logical implications offer an even more compact
possibility of representing a row reduced formal context without loss of
information. An implication $A\rightarrow B$ between attribute sets holds in a
formal context $(G,M,I)$, if an object $g\in G$ having all attributes $a\in A$
(premise) has also the attributes $b\in B$ (conclusion). This is expressed by
“every object intent respects $A\rightarrow B$” or $\forall g\in G\colon
g^{\prime}\models A\rightarrow B$. The attribute exploration algorithm
generates a special set of attribute implications, the stem base. Their
premises are pseudo-intents:
###### Definition 2.2.1.
[41, Definition 40] Let $M$ be a finite set. $P\subseteq M$ is called a
Pseudo-Intent of $(G,M,I)$ if and only if $P\neq P^{\prime\prime}$ and
$Q^{\prime\prime}\subseteq P$ holds for every pseudo-intent $Q\subseteq P$,
$Q\neq P$.
This recursive definition starts with the empty set. Only the first condition
has to be checked: $\emptyset^{\prime}=G$ (no distinctive attribute is
required for an object), thus the closure $\emptyset^{\prime\prime}$ contains
the attributes common to all $g\in G$. Often they have not been made explicit
in the context under consideration, and the empty set is closed. In this case
also for the sets with one element only the closure must be tested, and so on.
The first non-closed set in a linear order of set inclusion is a pseudo-intent
$Q$. Then the second condition $Q^{\prime\prime}\subset P$ has to be tested
for every superset $P$. For an example of pseudo-intents (respectively pseudo-
closed sets) within the context of attribute exploration see p. 5.1.
The following theorem gives the theoretical foundation of attribute
exploration.
###### Theorem 2.2.2 (Duquenne-Guigues).
Given a formal context $(G,M,I)$, the set of implications
$\mathcal{L}:=\\{P\rightarrow P^{\prime\prime}\,|\,P\text{ pseudo-intent}\\}$
is sound, complete and non-redundant.
###### Proof.
By definition of the closure operator ′′, the implications in $\mathcal{L}$
respect all object intents. Thus, they hold in the underlying context
$(G,M,I)$ and $\mathcal{L}$ is sound. For the proof of completeness and non-
redundancy see [41, Theorem 8]. ∎
The set $\mathcal{L}$ is called stem base of a formal context. In general, its
implications are noted in the short form $P\rightarrow
P^{\prime\prime}\setminus P$, as $P\rightarrow P$ is trivial. Completeness
means that every implication holding in a given formal context $(G,M,I)$ can
be derived logically from $\mathcal{L}$. This property is lost, if a single
implication is removed from the stem base (non-redundancy). For complete
syntactic inference, the Armstrong rules 1, 2 (6.8) and 6 (6.13) can be used
[41, Proposition 21]. They are sound in the sense that every implication
proven by the implications in $\mathcal{L}$ and the Armstrong rules is
semantically valid, i.e. holds in $(G,M,I)$.
By reason of these strong properties (where non-redundancy is not necessary),
an object reduced formal context can be reconstructed from its stem base as
well as the order relation of the corresponding concept lattice. Thus, Figure
2.1 represents a Boolean lattice, i.e. its order relation is given by set
inclusion of the elements in the power set $\mathfrak{P}(M)$,
$M:=\\{\text{MMP1:=1, TIMP1:=1, MMP9:=1}\\}$. Hence, there is no restriction
on intents and the stem base is empty. For examples of correlations between
implications and a concept lattice see Section 7.3.
During the interactive attribute exploration algorithm [41, p. 85ff.], an
expert is asked about the general validity of basic implications $A\rightarrow
B$ between the attributes of a given formal context $(G,M,I)$. If the expert
rejects the statement, (s)he must provide a counterexample, i.e. a new object
of the context. If she accepts, the implication is added to the stem base of
the – possibly enlarged – context, which at the end is precisely the set
$\mathcal{L}$ of Theorem 2.2.2. In many applications, one is only interested
in the set of all implications of a fixed formal context. Then, no expert is
needed for a confirmation of the implications. Sometimes I refer to this
algorithm as “computing the stem base” of a formal context.
A counterexample has to be chosen carefully, since its object intent defines a
new closed set. It must correspond to the explored (mathematical or other)
reality, i.e. either a single object with this attribute set exists, or a
class of objects has exactly these attributes in common. Otherwise a valid
implication may be precluded between a pseudo-closed set and the larger,
correct intent. If the counterexample intent is chosen too large, this can be
corrected by new counterexamples. A counterexample contradicting already
accepted implications is immediately rejected by the implementations of the
algorithm.
In this work, mostly the Java implementation Concept Explorer [2] was used. It
handles large contexts, offers the possibility of lattice visualisation with
highlighting of filters and ideals, reads – among other formats – tabulator
separated *.txt or *.csv files, and its graphical user interface is easy to
use. The DOS and Linux command line tool ConImp [26] is restricted to 255
objects and attributes, respectively, but offers enlarged possibilities like
handling incomplete or background (cf. Chapter 6) knowledge.
### 2.3 Temporal Concept Analysis (TCA)
Figure 2.2: Visualisation by life tracks of the gene expression time series
from Table 2.1, for cells from osteoarthritis patients 190, 202, 205 (solid
arrows) and rheumatoid arthritis patients 220, 221, 87 (dashed arrows), at
$t=0,1,2,4$ and $12$ hours after stimulation with TNF. The cubus represents
the Hasse diagram $\mathfrak{B}(\mathbb{K}_{C})$ of the formal context C, the
space part of a CTSOT; it is identical to the concept lattice in Figure 2.1.
States – indicated by black circles – are object concepts of
$\mathfrak{B}(\mathbb{K}_{C})$, i.e. sets of measurements $(o,t),o\in O,t\in
T$ of cell cultures $O$ at specific time points $T$, where a common set of
genes is expressed and the remaining genes are not. For data analyses based on
similar TCA diagrams see [112].
In this section only a short impression of TCA is given, in order to compare
it with our independent approach with partly different purposes. Therefore,
this section may be skipped, or an intuitive understanding of Figure 2.2 may
be sufficient.
K.E. Wolff developed an FCA based “temporal conceptual granularity theory for
movements of general objects in abstract or ’real’ space and time” [108, p.
127]. Temporal Concept Analysis (TCA) is based on many-valued contexts
representing, e.g., several observed time series: row entries are actual
objects, i.e. pairs $(o,t)\in\Pi\subseteq O\times T$, where $o$ is called an
object (interpreted for example as a manufacturing machine, a metereological
station or a person), and $t$ a time granule (interpreted for example as a
time point or a time interval).
More exactly, a Conceptual Time System with Actual Objects and a Time Relation
(CTSOT) is defined as a pair $(\mathbf{T},\mathbf{C})$ of two many-valued
contexts on the same set $\Pi$ of actual objects, together with a relation
$R\subseteq\Pi\times\Pi$. The attributes discriminate between the time part
$\mathbf{T}$ and the event part or space part $\mathbf{C}$.
$\mathbb{K}_{TC}:=\mathbb{K}_{T}\mid\mathbb{K}_{C}$ is the apposition of the
respective derived, one-valued contexts. This leads to the definition of a
state as an object concept of $\mathbb{K}_{C}$, and of a situation as an
object concept of $\mathbb{K}_{TC}$.
The principal aim of TCA is the visualisation of temporal data within the
corresponding concept lattices $\mathfrak{B}(\mathbb{K}_{C})$ and
$\mathfrak{B}(\mathbb{K}_{TC})$. The object concept mapping
$\gamma\colon\Pi\rightarrow\mathfrak{B}(\mathbb{K}_{TC})$ yields the directed
graph of life tracks, connecting the object concepts $\gamma(o,t)$ of a single
object $o$ according to $R$. The Life Track Lemma [108, p. 139] gives
attention to a mapping
$\gamma_{C}\colon\Pi\rightarrow\mathfrak{B}(\mathbb{K}_{C})$ onto the state
instead of the situation lattice (compare Figure 2.2), as well as to mappings
onto lattices according to any other restriction of the attribute set (view).
It specifies how life tracks in $\mathfrak{B}(\mathbb{K}_{TC})$ may be mapped
onto life tracks in the sublattices, which are meet-preservingly embedded into
$\mathfrak{B}(\mathbb{K}_{TC})$.
Conceptual Semantic Systems (CSS) [107] include spatially distributed objects
covering yet quantum incertainty, but they are much too general for our
approach.
In [112], TCA is applied to the graphical analysis of gene expression data,
while the present work principally aims at temporal logic. Life tracks are a
structure supplementary to the underlying formal context and concept lattice.
In contrast, a single context representing the time relation $R$ is required
in order to apply attribute exploration. We also wanted to start from the most
general FCA framework to take advantage of the broad range of mathematical
results and of existing software. Therefore, we constructed a parallel
modelling approach that is based on automata theory (and similar approaches)
just like TCA. The mutual relation will be specified in Section 3.1.4.
## Chapter 3 Algebraic and logic process modelling
The present work aims at providing a framework that may be useful to validate
and further analyse process models formulated in very general classes of
languages. On the semantic level, this chapter gives an overview on automata,
Labelled Transition Systems with Attributes (LTSA, an extension of
semiautomata) and Kripke structures. In order to reveal connnections, they are
presented as different types of universal coalgebras. Focusing on the
syntactic level, I concentrate on A. S. Priors logic of time and Computation
Tree Logic (CTL). Since there is important research concerning connections of
FCA and Description Logics (DL), and DL relations may be considered as
temporal, the basic definitons and an explicitly temporal extension are
presented in Section 3.2.3. Section 9.1 discusses how the defined formal
contexts may be translated to a DL. In Section 3.3, further modelling
languages used in systems biology are mentioned, in particular Boolean
networks.
The present approach is based on LTSA, more exactly on Kripke structures,
since different actions are not distinguished. For simulations, Boolean
networks are used, and the temporal logic CTL for dynamic assertions.
Within this chapter, it is not possible to give a self-contained introduction
to the broad range of theories. It aims more at drawing connections which
might be interesting for readers familiar with a theory, thus at anchoring my
own approach defined without special presuppositions in Chapter 4. There,
references to the present chapter might be overread. To understand the
immediate background of Chapter 4, it should be sufficient to read the
introduction to Boolean networks in Section 3.3.2, the definitions of an LTSA
(3.1.7), of a Kripke structure (3.1.5) and of the CTL operators (Section
3.2.2), possibly also the paragraphs concerning their origin in propositional
tense logic (Section 3.2.1). The main part of this section discusses
philosophical ideas of A.S. Prior regarding the flux of events, their fixation
in a data frame, open future, freedom and limits of temporal knowledge. It is
an excursus fitting well to my view of FCA as a method aiming to support human
understanding and responsible discussion of the reach of data analyses.
### 3.1 An unifying approach: Universal coalgebras
In computer science the mathematical discipline of Universal Coalgebra
achieved large success as a common theory of state based systems,
generalising, e.g., automata and Kripke structures. The observable output of
such dynamic systems depends on an input as well as on an internal state,
where the input may change the state. As will be shown in important special
cases, this can be described by a set $S$ of states and a mapping from $S$ to
a combination of states and outputs.
An (universal) coalgebra is defined in the language of category theory, which
aims to describe structural similarities between mathematical theories. A
category $\mathfrak{C}$ consists of a class of objects (e.g. the class of
sets, groups or vector spaces) and a class of morphisms (e.g. homomorphisms):
For two objects $A,B\in\mathfrak{C}$, a set $\operatorname{Mor}(A,B)$ is
defined, and the axioms are satisfied (the very natural first axiom is
omitted) [69, p. 53]:
* CAT 2: For each object $A\in\mathfrak{C}$ there is a morphism $\operatorname{id}_{A}\in\operatorname{Mor}(A,A)$, the identical map on $A$.
* CAT 3: The class of morphisms is closed against composition. The law of composition is associative: If $A,B,C,D\in\mathfrak{C}$ and $f\in\operatorname{Mor}(A,B),\>g\in\operatorname{Mor}(B,C)$ and $h\in\operatorname{Mor}(C,D)$, then
$(h\circ g)\circ f=h\circ(g\circ f).$
A functor defines exactly how objects and morphisms of one category can be
transferred to another category. We only need covariant functors respecting
the direction of morphisms:
###### Definition 3.1.1.
[69, p. 62] A covariant functor $\Omega$ of a category $\mathfrak{C}$ into a
category $\mathfrak{D}$ is a rule which to each object $A\in\mathfrak{C}$
associates an object $\Omega(A)\in\mathfrak{D}$, and to each morphism
$f:A\rightarrow B$ associates a morphism
$\Omega(f)\colon\Omega(A)\rightarrow\Omega(B)$ so that:
* FUN 1. For all $A\in\mathfrak{C}$ we have $\Omega(\operatorname{id}_{A})=\operatorname{id}_{\Omega(A)}$.
* FUN 2. If $f\colon A\rightarrow B$ and $g\colon B\rightarrow C$ are two morphisms of $\mathfrak{C}$ then
$\Omega(g\circ f)=\Omega(g)\circ\Omega(f).$
Now a coalgebra is defined by means of a functor on the category of sets
alone, i.e. an endofunctor $\Omega\colon\text{{Set}}\rightarrow\text{{Set}}$.
It maps sets of states $S$ to sets of (in general) higher complexity,
including the Cartesian product of sets or sets of functions, like
$S^{\Sigma}:=\\{\Sigma\rightarrow S\\}$, for two sets $\Sigma$ and $S$.
###### Definition 3.1.2.
[44, Definition 3.0.1] Let a Type be an endofunctor
$\Omega\colon\textnormal{{Set}}\rightarrow\textnormal{{Set}}$. Then a
Coalgebra of Type $\Omega$ is a pair $\mathcal{A}=(S,\alpha_{S})$ consisting
of a set $S$ and a mapping
$S\overset{\alpha_{S}}{\longrightarrow}\Omega(S).$
In the following subsections, automata, Kripke structures and LTSA will be
presented as universal coalgebras. Since only basic structural similarities
are highlighted, set-theoretic morphisms mostly are not considered explicitly.
#### 3.1.1 Automata theory
###### Definition 3.1.3.
[44, 1.4] An Automaton is a tuple
$A:=\\{S,\Sigma,\delta,D,\gamma\\},\text{ with:}$
1. 1.
A set of States $S$.
2. 2.
A finite set of Input Symbols $\Sigma$.
3. 3.
A Transition Function $\delta\colon S\times\Sigma\rightarrow\mathfrak{P}(S)$.
4. 4.
A set of Data $D$.
5. 5.
An Output Function $\gamma\colon S\rightarrow D$.
In the case of a Deterministic Automaton, the transition function $\delta$
maps to the set of singletons identifiable with $S$. A Finite Automaton has a
finite set $S$.
The value of the transition function $\delta(s,e)$ can be interpreted as
denoting the possible states the automaton is in after reading the input $e$
while in the state $s$. However, states often cannot be observed directly, but
by means of the output function $\gamma$: Each internal state $s\in S$ can
only be observed by an external attribute $\gamma(s)\in D$.
In a main field of application only the paths are interesting that lead from a
fixed start state $s_{0}$ to a final (or accepting) state $s\in F\subseteq S$
($S$ finite). $F$ may be coded by its characteristic function $\gamma\colon
S\rightarrow D$ where $D:=\\{0,1\\}$. This type of automaton is also called
acceptor [44, p. 165]. Then, an automaton defines a language of all successful
words $(a_{1},...,a_{n})\in\Sigma^{n},\>n\in\mathbb{N}$, corresponding to a
path of transitions from $s_{0}$ to an accepting state $s_{n}$.
Figure 3.1: Transition diagram for the automaton example 3.1.4 (recognition of
a 01-substring), from [54, p. 48].
###### Example 3.1.4.
[54, p. 46-49]. A finite deterministic automaton accepts strings over an
alphabet $\Sigma:=\\{0,1\\}$, and the aim is to decide whether the string
contains the sequence $01$. Besides the final state $q_{1}$, indicating that
the substring has been found, there is the initial state $q_{0}$ (no input or
last input 1) and an intermediate state $q_{2}$ (most recent input 0). The
graph of Figure 3.1 represents the possible transitions depending on the next
input symbol. Explicitly, three types of input words are distinguished, each
being preceded by the corresponding state of the automaton:
$q_{0}\colon\underset{\geq 0}{\underbrace{1...1}},\qquad
q_{2}\colon\underset{\geq
0}{\underbrace{1...1}}\underset{>0}{\underbrace{0...0}},\qquad
q_{1}\colon(...)01(...)$
In order to make a clear difference to Kripke structures (Section 3.1.2), a
deterministic automaton is represented as a coalgebra: Let a functor
$\Omega\colon\textnormal{{Set}}\rightarrow\textnormal{{Set}}$ be defined by
$\Omega(S):=D\times S^{\Sigma}$ for a set $S$. For $f\colon S_{1}\rightarrow
S_{2}$, define $\Omega(f)\colon\Omega(S_{1})\rightarrow\Omega(S_{2})$ by
$\Omega(f)(d,\delta^{\prime}):=(d,f\circ\delta^{\prime})$, for all $d\in
D,\>\delta^{\prime}\in X^{\Sigma}$. The properties of a set functor are
fulfilled, since the constant functor, the power functor and the cross product
of functors are functors [44, Beispiel 2.4.2, 2.4.3 and 2.4.6]). Then a
deterministic automaton is a coalgebra $\mathcal{A}=(S,\alpha_{S})$ of type
$\Omega$ with
$\alpha_{S}\colon S\rightarrow\Omega(S):=D\times S^{\Sigma},$
where $\alpha_{S}(s)=(\gamma(s),\delta(s,\cdot))$, for all $s\in S$.
A nondeterministic automaton is identifiable with a coalgebra where
$\alpha_{S}\colon S\rightarrow\Omega(S):=D\times(\mathfrak{P}(S))^{\Sigma}.$
#### 3.1.2 Kripke structures
###### Definition 3.1.5.
A Kripke structure consists of a set of states $S$, a set of atomic
propositions $M$, an output function $\gamma\colon
S\rightarrow\mathfrak{P}(M)$ and a relation $R\subseteq S\times S$.
A Kripke structure may be considered as a special case of a nondeterministic
automaton: With the trivial alphabet $\Sigma=\\{\text{update}\\}$, i.e.,
without a special input, the transition function gets $\delta_{R}\colon
S\rightarrow\mathfrak{P}(S)$, which can be identified with a relation
$R_{\delta}\subseteq S\times S$ as follows:
$\displaystyle R_{\delta}$ $\displaystyle:=\\{(s,s^{\prime})\mid s\in
S,s^{\prime}\in\delta_{R}(s)\\}$ $\displaystyle\delta_{R}$
$\displaystyle:\quad s\mapsto\\{s^{\prime}\in S\mid(s,s^{\prime})\in
R_{\delta}\\}=:[s]R_{\delta}$
The output function is given over a set $M$ by letting $D:=\mathfrak{P}(M)$,
i.e. $\gamma\colon S\rightarrow\mathfrak{P}(M)$. An arbitrary set $M$ may be
considered as a set of atomic propositions. Then $\gamma(s)$ is the set of
atomic propositions being true in the state $s$.
Accordingly, a Kripke structure is a universal coalgebra with
$\alpha_{S}\colon
S\rightarrow\Omega(S):=\mathfrak{P}(M)\times\mathfrak{P}(S).$
###### Example 3.1.6.
(compare [44, p. 167]). Due to the number of interacting measuring or control
devices with observable output $M$, a technical or engineering system may not
be predictable in detail, but the set of allowed transitions $R$ may be
restricted indirectly. Thus, in a computer program with parallel processes,
$R$ constrains the transitions between states within and between processes.
States are defined by an assignment of the variables declared for a process.
Such variable assignments are an example of atomic propositions $m\in M$
(compare Table 3.1), and allowed transitions are given by Boolean expressions
over $M$ (compound attributes in the language of FCA [38, p. 101]) relating an
input to an output state. If the precondition applies to the input state, then
the postcondition has to be true in the subsequent state, as in the example
describing the main steps of the German legislation process:
bundestag.vote AND bundesrat.vote AND NOT president.veto ==> law.published.
| input.0 | input.1 | position.start | position.final
---|---|---|---|---
$q_{0}^{a}$ | | | x |
$q_{0}^{b}$ | | x | x |
$q_{1}^{a}$ | x | | | x
$q_{1}^{b}$ | | x | | x
$q_{2}$ | x | | |
Table 3.1: Assignment of atomic propositions $\gamma\colon
S\rightarrow\mathfrak{P}(M)$ for the Kripke structure modelling Example 3.1.4
represented as a formal context $(S,M,I)$. It is derived by nominal scaling
from the many-valued context $(S,\tilde{M},W,J)$, where
$\tilde{M}:=\\{\text{input},\text{position}\\}$ and
$W:=\\{0,1,\text{start},\text{final}\\}$. Atomic propositions are mappings
$m\in M\subseteq\\{\tilde{M}\rightarrow W\\}$, i.e. variable assignments.
Compared to the acceptor type of an automaton, in a Kripke structure a state
may be mapped to a set of output attributes, not only to single values like
final state or no final state. An advantage of a Kripke structure is the
possibility of a differentiate state description by a large number of
attributes, which will be important for our main biological application. In
Example 3.1.4, instead of “colouring” a state by $D=\\{\text{final, not
final}\\}$, attribute combinations from
$M:=\\{0,1,\text{start},\text{final}\\}$ may be assigned. Table 3.1 represents
the output function $\gamma:S\rightarrow\mathfrak{P}(M)$ as a formal context,
which will be named state context (Definition 4.2.1). $0$ and $1$ describe the
input leading to a state, hence only the most recently read value of the input
string is relevant. In Kripke structures, input strings are not considered
explicitly, since $\Sigma$ contains only one element. However, a part of this
information could be preserved by remembering two or more input values as
attributes, for instance by $\bigcup_{i\in I}\\{0_{i},1_{i}\\}\subseteq M$. In
our example, the defined set of atomic propositions $M$ is sufficient to
distinguish the three states and to get valuable information regarding the
dynamic system (cf. the interpretation of the states $q_{0},q_{1},q_{2}$ in
Example 3.1.4).
A relation $R^{\prime}$ is given by the transitions defined by the automaton
graph (Figure 3.1):
$R^{\prime}=\\{(q_{0}^{a},q_{0}^{b}),(q_{0}^{a},q_{2}),(q_{0}^{b},q_{0}^{b}),(q_{0}^{b},q_{2}),(q_{2},q_{2}),(q_{2},q_{1}^{b}),$
$(q_{1}^{a},q_{1}^{a}),(q_{1}^{a},q_{1}^{b}),(q_{1}^{b},q_{1}^{a}),(q_{1}^{b},q_{1}^{b})\\}.$
Inversely, in this example the transition graph may be reconstructed from the
Kripke structure, together with some drawing conventions. For this purpose, it
might be desirable to have an equivalent of the original automaton states:
$\\{q_{0}\\}$ and $\\{q_{1}\\}$ are concept extents, whereas $\\{q_{2}\\}$ is
given as extent of a supplementary attribute “intermediate”.
#### 3.1.3 Labelled Transition Systems with Attributes (LTSA)
For our FCA process modelling, we start from the definition of LTSA in [40,
Definition 1] generalising abstract automata. Labelled Transition Systems
(LTS) or State transition systems are finite state automata, where all states
are final states, or semiautomata. They are also used in operational semantics
and may be described by process algebras.111Keijo Heljanko, Networks and
Processes: Process Algebra, 2004. www.fmi.uni-stuttgart.de/
szs/teaching/ws0304/nets/slides20.ps The notion of LTSA complements this
structure by state attributes or atomic propositions in the language of Kripke
structures:
###### Definition 3.1.7.
A Labelled Transition System with Attributes (LTSA) is a 5-tuple $(S,M,I,A,R)$
with
1. 1.
$S$ being a set of states.
2. 2.
$M$ being a set of state attributes.
3. 3.
$I\subseteq S\times M$ being a relation.
4. 4.
$A$ being a finite set of actions.
5. 5.
$R\subseteq(S\times A\times S)$ being a set of transitions, where
$(s_{1},a,s_{2})\in R$ means “action $a$ can cause the transition from state
$s_{1}$ to state $s_{2}$”.
###### Proposition 3.1.8.
Every automaton can be represented by an LTSA, and vice versa.
###### Proof.
An LTSA is a an automaton according to Definition 3.1.3, with the
specialisation $D:=\mathfrak{P}(M)$ for the output function $\gamma\colon
S\rightarrow D,$ $s\mapsto\\{m\in M\mid(s,m)\in I\\}$. For the rest,
$\Sigma:=A$, and $\delta\colon S\times A\rightarrow\mathfrak{P}(S)$ is given
by $(s,a)\mapsto\\{s^{\prime}\in S\mid(s,a,s^{\prime})\in R\\}$. (Compare
[106, p. 343].)
On the other hand, each automaton can be translated into an LTSA via
$A:=\Sigma$, $M:=D$ and $R:=\\{(s^{in},a,s^{out})\mid s^{in}\in
S,\>a\in\Sigma,\>s^{out}\in\delta(s^{in},a)\subseteq\mathfrak{P}(S)\\}$; the
relation $I$ is given by the function $\gamma\colon S\rightarrow
D\cong\\{\\{d\\}\mid d\in D\\}\subseteq\mathfrak{P}(D)$. ∎
By the proposition, an LTSA is also a universal coalgebra:
$S\rightarrow\Omega(S):=\mathfrak{P}(M)\times(\mathfrak{P}(S))^{A}$
In our FCA model, start and final states are not considered explicitly.
Investigating the attribute logic we are not mainly interested in automata as
acceptors, i.e. in testing allowed languages. The definition in [40] is
slightly changed assuming a finite set of actions according to automata
theory. Infinite words over the alphabet $\Sigma:=A$ are allowed, but an
infinite set of actions is not very meaningful.
#### 3.1.4 TCA – LTSA – automata theory
There is also a strong relation to TCA: An LTSA and especially an automaton
can be described by and reconstructed from a CTSOT. In [106, 2.2], an
automaton is defined like an acceptor, but slightly more general: a set of
start states is admitted, thus an output function $\gamma\colon S\rightarrow
D:=\\{\text{start, intermediate, final}\\}$. The main idea of the Map
Reconstruction Theorem [106, 4.2] is taking the set of actions (plus “missing
value”) as attributes of the time part $\mathbf{T}$ with nominal scaling. Then
given an LTSA $L$, there is an isomorphism from $L$ onto a state-LTSA derived
from a CTSOT, so that each path of $L$ (given by the transitions $R$) is
mapped onto a life track.
### 3.2 Temporal logics
The atomic propositions $M$ of a Kripke structure and the transition relation
$R$ define the semantics of a dynamic system. Based on it, a multitude of
logics have been developed in order to reason about temporal properties of the
system. Within the framework of the present thesis two classical approaches
and a temporal extension of description logics are outlined, which have been
developed and investigated in many directions more recently.
#### 3.2.1 Propositional tense logic
A very important contribution to the modern logic of time – including concepts
and reasoning – was made by A.N. Prior (1914 - 1969) based on philosophical
traditions of antiquity and the Middle Ages. He started from J.M.E. McTaggarts
(1866 - 1925) distinction between the A- and B-series conceptions of time
(which lead McTaggart to a famous paradox and the refutation of the reality of
time): The A-concepts past, present and future are more fundamental for a
proper understanding of time than the B-series conception of a set of instants
organised by the earlier-later relation. Prior also rejected the latter static
view of time, intending to substantiate the notion of freedom:
> I believe that what we see as a progress of events is a progress of events,
> a coming to pass of one thing after another, and not just a timeless
> tapestry with everything stuck there for good and all. (A.N. Prior, cited
> after [77, p. 69])
Events that have become past are “out of our reach” and unchangeable, whereas
the future is to some extent open and depends on the decision of a free agent.
Furthermore, Prior considered B-theory as a reduction of reality since the
notion of the present, the Now, disappears.
In both views, well formed formula are composed by arbitrary propositional
variables, $\wedge$, $\sim$ (negation), $F$ (“in the future”) and $P$ (“in the
past”). Yet like in one tradition of modern philosophy of language, Prior did
not assume a sharp distinction between an object language and a metalanguage.
Accordingly, there is no model, e.g. a Kripke structure (denoted by
$(\text{TIME},<,\nu)$) as a second level. It would be a tenseless metalanguage
or “timeless tapestry” since it is based on a set of instants or durations.
Instead of such a reification of instants, Prior introduced a special type of
propositions, Instant-propositions or World-state propositions $t$. This makes
it possible to define formula $T(t,\varphi)$ meaning that a tense-logical
formula $\varphi$ is true at time $t$. Instant propositions are defined
axiomatically in terms of the tense-logical language itself, together with a
necessity operator $\Box$ and a possibility operator
$\Diamond:=\,\sim\\!\Box\sim$, as well as standard quantification $\exists$
(and thus $\forall$). In this way, instants or times are treated as artificial
constructs. They are replaced by the conjunction of a maximal consistent set
of propositions that may be said to be true at $t$. Thus, Prior adapts the
notion of possible worlds to time.222Compare John Wood, Course Modal Logic,
University of British Columbia, Spring 2007, Note 23
[http://www.johnwoods.ca/Courses/Phil322-07/]. His approach has also been
followed by hybrid logics. [77, p. 70f.]
He developed a theory of possibility and indeterminism based on the notion of
branching time, which had been suggested in a letter from Saul Kripke in 1958.
Branching time is representable by a tree, where the present is a node of
“rank 0”, and the possible future states at the following moments have a
higher rank, i.e. depth. Finally, Prior incorporated this idea into the
concept of time itself by introducing the notion of chronicles or histories,
i.e. maximal linearly ordered subsets in $(\text{TIME},<)$, where TIME is the
set of instant-propositions.
Prior distinguished two models of branching time, inspired by William of
Ockham (ca. 1285 - 1349) and Charles S. Peirce (1839 - 1914), respectively. In
the Ockhamistic model, the operators $F$ (“tomorrow”), $\Diamond F$ (“possibly
tomorrow”) and $\Box F$ (“necessarily tomorrow”) are distinguishable. In the
Peircean view, however, “tomorrow” is identified with “necessarily tomorrow”,
since Peirce emphasised the difference between future and past. There is no
“plain” or “true future” and no factual, non-necessary statements concerning
the future make sense. Consequently, for an arbitrary formula $q$ only in the
Ockhamistic system $q\rightarrow HFq$ is a theorem, with
$H:=\,\sim\\!P\\!\sim$ (“at all times in the past”). In both systems, there
are no alternative pasts but a single chronicle in the past of an instant-
proposition $t$. Hence, the theorem does not hold in the Peircean system but
only for $F\neq\Box F$, since $\Box F$ refers to all possible chronicles. [77,
p. 72-77]
Priors favorised A-theory is “politically correct” within the context of
contemporary philosophy, but it is too sophisticated within the framework of
this study. Instead we remain with the usual model-theoretic, B-theoretic
approach to time, since FCA is based on data, a fundamental concept of modern
science. A data frame of observations representing – or statistically
interpreted as – a deterministic time series is a “timeless tapestry”. For the
observation time, future is considered as retrospective or “plain”, and the
observations are extrapolated to future by postulating natural laws. Prior
reminds us that this is a simplification. Sometimes the whole background of
data analysis and theory building should be made conscious again, like
abduction, induction, falsification, paradigm change and historical
development of scientific notions.
> But even in this flux there is a pattern, and this pattern I try to trace
> with my tense-logic; and it is because this pattern exists that men have
> been able to construct their seemingly timeless frame of dates. Dates, like
> classes, are a wonderful and tremendously useful invention, but they are an
> invention; the reality is things acting. (A.N. Prior, Bodleian Library, MS
> in box 6, 1 sheet, no title. Cited after [77, p. 78])
We follow an intermediate approach: Within the FCA model for transitions that
will be developed in Chapter 4 we include nondeterminism, hence future is
regarded as open. Furthermore, we decided for throughout quantification over
paths according to the Peircean future operators (but also for an Ockhamistic
model, the notion of a chronicle could be defined by a supplementary attribute
of the state context (Definition 4.2.1). In this way, there is no “true
future”, only possibility and necessity are considered. On the other side, we
follow a data-driven approach but try to be cautious: Analysis results depend
on specific experimental conditions, preprocessing, definition of thresholds
(e.g. for gene expression up- or downregulation) or choice of algorithms.
Hence, necessity should not be judged by given data only, but by all existing
knowledge. Attribute exploration enforces the role of the expert, who ideally
should interprete necessity before the background of all events possible
according to the state of science. In a further step of reflection, the
resulting temporal stem base can be considered as a clear and concise
knowledge representation, which by determining the domain of interest also
defines its limits as well as those of the present understanding of temporal
reality. Against this background, a clear definition and validation of a
temporal data frame implies the respect for the potentially infinite
complexity of nature and life.
#### 3.2.2 Computation Tree Logic (CTL)
Current temporal logics include Interval Temporal Logic (ITL) and
$\mu$-calculus, of which important subsets are Linear Temporal Logic (LTL) and
Computation Tree Logic (CTL). Thus, like all approaches considered here, CTL
abstracts from duration values; the basic unity is an event corresponding to a
state. CTL is able to describe properties of nondeterministic transition
systems (with branching time) and extends propositional or first order logic
with the following path quantifiers and temporal operators defined formally in
Section 4.5:
* •
$A$: “for all transition paths” (corresponding to Priors necessity operator
$\Box$)
* •
$E$: “for some (existing) transition path” (corresponding to
$\Diamond:=\,\sim\\!\Box\sim$)
* •
$F$: “eventually (finally) in the future”
* •
$G$: “always (generally) in the future”
* •
$X$: “next time”
* •
$U$: “until”
A safety property specifying that some situation described by a formula $\phi$
can never happen is expressed by the CTL formula $AG\neg\phi$, i.e. on all
paths $\phi$ is always false. A liveness property specifying that something
good $\phi$ will eventually happen is expressed by the formula $AF\phi$.
State formulas $\phi$ are evaluated on (arbirary) states, whereas path
formulas $\psi$ are evaluated on single paths. With a set of atomic
propositions $AP$, CTL has the following grammar, including ordinary Boolean
connectives [29, 4.1]:
$\displaystyle\phi$ $\displaystyle:=\alpha\in AP\mid E\psi\mid A\psi$
$\displaystyle\psi$ $\displaystyle:=\phi\mid X\psi\mid F\psi\mid G\psi\mid\psi
U\psi$
While in CTL∗ arbitrary state and path formulas are admitted, in CTL a path
formula $\psi$ has to be preceeded by a path quantifier $E$ or $A$. Hence, an
Ockhamistic model of time (in Priors understanding) cannot be expressed in
CTL.
#### 3.2.3 Description logics
##### General definitions
Description Logics (DLs) are a family of knowledge representation formalisms
with a broad range of applications, such as data mining, natural language
processing, semantic web or ontologies. Similar to FCA, a principal aim of a
DL is to define conceptual hierarchies, and there are attempts aiming at the
application of attribute exploration to construct DL knowledge bases [19]
[20].
Concept descriptions are built starting from a set $N_{C}$ of concept names
(unary predicates) and a set $N_{R}$ of role names (binary predicates) with
the aid of concept constructors specific for the language. Important
constructors are the conjunction $C\sqcap D$ and the disjunction $C\sqcup D$,
where $C,D$ are concept names or more complex concept descriptions. If DL
concepts are given as FCA concepts, constructors are identical to the infimum
or supremum of two formal concepts. Besides negation, role restrictions are
used, e.g. $\exists r.C\>(r\in N_{R})$ meaning, e.g., the concept of having a
French team member for a role $r$: (FootballTeam, Member) and a concept
description $C$ “French nationality”. Finally, individual names are assembled
in a set $N_{I}$ referring to elements of the domain $\Delta^{I}$ by means of
which the semantics of a DL is given (Table 3.2).
A DL knowledge base consists of a TBox and a ABox. The TBox is a finite set of
general concept inclusions (GCIs) $C\sqsubseteq D$ expressing a subconcept-
superconcept relation. Concept definitions $C\equiv D$ abbreviate two GCIs
holding for both directions. The $ABox$ assigns concepts and roles to
individual names.
Standard DL | Syntax | Semantics
---|---|---
Basic sets | |
Concept | $C\in N_{C}$ | $C^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}$
Empty concept | $\bot$ | $\bot^{\mathcal{I}}=\emptyset\subset\Delta^{\mathcal{I}}$
Most general concept | $\top$ | $\top^{\mathcal{I}}=\Delta^{\mathcal{I}}$
Role | $r\in N_{R}$ | $r^{\mathcal{I}}\subseteq\Delta^{\mathcal{I}}\times\Delta^{\mathcal{I}}$
Individual name | $a\in N_{I}$ | $a^{\mathcal{I}}\in\Delta^{\mathcal{I}}$
Constructors | |
Negation | $\neg C$ | $\Delta^{\mathcal{I}}\setminus C^{\mathcal{I}}$
Conjunction | $C\sqcap D$ | $C^{\mathcal{I}}\cap D^{\mathcal{I}}$
Disjunction | $C\sqcup D$ | $C^{\mathcal{I}}\cup D^{\mathcal{I}}$
Existential restriction | $\exists r.C$ | $\\{x\in\Delta^{\mathcal{I}}\mid\exists y\in\Delta^{\mathcal{I}}\colon(x,y)\in r^{\mathcal{I}}\wedge y\in C^{\mathcal{I}}\\}$
General restriction | $\forall r.C$ | $\\{x\in\Delta^{\mathcal{I}}\mid\forall(x,y)\in r^{\mathcal{I}}\colon y\in C^{\mathcal{I}}\\}$
TBox | |
GCI | $C\sqsubseteq D$ | $C^{\mathcal{I}}\subseteq D^{\mathcal{I}}$
ABox | |
Concept assertion | $C(a)$ | $a^{\mathcal{I}}\in C^{\mathcal{I}}$
Role assertion | $r(a,b)$ | $(a^{\mathcal{I}},b^{\mathcal{I}})\in r^{\mathcal{I}}$
Table 3.2: Semantics of DL concept descriptions, TBoxes and ABoxes in terms of
an interpretation $\mathcal{I}:=(\Delta^{\mathcal{I}},\cdot^{\mathcal{I}})$,
where $\Delta^{I}$ is a nonempty set, and the interpretation function
$\cdot^{\mathcal{I}}$ is defined as above (compare [20, p. 4f.]).
##### Temporal extensions of description logics
A basic possibility of applying a DL to dynamic processes is to interprete the
domain as a set of states and to describe transitions by roles nextState or
reachableState (see Section 9.1, where a translation of our approach into the
language of DL will be discussed.)
There are also several extensions of DL by operators from the temporal logic
LTL together with the definition of an appropriate semantics, e.g. in [17,
6.2.4]. I will delineate briefly the more detailed proposal of the temporal
extension $\mathcal{TDL}-Lite_{bool}$ [15].
Besides the usual global roles $T_{i},\>i\in I$, time-dependent local roles
$P_{i}$ are introduced, denoting a one step transition like nextState. The
semantics of a role $P_{i}^{-}$ is the inverse relation on the domain (the
state set) $\Delta$. Furthermore, the temporal operators $\bigcirc$ (“at the
next moment”) and $\mathcal{U}$ (“until”) are introduced. The semantics of the
non-standard operators is listed in Table 3.3 with an interpretation according
to Definition 3.2.1. The complete syntax is defined as follows:
$\displaystyle R:=\>$ $\displaystyle P_{i}\mid P_{i}^{-}\mid T_{i}\mid
T_{i}^{-}$ $\displaystyle B:=\>$ $\displaystyle\bot\mid A_{i}\mid\,\geq q\,R$
$\displaystyle C:=\>$ $\displaystyle B\mid\neg C\mid C_{1}\sqcap
C_{2}\mid\bigcirc C\mid C_{1}\,\mathcal{U}\,C_{2}.$
###### Definition 3.2.1.
Given a nonempty set (domain) $\Delta$, object names $a_{i}$, concept names
$C_{i}$, local role names $P_{i}$ and global role names $T_{i}$, $i\in I$, the
semantics of $\mathcal{TDL}-Lite_{bool}$ is given by an Interpretation
Function $\mathcal{I}$:
$\mathcal{I}(n):=(\Delta,a_{0}^{\mathcal{I}(n)},...,A_{0}^{\mathcal{I}(n)},...,P_{0}^{\mathcal{I}(n)},...,T_{0}^{\mathcal{I}(n)},...),$
where
$n\in\mathbb{N},\>a_{i}^{\mathcal{I}(n)}\in\Delta,\>A_{i}^{\mathcal{I}(n)}\subseteq\Delta,\>P_{i}^{\mathcal{I}(n)}\subseteq\Delta\times\Delta,\>(i\in
I)$, and
$a_{i}^{\mathcal{I}(n)}=a_{i}^{\mathcal{I}(m)},\>T_{i}^{\mathcal{I}(n)}=T_{i}^{\mathcal{I}(m)}$
for all $n,m\in\mathbb{N}$.
Finally, the usual unique name assumption is made: $a_{i}^{\mathcal{I}(n)}\neq
a_{j}^{\mathcal{I}(n)}$ for all $i\neq j$. The two operators $\bigcirc$ and
$\mathcal{U}$ are sufficient to define other temporal operators:
$F\,C\equiv\top\,\mathcal{U}\,C$ (“some time in the future”) and
$G\,C\equiv\neg F\,\neg C$ (“always in the future”).
$\mathbf{TDL-Lite_{bool}}$ | Syntax | Semantics
---|---|---
At the next moment | $\bigcirc$ $C$ | $\bigcirc\,C^{\mathcal{I}(n)}:=C^{\mathcal{I}(n+1)}$
Until | $C_{1}\,\mathcal{U}\,C_{2}$ | $(C_{1}\,\mathcal{U}\,C_{2})^{\mathcal{I}(n)}:=\bigcup_{k>n}\huge(C_{2}^{\mathcal{I}(k)}\cap\,\bigcap_{n<m<k}C_{1}^{\mathcal{I}(m)}\huge)$
Inverse role | $R^{-}$ | $(R^{-})^{\mathcal{I}(n)}:=\\{(y,x)\mid(x,y)\in R^{\mathcal{I}(n)}\\}$
Related states | $\geq q\,R$ | $(\geq qR)^{\mathcal{I}(n)}:=\\{x\in\Delta\mid|\\{y\mid(x,y)\in R^{\mathcal{I}(n)}\\}|\geq q\\}$
Table 3.3: Semantics of the supplementary operators in $TDL-Lite_{bool}$.
### 3.3 Systems biology
As models for regulatory networks, linear or nonlinear ordinary differential
equations are often used. Partial differential equations additionally make it
possible to model spatial behaviour, e.g. cell differentiation and movement.
Those models are used primarily for simulation and prediction and offer
possibilities of subsequent analyses, e.g. of stability or bifurcation.
#### 3.3.1 Discrete models
Methods of symbolic computation allow for further and differentiated analyses
by logical queries. They are closer to the thinking of human experts and can
also be applied if quantitative data is sparse or only the qualitative
behaviour is known [29, Introduction] [50]. For instance, methods of software
or hardware verification have been adapted to systems biology, like the
$\pi$-calculus, a process algebra for concurrent computation. Molecules are
represented as processes in which they participate and interactions as
communication channels. The $\pi$-calculus is used for simulation and
verification of assertions like “Will a signal reach a particular molecule¿‘
[82].
CTL was used in [29] to analyse protein-protein and protein-DNA interaction
networks. The approach developed in the following chapters is based on CTL and
Boolean networks.
#### 3.3.2 Boolean networks
Boolean networks (BN) are often applied to the analysis of gene regulation. By
abstraction only two expression levels off and on (0/1 or -/+) are considered.
This is justified since there exist relatively fixed thresholds of activation
for many genes [86]. Also in continuous models the dynamics are often
approximated by a steep sigmoid function (e.g.
$f(t):=\frac{1}{1+\operatorname{e}^{-t}})$. Moreover, a switch-like behaviour
may be strengthened by a positive feedback of a transcription factor on its
own expression [60, p. 14797]. The classical approach of Boolean networks [61]
[59] is able to capture essential dynamic aspects of regulatory networks and
scales up well to larger sets of genes. Boolean networks require time-series
data as input (reverse engineering) and generate such data as output
(simulation). They can be represented as directed graphs with nodes labelled
by Boolean functions, which determine one of two attribute values 0 or 1 for
each entity (e.g. gene) after one time step (output) given the values of the
entities at a given moment (input). Boolean networks are widely used in
molecular biology for logical analysis and simulation of medium or large scale
networks [66] [91]. For example, Kervizic et al. developed a method for the
cholesterol regulatory pathway in 33 species which eliminates spurious cycles
in a synchronous Boolean network model [63]. A formal definition within our
conceptual framework will be given in 4.3.2.
## Chapter 4 Modelling discrete temporal transitions by FCA
Our intention was to develop an FCA approach into which different types of
process models may be translated. In the following will be demonstrated how
the types of universal coalgebras presented in Section 3.1 are representable
by (a family of) transition contexts (Section 4.3). First, a basic state
context will be defined, then in Section 4.4 a transitive context which makes
the information related to reachable states explicit. Finally, attributes from
the temporal logic CTL are integrated into a temporal context.
### 4.1 Example: Installing a wireless device
In order to illustrate the definitions, the method and possible applications,
I introduce a very simple example. More realistic applications to systems
biology will be described in Chapters 7 and 8.
A Linux (Ubuntu) help page aims at guiding a user through the process of
installing a wireless card and establishing an internet connection. The
definition of formal contexts and attribute exploration (which will be
described in Chapter 5) may support a good structure of this page (e.g. by
hyperlinks to subsequent steps) and can prevent to forget occurring cases. It
might even be used to determine the process logic with the purpose of
establishing an expert system.
The formal context of Table 4.1 relates states to attributes indicating which
of four main steps of the installation process (including an alternative) are
accomplished. In Definition 4.2.1 this type of contexts will be called state
context and referred to an LTSA.
| driver.linux | ndiswrapper | driver.windows | connection
---|---|---|---|---
$s_{0}$ | | | |
$s_{10}$ | $\times$ | | |
$s_{11}$ | | $\times$ | |
$s_{20}$ | $\times$ | | | $\times$
$s_{21}$ | | $\times$ | $\times$ |
$s_{31}$ | | $\times$ | $\times$ | $\times$
Table 4.1: Formal context indicating in the columns (attributes) which of four
main steps of a wireless card installation process are accomplished. The row
names (objects) denote succeeding states. The indices suggest a branching
after the initial state $s_{0}$ into procedures for a native Linux driver
(states $s_{i0}$) or for a driver originally developed for Windows operating
systems (states $s_{i1}$).
* •
driver.linux: Newer versions of Ubuntu provide full “out of the box” support
for several wireless cards. In other cases, the driver has to be downloaded
manually, unpacked to an appropriate directory and compiled.
* •
driver.windows: Often, no open source driver exists; then the Windows driver
can be used.
* •
ndiswrapper: The Linux module ndiswrapper has been developed with the purpose
of using a Windows driver. Since it does not belong to the basic Ubuntu
distributions, it must be installed first.
* •
connection: Finally, the individual connection data is entered (usually ESSID
and password for the router). At this basic stage of process modelling,
connection is the final state and signifies overall success, i.e. an
established internet connection.
### 4.2 The state context $\mathbb{K}_{s}$ and some useful scalings
In order to investigate a process, the occurring states have to be defined
first. We do this by means of their attributes, i.e. by a formal context.
###### Definition 4.2.1.
[40, p. 147] The formal context $\mathbb{K}_{s}:=(S,M,I)$ with the state set
$S$, the attribute set $M$ and relation $I\subseteq S\times M$ from an LTSA
$(S,M,I,A,R)$ is called the State Context of this LTSA.
For an example see Table 4.1. A non clarified state context may contain
diverse states indiscernable by the attributes, related to different time
points or observations. Then, the information regarding time granules may be
coded in the object names (compare Table 2.1). As for the attributes, we are
focusing primarily on the logic of the state space, i.e. of
$\mathfrak{B}(\mathbb{K}_{C})$ in TCA. In (biological) regulatory networks,
one is more interested in what happens before or after a certain class of
states, and less in exact time points. However, a coarse granularity of time
could be useful to describe, e.g., early and late activation of gene
expression. For this purpose, our framework may be easily applied to the
situation space $\mathfrak{B}(\mathbb{K}_{TC})$ by introducing a supplementary
many-valued attribute “time point” or “time interval”.
If the state context is clarified, a state is attribute defined (i.e.
unambigously identifiable by its attributes). If further nominal scaling (p.
2.1) is applied, a unique value is assigned to each attribute or variable.
Then, a state may be identified with a function $s\in F^{E}:=\\{E\rightarrow
F\\}$ with
* •
The universe $E$. The elements of $E$ will be called entities. They represent
the objects of the world which we are interested in (installation steps,
measuring devices, genes, etc.).
* •
The set $F$ (fluents) denotes changing properties of the entities. It is the
union of the scale values, for all $e\in E$. This descriptive term is adopted
from the fluent calculus, an agent based modelling and reasoning method [95].
With these restrictions, Definition 4.2.1 is equivalent to
###### Definition 4.2.2.
Given sets $S$ (states), $E$ (entities), $F$ (fluents) and a function
$\gamma:S\rightarrow F^{E}$, a state context is a formal context $(S,M,I)$
with $M\subseteq E\times F$. Its relation $I$ is given as
$s\,I\,(e,f)\Leftrightarrow\gamma(s)(e)=f$, for all $s\in S,\>e\in E$ and
$f\in F$.
Thus, in the language of automata theory and Kripke structures, the output
function $\gamma:S\rightarrow D$ maps to the data set
$D:=F^{E}\subseteq\mathfrak{P}(E\times F)$. A state is a function name and a
row of the context defines the mapping.
Besides nominal or dichotomic (for $F:=\\{0,1\\}$) scaling as in Definition
4.2.2, different scalings may be useful, if the state context is derived from
a many-valued context $(S,E,F,J)$. Biordinal scaling ([41, Definition 31],
Table 4.2) differentiates low and high measured values into several classes
according to thresholds. Simultaneously a coarser and finer “clustering” of
observed values may be expressed, as well as imprecise knowledge: Intermediate
scale values can be represented without loss of information for the extreme
values. This is biologically relevant, if for instance a transcription factor
activates or inhibits different genes at different expression levels.
A similar scale [38, Figure 3] may also be appropriate in the case of possible
imprecise measuring or if no precise threshold of effectiveness (“high”) is
known. In addition to “low” (e.g. $\leq 300$) and “high” ($\geq 600$), this
scale contains the attributes “not low” ($>300$) and “not high” ($<600$)
expressing intermediate values. Of course, a scale can have even more
discretisation steps (scale attributes).
a) ETS1 SMAD4 $s_{0}$ 280 305 $s_{1}$ 345 567 $s_{2}$ 628 410 b) $\leq 300$
$\leq 450$ $>450$ $\geq 600$ 280 $\times$ $\times$ 305 $\times$ 345 $\times$
410 $\times$ 567 $\times$ 628 $\times$ $\times$
c) ETS1$\leq 300$ ETS1$\leq 450$ ETS1$\geq 450$ ETS1$\geq 600$ SMAD4$\leq
300$ SMAD4$\leq 450$ SMAD4$>450$ SMAD4$\geq 600$ $s_{0}$ $\times$ $\times$
$\times$ $s_{1}$ $\times$ $\times$ $s_{2}$ $\times$ $\times$ $\times$
Table 4.2: A small part of a single gene expression time series for cells
stimulated with TGF$\beta$1; the complete data set will be analysed in Chapter
8. The states $s_{0},s_{1}$ and $s_{2}$ represent mRNA measurements of the
transcription factors ETS1 and SMAD4 at three time points. a): original data,
considered as a many-valued context (Definition 2.1.2). b): discretising
biordinal scale (Definition 2.1.4). c): derived one-valued context.
### 4.3 The transition context $\mathbb{K}_{t}$
In order to express dynamics, we need a supplementary structure: a relation
$R\subseteq S\times S$ indicating temporal transitions between the states. The
output function $\gamma\colon S\rightarrow\mathfrak{P}(M)$ is representable by
a state context $(S,M,I)$ (each row defines $s\mapsto\gamma(s)$, for $s\in
S$). Since $R$ is in one-to-one correspondence to a transition function
$\delta\colon S\rightarrow\mathfrak{P}(S)$, we have a Kripke structure. It is
expressed as a unique mathematical structure (more integrated than a tuple of
sets and maps like in Definition 3.1.3), following an approach in [40]. In
this work, an action context of an LTSA has been introduced, representing the
Kripke structure for each action $a\in A$ by a unique formal context and
allowing attribute exploration of dynamic properties. Moreover, it follows
from the definitions in Chapter 3 that automata and LTSA are representable by
a family of action contexts ${(\mathbb{K}_{t}^{a})}_{a\in A}$, which here are
called transition contexts (Theorem 5.3.1). We concentrate on single
transition contexts where the LTSA relation $R\subseteq S\times\\{a\\}\times
S$ is identifiable with $R\subseteq S\times S$.
###### Definition 4.3.1.
Given a state context $\mathbb{K}_{s}=(S,M,I)$ and a relation $R\subseteq
S\times S$, the transition context $\mathbb{K}_{t}$ of $\mathbb{K}_{s}$ with
respect to $R$ is the context $(R,M\times\\{in,out\\},\nabla)$ with relation
$\nabla$:
$(s^{in},s^{out})\nabla(m,i):\Leftrightarrow s^{i}I\,m\qquad\text{for all
}m\in M,\>i\in\\{in,out\\}\text{ and }(s^{in},s^{out})\in R.$
| driver.linux.in | ndiswrapper.in | driver.windows.in | connection.in | driver.linux.out | ndiswrapper.out | driver.windows.out | connection.out
---|---|---|---|---|---|---|---|---
$(s_{0},s_{10})$ | | | | | $\times$ | | |
$(s_{0},s_{11})$ | | | | | | $\times$ | |
$(s_{10},s_{20})$ | $\times$ | | | | $\times$ | | | $\times$
$(s_{11},s_{21})$ | | $\times$ | | | | $\times$ | $\times$ |
$(s_{21},s_{31})$ | | $\times$ | $\times$ | | | $\times$ | $\times$ | $\times$
$(s_{20},s_{20})$ | $\times$ | | | $\times$ | $\times$ | | | $\times$
$(s_{31},s_{31})$ | | $\times$ | $\times$ | $\times$ | | $\times$ | $\times$ | $\times$
counterEx1 | | | | | | | $\times$ |
counterEx2 | | | $\times$ | | | $\times$ | $\times$ |
Table 4.3: Transition context related to four main steps of the Linux
installation process for a wireless card. The objects are pairs of succeeding
states from Table 4.1. The four left columns denote attributes of the input,
the right columns of the output states. The last two rows are counterexamples
introduced during attribute exploration (cf. Section 5.1).
Thus, a transition context is a subcontext of the semiproduct
$\mathbb{K}_{s}\>\begin{sideways}$\bowtie$\end{sideways}\;\mathbb{K}_{s}$
(Definition 2.1.6) of a state context with itself. It may be regarded as the
context derived from the many-valued context $(R,\\{\text{in,out}\\},S,J)$:
| in | out
---|---|---
$t_{0}$ | $s_{0}$ | $s_{1}$
… | |
$t_{n}$ | $s_{i}$ | $s_{j}$
by scaling both attributes with $\mathbb{K}_{s}$
($t_{0},...,t_{n},\>n\in\mathbb{N}_{0}$ are transitions,
$i,j\in\\{0,1,...,{|S|-1}\\}$). Hence, a transition context is derived by
replacing the attributes by the rows of $\mathbb{K}_{s}$ for the input and
output state of the respective transition.
Transitions may reflect observations repeated at different time points, or
they may be generated by a dynamic model. In this respect, we focus on BN
(Section 3.3.2). Dichotomic scaling will be applied, if $0$ is regarded as
explicit attribute and implications with this attribute are meaningful. For
instance, both low and high expression values of different genes or even of
the same gene can have effects.
###### Definition 4.3.2.
Let $E$ be an arbitrary set of entities and $F:=\\{0,1\\}$ a set of fluents. A
transition function $\delta\colon F^{E}\rightarrow F^{E}$ is called a Boolean
Network.
Given $E$ and $F$, $F^{E}$ is the set of all possible state descriptions
(compare Definition 4.2.2). Together with an injective (i.e., $s\in S$ is
attribute defined) output function $\gamma\colon S\rightarrow F^{E}$, a BN
defines a transition function $\delta^{\prime}\colon S\rightarrow S$ by
$\delta^{\prime}(s):=(\gamma^{-1}\circ\delta\circ\gamma)(s)\>(s\in S)$, hence
a deterministic Kripke structure. $\delta^{\prime}$ is well-defined, if the
state set is chosen large enough, i.e. if all state descriptions generated by
$\delta$ correspond again to a state $s\in S$: $\forall s\in
S\colon(\delta\circ\gamma)(s)\in\gamma(S)$.
For $|E|=n$, a BN is an $n$-ary Boolean function and
$F^{E}\cong\\{0,1\\}^{n}$. For ease of notation, $s\in S$ is identified with
$\gamma(s)\in F^{E}$. With states $s\in\\{0,1\\}^{n}$ and coordinate functions
$\delta_{j}\colon\\{0,1\\}^{n}\rightarrow\\{0,1\\},\>j=1,\dots n$, the
transition function $\delta$ is given by
$\delta(s):=\begin{pmatrix}\delta_{1}(s)\\\ \vdots\\\
\delta_{n}(s)\end{pmatrix}$
A BN is representable by a directed graph where only the edges $(i,j)$ from
influencing entities to an output node are drawn:
$\exists\,s\in\\{0,1\\}^{n}\colon\delta_{j}(s\mid s_{i}=0)\neq\delta_{j}(s\mid
s_{i}=1)$.111The notation $(s\mid s_{i}=0)$ means: In the tuple
$s=(s_{0},\dots,s_{n-1})\in\\{0,1\\}^{n}$ the entry
$s_{i},i\in\\{0,\dots,n\\}$ is replaced by 0. The nodes are labelled by the
coordinate functions $\delta_{j}$.
A BN generates a dynamic simulation, i.e. a process, by repeated application
of $\delta^{\prime}$ to a set of start states $S^{start}\subseteq S$. After
each discrete time step, all component functions may be updated simultaneously
or with specific time delays (synchronous or asynchronous BN). Boolean
networks may be generalised in order to include nondeterminism. Then,
different output states are generated from a single input state (compare
[110], Section 7.2 and Table 7.1).
### 4.4 The transitive context $\mathbb{K}_{tt}$
It appears promising to consider the transitive closure
$t(R)=\bigcup_{n\in\mathbb{N}}R^{n}$, i.e. $(s_{0},s_{1})\in t(R)$ for two
elements $s_{0}$ and $s_{1}$ of $S$, if there exist
$\alpha_{0},\alpha_{1},...,\alpha_{n}\in S$ with
$\alpha_{0}=s_{0},\alpha_{n}=s_{1}$ and $(\alpha_{r},\alpha_{r+1})\in R$ for
all $0\leq r<n$. That means, the state $s_{1}$ emerges from $s_{0}$ by some
transition sequence of arbitrary finite length. A transitive context contains
explicit information regarding reachable states.
###### Definition 4.4.1.
The Transitive context $\mathbb{K}_{tt}$ of a given transition context
$\mathbb{K}_{t}:=(R,M\times\\{in,out\\},\nabla)$ is the formal context with
object set $t(R)$, the transitive closure of $R$. $\nabla$ is extended
accordingly:
$\mathbb{K}_{tt}:=(t(R),M\times\\{in,out\\},\nabla).$
| driver.linux.in | ndiswrapper.in | driver.windows.in | connection.in | driver.linux.out | ndiswrapper.out | driver.windows.out | connection.out
---|---|---|---|---|---|---|---|---
$(s_{0},s_{10})$ | | | | | $\times$ | | |
$(s_{0},s_{20})$ | | | | | $\times$ | | | $\times$
$(s_{0},s_{11})$ | | | | | | $\times$ | |
$(s_{0},s_{21})$ | | | | | | $\times$ | $\times$ |
$(s_{0},s_{31})$ | | | | | | $\times$ | $\times$ | $\times$
$(s_{10},s_{20})$ | $\times$ | | | | $\times$ | | | $\times$
$(s_{11},s_{21})$ | | $\times$ | | | | $\times$ | $\times$ |
… | | | | | | | |
Table 4.4: Transitive context derived from the transition context of Table
4.3. Now the objects are transitions, which relate an input to an output state
succeeding after an arbitrary number of time steps.
### 4.5 The temporal context $\mathbb{K}_{tmp}$
A transition context represents a Kripke structure by which the semantics of a
temporal logic is given. It thus generates a new temporal context: the
attributes of the underlying state context are extended by the set of atomar
propositions formed with the original attributes and three operators from
temporal logic. Using the corresponding transitive context for definitions
will show to be more convenient in important cases like deterministic
processes.
The language CTL is chosen since it is quite universal and includes
nondeterminism. A major restriction, however, is that CTL does not provide
operators related to the past. First, we do not want to enlarge the number of
operators within this basic study and therefore will even consider a subset of
CTL operators explicitly. More importantly, the principal aim of this thesis
are applications related to user guidance, process control or predictions.
Finally, in the transition and transitive contexts, output-input implications
related to the past hold. Hence, in principle past operators could be defined
analogously to the following introduction of future operators. In this case,
the asymmetry of time should be kept in mind, usually assumed in temporal
logics and highlighted by Prior. Since there is no “retrospective
nondeterminism”, only the deterministic versions of the subsequent definitions
should be adapted. In a nondeterministic transition context, this is not quite
natural and needs some technical efforts.
In order to emphasise the structural difference, in CTL, of the path and
temporal operators, I use Priors (as well as the modal logic) notation
$\Diamond$ for the possibility operator (corresponding to $E$) and $\Box$ for
the necessity operator (corresponding to $A$). As for the temporal operators,
we focus on $F$ (“eventually”), $G$ (“always”) and $\neg F$ (“never”).
###### Definition 4.5.1.
Let $R\subseteq S\times S$ be a relation. Then a Path is a finite sequence
$\pi:=(s_{0},\ldots,s_{n})\in S^{n+1}\>(n\in\mathbb{N})$ or an infinite
sequence $\pi:=(s_{i})_{i\in\mathbb{N}_{0}}$, so that $(s_{i},s_{i+1})\in R$
for all $0\leq i<n$ or $i\in\mathbb{N}_{0}$, respectively. If $\pi$ is finite,
it is required to be maximal: $\nexists\,s_{n+1}\in S\colon(s_{n},s_{n+1})\in
R$. The set of all paths $\pi$ generated by $R$ is denoted by $Seq_{R}$.
In CTL usually infinite paths are required, hence a total relation:
$\forall{s\in S}\>\exists\,{s^{\prime}\in S}\colon(s,s^{\prime})\in R$ [29,
Section 4.1]. If there are final states $s_{F}$ like for an automaton,
assuming $(s_{F},s_{F})\in R$ makes the relation total. In applications,
however, observations or predictions are often incomplete. Then we accept that
the relation $R$ is not total.
###### Definition 4.5.2.
Given a state context $\mathbb{K}_{s}=(S,M,I)$ and a relation $R\subseteq
S\times S$, a temporal context is defined as
$\mathbb{K}_{tmp}:=(S^{in},M\cup\,T,I^{in}\cup I_{T})$. The state set is
restricted to the set of input states $S^{in}:=\\{s\in S\mid\exists
s^{\prime}\in S\colon(s,s^{\prime})\in R\\}$, $I$ to $I^{in}$ correspondingly,
whereas the attribute set is extended by $T:=\\{\Diamond Fm\mid\>m\in
M\\}\,\cup\,\\{\Box Fm\mid\>m\in M\\}\,\cup\,\\{\Diamond Gm\mid\>m\in
M\\}\,\cup\,\\{\Box Gm\mid\>m\in M\\}\,\cup\,\\{\Diamond\neg Fm\mid\>m\in
M\\}\,\cup\,\\{\Box\neg Fm\mid\>m\in M\\}$. Let $s\in S^{in}$ and
$Seq_{R}^{s}:=\\{\pi\in Seq_{R}\mid s_{0}=s\\}$. The relation $I_{T}$ is
defined as follows:
$\displaystyle s\>I_{T}\>\Diamond Fm:\Leftrightarrow\>$
$\displaystyle\exists\,\pi\in
Seq^{s}_{R}\>\exists\,i\in\mathbb{N}:\>(s_{i},m)\in I$ (4.1) $\displaystyle
s\>I_{T}\>\Box Gm:\Leftrightarrow\>$ $\displaystyle\forall\,\pi\in
Seq^{s}_{R}\>\forall\,i\in\mathbb{N}:\>(s_{i},m)\in I$ (4.2) $\displaystyle
s\>I_{T}\>\Box\neg Fm:\Leftrightarrow\>$ $\displaystyle\forall\,\pi\in
Seq^{s}_{R}\>\forall\,i\in\mathbb{N}:\>(s_{i},m)\notin I$ (4.3) $\displaystyle
s\>I_{T}\>\Box Fm:\Leftrightarrow\>$ $\displaystyle\forall\,\pi\in
Seq^{s}_{R}\>\exists\,i\in\mathbb{N}:\>(s_{i},m)\in I$ (4.4) $\displaystyle
s\>I_{T}\>\Diamond Gm:\Leftrightarrow\>$ $\displaystyle\exists\,\pi\in
Seq^{s}_{R}\>\forall\,i\in\mathbb{N}:\>(s_{i},m)\in I$ (4.5) $\displaystyle
s\>I_{T}\>\Diamond\neg Fm:\Leftrightarrow\>$ $\displaystyle\exists\,\pi\in
Seq^{s}_{R}\>\forall\,i\in\mathbb{N}:\>(s_{i},m)\notin I$ (4.6)
For $B\subseteq M$, set $\Diamond FB$ := $\\{\Diamond Fm_{1},...,\Diamond
Fm_{n}\mid m_{1},...,m_{n}\in B\\}$, and so forth.
$\mathbb{K}_{tmp}$ is the apposition $\mathbb{K}^{in}_{s}\mid\mathbb{K}_{T}$
of a state context $\mathbb{K}^{in}_{s}:=(S^{in},M,I^{in})$ and
$\mathbb{K}_{T}:=(S^{in},T,I_{T})$. It should be kept in mind that sets of
temporal attributes may relate to different paths. We understand
$0\notin\mathbb{N}$, hence the definitions refer to states subsequent to $s$
and have a clear dynamic meaning. This is in accordance to Definition 4.5.1,
which does not admit sequences $(s_{0})$.
| driver.linux | ndiswrapper | driver.windows | connection | $\Diamond F\,$driver.linux | $\Box\,G\,$driver.linux | $\Box\,\neg F\,$driver.linux | $\Box\,F\,$driver.linux | $\Diamond\,G\,$driver.linux | $\Diamond\,\neg F\,$driver.linux | … | $\Diamond\,F\,$connection | $\Box\,G\,$connection | $\Box\,\neg F\,$connection | $\Box\,F\,$connnection | $\Diamond\,G\,$connection | $\Diamond\,\neg F\,$connnection
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$s_{0}$ | | | | | $\times$ | | | | $\times$ | $\times$ | | $\times$ | | | $\times$ | |
$s_{10}$ | $\times$ | | | | $\times$ | $\times$ | | $\times$ | $\times$ | | | $\times$ | $\times$ | | $\times$ | $\times$ |
$s_{11}$ | | $\times$ | | | | | $\times$ | | | $\times$ | | $\times$ | | | $\times$ | |
$s_{20}$ | $\times$ | | | $\times$ | $\times$ | $\times$ | | $\times$ | $\times$ | | | $\times$ | $\times$ | | $\times$ | $\times$ |
$s_{21}$ | | $\times$ | $\times$ | | | | $\times$ | | | $\times$ | | $\times$ | $\times$ | | $\times$ | $\times$ |
$s_{31}$ | | $\times$ | $\times$ | $\times$ | | | $\times$ | | | $\times$ | | $\times$ | $\times$ | | $\times$ | $\times$ |
Table 4.5: Temporal context (Definition 4.5.2) extending the state context of
Table 4.1 by attributes from temporal logic. $\Diamond\colon\text{for one
path}$, $\Box\colon\text{for any path}$, $F\colon\text{eventually}$,
$G\colon\text{always}$, $\neg F\colon\text{never}$. More specifically, the
last three predicates will be used for $\Diamond F$, $\Box\,G$ or $\Box\,\neg
F$, respectively.
As we admit finite paths and do not presuppose a total relation $R$, the
objects of $\mathbb{K}_{tmp}$ are restricted to input states in order to have
an exact correspondence to $\mathbb{K}_{tt}$. In Chapter 6, inferences between
implications of the two formal contexts will be investigated.
The temporal attributes may also be defined by $\Diamond$, $F$ and $\neg$
only. For all $s\in S$ and all $m\in M$ holds:
$s\,I_{T}\,\square\neg Fm\Leftrightarrow s\,I_{T}\,\neg\Diamond Fm$
Thus, one attribute can be expressed by the absence of the other. Both are
needed in the case of dichotomically scaling $\Diamond Fm$, i.e. if one is
interested in implications with the attribute $\neg\Diamond Fm$, or vice
versa. The same holds for $\Box Fm\Leftrightarrow\neg\Diamond\neg Fm$ and
$\Box Gm\Leftrightarrow\neg\Diamond F\neg m.$
Since a path is defined by $R$, at least partial knowledge of $\mathbb{K}_{t}$
is required to decide if the temporal attributes hold. $\mathbb{K}_{t}$ cannot
be reconstructed unambigously from a transitive context $\mathbb{K}_{tt}$
(whereas the inverse holds, of course). However, in important cases knowledge
of the transitive context is sufficient. It contains information regarding
reachable states, while the paths do not have to be known.
###### Remark 4.5.3.
The first three temporal attributes may be decided more conveniently by the
transitive context $\mathbb{K}_{tt}=(t(R),M\times\\{in,out\\},\nabla)$
generated by $\mathbb{K}_{t}$. For all $s\in S^{in}$ and all $m\in M$ holds:
$\displaystyle s\>I_{T}\>\Diamond Fm\Leftrightarrow\>$
$\displaystyle\exists\,(s^{in},s^{out})\in t(R)\colon
s=s^{in}\wedge((s^{in},s^{out}),m^{out})\in\nabla$ $\displaystyle
s\>I_{T}\>\Box Gm\Leftrightarrow\>$ $\displaystyle\forall\,(s^{in},s^{out})\in
t(R)\text{ with }s=s^{in}\colon((s^{in},s^{out}),m^{out})\in\nabla$
$\displaystyle s\>I_{T}\>\Box\neg Fm\Leftrightarrow\>$
$\displaystyle\forall\,(s^{in},s^{out})\in t(R)\text{ with
}s=s^{in}\colon((s^{in},s^{out}),m^{out})\notin\nabla.$
As $s\in S^{in}$ is presupposed, $s\,I_{T}\,\Box Gm$ implies
$s\,I_{T}\,\Diamond Fm$.
In the following, “eventually”, “always” and “never” will have the specific
semantics given by the three properties, which only makes a difference for
nondeterminism. In the deterministic case, a single path exists starting from
an arbitrary state $s_{0}\in S^{in}$. Therefore, also the other attributes are
definable by $\mathbb{K}_{tt}$, with $\Box Fm=\Diamond Fm,\Diamond Gm=\Box Gm$
and $\Diamond\neg Fm=\Box\neg Fm$.
$\Box G\neg m$ expresses a safety property, $\Box Fm$ a liveness property (see
Section 3.2.2). In the following applications we focus on the properties
definable by the transitive context, thus on $\Diamond Fm$ instead of $\Box
Fm$. The property is adequate to stochastic data like in biology and
interesting as negation of safety.
The remaining operators from CTL – $X$ (“next”) and $U$ (“until”) – are
definable as follows, for $m\in M$, $B\subseteq M$ and for all $s\in S^{in}$:
$\displaystyle s\>I_{T}\>\Diamond\,Xm:\Leftrightarrow\>$
$\displaystyle\exists\,\pi\in Seq^{s}_{R}\colon(s_{1},m)\in I$
$\displaystyle\Leftrightarrow\>$ $\displaystyle\exists\,(s_{in},s_{out})\in
R\colon(s_{in}=s)\wedge(s^{in},s^{out})\,\nabla\,m^{out}$ $\displaystyle
s\>I_{T}\>\Diamond\,m\,UB:\Leftrightarrow\>$ $\displaystyle\exists\,\pi\in
Seq^{s}_{R}\>\exists\,j\in\mathbb{N}_{0}\colon(s_{j}\in
B^{I})\wedge(\forall\,0\leq i<j\colon(s_{i},m)\in I).$
With the condition $i<j$, $m\,UB$ is satisfied if the start state of $\pi$ has
all attributes in $B$, i.e. if those attributes are contained in the object
intent $s^{\prime}$. The definitions for the necessity operator $\Box$ are
analogous.
All CTL operators may be constructed with the aid of $\Diamond$, $X$, $U$ and
$\neg$, e.g. $\Diamond\,B=\Diamond\,\top\,UB$ (compare [15, p. 2]). However,
the two operators will not be examined below. $X$ is expressed by a transition
context and $U$ enlarges the set of temporal attributes $T$ excessively, since
it connects two attribute sets $\\{m\\},B\subseteq M$.
## Chapter 5 Using attribute exploration of the defined formal contexts
Attribute exploration of discrete temporal transitions aims at unfolding the
dynamics of a process by investigating its rules. In the last chapter, four
data structures were defined: A state context $\mathbb{K}_{s}$ describes by
observable attributes the different states occuring during a temporal
development. It is enlarged to a context $\mathbb{K}_{tmp}$ by attributes
constructed with operators from temporal logic. Their semantics is given by a
transition context $\mathbb{K}_{t}$. For deterministic processes the related
transitive context $\mathbb{K}_{tt}$ is sufficient. It expresses knowledge
about reachable states and their attributes.
In order to give an intuition of the methodic possibilities, I start with the
wireless card example of Section 4.1. In this example, the proposed
implications are decidable by a human basically experienced with a Linux
operating system. In Section 5.2, more structured interactions between theory
and observations will be presented. One purpose is to relate observations to
existing knowledge by a scientist. Inversely, model predictions can be
validated empirically. I propose a procedure which will be applied in the main
biological example of Chapter 8. In Chapter 7, the dynamics of a biological
model from literature are analysed by computing the stem base of the related
transitive context. In Section 5.3, the universal applicability of our
approach to three important classes of process models is summarised in a
theorem: LTSA, Kripke structures and automata are representable by a family of
transition contexts.
Given a set of observations related to the input-output-behaviour, the
attribute exploration algorithm asks if those observations are complete. Next
Closure, the core of the algorithm, finds the next pseudo-closed set with
respect to the lectic order of the attributes [41, p. 66f., 85]; regarding the
supplementary criterion of set inclusion it is the smallest one. If the
corresponding implication is rejected, the counterexample represents a minimal
missing observation in this sense. Furthermore, the number of newly introduced
observations is generally small, in correspondence to the minimality of the
stem base.
### 5.1 Example
The stem base of the state context (Table 4.1) is simple:
1 < 2 > driver.windows ==> ndiswrapper;
2 < 1 > ndiswrapper, connection ==> driver.windows;
3 < 0 > driver.linux, ndiswrapper ==> driver.windows, connection;
The first two implications express that a Windows driver and the wrapper
module have to be installed together, implicating temporal priority of the
latter. For an open source driver, no further statement is possible. The
attribute driver.linux may only occur together with connection, but there is
no dependency in any direction. The third implication signifies the exclusion
of the Windows and Linux procedures: all remaining attributes occur in the
conclusion, and the extent of the premise as well as of the complete attribute
set is empty. For implications, in contrast to less strict association rules,
the cardinality of this extent is the support of the rule (see p. 5.2). It is
indicated in brackets <$\,\cdot$ >. This type of implications is mostly noted
with the $\bot$ symbol: driver.linux, ndiswrapper ==> $\bot$.
The transition context describes immediately succeeding installation steps
$(s^{in},s^{out})\in R$ by their respective attributes, i.e. by single
performed procedures. Accordingly, static implications of $\mathbb{K}_{s}$ are
found again between attributes referring to $s^{in}$ or $s^{out}$ solely. New
information regarding dynamics is obtained by “mixed” implications with input
and output attributes. The exploration of the transition context (Table 4.3)
leads to two counterexamples:
* •
driver.windows.out $\rightarrow$ ndiswrapper.in, ndiswrapper.out: The proposed
implication raises the question if the order can also be changed: A user might
download a Windows driver first and then s/he learns that s/he needs the
ndiswrapper module. This procedure is possible, since ndiswrapper copies the
driver files into its own directory anyway. Therefore, the first
counterexample in Table 4.3 is introduced.
* •
It is important to correct the later implication ndiswrapper.out,
driver.windows.out $\rightarrow$ ndiswrapper.in and to add counterexample 2
with driver.windows.in instead of ndiswrapper.in.
The stem base of the transition context is the following:
1 < 3 > ndiswrapper.in ==> ndiswrapper.out, driver.windows.out;
2 < 3 > driver.windows.in ==> ndiswrapper.out, driver.windows.out;
3 < 2 > driver.linux.out, connection.out ==> driver.linux.in;
4 < 2 > ndiswrapper.out, connection.out ==> ndiswrapper.in,
driver.windows.in, driver.windows.out;
5 < 2 > driver.windows.out, connection.out ==> ndiswrapper.in,
driver.windows.in, ndiswrapper.out;
6 < 2 > driver.linux.in ==> driver.linux.out, connection.out;
7 < 2 > connection.in ==> connection.out;
8 < 2 > ndiswrapper.in, driver.windows.in, ndiswrapper.out,
driver.windows.out ==> connection.out;
9 < 0 > driver.linux.out, ndiswrapper.out ==> driver.linux.in,
ndiswrapper.in, driver.windows.in, connection.in, driver.windows.out,
connection.out;
10 < 0 > driver.linux.out, driver.windows.out ==> driver.linux.in,
ndiswrapper.in, driver.windows.in, connection.in, ndiswrapper.out,
connection.out;
The first two implications describe the alternative paths of the Windows
driver installation, the third to fifth preconditions for entering the
connection data, combined in the latter two with an (output) state
implication. Implication 6 and 8 define the last step in the Linux and Windows
driver installation process, respectively. Implication 7 marks the state with
connection attribute as the final state, or a steady state in engineering,
physical or biological systems. The last two implications again express the
exclusion of the alternative paths for a Linux and Windows driver.
A state $s_{12}$ with single attribute driver.windows could be found as
counterexample during the exploration of the state context already (Table
4.1), if one considers the complete dynamics. Then, the counterexamples of the
transition context will be introduced before its exploration as the
transitions $(s_{0},s_{12})$ and $(s_{12},s_{21})$ with the new state.
Consequently, no other counterexamples need to be added during the exploration
of $\mathbb{K}_{t}$. With $s_{12}$, the stem base of $\mathbb{K}_{s}$ is:
1 < 1 > ndiswrapper, connection ==> driver.windows;
2 < 1 > driver.windows, connection ==> ndiswrapper;
3 < 0 > driver.linux, ndiswrapper ==> driver.windows, connection;
4 < 0 > driver.linux, driver.windows ==> ndiswrapper, connection;
Implication 4 is new, since now {driver.windows} is closed as object intent:
only the driver may be installed. Then, all subsets of the premise (including
$\emptyset$)111All objects (states) have the empty attribute set in common,
but no other attribute; there is no implication $\emptyset\rightarrow...$ are
closed, and the condition of a pseudo-intent is trivially fulfilled. In the
first version of the stem base, however, {driver.linux, driver.windows} does
not contain the closure of the pseudo-intent {driver.windows}, indicated by
implication 1. Hence, the attribute set is not pseudo-closed and there is no
corresponding implication in the previous stem base.
If the enlarged stem base of $\mathbb{K}_{s}$ is entered as background
knowledge prior to an exploration of $\mathbb{K}_{t}$ (compare Section 6.1),
pure input or output implications can be decided automatically, like $3\text{
(in }\mathbb{K}_{s})\models 9\text{ (in }\mathbb{K}_{t})$, $4\text{ (in
}\mathbb{K}_{s})\models 10\text{ (in }\mathbb{K}_{t})$, or ndiswrapper.out,
connection.out ==> driver.windows.out (output part of implication 4).
Without the implications derivable from those of the (corrected) state
context, the stem base of the transitive context (Table 4.4) is:
1 < 4 > ndiswrapper.in ==> ndiswrapper.out, driver.windows.out;
2 < 4 > driver.windows.in ==> ndiswrapper.out, driver.windows.out;
3 < 2 > driver.linux.in ==> driver.linux.out, connection.out;
4 < 2 > connection.in ==> connection.out;
5 < 2 > ndiswrapper.in, driver.windows.in, ndiswrapper.out,
driver.windows.out ==> connection.out;
6 < 1 > connection.in, driver.linux.out, connection.out
==> driver.linux.in;
7 < 1 > connection.in, ndiswrapper.out, driver.windows.out,
connection.out ==> ndiswrapper.in, driver.windows.in;
It expresses the following new facts:
* •
Implications 1 to 4 are identical to the implications 1, 2, 6 and 7 of
$\mathbb{K}_{t}$, but have different semantics: The attributes of a state
remain those of all subsequent states, which signifies that a success cannot
be destroyed by a further action. This seems to be realistic with the
assumption of a sufficiently experienced user. For the same input and output
attribute this meaning is already implicit in the transition context, e.g.
driver.linux.in $\rightarrow$ driver.linux.out.
* •
Implication 6 connects input and output states in a more complicated manner
(connection.out follows from implication 4): If after an input state with
attribute connection the Linux driver is installed, it must have been
installed already at this input state. Thus, the temporal order of the two
installation steps is fixed. Implication 6 replaces 3 in $\mathbb{K}_{t}$.
$(s_{10},s_{20})^{\prime}$ = {driver.linux.in, driver.linux.out,
connection.out} is no more generated by the former premise {driver.linux.out,
connection.out}, but this attribute set is closed as object intent of the new
transition $(s_{0},s_{20})$ with the start state $s_{0}$ where
$s_{0}^{\prime}=\emptyset$:
$(s_{0},s_{20})^{\prime}=\text{\\{{driver.linux.out,
connection.out}\\}}\varsubsetneq(s_{10},s_{20})^{\prime}\varsubsetneq(s_{20},s_{20})^{\prime}$
With the supplementary condition connection.in implication 6 refers to
$(s_{20},s_{20})$ only and describes a property of the steady state $s_{20}$.
Implication 7 is analogous for the Windows driver path.
The first of 18 implications in the stem base of the temporal context are,
with ev(entually)$:=\Diamond F$, alw(ays)$:=\Box\,G$ and nev(er)$:=\Box\,\neg
F$:
1 < 14 > { } ==> ev(connection);
2 < 12 > ev(driver.windows) ==> ev(ndiswrapper);
3 < 12 > ev(ndiswrapper) ==> ev(driver.windows);
4 < 2 > connection ==> alw(connection);
5 < 4 > driver.windows ==> ev(ndiswrapper), ev(driver.windows),
alw(ndiswrapper), alw(driver.windows), nev(driver.linux);
6 < 4 > ndiswrapper ==> ev(ndiswrapper), ev(driver.windows),
alw(ndiswrapper), alw(driver.windows), nev(driver.linux);
...
Implications 2 and 3 indicate that the two steps driver.windows and
ndiswrapper have to be taken on the Windows path in order to establish the
connection, similarly implications 5 and 6. Those mean again: Once ndiswrapper
or a windows driver is installed, it remains installed; also the Windows and
Linux paths are exclusive. The first and fourth implication express a hopeful
view: An internet connection will be established some day (and by some way) –
and then it will hold forever…
Since users sometimes make bad experiences with the compatibility of hardware
and software or with their internet provider, more realistic counterexamples
could be entered into the context. However, they only make sense together with
the introduction of new attributes, e.g. line for the existence or breakdown
of the external internet connection. In this case, the previously explored
implications cannot be used further. After all, even if several implications
are removed, only a defined part of the stem base has to be computed again
[20, p.14]. A better solution is to interpret connection as only entering the
personal connection data and to define a new attribute success, corresponding
to the final state of an automaton. Then, no new transitions $\notin R$ with
connection.in but not connection.out are introduced, contradicting the
implication connection.in $\rightarrow$ connection.out in $\mathbb{K}_{tt}$,
which corresponds to implication 4 in the stem base of $\mathbb{K}_{tmp}$.
In order to refine the process description, new attributes may be introduced,
for example device recognition with sudo lshw -C network at the beginning,
resulting in the alternative actions device.on.out (turning the wireless
device on by a hardware switch or even by Windows), ndiswrapper.out,
driver.linux.out or
connection.out.222https://help.ubuntu.com/9.04/internet/C/troubleshooting-
wireless.html# troubleshooting-wireless-device Furthermore, existing
attributes may be splitted, for instance in ndiswrapper.graphical and
ndiswrapper.commandline.333https://help.ubuntu.com/community/WifiDocs/Driver/Ndiswrapper#Installing
Windows driver
In general, newly added attributes should be chosen carefully to avoid a
change of the meaning of the first attributes. Then, the implications of all
contexts hold in the enlarged contexts. For the reader familiar with FCA, the
following remark expresses this demand more formally.
###### Remark 5.1.1.
For a transition context $\mathbb{K}_{t}:=(R,(M_{1}\cup
M_{2})\times\\{in,out\\},\nabla)$ (or a transitive context $\mathbb{K}_{tt}$),
implications of the original attribute set $M_{1}$ are preserved, if the set
of transitions $R$ is not changed and the restrictions of the incidence
relation $\nabla$ of the enlarged context $\mathbb{K}_{t}$ to
$R\times(M_{1}\times\\{in,out\\})$ and $R\times(M_{2}\times\\{in,out\\})$
yield the incidence relations of the partial contexts
$\mathbb{K}_{t}^{1}:=(R,M_{1}\times\\{in,out\\},\nabla_{1})$ and
$\mathbb{K}_{t}^{2}:=(R,M_{2}\times\\{in,out\\},\nabla_{2})$. Then, a
supremum-preserving order embedding of the lattice
$\mathfrak{B}(\mathbb{K}_{t})$ into the direct product of
$\mathfrak{B}(\mathbb{K}_{t}^{1})$ and $\mathfrak{B}(\mathbb{K}_{t}^{2})$ is
defined. By reason of the supremum condition, an implication of
$\mathbb{K}_{t}^{1}$ (and $\mathbb{K}_{t}^{2}$) is also an implication of
$\mathbb{K}_{t}$, since the conclusion states that the respective attribute
concepts and all infima are greater or equal to the concept generated by the
premise. The concept lattice of $\mathbb{K}_{t}$ can be visualised by a nested
line diagram, where copies of $\mathbb{K}_{t}^{2}$ and the embedding are
represented in each node of $\mathbb{K}_{t}^{1}$, as a kind of zooming into
more differentiated object descriptions. [41, p. 77ff. and Theorem 7]
An order embedding is also given, if an original context
$\mathbb{K}_{t}^{1}:=(R^{\prime},M_{1}\times\\{in,out\\},\nabla_{1}),\>R^{\prime}\subseteq
R$ is enlarged by all transitions in $R\setminus R^{\prime}$ without changing
the set of object intents, i.e. if clarifying the enlarged context results in
$\mathbb{K}_{t}^{1}$. Then, new attributes from $\mathbb{K}_{t}^{2}$ are
introduced that discriminate the supplementary from the previous transitions.
### 5.2 Integration of knowledge and data
The defined mathematical structures may be used in various ways. In Chapter 7,
a transitive context is generated from a BN. Then, the computed stem base is
searched for implications which make the temporal behaviour explicit and give
new insight. Furthermore, experimental time series can be evaluated by
comparison with existing knowledge, i.e. implications are generalised or
rejected supposing outliers or by reason of special conditions. Inversely, for
the ECM application in Chapter 8 we developed a procedure starting from
knowledge:
1. 1.
Discretise a set of time series of (gene expression) measurements and
transform it to an observed transition context $\mathbb{K}_{t}^{obs}$.
2. 2.
For a set of interesting genes translate interactions from biological
literature and databases into a Boolean network. This step could be supported
by text mining software.
3. 3.
Construct the transition context $\mathbb{K}_{t}$ by a simulation starting
from a set of states, e.g. the initial states of $\mathbb{K}_{t}^{obs}$ or all
states (for small networks).
4. 4.
Derive the respective transitive contexts $\mathbb{K}_{tt}$ and
$\mathbb{K}^{obs}_{tt}$.
5. 5.
Perform attribute exploration of $\mathbb{K}_{tt}$. Decide about an
implication $A\rightarrow B$, $A,B\subseteq M$, by checking its validity in
$\mathbb{K}_{tt}^{obs}$ and/or by searching for supplementary knowledge.
Possibly provide a counterexample from $\mathbb{K}^{obs}_{tt}$.
6. 6.
Answer logical queries from the modified context $\mathbb{K}_{tt}$ and from
its stem base.
In step 5 automatic decision criteria could be thresholds of support
$q=|(A\cup B)^{\prime}|$ and confidence $p=\frac{|(A\cup
B)^{\prime}|}{|A^{\prime}|}$ for an association rule in
$\mathbb{K}^{obs}_{tt}$ (which coincides with the respective implication for
$p=1$). A weak criterion is to reject only implications with support 0 (but if
no object in $\mathbb{K}^{obs}_{tt}$ has all attributes from A, the
implication is not violated). In [110] and the main part of Chapter 8, a
strong criterion is applied: implications of $\mathbb{K}_{tt}$ have to be
valid also in the observed context ($p=1$). This is equivalent to an
exploration of the union of the two contexts. Moreover, in Section 8.3.9
attribute exploration is performed where the human expert can use support and
confidence as two among several decision criteria.
I implemented the steps 1, 3 and 4 in the scripting language R designed for
data analysis and statistical purposes [6]. For step 5, I used the Java tool
Concept Explorer [2]. The output was translated with R into a Prolog knowledge
base, which can be queried according to step 6.
### 5.3 Transition contexts and automata
As introduced in Section 4.3, a transition context represents a Kripke
structure and a single context from the action context family of an LTSA. In
Chapter 3 the correspondence of an LTSA and an automaton was mentioned. The
following theorem summarises these relationships:
###### Theorem 5.3.1.
With sets $S$ (states), $M$ (attributes) and $A$ (actions), let
$\mathcal{A}:=(S,\alpha_{S})$ be a universal coalgebra of type $\Omega$, where
$\alpha_{S}\colon
S\rightarrow\Omega(S):=\mathfrak{P}(M)\times(\mathfrak{P}(S))^{A}.$
The component maps of $\alpha_{S}$ are denoted by $\gamma\colon
S\rightarrow\mathfrak{P}(M)$ and $\delta\colon
S\rightarrow(\mathfrak{P}(S))^{A}$, thus for all $s\in S$
$\alpha_{S}(s)=(\gamma(s),\delta(s)),$
where $\delta(s)\colon A\rightarrow\mathfrak{P}(S)$.
Then $\mathcal{A}$ is representable by a family of transition contexts
$(\mathbb{K}_{t}^{a})_{a\in A}$, if and only if for all $s\in S$ one of the
following conditions holds:
1. 1.
$\exists\,a\in A\colon\delta(s)(a)\neq\emptyset$.
2. 2.
$\exists\,a\in A\>\exists\,s^{\prime}\in S\colon s\in\delta(s^{\prime})(a).$
###### Proof.
According to Proposition 3.1.8, an LTSA is a universal coalgebra of type
$\Omega$, and every such coalgebra can be represented as an appropriate LTSA.
Its transition relation $R\subseteq S\times A\times S$ (Definition 3.1.7)
corresponds to the component map $\delta\colon
S\rightarrow(\mathfrak{P}(S))^{A}$. It is representable by a family
$(R_{a})_{a\in A}$, where $R_{a}:=\\{(x,y)\mid(x,a,y)\in R\\}\subseteq S\times
S$. Each relation $R_{a}$ is the object set of a transition context (or action
context) $\mathbb{K}_{t}^{a}$, so $R$ can be represented as a family
$(\mathbb{K}_{t}^{a})_{a\in A}$.
The supplementary conditions are equivalent to
$\forall s\in S\>\exists\,a\in A\>\exists\,s^{\prime}\in S:(s,a,s^{\prime})\in
R\vee(s^{\prime},a,s)\in R.$
Hence, each state is either an input or an output state for some transition
context from $(\mathbb{K}_{t}^{a})_{a\in A}$. Then the state context, i.e. the
map $\gamma\colon S\rightarrow\mathfrak{P}(M)$ can be reconstructed. The
condition is necessary, since transition contexts only contain information
regarding states occuring in a relational pair. ∎
Besides LTSAs, Kripke structures – with $A:=\\{\text{update}\\}$ – are such
types of coalgebras (Section 3.1.2). By Proposition 3.1.8 an automaton is
identifiable with an LTSA, hence it can also be represented by a family of
transition contexts.
It is more convenient to have explicit information regarding $\gamma\colon
S\rightarrow\mathfrak{P}(M)$ by a state context $\mathbb{K}_{s}$. In this
case, $\mathcal{A}$ will be represented by a pair
$(\mathbb{K}_{s},(\mathbb{K}_{t})_{a\in A})$ and the supplementary conditions
are not required.
In principle, formal contexts may have an infinite attribute set $M$, like
LTSAs or automata (set of data $D$). However, this has restricted practical
bearing, since for example there are termination problems of attribute
exploration. Nevertheless, for attributes defined as DL concepts, S. Rudolph
[87] as well as F. Baader and F. Distel [18] proved by different approaches
that the algorithm terminates, if a finite model exists (compare p. 9.2). It
may be given as a transition context.
It could be theoretically interesting to define a category of transition
contexts and to investigate if it is equivalent to a category of Kripke
structures. If the morphisms are chosen accordingly, this should be obvious.
However, there are some technical problems, since within a Kripke structure
the state context is given explicitly, whereas states may occur only in the
first or second component of the transition relation $R$, as indicated in the
proof of the theorem.
In order to show categorial equivalence, it is sufficient to prove that there
is a full, faithful and essentially surjective functor in one direction.
Regarding automata, surely only one or two of these properties are given.
Constructing the respective functors could permit to transfer mathematical
results from one category (or theory) to the other. Yet, for our purposes it
is adequate to state the above structural paralleles.
Figure 5.1: Transitive context of the automaton example 3.1.4 shown in Figure
3.1 (Concept Explorer screenshot).
Finally an example is given how attribute exploration might be used for the
analysis of an automaton. The stem base of the transitive context (Figure 5.1)
according to example 3.1.4 is rather simple:
1 < 4 > final_in ==> final_out;
2 < 2 > start_out ==> start_in 1_out;
3 < 2 > 0_in 1_out ==> final_out;
4 < 0 > start_in 1_out start_out final_out ==> 0_in 1_in final_in 0_out;
5 < 0 > 0_in 1_in ==> start_in final_in 0_out 1_out start_out final_out;
6 < 0 > 0_in start_in ==> 1_in final_in 0_out 1_out start_out final_out;
7 < 0 > 0_out 1_out ==> 0_in 1_in start_in final_in start_out final_out;
8 < 0 > start_in final_in final_out ==> 0_in 1_in 0_out 1_out start_out;
Implication 1 signifies that a string is recognised as soon as the first
01-substring is entered, 3 expresses the final condition, 2 that an input 1 is
necessary to remain in a start state. In implications 4-8 all attributes occur
and the support is 0. They express that states must not be start and final at
the same time, 0 and 1 are exclusive, and 0 is not possible in the start
state, but an empty input or 1. These implications could have been entered as
background knowledge, but an attribute exploration helps not to forget one of
the implications, e.g. 6. For this example they are rather obvious for a
human, but in a computer program for automatic reasoning every piece of
information has to be made explicit.
## Chapter 6 Inference rules for the integration of background knowledge
This chapter presents a method to derive inference rules integrating already
acquired knowledge into the successive exploration of the four formal
contexts. This method uses attribute exploration again on a higher level by
means of a test context defined in Section 6.2.2.
A reasonable order is to first explore a state context $\mathbb{K}_{s}$, then
the related transitive, transition and temporal contexts $\mathbb{K}_{tt}$,
$\mathbb{K}_{t}$ and $\mathbb{K}_{tmp}$ (Section 6.1). We concentrate on
inference rules for implications of $\mathbb{K}_{tt}$, which defines the
semantics of three temporal attributes, or of all in the case of a
deterministic process. Nevertheless, the stem base of $\mathbb{K}_{tmp}$ is
not completely derivable from the stem base of $\mathbb{K}_{tt}$; a subsequent
exploration of $\mathbb{K}_{tmp}$ creates new information (Section 6.2.1).
Inference rules are proven for two important classes of implications:
* •
Section 6.3: The premises are one or two “homogeneous” attribute sets, e.g.
$X^{in}\wedge\,Y^{out}$ (in $\mathbb{K}_{tt}$) or $\operatorname{ev}{Y}$ (in
$\mathbb{K}_{tmp}$), $X,Y\subseteq M$. Due to computational limits, the test
context had to be constructed with $|M|=3$, but most of the discovered
inference rules are general.
* •
Section 6.4: The many-valued attributes $e\in E$ take Boolean values $f\in
F=\\{0,1\\}$ (compare Definition 4.2.2). The implications have one homogeneous
attribute set as premise and conclusion, respectively.
### 6.1 A hierarchy of formal contexts
The four defined formal contexts are closely related. $\mathbb{K}_{t}$ is a
subcontext of $\mathbb{K}_{tt}$, since transitions according to the transitive
closure of $R$ are added. Here subcontext means that the attribute set is
equal and the object set is a subset of the object set of the larger context
(of course equality is possible). The same holds for the sets of intents.
$\mathbb{K}_{t}^{in/out}$ and $\mathbb{K}_{tt}^{in/out}$, the input and output
parts of a transition and a transitive context, are equal after object
clarifying and identification of a transition $(s^{in},s^{out})$ with the
state $s^{in}$ and $s^{out}$, respectively. Modified that way and after
identification of an attribute $a^{in}/a^{out}$ with $a$,
$\mathbb{K}_{t}^{in/out}$ and $\mathbb{K}_{tt}^{in/out}$ are subcontexts of
the scale $\mathbb{K}_{s}$.
$\operatorname{Imp}\mathbb{K}$ denotes the set of attribute implications that
hold in a context $\mathbb{K}$. Since implications valid for a formal context
are less restricted by a smaller set of intents, the inverse subset relations
hold for the respective implication sets:
$\operatorname{Imp}\mathbb{K}_{s}\subseteq\operatorname{Imp}\mathbb{K}_{tt}^{in/out}=\operatorname{Imp}\mathbb{K}_{t}^{in/out}$
(6.1)
$\operatorname{Imp}\mathbb{K}_{tt}\subseteq\operatorname{Imp}\mathbb{K}_{t}$
(6.2)
However, implications between sets of input and output attributes of a
transition and a transitive context have different semantics. They relate to
single transitions and sequences of transitions, respectively.
The subset relations can be proven more formally before the background of
model theory.
#### 6.1.1 Excursus: Model theory and Galois connections
The two derivation operators ′ of a formal context (Definiton 2.1.1) define
maps $\sigma\colon\mathfrak{P}(G)\rightarrow\mathfrak{P}(M)$ and
$\tau\colon\mathfrak{P}(M)\rightarrow\mathfrak{P}(G)$ between power sets
ordered by set inclusion. They constitute a Galois connection between $G$ and
$M$.
###### Definition 6.1.1.
[56, Definition 3.1.3] A Galois connection between the sets $G$ and $M$ is a
pair of maps
$\sigma\colon\mathfrak{P}(G)\rightarrow\mathfrak{P}(M)\qquad\text{and}\qquad\tau\colon\mathfrak{P}(M)\rightarrow\mathfrak{P}(G),$
so that for all $X,X_{1},X_{2}\subseteq G$ and all $Y,Y_{1},Y_{2}\subseteq M$
the subsequent conditions hold:
1. 1.
$X_{1}\subseteq X_{2}\Rightarrow\sigma(X_{2})\subseteq\sigma(X_{1})$
2. 2.
$Y_{1}\subseteq Y_{2}\Rightarrow\tau(Y_{2})\subseteq\tau(Y_{1})$
3. 3.
$X\subseteq\tau\sigma(X)\text{ and }Y\subseteq\sigma\tau(Y)$.
Now this definition will be applied to the formal context
$(\mathfrak{P}(M),\mathfrak{P}(M)\times\mathfrak{P}(M),\models).$
The attributes in $\mathfrak{P}(M)\times\mathfrak{P}(M)$ are interpreted as
implications $\alpha\colon A\rightarrow B,\>A,B\subseteq M$. The incidence
means that $X\in\mathfrak{P}(M)$ respects the implication $\alpha$, i.e.
$X\models\alpha\Leftrightarrow A\not\subseteq X\text{ or }B\subseteq X$ [38,
p. 110-112]. $X$ is called a model of $\alpha$.
For subsets $\mathcal{F}\subseteq\mathfrak{P}(M)\times\mathfrak{P}(M)$ and
$\mathcal{S}\subseteq\mathfrak{P}(M)$ let
$\operatorname{Mod}\mathcal{F}:=\\{X\subseteq M\mid X\models\alpha\text{ for
all }\alpha\in\mathcal{F}\\}$
be the set of models of $\mathcal{F}$ and
$\operatorname{Imp}\mathcal{S}:=\\{\alpha\in\mathfrak{P}(M)\times\mathfrak{P}(M)\mid
X\models\alpha\text{ for all }X\in\mathcal{S}\\}.$
The operators $\operatorname{Imp}$ and $\operatorname{Mod}$ define a Galois
connection between subsets of $M$ and implications over $M$, i.e. between
$\mathfrak{P}(M)$ and $\mathfrak{P}(M)\times\mathfrak{P}(M)$ (compare [37]).
For every Galois connection the compositions $\tau\sigma$ and $\sigma\tau$
define a closure operator on $G$ and $M$, respectively (p. 2.1.7). Here,
$G:=\mathfrak{P}(M)$, $M:=\mathfrak{P}(M)\times\mathfrak{P}(M)$,
$\sigma:=\operatorname{Imp}$ and $\tau:=\operatorname{Mod}$. The extents and
intents of $(\mathfrak{P}(M),\mathfrak{P}(M)\times\mathfrak{P}(M),\models)$
are Galois closed sets. This closure means that the extents are precisely the
closure systems on $M$ (they are closed under arbitrary intersections), the
intents are the respective implicational theories [38, p. 110]. Thus,
$\operatorname{Mod}\operatorname{Imp}\,\mathcal{S}$ is a closure system on
$M$. It can be represented as the system of intents of a formal context
$\mathbb{K}=(G,M,I)$. Then, $\operatorname{Imp}\mathcal{S}$ is the set of
implications holding in $\mathbb{K}$, or the implicational theory generated by
the stem base of $\mathbb{K}$.
Returning to $\mathbb{K}_{s}$ and $\mathbb{K}_{t}^{in/out}$ identified with a
subcontext of $\mathbb{K}_{s}$, let $\mathcal{S}_{1}$ be the set of object
intents of $\mathbb{K}_{s}$ and $\mathcal{S}_{2}$ the set of object intents of
$\mathbb{K}_{t}^{in/out}$. Then $\operatorname{Imp}\mathcal{S}_{1}$ is the
implicational theory of $\mathbb{K}_{s}$, $\operatorname{Imp}\mathcal{S}_{2}$
that of $\mathbb{K}_{t}^{in/out}$.
$\operatorname{Mod}\operatorname{Imp}\mathcal{S}_{i}$ are the respective
systems of all intents. Since $\mathcal{S}_{2}\subseteq\mathcal{S}_{1}$, we
obtain by virtue of the Galois connection between $\mathfrak{P}(M)$ and
$\mathfrak{P}(M)\times\mathfrak{P}(M)$ (property 1.):
$\operatorname{Imp}\mathbb{K}_{s}=\operatorname{Imp}\mathcal{S}_{1}\subseteq\operatorname{Imp}\mathcal{S}_{2}=\operatorname{Imp}\mathbb{K}_{t}^{in/out}.$
The inclusion $\eqref{eq:impKttKt}$ is proven analogously.
#### 6.1.2 Background knowledge for the successive exploration of
$\mathbb{K}_{s},\mathbb{K}_{tt},\mathbb{K}_{t}$ and $\mathbb{K}_{tmp}$
By entering background knowledge (not necessarily implications) prior to an
attribute exploration, the algorithm may be shortened considerably [38]. Thus,
a well structured analysis of a dynamic system should first explore the set of
possible states, then the long-term dynamics expressed by the transitive
context. If a more fine-grained investigation is desired, then also the
transiton context may be explored. Finally, the temporal context should be
computed out of $\mathbb{K}_{t}$, respectively $\mathbb{K}_{tt}$, if the
transitive context is sufficient to determine the semantics of the interesting
attributes (compare Remark 4.5.3). Then the stem base of $\mathbb{K}_{s}$
serves as background knowledge for the exploration of $\mathbb{K}_{tt}$ and
the resulting stem base can be used for the exploration of $\mathbb{K}_{t}$.
Given a state context $\mathbb{K}_{s}$ as scale for a transition as well as a
transitive context (p. 4.3), exploration means determining the attribute logic
with respect to a set of background sequents $\bigwedge A\rightarrow\bigvee
B,\>A,B\subseteq M,$ valid in $\mathbb{K}_{s}$ [38, 104f.]. A formal context
is determined by its stem base up to object reduction and by a set of sequents
up to object clarification. The stem base of $\mathbb{K}_{s}$ excludes only
object intents but does not determine which intents are object intents. It can
be used as background knowledge instead of computing the sequents (possibly
additionally to an already performed attribute exploration), if one is only
interested in the implicational logic of the transition context and does not
need to fix occurring states positively. However, if partially defined states
should be excluded, in general sequents of the structure $\emptyset\multimap
B,B\subseteq M$ are necessary. They can express that for each attribute a
scale value exists, for instance $\top\rightarrow m.0\vee m.1$ for a
dichotomic attribute $m\in M$.
As will be shown in Section 6.2.1, generally the stem base of
$\mathbb{K}_{tt}$ does not determine $\mathbb{K}_{tmp}$ uniquely. However, I
will prove semantic inference rules in order to integrate respective
background knowledge into the exploration of a temporal context. Since
$\mathbb{K}_{tmp}$ extends the attribute set of $\mathbb{K}_{s}$ and restricts
its object set to the set of input states, the implications of
$\mathbb{K}_{s}$ remain valid but are possibly “slightly” too restrictive.
Nevertheless, they can serve as background knowledge for the exploration of
$\mathbb{K}_{tmp}$.
Finally, $\operatorname{Imp}\mathbb{K}_{tt}$ is partly determined by
$\operatorname{Imp}\mathbb{K}_{t}$, e.g.
$A_{in}\rightarrow B_{out},\>B_{in}\rightarrow C_{out}\>(\text{in
}\mathbb{K}_{t})\models A_{in}\rightarrow C_{out}\>(\text{in
}\mathbb{K}_{tt}).$
Hence, it could also make sense to first explore $\mathbb{K}_{t}$ and to
integrate the information into the exploration of $\mathbb{K}_{tt}$ by
inference rules. I will not investigate this rare case further.
### 6.2 Transitive and temporal contexts: A calculus for implications between
temporal atomic propositions
I searched for first order logic background formula in order to use the
results of an attribute exploration of $\mathbb{K}_{tt}$ for the exploration
of $\mathbb{K}_{tmp}$. Then the implications of the latter context are
derivable from this background knowledge and a reduced set of new
implications. The developed method also generates rules between implications
of $\mathbb{K}_{tmp}$ alone. Therefore, during an exploration of
$\mathbb{K}_{tmp}$ implications can be decided automatically based on the stem
base of $\mathbb{K}_{tt}$ and on already accepted implications. The second
type of inferences exploits implications between temporal attributes following
from their definitions, like $\Box Gm\rightarrow\Diamond Fm$.
Prior to searching inference rules systematically by means of a test context,
the subsequent proposition summarises inference rules between implications of
$\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ that follow immediately from the
definition of the temporal attributes by $\mathbb{K}_{tt}$ (Remark 4.5.3):
###### Proposition 6.2.1.
Let $\mathbb{K}_{tmp}=(S^{in},M\cup T,I^{in}\cup I_{T})$ be a temporal
context, $m,m_{1}$ and $m_{2}\in M$ and $B\subseteq M$. Suppose the relation
$t(R)\subseteq S\times S$ is the object set of the related transitive context
$\mathbb{K}_{tt}=(t(R),M\times\\{in,out\\},\nabla)$. Then the following
entailments between implications of both contexts are valid:
$\displaystyle B^{in}\rightarrow m^{out}\>(\text{in }\mathbb{K}_{tt})$
$\displaystyle\equiv B\rightarrow\Box\,Gm\>(\text{in }\mathbb{K}_{tmp})$ (6.3)
$\displaystyle B^{in}\rightarrow m^{out}\>(\text{in }\mathbb{K}_{tt})$
$\displaystyle\models B\rightarrow\Diamond Fm\>(\text{in }\mathbb{K}_{tmp})$
(6.4) $\displaystyle B^{out}\rightarrow m^{out}\>(\text{in }\mathbb{K}_{tt})$
$\displaystyle\models\Box\,GB\rightarrow\Box\,Gm\>(\text{in
}\mathbb{K}_{tmp})$ (6.5) $\displaystyle m_{1}^{out}\rightarrow
m_{2}^{out}\>(\text{in }\mathbb{K}_{tt})$ $\displaystyle\models\Diamond
Fm_{1}\rightarrow\Diamond Fm_{2}\>(\text{in}\>\mathbb{K}_{tmp})$ (6.6)
If there is no transition $(s^{in},s^{out})\in t(R)$ with extents
$(s^{in})^{I}=(s^{out})^{I}=M$, the inference holds:
$B^{in}\cup m^{out}\rightarrow\bot\>(\text{in }\mathbb{K}_{tt})\models
B\rightarrow\Box\,\neg Fm\>(\text{in }\mathbb{K}_{tmp})$ (6.7)
###### Proof.
We remind the reader that for relations $R\subseteq A\times B$ and $a\in A$
the notation $[a]R$ means $[a]R:=\\{b\in B\mid(a,b)\in R\\}$. First we show
item (6.5).
$\displaystyle B^{out}\rightarrow m^{out}\>(\text{in }\mathbb{K}_{tt})$
$\displaystyle\Leftrightarrow$ $\displaystyle\forall\,(s^{in},s^{out})\in
t(R)\colon s^{out}\in B^{I}\Rightarrow s^{out}Im$ $\displaystyle\Rightarrow$
$\displaystyle\forall\,s^{in}\in
S^{in}\colon\text{\large(}\forall\,(s^{in},s^{out})\in t(R)\colon s^{out}\in
B^{I}\text{\large)}\Rightarrow\text{\large(}\forall\,(s^{in},{s^{out}})\in
t(R)\colon s^{out}Im\text{\large)}$ $\displaystyle\Leftrightarrow$
$\displaystyle\Box\,GB\rightarrow\Box\,Gm\>(\text{in }\mathbb{K}_{tmp}).$
Now we show item (6.6).
$\displaystyle m_{1}^{out}\rightarrow m_{2}^{out}\>(\text{in
}\mathbb{K}_{tt})$ $\displaystyle\Leftrightarrow\>$
$\displaystyle\forall\,(s^{in},s^{out})\in t(R)\colon(s^{in},s^{out})\nabla
m_{1}^{out}\Rightarrow(s^{in},s^{out})\nabla m_{2}^{out}$
$\displaystyle\Leftrightarrow\>$ $\displaystyle\forall\,(s^{in},s^{out})\in
t(R)\colon s^{out}Im_{1}\Rightarrow s^{out}Im_{2}\>(\text{in
$\mathbb{K}_{s}$})$ $\displaystyle\Leftrightarrow\>$
$\displaystyle\forall\,s^{in}\in
S^{in},\>\forall\,s^{out}\in[s^{in}]t(R)\colon s^{out}\in m_{1}^{I}\Rightarrow
s^{out}Im_{2}$
$\displaystyle\overset{{s^{out}}^{\prime}:=s^{out}}{\Rightarrow}\>$
$\displaystyle\forall\,s^{in}\in
S^{in}\colon\text{\large(}\exists\,s^{out}\in[s^{in}]t(R)\colon s^{out}\in
m_{1}^{I}\text{\large)}\Rightarrow\text{\large(}\exists\,{s^{out}}^{\prime}\in[s^{in}]t(R)\colon{s^{out}}^{\prime}Im_{2}\text{\large)}$
$\displaystyle\Leftrightarrow\>$ $\displaystyle\Diamond
Fm_{1}\rightarrow\Diamond Fm_{2}\>(\text{in }\mathbb{K}_{tmp}).$
The remaining equivalences are proven analogously. The supplementary condition
for $\eqref{prop:5}$ ensures that $\bot^{I}=M^{I}=\emptyset$. Hence, the
presupposed implication signifies exclusion of $B^{in}$ and $m^{out}$, or
$B^{I}=\emptyset$. In the latter case the inferred conclusion is trivially
valid. ∎
The restriction of $\mathbb{K}_{tmp}$ to an object set $S^{in}$ (Definition
4.5.2) is necessary for the proofs. Regarding (6.4), consider e.g.
$S=\\{0,1\\}$, $s=0$, $R=\\{(1,1)\\}$ and $M=\\{m,n\\}$, $B=\\{n\\}$ and
$I=\\{(0,n),(1,m)\\}$. Then $B^{in}=\\{n^{in}\\}\not\subseteq(1,1)^{\nabla}$,
so the left-hand side of the entailment is trivially true. However,
$B\subseteq 0^{I}$ but $(0,\Diamond Fm)\notin I_{T}$, since
$Seq^{s}_{R}=\emptyset$ for $s=0$.
#### 6.2.1 Incomplete determination of $\mathbb{K}_{tmp}$ by the stem base of
$\mathbb{K}_{tt}$
For $\Diamond F$, $\Box\,G$ and $\Box\,\neg F$, the complete dynamics of the
investigated system, thus $\mathbb{K}_{tmp}$, is determined by the respective
transitive (or transition) context, but there is no one-to-one correspondence
between the stem bases of $\mathbb{K}_{tmp}$ and $\mathbb{K}_{tt}$. Reducible
transitions may exist that prevent a temporal implication, for
nondeterministic processes or a data set consisting of different deterministic
time series. Omitting them does not change the stem base of $\mathbb{K}_{tt}$.
For an example, see Table 6.1.
Transition | $a^{in}$ | $b^{in}$ | $c^{in}$ | $a^{out}$ | $b^{out}$ | $c^{out}$
---|---|---|---|---|---|---
$(s_{0}^{in},s_{1}^{out})$ | $\times$ | $\times$ | | | $\times$ | $\times$
$(s_{0}^{in},s_{2}^{out})$ | $\times$ | $\times$ | | $\times$ | | $\times$
$(s_{1}^{in},s_{2}^{out})$ | | $\times$ | $\times$ | $\times$ | | $\times$
$(s_{3}^{in},s_{4}^{out})$ | | $\times$ | | | | $\times$
Table 6.1: The transitive context $\mathbb{K}_{tt}$ for two deterministic time
series $s_{0}-s_{1}-s_{2}$ and $s_{3}-s_{4}$.
The stem base of this transitive context is:
1. 1.
$\top\rightarrow b^{in},c^{out}$
2. 2.
$b^{in},c^{in},c^{out}\rightarrow a^{out}$
3. 3.
$b^{in},b^{out},c^{out}\rightarrow a^{in}$
1. 4.
$a^{in},b^{in},c^{in},a^{out},c^{out}\rightarrow b^{out}$
2. 5.
$a^{in},b^{in},a^{out},b^{out},c^{out}\rightarrow c^{in}$
$(s_{3}^{in},s_{4}^{out})$ is a reducible object, because for example:
$\displaystyle(s_{3}^{in},s_{4}^{out})^{\prime}$
$\displaystyle=\\{b^{in},c^{out}\\}=\\{a^{in},b^{in},b^{out},c^{out}\\}\cap\\{b^{in},c^{in},a^{out},c^{out}\\}$
$\displaystyle=(s_{0}^{in},s_{1}^{out})^{\prime}\cap(s_{1}^{in},s_{2}^{out})^{\prime}$
Therefore, the context without this transition has the same stem base.
However, only without this observation $\top\rightarrow\Diamond Fa$ holds in
$\mathbb{K}_{tmp}$.
The following proposition concerns a linear order $R$ and specifies a special
case where the temporal context is determined by the stem base of the
transitive context: If it is known that only transitions with the first (or
only with the last) transition are reducible, not all of them can be reduced
and the order relation can be reconstructed.
###### Proposition 6.2.2.
Let $\mathbb{K}_{tt}=(t(R),M\times\\{in,out\\},\nabla)$ be a transitive
context with a linear order $R\subseteq S\times S$, $R\neq\emptyset$. Then the
removal from $R$ of all transitions with the first (last) state changes the
stem base of $\mathbb{K}_{tt}$. In general however, the following changes
might not affect the stem base of $\mathbb{K}_{tt}$, but result in a different
stem base of the temporal context $\mathbb{K}_{tmp}$ defined by
$\mathbb{K}_{tt}$:
1. 1.
The removal of a state and all related transitions that is not the first or
last state.
2. 2.
A switch of the order relation of two states.
###### Proof.
The implication logic determines $\mathbb{K}_{tt}$ up to object reduction. If
$\mathbb{K}_{tt}$ consists of a single transition, it is reducible only if
$(s_{0}^{in},s_{1}^{out})=(M\times\\{in,out\\})^{\prime}=\emptyset^{\prime}$.
But since the resulting concept lattice is isomorphic to the one element
lattice,
$\mathfrak{B}(\emptyset,\emptyset,\emptyset)\cong\mathfrak{B}((s_{0}^{in},s_{1}^{out}),\emptyset,\emptyset)$,
this case could be considered as attribute reduction and therefore excluded
(attribute reduction results in a different stem base).
A state $s_{k}$ can be removed from the linear order, if all transitions
$(s_{i},s_{k})$ and $(s_{k},s_{j})$ are reducible. Then all $s_{i}$ have to be
intersections of input, all $s_{j}$ intersections of output state intents,
which cannot be generelly excluded.
However, if $k=0$, only the output parts of the respective transitions are
available for intersection. With the order relation of intersection, these
intents build a lattice, and each nonempty lattice has $\bigvee$-irreducible
elements. Hence, a corresponding transition $(s_{0},s_{j})$ is irreducible,
and the linear order may be reconstructed. The same argument holds for a final
output state $s_{k}$: There exist irreducible $(s_{i},s_{k})$.
Changing the order of two states may result in the same stem base of
$\mathbb{K}_{tt}$, but generate a temporal context with different
implicational logic: In the transitive context derivable from the transition
context of Table 6.2, the transition $(s_{5},s_{6})$ is reducible. Changing
only the order of $s_{5}$ and $s_{6}$ results in the same transitive context,
with the exception of transition $(s_{6},s_{5})$. It is also reducible to the
same context. However, the corresponding temporal contexts are different. The
implication $\Diamond Fa\rightarrow\Diamond Fb$ only holds in the original,
$\Diamond Fb\rightarrow\Diamond Fa$ only in the modified temporal context. ∎
Transition | $a^{in}$ | $b^{in}$ | $c^{in}$ | $d^{in}$ | $a^{out}$ | $b^{out}$ | $c^{out}$ | $d^{out}$
---|---|---|---|---|---|---|---|---
$(s_{0}^{in},s_{1}^{out})$ | $\times$ | | | $\times$ | | $\times$ | | $\times$
$(s_{1}^{in},s_{2}^{out})$ | | $\times$ | | $\times$ | $\times$ | | $\times$ |
$(s_{2}^{in},s_{3}^{out})$ | $\times$ | | $\times$ | | | $\times$ | $\times$ |
$(s_{3}^{in},s_{4}^{out})$ | | $\times$ | $\times$ | | $\times$ | $\times$ | | $\times$
$(s_{4}^{in},s_{5}^{out})$ | $\times$ | $\times$ | | $\times$ | $\times$ | | |
$(s_{5}^{in},s_{6}^{out})$ | $\times$ | | | | | $\times$ | |
$(s_{6}^{in},s_{7}^{out})$ | | $\times$ | | | | | |
Table 6.2: Transition context for a linearly ordered state set. If the order
of $s_{5}$ and $s_{6}$ is exchanged, the stem base of the respective reduced
$\mathbb{K}_{tt}$ remains the same, but a different $\mathbb{K}_{tmp}$ is
generated.
Nevertheless, note that there exist cases 1. and 2. where $\mathbb{K}_{tmp}$
is reconstructable from the stem base of $\mathbb{K}_{tt}$. For instance, if
the order is changed at an early point, this may result in the same temporal
context. In Table 6.3, the object $(s_{0},s_{1})$ is reducible. Thus,
supposing a linear order the relation of $s_{0}$ and $s_{1}$ is not reflected
by the stem base. In general, the inverse relation may induce different
temporal attributes. In this example, however, the context with
$(s_{1},s_{0})$ as first transition has the same stem base, but also the same
temporal context $\mathbb{K}_{tmp}$ (only the attributes $b^{in}$ and
$b^{out}$ are changed in the first row).
Transition | $a^{in}$ | $b^{in}$ | $c^{in}$ | $a^{out}$ | $b^{out}$ | $c^{out}$
---|---|---|---|---|---|---
$(s_{0}^{in},s_{1}^{out})$ | | $\times$ | | | |
$(s_{0}^{in},s_{2}^{out})$ | | $\times$ | | $\times$ | $\times$ |
$(s_{0}^{in},s_{3}^{out})$ | | $\times$ | | | $\times$ | $\times$
$(s_{0}^{in},s_{4}^{out})$ | | $\times$ | | | | $\times$
$(s_{1}^{in},s_{2}^{out})$ | | | | $\times$ | $\times$ |
$(s_{1}^{in},s_{3}^{out})$ | | | | | $\times$ | $\times$
$(s_{1}^{in},s_{4}^{out})$ | | | | | | $\times$
$(s_{2}^{in},s_{3}^{out})$ | $\times$ | $\times$ | | | $\times$ | $\times$
$(s_{2}^{in},s_{4}^{out})$ | $\times$ | $\times$ | | | | $\times$
$(s_{3}^{in},s_{4}^{out})$ | | $\times$ | $\times$ | | | $\times$
Table 6.3: The transitive context for a linear order $s_{0}\leq s_{1}\leq
s_{2}\leq s_{3}\leq s_{4}$. $(s_{0},s_{1})$ is reducible, as well as
$(s_{1},s_{0})$ in a context with this transition instead, but both formal
contexts generate the same temporal context.
###### Remark 6.2.3.
Implications of the form $B\rightarrow\Box\neg Fm\text{ in }\mathbb{K}_{tmp}$
may be derived from implications $B^{in}\cup m^{out}\rightarrow\bot\text{ in
}\mathbb{K}_{tt}$ (compare Proposition 6.2.1). However, implications with
negated attributes in the premise can only be decided using the stem base of
$\mathbb{K}_{tt}$ in the case of dichotomic scaling. Else, there is no
explicit information regarding the non-occurrence of an attribute. In the
example of Table 6.2, the only implication in the stem base of
$\mathbb{K}_{tt}$ with $a^{out},b^{out}$ in the premise is
$a^{out},b^{out}\rightarrow d^{out}$. With
$b^{out},c^{out},d^{out}\rightarrow\bot$,
$\Box\,Ga,\Box\,Gb\rightarrow\Box\,\neg Fc$ is derivable, but there is no
inference for deriving from $\mathbb{K}_{tt}$ the valid implication
$\Box\,Fa,\Box\,Fb,\Box\,\neg Fc\rightarrow b$, which is supported by
$\\{s_{3}\\}$ in $\mathbb{K}_{tmp}$. This can be easily seen, since there is
no $\mathbb{K}_{tt}$ implication with $b^{in}$ in the conclusion. The same
holds for the possibility operator $\Diamond$, since it is equivalent to the
necessity operator in the case of deterministic time series.
#### 6.2.2 The test context
In order to get a complete overview on valid “pure” or “mixed” entailments
between implications of $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$, we performed
attribute exploration of the following test context: Given fixed sets $S$ and
$M$, the attributes are implication forms $IF$ in variables $V:=\\{X,Y,Z\\}$,
the objects are all possible $\mathbb{K}_{tt}$ respectively the corresponding
$\mathbb{K}_{tmp}$ (Table 6.4) together with a variable assignment, and the
incidence relation means: “With the respective values of the set variables,
the implication holds in the context.” More precisely, the objects are
appositions ($\mathbb{K}_{tt}|\mathbb{K}^{\prime}_{tmp})$, where
$\mathbb{K}^{\prime}_{tmp}$ proceeds from $\mathbb{K}_{tmp}$ by replacing an
object $s^{in}$ of the latter by all transitions $(s^{in},s^{out})\in t(R)$.
This state context $\mathbb{K}_{tmp}^{\prime}$ is not object clarified, i.e.
redundant. The attributes are given by $(s^{in},s^{out})^{\prime}=s^{in}$′.
According to Definition 4.5.2 of a temporal context, the last state of a
finite path is omitted.
The investigated temporal operators are restricted to
$\operatorname{ev}:=\Diamond F$, $\operatorname{alw}:=\Box G$ and
$\operatorname{nev}:=\Box\neg F$ decidable by $\mathbb{K}_{tt}$, where the
same quantifier $\exists$ or $\forall$ applies to paths and transitions
(Definition 4.5.2).
Thus, we performed attribute exploration of the following test context:
$\mathbb{K}_{test}=(\\{\alpha:\\{X,Y,Z\\}\rightarrow\mathfrak{P}(M)\setminus\emptyset\\}/S_{M}\times\\{\mathbb{K}_{tt}|\mathbb{K}_{tmp}^{\prime}\\},IF,\models)$
Variable assignments $\operatorname{mod}S_{M}$ (the symmetric group on $M$)
are sufficient, because also $\\{\mathbb{K}_{tt}|\mathbb{K}_{tmp}^{\prime}\\}$
is symmetric in the attributes $M$.
| $a^{in}$ | $b^{in}$ | $c^{in}$ | $a^{out}$ | $b^{out}$ | $c^{out}$ | $\operatorname{ev}a$ | $\operatorname{ev}b$ | $\operatorname{ev}c$ | $\operatorname{alw}a\>$ | $\operatorname{alw}b$ | $\operatorname{alw}c$ | $\operatorname{nev}a$ | $\operatorname{nev}b$ | $\operatorname{nev}c$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$(s_{0}^{in},s_{1}^{out})$ | $\times$ | | | $\times$ | $\times$ | | $\times$ | $\times$ | $\times$ | | | | | |
$(s_{0}^{in},s_{2}^{out})$ | $\times$ | | | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | | | | | |
$(s_{0}^{in},s_{3}^{out})$ | $\times$ | | | | | $\times$ | $\times$ | $\times$ | $\times$ | | | | | |
$(s_{1}^{in},s_{2}^{out})$ | $\times$ | $\times$ | | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | | | $\times$ | | |
$(s_{1}^{in},s_{3}^{out})$ | $\times$ | $\times$ | | | | $\times$ | $\times$ | $\times$ | $\times$ | | | $\times$ | | |
$(s_{2}^{in},s_{3}^{out})$ | $\times$ | $\times$ | $\times$ | | | $\times$ | | | $\times$ | | | $\times$ | $\times$ | $\times$ |
Table 6.4: A single object of the test context: the apposition of the
transitive and temporal contexts corresponding to the time series $a-ab-
abc-c.$
Since it was not feasible to investigate all possible implications, we chose
the following classes with at most two premises (with different variables $X$,
$Y$ or the constant $\top$) and a single conclusion. Many implications with
more than two different attribute sets in the premise can be derived by the
second and third Armstrong rule.
| $\mathbb{K}_{tt}$ | $\mathbb{K}_{tmp}$
---|---|---
1 or 2 premises | $X^{in},Y^{out},\top$ | $X,\,\Diamond FY,\,\Box GY,\,\Box\neg FY,\top$
1 conclusion | $Z^{in},Z^{out},\bot$ | $Z,\,\Diamond FZ,\,\Box\,GZ,\,\Box\neg FZ,\bot$
The sets were supposed to be nonempty. Thus, $\top:=\emptyset$ was considered
explicitly, in order to avoid redundancies like $\Diamond F\emptyset=\Box
G\emptyset$. Tautologic or unsatisfiable implication forms were not
considered, like $\Box GY\rightarrow\Diamond FY$ or $\Diamond FY,\>\Box\neg
FY\rightarrow\bot$.
### 6.3 Inference rules for a three element attribute set $M$
Even with computational optimisation, it took 10 days, partly on two desktop
computers, to generate the test context with $|M|=3$. Since computation time
is exponential in $|M|$, it is not feasible to compute the test context for
larger attribute sets. For $M=\\{a,b,c\\}$, we generated all deterministic
time series (linear orders) including cycles.
If the 81 inference rules of the stem base are proven, the hypothesis is
confirmed that this attribute set is general enough. However, this was only
possible for rules 1 to 56. For rule 57, 59 and 65, a supplementary condition
is necessary restricting the allowed transitions (Propositions 6.3.6 and
6.3.7). Since during the computation of a stem base the generated rules depend
on the previous, I proved only several further rules resulting in a large, but
not complete set of inference rules for transitive and temporal context.
Further research is needed: Expert centered attribute exploration should be
performed by introducing counterexamples to rules that are not generalisable
and by proving the remaining or new implications. Then the Theorem of
Duquenne-Guigues 2.2.2 ensures the completeness of the stem base rules for
drawing all rule-like conclusions between the implications investigated as
attributes of the test context.
A further purpose of the following proofs is to determine presuppostions as
generally as possible. It turned out that most rules are also valid for
nondeterminstic processes.
As indicated by the used sofware ConImp [26], we have to prove only the parts
of the premise and conclusion not following from previous implications.
Several implications follow trivially, since $\bot$ means “every attribute
occurs”, for instance
$\operatorname{alw}Y\rightarrow\bot\>\models\operatorname{alw}Y\rightarrow
Z^{in}$.
In a first step, general propositions are proven for example inferences. The
attached file testCxt.pro – the output of the test context exploration –
contains all 81 inference rules. Their indices are listed in Table 6.5
together with the propositions necessary for the proofs (sometimes easy
consequences are needed, or analogous applications of the proofs).
For the subsequent propositions, the following presuppositions are made:
$\mathbb{K}_{tt}=(t(R),M\times\\{in,out\\},\nabla)$ with $R\subseteq S\times
S$ and $\mathbb{K}_{tmp}=(S^{in},M\cup T,I^{in}\cup I_{T})$ are formal
contexts generated by the same process. $B$, $X$, $Y$ and $Z\subseteq M$ are
their basic attribute sets. If the process is not qualified, it may be
deterministic or nondeterministic. $i,j,k\in\mathbb{N}_{0}$ are indices from
sets according to the number of transitions of a path. Implications with
attribute sets indexed by $.^{in}$ or $.^{out}$ refer to a transitive context
and the left part of the apposition
$\mathbb{K}_{tt}\mid\mathbb{K}_{tmp}^{\prime}$, the remaining attributes to
$\mathbb{K}_{tmp}^{\prime}$ and therefore to the temporal context
$\mathbb{K}_{tmp}$ generated by $\mathbb{K}_{tt}$.
###### Proposition 6.3.1.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
process, the following inference rules hold by the second Armstrong rule.
$\displaystyle 11:\>$ $\displaystyle\operatorname{alw}Y\rightarrow\bot$
$\displaystyle\models\>$ $\displaystyle
X\wedge\operatorname{alw}Y\rightarrow\bot,\>\operatorname{ev}Y\wedge\operatorname{alw}Y\rightarrow\bot,$
$\displaystyle X\wedge\operatorname{nev}Y\rightarrow\operatorname{nev}Z$
$\displaystyle 5:\>$ $\displaystyle\top\rightarrow Z^{in}$
$\displaystyle\models\>$ $\displaystyle X^{in}\rightarrow
Z^{in},\>Y^{out}\rightarrow Z^{in},\>\operatorname{ev}Y\rightarrow
Z,\>X^{in}\wedge Y^{out}\rightarrow Z^{in},...$
###### Proof.
Rule 11 follows by the second Armstrong rule, i.e. an expansion of the premise
$P$, with attributes $A_{i}\subseteq M\cup(M\times\\{in,out\\})\cup T$ (where
attributes of $\mathbb{K}_{tmp}$ are in $M$ or $T$):
$\displaystyle P:\underline{\qquad A_{2}\>}$
$\displaystyle\underline{\>\rightarrow A_{3}}$ (6.8) $\displaystyle
C:A_{1}\wedge A_{2}$ $\displaystyle\rightarrow A_{3}$
In rule 5, $A_{2}=\emptyset$. With the presupposed implication of
$\mathbb{K}_{tt}$, all implications of $\mathbb{K}_{tt}$ with the conclusion
$Z^{in}$ and all implications of $\mathbb{K}_{tmp}$ with the conclusion $Z$
can be inferred, since the extents are equal: $Z^{in}=Z\subseteq
M\Rightarrow(Z^{in})^{I^{in}}=Z^{I^{in}}$ (remember that in $\mathbb{K}_{tmp}$
the object set is restricted to the set of input states $S^{in}$, consequently
the relation $I$). By further applications of the second Armstrong rule,
implications with arbitrarily enlarged premises are shown to be valid. ∎
###### Proposition 6.3.2.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
process, the following inference rules hold by definition of the temporal
operators $\operatorname{ev},\>\operatorname{alw}$ and $\operatorname{nev}$:
$\displaystyle 1.\>$ $\displaystyle 26\colon\>$
$\displaystyle\operatorname{ev}Y\wedge\operatorname{alw}Y\rightarrow\bot\qquad$
$\displaystyle\models$ $\displaystyle\operatorname{alw}Y\rightarrow\bot$
$\displaystyle 27\colon\>$
$\displaystyle\operatorname{ev}Y\wedge\operatorname{alw}Y\rightarrow\operatorname{nev}Z\qquad$
$\displaystyle\models$
$\displaystyle\operatorname{alw}Y\rightarrow\operatorname{nev}Z$
$\displaystyle 2.\>$ $\displaystyle 41:\>$ $\displaystyle
X\wedge\operatorname{ev}Y\rightarrow\bot\qquad$ $\displaystyle\models$
$\displaystyle X^{in}\wedge Y^{out}\rightarrow\bot$ $\displaystyle 44:\>$
$\displaystyle X\wedge\operatorname{ev}Y\rightarrow\operatorname{ev}Z\qquad$
$\displaystyle\models$ $\displaystyle
X\wedge\operatorname{alw}Y\rightarrow\operatorname{ev}Z$ $\displaystyle 46:\>$
$\displaystyle X\wedge\operatorname{ev}Y\rightarrow Z^{in}\qquad$
$\displaystyle\models$ $\displaystyle X^{in}\wedge Y^{out}\rightarrow
Z^{in},\>X\wedge\operatorname{alw}Y\rightarrow Z$ $\displaystyle 3.\>$
$\displaystyle 32:\>$ $\displaystyle
X\wedge\operatorname{nev}Y\rightarrow\operatorname{alw}Z\qquad$
$\displaystyle\models$ $\displaystyle
X\wedge\operatorname{nev}Y\rightarrow\operatorname{ev}Z$ $\displaystyle 37:\>$
$\displaystyle X\wedge\operatorname{alw}Y\rightarrow\operatorname{alw}Z\qquad$
$\displaystyle\models$ $\displaystyle
X\wedge\operatorname{alw}Y\rightarrow\operatorname{ev}Z$ $\displaystyle 43:\>$
$\displaystyle X\wedge\operatorname{ev}Y\rightarrow\operatorname{alw}Z\qquad$
$\displaystyle\models$ $\displaystyle X^{in}\wedge Y^{out}\rightarrow
Z^{out},\>X\wedge\operatorname{alw}Y\rightarrow\operatorname{alw}Z.$
###### Proof.
Let $D$ be an implication valid in the test context that follows immediately
from the definitions. Examples are the subsequent implications of
$\mathbb{K}_{tmp}$ (6.9) or with attributes of $\mathbb{K}_{tmp}$ and
$\mathbb{K}_{tt}$:
$\displaystyle\operatorname{alw}B$
$\displaystyle\rightarrow\operatorname{ev}B$ (6.9)
$\displaystyle\operatorname{alw}B$ $\displaystyle\rightarrow B^{out}$ (6.10)
$\displaystyle B^{out}$ $\displaystyle\rightarrow\operatorname{ev}B.$ (6.11)
We distinguish the following inference modi, with attribute sets
$A_{i}\subseteq M\cup(M\times\\{in,out\\})\cup T$. They are special cases of
the third Armstrong rule that expresses generalised transitivity.
1. 1.
$\displaystyle P:A_{1}\wedge A_{2}$ $\displaystyle\rightarrow A_{3}$ (6.12)
$\displaystyle D:\underline{\qquad\>\,A_{2}}$
$\displaystyle\underline{\>\rightarrow A_{1}}$ $\displaystyle
C:\qquad\>\,A_{2}$ $\displaystyle\rightarrow A_{3}$
With $A_{1}:=\operatorname{ev}Y,\>A_{2}:=\operatorname{alw}Y$ and
$A_{3}:=\bot$, $P\models C$ is inference rule 26. Rule 27 and others follow
with different attribute sets $A_{3}$ (see Table 6.5, where also the used
implications $D$ are indicated in the column “Definition”).
2. 2.
In 41, 44 and 46, a part of the premise is replaced by a stronger assumption.
Here D is either $\operatorname{alw}Y\rightarrow\operatorname{ev}Y$ or
$Y^{out}\rightarrow\operatorname{ev}Y$:
$\displaystyle P:A_{1}\wedge A_{2}$ $\displaystyle\rightarrow A_{3}$ (6.13)
$\displaystyle D:\underline{\qquad\;A_{4}}$
$\displaystyle\underline{\>\rightarrow A_{2}}$ $\displaystyle C:A_{1}\wedge
A_{4}$ $\displaystyle\rightarrow A_{3}$
3. 3.
In 32 and 37, $D\colon\operatorname{alw}Y\rightarrow\operatorname{ev}Y$ is
applied to the conclusion of the presupposed rule (ordinary transitive
inference):
$\displaystyle P:A_{1}\wedge A_{2}$ $\displaystyle\rightarrow A_{3}$ (6.14)
$\displaystyle D:\underline{\qquad\;A_{3}}$
$\displaystyle\underline{\>\rightarrow A_{4}}$ $\displaystyle C:A_{1}\wedge
A_{2}$ $\displaystyle\rightarrow A_{4}$
In 43, $D\colon\operatorname{alw}Z\rightarrow Z^{out}$ is used, additionally
$\operatorname{alw}Y\rightarrow\operatorname{ev}Y$ and
$Y^{out}\rightarrow\operatorname{ev}Y$ according to inference pattern (6.13).
∎
If the inference rules are implemented as an extension of an attribute
exploration software, it has to be decided if the rules according to this
proposition are an immediate input to a reasoner. Alternatively, implications
as (6.9), (6.10) and (6.11) could be assembled, together with an
implementation of the Armstrong rules for attribute exploration itself.
Proposition 6.3.1 may have little practical relevance: If an implication
$A_{2}\rightarrow A_{3}$ is accepted, $A_{1}\cup A_{2}$ does not contain the
closure $A_{3}$ of the pseudo-intent $A_{2}$. Therefore, it is no pseudo-
intent and cannot be proposed as implication of the stem base. At least the
proposition could be useful for “mixed” implications: rule 5 translates the
implication $\top\rightarrow Z^{in}$ of $\mathbb{K}_{tt}$ into implications of
$\mathbb{K}_{tmp}$. However, I will not discuss such issues of practical
applicability further.
###### Proposition 6.3.3.
Let $\mathbb{K}_{tmp}$ be a temporal context and $\mathbb{K}_{tt}$ the related
transitive context. Then the entailment between implications of
$\mathbb{K}_{tmp}$ and $\mathbb{K}_{tt}$ is valid:
$53\colon X^{in}\rightarrow Z^{out}\models X\rightarrow\operatorname{alw}Z.$
###### Proof.
The rule follows from Proposition 6.2.1 (6.3) applied to every $z\in Z$. ∎
If the inferences of Proposition 6.2.1 are entered as background knowledge for
the exploration of the test context (with variants for the set variables $X,Y$
and $Z$), rule 53 and a few others can be decided automatically.
###### Proposition 6.3.4.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
process, the following inference rules hold by exclusion of
$\operatorname{ev}/\operatorname{alw}$ and $\operatorname{nev}$:
$\displaystyle\;1:\>\
\top\models\>\operatorname{ev}Y\wedge\operatorname{nev}Y\rightarrow\bot,\>\operatorname{alw}Y\wedge\operatorname{nev}Y\rightarrow\bot$
(6.15) $\displaystyle
38:\>X\wedge\operatorname{alw}Y\rightarrow\operatorname{ev}Z,\>X\wedge\operatorname{alw}Y\rightarrow\operatorname{nev}Z\models\>X\wedge\operatorname{alw}Y\rightarrow\bot$
(6.16)
###### Proof.
By definition, $\operatorname{ev}y$ and $\operatorname{nev}y$ cannot be both
attributes of a state $s\in S^{in}$, for all $y\in Y$. By pairwise exclusion,
the extent $(\operatorname{ev}Y\cup\,\operatorname{nev}Y)^{I_{T}}$ is empty;
this means $\operatorname{ev}Y\wedge\,\operatorname{nev}Y\rightarrow\bot$. The
second implication of rule 1 follows with $\eqref{eq:alwEv}$ and the third
Armstrong rule (6.13). 38 is inferred by $\eqref{eq:conPremise}$ with the
attribute set $A_{3}=\operatorname{ev}Z\,\cup\,\operatorname{nev}Z$. ∎
A surprising rule expressing more complex temporal relationships is the
following.
###### Proposition 6.3.5.
Let $\mathbb{K}_{tmp}$ be a temporal context of a deterministic process, i.e.
$\forall s\in S^{in}\colon|[s]R|:=|\\{s^{out}\in S\mid\exists(s,s^{out})\in
R\\}|=1$. Then the subsequent entailments between implications of
$\mathbb{K}_{tmp}$ are valid:
$\displaystyle 39\colon\operatorname{alw}Y\rightarrow
Z,\>\operatorname{alw}Y\rightarrow\operatorname{ev}Z$
$\displaystyle\models\operatorname{alw}Y\rightarrow\operatorname{alw}Z$
$\displaystyle 34\colon\operatorname{nev}Y\rightarrow
Z,\>\operatorname{nev}Y\rightarrow\operatorname{ev}Z$
$\displaystyle\models\operatorname{nev}Y\rightarrow\operatorname{alw}Z.$
###### Proof.
We consider an arbitrary transitive context
$\mathbb{K}_{tt}=(t(R),M\times\\{in,out\\},\nabla)$ generating
$\mathbb{K}_{tmp}$ and differentiate between two cases for a state
$s_{0}^{in}\in S^{in}$:
1. 1.
$\operatorname{alw}Y\nsubseteq(s_{0}^{in})^{I_{T}}$.
$\Rightarrow$ The implications and the entailment trivially hold.
2. 2.
$\operatorname{alw}Y\subseteq(s_{0}^{in})^{I_{T}}$. Then
$\operatorname{alw}Y$ $\rightarrow
Z,\>\operatorname{alw}Y\rightarrow\operatorname{ev}Z$ is supported by
$s_{0}^{in}$. $\Leftrightarrow$ $\forall\,(s_{0}^{in},s^{out})\in
t(R),\>\forall\,y\in Y\colon(s^{out},y)\in I\qquad\qquad(*)$ $\wedge$ $\forall
z\in Z\colon(s_{0}^{in},z)\in I$ $\wedge$ $\forall z\in
Z\>\exists\,(s_{0}^{in},s^{out})\in t(R)\colon(s^{out},z)\in I$. $\Rightarrow$
$\forall\,(s_{0}^{in},s^{out})\in
t(R)\colon\operatorname{alw}Y\subseteq(s_{0}^{out})^{I_{T}}$, if
$s^{out}\in\mathbb{K}_{tmp}$, i.e.
$\exists\,s^{out^{\prime}}\colon(s^{out},s^{out^{\prime}})\in t(R)$.
$\Rightarrow$ $\forall\,(s_{0}^{in},s^{out})\in t(R),\>\forall z\in
Z\colon(s^{out},z)\in I$: If $s^{out}$ is a final state, $(s^{out},z)\in I$
follows with $\operatorname{alw}Y\rightarrow\operatorname{ev}Z$ for an
immediately antecedent state and a deterministic process. $\Leftrightarrow$
$\operatorname{alw}Y\rightarrow\operatorname{alw}Z$ holds for $s_{0}^{in}$.
Like $\operatorname{alw}$, $\operatorname{nev}$ applies to every state
subsequent to $s_{0}^{in}$, if defined and
$\operatorname{nev}Y\subseteq(s_{0}^{in})^{I_{T}}$. Therefore, the entailment
for $\operatorname{nev}Y$ is proven in the same way with $(s^{out},y)\notin I$
in statement $(*)$. ∎
Rule | Inference modus | Definition
---|---|---
1. | Proposition 6.3.4, (6.13) | (6.9)
2. | (6.8) |
3. | (6.14), (6.8) | (6.9), (6.10), (6.11)
4. | (6.8) |
5. | (6.8) |
6. | trivial, (6.8) |
7. | (6.8) |
8. | (6.14), (6.8) | (6.9)
9. | (6.8) |
10. | (6.8) |
11. | trivial, (6.8) |
12. | (6.8) |
13. | (6.14), (6.8) | (6.9)
14. | (6.8) |
15. | (6.8) |
16. | trivial, (6.8) |
17. | (6.13), (6.8) | (6.9)
18. | (6.13), (6.14), (6.8) | (6.9), (6.10), (6.11)
19. | (6.13), (6.8) | (6.9)
20. | (6.13), (6.8) | (6.9), (6.11)
21. | trivial, (6.8) |
22. | (6.8) |
23. | (6.14), (6.8) | (6.9), (6.10)
24. | (6.8) |
25. | (6.8) |
26. | (6.12) | (6.9)
27. | (6.12) | (6.9)
28. | (6.12) | (6.9)
29. | (6.12) | (6.9)
30. | (6.12) | (6.9)
31. | trivial |
32. | (6.14) | (6.9)
33. | (6.14), (6.15) |
34. | Proposition 6.3.5 |
35. | (6.14), (6.15) |
36. | trivial |
37. | (6.14) | (6.9)
38. | Proposition 6.3.4 |
39. | Proposition 6.3.5 | (6.9)
40. | (6.14), (6.15) |
41. | trivial, (6.13) | (6.9), (6.11)
42. | (6.13) | (6.9)
43. | (6.13), (6.14) | (6.9), (6.10), (6.11)
44. | (6.13) | (6.9)
45. | (6.14), (6.15) |
46. | (6.13) | (6.9), (6.11)
47. | Proposition 6.3.3 |
Rule | Inference modus | Definition
---|---|---
48. | identical |
49. | trivial, (6.8) |
50. | (6.13), (6.8), Proposition 6.3.3 | (6.10)
51. | (6.8) |
52. | identical |
53. | Proposition 6.3.3 |
54. | identical |
55. | trivial |
56. | (6.13), Proposition 6.3.3 | (6.10)
57. | Proposition 6.3.6 with supplementary condition |
58. | (6.13) | (6.10)
59. | 6.3.7 with supplementary condition |
65. | 6.3.7 with supplementary condition |
68. | 6.3.8 |
Table 6.5: Inference rules for implications of $\mathbb{K}_{tt}$ and
$\mathbb{K}_{tmp}$. The second column lists the inference modi or propositions
required for the proofs, the third column implications following from the
definitions of a temporal context.
###### Proposition 6.3.6.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
process, the following inference rule holds, if $\forall(s^{in},s^{out})\in
R,\forall y\in Y\colon y\in(s^{out})^{I}\Rightarrow Y\subseteq(s^{out})^{I}$,
especially for $|Y|=1\\!:$
$57\colon X^{in}\wedge Y^{out}\rightarrow
Z^{out},\>X\wedge\operatorname{ev}Y\rightarrow\operatorname{nev}Z\models
X\wedge\operatorname{ev}Y\rightarrow\bot.$
###### Proof.
Suppose there is a state $s_{0}^{in}\in S^{in}$ with attributes $X$ and
$\operatorname{ev}Y$. Then a transition $(s_{0}^{in},s^{out})$ exists in
$\mathbb{K}_{tt}$ with $Y\subseteq(s^{out})^{I}$. For this step the condition
of the proposition is important: A single transition with the property has to
exist, the $y\in Y$ must not be attributes of subsequent states or of states
belonging to different paths. By the first presupposed implication, then also
$Z\subseteq(s^{out})^{I}$. This contradicts the second implication. Hence,
there is no state with attributes $X$ and $\operatorname{ev}Y$ and the second
implication trivially holds, i.e. $X\wedge\operatorname{ev}Y\rightarrow\bot$.
∎
Without the condition for the transitions $(s^{in},s^{out})\in R$, the
inference is valid:
$X^{in}\wedge Y^{out}\rightarrow
Z^{out},\>X\wedge\operatorname{ev}Y\rightarrow\operatorname{nev}Z\models
X^{in}\wedge Y^{out}\rightarrow\bot.$
This is the first inference rule which could not be generally proven during
the attribute exploration of the test context restricted to a three element
attribute set. The supplementary condition mostly restricts $Y$ to a one
element set. This contradicts the aim of proving rules for implications with
set variables, so the condition was not presupposed generally. As mentioned in
the introduction of this section, an alternative would be an expert centered
exploration, where counterexamples are introduced. However, this is out of the
scope of this thesis. Nevertheless, I proved three further rules which express
interesting dynamic dependencies.
###### Proposition 6.3.7.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
process, the following inference rules holds, if $\forall\,(s^{in},s^{out})\in
R,\forall\,y\in Y\colon y\in(s^{out})^{I}\Rightarrow Y\subseteq(s^{out})^{I}$,
especially for $|Y|=1\\!:$
$65:Y^{out}\rightarrow
Z^{in},\>\operatorname{ev}Y\rightarrow\operatorname{nev}Z\models\operatorname{ev}Y\rightarrow
Z$ $59:X^{in}\wedge Y^{out}\rightarrow
Z^{in},\>X\wedge\operatorname{ev}Y\rightarrow\operatorname{nev}Z\models\>X\wedge\operatorname{ev}Y\rightarrow
Z.$
###### Proof.
If $Y^{out}\rightarrow Z^{in}$ is supported by any, then by the immediate
preceding transition $(s_{k},s_{k+1})$ and by the transitions
$(s_{k-i},s_{k+1})$ with all previous input states. All underlying paths have
the structure:
$Z-\cdots-Z-Y-\cdots$
Since $\operatorname{ev}Y$ is an attribute of the first state of a path,
$\operatorname{ev}Y\rightarrow\operatorname{nev}Z$ requires the structure:
$Z-Y\overline{Z}-\overline{Z}-\cdots$
Thus, only the first state has the attribute $\operatorname{ev}Y$ and also the
attributes in $Z=Z^{in}$ as demonstrated.
The condition makes sure that there is no other path with a state
$s\in(\operatorname{ev}Y)^{I_{T}}$, but there is no subsequent state
$s^{\prime}\in Y^{I}$. If there is no transition with $Y^{out}$ at all, the
condition implies that $\operatorname{ev}Y$ does not apply to any state, and
the inferred implication trivially holds.
Rule 59 is proven analogously. ∎
###### Proposition 6.3.8.
For all $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ generated by the same
deterministic process, the inference rule holds:
$68:\top\rightarrow
Z^{in},\>X^{in}\wedge\,\operatorname{alw}Y\rightarrow\operatorname{ev}Z\models
X^{in}\wedge\,\operatorname{ev}Y\rightarrow\operatorname{ev}Z.$
###### Proof.
By $\top\rightarrow Z^{in}$, all but the final states have attributes $Z$. The
same holds for the corresponding output states of a transition. For the last
input state of a path and a deterministic process, $\operatorname{ev}Y$ and
$\operatorname{alw}Y$ have the same meaning, thus the inferred implication
holds, too. ∎
### 6.4 Inference rules for Boolean attributes
In order to get a complete overview on valid entailments for Boolean attribute
values, we performed manual attribute exploration of a test context similar to
Section 6.2.2 with attributes $IF$ as listed below. Formal contexts are
considered where the attributes are dichotomically scaled, i.e.
$\mathbb{K}_{tmp}$ is an extension of a state context according to Definition
4.2.2: $\mathbb{K}_{s}=(S,M,I)$ and $M\subseteq E\times F$. Now
$F:=\\{0,1\\}$. The attributes of $\mathbb{K}_{tt}$ and $\mathbb{K}_{tmp}$ are
constructed as usual from $M$. In the subsequent implications, the sets
$B_{0},B_{1},C_{0},C_{1}$ and $C$ are nonempty subsets of $M$. The indices
express which of the fluents is assigned to the entities: $m=(e,f)\in
B_{0}\text{ or }C_{0}\Rightarrow\gamma(s)(e)=0$ and $m\in B_{1}\text{ or
}C_{1}\Rightarrow\gamma(s)(e)=1$. We suppose that all states and transitions
are completely defined.
1. 1.
$B^{in}\rightarrow C^{in}$
2. 2.
$B^{in}\rightarrow C_{0}^{out}\equiv B^{in}\rightarrow$
$\operatorname{nev}C_{1}$
3. 3.
$B^{in}\rightarrow C_{1}^{out}\equiv B^{in}\rightarrow$
$\operatorname{alw}C_{1}$
4. 4.
$B^{in}\rightarrow$ $\operatorname{ev}C_{1}$
5. 5.
$B_{0}^{out}\rightarrow C^{in}$
6. 6.
$B_{1}^{out}\rightarrow C^{in}$
7. 7.
$\operatorname{ev}B_{1}$ $\rightarrow C$
8. 8.
$\operatorname{alw}B_{1}$ $\rightarrow C$
9. 9.
$\operatorname{nev}B_{1}$ $\rightarrow C$
10. 10.
$B_{0}^{out}\rightarrow C_{0}^{out}$
11. 11.
$B_{0}^{out}\rightarrow C_{1}^{out}$
1. 12.
$B_{1}^{out}\rightarrow C_{0}^{out}$
2. 13.
$B_{1}^{out}\rightarrow C_{1}^{out}$
3. 14.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$
4. 15.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$
5. 16.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$
6. 17.
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$
7. 18.
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$
8. 19.
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$
9. 20.
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$
10. 21.
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$
11. 22.
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$
The equivalences in 2. and 3. follow from Proposition 6.2.1 (6.3). Since the
implications comprising input attributes are independent from those related
only to output attributes, attribute exploration was performed separately for
the first 9 and the remaining 13 implications. Results for the second part are
shown here.
The exploration with ConImp started from a hypothetical context as single
object of the test context, where no implications are valid. Before, 25 known
entailments were added as background rules (BR) like those of Proposition
6.2.1 or other rules following from the definitions like
$\operatorname{alw}B_{1}\rightarrow\operatorname{ev}B_{1}$. A counterexample
had to be chosen carefully, since an object not having its maximal attribute
set might preclude a valid entailment (compare p. 2.2). The exploration
resulted in the following stem base of only 14 entailments. Most of them are
background rules (they are accepted automatically during the exploration), but
not all of these are needed in order to derive all valid entailments between
the chosen implications. This demonstrates the effectivity and minimality of
the algorithm. Entailments 5., 6., 7. and 10. were new findings.
1. 1.
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$ $\models$
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ (BR 1)
2. 2.
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$,
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$
$\models\bot$ (BR 11)
3. 3.
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$ $\models$
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ (BR 2)
4. 4.
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$,
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$
$\models\bot$ (BR 14)
5. 5.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$,
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$ $\models$
$B_{0}^{out}$ $\rightarrow$ $C_{0}^{out}$, $B_{1}^{out}$ $\rightarrow$
$C_{0}^{out}$
6. 6.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$,
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$ $\models$
$B_{1}^{out}$ $\rightarrow$ $C_{0}^{out}$
7. 7.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{nev}C_{1}$,
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ $\models\bot$
8. 8.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$ $\models$
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$,
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$,
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{alw}C_{1}$ (BR 3)
9. 9.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ $\models$
$\operatorname{alw}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ (BR 4)
10. 10.
$\operatorname{ev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$,
$\operatorname{nev}B_{1}$ $\rightarrow$ $\operatorname{ev}C_{1}$ $\models$
$B_{0}^{out}$ $\rightarrow$ $C_{1}^{out}$, $B_{1}^{out}$ $\rightarrow$
$C_{1}^{out}$, $\operatorname{ev}B_{1}$ $\rightarrow$
$\operatorname{alw}C_{1}$
11. 11.
$B_{1}^{out}$ $\rightarrow$ $C_{1}^{out}$ $\models$ $\operatorname{ev}B_{1}$
$\rightarrow$ $\operatorname{ev}C_{1}$, $\operatorname{alw}B_{1}$
$\rightarrow$ $\operatorname{ev}C_{1}$, $\operatorname{alw}B_{1}$
$\rightarrow$ $\operatorname{alw}C_{1}$
(BR 4, BR 5 $\Leftarrow$ Proposition 6.2.1 (6.5) (6.6))
12. 12.
$B_{1}^{out}$ $\rightarrow$ $C_{0}^{out}$ $\models$ $\operatorname{alw}B_{1}$
$\rightarrow$ $\operatorname{nev}C_{1}$ (BR 9 $\Leftarrow$ Proposition 6.2.1
(6.5))
13. 13.
$B_{0}^{out}$ $\rightarrow$ $C_{1}^{out}$ $\models$ $\operatorname{nev}B_{1}$
$\rightarrow$ $\operatorname{ev}C_{1}$, $\operatorname{nev}B_{1}$
$\rightarrow$ $\operatorname{alw}C_{1}$
(BR 1, 10 $\Leftarrow$ Proposition 6.2.1 (6.5))
14. 14.
$B_{0}^{out}$ $\rightarrow$ $C_{0}^{out}$ $\models$ $\operatorname{nev}B_{1}$
$\rightarrow$ $\operatorname{nev}C_{1}$ (BR 6 $\Leftarrow$ Proposition 6.2.1
(6.5))
It remains to prove the rules of this stem base, which is straightforward from
the definitions. I am giving some hints.
BR 1, 2, 3 and 4 are based on
$\operatorname{alw}A\rightarrow\operatorname{ev}A$, $A\subseteq M$, and BR 11
and 14 on
$\operatorname{nev}C_{1}\wedge\operatorname{ev}C_{1}\rightarrow\bot$. Hence,
in BR 14 $\operatorname{alw}B_{1}\rightarrow\bot$ follows in the underlying
contexts. This implication has not been considered explicitly, but it is
presupposed that other implications with $\operatorname{alw}B_{1}$ do not hold
in this case. Their common extent is empty in the test context and
$\emptyset^{\prime}=\bot$ follows.
7.: By the third Armstrong rule (6.13) and
$\operatorname{alw}B_{1}\rightarrow\operatorname{ev}B_{1}$,
$\operatorname{alw}B_{1}\rightarrow(\operatorname{nev}C_{1}\wedge\operatorname{ev}C_{1})$
follows from the presupposed implications. This is precisely the premise of
rule 4 (BR 14).
10.: Inversely, in all possible cases the states / transitions have the
attribute $\operatorname{ev}C_{1}$ and therefore also
$\operatorname{alw}C_{1}$ and $C_{1}^{out}$. This means explicitly:
$\top\rightarrow\operatorname{ev}C_{1}$,
$\top\rightarrow\operatorname{alw}C_{1}$, $\top\rightarrow C_{1}^{out}$. 5. is
a parallel rule concerning $\operatorname{nev}C_{1}$. Rules 7. and 10. suggest
that implications with empty premise $\top$ or with conclusion $\bot$ should
be considered explicitly.
11.: This rule has to be modified slightly for the first inference, according
to a restriction of Proposition 6.2.1 after this exploration had been
finished:
$m_{1}^{out}\rightarrow
C_{1}^{out}\models\operatorname{ev}m_{1}\rightarrow\operatorname{ev}C_{1}\>(m_{1}=(e,1)\in
M).$
As a conclusion it can be stated that we have derived another set of rules
(for Boolean attributes), which can shorten attribute exploration and narrow
the decisions of an expert to really new implications. This rule set
represents a sound and complete entailment calculus for the selected class of
implications for transition and state contexts under the condition of
Proposition 6.3.6:
$\forall(s^{in},s^{out})\in R,\forall m\in B_{1}\colon
m\in(s^{out})^{I}\Rightarrow B_{1}\subseteq(s^{out})^{I},$
in particular if $|B_{1}|=1$.
### 6.5 Overview of the implemented R scripts
Concept Explorer [2] and ConImp [26] were used for attribute exploration. R
scripts were programmed for the following tasks:
* •
First, thanks to Mike Behrisch for generating all deterministic time series
(including cycles) for 1 to 3 attributes, by his C++ program AllinOne1.1.cpp.
The output files are stored as inferenceRules/ergebnis2-3/ergebnis1.xt,
.../ergebnis2.txt, .../ergebnis.3.txt.
* •
generateTestContext3.2.r uses:
* –
The file generateKttKtmpFct1.0.r contains a function generating an apposition
of $\mathbb{K}_{tt}$ (or $\mathbb{K}_{t}$) and $\mathbb{K}_{tmp}^{\prime}$.
* –
The function testImpFct2.0.r decides about he validity of an implication in an
arbitrary one-valued formal context.
Felix Steinbeck optimised the code so that the script terminated in yet
acceptable time.
* •
startPartContext.r: R script for the serial computation of the test context.
* •
The output files were object reduced with ConExp and concatenated to
testContext3AttrImpForms_noEmptyset3.2_complete.txt, a tabulator separated
text file. “noEmptyset” indicates that the empty premise or conclusion were
considered explicitly and only nonempty sets were assigned to the variables
$X,Y$ and $Z$.
* •
testCxt.cxt: The test context converted by ConExp to the Burmeister format
readable by ConImp.
* •
testContext3AttrImpForms_noEmptyset3.2_imp.txt: Stem base (inference rules) as
computed by ConExp.
* •
testCxt.pro (text file): Stem base as computed by ConImp, with short form of
rules highlighting attributes (implicaton forms) not following from previous
rules.
* •
tCxtImp.duq: Stem base stored in ConImp format.
For details, see the comments within the R files on the attached CD.
## Chapter 7 Gene regulatory networks I: Analysis of a Boolean network from
literature
### 7.1 Gene regulatory networks
Figure 7.1: Gene expression: the two phases of transcription and translation
[www.scientificpsychic.com/fitness/aminoacids1.html].
To understand normal and destructive cellular reactions, systems biology has
developed models describing processes at the molecular level. A fundamental
process is gene expression, which is a sequence of two phases, transcription
and translation (Figure 7.1): i) during transcription, messenger ribonucleic
acid (mRNA) is produced according to the genetic DNA template (i.e., a DNA
sequence coding for a single protein); and ii) during translation, proteins
are built from amino acids using these mRNA templates. [9]
Proteins are the main regulators of living organisms; they activate almost
every chemical reaction within an organism as enzymes, build new structures
during cell division or transduce biochemical signals from the cell surface to
the cytoplasm and the nucleus, e.g., by phosphorylation of target proteins.
These signals can activate another class of proteins, the transcription
factors, which bind to the DNA and thus are able to initiate, enhance or
repress transcription. [9]
Mathematical and computational models may assist biologists in further
research activities by generating predictions and hypotheses that can be
experimentally tested. Network models, generated on the basis of extracted
information and/or experimental data, facilitate the analysis of interactions
among different key molecules and provide new insight into complex biological
pathways and interactions (for an overview of methods see [88] and [50]).
In the present chapter, the proposed method is applied to the analysis of a
gene regulatory network assembled from literature and database information in
[31] and transformed to a Petri net as well as a Boolean network in [91].
### 7.2 Sporulation in Bacillus subtilis
B. subtilis is a gram positive soil bacterium. Under extreme environmental
stress, it produces a single endospore, which can survive ultraviolet or gamma
radiation, acid, hours of boiling or long periods of starvation. The bacterium
leaves the vegetative growth phase in favour of a dramatically changed and
highly energy consuming behaviour, and it dies at the end of the sporulation
process. This corresponds to a switch between two clearly distinguished
genetic programs, which are complex but comparatively well understood.
By literature and database search, de Jong et al. [31] identified 12 main
regulators, constructed a model of piecewise linear differential equations and
obtained realistic simulation results. An exogenous signal (starvation)
triggers the phosphorylation of the transcription factor Spo0A to Spo0AP by
the kinase KinA. This process is reversible by the phosphatase Spo0E. Spo0AP
is necessary to transcribe SigF, which regulates genes initiating sporulation
and therefore is an output of the model. The interplay with other
transcription factors AbrB, Hpr, SigA, SigF, SigH and SinR is graphically
represented in [31, Figure 3]. SinI inactivates SinR by binding to it. SigA
and Signal are considered as an input to the model and are always on. Table
7.1 lists the Boolean equations building the model in [91]. They exhibit a
certain degree of nondeterminism, since the functions for the off fluents
sometimes are not the negation of the on functions. This accounts for
incomplete or inconsistent knowledge. In the case of state transitions
determined by $k$ conflicting function pairs, $2^{k}$ output states were
generated.
AbrB | = | SigA $\overline{\text{AbrB}}$ $\overline{\text{Spo0AP}}$
---|---|---
$\overline{\text{AbrB}}$ | = | $\overline{\text{SigA}}$ \+ AbrB + Spo0AP
SigF | = | (SigH Spo0AP $\overline{\text{SinR}})$ \+ (SigH Spo0AP SinI)
$\overline{\text{SigF}}$ | = | (SinR $\overline{\text{SinI}}$) + $\overline{\text{SigH}}$ \+ $\overline{\text{Spo0AP}}$
KinA | = | SigH $\overline{\text{Spo0AP}}$
$\overline{\text{KinA}}$ | = | $\overline{\text{SigH}}$ \+ Spo0AP
Spo0A | = | (SigH $\overline{\text{Spo0AP}}$) + (SigA $\overline{\text{Spo0AP}}$)
$\overline{\text{Spo0A}}$ | = | ($\overline{\text{SigA}}$ SinR $\overline{\text{SinI}}$) + ($\overline{\text{SigH}}$ $\overline{\text{SigA}}$ ) + Spo0AP
Spo0AP | = | Signal Spo0A $\overline{\text{Spo0E}}$ KinA
$\overline{\text{Spo0AP}}$ | = | $\overline{\text{Signal}}$ \+ $\overline{\text{Spo0A}}$ \+ Spo0E + $\overline{\text{KinA}}$
Spo0E | = | SigA $\overline{\text{AbrB}}$
$\overline{\text{Spo0E}}$ | = | $\overline{\text{SigA}}$ \+ AbrB
SigH | = | SigA $\overline{\text{AbrB}}$
$\overline{\text{SigH}}$ | = | $\overline{\text{SigA}}$ \+ AbrB
Hpr | = | SigA AbrB $\overline{\text{Spo0AP}}$
$\overline{\text{Hpr}}$ | = | $\overline{\text{SigA}}$ \+ $\overline{\text{AbrB}}$ \+ Spo0AP
SinR | = | (SigA $\overline{\text{AbrB}}$ $\overline{\text{Hpr}}$ $\overline{\text{SinR}}$ $\overline{\text{SinI}}$ Spo0AP) +
| | (SigA $\overline{\text{AbrB}}$ $\overline{\text{Hpr}}$ SinR SinI Spo0AP)
$\overline{\text{SinR}}$ | = | $\overline{\text{SigA}}$ \+ AbrB + Hpr + (SinR $\overline{\text{SinI}}$) + ($\overline{\text{SinR}}$ SinI) + $\overline{\text{Spo0AP}}$)
SinI | = | SinR
$\overline{\text{SinI}}$ | = | $\overline{\text{SinR}}$
SigA | = | TRUE (input to the model)
Signal | = | TRUE or FALSE (constant, depending on the initial state)
Table 7.1: Boolean rules for the nutritional stress response regulatory
network, derived in [91] from [31]. $\overline{x}\hat{=}\neg
x,\>x+y\,\hat{=}\,x\vee y,\>xy\,\hat{=}\,x\wedge y$.
Since the validation of the model by data and experimental literature has been
done before in [31], we analysed pure knowledge-based simulations. For that
reason, in step 5 of the protocol (p. 5.2), the stem base is computed
automatically without further confirmation by an expert. The developed R [6]
scripts for simulation, for generating the transitive contexts and for
converting a stem base into Prolog format are indicated in Section 8.2.7.
### 7.3 Simulation starting from a state typical for the vegetative growth
phase
We performed supplementary analyses of the transitions starting from a typical
state without the starvation signal [91, Table 4].
Transition | KinAin | Spo0Ain | Spo0APin | AbrBin | Spo0Ein | SigHin | Hprin | KinAout | Spo0Aout | Spo0APout | AbrBout | Spo0Eout | SigHout | Hprout
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$(s_{0}^{in},s_{1}^{out})$ | - | + | - | - | - | - | + | - | + | - | + | + | + | -
$(s_{0}^{in},s_{2}^{out})$ | - | + | - | - | - | - | + | + | + | - | - | - | - | +
$(s_{1}^{in},s_{1}^{out})$ | - | + | - | + | + | + | - | - | + | - | + | + | + | -
$(s_{1}^{in},s_{2}^{out})$ | - | + | - | + | + | + | - | + | + | - | - | - | - | +
$(s_{2}^{in},s_{1}^{out})$ | + | + | - | - | - | - | + | - | + | - | + | + | + | -
$(s_{2}^{in},s_{2}^{out})$ | + | + | - | - | - | - | + | + | + | - | - | - | - | +
Table 7.2: The (many-valued) transitive context corresponding to a simulation
starting from a B. subtilis state without nutritional stress [91, Table 4]. +:
on, -: off. Only the attributes are listed that are changing during the
simulation as well as Spo0A and Spo0AP.
The concept lattice for the resulting transitive context (Table 7.2, with only
a part of the attribute set only) is shown in Figure 7.2.
Figure 7.2: Concept lattice (computed and drawn with Concept Explorer [2])
representing a simulation without nutritional stress. Signal: starvation;
AbrB, Hpr, SigA, SigF, SigH, SinR, Spo0A (phosporylated form Spo0AP):
transcription factors – SigF initiates sporulation; KinA: kinase; Spo0E:
phosphatase; SinI inactivates SinR by binding to it. $i$-$j$ indicates a
transition ($s_{i}^{in},s_{j}^{out})$. Bold / blue lines: Filter
(superconcepts) and ideal (subconcepts) of the concept
({($s_{0}^{in},s_{1}^{out}),(s_{0}^{in},s_{2}^{out}),(s_{2}^{in},s_{1}^{out}),(s_{2}^{in},s_{2}^{out})$},
{AbrB.in.off, SigH.in.off, Spo0E.in.off, Hpr.in.on})
The larger circles at the bottom represent object concepts; their extents are
the four single transitions with the input state at $t=0$ or $t=2$, and the
intents are all attributes above a concept. Thus for instance, the two latter
transitions have the attribute KinA.in.on in common, designating the
respective concept. Moreover, they are distinguished unambigously from other
sets of transitions by this attribute – the concept is generated by
“KinA.in.on”.
Implications of the stem base can be read from the lattice. For instance there
are implications between the generators of a concept:
$<4>\text{ AbrB.in.off $\rightarrow$ SigH.in.off, Spo0E.in.off, Hpr.in.on}$
(7.1)
Analogous implications hold for the attributes of the conclusion, and there
are implications between attributes of sub- and superconcepts. $<4>$ indicates
that the rule is supported by four transitions.
The bottom concept has an empty extent. Its intent is the set of attributes
never occuring during this simulation comprising only three time steps. The
top concept does not have an empty intent – as it is often the case –, but it
consists of 10 attributes common to all 6 transitions. The corresponding rule
has an empty body ($\top$):
$\displaystyle<6>\>\top\rightarrow$ Signal.in.off, SigA.in.on, SigF.in.off,
Spo0A.in.on, Spo0AP.in.off, (7.2) SinR.in.off, SinI.in.off, Signal.out.off,
SigA.out.on, SigF.out.off, Spo0A.out.on, Spo0AP.out.off, SinR.out.off,
SinI.out.off
The transitive context generated by a simulation following nutritional stress
has about 20 transitions, 500 concepts and 50 implications. In such a case it
is more convenient to query the implicational knowledge base. But also for the
visualisation of large concept hierarchies, there exist more sophisticated
tools like the ToscanaJ suite [7].
### 7.4 Analysis of all possible transitions
In order to analyse the dynamics of the B. subtilis network exhaustively, I
generated 4224 transitions from all possible $2^{12}=4096$ initial states
(thus the rules are nearly deterministic). There were 11.700 transitions in
the transitive context, from which the stem base was computed containing 524
implications with support $>$ 0, but $11.023.494\approx 2^{24}$ concepts.
It was not feasible to provide biological evidence for a larger part of the
implications, within the scope of this methodological study. This could be
done by literature search, especially automatic text mining, by new specific
experiments or by comparison with observed time series ([110, 3.2], Chapter
8). Instead examples will be given for classes of implications that can be
validated or falsified during attribute exploration in specific ways. I start
with the examples of [91, 4.3].
* •
“For example, we know that in the absence of nutritional stress, sporulation
should never be initiated [31]. We can use model checking to show this holds
in our model by proving that no reachable state exists with SigF = 1 starting
from any initial state in which Signal = 0, SigF = 0 and Spo0AP = 0.” [91, p.
341] This is equivalent to the rule following from the stem base:
$\text{Signal.in.off, SigF.in.off,
Spo0AP.in.off}\rightarrow\text{SigF.out.off}$ (7.3)
* •
SigF.out.on $\rightarrow$ KinA.out.off, Spo0A.out.off, Hpr.out.off,
AbrB.out.off:
Spo0AP is reported to activate the production of SigF but also to repress its
own production (mutual exclusion). [31]
* •
SigH.out.off $\rightarrow$ AbrB.out.off, Spo0E.out.off, SinR.out.off,
SinI.out.off
All these genes are regulated
$\overline{gene.out}=\overline{\text{SigA.in}}+\text{AbrB.in}$ (+ …).
In our approach, such dependencies and mutual exclusions can be checked
systematically. We searched the stem base for further interesting and simple
implications:
$\displaystyle<4500>\text{Spo0AP.in.on,
KinA.out.off}\rightarrow\text{Hpr.out.off}$ (7.4)
$\displaystyle<4212>\text{SigH.in.on.
KinA.out.off}\rightarrow\text{Hpr.out.off}$ (7.5)
$\displaystyle<3972>\text{AbrB.in.off,
KinA.out.off}\rightarrow\text{Hpr.out.off}$ (7.6)
$\overline{\text{Hpr}}$ and $\overline{\text{KinA}}$ are determined by
different Boolean functions, but they are coregulated in all states emerging
from any input state characterised by the single attributes Spo0AP.on, SigH.on
or AbrB.on.
$\displaystyle<3904>\text{AbrB.out.on }\rightarrow$ SigA.in.on, SigA.out.on,
SigF.out.off, (7.7) Spo0A.out.on, Spo0E.out.on, SigH.out.on, Hpr.out.off,
SinR.out.off, SinI.out.off
AbrB is an important “marker” for the regulation of many genes, which is
understandable from the Boolean rules with hindsight. By a PubMed query, a
confirmation was found for the downregulation of SigF together with the
upregulation of AbrB [96].
Finally, we entered sets of interesting attributes into the Prolog knowledge
base, such that a derived implication was computed, and accordingly the
closure of the attribute set. In order to avoid infinite regresses, tabled
resolution has been applied, which is implemented in the Prolog extension XSB
[http://xsb.sourceforge.net]. Complementary to (7.3), we searched after
conditions for an eventual switch towards sporulation (SigF.out.on). If the
premise is entered as set of facts off(sigF.in), off(Spo0AP.in) and
on(sigF.out), then with the implications of the stem base the conclusion is
returned as a set of derived facts. For the analysed complete simulation, the
conditions Signal = SigA = TRUE had been dropped, but they were supposed to be
constant (see the first four implications, which in Prolog are read from right
to left).
:- table off/1.
:- table on/1.
off(sigF.in).
off(Spo0AP.in).
on(sigF.out).
on(signal.in) :- on(signal.out).
on(signal.out) :- on(signal.in).
on(sigA.in) :- on(sigA.out).
on(sigA.out) :- on(sigA.in).
off(abrB.out) :- on(sigF.out).
off(kinA.out) :- on(sigF.out).
off(Spo0AP.out) :- off(spo0A.in), on(sigF.out).
on(signal) :- off(sigH.in), on(sigF.out).
...
The subsequent implication was found:
SigF.in.off, Spo0AP.in.off, SigF.out.on (7.8) $\displaystyle\rightarrow$
Signal.in.on, Signal.out.on, SigA.in.on, SigA.out.on, Spo0AP.out.off,
Spo0A.out.off, AbrB.out.off, KinA.out.off, Hpr.out.off.
The latter four attributes follow immediately from the Boolean rules, but
Spo0AP.out.off depends in a more complex manner on the premises. It is also
noteworthy that the class of input states developing to a final state with
attribute SigF.out.on is characterised only by the attributes Signal.in.on and
the ubiquitous transcription factor SigA.in.on, i.e. the initial presence or
absence of no other gene is necessary for the initiation of sporulation.
## Chapter 8 Gene regulatory networks II: Adapted Boolean network models for
extracellular matrix formation
### 8.1 Biomedical and bioinformatics background
Rheumatoid arthritis (RA) is a systemic (whole body) autoimmune disease, the
causes of which are incompletely elucidated so far. Autoantibodies are
directed against citrullinated proteins, where one or several arginine amino
acids are post-translationally modified. Susceptibility factors are the
genotype of the antigen presenting major histocompatibility complex HLA-DRB1,
smoking, and there is a higher risk for women pointing at hormonal factors.
The immune disorder might also be triggered by a viral or bacterial infection.
[101]
Our focus is a better understanding of the symptom generating processes and
the therapy against the cytokine TNF$\alpha$. RA is characterised by chronic
inflammation and destruction of multiple joints perpetuated by the synovial
membrane (SM). A major component of the inflamed SM (also called pannus
tissue) are activated, semi-transformed synovial fibroblasts (SFB, or
synoviocytes) [8] [55] [57] [65] [90]. In normal joints, SFB show a balanced
expression of proteins, regulating the formation and degradation of the
extracellular matrix (ECM), a fibrous structure providing support to the cells
(besides other functions). In RA, however, SFB are known for predominant
expression and secretion of pro-inflammatory cytokines and tissue-degrading
enzymes [57] [65], thus maintaining joint inflammation and degradation of ECM
components of cartilage and bone, which are also invaded by SFB. In addition,
enhanced formation of soft ECM components such as collagens in the affected
joints (an attempt of wound healing resulting in fibrosis) is also driven by
SFB [81].
Figure 8.1: The knee joint: normal morphology (left) and schematic
representation of rheumatoid arthritis features (right) [90, Figure 1].
Central transcription factors (TFs) involved as key players in RA pathogenesis
[34] [115] and in the activation of SFB in RA patients are AP‑1, NF-$\kappa$B,
ETS‑1, and SMADs [13] [16] [34] [47]. These TFs show binding activity for
their cognate recognition sites in the promoters of inflammation-related
cytokines (e.g., TNF$\alpha$, IL1$\beta$, IL6 [8]) and matrix-degrading target
genes [34] [48] [76] [104] [115], e.g., collagenase (MMP‑1) [8] and
stromelysin1 (MMP‑3) [76]. The latter two show high expression levels in RA
[35] [43] [73] and contribute to tissue degradation [103] by destruction of
ECM components, including aggrecan or collagen types IV, X, and XI [102]
[113].
Mathematical and computational models are of particular importance in the
context of rheumatic diseases and cartilage/bone metabolism, since the
development of new and/or adapted molecular therapies depends on the
understanding of superordinate pathway interrelationships in the pro-
inflammatory microenvironment of the joint [57]. Therefore, we developed a
method for simulating the temporal behaviour of regulatory and signalling
networks. It was used to create two gene regulatory networks emulating ECM
formation and destruction, based on literature information about SFB on the
one hand and on experimental data on the other, which we obtained from
TGF$\beta$1 or TNF$\alpha$ stimulated SFB. As motivated in Section 3.3.2, we
applied Boolean network architecture for modelling. Using attribute
exploration, the simulation results and the observed time series were further
integrated in a fine-tuned and automated manner resulting in sets of rules
that determine system dynamics.
For our analysis we used a collection of 18 genes, which can be classified
into five functional groups, sufficient to create a self-contained regulatory
network of ECM maintenance: (1) structural proteins which are the target
molecules (i.e., the collagen-forming subunits COL1A1 and COL1A2); (2) enzymes
degrading them (i.e., the matrix metalloproteinases MMP1, -3, ‑9, and ‑13);
(3) molecules that inhibit these proteases (tissue inhibitor of
metalloproteinases TIMP1); (4) TFs (i.e., ETS1, FOS, JUN, JUNB, JUND, NKFB1)
and modulators acting as TFs (i.e., SMAD3, SMAD4, SMAD7) which are regulated
by (5) the external signalling molecules TNF$\alpha$ (TNF) and TGF$\beta$1
(TGFB1). These genes (Figure 8.2) are known to be expressed by SFB, except for
TNF and MMP9, for which the expression is still under question (compare p. 10,
8.3.7).
Figure 8.2: List of genes used in this analysis.
### 8.2 Methods
#### 8.2.1 Clinical data
##### Patients and samples
Synovial membrane samples were obtained following tissue excision upon joint
replacement/synovectomy from RA and osteoarthritis (OA) patients ($n$ = 3
each; Figure 8.3). Informed patient consent was obtained and the study was
approved by the ethics committees of the respective university. RA patients
were classified according to the American College of Rheumatology (ACR)
criteria [14, p. 234], OA patients according to the respective criteria for
osteoarthritis [11, p. 237].
The preparation of primary SFB from RA and OA patients was performed as
previously described [116, p. 235]. Briefly, the tissue samples were minced
and digested with trypsin/collagenase P. The resulting single cell suspension
was cultured for seven days. Non-adherent cells were removed by medium
exchange. SFB were then negatively clarified using
Dynabeads${}^{\text{\textregistered}}$ M‑450 CD14 and subsequently cultured
over 4 passages in DMEM containing 100 $\mu$g/ml gentamycin, 100 $\mu$g/ml
penicillin/streptomycin, 20 mM HEPES (all from PAA Laboratories, Coelbe,
Germany), and 10 % FCS (BioWhittaker-Lonza, Basel, Switzerland).
Figure 8.3: Clinical characteristics of the patients at the time of
synovectomy/sampling. 1Erythrocyte sedimentation rate, 2C-reactive protein,
normal range: $<$ 5 mg/l, 3American Rheumatism Association (now: American
College of Rheumatology) 4Methotrexate 5Non steroidal anti-inflammatory drugs.
##### Cell stimulation and isolation of total RNA
At the end of the fourth passage, SFB were stimulated by 10 ng/ml TGF$\beta$1
or TNF$\alpha$ (PeproTech, Hamburg, Germany) in serum-free DMEM for 0, 1, 2,
4, and 12 h. At each time point, the medium was removed and the cells were
harvested after treatment with trypsin (0.25% in versene; Invitrogen,
Karlsruhe, Germany). After washing with PBS, they were lysed with RLT buffer
(Qiagen, Hilden, Germany) and frozen at ‑70°C. Total RNA was isolated using
the RNeasy Kit (Qiagen) according to the supplier’s recommendation.
##### Microarray data analysis
Analysis of gene expression was performed using U133 Plus 2.0 RNA microarrays
(Affymetrix${}^{\text{\textregistered}}$, Santa Clara, CA, USA). Labelling of
RNA probes, hybridisation, and washing were carried out according to the
supplier’s instructions. Microarrays were analysed by laser scanning (Hewlett-
Packard Gene Scanner). Background-corrected signal intensities were determined
and normalised using the MAS 5.0 software
(Affymetrix${}^{\text{\textregistered}}$). For this purpose, arrays were
grouped according to the respective stimulus (TGF$\beta$1 and TNF$\alpha$,
$n=6$ each). The arrays in each group were normalised using quantile
normalisation [24, p. 236]. Original data from microarray analysis have been
deposited in NCBI Gene Expression Omnibus [3] and are accessible through GEO
series accession number GSE13837. A list of probe sets and all expression time
courses are provided in additional file 4.
#### 8.2.2 Creating network and Boolean functions
For the selection of genes and proteins involved in ECM maintenance and for
network generation, Boolean queries were performed in PubMed [5]. Articles
were selected containing information about relevant genes expressed in SFB and
involved in ECM maintenance. For information extraction, the abstracts were
screened and filtered manually for statements on healthy conditions only. This
knowledge-based collection yielded the set of gene candidates analysed in
detail. The final gene list is presented in Figure 8.2.
The genes were also analysed using Bibliosphere [1] and literature not
extracted from PubMed was added. Subsequently, information concerning
regulatory relationships was collected and transformed into short statements
serving as input relations (edges) for the network building program Cytoscape,
version 2.6.0 [89]. Contradictory literature information was resolved by
preferring facts applying to the target cell type (human fibroblasts) and/or
by comparison with experimental gene expression results from our and other
microarrray data (GSE1742 and GSE2624, see additional file 5). The complete
list of used statements and the respective literature basis can be found in
additional file 1. In a further step, simulation results were iteratively
compared to the experimental data in the present study, resulting in two
adapted Boolean networks which represent hypotheses about regulatory processes
initiated by TGF$\beta$1 and TNF$\alpha$.
#### 8.2.3 Data discretisation
Since we were interested in regulatory interactions, the fold-change of the
expression values was more important than absolute levels. Hence, we
discretised individual time series separately. The discretised data served to
verify or falsify the temporal dependencies predicted from the extracted
literature knowledge. For that reason, we wanted to conserve as many effects
on gene expression as possible and set weak criteria for up-regulation: if the
highest fold-change (i.e., the difference of $\operatorname{log}_{2}$ values)
between two arbitrary time points was larger than 1, then the time profile was
discretised to 0 or 1 by k-means clustering (100 iterations, vote of 25
restarts). We set the constant value 0 if: (i) the highest fold-change between
two arbitrary points in a time series was less than 1; (ii) the absolute
expression value was below the threshold of 100 for one probe set; or (iii)
the Affymetrix detection value $p$ indicating the reliability of the
measurement exceeded 0.05. In all other cases, the constant was set to 1.
Applying these criteria, also individual values were set to 0 (i.e., off)
following clustering.
#### 8.2.4 Principles of simulation
Using the deterministic Boolean network, simulations were generated using an
asynchronous update scheme based on the subsequent biologically-founded
assumptions. In order to simulate the time courses more realistically,
transcription and translation were separated, i.e., the left side of a Boolean
function (output) was considered as mRNA and the right side as TF and/or
stimulus (input). Unfortunately, time-resolved data for gene expression
events, mRNA, or protein half-life are scarce in the literature. Therefore,
time steps were selected based on general expert knowledge and comparison of
literature and experimental data, if available. For example, the duration of
transcription was generally set to 1 time unit, for NF-$\kappa$B it was set to
a doubled time period, reflecting its markedly prolonged response time before
expression compared to the immediate early response transcription factors AP-1
and ETS1 [25].
In summary, we selected the time steps as follows: transcription 1 (NFKB1: 2),
translation: 1, mRNA lifespan: 1, and protein lifespan: 2. Since TGF$\beta$1
and TNF$\alpha$ have to be released into the extracellular medium after
translation, they were assumed to take effect three time units after
induction. The starting conditions of the simulations were characterised by
the initially observed, discretised states, and an initial state was
introduced, for which the TFs were set to on. Supposing a steady state
situation before starting the stimulation with TGF$\beta$1 and TNF$\alpha$,
the protein levels at step 0 and 1 were defined according to that of the
corresponding mRNA, and, in addition, the respective stimulating protein was
set to on. The simulations were performed over twelve time units, roughly
corresponding to the twelve hour duration of the gene expression experiments.
#### 8.2.5 Creation of a temporal rule knowledge base
The sets of observed and simulated states $S^{obs}$ and $S^{sim}$ were
characterised by the expression levels of each gene, i.e., by a subset of
attributes $M = E \times F$, with entities or genes $E$, and the corresponding
values $F = \\{\text{off, on}\\}$. Hence, they were assembled into state
contexts (Definition 4.2.2) $\mathbb{K}_{s}^{obs}$ and $\mathbb{K}_{s}^{sim}$.
A state can be considered as a tuple $(f_{1},..., f_{n})$ with $f_{i} \in F,n
= |E|$.
The transitions after one time step define relations $R^{obs}\subseteq S^{obs}
\times S^{obs}$ and $R^{sim} \subseteq S^{sim} \times S^{sim}$ on the states.
Thus, in general multiple output states $s^{out}$ following an input state
$s^{in}$ are possible. However, this case rarely occurred, justifying the use
of a deterministic simulation procedure.
We computed the transitive closure of these relations, since we were
interested in all states emerging from a given one, within the observation or
simulation time. The data of all time series related to one stimulus was
assembled in the transitive contexts (Definition 4.4.1)
$\mathbb{K}_{tt}^{obs}$ and $\mathbb{K}_{tt}^{sim}$. These define relations
$I$ between objects (the transitions) and attributes (the discretised gene
expression levels in input and output states).
By attribute exploration, we compared the literature-based implications with
those merely derived from the data and applied a strong criterion:
implications of $\mathbb{K}_{tt}^{sim}$ had to be valid for all transitions of
the observed context $\mathbb{K}_{tt}^{obs}$. This is equivalent to an
exploration of the union of the two contexts, where every proposed implication
is accepted. Thus, the resulting stem base was computed automatically with the
Java tool Concept Explorer, which supports also expert centered attribute
exploration [2]. The other calculations were made with my own R [6] programs
(Section 8.2.7). Computing the 2713 (8785) TGF$\beta$1 (TNF$\alpha$) rules,
Concept Explorer ran 21 (30) minutes on a 2.66 GHz/2 GB computer.
#### 8.2.6 Expert analysis of transition rules
The calculated transition rules were screened manually, focussing on the
appearance and the temporal behaviour of the following features: (i)
constitutive vs. induced gene expression; (ii) co-expression vs. divergent
expression of mediators, TFs, and target genes; (iii) expression of
mediators/transcription factors vs. expression of target genes; (iv)
regulation of target gene expression based on the expression of different
transcription factors; (v) expression of individual genes vs. expression of
their functional groups; and (vi) discrepancies to the literature.
Subsequently, the extracted rules were assessed with respect to biological
coherence and relevance.
#### 8.2.7 Overview of the implemented R scripts
The following R scripts have been developed; most of them were also utilised
for the B. Subtilis network analysis of Chapter 7.
* •
discretise4.1.r.
Data discretisation as described above.
* •
simulationTnf.r, simulationTgf.r.
Simulations according to the Boolean networks.
(allStates.r: Simulations for the nondeterministic B. subtilis network,
starting from all possible initial states.)
* •
aposterioriTrans2.0.r.
Generate a transition and a transitive context from observed or simulated
data.
* •
selectGenes2.1.r.
Compute support and confidence of a rule in an (observed) transition context,
show transitions in the observed and simulated context with the premise
attributes.
Provides a decision criterion for expert attribute exploration.
* •
convert2Prolog.r.
Convert a stem base into Prolog format for queries.
#### 8.2.8 Additional files
In the data CD, the R scripts are included as well as the additional files
published with [109], at http://www.biomedcentral.com/content/supplementary/:
* •
1752-0509-3-77-S1.doc.
Literature used for the network construction. Each citation corresponds to one
edge in the regulatory network.
* •
7521-0509-3-77-S2.zip.
Cytoscape import file. Import this file into Cytoscape [89] to analyse the
gene regulatory network in more detail. It also includes external links for
the genes and references cited to GenBank, Uniprot, and PubMed.
* •
7521-0509-3-77-S3.zip.
Cytoscape import file. Open this file after importing the CYS file (provided
by Additional file 2) into Cytoscape [89] if the layout of the CYS file cannot
be displayed correctly with your Cytoscape version.
* •
7521-0509-3-77-S4.xls.
List of probe sets used, processed microarray data and visualisation of
expression time courses for the genes analysed. Raw data are deposited under
accession number GSE13837 at GEO [3].
* •
7521-0509-3-77-S5.xls.
Processed and visualisation of GEO Data. Data were extracted from GSE1742
(TGF$\beta$1) and GSE2624 (TNF$\alpha$) at GEO [3].
* •
7521-0509-3-77-S6.xls.
Discretised gene expression time courses.
* •
7521-0509-3-77-S7.xls.
Histograms of gene expression simulation. The simulations for TGF$\beta$1
(blue) and TNF$\alpha$ (red) were run for 12 time steps (x-axis) and for each
initial state derived from the patients’ data separately. A simulated
expression of 100% (y-axis) means that in all six cases the gene was on.
* •
7521-0509-3-77-S8.xls.
List of the top 500 knowledge base rules valid for the simulations as well as
for the data from stimulations with TGF$\beta$1 and TNF$\alpha$.
* •
ECMData.xls
Measured values for probesets, after normalization of mean and variance for
all TGFB1 and TNF chips, respectively. Selection of probesets, logarithmic
values of the geometric mean for one gene.
Moreover, the attached CD contains the discretised measured and simulated data
as formal contexts (*.txt files readable also with Concept Explorer [2]), as
well as the complete stem bases in *.txt and PROLOG format (see contents at
readme.txt).
### 8.3 Results and discussion
#### 8.3.1 Creating a regulatory network by information extraction from
literature
The available literature was screened for genes and proteins involved in ECM
maintenance and expressed in the lining layer SFB of the SM. In order to
derive a regulatory network, we comprehensively collected literature knowledge
related to the formation and degradation of ECM in human fibroblasts and
analysed it manually. We chose collagen type I, which is formed by the COL1A1
and COL1A2 gene products, as a connective tissue representative, several MMPs
as ECM-degrading enzymes, their inhibitors, and TFs regulating them. Finally,
we selected 18 genes (Figure 8.2) and the literature was screened again for
gene regulatory relations and interactions connecting them (see additional
file 1 for a complete list). Some contradictory literature findings were
resolved manually (see Section 8.3.5).
The resulting regulatory network is almost closed and represents the most
important ECM network functions. Here, we imply that the receptors for the
external signalling molecules are always available and functional in SFB.
Note, that TGF$\beta$1 (TGFB1) and TNF$\alpha$ (TNF) are the only entities
playing a dual role as both external signal molecules and target genes because
of their introduction into the simulation as starting effectors.
It turned out that the knowledge about gene regulatory events is limited and
that, to the best of our knowledge, the regulation of SMAD and SMAD expression
has not been fully characterised so far. The SMAD gene products seem to be
available in sufficient amounts and we were unable to find reports about their
regulated expression. In addition, not all influences of TGF$\beta$1 and
TNF$\alpha$ on gene expression could be described as direct effects of
transcription factors at the mRNA level because the important SMAD family
members act as regulators on the protein-protein interaction level. All
influences were included in the network at this point to avoid premature loss
of information.
Figure 8.4: Overview of the ECM network in hierarchical order. Regulatory
effects via TFs are shown as continuous lines, others as indirect effects as
dashed lines. Inhibition is marked by a red T-arrow, induction is illustrated
by black arrows. The external signals TGF$\beta$1 and TNF$\alpha$ are shown as
light grey circles, the internal SMAD signalling molecules as dark grey
squares, TFs are depicted as black (AP‑1 components) or white squares, and the
target genes are shown as white octagons. This picture was generated using
Cytoscape 2.6.0.
Although many TFs such as AP‑1 are also regulated at the protein level (e.g.,
by phosphorylation), those effects can be reflected simplistically by
regulatory processes at the transcriptional level. However, activating SMADs
as SMAD3 and SMAD4 are also regulated by inhibitory members of the SMAD family
(SMAD6 and SMAD7), which may counteract transcriptional activation and add an
extra level of complexity [85]. Therefore, SMAD7 was introduced into the
network as a TGF$\beta$1-dependent repressor of SMAD-dependent transcription.
In the case of SMAD3, we decided to subsume its influence under the SMAD4
effects because both are described to have nearly identical effects and act in
concert. Moreover, we could not find well-defined information about SMAD3
regulation. Hence, we added an inducing influence of SMAD4 on MMP13 (at
present only known for SMAD3) for keeping all the SMAD effects in the network.
The subunits of the homo- or heterodimer TF AP‑1, i.e., Jun, JunB, JunD, and
Fos (JUN, JUNB, JUND, FOS), determine its different regulatory activities (for
AP‑1 components see [51] and references therein). Therefore, we decided to
disassemble the transcriptional active entity AP‑1 into its subunits. In
contrast, for the dimeric TF NF‑$\kappa$B, which can be composed of the gene
products of NFKB1, NFKB2, RELA, RELB, and/or REL [79], we selected NFKB1 as
the representative gene with respect to our signalling framework. All the
genes and their interrelations were transferred into the program Cytoscape
[89] to visualise our network containing 19 nodes and 79 edges, respectively,
as shown in Figure 8.4. Detailed network examination is available through the
network description files (additional files 2 and 3), also providing external
links to GenBank, Uniprot, and PubMed for all edges and nodes.
Available tools for automatic text mining decide schematically, e.g., by pre-
built rules like co-occurrence of gene names and interaction verbs or pattern
matching, whereas a human expert is able to integrate unanticipated types of
information and to decide whether the paper confirms the investigated
situation. However, we used the tools Bibliosphere [1] and Pathway Studio [4]
in order to verify completeness and consistency of the assembled network (data
not shown).
#### 8.3.2 Boolean functions
Due to its capability for displaying dynamic dependencies between individual
parameters, a Boolean network is more specific than the graphical network in
Figure 8.4, which summarises isolated literature facts. In order to decide
about the connectives OR/AND, which represent causally determined relations
between different genes, cellular signalling processes were also considered.
In the case of a known transcriptional activation of any gene by the stimuli
TNF$\alpha$ or TGF$\beta$1 via a specific TF, this activation was represented
in the network using the term GENE.out = STIMULUS AND TF. Without such
evidence, these influences were connected by GENE.out = STIMULUS OR TF. Since
it is well known that the so-called SMAD pathway is activated by TGF$\beta$1
but not influenced by TNF$\alpha$ [97], we used the AND connection for SMAD3/4
and TGF$\beta$1, even if there was no explicit literature evidence for an
impact of TGF$\beta$1 onto the respective gene.
Another example for setting up the functions is the integration of: (i) the
known auto-regulatory transcriptional activation of JUN by TNF$\alpha$ via
JUN, and (ii) the activation of JUN via SMAD4 (TNF$\alpha$-independent) into
the single Boolean function 5 (compare Table 8.5 with Tables 8.7 and 8.8):
JUN.out = (TNF AND JUN) OR SMAD4. Based on the illustrated principles, the
Boolean functions characterising formation and remodelling of the ECM were
generated (Table 8.5).
Figure 8.5: Boolean rules based on literature information.
#### 8.3.3 Gene expression time courses following TGF$\beta$1 and TNF$\alpha$
stimulation
Figure 8.6: Gene expression time courses following TGF$\beta$1 or TNF$\alpha$
treatment. COL1A1 (A), JUNB (B), and SMAD7 (C) gene expression in response to
TGF$\beta$1 treatment (upper row); TNF$\alpha$ response (lower row) of NFKB1
(D), MMP1 (E), and SMAD7 (F). The average time course is shown as light red
curve without symbols, the data for individual samples are depicted in other
colours (OA1: blue, filled symbol; OA2: red, filled symbol; OA3: green, filled
symbol; RA1: purple, filled symbol; RA2: blue, open symbol; RA3: yellow, open
symbol). The time courses and the values calculated from the microarray
experiments for all analysed genes are included in additional file 4.
We analysed gene expression changes of SFB from patients with RA (3 patients)
or OA (3 patients) following TGF$\beta$1 and TNF$\alpha$ stimulation (Figure
8.3). Due to the strong stimuli, both groups of cells reacted in an almost
identical way, and we did not differentiate among them. In another study, for
example, OA cells were considered to be a disease control group [112].
Following pre-processing of the microarray data gained from U133 Plus 2.0
arrays, we extracted the data for probe sets related to our genes of interest
(see Methods). The data are available in the GEO database (GSE13837 at [3]).
For the following analyses we excluded values which exceeded the reliability
threshold of $p\leq 0.05$ for any patient at any time point (0, 1, 2, 4, 6,
and 12 hours). In Figure 8.6, some selected examples for the influence of
TGF$\beta$1 and TNF$\alpha$ on gene expression are presented. The time courses
of COL1A1 and JUNB expression are shown to illustrate the TGF$\beta$1 response
in SFB, and the TNF$\alpha$ response is illustrated by NFKB1 and MMP1
expression. SMAD7 expression data are also included for both treatments. The
data and the respective diagrams for all genes and both treatments can be
found in additional file 4.
For comparative purposes, we also analysed public data from the GEO database,
first, TGF$\beta$1 treated murine embryonic fibroblasts (GSE1742) and second,
TNF$\alpha$ stimulation of endothelial cells (HeLa, GSE2624). Following
prolonged TGF$\beta$1 treatment in murine cells, COL1A2, JUN and TIMP1 gene
expression increased, whereas FOS decreased. In contrast, FOS, JUN and JUNB
expression in HeLa cells rapidly increased following TNF$\alpha$ stimulation.
Unfortunately, no data about the protease genes were available in this dataset
(additional file 5). Even though cell type, experimental design and duration
of treatment differ from our experiments, they reflect the two general trends:
a positive effect on ECM formation by TGF$\beta$1 and a degradative influence
on ECM by TNF$\alpha$ (mediated at least in part by FOS and JUN), which is
consistent with our data. However, the evaluation of the complete data sets
revealed discrepancies between the expected expression profile of individual
genes and their time courses following stimulation in the experiment.
#### 8.3.4 Data discretisation
We developed a data discretisation method which appropriately captures
biologically relevant differences in gene expression levels. The individual
time profiles for each gene were separately discretised to the values 0 or 1
by $k$-means clustering [49], a method which is often applied for gene
expression time series. No improvements were observed when applying Ward’s
hierarchical clustering [100] or single linkage clustering as proposed in [32]
(data not shown). We introduced several supplementary criteria (see Methods),
e.g., the values of a time series were all discretised to the constant value 0
or 1, if the differences of all $\operatorname{log}_{2}$ values (fold-changes)
were less than 1 [72]. For the discretised data see additional file 6.
#### 8.3.5 Boolean functions adapted to the data
Figure 8.7: Revised Boolean functions for the simulation of TGF$\beta$1
stimulation. Rows marked by an asterisk indicate differences of the functions
for TGF$\beta$1 and TNF$\alpha$ stimulation, changes to Table 8.5 are
italicised and bold, and function numbers in bold indicate omitted (1, 2, 3,
8, 9, 11, 12, 14) or inserted (18) terms.
Simulations were generated using an asynchronous update scheme, assuming time
intervals – approximately equal to 1 h time steps – as follows: transcription
1 (NFKB1: 2), translation: 1, RNA lifespan: 1, and protein lifespan: 2. The
Boolean functions generated the transcriptional states according to the
functional influence of proteins (stimuli or TF); translation and mRNA/protein
degradation were computed from this output state according to the predefined
intervals (see Section 8.2.4).
As starting conditions of the simulations we chose the discretised initial
states derived from our experimental data. An additional initial state was
introduced in which solely the transcription factors were set to on, which
enables the model system to respond to the external stimulators TNF$\alpha$ or
TGF$\beta$1 immediately. The simulations were performed over twelve time
steps; however, we did not aim at an exact correspondence to the experimental
observation time of twelve hours, but tried to adjust the simulated time
courses to qualitative features such as early, intermediate or late up-
regulation. Improving the Boolean functions accordingly, the initially applied
literature-based information was completed by: (i) valid and specific
experimental information; (ii) knowledge and experience of biological experts;
and (iii) in some cases, a more focused and precise literature query. For a
comparison of the discretised observed time series and the final simulations
see the additional files 6 and 7. We developed several biologically
interesting and plausible data-independent hypotheses, for example, we
modelled the regulation of SMAD3/SMAD4 effects by a protein-protein
interaction with SMAD7.
Figure 8.8: Revised Boolean functions for the simulation of TNF$\alpha$
stimulation. Rows marked by an asterisk indicate differences of the functions
for TGF$\beta$1 and TNF$\alpha$ stimulation, changes to Table 8.5 are
italicised and bold, and function numbers in bold indicate omitted (1, 2, 8,
9, 11, 12, 13, 14) or inserted (18) terms.
The resulting optimised Boolean network with the revised Boolean functions
(Tables 8.7 and 8.8) represents an enhanced ECM model, roughly matching the
given biological conditions and extensively exceeding the present
possibilities of automatic methods such as text mining, symbolic computation
or machine learning. Considering the additional information available, we
accepted these biologically reasonable changes:
1. 1.
In the case of TNF$\alpha$ (or TGF$\beta$1) stimulation, the production and
secretion of TGF$\beta$1 (or TNF$\alpha$) by SFB should not contradict the
influence of the abundant stimulating protein TNF$\alpha$ (or TGF$\beta$1). In
these cases (e.g., for the COL1A1.out function in Table 8.8, and for the
MMP1.out function in Table 8.7) we removed TGFB1 (TNF) AND (…) from the
Boolean function term. This adjustment did not always change the simulation
result, since, for example, TNF was always off following TGF$\beta$1
stimulation (numbers of the Boolean functions (BF) affected: 1, 2, 3, 8, 9,
11, 12 and 14).
2. 2.
Down-regulation of gene expression is an essential biological principle. For
that reason we had to introduce a time-limited inactivation mechanism which
could not be derived from the literature because information regarding down-
regulatory mechanisms is very restricted. Moreover, complex and variable
mechanisms were hard to model, e.g., JUN down-regulation which is driven by:
(i) inactivation of the TF protein itself; (ii) a general shift in the
composition of the TF AP‑1, resulting in a reduced amount of TF enhancing JUN
transcription; and (iii) binding/inactivation of JUN by other proteins.
Therefore, a time-limited mRNA inactivation was introduced for JUN, JUNB, FOS
and ETS1. Accordingly, an inactivating rule was created: if these TFs are
expressed at $t > 0$, they will be set to off at $t+3$ and afterwards (no. of
BF affected: 18). In addition, at that step we included an inhibition of
TGFB1/SMAD4 signalling-based target gene expression by integrating a
SMAD4-inhibiting signal (i.e., SMAD7, included as AND NOT SMAD7) guaranteeing
the subsequent inactivation of TGF$\beta$1-related gene expression (BF
affected: 3, 5, 6, 14 and 15). JUND is constitutively expressed at an
intermediate level, which is consistent with GEO (GSE1742 and GSE2624) and our
own data, as well as with the literature [52]. For NFKB1 transcription, an
inhibitory effect was not implemented, since the activity of NF‑$\kappa$B at
the protein level is controlled by interaction with several IKB proteins [46]
which were not included in our ECM network model.
3. 3.
SMAD4 induction is not dependent on TGF$\beta$1 stimulation, because it is
constitutively expressed (i.e., always TRUE, BF affected: 13). However,
without TGF$\beta$1-mediated phosphorylation, SMAD4 is not activated at the
protein level and shows no transcriptional activity, even though
constitutively expressed. Therefore, we amended the term SMAD4 to TGF AND
SMAD4 in order to represent the necessity of TGF$\beta$1 for SMAD4 activation
(BF affected: 3, 5).
4. 4.
We considered the relation ETS1 AND NFKB1 for the target genes [68] instead of
assuming alternative pathways by ETS1 OR NFKB1 because regulation by
NF‑$\kappa$B seems to be dependent on ETS1, and the MMPs, for example, require
both factors [74] (BF for TGF stimulation affected: 9, 10 and 17, Table 8.7;
BF for TNF stimulation affected: 2, 8, 9, 10 and 17, Table 8.8).
5. 5.
Since the inhibition of JUND expression by FOS is only observed in the case of
a concomitant JUND-based positive feedback, the inhibitory effect of FOS has
been restricted to this case [23] (BF affected: 7).
6. 6.
Since a TF should not necessarily be required for its own expression (positive
feedback), in the case of JUND (and also NFKB1) the AND connective was changed
to OR. The revision of this function prevents the absence of JUND expression
following TGF$\beta$1 stimulation (BF affected: 7, 12).
7. 7.
Concerning the regulation of MMP1 expression by FOS, there were contradictory
findings in the literature [22], [105]. We decided for an inhibitory influence
of FOS following TGF$\beta$1 stimulation, because otherwise MMP1 would have
been permanently down-regulated by TGF$\beta$1 during the simulation (BF
affected: 8, Table 8.7).
8. 8.
The Boolean function MMP3.out = (…) OR TGFB1 was in obvious contradiction to
the data of the present study, thus, the term OR TGFB1 was deleted. The same
was done for the MMP13 function (BF affected: 9, 11).
9. 9.
In the case of NFKB1, the absence of TNF$\alpha$ following TGF$\beta$1
stimulation had no decisive influence (NFKB1 was not always off). For that
reason, we changed NFKB1.out = TNF AND (ETS1 OR NFKB1) to NFKB1.out = TNF OR
ETS1 OR NFKB1 (BF affected: 12, Table 8.7).
10. 10.
However, concerning the expression of TNF itself, the necessity of a positive
feedback could explain its complete absence following TGF$\beta$1 stimulation.
On the other hand, TNF was expressed at some time points following TNF$\alpha$
stimulation, whereas it is commonly assumed that fibroblasts do not express
TNF (BF not changed: 17).
In summary, we adjusted the set of BF obtained by adaptation to the gene
expression data measured under two experimental conditions (TNF$\alpha$ and
TGF$\beta$1 stimulation), in order to create an appropriate set of BF
representing the existing knowledge about naturally occurring
interrelationships as accurately as possible.
#### 8.3.6 Computing temporal rules by attribute exploration
For each stimulus, the observed and the final simulated time series were
translated and merged into a single transitive context $\mathbb{K}_{tt}$ (see
Definition 4.4.1 and the Methods Section 8.2.5). States are defined by the
value on or off for each gene, and transitions were computed by linking an
occurring input state to an arbitrary future (output) state of the simulation
or observation. The set of all these transitions, represented by
$\mathbb{K}_{tt}$, was analysed by the automatic, non-interactive version of
the attribute exploration algorithm.
The implications of the resulting stem base are temporal rules expressing
hypotheses about attributes of states (e.g., co-regulation or mutual exclusion
of gene expression) or system dynamics, which are supported by pre-existing
knowledge and by the analysed data. Since an implication holds for the
transitions between all temporally related states, a rule such as GENE1.in.out
$\rightarrow$ GENE2.out.on means: if gene1 is expressed, gene2 always will be
up-regulated in the future, at all subsequent observation time points and
simulation steps. Due to this semantics, the implications neither depend on
the correspondence of a simulation time step to a specific observation
interval, nor on prior knowledge about time periods of direct or indirect
transcriptional effects. Within the large knowledge bases for TGF$\beta$1
(2713 rules) and TNF$\alpha$ (8785 rules) stimulation, the most frequent and
simple temporal rules were considered and analysed for dependencies between
stimuli, induced TFs and their target genes.
#### 8.3.7 Results of the attribute exploration
##### Stimulation with TNF$\alpha$
Regarding stimulation with TNF$\alpha$, a coordinated down-regulation of the
two TF SMAD7 (inhibitor of TGF$\beta$1/SMAD4 signalling) and ETS1 emerges, as
indicated by the rules 33, 114, 135, 144, 157, and 186 (see combTransTnf.txt
or additional file 8). For example, rule 186:
$<90>\text{COL1A1.out.off, ETS1.out.off}\rightarrow\text{SMAD7.out.off}$
has the meaning: in all simulated and observed states characterised by the
absence of COL1A1 and ETS1, SMAD7 is also off. $<90>$ stands for the support
of the rule, i.e., the number of transitions (90 out of 294) that actually
have the attributes of the premise. Rules 114, 135, 144, 157 and 186 indicate:
if the TNF$\alpha$-dependent genes are not induced (ETS1 as mediator), then
simultaneously the expression of TGF$\beta$1-dependent genes is enabled (SMAD7
is off). This suggests that TNF$\alpha$ and TGF$\beta$1 may act as antagonists
in SFB, as described in [84], [98].
The expression of NFKB1, which is also induced by TNF$\alpha$, proceeds
conversely to that of ETS1 and SMAD7 (rules 34, 45, 70, 71, 134, 144, 154, 157
and 173) reflecting the different targets of NF-$\kappa$B and SMAD7. The
antagonistic expression pattern of NFKB1 and SMAD7 appears indirectly in rule
33, where the two genes show up in the premise of a rule with high support:
$<150>(...)\>\text{NFKB1.out.on, SMAD7.out.off}\rightarrow\text{ETS1.out.off}$
Regarding this rule, it is interesting that ETS1 always acts in the same
direction as NF-$\kappa$B, according to the network derived from the
literature (Figure 8.4). In the adapted network (Table 8.8), we assumed a
necessary cooperation (i.e., an AND connective) for the positive regulation of
ETS1, MMP1, MMP3, MMP9, and TNF, as well as for the inhibition of COL1A1 and
COL1A2. Thus, rule 33 further suggests that the coordinated action of
NF-$\kappa$B and ETS1 is turned off in states which are characterised by
supplementary conditions as SMAD7.out.off.
The generated rules adequately reflect the major influence of the TF AP‑1 in
the TNF$\alpha$ system: the expression of prominent targets, such as COL1A1,
MMP1, and MMP3, depends on JUN (rules 211 and 258) and/or FOS (rule 204), with
JUN as the key player. These rules connect input and output states and thus
their semantics is directly related to dynamics, as seen in rule 211:
$<87>\text{TGFB1.in.on, TIMP1.in.on, ETS1.in.on,
JUN.in.on}\rightarrow\text{MMP1.out.on}$
making this strong statement: if ETS1 and JUN are on, MMP1 will always be up-
regulated in the future (at least within the time frame of 12 hours).
Sometimes, MMP1 is expressed simultaneously or before ETS1 and JUN. In the
simulation, MMP1 was always on in the output state and from time point 2 h in
the data. An exception can be found for the experimental results from OA
sample OA3 (Figure 8.3), where MMP1 is off after 12 h. This is the reason for
the computation of the auxiliary conditions TGFB1.in.on and TIMP1.in.on in
rule 211.
Concerning the regulation of target genes, the expression of MMP1, MMP3, and
MMP13 is co-regulated (rules 35, 63, 82, 86 and 176), while MMP9 is expressed
independently (rules 24 and 35). There is a contradiction between the
simulation and the data: in the observed experimental time series, MMP13 is
always off, whereas the Boolean network predicts an up-regulation similar to
MMP1 and MMP3. This unexpected absence of predicted MMP13 expression may be an
indication for a more complex regulation of MMP13 transcription, exceeding the
already known regulatory interrelations. Therefore, the MMP13 promoter and
further enhancer/repressor sequences should be targeted for a more pronounced
structural and functional analysis. For MMP9, the simulation and the
experimental data are in good agreement: the gene is off in most, but not all
states. However, since the expression of MMP9 by (S)FB is discussed
controversially in the literature (see [92] and [114] vs. [42]), the
calculated expression of MMP9 by fibroblasts – at least at a limited number of
time points – supports the majority of studies, reporting detectable MMP9 mRNA
amounts in (S)FB.
Several rules unanimously indicate the co-expression of the ECM-forming genes
COL1A1 and COL1A2 (rules 87, 88 and 95), but contradictory rules occur
concerning their expression profile in comparison to the MMPs. COL1A1 and
COL1A2 seem to be co-expressed with MMP1 (rules 90 and 176), for COL1A2,
however, a certain co-expression with MMP9 is calculated as well (rules 76 and
77), which conflicts with the opposing expression of MMP1 and MMP9 (see
above). Therefore, the expression of collagens does often, but not necessarily
always correlate with the expression of MMPs. This reflects the imbalance
between MMP-dependent destruction and collagen-driven regeneration/fibrosis of
ECM in the joints in inflammatory RA.
The calculated knowledge base also contains a further unexpected correlation.
According to rule 166:
$\displaystyle<94>\quad\>$ FOS.in.off, TIMP1.in.on, SMAD7.out.off
$\displaystyle\quad\rightarrow\>$ TGFB1.in.on, MMP1.out.on, TGFB1.out.on
and rule 188, the expression of MMP1 may also be induced in the absence of FOS
(e.g., by JUN-containing AP-1 complexes), indicating that the regulation of
MMP1 does not predominantly depend on FOS as proposed in the literature [104],
[94]. This result may point to the influence of other TFs, e.g., NF-$\kappa$B,
ETS1, or AP-1 complexes containing JUN, which may indeed be able to induce
target gene expression in the absence of FOS.
##### Stimulation with TGF$\beta$1
For the stimulation with TGF$\beta$1, we had a total number of 341
transitions. The SMADs play a major role for the expression of
TGF$\beta$1-dependent target genes, as reflected by various classes of rules
containing SMAD4 and/or SMAD7 (see combTransTgf.txt or additional file 8). For
example, SMAD4 can be involved in the expression of COL1A1, see rule 15 (and
also rules 21, 26 and 30):
$<239>\text{ETS1.out.off}\rightarrow\text{SMAD4.in.on, COL1A1.out.on,
SMAD4.out.on}$
This also suggests an antagonistic behaviour of ETS1 and SMAD4: if ETS1 was
off, then SMAD4 was on, as well as in all previous states. Rules 52 and 57
suggest a dependency of MMP1 on SMAD4. However, this seems to be one amongst
many other influences (or could be a non-influencing coincidence), since SMAD4
was permanently on during simulation and experimental stimulation with
TGF$\beta$1 (exception: sample RA3 at time point 2h).
The expression of MMP9 is neither induced by SMAD4 (rules 7, 24 and 41) nor by
any other TF, indicating that MMP9 is not influenced by TGF$\beta$1\. The fact
that TGF$\beta$1 obviously does not induce MMP9 (but other MMPs) agrees with
findings reported previously [42] and represents a clear contrast to the MMP
expression profiles following TNF$\alpha$ stimulation.
A further case of an antagonistic expression pattern was calculated for MMPs
and COL1A1 (rules 21, 30, 36, 41, 54, and 60), for example, in rule 54:
$\displaystyle<170>\quad\>$ SMAD4.in.on, MMP3.out.off, MMP9.out.off,
MMP13.out.off, (…) $\displaystyle\quad\rightarrow\>$ COL1A1.out.on
Antagonistic expression profiles also can be observed for SMAD4 and other TFs,
e.g., JUN and JUNB (rules 12, 39) or ETS1 (rule 15, see above). The variety of
TF combinations found, even following the same stimulus, exceeds the
possibilities of conventional TF studies because stimulation experiments are
generally restricted to a selected set of read-out parameters (e.g., the
expression of single TFs or target genes) which are not able to reflect the
multiplicity of different effects in the cell.
Following stimulation with TGF$\beta$1, interestingly COL1A2 appears to be
constitutively expressed since its status is always calculated as on (rule 1).
Therefore, for the formation of collagen I, which contains COL1A1 and COL1A2
chains, COL1A1 expression seems to be the critical switch.
##### TGF$\beta$1 versus TNF$\alpha$ effects
The calculated results impressively illustrate that TGF$\beta$1 and
TNF$\alpha$ stimulation are mediated via separate signal transduction
pathways, leading to the expression and activation of different TFs. In
general, ETS1 and NFKB1 are induced predominantly by TNF$\alpha$, whereas SMAD
expression depends on TGF$\beta$1 (represented by differential expression
profiles of ETS1 and SMAD4). JUN and FOS, however, strikingly respond to both
stimuli. This defined pattern results in the expression of target genes with
opposing roles. TGF$\beta$1 positively regulates the enhanced formation of ECM
components, whereas TNF$\alpha$ is strongly involved in the expression of ECM-
degrading enzymes. This was the main reason for a discriminative revision of
the BF for TNF$\alpha$ and TGF$\beta$1 (Tables 8.7 and 8.8). Six BF were found
to be differently adjusted (BF 1, 2, 3, 8, 12 and 14) which concern either the
key players for ECM destruction (MMP1; BF 8), ECM formation (COL1A1 and
COL1A2; BF 1 and 2) or important regulatory genes (ETS1, NFKB1, SMAD7). This
may indicate that the differential effects on ECM induced by TNF$\alpha$ or
TGF$\beta$1 are mainly mediated via ETS1 (BF 3), NFKB1 (BF 12, especially in
the TNF$\alpha$ pathway), or SMAD7 (BF 14, especially in the TGF$\beta$1
pathway) identifying ETS1- and NFKB1-associated pathways as the major
TNF$\alpha$-induced pro-inflammatory/pro-destructive signalling modules in
rheumatic diseases, whereas TGF$\beta$1-driven and SMAD7-related signalling
appears prominently involved in fibrosis.
#### 8.3.8 Querying the knowledge base
The minimal rule set gave many new insights, and further queries can be
addressed by accessing the TNF$\alpha$ and TGF$\beta$1 knowledge bases in one
of two ways: (i) the Excel file containing the transition rules for structured
searches within the rule sets (see additional file 8 containing the top 500
transition rules, impCombTransTgf.txt and impCombTransTnf.txt for complete
lists); and (ii) the stem base in PROLOG format for queries concerning
logically implied rules as in Section 7.4 for the B. subtilis simulations
(impCombTransTgf.P and impCombTransTnf.P).
#### 8.3.9 Expert centered attribute exploration
Resuming the presented study published in [109], I investigated in detail four
genes showing strong changes over the time steps of the simulation modelling
TGF$\beta$1 stimulation, by interactive attribute exploration:
1. 1.
MMP13 is up- and downregulated at one or two time points during the
simulation, but it was never expressed significantly in the microarry
experiments. A human expert can partly solve this contradiction and decide for
each relative implication to which of the two knowledge sources more
plausibility is attributed.
2. 2.
I was also interested in the interplay of three TF belonging to different
pathways, but connected in the Boolean formula for JUNB (BF 6 of Table 8.7):
JUNB.out = (TGFB1 $\wedge$ NFKB1 $\wedge$ SMAD4 $\wedge\>\neg$SMAD7) $\vee$
(TNF $\wedge$ NFKB1)
JUNB represents the AP1 complex playing an important role in the investigated
biological context.
3. 3.
NFKB1 is known to activate MMP13.
4. 4.
Whereas JUNB and NFKB1 generally belong to TNF$\alpha$ pathways, SMAD7
inhibits TGF$\beta$1 signaling via SMAD4. The inhibition of SMAD7 expression
by TNF$\alpha$ connects both pathways.
During attribute exploration of the simulated transitive context, the
following implications were accepted, or an observed transition was introduced
as counterexample, which differed little from a simulated transition.
Counterexamples are indicated as binary numbers designating the values of the
attributes JUNB.in, MMP13.in, NFKB1.in, SMAD7.in, JUNB.out, MMP13.out,
NFKB1.out and SMAD7.out. Important decision criteria were support (number of
transitions with the premise attributes / 72 overall transitions) and
confidence (transitions with conclusion attributes / premise transitions) of
the implication in the observed transitive context. For this purpose, the R
script selectGenes2.1.r was used. In the protocol of the exploration, I omit
background implications expressing dichotomic scaling like “gene.in.on,
gene.in.out $\rightarrow\bot$”. The resulting context is
exploredTransTgf_regulationMMP13.txt, the complete stem base is documented at
exploredTransTgf_regulationMMP13_inf.txt.
1. 1.
SMAD7.out.off $\rightarrow$ JUNB.out.off, MMP13.out.off, NFKB1.out.on.
The implication was rejected by reason of a week confidence 13/30. The
counterexample 1001 0000 was introduced, i.e. a transition where JUNB.in and
SMAD7.in are on and the other attributes off.
2. 2.
SMAD7.out.off $\rightarrow$ JUNB.out.off, MMP13.out.off.
Accepted with support 30/72 and confidence 23/30.
3. 3.
MMP13.out.on $\rightarrow$ SMAD7.out.on.
The implication was accepted, since there is no information in the observed
context: MMP13 is always off, hence the support is 0.
4. 4.
JUNB.out.on $\rightarrow$ SMAD7.out.on.
Accepted with confidence 29/36. The implication is biologically interesting:
If TNF$\alpha$ signaling via JUNB is enabled, TGF$\beta$1 signaling is
inhibited via SMAD7 – in spite of its reported inhibition, in turn, by
TNF$\alpha$ (BF 14 of Table 8.5). However, TNF$\alpha$ was supposed to have a
minor influence compared to the stimulus TGF$\beta$1 (BF 14 of Table 8.7),
which points at more complex regulations.
5. 5.
JUNB.out.on, SMAD7.out.on, NFKB1.out.off $\rightarrow$ NFKB1.in.on,
JUNB.in.off, MMP13.in.off, SMAD7.in.off.
Despite a confidence 0/4, the complicated, little expressive implication has
been accepted. The latter fact is reflected in the small support of 4/72.
6. 6.
JUNB.out.off, MMP13.out.off, NFKB1.out.off, SMAD7.out.off $\rightarrow$
JUNB.in.on, MMP13.in.off, NFKB1.in.off, SMAD7.in.on.
Accepted as before.
7. 7.
SMAD7.in.on, NFKB1.out.off $\rightarrow$ JUNB.out.off.
Accepted with confidence 6/7.
8. 8.
SMAD7.in.on, JUNB.out.on, SMAD7.out.on $\rightarrow$ NFKB1.out.on.
Accepted with confidence 12/13.
9. 9.
SMAD7.in.off $\rightarrow$ JUNB.in.off, MMP13.in.off.
The rule was accepted with confidence 32/32, i.e. it holds as a strict
implication also in the data. Like temporal rule 4, it underlines the
interdependency of TNF$\alpha$ and TGF$\beta$1 signaling.
10. 10.
NFKB1.in.on, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
The rule should be rejected by reason of a small support (5/72) and confidence
(0/5). As counterexample 0010 1000 was chosen. However, as several other
counterexamples, it violates previously accepted implications, thus the rule
has been accepted in this first run of exploration.
11. 11.
NFKB1.in.on, JUNB.out.off, MMP13.out.off, SMAD7.out.off $\rightarrow$
NFKB1.out.on.
Accepted with 7/12.
12. 12.
NFKB1.in.off, NFKB1.out.off $\rightarrow$ JUNB.out.off.
The confidence is 5/9, but the implication is accepted as plausible: JUNB
depends on NFKB1 (BF 6); if it remains off, then also JUNB.
13. 13.
NFKB1.in.off, JUNB.out.on, SMAD7.out.on $\rightarrow$ NFKB1.out.on.
The implication is rejected with confidence 2/6 and due to an almost inverted
premise compared to 11, but the same conclusion. Again, counterexample 3 (1001
1001) and others violate implication 12.
At this point, I restarted attribute exploration again with counterexamples
1-3, since the last two examples were judged to be important and should not be
omitted. They only had become obvious at this later point of exploration, but
the result should not depend on the order of the attributes.
1. 1.
SMAD7.out.off $\rightarrow$ MMP13.out.off.
Subsumed by implication 2 of the first run and accepted with confidence 30/30
(remember that MMP13 was always off in the observations).
2. 2.
MMP13.out.on $\rightarrow$ SMAD7.out.on.
Accepted with support 0 (identical to implication 3 of the first exploration
run).
3. 3.
NFKB1.out.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.out.off.
The confidence is not high (13/20) and the implication contradicts BF 6, if a
stable expression of NFKB1 and SMAD7 is given over several time steps. Hence
counterexample 4 (0000 1010) was introduced.
4. 4.
MMP13.out.off, NFKB1.out.off, SMAD7.out.off $\rightarrow$ MMP13.in.off.
Accepted by reason of the MMP13 measurements.
5. 5.
JUNB.out.on, NFKB1.out.off $\rightarrow$ MMP13.in.off.
Ditto.
6. 6.
JUNB.out.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.in.off,
MMP13.in.off, SMAD7.in.off.
Accepted with confidence 3/7, since I did not want to contradict such a
complicated, possibly artificial implication. The complexity should not be
augmented by an unreliable counterexample.
7. 7.
SMAD7.in.on, MMP13.out.on, NFKB1.out.off, SMAD7.out.on $\rightarrow$
JUNB.out.off.
Complicated and accepted.
8. 8.
SMAD7.in.on, MMP13.out.off, SMAD7.out.off $\rightarrow$ JUNB.out.off.
Accepted with confidence 14/18.
9. 9.
SMAD7.in.on, JUNB.out.on $\rightarrow$ SMAD7.out.on.
The implication is a restriction of rule 4 in the first run and was accepted
with confidence 13/17. It is unanticipated by the additional reason that JUNB
and SMAD7 are regulated differently. In the adapted network (Table 8.7), the
only common influencing genes are SMAD4 and SMAD7. Possibly, the inhibition of
both by SMAD7, suggested by SMAD7.in.on, could be the relevant influence on
the transitions supporting the implication, since SMAD4 is always on in the
simulations and the data, except for a single measurement.
10. 10.
SMAD7.in.on, JUNB.out.on, MMP13.out.on, SMAD7.out.on $\rightarrow$
NFKB1.out.on.
Complicated and accepted.
11. 11.
SMAD7.in.off $\rightarrow$ JUNB.in.off, MMP13.in.off.
The implication is identical to rule 9 of the first run and relates to the
inhibition of TGF$\beta$1 signaling via SMAD7 and the activation of
TNF$\alpha$ signaling via JUNB.
12. 12.
NFKB1.in.on, MMP13.out.off, NFKB1.out.on, SMAD7.out.off $\rightarrow$
JUNB.out.off.
Confidence 7/12, accepted as complicated.
13. 13.
NFKB1.in.on, JUNB.out.on, NFKB1.out.on $\rightarrow$ SMAD7.out.on.
Accepted with confidence 23/28, similar to implication 9.
14. 14.
NFKB1.in.on, JUNB.out.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
The restriction of implication 10 in the first run was accepted despite
confidence 0/5. There are few contradictory observations, and an antagonistic
expression is plausible by the BF 6, 12 and 14: JUNB and NFKB1 are activated
by NFKB1, SMAD7 is not. The implication constrains the coregulation of JUNB
and SMAD7 found in the previous and other implications. However, it remains
insecure and should be investigated by further experiments.
15. 15.
3 implications related to NFKB1 are accepted as complicated.
16. 18.
MMP13.in.on $\rightarrow$ SMAD7.in.on.
No support, accepted, similarly the next implication with MMP13.
17. 20.
7 further implications related to MMP13 are accepted, since they are less
expressive and a data control is not possible.
18. 27.
JUNB.in.on $\rightarrow$ SMAD7.in.on.
This again is a noteworthy, clear implication with highest confidence 38/38.
It underlines the discovered coregulation of the two genes (compare 9, 11 and
13, but also 14).
19. 28.
JUNB.in.off, JUNB.out.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on.
Confidence 0/4, but the number of counterexamples in the data is not
sufficient. The biological meaning is similar to implication 14, but this time
it might be considered as a hint on a still unknown influence of JUNB on its
own expression and that of NFKB1 and SMAD7 (where BF 14, however, reflects
inhibition).
20. 29.
SMAD7.in.on, JUNB.in.off, NFKB1.out.off $\rightarrow$ SMAD7.out.on,
JUNB.out.off.
No support in the data, accepted.
21. 30.
The 7 last implications concerning JUNB, together with other genes, are
accepted as too complicated or because of a small support in the set of
observated transitions.
The two rules with highest support in the resulting stem base relate MMP13
expression to SMAD7 upregulation, in the input and the output state:
$\displaystyle<104>\>$
$\displaystyle\>\text{MMP13.out.on}\rightarrow\>\text{SMAD7.out.on}$
$\displaystyle<89>\>$
$\displaystyle\>\text{MMP13.in.on}\rightarrow\>\text{SMAD7.in.on}$
In contrast, there is no implication with MMP13.on in the conclusion (already
for the simulated transitions). This reflects the weak empirical evidence for
MMP13 expression after Tgf$\beta$1 stimulation, which in turn is the reason
why there were no counterexamples to the two rules in the observed data. The
high support originates from the literature based prediction of expression
after Tgf$\beta$1 stimulation. It was no decision criterion during the
exploration aiming at qualitative, not quantitative relations. The rules can
be interpreted as follows: In the rare cases of MMP13 upregulation, it is
expected to be coregulated with SMAD7, but no positive conditions for its
expression are found.
## Chapter 9 Conclusion and outlook
In this thesis, an FCA framework for the description and analysis of discrete
processes was developed and investigated. The knowlege bases generated by
attribute exploration suppport automatic reasoning. So it is worthwhile to
sketch connections to DL, where fast reasoning software exists. The second
section of the present chapter outlines possibilities of solving open
mathematical and logical questions. Finally, the biological applications will
be discussed, in particular the study of the ECM network related to rheumatic
diseases.
### 9.1 Transfer to Description Logics
First I will discuss how our approach may be expressed in a standard DL and in
$\mathcal{TDL}-Lite_{Bool}$, a temporal extension by [15]. I will point at
advantages of remaining within the general framework of FCA, instead of using
the temporal expressivity of DL and attribute exploration adapted to the
construction of DL knowledge bases. Furthermore, reasons will be given to
concentrate on specific, practically relevant parts of temporal logic. I will
give hints to the embedding of the syntax and semantics given by the defined
formal contexts and their stem bases into $\mathcal{EL}$ (a DL designed for
reasoning about ontologies) and $\mathcal{TDL}-Lite_{Bool}$. It is out of the
coverage of this study to search for a DL corresponding to the expressivity of
the proposed formal contexts, not to mention its definition. However, we see a
potential of the integration of our ideas into a DL and of adapting attribute
exploration accordingly.
A transition or transitive context (analogously and easier a state context)
may be translated into a DL like $\mathcal{EL}$ as follows:
* •
The objects are elements of the model domain $\Delta:=S$ (states).
* •
Attributes $m\in M$ of the input states are considered as concept names
$C_{m}\in N_{C}$.
* •
Attributes of the output states are given by concept descriptions with the
role $n\in\mathcal{N}_{r}$ (transition to the next state of a path) or
$t\in\mathcal{N}_{r}$ (transition to an arbitrary subsequent state).
Then a formal context similar to [19, p. 159] will be defined. It corresponds
to a deterministic transition context $\mathbb{K}_{t}$ (compare Table 9.1 with
Table 4.3). Nondeterminism cannot be distinguished syntactically from a
deterministic transition to a state with all alternative attributes (the first
3 lines in Table 9.1 collapse), only semantically by the formal context. As in
Table 4.3, the states are labelled by the path they belong to, e.g.
$s_{00},s_{01},s_{02}$ are the initial states of the Linux driver installation
and the Windows process with ndiswrapper or driver.windows as next steps,
respectively. (At $s_{21}$ the two Windows paths coincide.)
| $driverLinux$ | $ndiswrapper$ | $driverWindows$ | $connection$ | $\exists\,n.driverLinux$ | $\exists\,n.ndiswrapper$ | $\exists\,n.driverWindows$ | $\exists\,n.connection$
---|---|---|---|---|---|---|---|---
$s_{00}$ | | | | | $\times$ | | |
$s_{01}$ | | | | | | $\times$ | |
$s_{02}$ | | | | | | | $\times$ |
$s_{10}$ | $\times$ | | | | $\times$ | | | $\times$
$s_{11}$ | | $\times$ | | | | $\times$ | $\times$ |
$s_{12}$ | | | $\times$ | | | $\times$ | $\times$ |
$s_{20}$ | $\times$ | | | $\times$ | $\times$ | | | $\times$
$s_{21}$ | | $\times$ | $\times$ | | | $\times$ | $\times$ | $\times$
$s_{31}$ | | $\times$ | $\times$ | $\times$ | | $\times$ | $\times$ | $\times$
Table 9.1: A transition context $\mathbb{K}_{t}$ in the language of
description logics.
A transitive context $\mathbb{K}_{tt}$ might be translated by means of the
role $t$. Then the information – expressed by $n\in\mathcal{N}_{r}$ – related
to individual subsequent states is lost. Instead, $\exists t.C_{m},\,m\in M$
indicates if a reachable state is in the extent of $\\{m\\}$ in
$\mathbb{K}_{s}$ or, in the language of DL, if $C_{m}$ is interpreted by the
respective reachable state. Thus, it makes sense to combine a
DL-$\mathbb{K}_{t}$ and a DL-$\mathbb{K}_{tt}$ (Table 9.2).
| $driverLinux$ | $ndiswrapper$ | $driverWindows$ | $connection$ | $\exists\,n.driverLinux$ | $\exists\,n.ndiswrapper$ | $\exists\,n.driverWindows$ | $\exists\,n.connection$ | $\exists\,t.driverLinux$ | $\exists\,t.ndiswrapper$ | $\exists\,t.driverWindows$ | $\exists\,t.connection$
---|---|---|---|---|---|---|---|---|---|---|---|---
$s_{00}$ | | | | | x | | | | x | | | x
$s_{01}$ | | | | | | x | | | | x | x | x
$s_{02}$ | | | | | | | x | | | x | x | x
$s_{10}$ | x | | | | x | | | x | x | | | x
$s_{11}$ | | x | | | | x | x | | | x | x | x
$s_{12}$ | | | x | | | x | x | | | x | x | x
$s_{20}$ | x | | | x | x | | | x | x | | | x
$s_{21}$ | | x | x | | | x | x | x | | x | x | x
$s_{31}$ | | x | x | x | | x | x | x | | x | x | x
Table 9.2: The apposition $\mathbb{K}_{t}\mid\mathbb{K}_{tt}^{out}$ of a
transition context and the output part of a transitive context in the language
of DL.
A transitive context is more expressive than its DL equivalent. Because the
objects of the context 9.2 are states, not transitions, only the information
regarding the next and any reachable state is kept. The following implication
of $\mathbb{K}_{tt}$ (p. 5.1) expresses: If the wrapper module and the driver
are and remain installed (at a second time point), then the connection data
has to be (is) entered:
ndiswrapper.in, driver.windows.in, ndiswrapper.out,
driver.windows.out $\rightarrow$ connection.out
Yet in the DL context of Table 9.2, already the less meaningful implication
holds $\top\rightarrow\exists t.connection$. It turns out that the role $t$
corresponds better to the operator $\operatorname{ev}$ of a temporal context.
The semantics for the $\mathbb{K}_{tmp}$ attributes from CTL is given by
$\mathbb{K}_{t}$, for $\Diamond Fm$, $\Box Gm$ and $\Box\neg Fm$ also by
$\mathbb{K}_{tt}$ (Remark 4.5.3). Two of the $\mathbb{K}_{tmp}$ attributes are
equivalent to the following concept descriptions:
$\displaystyle\Diamond Fm$ $\displaystyle\cong\exists\,t.C_{m}$
$\displaystyle\Box\neg Fm$ $\displaystyle\cong\neg\exists\,t.C_{m}$
If concept descriptions relate to states indexed by the path they belong to,
as in Table 9.2, the following temporal operator may also be expressed:
$\Diamond\neg Fm\cong\neg\exists\,t.C_{m}.$
S. Rudolph [87] and F. Baader / F. Distel [19] deal with a potentially
infinite attribute set, caused by nesting of roles like $\exists r.\exists
r.\cdots$ . In the case of cyclic concept descriptions, e.g., the potential
role depth (number of nested roles) is infinite. If such DL concepts are
interpreted temporally, they express properties of cyclic time developments.
S. Rudolph investigated attribute exploration of formal concepts with
increasing role depth (in the more expressive DL $\mathcal{FLE}$) and showed
the existence of a termination condition for finite models. However, there are
computational problems, since the number of attributes grows exponentially
with the depth. F. Distel [18] proved that a finite basis exists for the set
of all $\mathcal{EL}$-implications (or GCIs) holding in a finite model. In
[19], a single formal context with relational attributes has been defined; its
attribute exploration provides an algorithm to explicitly compute the
implicational base.
As mentioned before, the DL transition context of Table 9.1 is derived from
such a context. However, we restricted our framework to simple relational
attributes (role depth one). Since temporal attributes can also refer to
infinite paths, we do not need nested roles in this case. Roles of depth two
or three may still be meaningful to an expert, but the purpose of the present
work was not to develop a general framework for the conceptual exploration of
relations. If the number of relational attributes remains restricted and
almost all attributes occur in at least one of the output states, the set of
attributes $\exists r.C_{m},C_{m}\in\mathcal{N}_{C}$ may be fixed in advance.
Then general attribute exploration can be applied, and no algorithm handling a
growing set of attributes is needed as designed in [19].
Thus, standard FCA algorithms could be used. Nevertheless, our approach is
generalisable to more complicated relational attributes; also Relational
Concept Analysis (RCA) [45] may be applied. Inversely, attribute exploration
software using fast DL reasoners could be used for our special formal
contexts.
In the approach of [15] (see Section 3.2.3), a flow of time is represented by
$\mathbb{N}_{0}$, and the interpretation consists in a family of succeeding
situations $(\mathcal{I}_{n})_{n\in\mathbb{N}_{0}}$ defined by the
interpretation function $\mathcal{I}$. According to [18, p. 48 f.], the family
can be seen as a set of temporally changing models with a constant domain
$\Delta$ for the concept description language $(\mathcal{L},\mathcal{I})$
given by $\mathcal{TDL}-Lite_{Bool}$, and each model correponds to a formal
context. This is a more expressive framework than ours and may not easily be
translated into the defined formal contexts. On the other hand,
$\mathcal{TDL}-Lite_{Bool}$ is based on the linear time logic LTL, whereas
attributes of the temporal context are definable by the nondeterministic,
branching time logic CTL.
A state context $\mathbb{K}_{s}$ can be translated into
$\mathcal{TDL}-Lite_{Bool}$ as follows:
* •
The domain $\Delta$ does not correspond to the object set, but to the
attribute set $M$ of $\mathbb{K}_{s}$.
* •
The object names $a_{i}$ are in one-to-one correspondence to the attributes by
$a_{i}^{\mathcal{I}(n)},\>i\in\\{0,...,|M|-1\\},n\in\mathbb{N}_{0}$.
* •
The object intent of a state $s\in S$ could be identified with a single
concept name $A_{0}$. Then $A_{0}^{\mathcal{I}(n)}:=s^{\prime}$, but the whole
syntactic information concerning attributes holding in a state would be lost.
A better alternative is to define $|M|$ DL concepts $A_{i}$ by
$A_{i}^{\mathcal{I}(n)}:=a_{i}^{\mathcal{I}(n)}$, if
$a_{i}^{\mathcal{I}(n)}\in s^{\prime}$, else
$A_{i}^{\mathcal{I}(n)}:=\emptyset$. Then the assertion $A_{i}(a_{j})$ holds,
if $i=j$ and $a_{j}^{\mathcal{I}(n)}$ is contained in the intent of $s$.
* •
Then every line of a state context corresponds to an interpretation
$\mathcal{I}(n)_{C}$ without role names:
$\mathcal{I}(n)_{C}:=(\Delta,a_{0}^{\mathcal{I}(n)},...,A_{0}^{\mathcal{I}(n)},...),$
where the object intent is given by the current interpretation of the DL
concepts $A_{i}$. This is a bijection only if the state context is not
clarified.
This translation defines a concept description language
$(\mathcal{L},\mathcal{I})$ of which $\mathbb{K}_{s}$ is a model. GCIs
$C_{1}\sqsubseteq C_{2}$ hold iff $C_{1}^{\mathcal{I}(n)}\rightarrow
C_{2}^{\mathcal{I}(n)}$ holds in $\mathbb{K}_{s}$ for all
$n\in\mathbb{N}_{0}$, with $C_{1}:=A_{i}\sqcap...\sqcap
A_{j},\>C_{2}:=A_{i}\sqcap...\sqcap A_{j}\sqcap...\sqcap A_{k}$, for
$i<j<k\leq|M-1|$ (compare [18, p. 48-50]).
It is only mentioned here that the interpretation
$P_{i}^{\mathcal{I}(n)}\in\Delta\times\Delta$ of local roles $P_{i}$ may be
given by a transition context.
[15] investigate also a weak temporalisation of $\mathcal{EL}$. In
$\mathcal{TL}_{\mathcal{EL}}$ existential role restrictions are admitted, but
no local roles nor negation, and temporal operators are restricted to $F$
(eventually) for concept constructors and $G$ (always) for formula.
$C:=\top\mid A_{i}\mid C_{1}\sqcap C_{2}\mid\exists T_{i}.C\mid F\,C$
In this logic GCIs are satisfiable in the most general model (where all
concepts and roles are interpreted by the whole domain at every time point),
but it is undecidable whether a GCI is a consequence of a finite set of GCIs
[15, Theorem 11]. In our “pure” FCA framework however, we are safe that such
problems do not occur: For each formal context the specific implications are
decidable, even in linear time relative to the size of the stem base.
A detailed investigation of decidability and complexity issues of the
presented approach in comparison to different DLs is out of reach of this
thesis.
### 9.2 Open mathematical and logical questions
We have established the foundation to exploit manifold mathematical results of
FCA for the analysis of gene expression dynamics and of discrete temporal
transitions in general. In Chapter 6 mathematical questions related to
background knowledge and to the mutual dependency of the defined formal
contexts and their stem bases were solved. While it was not feasible to
develop a complete rule calculus for a large class of temporal implications,
the method was applied to special cases and possibilities of tackling the
general problem were discussed.
A related question should be further analysed: How can attribute exploration
be split into partial problems for these special contexts? For instance, one
could focus on a specific set of genes (Remark 5.1.1) or temporal attributes
first, which is understandable as a scaling (p. 4.3). Then, the decomposition
theory of concept lattices will be useful, which permits an elegant
description by means of the corresponding formal contexts [41, Chapter 4].
The price of the logical completeness of attribute exploration is its
computational complexity. Computation time strongly depends on the logical
structure of the context, and there exist cases where the size of the stem
base is exponential in the size of the input [67]. It was proved recently that
the basic step – recognising whether a subset of attributes is a pseudo-intent
– is coNP-complete [21]. However, deriving an implication from the stem base
is possible in linear time, related to the size of the base, and the Prolog
queries in Section 7.4 were very fast.
In addition to the integration of background knowledge, attribute exploration
can be simplified and shortened, if implications are decided without the
necessity to generate all possible transitions. For that purpose, model
checking could be a promising approach [33], [29], or the structural and
functional analysis of Boolean networks by an adaptation of metabolic network
methods in [66]. In this study, determining activators or inhibitors
corresponds to the kind of rules found by our method, and logical steady state
analysis indicates which species can be produced from the input set and which
not. Another direction of research would be to conclude dynamical properties
of Boolean networks from their structure and the transition functions, for
instance by regarding them as polynomial dynamical systems over finite fields
and by exploiting theoretical work in the context of cellular automata [70,
Section 4 and 6].
### 9.3 Assessment of the biological applications
The analyses in the ECM study (Chapter 8) were based on literature data
refering to healthy human SFB. These findings were fine-tuned and adapted to
gene expression time course data triggered by TGF$\beta$1 and TNF$\alpha$ in
SFB from RA and OA patients. Both the assembly of previous knowledge and the
adaptation of the Boolean functions gave detailed insights into disease-
related regulatory processes. To the best of our knowledge, this is the first
dynamical model of ECM formation and degradation by human SFB.
Manual adaptation of the network may be superseded by algorithms of network
inference [50] in order to achieve a scalability to larger networks. Then,
however, knowledge and data are integrated by criteria fixed in advance and
hardly controllable in their effects, not by flexible and open expert
decisions. Moreover, the relation of the validation by network inference and
by computing the stem base should be clarified, or an expert centered
attribute exploration should be applied.
The fidelity of the obtained temporal rules was reinforced by the comparison
of simulated and observed time series data, first manually, then automatically
by computing the stem base of the combined transitive context. One of the
strengths of the FCA method applied here is its ability to give a complete,
but minimal representation of a data set. This complete overview of temporal
rules enabled us to find new relationships. The most unexpected result is the
expression of TNF at some time points following TNF$\alpha$ stimulation,
whereas it is commonly assumed that SFB do not express TNF [27], [83].
Similarly, our experimental data as well as our simulation results support
MMP9 expression in SFB thus corroborating the majority of the literature
regarding expression of this protease [92], [114]. Here, it is important to
note that a contamination of the SFB population with macrophages (potentially
contributing to MMP9 production) can be excluded due to the SFB isolation
protocol, resulting in a pure SFB population [116]. We also found that MMP1
was induced in the absence of FOS after TNF$\alpha$ stimulation, whereas MMP13
was not expressed despite reports about its induction by NF-$\kappa$B, JUN or
FOS. These facts indicate that the regulation of MMP expression may be more
diverse than presently known and that it still represents a relevant research
target to elucidate the role of SFB in the pathophysiology of rheumatic
diseases.
Concerning the formation of collagen type I fibres by COL1A1 and COL1A2
proteins following the stimulation with TGF$\beta$1, a constitutive expression
of COL1A2 was calculated. Based on these data, COL1A1 has to be regarded as
the critical switch for the formation of collagen I. In contrast, the
corresponding literature generally postulates a co-regulation of both genes,
due to similarities in their promoters [58], [28]. This difference suggests
that the regulation of COL1A1 and COL1A2 may not have been fully elucidated so
far possibly pointing at COL1A1 as a more promising target for the exploration
of fibrosis.
Our analyses also show that TNF$\alpha$-induced signalling predominantly
results in the activation of ETS1 and NFKB1, whereas TGF$\beta$1-related
signal transduction is ultimately regulated via proteins of the SMAD family.
Defined intervention addressing these signalling modules, alone or in
combination with established therapies targeting TNF$\alpha$ (e.g.
etanercept), may therefore improve the efficiency and outcome of current anti-
rheumatic treatment [12]. Alternatively, the present results may be employed
to define subpopulations of RA patients in characteristic phases of RA (active
inflammatory early versus burnt-out/fibrotic late) and tailor anti-rheumatic
treatment to the particular needs of the respective phase [30].
The semantics of a transitive context implies rules to be valid for the
complete simulated and observed time interval. Thus, besides on co-regulation
and contrary regulation, the focus is on (positive or negative) attributes of
gene expression processes which will always hold after a given class of
states. Since in the ECM study always was restricted to the observation time
of 12 h, meaningful results were obtaned. This type of rules is even more
appropriate in cases of a dramatic, permanent switch of the cellular
behaviour, like for sporulation of B. subtilis (Chapter 7). An exploration of
a temporal context was sketched in Section 5.1 for the computer example. Also
in biological applications it will give supplementary insight, in particular
by the eventually attributes.
Exact logical rules (implications) are a precondition for the uniqueness and
minimality of the stem base, in contrast to association rule mining. Data
discretisation is one possibility of handling biological imprecion and
flexibility and to filter out noise resulting from measurements [71]. The
presented method could be developed further by integration of “blurred”
methods like fuzzy FCA [10] or rough sets [39]. Clustering methods could be
applied for data preprocessing.
Data discretisation inevitably causes loss of information. Carefully
evaluating the method, we tried to keep as much important information as
possible and set low thresholds for differential gene expression. A recently
developed FCA-based method avoids predefined discretisation but computes an
ordered set of interval pattern structures depending on the observed values
[62]. Thus, a data set may be described without loss of information or by
means of any desired granularity.
Because the mathematical framework of FCA is very general and open, many
adaptations of the presented methods are possible. According to the mainstream
of applied FCA research, they should aim at the reduction of complexity as
indicated. Then FCA methods can demonstrate their strength best: a clear
account and visualisation of a conceptual logic. Beside this, multifarious
refinements are possible, according to current approaches of modelling
dynamics within systems biology. For instance, the introduction of more fine-
grained expression levels is possible, e.g. in the sense of qualitative
reasoning [64].
However, despite the rough discretisation to two levels off and on, the
complexity of even relatively small networks such as our ECM network and the
completeness of the attribute exploration algorithm led to a large number of
temporal rules. High support of a rule (often correlated to its simplicity)
was used as an indicator for the most meaningful hypotheses about co-
regulation, mutual exclusion, and/or temporal dependencies not only between
single genes, but also between small sets of (functionally related) genes. The
inspection of the rule sets by experts should be aided by automatic
preselection. Iceberg concept lattices lead to a set of “important” rules with
high support, by taking advantage of the duality between the stem base and the
concept lattice of a formal contexts [93].
We applied also complete and in-depth investigation of a small set of
interesting genes by interactive attribute exploration. Using this procedure
for the knowledge base construction, single rules can be validated manually or
by a supporting computer program, or even new experiments can be suggested.
While for the automatic stem base computation we applied a strong validation
criterion requiring rules to hold for all simulated and observed transitions,
the expert can evaluate thresholds of support and confidence. This may reduce
noise or eliminate measurement errors. Importantly, these decision criteria
can be freely combined with relevant knowledge, and the method does not depend
on mathematical problems of association rule mining, since the expert decides
about strict implications. As a main result, several temporal rules were found
and confirmed, which express a coregulation of JUNB and SMAD7, i.e. an
activation of TNF$\alpha$ signalling together with an inhibition of
TGF$\beta$1 signalling (partly also for the inverse case). This is
biologically plausible in general, but was surprising regarding the Boolean
functions. Thus, by simulation, data and expert validation we discovered a
temporal invariant of the network.
Expert decisions are not always obvious. During the first run of the
exploration it turned out that too restrictive implications had been accepted
and therefore different plausible counterexamples could not be introduced. In
the small example, restarting the exploration with the counterexamples was no
major problem, but it would be helpful to keep correct implications. This is
possible if the implications accepted after the first error are checked again
and used as background knowledge for the further exploration [20, p. 14].
Corresponding to the stem base, but with the supplementary information of
extents (states or transitions), Hasse diagrams of the investigated formal
contexts like in Figure 7.2 give detailed (albeit less compact) insight into
gene regulatory processes. They should be generated for small subsets of
genes. Nested line diagrams as implemented in ToscanaJ [7] offer the
possibility to investigate achievable subsets of genes and to combine two of
them at a time according to a chain of questions.
Combining two well-developed algebraic, discrete and logical methods – Boolean
network construction and FCA – it was possible to include human expert
knowledge in all different phases (assembly of the network, adjustment to the
data, choice of relevant temporal rules and interactive attribute
exploration), with the exception of the challenging data discretisation step.
“Digital” and “analog” thinking are combined by our approach. Bioinformatic
algorithms are often complicated and – even if carefully evaluated – can
produce hardly interpretable results. Against this tendency, we wanted to
avoid uncritical trust in the objectivity of a method and let the main task of
interpretation with human experts. Hence, the main biological results are not
only single findings, but within the interdisciplinary collaboration the
medical, biological and bioinformatic scientists as well as myself were
inspired to complex reflections about regulatory processes. We got new
insights regarding existing knowledge and data, which are verifiable by
further experiments.
## Bibliography
* [1] Bibliosphere. http://www.genomatix.de/products/BiblioSphere/
BiblioSpherePE1.html.
* [2] Concept Explorer. http://sourceforge.net/projects/conexp/.
* [3] NCBI Gene Expression Omnibus. http://www.ncbi.nlm.nih.gov/geo/.
* [4] Pathway studio. http://www.ariadnegenomics.com.
* [5] Pubmed. http://www.ncbi.nlm.nih.gov/pubmed/.
* [6] R. A free software environment for statistical computing and graphics. http://www.r-project.org/.
* [7] ToscanaJ. An FCA suite. http://toscanaj.sourceforge.net/.
* [8] AM Abeles and MH Pillinger. The role of the synovial fibroblast in rheumatoid arthritis: cartilage destruction and the regulation of matrix metalloproteinases. Bull NYU Hosp Jt Dis, 64:20–24, 2006.
* [9] B. Alberts et al. Molecular Biology of the Cell, with CD-ROM. Taylor & Francis, London, 5th rev. ed. edition, 2008.
* [10] Christina Alcalde, Ana Burusco, and Ramón Fuentes-González. L-fuzzy concepts and linguistic variables in knowledge acquisition processes. In Marzena Kryszkiewicz and Sergei Obiedkov, editors, CLA-2010. Proceedings of the 7th International Conference on Concept Lattices and Their Applications, Sevilla, Spain, October 19-21, 2010., pages 50–59. University of Sevilla, 2010.
* [11] R Altman, E Asch, D Bloch, G Bole, D Borenstein, K Brandt, W Christy, TD Cooke, R Greenwald, and M Hochberg. Development of criteria for the classification and reporting of osteoarthritis. classification of osteoarthritis of the knee. diagnostic and therapeutic criteria committee of the american rheumatism association. Arthritis Rheum, 29:1039–1049, 1986.
* [12] ET Andreakos, BM Foxwell, FM Brennan, RN Maini, and M Feldmann. Cytokines and anti-cytokine biologicals in autoimmunity: present and future. Cytokine Growth Factor Rev, 13:299–313, 2002.
* [13] P Angel and M Karin. The role of Jun, Fos and the AP-1 complex in cell-proliferation and transformation. Biochim Biophys Acta, 1072:129–157, 1991.
* [14] FC Arnett, SM Edworthy, DA Bloch, DJ McShane, JF Fries, NS Cooper, LA Healey, SR Kaplan, MH Liang, and HS Luthra. The American Rheumatism Association 1987 revised criteria for the classification of rheumatoid arthritis. Arthritis Rheum, 31:315–324, 1988.
* [15] A. Artale, R. Kontchakov, C. Lutz, F. Wolter, and M. Zakharyaschev. Temporalising tractable description logics. In Proceedings of the Fourteenth International Symposium on Temporal Representation and Reasoning. IEEE Computer Society Press, 2007.
* [16] H Asahara, K Fujisawa, T Kobata, T Hasunuma, T Maeda, M Asanuma, N Ogawa, H Inoue, T Sumida, and K Nishioka. Direct evidence of high DNA binding activity of transcription factor AP-1 in rheumatoid arthritis synovium. Arthritis Rheum, 40:912–918, 1997.
* [17] F. Baader, D. McGuinness, D. Nardi, and P.F. Patel-Schneider. The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, 2003.
* [18] Franz Baader and Felix Distel. A finite basis for the set of $\mathcal{EL}$-implications holding in a finite model. In Raoul Medina and Sergei A. Obiedkov, editors, ICFCA, volume 4933 of Lecture Notes in Computer Science, pages 46–61. Springer, 2008\.
* [19] Franz Baader and Felix Distel. Exploring finite models in the description logic $\mathcal{EL}_{gfp}$. In Sébastien Ferré and Sebastian Rudolph, editors, ICFCA, volume 5548 of Lecture Notes in Computer Science, pages 146–161. Springer, 2009.
* [20] Franz Baader and Baris Sertkaya. Usability issues in description logic knowledge base completion. In Sébastien Ferré and Sebastian Rudolph, editors, ICFCA, volume 5548 of Lecture Notes in Computer Science, pages 1–21. Springer, 2009.
* [21] Mikhail A. Babin and Sergei O. Kuznetsov. Recognizing pseudo-intents is coNP-complete. In Marzena Kryszkiewicz and Sergei Obiedkov, editors, CLA-2010. Proceedings of the 7th International Conference on Concept Lattices and Their Applications, Sevilla, Spain, October 19-21, 2010., pages 294–301. University of Sevilla, 2010.
* [22] A Barchowsky, D Frleta, and MP Vincenti. Integration of the NF-kappaB and mitogen-activated protein kinase/AP-1 pathways at the collagenase-1 promoter: divergence of IL-1 and TNF-dependent signal transduction in rabbit primary synovial fibroblasts. Cytokine, 12:1469–1479, 2000.
* [23] I Berger and Y Shaul. c-Fos antagonizes the junD gene positive autoregulatory loop; a novel c-Fos role in promoter switching. Gene, 211:375–382, 1998.
* [24] BM Bolstad, RA Irizarry, M Astrand, and TP Speed. A comparison of normalization methods for high density oligonucleotide array data based on variance and bias. Bioinformatics, 19:185–193, 2003.
* [25] V Bours, J Villalobos, PR Burd, K Kelly, and U Siebenlist. Cloning of a mitogen-inducible gene encoding a $\kappa$B DNA-binding protein with homology to the rel oncogene and to cell-cycle motifs. Nature, 348:76–80, 1990.
* [26] Peter Burmeister. Formal Concept Analysis with ConImp: Introduction to the Basic Features. University of Technology, Darmstadt. http://www.mathematik.tu-darmstadt.de/burmeister/ConImpIntro.pdf, 2003.
* [27] GR Burmester, B Stuhlmuller, G Keyszer, and RW Kinne. Mononuclear phagocytes and rheumatoid synovitis. Mastermind or workhorse in arthritis? Arthritis Rheum, 40:5–18, 1997.
* [28] C Buttner, A Skupin, and EP Rieber. Transcriptional activation of the type I collagen genes COL1A1 and COL1A2 in fibroblasts by interleukin-4: analysis of the functional collagen promoter sequences. J Cell Physiol, 198:248–258, 2004.
* [29] Nathalie Chabrier-Rivier et al. Modeling and querying biomolecular interaction networks. Theoretical Computer Science, 325(1):25–44, 2004.
* [30] PG Conaghan, MJ Green, and P Emery. Established rheumatoid arthritis. Baillieres Best Pract Res Clin Rheumatol, 13:561–575, 1999.
* [31] H. de Jong et al. Qualitative simulation of the initiation of sporulation in Bacillus subtilis. Bulletin of Mathematical Biology, 66:261–299, 2004.
* [32] E. Dimitrova, J. McGee, and R. Laubenbacher. Discretization of time course data. http://polymath.vbi.vt.edu/discretization/DimitrovaMcGeeLaubenbacher.pdf, 2005\.
* [33] J. Esparza. Model checking using net unfoldings. Sci. Comput. Programm., 23:151–195, 1994.
* [34] GS Firestein and AM Manning. Signal transduction and transcription factors in rheumatic disease. Arthritis Rheum, 42:609–621, 1999.
* [35] GS Firestein, MM Paine, and BH Littman. Gene expression (collagenase, tissue inhibitor of metalloproteinases, complement, and HLA-DR) in rheumatoid arthritis and osteoarthritis synovium. Quantitative analysis and effect of intraarticular corticosteroids. Arthritis Rheum, 34:1094–1105, 1991.
* [36] B. Ganter, G. Stumme, and R. Wille. Formal Concept Analysis – Foundations and Applications, volume 3626 of LNAI. Springer, Heidelberg, 2005.
* [37] Bernhard Ganter. Attribute exploration with background knowledge. Theor Comput Sci, 217(2):215–233, 1999.
* [38] Bernhard Ganter. Contextual attribute logic of many-valued attributes. In B. Ganter, S. Stumme, and R. Wille, editors, Formal Concept Analysis – Foundations and Applications, number 3626 in LNAI, pages 101–113, Heidelberg, 2005. Springer.
* [39] Bernhard Ganter and Christian Meschke. A formal concept analysis approach to rough data tables. In RSFDGrC, pages 117–126, 2009.
* [40] Bernhard Ganter and Sebastian Rudolph. Formal concept analysis methods for dynamic conceptual graphs. In ICCS 2001, LNAI 2120, pages 143–156, Heidelberg, 2001. Springer.
* [41] Bernhard Ganter and Rudolf Wille. Formal Concept Analysis – Mathematical Foundations. Springer, Heidelberg, 1999.
* [42] F Gaultier, A Foucault-Bertaud, E Lamy, AL Ejeil, SM Dridi, N Piccardi, A Piccirilli, P Msika, G Godeau, and B Gogly. Effects of a vegetable extract from lupinus albus (LU105) on the production of matrix metalloproteinases (MMP1, MMP2, MMP9) and tissue inhibitor of metalloproteinases (TIMP1, TIMP2) by human gingival fibroblasts in culture. Clin Oral Investig, 7:198–205, 2003.
* [43] EM Gravallese, JM Darling, AL Ladd, JN Katz, and LH Glimcher. In situ hybridization studies of stromelysin and collagenase messenger RNA expression in rheumatoid synovium. Arthritis Rheum, 34:1076–1084, 1991.
* [44] H. P. Gumm. Universelle Coalgebra. In Thomas Ihringer, Allgemeine Algebra, pages 155–208. Heldermann, 2003.
* [45] Mohamed Rouane Hacene, Marianne Huchard, Amedeo Napoli, and Petko Valtchev. A proposal for combining formal concept analysis and description logics for mining relational data. In Sergei O. Kuznetsov and Stefan Schmidt, editors, ICFCA, volume 4390 of Lecture Notes in Computer Science, pages 51–65. Springer, 2007.
* [46] H Hacker and M Karin. Regulation and function of IKK and IKK-related kinases. Sci STKE, 2006:re13, 2006.
* [47] Z Han, DL Boyle, AM Manning, and GS Firestein. AP-1 and NF-kappaB regulation in rheumatoid arthritis and murine collagen-induced arthritis. Autoimmunity, 28:197–208, 1998.
* [48] ML Handel and L Girgis. Transcription factors. Best Pract Res Clin Rheumatol, 15:657–675, 2001.
* [49] JA Hartigan and MA Wong. A k-means clustering algorithm. Applied Statistics, 28:100–108, 1979.
* [50] Michael Hecker, Sandro Lambeck, Susanne Toepfer, Eugene van Someren, and Reinhard Guthke. Gene regulatory network inference: Data integration in dynamic models – a review. BioSystems, 96:86–103, 2009.
* [51] JM Hernandez, DH Floyd, KN Weilbaecher, PL Green, and K Boris-Lawrie. Multiple facets of junD gene expression are atypical among AP-1 family members. Oncogene, 27:4757–4767, 2008.
* [52] SI Hirai, RP Ryseck, F Mechta, R Bravo, and M Yaniv. Characterization of junD: a new member of the jun proto-oncogene family. Embo J, 8:1433–1439, 1989.
* [53] Jens Hollunder, Maik Friedel, Andreas Beyer, Christopher T. Workman, and Thomas Wilhelm. DASS: efficient discovery and p-value calculation of substructures in unordered data. Bioinformatics, 23(1):77–83, 2007.
* [54] John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison Wesley, Upper Saddle River, N.J., 2nd edition, 2001.
* [55] LC Huber, O Distler, I Tarner, RE Gay, S Gay, and T Pap. Synovial fibroblasts: key players in rheumatoid arthritis. Rheumatology (Oxford), 45:669–675, 2006.
* [56] Thomas Ihringer. Allgemeine Algebra. Heldermann, Lemgo, 2003.
* [57] E Karouzakis, M Neidhart, RE Gay, and S Gay. Molecular and cellular basis of rheumatoid joint destruction. Immunol Lett, 106:8–13, 2006.
* [58] G Karsenty and RW Park. Regulation of type I collagen genes expression. Int Rev Immunol, 12:177–185, 1995.
* [59] S.A. Kauffman. The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York, 1993.
* [60] Stuart Kauffman, Carsten Peterson, Björn Samuelsson, and Carl Troein. Random Boolean network models and the yeast transcriptional network. PNAS, 100(25):14796–14799, 2003.
* [61] Stuart A. Kauffman. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol, 22:437–467, 1969.
* [62] Mehdi Kaytoue, Sergei O. Kuznetsov, Amedeo Napoli, and Sébastien Duplessis. Mining gene expression data with pattern structures in formal concept analysis. Information Sciences, 181(10):1989 – 2001, 2011.
* [63] G Kervizic and L Corcos. Dynamical modeling of the cholesterol regulatory pathway with Boolean networks. BMC Syst Biol, 2:99, 2008.
* [64] Ross D. King, Simon M. Garrett, and George M. Coghill. On the use of qualitative reasoning to simulate and identify metabolic pathways. Bioinformatics, 21(9):2017–2026, 2005.
* [65] RW Kinne, E Palombo-Kinne, and F Emmrich. Activation of synovial fibroblasts in rheumatoid arthritis. Ann Rheum Dis, 54:501–504, 1995.
* [66] S. Klamt. A methodology for the structural and functional analysis of signaling and regulatory networks. BMC Bioinformatics, 7(56), 2006.
* [67] S.O. Kuznetsov and S.A. Obiedkov. Counting pseudo-intents and #P-completeness. In J. Schmid, editor, ICFCA 2006, volume 3874 of LNCS, pages 306–308, Heidelberg, 2006. Springer.
* [68] PF Lambert, MJ Ludford-Menting, NJ Deacon, I Kola, and RR Doherty. The Nfkb1 promoter is controlled by proteins of the Ets family. Mol Biol Cell, 8:313–323, 1997.
* [69] Serge Lang. Algebra. Springer, Heidelberg, 3rd edition, 2002.
* [70] R. Laubenbacher. Algebraic models in systems biology. In H. Anai and K. Horimoto, editors, Algebraic Biology 2005, Proceedings of the First International Conference on Algebraic Biology – Computer Algebra in Biology, pages 33–35, Tokyo, 2005. Universal Academy Press.
* [71] R Laubenbacher and B Stigler. A computational algebra approach to the reverse engineering of gene regulatory networks. J Theor Biol, 229:523–537, 2004.
* [72] S Martin, Z Zhang, A Martino, and JL Faulon. Boolean dynamics of genetic regulatory networks inferred from microarray time series data. Bioinformatics, 23:866–874, 2007.
* [73] SS McCachren. Expression of metalloproteinases and metalloproteinase inhibitor in human arthritic synovium. Arthritis Rheum, 34:1085–1093, 1991.
* [74] M Mizui, Y Isaka, Y Takabatake, Y Sato, H Kawachi, F Shimizu, S Takahara, T Ito, and E Imai. Transcription factor Ets-1 is essential for mesangial matrix remodeling. Kidney Int, 70:298–305, 2006.
* [75] Susanne Motameny, Beatrix Versmold, and Rita Schmutzler. Formal concept analysis for the identification of combinatorial biomarkers in breast cancer. In Raoul Medina and Sergei A. Obiedkov, editors, ICFCA, volume 4933 of Lecture Notes in Computer Science, pages 229–240. Springer, 2008\.
* [76] EP Newberry, D Willis, T Latifi, JM Boudreaux, and DA Towler. Fibroblast growth factor receptor signaling activates the human interstitial collagenase promoter via the bipartite Ets-AP1 element. Mol Endocrinol, 11:1129–1144, 1997.
* [77] Peter Øhrstrøm. Time and logic: A.N. Prior’s formal analysis of temporal concepts. In Sébastien Ferré and Sebastian Rudolph, editors, Formal Concept Analysis, 7th International Conference, ICFCA 2009, Darmstadt, Germany, May 21-24, 2009, Proceedings, volume 5548 of Lecture Notes in Computer Science, pages 66–81, Berlin Heidelberg, 2009. Springer.
* [78] Charles Sanders Peirce. How to make our ideas clear. In C. Hartshorne and P. Weiss, editors, Collected papers, Cambridge / Mass., 1931 - 1935. Harvard University Press.
* [79] SG Pereira and F Oakley. Nuclear factor-kappaB1: regulation and function. Int J Biochem Cell Biol, 40:1425–1430, 2008.
* [80] John L. Pfaltz and Christopher M. Taylor. Closed set mining of biological data. In BIOKDD02: Workshop on Data Mining in Bioinformatics, 2002.
* [81] AE Postlethwaite, MA Holness, H Katai, and R Raghow. Human fibroblasts synthesize elevated levels of extracellular matrix proteins in response to interleukin 4. J Clin Invest, 90:1479–1485, 1992.
* [82] Aviv Regev and Ehud Shapiro. The pi-calculus as an abstraction for biomolecular systems. In G. Ciobanu and G. Rozenberg, editors, Modelling in Molecular Biology, Natural Computing Series. Springer, 2004.
* [83] C Ritchlin. Fibroblast biology. Effector signals released by the synovial fibroblast in arthritis. Arthritis Res, 2:356–360, 2000.
* [84] G Rogler and T Andus. Cytokines in inflammatory bowel disease. World J Surg, 22:382–389, 1998.
* [85] S Ross and CS Hill. How the Smads regulate transcription. Int J Biochem Cell Biol, 40:383–408, 2008.
* [86] R Rossignol, B Faustin, C Rocher, M Malgat, JP Mazat, and T Letellier. Mitochondrial threshold effects. Biochem J, 370:751–762, 2003.
* [87] Sebastian Rudolph. Relational Exploration – Combining Description Logics and Formal Concept Analysis for Knowledge Specification. Universitätsverlag Karlsruhe, December 2006. Dissertation.
* [88] T Schlitt and A Brazma. Current approaches to gene regulatory network modelling. BMC Bioinformatics, 8(Suppl 6):S9, 2007.
* [89] P Shannon, A Markiel, O Ozier, NS Baliga, JT Wang, D Ramage, N Amin, B Schwikowski, and T Ideker. Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Res, 13:2498–2504, 2003.
* [90] JS Smolen and G Steiner. Therapeutic strategies for rheumatoid arthritis. Nat Rev Drug Discov, 2:473–488, 2003.
* [91] L. Jason Steggles, Richard Banks, Oliver Shaw, and Anil Wipat. Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach. Bioinformatics, 23(3):336–343, 2007.
* [92] KM Stuhlmeier. Mepacrine inhibits matrix metalloproteinases-1 (MMP-1) and MMP-9 activation in human fibroblast-like synoviocytes. J Rheumatol, 30:2330–2337, 2003.
* [93] G. Stumme, R. Taouil, Y. Bastide, N. Pasquier, and L. Lakhal. Computing iceberg concept lattices with TITANIC. Data and Knowledge Engineering, 42:189–222, 2002.
* [94] Y Sun, L Wenger, CE Brinckerhoff, RR Misra, and HS Cheung. Basic calcium phosphate crystals induce matrix metalloproteinase-1 through the Ras/mitogen-activated protein kinase/c-Fos/AP-1/metalloproteinase 1 pathway. Involvement of transcription factor binding sites AP-1 and PEA-3. J Biol Chem, 277:1544–1552, 2002.
* [95] Michael Thielscher. FLUX: A logic programming method for reasoning agents. Theory and Practice of Logic Programming, 5(4-5):533–565, 2005\.
* [96] C.A. Tomas et al. DNA array-based transcriptional analysis of asporogenous, nonsolventogenic clostridium acetobutylicum strains SKO1 and M5. J. Bacteriol., pages 4539–4547, 2003.
* [97] F Verrecchia and A Mauviel. Transforming growth factor-beta signaling through the Smad pathway: role in extracellular matrix gene expression and regulation. J Invest Dermatol, 118:211–215, 2002.
* [98] F Verrecchia, M Pessah, A Atfi, and A Mauviel. Tumor necrosis factor-alpha inhibits transforming growth factor-beta/Smad signaling in human dermal fibroblasts via AP-1 activation. J Biol Chem, 275:30226–30231, 2000.
* [99] Hartmut von Hentig. Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozeß. Klett, Stuttgart, 1972.
* [100] JH Ward. Hierarchical grouping to optimize an objective function. J Am Stat Assoc, 58:236–244, 1963.
* [101] Natalia Wegner et al. Peptidylarginine deiminase from Porphyromonas gingivalis citrullinates human fibrinogen and $\alpha$-Enolase. Implications for autoimmunity in rheumatoid arthritis. Arthritis and Rheumatism, 62(9):2662–2672, 2010.
* [102] HG Welgus, JJ Jeffrey, and AZ Eisen. The collagen substrate specificity of human skin fibroblast collagenase. J Biol Chem, 256:9511–9515, 1981.
* [103] Z Werb. Proteinases and matrix degradation. Textbook of Rheumatology, pages 300–321, 1989.
* [104] LA White and CE Brinckerhoff. Two activator protein-1 elements in the matrix metalloproteinase-1 promoter have different effects on transcription and bind Jun D, c-Fos, and Fra-2. Matrix Biol, 14:715–725, 1995.
* [105] LA White, TI Mitchell, and CE Brinckerhoff. Transforming growth factor beta inhibitory element in the rabbit matrix metalloproteinase-1 (collagenase-1) gene functions as a repressor of constitutive transcription. Biochim Biophys Acta, 1490:259–268, 2000.
* [106] Karl Erich Wolff. Interpretation of automata in temporal concept analysis. In Uta Priss, Dan Corbett, and Galia Angelova, editors, Proceedings of the 10th International Conference on Conceptual Structures (ICCS 2002), volume 2393 of Lecture Notes in Computer Science, pages 341–353. Springer, 2002.
* [107] K.E. Wolff. States of distributed objects in conceptual semantic systems. In F. Dau, M.-L. Mugnier, and G. Stumme, editors, Conceptual Structures: Common Semantics for Sharing Knowledge, number 3596 in LNAI, pages 250–266, Heidelberg, 2005. Springer.
* [108] K.E. Wolff. States, transitions, and life tracks in temporal concept analysis. In B. Ganter, S. Stumme, and R. Wille, editors, Formal Concept Analysis – Foundations and Applications, number 3626 in LNAI, pages 127–148, Heidelberg, 2005. Springer.
* [109] J. Wollbold, R. Huber, D. Pohlers, D. Koczan, R. Guthke, R. Kinne, and U. Gausmann. Adapted Boolean network models for extracellular matrix formation. BMC Systems Biology, 3(77):21 July 2009, 2009.
* [110] Johannes Wollbold. Attribute exploration of discrete temporal transitions. In Alain Gély et al., editor, Contributions to ICFCA 2007 – 5th International Conference on Formal Concept Analysis, pages 121–130, Clermont-Ferrand, 2007.
* [111] Johannes Wollbold, Reinhard Guthke, and Bernhard Ganter. Constructing a knowledge base for gene regulatory dynamics by formal concept analysis methods. In K. Horimoto et al., editor, Algebraic Biology 2008, LNCS 5147, pages 230–244, Heidelberg, 2008. Springer.
* [112] Johannes Wollbold, René Huber, and Karl Erich Wolff. Conceptual representation of gene expression processes. In Karl Erich Wolff, Dmitry E. Palchunov, Nikolay G. Zagoruiko, and Urs Andelfinger, editors, Knowledge Processing and Data Analysis. Proceedings of KONT/KPP 2007, LNAI 6581, pages 79–100, Heidelberg, 2011. Springer.
* [113] JJ Wu, MW Lark, LE Chun, and DR Eyre. Sites of stromelysin cleavage in collagen types II, IX, X, and XI of cartilage. J Biol Chem, 266:5625–5628, 1991.
* [114] M Xue, L March, PN Sambrook, and CJ Jackson. Differential regulation of matrix metalloproteinase 2 and matrix metalloproteinase 9 by activated protein C: relevance to inflammation in rheumatoid arthritis. Arthritis Rheum, 56:2864–2874, 2007.
* [115] Y Yamanishi and GS Firestein. Pathogenesis of rheumatoid arthritis: the role of synoviocytes. Rheum Dis Clin North Am, 27:355–371, 2001.
* [116] T Zimmermann, E Kunisch, R Pfeiffer, A Hirth, HD Stahl, U Sack, A Laube, E Liesaus, A Roth, and E Palombo-Kinne. Isolation and characterization of rheumatoid arthritis synovial fibroblasts from primary culture – primary culture cells markedly differ from fourth-passage cells. Arthritis Res, 3:72–76, 2001.
## Index
* A- and B-concepts of time §3.2.1
* ABox §3.2.3
* Always §4.5
* Armstrong rules §2.2, §6.3, Proposition 6.3.1
* Association rule §5.2
* Attribute Definition 2.1.1
* Attribute exploration Chapter 1, §2.2
* complexity §9.2
* Next Closure algorithm Chapter 5
* Automaton §3.1.1
* Acceptor §3.1.1
* Boolean network Chapter 1, §3.3.2, Definition 4.3.2, item 2, Table 7.1, §8.2.2, §9.2
* Branching time §3.2.1, §3.2.2
* Category §3.1
* Chronicle §3.2.1
* Closure operator §2.1, §6.1.1
* Closure system §6.1.1
* Closure system, closed set §2.1
* Coalgebra §3.1, Definition 3.1.2
* Completeness (logics) §2.2
* Computation Tree Logic (CTL) §3.2.2
* Concept Explorer §2.2, §6.5, Figure 7.2, Figure 7.2
* Concept lattice (hierarchy) Figure 2.1, §2.1
* Confidence §5.2
* ConImp §2.2, §6.4, §6.5
* Counterexample §2.2
* Description logics (DL) §3.2.3, §9.1
* Discretisation §2.1, §8.2.3, §8.3.4, §9.3
* Endofunctor §3.1
* Entity §3.3.2, 1st item
* Eventually §4.5
* Extent Figure 2.1
* Filter (lattice) Figure 7.2
* Fluent 2nd item
* Formal concept Definition 2.1.7
* Formal Concept Analysis (FCA) Chapter 2
* Formal context Definition 2.1.1
* apposition Definition 2.1.5, §2.3, §4.5, §6.2.2, Table 9.2
* clarified §2.1
* derived §2.1
* many-valued Definition 2.1.2, Table 4.2
* reduced §2.1
* Functor §3.1
* Galois connection §6.1.1
* Gene regulatory network §7.1, Figure 8.4, §8.2.2, §8.3.1, §9.3
* Hasse diagram §2.1, §9.3
* History §3.2.1
* Ideal (lattice) Figure 7.2
* Implication §2.2
* holds §2.2
* respected §2.2
* Intent Figure 2.1
* Kripke structure §3.1.2, §4.3
* Labelled Transition System with Attributes (LTSA) §3.1.3
* Lattice Figure 2.1
* Boolean §2.2
* complete Chapter 1, §2.1
* Lectic order Chapter 5
* Liveness §3.2.2, §4.5
* Model §3.2.1, §5.3, §6.1.1, §9.1
* checking 1st item, §9.2
* Morphism §3.1
* Nested line diagram Remark 5.1.1, §9.3
* Never §4.5
* Next §4.5
* Non-redundancy (logics) §2.2
* Object Definition 2.1.1
* Output function §3.1.1, §3.1.2, §4.2, §4.3
* Petri net Chapter 1, §7.1
* Process
* deterministic Definition 3.1.3, §4.3, §4.5, §6.3, §8.2.5
* gene regulatory §7.1
* nondeterministic §3.1.2, §3.2.1, §3.2.2, §6.3, §7.2, §9.1
* Prolog §5.2, §7.2, §7.4, §8.2.8
* Propositional tense logic §3.2.1
* Pseudo-intent §2.2, §5.1
* Relational Concept Analysis (RCA) §9.1
* Role (DL) §3.2.3, 3rd item, §9.1
* depth §9.1
* local §3.2.3, §9.1, §9.1
* Safety §3.2.2, §4.5
* Scaling Definition 2.1.4, §4.2, §9.2
* biordinal Table 4.2
* dichotomic Table 2.2, §4.3, §4.5, Remark 6.2.3, §8.3.9
* nominal §2.1, Table 3.1, §4.2
* Semantic inference §6.1.1
* Semantics Chapter 1, §2.2, Chapter 3, Table 3.2, §4.5, Chapter 5
* Semiautomaton §3.1.3
* Semiproduct of formal contexts Definition 2.1.6, §4.3
* Soundness (logics) §2.2
* State §2.3, item 1, §3.1, Example 3.1.4, Example 3.1.6
* final §3.1.1, §3.1.2, §3.1.3, §3.1.3, 4th item, §4.5, §5.1
* input Example 3.1.6, §4.3, Definition 4.5.2, 2nd item
* output Example 3.1.6, §4.3, §4.3, §7.2, §8.2.5, 3rd item
* steady 2nd item, §5.1, §8.2.4
* State context §4.2
* $\mathcal{TDL}-Lite_{Bool}$ §9.1
* Stem base Chapter 1, §2.2, §2.2, §2.2
* Support §5.1, §5.2, §9.3
* Syntax Chapter 1, Chapter 3, Table 3.2
* TBox §3.2.3
* Temporal Concept Analysis (TCA) §2.3
* Temporal context Definition 4.5.2
* Test context §6.2.2
* Theorem
* Map Reconstruction §3.1.4
* of Duquenne-Guigues Theorem 2.2.2, §6.3
* Transition Example 3.1.6
* Transition context §4.3
* DL Table 9.1
* Transition function §3.1.1, §3.1.2, §4.3, Definition 4.3.2
* Transitive context §4.4
* DL Table 9.2
* Until §4.5
* $\pi$-calculus §3.3.1
## Attachments
1. 1.
Ehrenwörtliche Erklärung
2. 2.
CD with supplementary files and the developed R scripts
## Ehrenwörtliche Erklärung
Hiermit erkläre ich ehrenwörtlich, dass mir die geltende Promotionsordnung der
Fakultät für Mathematik und Informatik der Friedrich-Schiller-Universität Jena
bekannt ist. Ich versichere, dass ich die vorliegende Arbeit selbständig, ohne
unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen
Hilfsmittel angefertigt habe. Die aus fremden Quellen direkt oder indirekt
übernommenen Gedanken sowie persönliche Mitteilungen sind als solche kenntlich
gemacht. Eigene Prüfungsarbeiten wurden nicht verwendet. Bei der Auswahl und
Auswertung des Materials sowie bei der Herstellung des Manuskripts haben mich
die in der Danksagung genannten Personen unterstützt. Ich habe nicht die Hilfe
von Vermittlungs- bzw. Beratungsdiensten in Anspruch genommen. Niemand hat von
mir unmittelbar oder mittelbar geldwerte Leistungen für Arbeiten erhalten, die
im Zusammenhang mit dem Inhalt der vorgelegten Arbeit stehen.
Weiterhin erkläre ich, dass ich diese Dissertation noch nicht als
Prüfungsarbeit für eine staatliche oder andere wissenschaftliche Prüfung
eingereicht habe. Auch habe ich noch keine Dissertation bei einer anderen
Hochschule eingereicht.
Erfurt, 9. Mai 2011 Johannes Wollbold
|
arxiv-papers
| 2012-04-09T21:23:04 |
2024-09-04T02:49:29.522952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Johannes Wollbold",
"submitter": "Johannes Wollbold",
"url": "https://arxiv.org/abs/1204.1995"
}
|
1204.1996
|
# ANALYTIC CONTINUATION OF WEIGHTED $q$-GENOCCHI NUMBERS AND POLYNOMIALS
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
and Aynur Gürsul University of Gaziantep, Faculty of Science and Arts,
Department of Mathematics, 27310 Gaziantep, TURKEY aynurgursul@hotmail.com
###### Abstract.
In the present paper, we analyse analytic continuation of weighted
$q$-Genocchi numbers and polynomials. A novel formula for weighted
$q$-Genocchi-Zeta function $\widetilde{\zeta}_{G,q}\left(s\mid\alpha\right)$
in terms of nested series of $\widetilde{\zeta}_{G,q}\left(n\mid\alpha\right)$
is derived. Moreover, we introduce a novel concept of dynamics of the zeros of
analytically continued weighted $q$-Genocchi polynomials.
###### Key words and phrases:
Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials,
weighted$\ q$-Genocchi numbers and polynomials, weighted $q$-Genocchi-Zeta
function.
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. INTRODUCTION
In this paper, we use notations like $\mathbb{N}$, $\mathbb{R}$ and
$\mathbb{C}$, where $\mathbb{N}$ denotes the set of natural numbers,
$\mathbb{R}$ denotes the field of real numbers and $\mathbb{C}$ also denotes
the set of complex numbers. When one talks of $q$-extension, $q$ is variously
considered as an indeterminate, a complex number or a p-adic number.
Throughout this work, we will assume that $q\in\mathbb{C}$ with
$\left|q\right|<1$. The $q$-integer symbol $\left[x:q\right]$ denotes as
$\left[x:q\right]=\frac{q^{x}-1}{q-1}\text{.}$
Firstly, analytic continuation of $q$-Euler numbers and polynomials was
investigated by Kim in [1]. He gave a new concept of dynamics of the zeros of
analytically continued $q$-Euler polynomials. Actually, we were motivated from
his excellent paper which is ”Analytic continuation of $q$-Euler numbers and
polynomials, Applied Mathematics Letters 21 (2008) 1320-1323.” We also procure
to analytic continuation of weighted $q$-Genocchi numbers and polynomials as
parallel to his article. Also, we give some interesting identities by using
generating function of weighted $q$-Genocchi polynomials.
## 2\. PROPERTIES OF THE WEIGHTED $q$-GENOCCHI NUMBERS AND POLYNOMIALS
For $\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, the
weighted $q$-Genocchi polynomials are defined by means of the following
generating function:
For $x\in\mathbb{C}$,
(2.1)
$\sum_{n=0}^{\infty}\widetilde{G}_{n,q}\left(x\mid\alpha\right)\frac{t^{n}}{n!}=\left[2:q\right]t\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}e^{t\left[n+x:q^{\alpha}\right]}\text{.}$
As a special case $x=0$ into (2.1),
$\widetilde{G}_{n,q}\left(0\mid\alpha\right):=\widetilde{G}_{n,q}\left(\alpha\right)$
are called weighted $q$-Genocchi numbers. By (2.1), we readily derive the
following
(2.2)
$\frac{\widetilde{G}_{n+1,q}\left(x\mid\alpha\right)}{n+1}=\frac{\left[2:q\right]}{\left[\alpha:q\right]^{n}\left(1-q\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{q^{\alpha
lx}}{1+q^{\alpha l+1}}\text{,}$
where $\binom{n}{l}$ is the binomial coefficient. By expression (2.1), we see
that
(2.3) $\widetilde{G}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\left(q^{\alpha
x}\widetilde{G}_{q}\left(\alpha\right)+\left[x:q^{\alpha}\right]\right)^{n}\text{,}$
with the usual convention of replacing
$\left(\widetilde{G}_{q}\left(\alpha\right)\right)^{n}$ by
$\widetilde{G}_{n,q}\left(\alpha\right)$ is used (for details, see [7], [8]).
Let $\widetilde{T}_{q}^{\left(\alpha\right)}\left(x,t\right)$ be the
generating function of weighted $q$-Genocchi polynomials as follows:
(2.4)
$\widetilde{T}_{q}^{\left(\alpha\right)}\left(x,t\right)=\sum_{n=0}^{\infty}\widetilde{G}_{n,q}\left(x\mid\alpha\right)\frac{t^{n}}{n!}\text{.}$
Then, we easily notice that
(2.5)
$\widetilde{T}_{q}^{\left(\alpha\right)}\left(x,t\right)=\left[2:q\right]t\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}e^{t\left[n+x:q^{\alpha}\right]}\text{.}$
From expressions (2.4) and (2.5), we procure the followings:
For $k$ (=even) and
$n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we have
(2.6)
$q^{k}\frac{\widetilde{G}_{n+1,q}\left(k\mid\alpha\right)}{n+1}-\frac{\widetilde{G}_{n+1,q}\left(\alpha\right)}{n+1}=\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{k-l-1}\left[l:q^{\alpha}\right]^{n}\text{.}$
For $k$ (=odd) and
$n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we have
(2.7)
$q^{k}\frac{\widetilde{G}_{n+1,q}\left(k\mid\alpha\right)}{n+1}+\frac{\widetilde{G}_{n+1,q}\left(\alpha\right)}{n+1}=\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{k-l-1}\left[l:q^{\alpha}\right]^{n}\text{.}$
Via Eq. (2.5), we easily obtain the following:
(2.8) $\widetilde{G}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{G}_{k,q}\left(\alpha\right)\left[x:q^{\alpha}\right]^{n-k}\text{.}$
From (2.6)-(2.8), we get the following:
$\displaystyle\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{k-l-1}\left[l:q^{\alpha}\right]^{n}$
$\displaystyle=$ $\displaystyle\left(q^{\alpha
kn}-1\right)\frac{\widetilde{G}_{n+1,q}\left(\alpha\right)}{n+1}+q^{-\alpha
k}\sum_{j=0}^{n}\frac{1}{n+1}\binom{n+1}{j}q^{\alpha
jk}\widetilde{G}_{k,q}\left(\alpha\right)\left[k:q^{\alpha}\right]^{n+1-k}\text{,}$
here $k$ is an even positive integer. If $k$ is an odd positive integer. Then,
we can derive the following equality:
$\displaystyle\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{k-l-1}\left[l:q^{\alpha}\right]^{n}$
$\displaystyle=$ $\displaystyle\left(q^{\alpha
kn}+1\right)\frac{\widetilde{G}_{n+1,q}\left(\alpha\right)}{n+1}+q^{-\alpha
k}\sum_{j=0}^{n}\frac{1}{n+1}\binom{n+1}{j}q^{\alpha
jk}\widetilde{G}_{k,q}\left(\alpha\right)\left[k:q^{\alpha}\right]^{n+1-k}\text{.}$
## 3\. WEIGHTED $q$-GENOCCHI-ZETA FUNCTION
The famous Genocchi polynomials were defined as
(3.1)
$\frac{2t}{e^{t}+1}e^{xt}=\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi\text{ cf. \cite[cite]{[\@@bibref{}{kim 4}{}{}]}.}$
For $s\in\mathbb{C}$, $x\in\mathbb{R}$ with $0\leq x<1$, Genocchi-Zeta
function are given by
(3.2)
$\zeta_{G}\left(s,x\right)=2\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{\left(n+x\right)^{s}}\text{,
}$
and
(3.3)
$\zeta_{G}\left(s\right)=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n}}{n^{s}}\text{.}$
By (3.1), (3.2) and (3.3), Genocchi-Zeta functions are related to the Genocchi
numbers as follows:
$\zeta_{G}\left(-n\right)=\frac{G_{n+1}}{n+1}\text{.}$
Moreover, it is simple to see
$\zeta_{G}\left(-n,x\right)=\frac{G_{n+1}\left(x\right)}{n+1}\text{.}$
The weighted $q$-Genocchi Hurwitz-Zeta type function are defined by
$\widetilde{\zeta}_{G,q}\left(s,x\mid\alpha\right)=\left[2:q\right]\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m+x:q^{\alpha}\right]^{s}}\text{
.}$
Similarly, weighted $q$-Genocchi-Zeta function are given by
$\widetilde{\zeta}_{G,q}\left(s\mid\alpha\right)=\left[2:q\right]\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{s}}\text{.}$
For $n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we
have
$\widetilde{\zeta}_{G,q}\left(-n\mid\alpha\right)=\frac{\widetilde{G}_{n+1,q}\left(\alpha\right)}{n+1}\text{.}$
We now consider the function $\widetilde{G}_{q}\left(n:\alpha\right)$ as the
analytic continuation of weighted $q$-Genocchi numbers. All the weighted
$q$-Genocchi numbers agree with $\widetilde{G}_{q}\left(n:\alpha\right)$, the
analytic continuation of weighted $q$-Genocchi numbers evaluated at $n$. For
$n\geq 0$,
$\widetilde{G}_{q}\left(n:\alpha\right)=\widetilde{G}_{n,q}\left(\alpha\right)$.
We can now state $\widetilde{G}{\acute{}}_{q}\left(s:\alpha\right)$ in terms
of $\widetilde{\zeta}{\acute{}}_{G,q}\left(s\mid\alpha\right)$, the derivative
of $\widetilde{\zeta}_{G,q}\left(s:\alpha\right)$
$\frac{\widetilde{G}_{q}\left(s+1:\alpha\right)}{s+1}=\widetilde{\zeta}_{G,q}\left(-s\mid\alpha\right)\text{,
}\frac{\widetilde{G}{\acute{}}_{q}\left(s+1:\alpha\right)}{s+1}=\widetilde{\zeta}{\acute{}}_{G,q}\left(-s\mid\alpha\right)\text{.}$
For $n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$
$\text{
}\frac{\widetilde{G}{\acute{}}_{q}\left(2n+1:\alpha\right)}{2n+1}=\widetilde{\zeta}{\acute{}}_{G,q}\left(-2n\mid\alpha\right)\text{.}$
This is suitable for the differential of the functional equation and so
supports the coherence of $\widetilde{G}_{q}\left(s:\alpha\right)$ and
$\widetilde{G}{\acute{}}_{q}\left(s:\alpha\right)$ with
$\widetilde{G}_{n,q}\left(\alpha\right)$ and
$\widetilde{\zeta}_{G,q}\left(s\mid\alpha\right)$. From the analytic
continuation of weighted $q$-Genocchi numbers, we derive as follows:
$\frac{\widetilde{G}_{q}\left(s+1:\alpha\right)}{s+1}=\widetilde{\zeta}_{G,q}\left(-s\mid\alpha\right)\text{
and
}\frac{\widetilde{G}_{q}\left(-s+1:\alpha\right)}{-s+1}=\widetilde{\zeta}_{G,q}\left(s\mid\alpha\right)\text{.}$
Moreover, we derive the following:
For $n\in\mathbb{N}-\left\\{1\right\\}$
$\frac{\widetilde{G}_{-n+1,q}\left(\alpha\right)}{-n+1}=\frac{\widetilde{G}_{q}\left(-n+1:\alpha\right)}{-n+1}=\widetilde{\zeta}_{G,q}\left(n\mid\alpha\right)\text{.}$
The curve $\widetilde{G}_{q}\left(s:a\right)$ review quickly the points
$\widetilde{G}_{-s,q}\left(\alpha\right)$ and grows $\sim n$ asymptotically
$\left(-n\right)\rightarrow-\infty$. The curve
$\widetilde{G}_{q}\left(s:a\right)$ review quickly the point
$\widetilde{G}_{q}\left(-s:a\right)$. Then, we procure the following:
$\displaystyle\lim_{n\rightarrow\infty}\frac{\widetilde{G}_{q}\left(-n+1:\alpha\right)}{-n+1}$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\widetilde{\zeta}_{G,q}\left(n\mid\alpha\right)=\lim_{n\rightarrow\infty}\left(\left[2:q\right]\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{n}}\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(-q\left[2:q\right]+\left[2:q\right]\sum_{m=2}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{n}}\right)=-q^{2}\left[2:q^{-1}\right]\text{.}$
From this, we easily note that
$\frac{\widetilde{G}_{q}\left(-n+1:\alpha\right)}{-n+1}=\widetilde{\zeta}_{G,q}\left(n\mid\alpha\right)\mapsto\frac{\widetilde{G}_{q}\left(-s+1:\alpha\right)}{-s+1}=\widetilde{\zeta}_{G,q}\left(s\mid\alpha\right)\text{.}$
## 4\. ANALYTIC CONTINUATION OF WEIGHTED $q$-GENOCCHI POLYNOMIALS
For coherence with the redefinition of
$\widetilde{G}_{n,q}\left(\alpha\right)=\widetilde{G}_{q}\left(n:\alpha\right)$,
we have
$\widetilde{G}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{G}_{k,q}\left(\alpha\right)\left[x:q^{\alpha}\right]^{n-k}\text{.}$
Let $\Gamma\left(s\right)$ be Euler-gamma function. Then the analytic
continuation can be get as
$\displaystyle n$ $\displaystyle\mapsto$ $\displaystyle s\in\mathbb{R}\text{,
}x\mapsto w\in\mathbb{C}\text{,}$
$\displaystyle\widetilde{G}_{n,q}\left(\alpha\right)$ $\displaystyle\mapsto$
$\displaystyle\widetilde{G}_{q}\left(k+s-\left[s\right]:\alpha\right)=\widetilde{\zeta}_{G,q}\left(-\left(k+s-\left[s\right]\right)\mid\alpha\right)\text{,}$
$\displaystyle\binom{n}{k}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)}\mapsto\frac{\Gamma\left(s+1\right)}{\Gamma\left(1+k+\left(s-\left[s\right]\right)\right)\Gamma\left(1+\left[s\right]-k\right)}$
$\displaystyle\widetilde{G}_{s,q}\left(w\mid\alpha\right)$
$\displaystyle\mapsto$
$\displaystyle\widetilde{G}_{q}\left(s,w:\alpha\right)=q^{-\alpha
w}\sum_{k=-1}^{\left[s\right]}\frac{\Gamma\left(s+1\right)\widetilde{G}_{q}\left(k+\left(s-\left[s\right]\right):\alpha\right)q^{\alpha
w\left(k+\left(s-\left[s\right]\right)\right)}}{\Gamma\left(1+k+\left(s-\left[s\right]\right)\right)\Gamma\left(1+\left[s\right]-k\right)}\left[w:q^{\alpha}\right]^{\left[s\right]-k}$
$\displaystyle=$ $\displaystyle q^{-\alpha
w}\sum_{k=0}^{\left[s\right]+1}\frac{\Gamma\left(s+1\right)\widetilde{G}_{q}\left(-1+k+\left(s-\left[s\right]\right):\alpha\right)q^{\alpha
w\left(k-1+\left(s-\left[s\right]\right)\right)}}{\Gamma\left(k+\left(s-\left[s\right]\right)\right)\Gamma\left(2+\left[s\right]-k\right)}\left[w:q^{\alpha}\right]^{\left[s\right]+1-k}\text{.}$
Here $\left[s\right]$ gives the integer part of s, and so $s-\left[s\right]$
gives the fractional part.
Deformation of the curve $\widetilde{G}_{q}\left(1,w:\alpha\right)$ into the
curve of $\widetilde{G}_{q}\left(2,w:\alpha\right)$ is by means of the real
analytic cotinuation $\widetilde{G}_{q}\left(s,w:\alpha\right)$, $1\leq s\leq
2$, $-0.5\leq w\leq 0.5$.
## References
* [1] T. Kim, Analytic continuation of $q$-Euler numbers and polynomials, Applied Mathematics Letters 21 (2008) 1320-1323.
* [2] T. Kim, On explicit formulas of $p$-adic $q$-$L$-functions, Kyushu J. Math. 43 (1994) 73–86.
* [3] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008) 598–608.
* [4] T. Kim, On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465.
* [5] T. Kim, On a $q$-analogue of the $p$-adic $\log$ gamma functions and related integrals, Journal of Number Theory 76 (1999) 320-329.
* [6] T. Kim, On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, J. Math. Anal. Appl. 331 (2007) 779–792.
* [7] S. Araci, D. Erdal, J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [8] S. Araci, M. Acikgoz and J. J. Seo, A study on the weighted $q$-Genocchi numbers and polynomials with their interpolation function, Honam Mathematical Journal (in press).
|
arxiv-papers
| 2012-04-09T21:32:28 |
2024-09-04T02:49:29.543438
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Aynur Gursul",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1204.1996"
}
|
1204.2040
|
# A New Reduction from Search SVP to Optimization SVP
Gengran Hu, Yanbin Pan
Key Laboratory of Mathematics Mechanization
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Beijing 100190, China
hudiran10@mails.gucas.ac.cn, panyanbin@amss.ac.cn
###### Abstract
It is well known that search SVP is equivalent to optimization SVP. However,
the former reduction from search SVP to optimization SVP by Kannan needs
polynomial times calls to the oracle that solves the optimization SVP. In this
paper, a new rank-preserving reduction is presented with only one call to the
optimization SVP oracle. It is obvious that the new reduction needs the least
calls, and improves Kannan’s classical result. What’s more, the idea also
leads a similar direct reduction from search CVP to optimization CVP with only
one call to the oracle.
Keywords: Search SVP, Optimization SVP, Lattice, Reduction.
## 1 Introduction
Given a matrix $B=(b_{ij})\in\mathbb{R}^{m\times n}$ with rank $n$, the
lattice $L(B)$ spanned by the columns of $B$ is
$L(B)=\\{\sum_{i=1}^{n}x_{i}b_{i}|x_{i}\in\mathbb{Z}\\},$
where $b_{i}$ is the $i$-th column of $B$. Lattice has many important
applications in cryptography. The shortest vector problem (SVP) and the
closest vector problem (CVP) are two of the most famous problems of lattice.
SVP refers to find the shortest non-zero vector in a given lattice. There are
three different variants of SVP:
1\. Search SVP: Given a lattice basis $B\in\mathbb{Z}^{m\times n}$, find
$v\in\mathcal{L}(B)$ such that $\|v\|=\lambda_{1}(\mathcal{L}(B))$, where
$\lambda_{1}(\mathcal{L}(B))$ is the length of the shortest non-zero vector in
$\mathcal{L}(B)$.
2\. Optimization SVP: Given a lattice basis $B\in\mathbb{Z}^{m\times n}$, find
$\lambda_{1}(\mathcal{L}(B))$.
3\. Decisional SVP: Given a lattice basis $B\in\mathbb{Z}^{m\times n}$ and a
rational $r\in\mathbb{Q}$, decide whether $\lambda_{1}(\mathcal{L}(B))\leq r$
or not.
It has been proved that the three problems are equivalent to each other (see
[2]). It is easy to check that the decisional SVP is as hard as the
optimization SVP and the optimization variant can be reduced to the search
variant.
In 1987, Kannan [1] also showed that the search variant can be reduced to the
optimization variant. The basic idea of his reduction is to recover the
integer coefficients of some shortest vector under the given lattice basis by
introducing small errors to the original lattice basis. However, his reduction
is a bit complex. It needs to call polynomial times optimization SVP oracle,
since it could not determine the signs of the shortest vector’s entries at one
time. It also needs oracle to solve optimization SVP for some lattices with
lower rank besides with the same rank as the original lattice.
In this paper, we propose a new rank-preserving reduction which can solve the
search SVP with only one call to the optimization SVP oracle. It is obvious
that there is no reduction with less calls than ours. Instead of recovering
the shortest vector directly as in [1], we first recover the integer
coefficients of some shortest vector under the given lattice basis, then
recover the shortest vector.
A similar direct reduction from search CVP to optimization CVP with only one
call also holds whereas some popular reductions [2, 3] usually takes
decisional CVP to bridge the search CVP and optimization CVP. The former
reduction from decisional CVP to optimization CVP needs one call to the
optimization CVP oracle, but it needs polynomial times calls to the decisional
CVP oracle to reduce search CVP to decisional CVP.
## 2 The New Reduction
For simplicity, we just give the new reduction for the full rank lattice, i.e.
$n=m$, as in [1]. It is easy to general the new reduction for the lattices
with rank $n<m$.
### 2.1 Some Notations
Given a lattice basis $B=(b_{ij})\in\mathbb{R}^{n\times n}$, let
$M(B)=\max|b_{ij}|$. For lattice $L(B)$, we define its SVP solution set
$S_{B}$ as:
$S_{B}=\\{x\in\mathbb{Z}^{n}|\|Bx\|=\lambda_{1}(\mathcal{L}(B))\\}$
Denote by $poly(n)$ the polynomial in $n$.
### 2.2 Some Lemmas
We need some lemmas to prove our main theorem.
###### Lemma 1.
For every positive integer $n$, there exist $n$ positive integers
$a_{1}<a_{2}<\ldots<a_{n}$ s.t. all the $a_{i}+a_{j}(i\leq j)$’s are distinct
and $a_{n}$ is bounded by $poly(n)$.
###### Proof.
We can take $a_{k}=(n^{2}+k-1)^{2}$ for $k=1,2,\cdots,n$. Suppose
$a_{i_{1}}+a_{j_{1}}=a_{i_{2}}+a_{j_{2}}$ for some $i_{1},j_{1},i_{2},j_{2}$,
we get
$(i_{1}-1)^{2}+(j_{1}-1)^{2}+2n^{2}((i_{1}-1)+(j_{1}-1))=(i_{2}-1)^{2}+(j_{2}-1)^{2}+2n^{2}((i_{2}-1)+(j_{2}-1))$.
Since $(i_{1}-1)^{2}+(j_{1}-1)^{2},(i_{2}-1)^{2}+(j_{2}-1)^{2}<2n^{2}$, we
have $(i_{1}-1)^{2}+(j_{1}-1)^{2}=(i_{2}-1)^{2}+(j_{2}-1)^{2}$ and
$i_{1}+j_{1}=i_{2}+j_{2}$, which leads $\\{i_{1},j_{1}\\}=\\{i_{2},j_{2}\\}$.
Hence all the $a_{i}+a_{j}(i\leq j)$’s are distinct. It is obvious that
$a_{n}\leq(n^{2}+n-1)^{2}$. ∎
###### Lemma 2.
Given positive odd integer $p>2$, and any positive integer $n$, which
satisfies $n=\sum_{i=0}^{k}n_{i}p^{i}$ where $|n_{i}|\leq\lfloor p/2\rfloor$,
then we can recover the coefficients $n_{i}$’s in polynomial time.
###### Proof.
We can recover $n_{0}$ by computing $a\equiv n\mbox{ mod }p$ and choose $a$ in
the interval from $-\lfloor p/2\rfloor$ to $\lfloor p/2\rfloor$. After
obtaining $n_{0}$, we get another integer $(n-n_{0}*p^{0})/p$. Recursively, we
can recover all the coefficients. This can be done in polynomial time
obviously. ∎
###### Lemma 3.
For bivariate polynomial $f(x,y)=xy$, given any lattice basis matrix
$B\in\mathbb{Z}^{n\times n}$, $\lambda_{1}(L(B))$ has an upper bound $f(M,n)$,
where $M=M(B)$. What’s more, for every $x\in S_{B}$, $|x_{i}|$
$(i=1,\cdots,n)$ has an upper bound $f(M^{n},n^{n})$.
###### Proof.
The length of any column of $B$ is an upper bound of $\lambda_{1}(L(B))$, so
$\lambda_{1}(L(B))\leq n^{1/2}M\leq nM$.
For $x\in S_{B}$, we let $y=Bx$, then $\|y\|=\lambda_{1}(L(B))\leq\sqrt{n}M$.
By Cramer’s rule, we know that
$x_{i}=\dfrac{\det(B^{(i)})}{\det(B)},$
where $B^{(i)}$ is formed by replacing the $i$-th column of $B$ by $y$. By
Hadamard’s inequality, $|\det(B^{(i)})|\leq n^{n/2}M^{n}\leq n^{n}M^{n}$. We
know $|\det(B)|\geq 1$ since $\det(B)$ is a non-zero integer. Hence
$|x_{i}|\leq n^{n}M^{n}$. ∎
### 2.3 The Main Theorem
###### Theorem 1.
Assume there exists an oracle $\mathcal{O}$ that can solve the optimization
SVP for any lattice $L(B^{\prime})$ with basis
$B^{\prime}\in\mathbb{Z}^{n\times n}$, then there is an algorithm that can
solve the search SVP for any lattice $L(B)$ with basis
$B\in\mathbb{Z}^{n\times n}$ with only one call to $\mathcal{O}$ in
$poly(\log_{2}{M},n,\log_{2}{n})$ time, where $M=M(B)$.
###### Proof.
The main steps of the algorithm are as below:
(1) Constructing a new lattice basis $B_{\epsilon}\in\mathbb{Z}^{n\times n}$.
We construct $B_{\epsilon}$ from the original lattice $B$:
$B_{\epsilon}=\epsilon_{n+1}B+\left(\begin{array}[]{cccc}\epsilon_{1}&\epsilon_{2}&\dots&\epsilon_{n}\\\
0&0&\dots&0\\\ \vdots&\vdots&&\vdots\\\ 0&0&\dots&0\\\ \end{array}\right)$
where the $\epsilon_{i}$ will be determined as below.
For any $x\in\mathbb{Z}^{n}$, we difine $c(x)=\sum_{i=1}^{n}b_{1i}x_{i}$. For
$x\in S_{B}$, by Lemma 3, $|x_{i}|$ has an upper bound $f(M^{n},n^{n})$. Let
$M_{1}=2f((M+1)^{n},n^{n})$. In addition, $\|Bx\|=\lambda_{1}(L(B))$ is
bounded by $f(M,n)$. Let $M_{2}=f(M+1,n)$. $|c(x)|$ is also bounded by $M_{2}$
since $|c(x)|\leq\|Bx\|$. We let
$p=2*\max{\\{M_{2}^{2},2M_{1}M_{2},2M_{1}^{2}\\}}+1.$
By Lemma 1, we can choose $n+1$ positive integers
$a_{1}<a_{2}<\ldots<a_{n+1}$, such that all the $a_{i}+a_{j}(i\leq j)$’s are
distinct where $a_{n+1}$ is bounded by $poly(n)$. Let
$\epsilon_{i}=p^{a_{i}}.$
We first show that
$|\det{(\frac{1}{\epsilon_{n+1}}B_{\epsilon})|\geq\frac{1}{2}}$, so
$B_{\epsilon}$ is indeed a lattice basis. Notice that
$\det{(\frac{1}{\epsilon_{n+1}}B_{\epsilon})}=\det(B)+\sum_{i=1}^{n}\alpha_{i}\frac{\epsilon_{i}}{\epsilon_{n+1}},$
where $\alpha_{i}$ is the cofactor of $B_{1i}$ in $B$. Since
$\frac{\epsilon_{i}}{\epsilon_{n+1}}\leq\frac{1}{p^{2}}$ and $|\alpha_{i}|\leq
M^{n-1}(n-1)^{n-1}$,
$|\sum_{i=1}^{n}\alpha_{i}\frac{\epsilon_{i}}{\epsilon_{n+1}}|\leq\frac{1}{p^{2}}M^{n-1}n^{n}<\frac{1}{2}$.
By the fact $\det(B)$ is a non-zero integer, we get
$|\det{(\frac{1}{\epsilon_{n+1}}B_{\epsilon})|\geq\frac{1}{2}}.$ (1)
We claim that $S_{B_{\epsilon}}\subseteq S_{B}$. Since
$S_{B_{\epsilon}}=S_{\frac{1}{\epsilon_{n+1}}B_{\epsilon}}$, it is enough to
prove $S_{\frac{1}{\epsilon_{n+1}}B_{\epsilon}}\subseteq S_{B}$.
For any $x\in S_{\frac{1}{\epsilon_{n+1}}B_{\epsilon}}$, by (1) and the proof
of Lemma 3, we know that $|x_{i}|\leq M_{1}$, $|c(x)|\leq M_{2}$. By the
choice of $p$, $x_{i}^{2},2c(x)x_{i},2x_{i}x_{j}$ are in the interval
$[-\lfloor p/2\rfloor,\lfloor p/2\rfloor]$. Together with the fact that
$\frac{\epsilon_{i}\epsilon_{j}}{\epsilon_{n+1}^{2}}(i\leq j)$’s are different
powers of $p$, we have
$\begin{array}[]{rcl}\lambda_{1}(L(\frac{1}{\epsilon_{n+1}}B_{\epsilon}))^{2}&=&\|\frac{1}{\epsilon_{n+1}}B_{\epsilon}x\|^{2}\\\
&=&\|Bx\|^{2}+\sum_{i=1}^{n}x_{i}^{2}(\frac{\epsilon_{i}}{\epsilon_{n+1}})^{2}+\sum_{i=1}^{n}2c(x)x_{i}\frac{\epsilon_{i}}{\epsilon_{n+1}}+\sum_{i<j}2x_{i}x_{j}\frac{\epsilon_{i}\epsilon_{j}}{\epsilon_{n+1}^{2}}\\\
&>&\|Bx\|^{2}-(\lfloor
p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}.\end{array}$ (2)
Similarly, for any $y\in S_{B}$, we have
$\begin{array}[]{rcl}\|\frac{1}{\epsilon_{n+1}}B_{\epsilon}y\|^{2}&=&\|By\|^{2}+\sum_{i=1}^{n}y_{i}^{2}(\frac{\epsilon_{i}}{\epsilon_{n+1}})^{2}+\sum_{i=1}^{n}2c(y)y_{i}\frac{\epsilon_{i}}{\epsilon_{n+1}}+\sum_{i<j}2y_{i}y_{j}\frac{\epsilon_{i}\epsilon_{j}}{\epsilon_{n+1}^{2}}\\\
&<&\lambda_{1}(L(B))^{2}+(\lfloor
p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}\end{array}$ (3)
Next, we prove $S_{\frac{1}{\epsilon_{n+1}}B_{\epsilon}}\subseteq S_{B}$.
Suppose there exists $x\in S_{\frac{1}{\epsilon_{n+1}}B_{\epsilon}}$ but
$x\not\in S_{B}$, then
$\|Bx\|^{2}\geq\lambda_{1}(L(B))^{2}+1.$ (4)
Notice that $\frac{\epsilon_{n}}{\epsilon_{n+1}}<\frac{1}{p^{2}}$, we have
$0<(\lfloor p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}<\frac{1}{2}$.
Together with (2), (3) and (4), we have
$\begin{array}[]{rcl}\lambda_{1}(L(\frac{1}{\epsilon_{n+1}}B_{\epsilon}))^{2}&>&\|Bx\|^{2}-(\lfloor
p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}\\\
&\geq&\lambda_{1}(L(B))^{2}+1-(\lfloor
p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}\\\
&>&\lambda_{1}(L(B))^{2}+(\lfloor
p/2\rfloor+1)\frac{\epsilon_{n}}{\epsilon_{n+1}}\\\
&>&\|\frac{1}{\epsilon_{n+1}}B_{\epsilon}y\|^{2},\end{array}$
which is an contradiction, since $\frac{1}{\epsilon_{n+1}}B_{\epsilon}y\in
L(\frac{1}{\epsilon_{n+1}}B_{\epsilon})$. Hence $S_{B_{\epsilon}}\subseteq
S_{B}$.
(2) Querying the oracle $\mathcal{O}$ with $B_{\epsilon}$ once, we get
$\lambda_{1}(\mathcal{L}(B_{\epsilon}))$.
So there exists $x=(x_{1},\ldots,x_{n})^{\mathrm{T}}\in
S_{B_{\epsilon}}\subseteq S_{B}$, such that
$\|Bx\|^{2}\epsilon_{n+1}^{2}+\sum_{i=1}^{n}x_{i}^{2}\epsilon_{i}^{2}+\sum_{i=1}^{n}2c(x)x_{i}\epsilon_{n+1}\epsilon_{i}+\sum_{i<j}2x_{i}x_{j}\epsilon_{i}\epsilon_{j}=\lambda_{1}(\mathcal{L}(B_{\epsilon}))^{2}$
(3) Recovering all the $x_{i}$’s and output $Bx$.
Since $x\in S_{B}$, every coefficient
$\|Bx\|^{2},x_{i}^{2},2c(x)x_{i},2x_{i}x_{j}$ is in the interval $[-\lfloor
p/2\rfloor,\lfloor p/2\rfloor]$ and $\epsilon_{i}\epsilon_{j}$ $(i\leq j)$’s
are different powers of $p$. Hence,
$\log_{2}{(\lambda_{1}(\mathcal{L}(B_{\epsilon})))}$ is bounded by
$poly(\log_{2}M,n,\log_{2}n)$. Furthermore, by Lemma 2, we can recover all the
coefficients in $poly(\log_{2}M,n,\log_{2}n)$ time. Especially, we can recover
all $x_{i}^{2}$ and $x_{i}x_{j}(i\neq j)$. Let $k=\min\\{i|x_{i}\neq 0\\}$. We
fix $x_{k}=\sqrt{x_{k}^{2}}>0$, and can recover all the remaining
$x_{j}=sign(x_{k}x_{j})\sqrt{x_{j}^{2}}$ according to $x_{j}^{2}$ and
$x_{k}x_{j}(k\neq j)$.
It is easy to check that the complexity of every step is bounded by
$poly(\log_{2}{M},n,\log_{2}n)$. ∎
###### Remark 1.
For any search CVP instant $(B,t)$, given an oracle which can solve the
optimization CVP, we can call the oracle with $(B_{\epsilon},\epsilon_{n+1}t)$
only once to solve the search CVP similarly.
## 3 Conclusions
In this paper, we give a new reduction from search SVP to optimization SVP
with only one call, which is the least, to the optimization SVP oracle. A
similar result for CVP also holds. However, it seems hard to apply the idea
for GapSVP or GapCVP, since the new reduction is also sensitive to the error.
## References
* [1] R. Kannan, Minkowski’s convex body theorem and integer programming, Mathematics of Operation Research, 12(3): 415-440, 1987.
* [2] D. Micciancio, S. Goldwasser, Complexity of Lattice Problems: A Cryptography Perspective, Kluwer Academic Publishes, 2002.
* [3] O. Regev, Lattices in computer science, Lecture notes of a course given in Tel Aviv University, 2004.
|
arxiv-papers
| 2012-04-10T04:23:24 |
2024-09-04T02:49:29.548566
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gengran Hu and Yanbin Pan",
"submitter": "Yanbin Pan",
"url": "https://arxiv.org/abs/1204.2040"
}
|
1204.2218
|
# Decoder for Nonbinary CWS Quantum Codes
Nolmar Melo Laboratório Nacional de Computação Científica
Petrópolis, RJ 25651-075, Brazil
Email: nolmar@lncc.br Douglas F.G. Santiago Universidade Federal dos Vales
do Jequitinhonha e Mucuri
Diamantina, MG 39100000, Brazil
Email: douglassant@gmail.com Renato Portugal Laboratório Nacional de
Computação Científica
Petrópolis, RJ 25651-075, Brazil
Email: portugal@lncc.br
###### Abstract
We present a decoder for nonbinary CWS quantum codes using the structure of
union codes. The decoder runs in two steps: first, we use a union of
stabilizer codes to detect a sequence of errors, and second, we build a new
code, called union code, that allows the error correction.
## I Introduction
Quantum computers is able to solve many hard problems in polynomial time and
to increase the speed of most algorithms [1, 2, 3, 4]. Decoherence problems
are inherent in these computers requiring the use of quantum error correcting
codes (QECCs) [5, 6, 7, 8].
A large class of good binary codes is known in literature [9, 10, 11, 12].
However, in order to build a quantum fault tolerant quantum computer,
concatenation of quantum codes plays a crucial role. The optimum concatenation
is obtained when nonbinary codes are used [12].
An important class of nonadditive codes, called CWS, has been studied recently
[13, 14, 15, 16, 17]. The framework of CWS codes generalizes the stabilizer
code formalism and was used to build some good nonadditive codes. The
codification of binary and nonbinary CWS codes is well known, whereas the
decodification is only known for binary CWS codes [17, 18]. In this paper, we
present an algorithm to decode nonbinary CWS codes, generalizing the procedure
described in Ref. [18].
This article is divided in the following parts: In Section II, the CWS codes
are briefly reviewed. In Section III, the theory of union codes are presented.
Such codes provide the basis for our decoder. In Section IV, measurement
operators for union codes are presented. In Section V, the main theorems are
proved. In Section VI, the nonbinary decoder is presented and an analysis of
the computational cost is performed. In Section VII, an example is worked out.
In Section VIII, the conclusions are presented.
## II CWS Codes
Nonbinary CWS codes use the generalized Pauli group $G_{d}$ over qudits [19,
20, 21, 14]. Let $\chi$ be the character of group $\mathbb{Z}_{d}$ in the unit
cycle of $\mathbb{C}$. By definition, $\chi(x)=\exp(\frac{i2\pi x}{d})$. We
denote $\omega=\chi(1)$. The action of $X$ over qudits is
$X\left|i\right>=\left|i+1\right>$ and the action of $Z$, which performs a
phase shift, is $Z\left|i\right>=\omega^{i}\left|i\right>$. In matrix form, we
have
$X=\left(\begin{array}[]{c|c}0&1\\\ \hline\cr I&0\end{array}\right)_{d\times
d},\;\;Z=\left(\begin{array}[]{ccccc}1&0&0&\dots&0\\\ 0&\omega&0&\dots&0\\\
0&0&\omega^{2}&\dots&0\\\ \vdots&\vdots&\vdots&\ddots&0\\\
0&0&0&\dots&\omega^{d-1}\end{array}\right).$
It is easy to check that $X^{d}=Z^{d}=I$ and $ZX=\omega XZ$. Pauli group
$G_{d}$ is generated by $\\{X,Z,\omega^{j}I\\}$, where $j=0,\dots,d-1$. The
Pauli group acting on $n$ qudits is $G_{d}^{n}=\otimes^{n}G_{d}$.
A nonbinary CWS code is a nontrivial vector subspace of $\mathbb{C}^{d^{n}}$.
If this subspace has dimension $K$, the code is denoted by $((n,K))_{d}$ and,
if $\delta$ is the minimum distance, the notation is $((n,K,\delta))_{d}$. A
CWS code is described by a subgroup $S$ of the Pauli group (called stabilizer
group) and a set of $K$ Pauli operators $W=\\{w_{l}\\}_{l=1}^{K}$ called word
operators.
Group $S$ stabilizes a single word, usually $\left|S\right>$ and, in the
binary case, we have $S=\langle g_{i},\dots,g_{m}\rangle$ with $m=n$, whereas
in the nonbinary case we have $m\geq n$. The generators of $S$ have the form
$g_{i}=X^{r_{i}}Z^{t_{i}}$, where $r_{i}$ and $t_{i}$ are vectors with entries
in $\mathbb{Z}_{d}$. We can build a matrix $[r|t]$ of dimensions $m\times 2n$,
which is useful for establishing a connection with a classical code.
A basis for the quantum code is $\\{w_{l}\left|S\right>;\;w_{l}\in W\\}$,
where $\left|S\right>$ is stabilized by group $S$. Note that for each
$w_{i}\in W$, $w_{l}\left|S\right>$ is stabilized by $w_{l}Sw_{l}^{\dagger}$.
Moreover, for $g_{k}\in S$, we have
$w_{l}g_{k}w_{l}^{\dagger}=\omega^{l_{k}}g_{k}$ $\left|w_{l}\right>$, so a
classical vector on $\mathbb{Z}_{d}$
$c_{l}=(l_{1},\dots,l_{m})$
can be associated to $w_{l}$.
The error correction conditions for quantum codes state that, in order to
detect a set of errors $\mathcal{E}$, it is necessary and sufficient that
$\left<\psi_{i}\right|E\left|\psi_{j}\right>=C_{E}\delta_{ij},$
for all $E\in\mathcal{E}$, where $\left|\psi_{i}\right>$ and
$\left|\psi_{j}\right>$ are in an orthonormal basis of the code. Note that
$C_{E}$ does not depend on $i$ and $j$. $E_{1},E_{2}\in\mathcal{E}$ are
correctable if and only if
$\left<\psi_{i}\right|E_{1}^{\dagger}E_{2}\left|\psi_{j}\right>=C_{E_{1}E_{2}}\delta_{ij}.$
Again, $C_{E_{1}E_{2}}$ does not depend on $i$ and $j$. An error is degenerate
if $C_{E}\neq 0$. Two distinct errors, $E_{1},E_{2}\in\mathcal{E}$, belong to
the same degeneracy class when their actions on the code are the same, that
is, $C_{E_{1}E_{2}}\neq 0$. A code is said degenerate when the error set has a
degenerate element [22].
It is enough to consider errors as operators in the Pauli group acting on the
code, that is, an error $E$ has the form $\alpha Z^{v}X^{u}$, where
$\alpha\in\mathbb{C}$ and $v,u\in\mathbb{Z}_{d}^{n}$. We can map errors to
classical vectors using function $\mathcal{C}\ell_{S}$ given by
$\mathcal{C}\ell_{S}(E)=\sum_{l=1}^{n}v_{l}r_{l}-u_{l}t_{l},$ (1)
where $r_{l}$ and $t_{l}$ are the columns of the matrix $[r|t]$. An error $E$
is detectable in the quantum code if and only if $\mathcal{C}\ell_{S}(E)$ is
detectable in the associated classical code and $\mathcal{C}\ell_{S}(E)\neq 0$
or $\forall l,\;w_{l}E=Ew_{l}$. An error is degenerate when
$\mathcal{C}\ell_{S}(E)=0$ and two distinct errors,
$E_{1},E_{2}\in\mathcal{E}$, belong to the same degeneracy class when
$\mathcal{C}\ell_{S}(E_{1})=\mathcal{C}\ell_{S}(E_{2})$ [15].
## III Union and USt Codes
Let $S$ be the stabilizer group of code $C=[[n,k,d^{\prime}]]_{d}$. Let
$\mathcal{C}_{C}$ be the centralizer of $S$ and $\mathcal{T}$ be a subset of a
transversal set of $G_{d}^{n}/\mathcal{C}_{C}$. Then, the set
$\mathcal{Q}=\bigoplus_{t\in\mathcal{T}}tC$ (2)
is a quantum code with parameters $((n,Kd^{k},d^{\prime\prime}))_{d}$, where
$K=\\#\mathcal{T}$ and $d^{\prime\prime}\leq d^{\prime}$.
Note that if $t_{1},t_{2}\in\mathcal{T}$ are distinct, then $t_{1}C\perp
t_{2}C$. In fact, $t_{1}^{\dagger}t_{2}\notin\mathcal{C}_{C}$, therefore there
exists $s\in S$ such that $st_{1}^{\dagger}t_{2}=\alpha t_{1}^{\dagger}t_{2}s$
and $\alpha\neq 1$, that obeys
$\left<i\right|t_{1}^{\dagger}t_{2}\left|j\right>=0$, where
$\left|i\right>,\left|j\right>\in C$. If $B=\\{\left|w_{i}\right>\\}$ is a
base for $C$, then $\bigcup_{t\in\mathcal{T}}tB$ is a basis for $\mathcal{Q}$.
$\mathcal{Q}$ is called Union Stabilizer Code(USt) [23, 24, 18, 25].
More general yet is what we call a quantum union code. Let $C_{1}$ and $C_{2}$
be two quantum codes with parameters $((n,K_{1},d_{1}))$ and
$((n,K_{2},d_{2}))$, respectively. Let $B_{1}$ ($B_{2}$) be an orthogonal
basis of $C_{1}$ ($C_{2}$). Suppose that $C_{1}\perp C_{2}$, then
$B=B_{1}\bigcup B_{2}$ also is a basis for a vector space, which is the union
code. Note that the union code is given by $C=C_{1}\oplus C_{2}.$ Note that a
USt code is also a union code [26].
## IV Projectors on Union Codes
In this section, we show how to find the projector of a union code. Let $M$,
$P(M)$ and $P_{M}$ be a measurement operator, the space stabilized by $M$, and
the orthogonal projector on $P(M)$, respectively. We have $M=2P_{M}-I$. We
also use the notation $P_{Q}$ for the projector of a generic code $Q$.
Let $M_{Q}$ be the measurement operator of code $Q$. If
$\\{\left|w_{1}\right>,\dots,\left|w_{k}\right>\\}$ is an orthogonal basis of
the code, then
$P_{Q}=\sum_{i=1}^{k}\left|w_{i}\right>\left<w_{i}\right|,$
and
$\displaystyle M_{Q}$ $\displaystyle=$ $\displaystyle
2\sum_{i=1}^{k}\left|w_{i}\right>\left<w_{i}\right|-I$ $\displaystyle=$
$\displaystyle-\prod_{i=1}^{k}(I-2\left|w_{i}\right>\left<w_{i}\right|).$
Suppose that $Q$ is a union code $Q=C_{1}\oplus C_{2}$ and let $P_{1}$ and
$P_{2}$ be the projectors on codes $C_{1}$ and $C_{2}$, respectively. Since
$C_{1}\perp C_{2}$, the projector on $Q$ is $P=P_{1}\oplus P_{2}$ and the
measurement operator is $M_{Q}=2P_{1}\oplus P_{2}-I$.
Suppose that the union code $Q$ has the form
$\mathcal{Q}=\bigoplus_{t\in T}tQ_{0},$ (3)
where $Q_{0}$ is a quantum code. How does one find the code measurement
operator using operator $M_{0}$ of code $Q_{0}$? We answer this question
below.
Let $M_{1}$ and $M_{2}$ be two commutative measurement operators, we define
the measurement operator $M_{1}\wedge M_{2}$ as the operator that stabilize
the space $P(M_{1})\bigcap P(M_{2})$. We have that the projector associated to
this operator satisfies $P_{M_{1}\wedge M_{2}}=P_{M_{1}}P_{M_{2}}$.
Let $A$ and $B$ be two vector subspaces of $\mathbb{C}^{m}$. Define the
following associative operation:
$A\triangle B=(A\cap B^{\perp})\oplus(A^{\perp}\cap B).$ (4)
Let $M_{1}$ and $M_{2}$ be two commutative measurement operators. The
measurement operator associated with the space $P(M_{1})\triangle P(M_{2})$ is
denoted by $M_{1}\boxplus M_{2}$, that is
$\displaystyle P(M_{1}\boxplus M_{2})$ $\displaystyle=$ $\displaystyle
P(M_{1})\triangle P(M_{2})$ (5) $\displaystyle=$ $\displaystyle(P(M_{1})\cap
P(M_{2})^{\perp})\oplus$ $\displaystyle(P(M_{1})^{\perp}\cap P(M_{2})).$
Note that $P_{M_{1}\boxplus
M_{2}}=P_{M_{1}}(I-P_{M_{2}}^{\perp})+P_{M_{2}}(I-P_{M_{1}}^{\perp})$ and
$\displaystyle M_{1}\boxplus M_{2}$ $\displaystyle=$ $\displaystyle
2P_{M_{1}\boxplus M_{2}}-I$ (6) $\displaystyle=$ $\displaystyle
2P_{M_{1}}(I-P_{M_{2}}^{\perp})+$
$\displaystyle\,P_{M_{2}}(I-P_{M_{1}}^{\perp}))-I$ $\displaystyle=$
$\displaystyle-(2P_{M_{1}}-I)(2P_{M_{2}}-I)$ $\displaystyle=$ $\displaystyle-
M_{1}M_{2}.$
Consider again code $\mathcal{Q}$ of equation (3). Let $Q_{0}$ be the
stabilizer and $S=\\{G_{i}\\}_{i=1}^{k}$ is a set of stabilizers of code
$Q_{0}$. Define $Q_{t}=tQ_{0}$ and $M_{it}=tG_{i}t^{\dagger}$. Note that
$M_{it}$ stabilizes $t\left|w\right>$, for $\left|w\right>\in Q_{0}$. We have
$Q_{t}=\bigcap_{i=1}^{k}P(M_{it})$. Then
$\mathcal{Q}=\bigoplus_{t\in
T}\bigcap_{i=1}^{k}P(M_{i}t)=\mathop{\bigtriangleup}_{t\in
T}\bigcap_{i=1}^{k}P(M_{it})$ (7)
and the associated measurement operator is
$M_{\mathcal{Q}}=\mathop{\boxplus}_{t\in T}\bigwedge_{i=1}^{k}M_{it}.$ (8)
When code $Q_{0}$ is not additive (stabilizer), we use the classical way to
build the projector for a vector space employing an orthonormal basis. Let
$B=\\{\left|w_{i}\right>\\}_{i=1}^{l}$ be an orthonormal basis of $Q_{0}$. The
set $\bigcup_{t\in T}tB$ is an orthonormal basis for the code $\mathcal{Q}$.
Therefore, the projector of $\mathcal{Q}$ is
$P_{\mathcal{Q}}=\sum_{t\in
T}\sum_{i=1}^{l}t\left|w_{i}\right>\left<w_{i}\right|t^{\dagger}.$ (9)
## V Measurements on Union and USt Codes
Let $\mathcal{E}$ be a set of correctable errors of a quantum code and
$D\subset\mathcal{E}$. Define the nondegenerate complement of $D$ in
$\mathcal{E}$ as the set
$\mathcal{E}_{\overline{D}}=\\{E\in\mathcal{E};C_{EF}=0,\;\;\forall F\in
D\\}.$ (10)
When $\mathcal{E}$ is nondegenerate, $\mathcal{E}_{\overline{D}}$ is exactly
the complement of $D$, that is,
$\mathcal{E}_{\overline{D}}=\mathcal{E}\setminus D$. Following we use the
notation $D(Q)=\\{h\left|\psi\right>;h\in D,\;\left|\psi\right>\in Q\\}$.
###### Theorem V.1.
Let $Q$ be a CWS code and $D$ a finite commutative group of correctable
errors. Then $Q_{D}=D(Q)$ is a USt code.
###### Proof.
Let $B=\\{\left|w_{1}\right>,\dots,\left|w_{k}\right>\\}$ be a basis for code
$Q$. Then $D(B)=\bigcup_{\alpha\in D}\alpha(B)=\bigcup_{\alpha\in
D}\bigcup_{i=1}^{k}\alpha\left|w_{i}\right>=\bigcup_{i=1}^{k}D(\left|w_{i}\right>)$.
Note that $D(\left|w_{i}\right>)$ is an additive CWS code, that is, a
stabilizer code. Now we will show that $D(\left|w_{i}\right>)$ and
$D(\left|w_{j}\right>)$ are mutually orthogonal, for $i\neq j$.
Let $\alpha_{1},\alpha_{2}\in D$. Then $\alpha_{1}^{\dagger}\alpha_{2}$ a
detectable error and the error correction conditions
$\left<w_{i}\right|\alpha_{1}^{\dagger}\alpha_{2}\left|w_{j}\right>=0$ imply
that $D(\left|w_{i}\right>)$ is orthogonal to $D(\left|w_{j}\right>)$.
We have $\left|w_{i}\right>=w_{i}\left|0\right>$ for word operators $w_{i}$,
and $w_{i}d_{j}=\alpha_{ij}d_{j}w_{i}$, where $d_{j}\in D$ and
$\alpha_{ij}\in\mathbb{C}$. Let $D_{i}=\\{\alpha_{ij}d_{j};d_{j}\in D\\}$. We
have that the codes generated by $D({\left|0\right>})$ and by
$D_{i}(\left|0\right>)$ are the same. Then
$\bigcup_{i=1}^{k}D(\left|w_{i}\right>)=\bigcup_{i=1}^{k}w_{i}D_{i}(\left|0\right>)$.
Let $Q_{0}$ be the code generated by $D(\left|0\right>)$. We have
$D(Q)=\bigoplus_{i=1}^{k}w_{i}Q_{0},$
that is, $D(Q)$ is USt. ∎
###### Theorem V.2.
Let $Q$ be a CWS code defined from a stabilizer set $S$ and by a classical
code $C$. Let $D$ be a commutative group, the elements of which are in the set
of correctable errors $\mathcal{E}$. Then the USt code $Q_{D}=D(Q)$ detects
all errors in $\mathcal{E}_{\overline{D}}$.
###### Proof.
Let $E\in\mathcal{E}_{\overline{D}}$, $\alpha_{1},\alpha_{2}\in D$ and
$\left|w_{i}\right>,\left|w_{j}\right>\in Q$ such that
$\alpha_{1}\left|w_{i}\right>\perp\alpha_{2}\left|w_{j}\right>$. Then
$\alpha_{1}^{\dagger}\alpha_{2}\in D\subset\mathcal{E}$ and
$\left<w_{i}\right|\alpha_{1}^{\dagger}E\alpha_{2}\left|w_{j}\right>=a\left<w_{i}\right|\alpha_{1}^{\dagger}\alpha_{2}E\left|w_{j}\right>=0,\;\;\;\
a\in\mathbb{C},$
that is, $E$ is detectable. ∎
We have described how to detect errors in USt codes for the nonbinary case,
generalizing the method given in Ref. [18] for binary codes. When working with
the latter case, it is enough to use the criteria defined in Theo. V.2.
However, for nonbinary codes, the above theorem is not enough. In order to
find the error in this case, which is the main contribution of this paper, we
make use of union codes.
###### Theorem V.3.
Let $Q$ be a CWS code and $D$ a commutative group. $D\setminus\\{I\\}$ is a
set of nondegenerate and correctable errors. Let $D_{l}\subset D$ be the set
$\\{d_{l_{1}}^{k_{1}}\cdots d_{l_{t}}^{k_{t}};0<k_{i}<d\;\forall i\\}$, where
$d_{l_{i}}$ are $t$ independent generators of $D$. If
$D_{l_{s}}^{r}=D_{l}\setminus\\{d_{l_{1}}^{k_{1}}\cdots d_{l_{s}}^{r}\cdots
d_{l_{t}}^{k_{t}};0<k_{i}<d\;\forall i\neq s\\}$, then the code
$\mathbb{D}(Q)$ where $\mathbb{D}=D_{l}\setminus D_{l_{s}}^{r}$ detects
$\mathcal{E}=D_{l_{s}}^{r}$.
###### Proof.
Let $\alpha_{1},\alpha_{2}\in\mathbb{D}$, we have that
$\alpha_{1}^{\dagger}\alpha_{2}=a_{1}d_{l_{1}}^{k_{1}}\cdots
d_{l_{s}}^{0}\cdots d_{l_{t}}^{k_{t}}$, $a_{1}\in\mathbb{C}$. Now let
$E\in\mathcal{E}$, $\alpha_{1}^{\dagger}\alpha_{2}E=d_{l_{1}}^{h_{1}}\cdots
d_{l_{s}}^{h_{s}}\cdots d_{l_{t}}^{h_{t}}$ with $h_{s}\neq 0$. Then
$\alpha_{1}^{\dagger}\alpha_{2}E\in D\setminus\\{I\\}$ ie
$\alpha_{1}^{\dagger}\alpha_{2}E$ is a nondegenerate error of code $Q$. So
$\left<w_{i}\right|\alpha_{1}^{\dagger}E\alpha_{2}\left|w_{j}\right>=\left<w_{i}\right|\alpha_{1}^{\dagger}\alpha_{2}E\left|w_{j}\right>=C_{\alpha_{1}^{\dagger}\alpha_{2}E}\delta_{ij}=0,$
where $\left|w_{i}\right>,\left|w_{j}\right>$ belongs to a orthonormal basis
of code $Q$, in other words, $\mathbb{D}(Q)$ detect $E$. ∎
## VI Decoding
In the previous section, we have showed that, given a quantum code and a set
of correctable errors, we can build another code, which includes the first,
detecting the errors. However, we want to correct errors and not only to
detect them.
Suppose that the set of correctable errors $\mathcal{E}$ can be decomposed
into a union of finite abelian groups and nondegenerate elements, that is,
$\mathcal{E}=\bigcup_{j=1}^{t}D_{j}$. Using Theo. V.2, we can find the group
that contains the error. To correct the error we use the following strategy:
Suppose that the error is in group $D_{j}$ and it has $m$ generators, that is,
$D_{j}=\langle d_{1},\dots,d_{m}\rangle$. Define $D_{j}^{l}$ as the group with
a generator less, that is, $D_{j}^{l}=\langle
d_{1},\dots,\widehat{d_{l}},\dots,d_{m}\rangle$. By performing $m$
measurements, we find out that the error has the form
$E=d_{i_{1}}^{k_{1}}\cdots d_{i_{t}}^{k_{t}},$ (11)
where $t\leq m$ and $0<k_{i}<d-1$. It remains to find the value of $k_{i}$.
We use Theo. V.3. Take $D_{j}$ to be the group, and $D_{l}$ the subset of
$D_{j}$, with all errors found above. For all $s\in\\{1,\dots,t\\}$ and
$r\in\\{1,\dots,d-1\\}$, we perform the error detection in code
$D_{l_{s}}^{r}(Q)$. This is the last step to find the error.
## VII Example
We will use the decoder above described in the family of codes $((5,d,3))_{d}$
with $d>3$ presented in Ref. [21], which are nonadditive CWS codes. This
family is described by the stabilizer group
$S=\langle
X_{1}Z_{2}Z_{5},Z_{1}X_{2}Z_{3},Z_{2}X_{3}Z_{4},Z_{3}X_{4}Z_{5},Z_{1}Z_{4}X_{5}\rangle$
and by the word operators set $W=\\{Z^{v_{j}},Z^{a},Z^{b}\\}$, where
$v_{j}=(j,j,j,j,j)$, for $j\not\in\\{2,d-1\\}$, $a=(2,-1,-1,2,-1)$ and
$b=(-1,2,2,-1,2)$. Those codes correct all weight-1 quantum errors. Let
$\mathcal{E}$ be the set of all weight-1 errors including $I$.
To prove that this code can correct every weight-1 error, we use function
$\mathcal{C}\ell_{S}$ and the classical code associated to set of word
operators. The classical code is given by $C=\\{v_{j},a,b\\}$ and
$\displaystyle\mathcal{C}\ell_{S}(\mathcal{E})$ $\displaystyle=$
$\displaystyle\\{(r,0,0,0,0),(0,r,0,0,0),(0,0,r,0,0),$
$\displaystyle(0,0,0,r,0),(0,0,0,0,r),(r,0,0,r,0),$
$\displaystyle(0,r,0,0,r),(r,0,r,0,0),(0,r,0,r,0),$
$\displaystyle(0,0,r,0,r),(0,0,r,h,r),(r,0,0,r,h),$
$\displaystyle(h,r,0,0,r),(r,h,r,0,0),(0,r,h,r,0);$ $\displaystyle 0\leq
r<d,\;\;0<h<d\\}.$
Note that the difference between any two vectors in $C$ can not be equal to
the sum of any two elements of $\mathcal{C}\ell_{S}(\mathcal{E})$. This shows
that $C$ can correct weight-1 errors.
As a consequence of the graph structure of the stabilizer, every nonbinary
Pauli error acting on $\left|0\right>$ can be equivalently replaced by some
qudit phase flip errors [21]. We may consider all the word operators $w_{i}$
in the format $w_{i}=Z^{c_{i}}$.
Now, write $\mathcal{E}=\bigcup_{i=1}^{5}D_{i}$, where $D_{i}$ are the groups
generated by $\\{Z^{Cl_{S}(X_{i})},\,Z^{Cl_{S}(Z_{i})}\\}$, $1\leq i\leq 5$.
These groups satisfy Theo. V.1 and V.2. So, the decoder can be applied and
with at most $5-1=4$ USt measurements in order to detect the group error
$D_{i}$, that is, to locate the error. After these measurements, we will
perform two more USt measurements to determine if some of the generators of
$D_{i}$ are missing in the expression of the error. Then, we construct the
groups $D_{i}^{1}=\langle Z^{Cl_{S}(X_{i})}\rangle$ and $D_{i}^{2}=\langle
Z^{Cl_{S}(Z_{i})}\rangle$, perform measurements according to the USts codes
$D_{i}^{1}(Q)$ and $D_{i}^{2}(Q)$. We can find out, for example, in the worst
case, that no generator of $D_{i}$ is missing in the expression of the error.
To obtain explicitly the power associated with the generators, we use Theo.
V.3. For each fixed $r_{1}$, $0<r_{1}<d$, consider the sets
$D_{r}^{1}=\\{(Z^{Cl_{S}(X_{i})})^{r_{1}}(Z^{Cl_{S}(Z_{i})})^{k}\\}$, $0<k<d$,
and the union code $D_{r}^{1}(Q)$. Performing a measurement in this union
code, we obtain power $r_{1}$ of the first generator. To obtain the second
power, we repeat the procedure with the set
$D_{r}^{1}=\\{(Z^{Cl_{S}(X_{i})})^{k}(Z^{Cl_{S}(Z_{i})})^{r_{2}}\\}$, $0<k<d$,
obtaining the expression of the error
$E=(Z^{Cl_{S}(X_{i})})^{r_{1}}(Z^{Cl_{S}(Z_{i})})^{r_{2}}$. The number of
union code measurements is $2d-2$, because we perform $d-1$ measurements to
obtain power $r_{1}$ and $d-1$ measurements for power $r_{2}$.
## VIII Conclusion
The formalism of CWS codes is a procedure to find both, additive and
nonadditive codes, which generalizes the formalism of stabilizer codes. It was
used to build some optimal nonadditive codes, such as codes ((9,12,3)) and
((10,24,3)) [17]. A decoding procedure for binary codes of this class was
described recently [18].
We have described a decoding procedure for the nonbinary case. Part of the
procedure is a straightforward generalization of the binary case, using union
of stabilizer codes (USt). In the binary case, dealing with USts codes is
enough, whereas in the nonbinary case, after finding the group error $D$ and
$D_{l}\subset D=\\{d_{l_{1}}^{k_{1}}\cdots
d_{l_{t}}^{k_{t}};0<k_{i}<d\;\forall i\\}$, where $d_{l_{i}}$ are $t$
independent generators of $D$, we have to use union code measurements to
obtain $k_{i}$ using Theo. V.3.
## Acknowledgement
We thank CNPq’s financial support.
## References
* [1] P. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in _Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on_ , nov 1994, pp. 124 –134.
* [2] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in _Proceedings of the twenty-eighth annual ACM symposium on Theory of computing_ , ser. STOC ’96. New York, NY, USA: ACM, 1996, pp. 212–219. [Online]. Available: http://doi.acm.org/10.1145/237814.237866
* [3] M. Mosca, “Quantum algorithms,” in _Encyclopedia of Complexity and Systems Science_ , 2009, pp. 7088–7118.
* [4] A. M. Childs and W. van Dam, “Quantum algorithms for algebraic problems,” _Rev. Mod. Phys._ , vol. 82, pp. 1–52, Jan 2010. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.82.1
* [5] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist,” _Phys. Rev. A_ , vol. 54, pp. 1098–1105, Aug 1996. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.54.1098
* [6] A. M. Steane, “Simple quantum error-correcting codes,” _Phys. Rev. A_ , vol. 54, pp. 4741–4751, Dec 1996. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.54.4741
* [7] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum error correcting code,” _Phys. Rev. Lett._ , vol. 77, pp. 198–201, Jul 1996. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.77.198
* [8] D. Gottesman, “Class of quantum error-correcting codes saturating the quantum hamming bound,” _Phys. Rev. A_ , vol. 54, pp. 1862–1868, Sep 1996. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.54.1862
* [9] E. Knill and R. Laflamme, “Concatenated quantum codes,” _eprint arXiv:quant-ph/9608012_ , Aug. 1996.
* [10] M. Grassl, P. Shor, and B. Zeng, “Generalized concatenation for quantum codes,” in _Information Theory, 2009. ISIT 2009. IEEE International Symposium on_ , 28 2009-july 3 2009, pp. 953 –957.
* [11] M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, “Generalized concatenated quantum codes,” _Phys. Rev. A_ , vol. 79, p. 050306, May 2009\. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.79.050306
* [12] S. Beigi, I. Chuang, M. Grassl, P. Shor, and B. Zeng, “Graph concatenation for quantum codes,” _Journal of Mathematical Physics_ , vol. 52, no. 2, p. 022201, 2011. [Online]. Available: http://link.aip.org/link/?JMP/52/022201/1
* [13] I. Chuang, A. Cross, G. Smith, J. Smolin, and B. Zeng, “Codeword stabilized quantum codes: Algorithm and structure,” _Journal of Mathematical Physics_ , vol. 50, no. 4, p. 042109, 2009. [Online]. Available: http://link.aip.org/link/?JMP/50/042109/1
* [14] X. Chen, B. Zeng, and I. L. Chuang, “Nonbinary codeword-stabilized quantum codes,” _Phys. Rev. A_ , vol. 78, p. 062315, Dec 2008. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.78.062315
* [15] A. Cross, G. Smith, J. Smolin, and B. Zeng, “Codeword stabilized quantum codes,” in _Information Theory, 2008. ISIT 2008. IEEE International Symposium on_ , july 2008, pp. 364 –368.
* [16] J. A. Smolin, G. Smith, and S. Wehner, “Simple family of nonadditive quantum codes,” _Phys. Rev. Lett._ , vol. 99, p. 130505, Sep 2007. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.99.130505
* [17] S. Yu, Q. Chen, C. H. Lai, and C. H. Oh, “Nonadditive quantum error-correcting code,” _Phys. Rev. Lett._ , vol. 101, p. 090501, Aug 2008. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.101.090501
* [18] Y. Li, I. Dumer, M. Grassl, and L. P. Pryadko, “Structured error recovery for code-word-stabilized quantum codes,” _Phys. Rev. A_ , vol. 81, p. 052337, May 2010. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.81.052337
* [19] A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes,” _Information Theory, IEEE Transactions on_ , vol. 47, no. 7, pp. 3065 –3072, nov 2001.
* [20] A. Ketkar, A. Klappenecker, S. Kumar, and P. Sarvepalli, “Nonbinary stabilizer codes over finite fields,” _Information Theory, IEEE Transactions on_ , vol. 52, no. 11, pp. 4892 –4914, nov. 2006.
* [21] D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, “Graphical nonbinary quantum error-correcting codes,” _Phys. Rev. A_ , vol. 78, p. 012306, Jul 2008. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.78.012306
* [22] E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” _Phys. Rev. A_ , vol. 55, pp. 900–911, Feb 1997. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.55.900
* [23] M. Grassl and M. Rotteler, “Quantum goethals-preparata codes,” in _Information Theory, 2008. ISIT 2008. IEEE International Symposium on_ , july 2008, pp. 300 –304.
* [24] ——, “Non-additive quantum codes from goethals and preparata codes,” in _Information Theory Workshop, 2008. ITW ’08. IEEE_ , may 2008, pp. 396 –400.
* [25] Y. Li, I. Dumer, and L. P. Pryadko, “Clustered error correction of codeword-stabilized quantum codes,” _Phys. Rev. Lett._ , vol. 104, p. 190501, May 2010. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.104.190501
* [26] M. Grassl and T. Beth, “A note on non-additive quantum codes,” _eprint arXiv:quant-ph/9703016_ , Mar. 1997.
|
arxiv-papers
| 2012-04-10T17:03:56 |
2024-09-04T02:49:29.557489
|
{
"license": "Public Domain",
"authors": "Nolmar Melo and Douglas F.G. Santiago and Renato Portugal",
"submitter": "Nolmar Melo",
"url": "https://arxiv.org/abs/1204.2218"
}
|
1204.2231
|
# Investigating Keyphrase Indexing with Text Denoising
Rushdi Shams
Robert E. Mercer
rshams@csd.uwo.ca mercer@csd.uwo.ca
###### Abstract
In this paper, we report on indexing performance by a state-of-the-art
keyphrase indexer, Maui, when paired with a text extraction procedure called
text denoising. Text denoising is a method that extracts the denoised text,
comprising the content-rich sentences, from full texts. The performance of the
keyphrase indexer is demonstrated on three standard corpora collected from
three domains, namely food and agriculture, high energy physics, and
biomedical science. Maui is trained using the full texts and denoised texts.
The indexer, using its trained models, then extracts keyphrases from test sets
comprising full texts, and their denoised and noise parts (i.e., the part of
texts that remains after denoising). Experimental findings show that against a
gold standard, the denoised-text-trained indexer indexing full texts, performs
either better than or as good as its benchmark performance produced by a full-
text-trained indexer indexing full texts.
###### category:
H.3.1 Information Storage and Retrieval Content Analysis and Indexing
###### keywords:
Indexing method
###### category:
H.3.3 Information Storage and Retrieval Information Search and Retrieval
###### category:
H.3.4 Information Storage and Retrieval Systems and Software
###### keywords:
Performance evaluation (efficiency and effectiveness)
###### keywords:
Keyphrase extraction, topic extraction, indexing, text denoising, keyphrase
indexer, machine learning model, fog index
††terms: Experimentation, Performance.
## 1 Introduction
Since they provide high-level descriptions of document contents, keyphrases
serve as the meta-descriptions as well as a means to effective document
retrieval from digital libraries. Other reasons to use keyphrases include but
are not limited to document similarity measure, classification and clustering,
topic search, web tag clouds and document summarization [5].
Today, automatic keyphrase indexing is a popular notion which eliminates
several drawbacks of manual indexing such as conflicting time and effort, and
poor choice of keyphrases. Among the automatic keyphrase indexers, several are
tested across domains [5][6][9][11][15][16] while many are domain-specific
[3]. Most of these indexers are trained with full documents using algorithms
like Naïve Bayes and Bagging to extract keyphrases from full-text test
documents. A revealing experiment by Witten et al. Witten-et-al:1999
demonstrates that the performance of the indexers depends not only on these
features but also on document size. As they apply their full-text trained
Keyphrase Extraction Algorithm (hereinafter, KEA) on paper abstracts and
compare against a gold standard, they find its performance on these reduced
texts somewhat inferior and not competitive to that on full texts. The authors
concluded that this anomaly was unequivocal as fewer author-assigned
keyphrases appear in the chosen reduced texts than in the entire document.
Text Denoising is a method proposed by Shams and Mercer Shams:2011 which
reduces the amount of text in biomedical papers to 30% of the original. This
30% of the text is selected based on the Fog Index readability score [4] and
is called denoised text; the remaining text is called the noise text. In this
introductory work denoised text is shown to be the more content-rich portion
of the full text as it contains most of the biomedical concepts that are
explicitly or implicitly connected with biomedical relations. Although tests
have been carried out only with biomedical research articles, the authors
conclude that Fog Index can be a useful indicator of content richness for
other genres and different purposes although the threshold of 30% might need
to be reconsidered.
In this paper, we report on the performance of a state-of-the-art keyphrase
indexer named Maui [9] when paired with text denoising. We use three standard
full text corpora from the food and agriculture, high energy physics, and
biomedical science domains. From each corpus, we develop training sets
comprising full texts and their denoised parts. The test sets are composed of
full texts, and their denoised and noise parts. For training and testing each
dataset, we use a standard 10-fold cross validation. We show experimentally
that although a threshold of 30% performs well for biomedical relation
extraction, it is 70% for keyphrase indexing. To evaluate Maui, we use
quantitative measures like precision, recall and F-score as well as
qualitative measures like inter-indexer agreements. Experimental results show
that Maui, with denoised texts, performs either better or comparably to its
benchmark performance—those with full-text trained models to extract
keyphrases from full-text test sets.
The remainder of this paper provides background on text denoising and the
keyphrase indexer, Maui (Section 2), and discusses the methods for training
and testing the indexer (Section 3), an analysis of the results (Section 4),
and ends with some concluding remarks (Section 5).
## 2 Background
In this section, we briefly discuss the text denoising method as well as Maui,
the keyphrase indexer.
### 2.1 Text Denoising
In their paper, Witten et al. [5] have demonstrated that the performance of
the indexer KEA has been reduced when extracting keyphrases from paper
abstracts. Similarly, the performance of biomedical relation miners that
attempt to extract relations among drugs, chemicals, diseases, genes and
proteins from paper abstracts is such that a number of biomedical ontologies
like OMIM (Online Mendelian Inheritance in Man) and GO (Gene Ontology) use
human annotators to extract relations from full texts. This procedure is time-
consuming as well as error-prone. To overcome these shortcomings, Shams and
Mercer Shams:2011 proposed a method that identifies those areas within a text,
called denoised text, where content information, such as biomedical relations,
is more likely to occur. The authors suggested that the describing of
biomedical relations lengthens sentences and increases the use of polysyllabic
words. Some readability indexes, the Fog Index [4] in particular, are based on
these two factors. They proceeded to use Fog Index to measure sentence
readability and showed experimentally that the 30% of the sentences which had
the lowest-readability, the denoised part of a text, contained the relations
of interest.
Figure 1 illustrates the text denoising method applied to biomedical texts.
Text Denoising has been evaluated with a corpus comprising 24 full texts that
describe four related pairs of disease and chemical components. This method
extracted pairs of biomedical concepts from the denoised part of the texts of
which about 75 percent are reported as related according to the Unified
Medical Language System’s (UMLS) semantic relations network. It is noteworthy
that the rest of the text, called noise text, did not contain any related
biomedical concepts of interest.
Figure 1: Text denoising and connected concept extraction method described by
Shams and Mercer [14]
### 2.2 Maui
Automatic keyphrase indexing has been in practice for half a century [8], but
until recently the performance was not notable.
Maui111http://code.google.com/p/maui-indexer/ [9] is the final successor of a
legacy of keyphrase indexers and inherits from and builds upon both of its
predecessors KEA [5][16] and KEA++222http://www.nzdl.org/Kea/ [11]. Maui uses
$13$ features (among them are tf$\times$idf and first occurrence (i.e., the
number of words preceding a keyphrase normalized by the total number of words
in a document) inherited from KEA; node degree (i.e., the number of
connections between a candidate phrase and the other candidates in the SKOS
hierarchy) and keyphrase length inherited from KEA++) to develop machine
learning models and extract keyphrases. Maui’s own features include tf, idf,
last occurrence, spreads (i.e., number of words between first and last
occurrence of a phrase), semantic relatedness and generality (i.e., position
in the vocabulary hierarchy).
Furthermore, Maui uses a feature named keyphraseness (i.e., the probability
that a candidate phrase is present in the training documents) for domain-
specific keyphrase extraction. Frank et al. [5] reported that the use of this
feature increases indexers’ performances if their training and test documents
are retrieved from the same domain. Maui, moreover, can be incorporated with
any domain-specific controlled vocabulary written in
SKOS333http://www.w3.org/2001/sw/wiki/SKOS/Datasets (Simple Knowledge
Organization System) hierarchical format. Besides domain-specific keyphrase
indexing, Maui has the capability for both free-text indexing and indexing for
any domain that lacks a controlled vocabulary. For the latter case, Maui uses
Wikipedia as a domain-independent controlled vocabulary.
Maui has been evaluated with texts from three different domains: food and
agriculture, nuclear physics, and biomedical science. Although the performance
is not comparable to that of English, it has also been tested with texts
written in French and Spanish. Experimental outcomes show that Maui
outperforms its predecessors in both cases [9]. Moreover, it performs
significantly better than Medical Text Indexer
(MTI444http://skr.nlm.nih.gov/SKR_API/index.shtml) [3] and
BibClassify555http://invenio-demo.cern.ch/help/hacking/bibclassify-admin-guide
[12] for indexing biomedical and physics texts, respectively.
Figure 2: Keyphrase extraction from k-th fold
## 3 Methodology
In this section, we describe the datasets, training and testing procedure,
performance measures, and the means to find the appropriate text denoising
threshold for keyphrase indexing.
We use the datasets and follow the experimental protocols set by Medelyan [9]
except that we train Maui not only on full texts but also on their denoised
parts and test it on full texts as well as their denoised and noise parts.
### 3.1 Datasets
In this experiment, to train and test Maui, we use three standard corpora of
full texts and keyphrases associated with them from three different domains.
These corpora were collected by Medelyan [9] during her doctoral research.
The first dataset, which is referred to as FAO-780, contains $780$ full-text
documents and their keyphrases. The documents have been selected randomly from
the Food and Agriculture Organization (FAO) data repository. The dataset
contains about $24$ million words ($30,800$ words on average per document) and
$6,225$ keyphrases ($8$ keyphrases on average per document, ranging from $2$
to $23$). An in-depth analysis of the documents reveals that the set is
composed of both research articles and experimental reports.
Our second dataset comprises $290$ full-text documents and their keyphrases on
high energy physics randomly collected from the European Organization for
Nuclear Research (abbreviated as CERN) document server and thus named
CERN-290. Each document contained therein has an average of $6,300$ words and
$7$ keyphrases. CERN-290 is the smallest dataset that we use in this
experiment and contains mainly experimental reports.
We also used the NLM-500 corpus, collected by the NLM Indexing Initiative [2]
during the development of MTI, which consists of $500$ biomedical research
articles. This corpus has documents with average length of $4,500$ words and
an average number of assigned keyphrases of $15$ ranging from $2$ to $30$. The
contents of the dataset are mainly scholarly research articles collected from
the National Library of Medicine repository.
### 3.2 Training and Testing
In our first attempts at pairing Maui with reduced texts, we noted that Witten
et al. [16], using the Computer Science Technical Reports (CSTR) corpus,
showed that any training set containing more than $25$ documents has very
little effect on the indexer’s performance. In our initial experiments with
Maui we followed this protocol by randomly choosing $25$ training and $100$
test documents from the NLM-500 corpus. Against a gold standard—author-
assigned keyphrases for the $100$ test documents—we measured Maui’s precision,
recall and F-score. From these experiments, we have seen that
* •
although the performance of Maui with the denoised text trained model is
better than that with full-text trained model, the improvement is not
statistically significant and the improvement does not reflect on the entire
population, and
* •
Maui’s performance improves if we increase the text denoising threshold from
30% to 40%.
The first observation indicates that the methods followed by Witten et al.
[16] can be effective for certain domains but are not an effective means for
many others while the latter indicates that for keyphrase indexing, the text
denoising threshold is not 30%.
Therefore, we decided to use a more conventional k-fold experimental approach.
We followed the experimental procedure illustrated in Figure 2. We consider
the full texts from each dataset and divide them randomly into 10 equal-sized
folds where the documents in one fold do not overlap with the others. In
addition, we keep the denoised and noise parts of each fold. Then, we apply a
standard 10-fold cross validation to train and test Maui. To generate each
pair, we keep one of the $10$ folds out as our testing set and combine the
rest of the $9$ folds as our training set. Doing this $10$ times, each time
leaving out a different one from the $10$ folds as a testing set, we get $10$
pairs. We train Maui on the training sets comprising full texts and denoised
texts from each fold. In this way, we develop $20$ trained models for the
entire $10$ folds. The models the indexers develop from full texts are called
full-text trained models and those that are developed from denoised texts are
called denoised text trained models.
As the trained models are created, the indexers then apply them, k-th full-
text trained model and k-th denoised text trained model, to extract keyphrases
from the k-th test set composed of full texts, and their denoised and noise
parts. According to the average number of keyphrases in every document, we had
the indexers extract $8$ keyphrases, $7$ keyphrases, and $15$ keyphrases for
each document in the FAO-780, CERN-290, and NLM-500 test sets, respectively.
The extracted keyphrases are then compared against a gold standard which are
the author assigned keyphrases associated with the test documents. The testing
has been carried out for the rest of the folds and the performance measures
described in section 3.3 are then averaged.
It is noteworthy that during training and testing, we used controlled
vocabularies for the respective domains, and during testing we set the minimum
and maximum length of the keyphrases to be extracted per document to $1$ and
$5$, respectively, as this is the default setting of Maui.
(a) Error rates for different denoising thresholds with denoised texts
(b) Error rates for different denoising thresholds with noise texts
Figure 3: Text denoising threshold for FAO-780 dataset
### 3.3 Performance Measures
In this experiment, we use the conventional quantitative measures for
performance evaluation—precision, recall and F-score. In addition, we use
three inter-indexing agreement measures popularly used for qualitative
indexing assessment [10]. The measures are called Hooper’s (H), Roll-ing’s (R)
and Cosine (C) inter-indexing agreements. The common property of these
agreement measures is that they provide the number of correct keyphrases in
relation to the size of the two sets of keyphrases being compared. We briefly
summarize these agreement measures for the reader’s convenience. If M and N
are the number of idiosyncratic keyphrases assigned by two indexers and O is
the number of phrases two indexers have in common, then Hooper’s measure [7]
is
$H(\textit{indexer}_{1},\textit{indexer}_{2})=\frac{O}{M+N-O}.$
Similarly, Rolling’s measure [13] is defined as
$R(\textit{indexer}_{1},\textit{indexer}_{2})=\frac{2{\cdot}O}{M+N}.$
Cosine measure uses the geometric mean instead of Rolling’s arithmetic mean.
Thus, Cosine measure can be written as
$C(\textit{indexer}_{1},\textit{indexer}_{2})=\frac{O}{\sqrt{M{\cdot}N}}.$
The last two measures are almost identical unless the sets radically vary. It
can be noted that Hooper’s and Rolling’s measures are identical to Jaccard’s
coefficient and the Dice coefficient, respectively, which are used to measure
statistical similarity between two sets. The closer the agreement measures are
to 1, the more the indexers agree on extracted keyphrases.
We also calculate the error rates for every cross validation. The error rate
is defined as
$E=\frac{\textit{FP}+\textit{FN}}{N},$
where FP and FN are the number of false positives and false negatives,
respectively and N is the total number of instances in the test sets. The
reasons for using the error rates are twofold, first, to find a text denoising
threshold described in Section 3.4 and second, to measure a 10-fold cross
validated paired t-test [1] to report significant improvement of Maui when
paired with denoised texts. For any two given sets of results, we consider
their error rates to calculate a paired t-value. If this calculated t-value
lies outside $\pm 2.26$ with a degree of freedom $9$, then the difference
between the set whose results have the lower error rate and the other set is
said to be statistically significant at significance level $\alpha$ = $0.05$.
(a) Error rates for different denoising thresholds with denoised texts
(b) Error rates for different denoising thresholds with noise texts
Figure 4: Text denoising threshold for CERN-290 dataset
(a) Error rates for different denoising thresholds with denoised texts
(b) Error rates for different denoising thresholds with noise texts
Figure 5: Text denoising threshold for NLM-500 dataset
### 3.4 Text Denoising Threshold
To find the appropriate text denoising threshold for keyphrase indexing, we
evaluate Maui’s performance on each dataset by increasing the text denoising
threshold in increments of 10% from 30% to 90%. As we vary the threshold, we
plot the error rates of Maui on different test sets. Because Maui applies a
supervised learning algorithm to develop its trained models, the best-fitted
model should be where the test error has its global minimum. Therefore, the
objective of this plotting is to discover the global minimum with its Full-
text and Denoised-text trained models. This global minimum will eventually be
the denoising threshold.
As an example, Figure 3 shows the error rates for different denoising
thresholds. It is notable that when we use the Denoised-text trained models to
extract keyphrases from either of the test sets, or use the Full-text trained
models on Denoised-text test sets, Maui has its global minimum at 70%
denoising (Figure 3a). From this point on, the error rate increases and thus
indicates an overfitting in Maui’s models. Figure 3b, on the other hand, shows
that no matter which trained model is used, full-text or denoised, the error
rate for noise test sets increases with increasing thresholds. This indicates
that noise texts are not content-rich as Maui fails to extract a substantial
number of keyphrases from them. Figure 3 shows that Maui’s best performing
pair is Denoised-Full—those models that are trained with denoised texts for
keyphrase extraction from full texts.
Similarly, Maui’s best-fitted models with denoised texts for the CERN-290
dataset are also at the 70% threshold (Figure 4a). Maui’s models—full-text or
denoised—experience overfitting after this threshold. Similar to what we
observed for the FAO-780 dataset, Maui has no improvement with either of its
trained models on the noise test set (Figure 4b). Maui best performs on the
CERN-290 dataset with its full-text trained models to extract keyphrases from
denoised texts (Figure 4), unlike that on FAO-780.
Maui’s best-fitted models with denoised texts for the NLM-500 corpus are also
at 70% threshold, except for the full-text trained model on denoised-text test
sets, (Figure 5a). In fact, Maui does not have any global minimum when it
applies a full-text trained model on denoised test sets until its denoising
threshold is set at 90%. Maui’s performance with noise texts from this domain
is similar to that from the other domains (Figure 5b). Like its performance on
FAO-780 dataset, Maui best performs on full text test sets with its denoised
text trained model for NLM-500.
These observations lead us to set the denoising threshold at 70%. At this
threshold, Maui predicts keyphrases from unseen test examples most accurately.
| Benchmark Performance |
---|---|---
Trained Model | Test Set | Precision | Recall | F-score | Precision | Recall | F-score | t value
Denoised Text | Denoised Text | 30.02 | 32.92 | 31.36 | 30.56 | 33.47 | 31.86 | 2.23
Denoised Text | Full Text | 30.49 | 33.50 | 31.87 | 2.76
Full Text | Denoised Text | 30.48 | 32.96 | 31.63 | 1.81
(a) Maui’s performance on FAO-780 dataset with 70% of the texts
| Benchmark Performance |
---|---|---
Trained Model | Test Set | Precision | Recall | F-score | Precision | Recall | F-score | t value
Denoised Text | Denoised Text | 24.38 | 25.33 | 24.79 | 24.58 | 25.56 | 24.99 | 2.16
Denoised Text | Full Text | 23.99 | 24.95 | 24.42 | 2.26
Full Text | Denoised Text | 24.38 | 25.40 | 24.82 | 1.31
(b) Maui’s performance on CERN-290 dataset with 70% of the texts
| Benchmark Performance |
---|---|---
Trained Model | Test Set | Precision | Recall | F-score | Precision | Recall | F-score | t value
Denoised Text | Denoised Text | 29.14 | 32.36 | 30.66 | 29.69 | 32.74 | 31.13 | 2.01
Denoised Text | Full Text | 29.96 | 33.22 | 31.50 | 3.52
Full Text | Denoised Text | 28.99 | 32.00 | 30.40 | 1.85
(c) Maui’s performance on NLM-500 dataset with 70% of the texts
Table 1: Precision, recall and F-score of Maui with text denoising
| Benchmark Performance
---|---
Trained Model | Test Set | Hooper | Rolling | Cosine | Hooper | Rolling | Cosine
Denoised Text | Denoised Text | 0.18 | 0.29 | 0.30 | 0.18 | 0.30 | 0.31
Denoised Text | Full Text | 0.18 | 0.30 | 0.31
Full Text | Denoised Text | 0.18 | 0.30 | 0.30
(a) Maui’s indexing agreements on FAO-780 dataset with 70% of the texts
| Benchmark Performance
---|---
Trained Model | Test Set | Hooper | Rolling | Cosine | Hooper | Rolling | Cosine
Denoised Text | Denoised Text | 0.14 | 0.24 | 0.24 | 0.14 | 0.24 | 0.24
Denoised Text | Full Text | 0.14 | 0.24 | 0.24
Full Text | Denoised Text | 0.14 | 0.24 | 0.24
(b) Maui’s indexing agreements on CERN-290 dataset with 70% of the texts
| Benchmark Performance
---|---
Trained Model | Test Set | Hooper | Rolling | Cosine | Hooper | Rolling | Cosine
Denoised Text | Denoised Text | 0.18 | 0.30 | 0.30 | 0.18 | 0.30 | 0.31
Denoised Text | Full Text | 0.19 | 0.31 | 0.31
Full Text | Denoised Text | 0.18 | 0.30 | 0.30
(c) Maui’s indexing agreements on NLM-500 dataset with 70% of the texts
Table 2: Inter-indexing agreements of Maui with text denoising
## 4 Results and Discussions
In this section, we discuss the performance of Maui with text denoising and
compare this with its benchmark performance.
Table 1a shows the precision, recall and F-score of Maui with denoised texts
and its benchmark performance on the FAO-780 dataset. Maui, as it uses its
denoised text and full-text trained models on denoised text test sets,
achieves F-scores of $31.36$ and $31.63$, respectively, compared to its
benchmark F-score of $31.86$. By applying the 10-fold cross validation t-test,
we see that for these two cases, the t-values are $2.23$ and $1.81$,
respectively, which means that the differences between the F-scores are not
statistically significant at $\alpha$ = $0.05$. In other words, the benchmark
performance of Maui cannot be said to be different than that with text
denoising. On the other hand, Maui’s F-score with its denoised text trained
model on full-text keyphrase extraction is $31.87$. A significance test shows
that its t-value is $2.76$ which indicates that it is different at a
significance level of $\alpha$ = $0.02$. So, with 98% confidence we can say
that the result is better than the benchmark performance. In addition, from
Table 2a, we can see that Maui’s agreements with the gold standards are as
good as the benchmark agreements. This demonstrates that the indexing quality
of Maui has not been compromised with text denoising.
For CERN-290 dataset, although Maui could not outperform its benchmark F-score
of $24.99$, none of the t-values are significant at $\alpha$ = $0.05$. In
other words, its performance with denoised texts cannot be said to be
different than its benchmark performance with a 95% confidence level. Its best
performance with denoised texts is when it uses the full-text trained model to
extract keyphrases from denoised texts (F-score of $24.82$). Maui’s
performance details on the CERN-290 dataset are given in Table 1b.
It is noteworthy that the performance of BibClassify, a specialized keyphrase
indexer developed by CERN for physics documents, on the CERN-290 documents is
$15.40$ precision, $24.3$ recall and $18.80$ F-score [9]. If we compare this
performance with Maui (Table 1b), then we can see that using 70% of the text,
Maui outperforms BibClassify.
Interestingly enough, although Maui agrees less with the gold standard
keyphrases for CERN-290 dataset than that for FAO-780, its agreements on
keyphrases with denoised texts are as good as its benchmark performance. The
details for its inter-indexing agreement measures on the CERN-290 dataset are
listed in Table 2b. Like its performance on FAO-780, we see that Maui extracts
quality keyphrases from physics documents when paired with text denoising
compared to that with full texts.
Maui’s best performance for the NLM-500 corpus is with a denoised text trained
model on full texts. Its F-score of $31.50$ outperforms the benchmark F-score
of $31.13$ at the significance level of $\alpha$ = $0.05$. However, although
its other two F-scores with denoised texts is somewhat lower than its
benchmark F-score, they are not statistically significant with t-values of
$2.01$ and $1.85$ (Table 1c). Maui’s inter-indexing agreement on NLM-500 is
somewhat similar to that on FAO-780 except that it agrees more with NLM-500
gold standards than its benchmark performance (Table 2c).
## 5 Conclusions
In this experiment, we consider full texts as well as their denoised and noise
parts from different domains like food and agriculture, physics, and
biomedical science. For each genre of texts, we have seen that Maui’s trained
models overfit if we set denoising threshold beyond 70%. Considering this as
our denoising threshold for keyphrase indexing, we test Maui on these texts,
full and reduced.
From its experimental results, we show in this paper that text denoising
improves Maui’s performance for the biomedical science texts, or it allows
Maui to perform as good as its benchmark performance on the food and
agriculture, and the physics texts. It does so by reducing Maui’s training and
test sets to 70%. For instance, the FAO-780 dataset of $24$ million words has
been reduced by text denoising to a set of $17$ million words and Maui does
not perform poorer than its benchmark performance. In other words, the $7$
million removed words are not potential candidates as keyphrases. Text
denoising, as expected, left out the words from being considered so.
Although there are some cases where Maui, when paired with text denoising,
experiences marginally lower F-score than its benchmark, indexing agreement
measures show that its indexing quality has never been compromised; it
extracts even better quality keyphrases from biomedical texts than its
benchmark.
It is noteworthy that during this experiment, we did not change the way Maui
works rather we tested it to see its response on a set of reduced texts.
Therefore, our experimental findings reveal that
* •
document size, per se, does not have the suggested effect on keyphrase
indexing—it is the content richness that plays the key role in indexing,
* •
text denoising produces a content-rich set of sentences which can improve
indexer performance,
* •
the noise texts, i.e., the removed text, do not improve indexing rather they
increase the error rates,
* •
text denoising is useful not only for biomedical relation extraction but also
for keyphrase indexing, and
* •
text denoising can be used for different domains other than biomedical science
With these results in mind and recalling that there are other indexers that
use different features to train their machine learning models, we are
interested in further investigating the effect of text denoising on them. In
addition, when paired with text denoising, Maui performs better on biomedical
texts than texts from agriculture and physics. Because it has been originally
developed for relation mining in biomedical texts, we are also interested to
explore the reasons behind the success of text denoising for keyphrase
indexing in this domain. These investigations are left for future work.
## 6 Acknowledgments
This work was partially funded through a Natural Sciences and Engineering
Research Council of Canada (NSERC) Discovery Grant to Robert E. Mercer. Thanks
are due to Olena Medelyan for providing the datasets and Maui– the indexer and
her time to time responses regarding our queries. We also acknowledge Ian H.
Witten for his experienced remarks on experiment setup and results.
## References
* [1] E. Alpaydin. Assessing and comparing classification algorithms. In Introduction to Machine Learning, pages 342–343. The MIT Press, Cambridge, UK, 2004.
* [2] A. Aronson, O. Bodenreider, H. Chang, S. Humphrey, J. Mork, S. Nelson, T. Rindflesch, and W. Wilbur. The nlm indexing initiative. Proceedings of American Medical Informative Association (AMIA) Symposium, 2000.
* [3] A. R. Aronson, J. G. Mork, C. W. Gay, S. M. Humphrey, and W. J. Rogers. The nlm indexing initiative’s medical text indexer. In Proceedings of the 11th World Congress on Medical Informatics Demner-Fushman and Lin Answering Clinical Questions (MEDINFO 2004), pages 268–272, San Francisco, USA, 2004.
* [4] R. G. Fog. Fog index after twenty years. Journal of Business Communication, 6:3–13, 1969.
* [5] E. Frank, G. Paynter, I. Witten, and C. Gutwin. Domain-specific keyphrase extraction. In Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI’99), pages 668–673, Stockholm, Sweden, 1999.
* [6] K. Frantzi, S. Ananiadou, and J. Tsujii. The c-value/nc-value method of automatic recognition for multi-word terms. In Proceedings of the 2nd European Conference on Research and Advanced Technology for Digital Libraries (ECDL’98), pages 585–604, London, UK, 1998.
* [7] R. S. Hooper. Indexer consistency tests: Origin, measurements, results and utilization. IBM, Bethesda, 1965.
* [8] H. P. Luhn. Keyword in context index for technical literature (kwic index). IBM Report no. RC 127, 1959.
* [9] O. Medelyan. Human-competitive automatic topic indexing. PhD thesis, University of Waikato, New Zealand, 2009.
* [10] O. Medelyan and I. Witten. Measuring inter-indexer consistency using a thesaurus. In Proceedings of the 6th ACM/IEEE-CS Joint Conference on Digital libraries (JCDL-2006), pages 274–275, NC, USA, 2006. ACM Press.
* [11] O. Medelyan and I. Witten. Domain-independent automatic keyphrase indexing with small training sets. Journal of the American Society for Information Science and Technology (JASIST), 59:1026–1040, 2008.
* [12] A. Pepe and J. Yeomans. Protocols for scholarly communication. Astronomical Society of Pacific Conference Series, 377, 2008.
* [13] L. Rolling. Indexing consistency, quality and efficiency. Information Processing and Management, 17:69–76, 1981.
* [14] R. Shams and R. E. Mercer. Extracting connected concepts from biomedical texts using fog index. Procedia - Social and Behavioral Sciences, 27:70 – 76, 2011. Elsevier Science.
* [15] P. Turney. Learning algorithms for keyphrase extraction. Information Retreival, 2:303–336, 2000.
* [16] I. Witten, G. Paynter, E. Frank, C. Gutwin, and C. Nevill-Manning. Kea: Practical automatic keyphrase extraction. In Proceedings of the 4th ACM conference on Digital Libraries, pages 254–255. ACM Press, 1999.
|
arxiv-papers
| 2012-04-10T17:57:42 |
2024-09-04T02:49:29.563030
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rushdi Shams and Robert E. Mercer",
"submitter": "Rushdi Shams Mr",
"url": "https://arxiv.org/abs/1204.2231"
}
|
1204.2235
|
# Publishing Identifiable Experiment Code And Configuration Is Important, Good
and Easy
Richard T. Vaughan Jens Wawerla
Simon Fraser University
{vaughan,wawerla}@sfu.ca
(10 November 2009)
## 1 Introduction
A few months ago, a graduate student in another country called me (Vaughan) to
ask for the source code of one of my multi-robot simulation experiments. The
student had an idea for a modification that she thought would improve the
system’s performance. By the standards of scientific practice this was a
perfectly reasonable request and I felt obliged to give it to her. With our
original code, the student could (i) re-run our experiments to verify that we
reported the results correctly; (ii) inspect the code to make sure that it
actually implements the algorithm described in our paper; (iii) change
parameters and initial conditions to make sure our results were not a fluke of
the particular experimental setting; (iv) modify the robot controllers and
quantitatively compare her new method with our originals. It would cost me
nothing to make her a copy of our code, and her methodology would be
impeccable. Why then do we read so few papers using this methodology?
It turned out to be impossible to identify exactly which code was used to
perform the experiments in our years-old paper. We had not labeled the source
code at that moment, and it had subsequently been modified. All the code was
under version control, so we could obtain approximately the right code by
looking at revision dates. But having only approximately the right code
strictly invalidates the replication of the experiments. The user has no way
of knowing what the differences are between the code she has and the code we
used. So we were able to offer the requesting student some code that may or
may not be that used in the paper. This was better than nothing, but not good
enough, and we suspect this is quite typical in our community.
This disappointing episode was a wake-up call for me, and our group has been
discussing how we can make sure this doesn’t happen again. We propose to
routinely publish the exact code for each experiment that we use to justify
any claims at all.
This paper explains why we think complete experiment publication is important
and why it is good for the originating researchers as well as subsequent
users. The second half of the paper examines current standards of scientific
data and code publications, and cites some evidence of its benefits. But first
we present our protocol for publishing identifiable code and show how easy it
is to do.
## 2 Publishing code is easy
### 2.1 What to do
1. 1.
The complete and exact source code, build scripts, configuration files, maps,
log analysis scripts, list of critical external dependencies, details of run-
time environment and any other instructions and resources necessary for a
skilled researcher to replicate the experiment should be packaged and placed
in public for free and anonymous download by any reader.
2. 2.
The source package should be labeled with a textual identifier that is
specific to the version described in the paper, and the paper should contain
the identifier.
### 2.2 Doing it easily
This can be easily and cheaply achieved as follows. The code is assembled into
an archive file (tarball, gzip, etc), and a digital signature is obtained
using a cryptographic hash function such as SHA1[24]. The archive is published
at some reliable Internet host, and its URI and signature are published in the
paper. The archive can also be linked from the authors’ web publication list.
Upon downloading the file, the user can determine that the archive matches the
signature in the paper. Use of a good hash identifier makes it very difficult
for authors to modify the code by mistake or design, without this being
detectable by the user/reader.
Modern software development tools make for an even easier process. Revision
control systems like Git111http://git-scm.com/ automatically generate a SHA1
cryptographic hash key for each committed version, such that there is a low
probability of any two packages or versions having the same key. The entire
revision control database can be easily cloned from an URI, and users can
check out the correct version by its signature, while still having access to
later versions. The differences between versions are easy to inspect using
Git’s tools. We have chosen this approach, and are hosting our Git
repositories at the independent host GitHub222http://github.com. While
GitHub’s tools and convenience are currently compelling, the ideal host would
be a reliable and long-lived independent institution such as a university,
national library or professional organization such as the IEEE.
The idea of identifying a code package with a crypotgraphic hash is not new –
on the contrary the standard way to distribute packages on the Internet such
that the downloader can trust that they received the “real thing” is for the
author to post a web page containing a URI to an archive file along with the
SHA1 hash of the same file. Our contribution here is to recommend this
practice to our community as a means to achieve the benefits of trustworthy
code publication.
## 3 Publishing code is important
### 3.1 Falsifiability and shared artifacts
Publishing the actual experiment alongside the paper which describes and
interprets it increases the scientific and practical value of the work. It
goes a long way to solving a problem our field faces from a philosophy of
science point of view: the fact that we are a synthetic science that creates
and studies artifacts, rather than a natural science that studies an extant
universe common to all scientists. By reproducing and sharing our artifacts we
synthesize a common environment.
Scientific claims are required to be falsifiable. If I make a claim in a paper
about a system I created, and to which you do not have access, my claim is not
falsifiable in practice. My claims are more scientifically valuable if I make
them as easy to falsify as possible, which I can achieve by publishing the
artifacts.
### 3.2 Repeatability and quantitative comparison
In the natural sciences experimental results gain credibility after they are
independently repeated at least once. In order to be able to repeat an
experiment, we often require many details that are not available in the paper.
As we are often able to make the exact and entire experiment available for
replication at negligible cost, we can achieve the best possible
repeatability.
Of course, we can not prove experiments are correct and while simply re-
running a program is not a strong validation, even this alone can show up
mistakes. A stronger validation is obtained by completely re-implementing the
code, or the important parts of it, but by testing the new version using the
original setting, as determined by inspecting the original code, we can
improve our confidence in the results.
As in all of science, much work in robotics can be considered incremental
improvement over the work of another. This usually requires reimplementing the
original experiment from natural language and formal mathematical
descriptions. This re–implementation step usually allows only qualitative
comparison, since the details of parameters and initial conditions, etc, are
rarely published. It can also be a source of error and raises question such as
“did the new author really find the very best parameter set?”. Experiments
made public in an executable form will improve fairness to the original author
and will allow quantitative comparison of results.
A second level of repeatability is available to us. Components of experiments
can be re-used in different experiments and settings. If the component
performs as expected in this new setting, our confidence in it increases. In
fact this re-use of code is a cheap way of reproducing experiments.
## 4 Publishing code is good
### 4.1 Efficiency
Having access to data sets and software implementations increases the
efficiency of the scientific process in several ways. In the case of
incremental work, it saves a great deal of re-implementation effort. While the
use of middleware like Player[8], ROS333http://www.ros.org, and Microsoft
Robotics Developer Studio444http://msdn.microsoft.com/robotics has increased
the rate of code reuse in recent years, these systems focus on low-level
components and it is still unusual for a robot controller or an implementation
of an algorithm to be substantially reused. Making code available by default
would encourage reuse, particularly if the code is of good quality.
### 4.2 Quality
The quality of a research contribution is a function of the soundness and
originality of its theoretical foundation, the depth of analysis given and the
clarity and thoroughness of its presentation. It is assumed that the software
that produces the results is correct. Yet it is all too easy to make
implementation mistakes that grossly influence the outcome of an experiment.
Even when a paper presents a complete formal algorithm, discrepancies between
the description and the implementation that produced the results are possible.
Such discrepancies are impossible to detect without access to the source code.
We can very easily make code available for peer review, and so we should.
Further, it is often argued that well written and documented software has
fewer bugs. Developing software with the expectation that it will be peer
reviewed and reused is likely to cause roboticists to write better code, thus
increasing the overall quality of the work even before external review. We
should write code as we write papers: to be read and understood; to contribute
to knowledge.
## 5 Issues and objections
Achieving code publication requires a number of issues to be addressed. Some
of the most significant are:
1. 1.
“I object! All that extra work takes too long….”
There are three arguments. First, while producing peer-reviewable code may
feel like it takes longer, the additional discipline and code review should
result in improved code quality. By reducing bug-hunting and re-runs of faulty
experiments, the experimenter could actually save time compared to a typical
messy code base. Second, starting with others’ published code saves time in
the first place. Third, extra work is justified by the methodological
advantages: the main role of the “extra” work is to improve the quality and
usefulness of the research results, thus it should not be considered overhead.
Some documentation is usually required for code to be usable. Extensive end-
user documentation can be very costly to produce, but often just a few notes
can be sufficient to guide a colleague through reproducing and understanding
an experiment. Documentation, particularly when written in-line with the code,
can also help the author’s confidence in its correctness. This idea is well
developed in Knuth’s literate programming model[1].
2. 2.
Offline vs. online For work that does not need to interactively control the
robot (e.g. offline SLAM) logs of original sensor data along with the
algorithm code allow perfect experiment reproduction. The Radish repository
exists to curate such datasets, and some of its maps have become familiar
555http://radish.sourceforge.net. Sharing interactive robot code is more
troublesome.
3. 3.
Real world and unique robots: For interactive controller experiments, complete
reproduction may be straightforward when experiments are done in simulation
only. Yet real-robot experiments are essential. The arguments for experiment
source sharing still hold for real robot systems, and the value of the work is
maximized if the authors facilitate replication and extension. This can be
done by using a well-known robot e.g. Pioneer, Khepera, which can be assumed
to be widely available in research labs around the world. Well-known robots
also have the advantage that respected simulation models are readily
available.
If a custom robot is essential, we suggest providing either (i) a model for a
well-known simulator, or (ii) a dedicated simulator including source code.
Also when using custom hardware, using a well-known and open API for
controller code, (e.g. Player, ROS) makes porting to another robot or
simulator as easy as currently possible. Ideally, in all cases where a
simulation can produce similar results to the real robot with a reasonable
amount of effort that simulation should be provided.
Work that uses novel mechatronics will not be reproducible at the very low
cost of software-only or common-robot systems, but we suggest that engineering
drawings, CAD models, materials and machining details can be published using
the same methodology. The goal is to minimize the cost of reproduction. Work
that does not require novel mechatronics should use a well known hardware and
software platform. Section 8 below discusses a recent major effort to
bootstrap a widely-used platform.
4. 4.
Licensing: The free reading, copying, modification and subsequent
redistribution of modified code is absolutely required. In most jurisdictions
copyright law automatically applies, so the code must be explicitly licensed
to allow redistribution. The community already makes extensive use of Free and
Open Source Software, so we have experience with suitable licenses. Licences
that prohibit commercial uses may be acceptable for traditional academic
purposes, but clearly make the code less valuable for some users (the
difference in value being what they are willing to pay to use the code under a
commercial license).
5. 5.
Trade secrets and competitive advantage: Some authors feel that since their
code is precious, by “giving it away” they give away their competitive
advantage. If a “competing” lab needs six months to replicate my experiment, I
can get further ahead in the meantime. While this position is tempting for the
individual, we are seeking advantages in efficiency and quality for the entire
community, including our taxpayer-supported funding agencies. Companies are
under no obligation to serve the community, but they can get the benefits
described above by first protecting their ideas with patents before
publication. If groups withhold their code for their own interest and against
the interest of the community, their work is manifestly less valuable than it
could be, and should be evaluated accordingly. Conversely, releasing high
quality code should enhance a group’s reputation and success rates. This
provides a feedback mechanism that reinforces code publication.
## 6 Encouraging Code Publication
How can the publication of source code be made a community norm? Assuming the
existence of a few suitable protocols, how can researchers be encouraged to
use them? Though we believe the research quality and efficiency benefits
should persuade many researchers, achieving such a large cultural change is
likely to require activism at various levels in the community.
At the most executive level, organizations such as the IEEE could make paper
publication conditional on code publication, perhaps with exceptions in
extenuating circumstances. Such a policy seems impossibly heavy-handed at the
moment, though it might be possible for individual journals and conferences.
Perhaps a new journal or conference could adopt this strategy as a
differentiating feature: if the arguments above are true, such a venue could
expect to become disproportionately influential. We cite some evidence of this
effect from other fields below.
If requiring code publication seems too ambitious, it is straightforward to
prefer it. Publishers, editors, program committees and individual reviewers
can state that, all else being equal, submissions that provide code are
preferred over those that do not. In practice, editors would need to advise
reviewers on the weighting of this preference, as with any other major
criteria.
One simple concrete proposal is that the major conferences offer a new prize
for “best” (in quality, novelty or significance) published code, along with
the usual best paper and service prizes. This would be a low cost, high
visibility measure that recognizes this as a new and significant way to
contribute to the community.
At the most grassroots level, professors can expect their students to back up
all written work with published code. Generations of grad students are short,
and norms can be quickly established by generational change.
Generational change may happen last to tenure and promotion committees. We
hope that academic departments will gradually come to recognize code
publication as a valuable academic activity. However, we have argued above
(and will provide evidence below) that papers with code have more impact than
papers without code. Better quality, higher citation rates and good community
visibility leading to stronger referee letters are benefits that are already
working under the traditional evaluation criteria.
We have argued above (Sections 4.1, 5 points 1 & 2) that increased code re-use
can make work more efficient, so the candidate’s number of papers need not be
reduced, but if this is a concern, the following idea could help boost
publication rates. When an experiment is substantially re-used and the
modifications reported, the original author could be named as a co-author on
the new paper. This is not appropriate for middleware and simulation platform
code (e.g. Player and Stage), where normal citation is enough, but rather when
the code that embodies the idea of a specific experiment is inherited. This is
a way of rewarding production of impactful experimental infrastructure, and is
analagous to the long author lists seen in for example genomics and astronomy,
where infrastructure – much of which is now software – is recognized as
extremely valuable.
## 7 The Trend Toward Experiment Publishing
We have argued that publishing code and experimental data is important for the
robotics research community, is good for researchers and easy to do. Yet it is
not standard practice in our field. The idea of publishing experimental data
and other artifacts beyond finished papers is not new but it seems to be
becoming popular. Here we survey some government policies, practice in other
scientific disciplines and editorial policies of high impact journals.
### 7.1 Government and Funding Agency Policies
The US National Science Foundation (NSF)…
> …expects investigators to share with other researchers, at no more than
> incremental cost and within a reasonable time, the data, samples, physical
> collections and other supporting materials created or gathered in the course
> of the work. It also encourages awardees to share software and inventions or
> otherwise act to make the innovations they embody widely useful and
> usable.[12]
Since October 2003 the US National Institutes of Health (NIH) has required
grant applications for $500K per year and above to include a plan for data
sharing or a statement why data sharing is not possible [11]. While the form
of data sharing is not considered during the proposal assessment, the NIH
sends a clear signal to encourage publication of data.
The 2003 Berlin Declaration on Open Access to Knowledge [2] may come to be
seen as an important milestone. At the time of writing the declaration has
been signed by 264 funding agencies, universities and research organizations,
including CERN, the Chinese Academy of Sciences, the Indian National Science
Academy, and the German Research Foundation. The declaration states:
> A complete version of the work and all supplemental materials […] in an
> appropriate standard electronic format is deposited (and thus published) in
> at least one online repository using suitable technical standards (such as
> the Open Archive definitions) that is supported and maintained by an
> academic institution, scholarly society, government agency, or other well
> established organization that seeks to enable open access, unrestricted
> distribution, inter operability, and long-term archiving. [2]
In a 2004 statement by the Organization for Economic Co-operation and
Development (OECD) numerous governments including those of North America and
Europe agreed on a declaration on access to research data from public funding.
The OECD recognizes that
> an optimum international exchange of data, information and knowledge
> contributes decisively to the advancement of scientific research and
> innovation. […] Open access to, and unrestricted use of, data promotes
> scientific progress and facilitates the training of researchers. [20].
In 2007 the US government passed the “America COMPETES Act” [26] requiring
federal civilian agencies that conduct scientific research to openly exchange
data and results with other agencies, policymakers and the public.
All of these national and international governmental efforts are aimed at
improving the quality and efficiency of the science performed at public
expense, and each requires or requests that experimental data and artifacts
are shared.
### 7.2 Practice in non-robotics disciplines
According to Nielsen [18] since 1991 physicists have made extensive use of the
preprint server arXiv, which makes papers freely available at the same time as
they are submitted to a journal for publication. Nielsen views arXiv as an
important tool to speed up the transfer of knowledge, but goes further by
calling for the next generation of openness in science by “… making more types
of content available than just scientific papers; allowing creative reuse and
modification of existing work through more open licensing.”.
Arguably the life sciences are leading the trend. For example a US National
Academy of Sciences document on life science best practice [4] requires
authors to be consistent with the principles of publication. This means that
anything that is central to a paper is to be made available in a way that
enables replication, verification and furtherance of science. When it comes to
publishing algorithms, these guidelines are very explicit:
> …if the intricacies of the algorithm make it difficult to describe in a
> publication, the author could provide an outline of it in the paper and make
> the source code […] available to investigators… [4]
In epidemiology usually more is at stake than in everyday robotics.
Epidemiological findings often influence policymakers, thus society requires
highly reliable results from this field. Peng et al. [21] acknowledge the
sensitive nature of this kind of work, and argue that reproducibility is the
minimum standard for epidemiological research. Reproducibility allows
independent investigators to subject the original data to their own analysis
and interpretation. To enable reproducibility Peng
> …calls for data sets and software to be made available for 1) verifying
> published findings, 2) conducting alternative analyzes of the same data, 3)
> eliminating uninformed criticisms that do not stand up to existing data, and
> 4) expediting the interchange of ideas among investigators. [21]
### 7.3 Journals
While scientists like Leonardo da Vinci, Galileo Galilei and Christiaan
Huygens kept their discoveries secret [18], modern science is characterized by
publication. Britain’s Royal Society mandated peer-review of published
scientific articles in its Philosophical Transactions, first published in
1665\. This policy reflected the philosphy of the Royal Society, expressed in
their motto Nullius in Verba (nothing in words / take nobody’s word for it),
that scientific claims are only valid if reproducible. Since then, peer-
reviewed journals have been the most important way to communicate scientific
results. Now, as the cost of distributing large amounts of digital data
becomes very low - a small fraction of the total cost of an experiment, or of
paper publication - many journals require or at least encourage publication of
data, samples, code and detailed method descriptions alongside the traditional
paper.
#### 7.3.1 Science
In 2004 and 2005 Science published two stem cell papers (Science 303, 1669
(2004) and Science 308, 1777 (2005)) which were later discovered to be
fraudulent and retracted by the journal. As a consequence Science enlisted the
help of an outside committee to investigate the handling of the two papers and
suggest improvements to the editorial process of their journal. The committee
concluded that Science had correctly followed their policies and that no
procedure could protect against deliberate fraud [9]. Interesting in the
context of our paper is the committee’s recommendation for improving the
editorial process “Science should have substantially stricter requirements
about reporting the primary data”. Today Science requires that
* •
large data sets are deposited in approved public databases prior to
publication and an accession number is included in the published paper.
* •
all data necessary to understand, assess, and extend the conclusions of the
paper must be made available to the reader, flowing the policies of [12] and
[4].
* •
all reasonable requests for sharing materials are to be fulfilled. [3]
#### 7.3.2 Nature
The Nature Publishing Group’s policy on availability of data for all their
Nature publications is very similar:
> An inherent principle of publication is that others should be able to
> replicate and build upon the authors’ published claims. Therefore, a
> condition of publication in a Nature journal is that authors are required to
> make material, data and associated protocols promptly available to readers
> without preconditions. [13]
As with Science, Nature requires depositing dataset in publicly accessible
databases. Of high interest in relation to our paper is Nature’s policy on
sharing biological materials. It reads
> For materials such as mutant strains and cell lines, the Nature journals
> require authors to use established public repositories whenever possible […]
> and provide accession numbers in the manuscript. [13]
Nature recently highlighted the importance of this issue on the cover of an
issue that featured a related editorial and three news/opinion articles. The
editorial described “data’s shameful neglect”, calling for funding agencies to
boost support for (and pressure on) researchers to make data available [14].
Various reasons why many researchers choose not to share, despite the
existence of purpose-built infrastructure, are discussed in the context of the
perceived failure of a digital archive project at the University of
Rochester[17]. A distinction is made between the issues of pre-publication
[16] and post-publication [15] sharing of data and tools. We are advocating
simultaneous sharing and publication, which is a special case of post-
publication sharing.
#### 7.3.3 Others
Other journals like Nucleic Acids Research [19] and the Public Library of
Science journal series [23] have very similar policies on data sharing and
access to research material.
A less rigorous procedure is employed by the Annals of Internal Medicine. To
foster reproducible research and to enhance trust in scientific results of
publications, the journal encourages authors to make their data publicly
available and mandates authors to include a statement of whether materials are
being made available or not and if under which conditions [10].
#### 7.3.4 Robotics
The journal Autonomous Robots appears to have no stated policy on code and
data publication, though it does support the bundling of supplementary
material such as videos and data spreadsheets. The IEEE Transactions on
Robotics is similar, with no policy recommending data or code sharing.
However, it does mention these explicitly in the “multimedia” instructions:
> Multimedia can be “playable” files […] or “dataset” files (e.g., raw data
> with programs to manipulate them). Such material is intended to enhance the
> contents of a paper, both in clarity and in added value.
### 7.4 Impact on citation rates
Citation counts are commonly used to assess the impact of an author’s work
[5]. A 2007 meta-study of cancer micorarray clinical trials revealed that
papers which shared their mircoarray data were cited about 70% more frequently
[22] than those that did not. If this effect generalizes to robotics, authors
would have an interest in publishing code in order to boost their citation
counts. An increase in citations can also be achieved by publishing in open-
access journals [6].
### 7.5 Related efforts
A system for sharing and reproducing computations is proposed by Schwab et al.
[25], who use their system, based on GNU make, as the principal means for
organizing and transferring scientific computations in their geophysics
laboratory. The motivation for ReDoc is essentially the same as that described
in this paper. However, perhaps unfortunately, this tool does not appear to
have made a large impact in computer science so far. ReDoc is clever and
powerful, but requires the user to learn to read special Make macros. The
method we advocate in this paper is more simple and does not prescribe a
particular build system, and can be thought of as a subset of the ReDoc
workflow.
## 8 Sharing in Robotics
Various free-to-use middleware and simulartion platforms for robotics have
been used over the last fifteen years, of which Carmen[7], Player[8] and
recently ROS666http://ros.org are probably the most influential in research.
Much of the code-sharing that has occurred is facilitated by one of these
three platforms, each of which is the result of many hours of work.
ROS is currently the platform for a major effort to bootstrap a community
based around common resources. In 2010 Willow Garage777http://willowgarage.com
used their unusual resources to provide ten groups with highly capable PR2
mobile manipulator robots running ROS software under the condition that
resulting code be made available for later users. While the PR2 is too
expensive to be ubiquitous, Willow Garage publishes a family of simulation
models of their robot based on well-known and free platforms. ROS goes much
further than Carmen and Player in encapsulating algorithms, including recent
SLAM, vision and control methods, and making them ready to use as components
of a downstream system. While ROS is open source and has maby contributors,
Willow Garage curates the project and does significant development.
Willow Garage’s expensive effort is the first to provide state-of-the-art
mechatronics and system engineering along with free software tools, and is
explicitly intended to accelerate research progress by providing resources
shared resources to the community. While there will always be central place
for custom robots in research, more resources from components to complete
systems will become available in future. We can leverage the large investment
in open source software and projects like the PR2 by shifting to code sharing
as standard practice.
## 9 Conclusion
We have shown that meta-government organizations like the OECD see scientific
data exchange as an important tool for the efficient advancement of scientific
research. We have also argued that code is a form of data that is particularly
important for robotics research. Thus making experimental data including code
and configurations publicly available is important for the progress of
robotics.
Building upon other peoples work is an integral part of the scientific
process. Increasing the efficiency of this process increases community
productivity. We suggest that making experimental code identifiable will be
helpful. We have also argued that the direct and indirect effects of
publishing identifiable code are good for the researcher that shares, as well
as for the wider community.
Other research fields, especially life sciences, have strong requirements to
share data sets and provide free access to supporting materials alongside with
traditional paper publications. In the case of Nature a submission of
biological material to a public repository may be required. In robotics, while
sharing physical robots may be prohibitively expensive, sharing digital
resources takes little time or treasure. Freely available infrastructure
allows the upload of a complete experiment (software, build scripts, data,
analysis scripts etc.) to a public repository in a few seconds with a few
button presses. Code sharing in robotics is easy.
The issues discussed here are not new, and a subset of robotics researchers
does publish source code implementations of their algorithms, to the benefit
of everyone. The main contribution here is to point out the importance of
distributing uniquely identifiable versions and not just the latest and “best”
version. We have argued that this methodological issue is important, that
originators and subsequent users can both benefit, and suggested an easy-to-
follow publishing protocol. Our group will follow this protocol and observe
its effects.
## Acknowledgements
Thanks to Greg Mori, Alex Couture-Beil, Yaroslav Litus, Brian Gerkey, Gaurav
Sukhatme, and the organizers and attendees of the RSS’09 Workshop on
Methodology in Experimental Robotics for useful discussions on this issue.
## References
* [1] D.E. Knuth. Literate Programming. The Computer Journal (British Computer Society), 27 (2) 97–111, 1984.
* [2] Berlin declaration on open access to knowledge in sciences and humanities, 2003\. http://oa.mpg.de/openaccess-berlin/berlin_declaration.pdf [Online; accessed 09-June 2009].
* [3] Science: General information for authors, 2009. http://www.sciencemag.org/about/authors/prep/gen_info.dtl#datadep [Online; accessed 10-June-2009].
* [4] T. R. Cech et al. Sharing Publication-Related Data and Materials: Responsibilities of Authorship in the Life Sciences. National Academy of Sciences, 2003.
* [5] A. M. Diamond Jr. What is a citation worth. Journal of Human Resources, 21(2):200–215, 1986.
* [6] G. Eysenbach. Citation advantage of open access articles. PLoS Biology, 4(5):e157, 2006.
* [7] M. Montemerlo, N. Roy, and S. Thrun. Perspectives on standardization in mobile robot programming: The Carnegie Mellon navigation (CARMEN) toolkit. In IEEE/RSJ Proceedings of the International Conference on Intelligent Robots and Systems, pages XX–YY, 2003.
* [8] B. P. Gerkey, R. T. Vaughan, K. Stoy, A. Howard, G. S. Sukhatme, and M. J. Matarić. Most valuable player: A robot device server for distributed control. In IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 1226 – 1231, 2001.
* [9] D. Kennedy. Responging to fraud. Science, 314(5804):1353, December 2006.
* [10] C. Laine, S. N. Goodman, M. E. Griswold, and H. C. Sox. Reproducible research: moving toward research the public can really trust. Annals of Internal Medicine, 146(6):450–454, March 2007.
* [11] National Institues of Health Office of Extramural Research. Nih data sharing policy, 2003. http://grants.nih.gov/grants/policy/data_sharing/ [Online; assessed 09-June-2009].
* [12] National Science Foundation. Grant general conditions, 2001. http://www.nsf.gov/pubs/2001/gc101/gc101rev1.pdf [Online; accessed 09-June-2009].
* [13] Nature Editorial policies Guide to publication policies of the nature journals, 2009. http://www.nature.com/authors/gta.pdf [Online; accessed 09-June-2009].
* [14] Nature Editorial Data’s shameful neglect. Nature, 461(7271):145, 10 September 2009.
* [15] Schofield, P., Bubela, T., Weaver, T., Portilla, L., Brown, S., Hancock, J., Einhorn, D., Tocchini-Valentini, G., Hrabe de Angelis, M., and Rosenthal, N. Post-publication sharing of data and tools. Nature, 461(7271):171-173, 10 September 2009.
* [16] Toronto International Data Release Workshop Authors Prepublication data sharing. Nature, 461(7271):168-170, 10 September 2009.
* [17] Bryn Nelson Data sharing: Empty archives Nature, 461(7271):160-163, 10 September 2009.
* [18] M. Nielsen. Doing science in the open. Physicsworld, 22(5):30, 2009.
* [19] Nucleic Acids Research. General policies of the journal, 2009. http://www.oxfordjournals.org/our_journals/nar/for_authors/ed_policy.ht%ml [Online; accessed 09-June-2009].
* [20] Organisation for Economic Co-operation and Development. Science, technology and innovation for the 21st. century. meeting of the OECD commitee for scientific and technological policy at ministerial level, January 2004. http://www.oecd.org/document/0,2340,en_2649_34487_25998799_1_1_1_1,00.h%tml [Online; accessed 09-June-2009].
* [21] R. D. Peng, F. Dominici, and S. L. Zeger. Reproducible epidemiologic research.
* [22] H. A. Piwowar, R. S. Day, and D. B. Fridsma. Sharing detailed research data is associated with increased citation rate. PLoS ONE, 2(3):e308, 2007.
* [23] Public Library of Science (PLoS). Editorial and publishing policies, 2009. http://www.plosone.org/static/policies.action [Online; accessed 10-June-2009].
* [24] T. Hansen and G. Wollman RFC3174: US Secure Hash Algorithm 1 (SHA1). Technical report, Internet Engineering Task Force: Network Working Group, 2001.
* [25] M. Schwab, M. Karrenbach, and J. Clearbout. Making scientific computations reproducible. Computing in Science and Engineering, 2(6):61–67, 2000.
* [26] US Congress. America COMPETES Act, 2007. http://commdocs.house.gov/reports/110/h2272.pdf [Online; accessed 09-June 2009].
* [27] X. Wang and H. Yu. How to break md5 and other hash functions. In Eurocrypt 2005, volume 3494, pages 19–35. Lecture Notes in Computer Science, May 2005.
|
arxiv-papers
| 2012-04-10T18:07:39 |
2024-09-04T02:49:29.569519
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Richard Vaughan and Jens Wawerla",
"submitter": "Richard Vaughan",
"url": "https://arxiv.org/abs/1204.2235"
}
|
1204.2325
|
# A weighted $L_{p}$-theory for second-order elliptic and parabolic partial
differential systems on a half space
Kyeong-Hun Kim$\,{}^{1}$, Kijung Lee$\,{}^{2}$
$\;{}^{1}\,$Department of Mathematics, Korea University
Seoul, 136-701, Korea
$\;{}^{2}\,$Department of Mathematics, Ajou University
Suwon, 443-749, Korea
###### Abstract
In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem
and sharp function estimations in weighted Sobolev spaces. We also provide
uniqueness and existence results for second-order elliptic and parabolic
partial differential systems in weighed Sobolev spaces.
Keywords: Fefferman-Stein theorem, Hardy-Littlewood theorem, Weighted Sobolev
spaces, Sharp function estimations, $L_{p}$-theory, Elliptic partial
differential systems, Parabolic partial differential systems.
AMS 2000 subject classifications: 42B37, 35K45, 35J57.
\- Dedicated to 70th birthday of N.V. Krylov
## 1 Introduction
In this article we consider the elliptic system
$\displaystyle\sum_{i,j=1}^{d}\sum_{r=1}^{d_{1}}a^{ij}_{kr}u^{r}_{x^{i}x^{j}}(x)=f^{k}(x),\quad(k=1,2,\cdots,d_{1})$
(1.1)
and the parabolic system
$\displaystyle
u^{k}_{t}(t,x)=\sum_{i,j=1}^{d}\sum_{r=1}^{d_{1}}a^{ij}_{kr}(t)u^{r}_{x^{i}x^{j}}(t,x)+f^{k}(t,x),\quad(k=1,2,\cdots,d_{1})$
(1.2)
defined for $t>0$ and $x\in\mathbb{R}^{d}_{+}$.
In the study of partial differential equations (PDEs) or of partial
differential systems (PDSs) regularity theory play the key role of describing
essential relations between input data and the unknown solutions; the sharper
the theory is, the more understanding of the relations we get.
The primary goals of this article are to introduce some new mathematical tools
and ideas which are useful in the study of systems in $L_{p}$-spaces involving
weights and to provide another nice regularity theory for these systems.
In this article we use weighted Sobolev spaces for the unknown function
$u=(u^{1},\cdots,u^{d_{1}})$ and the inputs $f^{k}$. The need to introduce
weights comes from, for instance, the theory of stochastic partial
differential equations (SPDEs) or stochastic partial differential systems
(SPDSs), where a Hölder space approach does not allow us to obtain results of
reasonable generality and Sobolev spaces without weights are trivially
inappropriate (see [14] for details). To study such stochastic systems one has
to develop a nice regularity theory for the corresponding deterministic
systems in advance. Also Sobolev spaces with weights are very useful in
treating degenerate elliptic and parabolic equations (see, for instance, [16])
and in studying equations defined on non-smooth domains such as domains with
wedges (see, for instance, [5, 16, 18]).
In principle there are three main methods for $L_{p}$-theory: multiplier
theory, Calderón-Zygmund theory and the pointwise estimate using sharp
functions. Multiplier theory fits well when the principal operator is almost
Laplacian and the equation under consideration is defined on the entire space,
and Calderón-Zygmund theory works well when there exists an integral
representation of solutions and the integral is taken over $\mathbb{R}^{n}$
for some $n$. However, these two methods do not fit our case since we are
dealing with weighted $L_{p}$-theories for systems (1.1) and (1.1) defined on
a half space. Thus we use an approach based on pointwise estimates of the
sharp function of second order derivatives, but unlike the standard theory
(for instance, [13]) we need to use the weighted version. The elaboration of
this approach is one of our main results.
We also mention that if $d_{1}=1$ then weighted $L_{p}$-theories for single
equations defined on a half space can be constructed based on integration by
parts without relying on sharp function estimations (see the proof of Lemma
4.8 and Lemma 6.3 of [10]). However it seems that the arguments in the proof
of Lemma 4.8 and Lemma 6.3 of [10] cannot be reproduced for $L_{p}$-theory of
systems unless $p=2$ and some stronger algebraic conditions on $A^{ij}$ are
additionally assumed.
Interestingly, we discovered some very useful tools in the perspective of
linear Partial differential equations/systems theory. Even though, in this
article, we only consider the systems with coefficients independent of $x$,
the sharp function estimates and the tools used to derive them will naturally
lead to many subsequent works studying, for instance, elliptic and parabolic
equations and systems with discontinuous coefficients defined in an arbitrary
domain $U$ of $\mathbb{R}^{d}$. In this context, we refer the readers to very
extensive literature [13] and recent articles [1, 2, 3, 7, 6] (also see the
references therein), where (standard) $L_{p}$-theories are constructed for
single equations with VMO (or small BMO)-coefficients.
The article is organized as follows. In section 2 we prove the Fefferman-Stein
theorem and Hardy Littiewood theorem with our special weights; the proofs are
quite elementary. In section 3 we introduce weighted Sobolve spaces and
formulate our regularity results for the systems, Theorem 3.10 and Theorem
3.13. The _useful_ tools and ideas for proving Theorem 3.10 and Theorem 3.13
are in section 4 and 5; the local estimations and the sharp function
estimations. Finally Theorem 3.10 and Theorem 3.13 are proved in section 6.
As usual $\mathbb{R}^{d}$ stands for the Euclidean space of points
$x=(x^{1},...,x^{d})$ and
$\mathbb{R}^{d}_{+}=\\{x\in\mathbb{R}^{d}:x^{1}>0\\}$. For $i=1,...,d$, multi-
indices $\alpha=(\alpha_{1},...,\alpha_{d})$, $\alpha_{i}\in\\{0,1,2,...\\}$,
and functions $u(x)$ we set
$u_{x^{i}}=\frac{\partial u}{\partial x^{i}}=D_{i}u,\quad
D^{\alpha}u=D_{1}^{\alpha_{1}}\cdot...\cdot
D^{\alpha_{d}}_{d}u,\quad|\alpha|=\alpha_{1}+...+\alpha_{d}.$
By $\delta^{kr}$ we denote the Kronecker delta on the indices $k,r$. If we
write $N=N(\cdots)$, this means that the constant $N$ depends only on what are
in parenthesis.
The authors are sincerely grateful to Ildoo Kim for finding few errors in the
earlier version of this article.
## 2 F-S and H-L theorems in weighted $L_{p}$-spaces
Denote
$\Omega=\mathbb{R}\times\mathbb{R}^{d}_{+}:=\\{(t,x)=(t,x^{1},x^{2},\ldots,x^{d})\;:x^{1}>0\\}.$
Also, by $\mathcal{B}(\mathbb{R}^{d}_{+})$ and $\mathcal{B}(\Omega)$ we denote
the Borel $\sigma$-algebra on $\mathbb{R}^{d}_{+}$ and $\Omega$ respectively.
Fix $\alpha\in(-1,\infty)$ and define the weighted measures
$\nu(dx)=\nu_{\alpha}(dx)=(x^{1})^{\alpha}dx,\quad
d\mu=\mu_{\alpha}(dtdx):=\nu_{\alpha}(dx)dt.$
Then $(\mathbb{R}^{d}_{+},\mathcal{B}(\mathbb{R}^{d}_{+}),\nu)$ and
$(\Omega,\mathcal{B}(\Omega),\mu)$ are measure spaces with
$\nu(\mathbb{R}^{d}_{+})=\mu(\Omega)=\infty$. Let $p\in[1,\infty)$ and
$L_{p}(\Omega,\mu)=L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
($L_{p}(\mathbb{R}^{d}_{+},\nu)$ resp.) be the collection of Borel-measurable
functions $u=(u^{1},\ldots,u^{d_{1}})$ defined on $\Omega$ (on
$\mathbb{R}^{d}_{+}$ resp.) satisfying
$\displaystyle\|u\|^{p}_{L_{p}(\Omega,\mu)}:=\int_{\Omega}|u|^{p}d\mu<\infty,\quad\quad\left(\|u\|^{p}_{L_{p}(\mathbb{R}^{d}_{+},\nu)}:=\int_{\mathbb{R}^{d}_{+}}|u|^{p}\nu(dx)<\infty,\text{respectively}\right).$
Denote
$\mathcal{B}^{0}(\Omega):=\\{C\in\mathcal{B}(\Omega)\;:\;|C|:=\mu(C)<\infty\\},\quad\mathcal{B}^{0}(\mathbb{R}^{d}_{+}):=\\{D\in\mathcal{B}(\mathbb{R}^{d}_{+})\;:\;|D|:=\nu(D)<\infty\\}.$
We say $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$ if $fI_{C}\in
L_{1}(\Omega,\mu)$ for any $C\in\mathcal{B}^{0}(\Omega)$, where $I_{C}$ is the
indicator function of $C$. For $f=(f^{1},\ldots,f^{d_{1}})\in
L_{1}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $C\in\mathcal{B}^{0}(\Omega)$ we
define
$\displaystyle
f_{C}:=\frac{1}{|C|}\int_{C}fd\mu=-\int_{C}fd\mu=\left(-\int_{C}f^{1}d\mu,\ldots,-\int_{C}f^{d_{1}}d\mu\right).$
Similarly write $h\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$ if
$hI_{D}\in L_{1}(\mathbb{R}^{d}_{+},\nu)$ for any
$D\in\mathcal{B}^{0}(\mathbb{R}^{d}_{+})$, and define
$h_{D}:=\frac{1}{|D|}\int_{D}h\nu(dx)=-\int_{D}h\nu(dx)=\left(-\int_{D}h^{1}\nu(dx),\ldots,-\int_{D}h^{d_{1}}\nu(dx)\right).$
Let $(\mathbb{C}_{n},n\in\mathbb{Z})$ denote the filtration of the partitions
of $\bar{\Omega}$ defined by
$\displaystyle\mathbb{C}_{n}=\Big{\\{}\Big{[}\frac{i_{0}}{4^{n}},\frac{i_{0}+1}{4^{n}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}\;:\;i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z},\;i_{1}\in\\{0\\}\cup\mathbb{N}\Big{\\}},$
and $(\mathbb{D}_{n},n\in\mathbb{Z})$ be the corresponding filtration of the
partitions of $\bar{\mathbb{R}}^{d}_{+}$, that is,
$\displaystyle\mathbb{D}_{n}:=\Big{\\{}\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}\;:\;i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z},\;i_{1}\in\\{0\\}\cup\mathbb{N}\Big{\\}}.$
For any $(t,x)\in\Omega$, by $C_{n}(t,x)$ ($D_{n}(x)$ resp.) we denote the
unique cube in $\mathbb{C}_{n}$ (in $\mathbb{D}_{n}$ resp.) containing $(t,x)$
($x$ respectively). Let $\mathbb{L}=\mathbb{L}(\Omega)$ (resp.
$\mathbb{L}(\mathbb{R}^{d}_{+})$) denote the set of
$\mathbb{R}^{d_{1}}$-valued continuous functions with compact support in
$\Omega$ ( in $\mathbb{R}^{d}_{+}$ respectively).
###### Lemma 2.1.
(i) We have $\inf_{C\in\mathbb{C}_{n}}|C|\to\infty$ as $n\to-\infty$ and, for
any $f\in\mathbb{L}(\Omega)$, $\lim_{n\to\infty}f_{C_{n}(t,x)}=f(t,x)$ holds
for any $(t,x)\in\Omega$.
(ii) We have $\inf_{D\in\mathbb{D}_{n}}|D|\to\infty$ as $n\to-\infty$ and, for
any $f\in\mathbb{L}(\mathbb{R}^{d}_{+})$, $\lim_{n\to\infty}f_{D_{n}(x)}=f(x)$
holds for any $x\in\mathbb{R}^{d}_{+}$.
###### Proof.
It is obvious since $f$ is continuous. ∎
###### Lemma 2.2.
(i) For any $C\in\mathbb{C}_{n}$ there exists a unique
$C^{\prime}\in\mathbb{C}_{n-1}$ such that $C\subset C^{\prime}$ and
$\displaystyle\frac{|C^{\prime}|}{|C|}\leq N(\alpha)<\infty.$
(ii) For any $D\in\mathbb{D}_{n}$ there exists a unique
$D^{\prime}\in\mathbb{D}_{n-1}$ such that $D\subset D^{\prime}$ and
$\displaystyle\frac{|D^{\prime}|}{|D|}\leq N(\alpha)<\infty.$
###### Proof.
We only prove (i). Since $\mathbb{C}_{n-1}$ is a partition of $\Omega$, only
one member of it contains $C$; we call it $C^{\prime}$. Let
$\displaystyle
C^{\prime}=\Big{[}\frac{i_{0}}{4^{n-1}},\frac{i_{0}+1}{4^{n-1}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n-1}},\frac{i_{1}+1}{2^{n-1}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n-1}},\frac{i_{d}+1}{2^{n-1}}\Big{)}.$
Then we have
$\displaystyle|C^{\prime}|=\mu(C^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2^{(d+1)(n-1)}}\int^{\frac{i_{1}+1}{2^{n-1}}}_{\frac{i_{1}}{2^{n-1}}}(x^{1})^{\alpha}dx^{1}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{(d+1)(n-1)}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}\Bigg{]}.$
Note that $C$ is one of $4\cdot 2^{d}$ cubes belonging to $\mathbb{C}_{n}$
inside $C^{\prime}$ and by the location of $C$ we have either
$\displaystyle|C|=\frac{1}{2^{(d+1)n}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}\Bigg{]}$
(2.1)
or
$\displaystyle|C|=\frac{1}{2^{(d+1)n}}\cdot\frac{1}{\alpha+1}\Bigg{[}\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}\Bigg{]}.$
(2.2)
Case 1: Let $i_{1}\geq 1$ and $\alpha\geq 0$. Denoting
$\displaystyle a=\frac{i_{1}+1}{2^{n-1}},\quad b=\frac{i_{1}}{2^{n-1}},\quad
c=\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}},\quad\phi(x)=x^{\alpha+1},$
we get
$\displaystyle\frac{|C^{\prime}|}{|C|}$ $\displaystyle=$ $\displaystyle
2^{d+1}\cdot\frac{\phi(a)-\phi(b)}{\phi(a)-\phi(c)}\quad\textrm{or}\quad
2^{d+1}\cdot\frac{\phi(a)-\phi(b)}{\phi(c)-\phi(b)}$ (2.3) $\displaystyle=$
$\displaystyle
2^{d+1}\left(1+\frac{\phi(c)-\phi(b)}{\phi(a)-\phi(c)}\right)\quad\textrm{or}\quad
2^{d+1}\left(1+\frac{\phi(a)-\phi(c)}{\phi(c)-\phi(b)}\right)$
$\displaystyle=$ $\displaystyle
2^{d+1}\left(1+\frac{\phi^{\prime}(\beta)}{\phi^{\prime}(\alpha)}\right)\quad\textrm{or}\quad
2^{d+1}\left(1+\frac{\phi^{\prime}(\alpha)}{\phi^{\prime}(\beta)}\right),$
where $\alpha,\beta$ are some numbers satisfying $b<\beta<c<\alpha<a$; we used
mean value theorem. Since $\alpha+1>1$, the function $\phi$ is convex and
increasing on $(0,\infty)$. Hence, we have
$\displaystyle\frac{\phi^{\prime}(\beta)}{\phi^{\prime}(\alpha)}\leq
1,\quad\quad\frac{\phi^{\prime}(\alpha)}{\phi^{\prime}(\beta)}\leq\frac{\phi^{\prime}(a)}{\phi^{\prime}(b)}=\frac{a^{\alpha}}{b^{\alpha}}=\left(\frac{i_{1}+1}{i_{1}}\right)^{\alpha}\leq
2^{\alpha},$
and therefore
$\displaystyle\frac{|C^{\prime}|}{|C|}\leq 2^{d+1}(1+2^{\alpha})\leq
2^{\alpha+d+2}.$
Case 2: Assume $i_{1}=0$ and $\alpha\geq 0$. By similar but simpler
calculation we obtain
$\displaystyle\frac{|C^{\prime}|}{|C|}\leq 2^{\alpha+d+2}.$
Case 3: Assume $\alpha\in(-1,0)$. If $|C|$ is given as in (2.2), then since
$\phi(x)$ is concave,
$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}\leq
2.$
Let $|C|$ be given as in (2.1). If $i_{1}=0$, then
$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}}=\frac{2^{\alpha+1}}{2^{\alpha+1}-1},$
and if $i_{1}\geq 1$ then since $\phi$ is concave and $\phi^{\prime}$ is
positive on $(0,\infty)$
$\frac{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}}{2^{n-1}}\right)^{\alpha+1}}{\left(\frac{i_{1}+1}{2^{n-1}}\right)^{\alpha+1}-\left(\frac{i_{1}+1}{2^{n-1}}-\frac{1}{2^{n}}\right)^{\alpha+1}}\leq\frac{2^{-n+1}\phi^{\prime}(\frac{i_{1}}{2^{n-1}})}{2^{-n}\phi^{\prime}(\frac{i_{1}+1}{2^{n-1}})}\leq
2^{1-\alpha}.$
The lemma is proved. ∎
###### Remark 2.3.
(i) By Lemma 2.1, Lemma 2.2 and the outline of Section 3.1, 3.2 of [13] we get
Lemma 2.5, Theorem 2.7 and Theorem 2.8 below for free.
(ii) if $C_{n}\in\mathbb{C}_{n}$ and $C_{m}\in\mathbb{C}_{m}$ with $n\leq m$,
then $C_{n}\cap C_{m}=C_{m}$ or $\emptyset$.
###### Definition 2.4.
We call $\tau=\tau(x)\in\mathbb{Z}\cup\\{\infty\\}$ a _stopping time_ if
$\\{x:\tau(x)=n\\}=\emptyset$ or union of some elements in $\mathbb{C}_{n}$
for each $n\in\mathbb{Z}$.
For $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$, $h\in
L_{1,loc}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$ and $n\in\mathbb{Z}$ we
define
$\displaystyle
f_{|n}(t,x):=\frac{1}{\mu(C_{n}(t,x))}\int_{C_{n}(t,x)}f(s,y)\mu(dsdy)=-\int_{C_{n}(t,x)}f(s,y)\mu(dsdy),$
$\displaystyle
h_{|n}(x):=\frac{1}{\nu(D_{n}(t,x))}\int_{D_{n}(t,x)}h(y)\nu(dy)=-\int_{D_{n}(x)}h(y)\nu(dy),$
and
$\displaystyle
f_{|\tau}(t,x):=f_{|\tau(t,x)}(t,x)\quad\mathrm{if}\;\;\tau(t,x)\neq\infty;\quad
f_{|\tau}(t,x):=f(t,x)\quad\mathrm{if}\;\;\tau(t,x)=\infty.$
###### Lemma 2.5.
Let $\\{\mathbb{C}_{n}\;:\;n\in\mathbb{Z}\\}$ be a filtration of partitions of
$\bar{\Omega}$.
(i) Let $g\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{1})$, $g\geq 0$ and let $\tau$
be a stopping time. Then
$\displaystyle\int_{\Omega}g_{|\tau}(t,x)I_{\tau<\infty}(t,x)\mu(dtdx)$
$\displaystyle=$
$\displaystyle\int_{\Omega}g(t,x)I_{\tau<\infty}(t,x)\mu(dtdx),$
$\displaystyle\int_{\Omega}g_{|\tau}(t,x)\mu(dtdx)$ $\displaystyle=$
$\displaystyle\int_{\Omega}g(t,x)\mu(dtdx).$
(ii) Let $g\in L_{1}(\Omega,\mu;\mathbb{R}^{1})$, $g\geq 0$ and let
$\lambda>0$ be a constant. Then
$\displaystyle\tau(t,x):=\inf\\{n:g_{|n}(t,x)>\lambda\\}\quad\quad(\inf\emptyset:=\infty)$
is a stopping time. Furthermore, we have
$\displaystyle 0\leq g_{|\tau}(t,x)I_{\tau<\infty}\leq
N_{0}\lambda,\quad|\\{(t,x):\tau(t,x)<\infty\\}|\leq\lambda^{-1}\int_{\Omega}g(t,x)I_{\tau<\infty}\mu(dtdx).$
###### Remark 2.6.
$($Riesz-Calderón-Zygmund decomposition$)$ Any $g\in
L_{1}(\Omega,\mu;\mathbb{R}^{1})$ is decomposed by
$\displaystyle g=\xi+\eta,$
where $\xi=g-g_{|\tau}$,
$\eta=g_{|\tau}=g_{|\tau}\;I_{\tau<\infty}+g_{|\tau}\;I_{\tau=\infty}$.
Moreover, we have (i) $\eta\leq N_{0}\lambda$ a.e. (ii)
$|\\{(t,x):\xi(t,x)\neq 0\\}|\leq\lambda^{-1}\|g\|_{L_{1}(\Omega,\mu)}$ (iii)
$\xi_{|\tau}=0$.
Now, for $f\in L_{1,loc}(\Omega,\mu;\mathbb{R}^{d_{1}})$ we define the maximal
function
$\displaystyle\mathcal{M}f(t,x):=\left(\sup_{n<\infty}|f^{1}|_{|n}(t,x),\;\ldots\;,\sup_{n<\infty}|f^{d_{1}}|_{|n}(t,x)\right)$
and the sharp function
$\displaystyle
f^{\\#}(t,x)=\left(\sup_{n<\infty}-\int_{C_{n}(t,x)}|f^{1}(s,y)-f^{1}_{|n}(s,y)|\mu(dsdy),\,\ldots,\,\sup_{n<\infty}-\int_{C_{n}(t,x)}|f^{d_{1}}(s,y)-f^{d_{1}}_{|n}(s,y)|\mu(dsdy)\right).$
We define $\mathcal{M}h(x)$ and $h^{\\#}(x)$ similarly for functions
$h=h(x)\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$.
###### Theorem 2.7.
Let $p\in(1,\infty)$. Then for any $f\in L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
and $h\in L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$, we have
$\displaystyle\|\mathcal{M}f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|\mathcal{M}h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$
where $N=N(\theta,p,d,d_{1})$.
###### Theorem 2.8.
Let $p\in(1,\infty)$. Then for any $f\in L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$
and $h\in L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})$ we have
$\displaystyle\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f^{\\#}\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h^{\\#}\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$
where $N=N(\theta,p,d,d_{1})$.
We investigate the relation between our maximal and sharp functions and more
general ones. Let $B^{\prime}_{r}(x^{\prime})$ denote the open ball in
$\mathbb{R}^{d-1}$ of radius $r$ with center $x^{\prime}$. For
$x=(x^{1},x^{\prime})\in\mathbb{R}^{d}_{+}$ and $t\in\mathbb{R}$, denote
$B_{r}(x)=B_{r}(x^{1},x^{\prime})=(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime}),\quad Q_{r}(t,x):=(t,t+r^{2})\times B_{r}(x)$
and $\mathbb{Q}$ be the collection of all such open sets
$Q_{r}(t,x)\subset\Omega$. For $f\in L_{1,loc}(\Omega,\mu:\mathbb{R}^{d_{1}})$
we define
$\displaystyle
f^{i}_{Q}=-\int_{Q}f^{i}\;d\mu,\quad\mathbb{M}f^{i}(t,x)=\sup_{(t,x)\in
Q}-\int_{Q}f^{i}d\mu,\quad(f^{i})^{\sharp}(t,x)=\sup_{(t,x)\in
Q}-\int_{Q}|f^{i}-f^{i}_{Q}|d\mu,\quad i=1,\ldots,d_{1},$
where the supremum is taken for all $Q\in\mathbb{Q}$ containing $(t,x)$.
Denote
$\mathbb{M}f:=(\mathbb{M}f^{1},\ldots,\mathbb{M}f^{d_{1}}),\quad
f^{\sharp}:=((f^{1})^{\sharp},\ldots,(f^{d_{1}})^{\sharp}).$
For functions $h\in L_{1,loc}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$, the
functions $\mathbb{M}h(x)$ and $(h)^{\sharp}(x)$ are defined similarly.
###### Lemma 2.9.
For a scalar function $g=g(t,x)$ and $h=h(x)$ we have
$\displaystyle g^{\\#}(t,x)\leq N\;g^{\sharp}(t,x),\quad\quad h^{\\#}(x)\leq
N\;h^{\sharp}(x)$
where $N=N(\theta,p,d)$.
###### Proof.
We only prove the first assertion. For $(t,x)\in\Omega$, denote the
corresponding unique cube $C_{n}(t,x)\in\mathbb{C}_{n}$ by
$\displaystyle\Big{[}\frac{i_{0}}{4^{n}},\frac{i_{0}+1}{4^{n}}\Big{)}\times\Big{[}\frac{i_{1}}{2^{n}},\frac{i_{1}+1}{2^{n}}\Big{)}\times\cdots\times\Big{[}\frac{i_{d}}{2^{n}},\frac{i_{d}+1}{2^{n}}\Big{)}$
where $i_{0},i_{2},\ldots,i_{d}\in\mathbb{Z}$ and
$i_{1}\in\\{0\\}\cup\mathbb{N}$. We define
$Q_{(n)}(t,x):=Q_{\frac{d}{2^{n}}}(t^{*},x^{*})$ with
$t^{*}=\frac{i_{0}}{4^{n}}$ and
$x^{*}=(\frac{i_{1}+d}{2^{n}},\frac{i_{2}}{2^{n}},\ldots,\frac{i_{d}}{2^{n}})$.
We have $(t,x)\in C_{n}(t,x)\subset\overline{Q_{(n)}(t,x)}$ and
$\displaystyle\frac{|Q_{(n)}(t,x)|}{|C_{n}(t,x)|}=N(d)\cdot\frac{(i_{1}+2d)^{\alpha+1}-i_{1}^{\alpha+1}}{(i_{1}+1)^{\alpha+1}-i_{1}^{\alpha+1}}$
(2.4)
by simple calculation. If $i_{1}=0$, (2.4) is $N(d)(2d)^{\alpha+1}$; if
$i_{1}\geq 1$ and $\alpha\geq 0$ then (2.4) is less than or equal to
$\displaystyle N(d)\cdot(2d)\left(\frac{i_{1}+2d}{i_{1}}\right)^{\alpha}\leq
N(d)\cdot(2d)\cdot(1+2d)^{\alpha},$
by mean value theorem. If $\alpha\in(-1,0)$ then we use the concavity of
$x^{\alpha+1}$ to prove that (2.4) is less then $N(d)(2d)^{\alpha+1}$. The
lemma is proved. ∎
Lemma 2.9 and Theorem 2.8 imply the following version of Fefferman-Stein
theorem:
###### Theorem 2.10.
$($Fefferman-Stein$)$ Let $p\in(1,\infty)$. Then for any $f\in
L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $h\in
L_{p}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$, we have
$\displaystyle\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f^{\sharp}\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\quad\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h^{\sharp}\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}$
where $N=N(\theta,p,d,d_{1})$.
The following lemma will be used in the proof of Theorem 2.12 below.
###### Lemma 2.11.
Let $\alpha>-1$ and $\phi(x)=x^{\alpha+1}$ on $x>0$. Then for any $x>0$ and
$r>0$ we have
$\displaystyle\frac{\phi(x+2r)-\phi(x+r)}{\phi(x+r)-\phi(x)}\;\leq\;2^{\alpha+1}.$
###### Proof.
If $\alpha\in(-1,0]$ the claim is obvious since $\phi$ is concave.
Assume $\alpha>0$, fix $r>0$, and define
$\displaystyle f(x):=\frac{\phi(x+2r)-\phi(x+r)}{\phi(x+r)-\phi(x)}.$
We show that $f^{\prime}(x)\leq 0$ for $x>0$ so that $f(x)\leq
f(0)=2^{\alpha+1}-1$; note that $f(0)$ does not depend on $r$. A simple
calculation shows
$\displaystyle
f^{\prime}(x)=r(\alpha+1)\cdot\frac{2(x+2r)^{\alpha}x^{\alpha}-(x+2r)^{\alpha}(x+r)^{\alpha}-(x+r)^{\alpha}x^{\alpha}}{((x+r)^{\alpha+1}-x^{\alpha+1})^{2}}.$
(2.5)
The numerator in (2.5) is
$\displaystyle 2\cdot
x^{\alpha}(x+r)^{\alpha}(x+2r)^{\alpha}\cdot\left[(x+r)^{-\alpha}-\frac{x^{-\alpha}+(x+2r)^{-\alpha}}{2}\right].$
(2.6)
Since the function $x^{-\alpha}$ is convex and $x+r$ is the midpoint of $x$
and $x+2r$, the square bracket in (2.6) is non-positive and so is
$f^{\prime}(x)$. The lemma is proved. ∎
###### Theorem 2.12.
$($Hardy-Littlewood$)$ Let $p\in(1,\infty)$. Then for $f\in
L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})$ and $h\in
L_{p}(\mathbb{R}^{d}_{+},\nu,\mathbb{R}^{d_{1}})$ we have
$\displaystyle\|\mathbb{M}f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})}\leq
N\|f\|_{L_{p}(\Omega,\mu;\mathbb{R}^{d_{1}})},\quad\|\mathbb{M}h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}\leq
N\|h\|_{L_{p}(\mathbb{R}^{d}_{+},\nu;\mathbb{R}^{d_{1}})}.$
###### Proof.
Again we only proof the first assertion. We follow the outline for the proof
of Theorem 3.3.2 which does not involve a weight in the norm. Without loss of
generality we assume $d_{1}=1$ and $g:=f\geq 0$.
For $\lambda>0$, denote $A(\lambda):=\\{(t,x):\mathbb{M}g(t,x)>\lambda\\}$.
Then since $\mathbb{M}g$ is lower semi-continuous, $A(\lambda)$ is open. To
prove the theorem it is enough to show that for any $\lambda>0$ and compact
set $K\subset A(\lambda)$
$\displaystyle|K|\leq\frac{N}{\lambda}\int_{\Omega}I_{A(\lambda)}(t,x)g(t,x)\mu(dtdx),$
where $N=N(\theta,p,d)$. For the details see the proof of Theorem 3.3.2 of
[13].
For any $(t,x)\in K$ there exists $Q$ containing $(t,x)$ such that
$\int_{Q}gd\mu>\lambda|Q|$. Also, we observe that $Q\subset A(\lambda)$ and
there exists a finite cover $\\{Q_{1},\ldots,Q_{n}\\}$ of $K$ such that
$\displaystyle\int_{Q_{i}}gd\mu>\lambda|Q_{i}|.$
For $Q=(t-\frac{1}{2}r^{2},t+\frac{1}{2}r^{2})\times(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime})\in\mathbb{Q}$, denote
$3Q:=\big{(}t-\frac{3}{2}r^{2},t+\frac{3}{2}r^{2}\big{)}\times(x^{1}-3r,x^{1}+3r)\times
B^{\prime}_{3r}(x^{\prime}).$
When $Q$ is close to the boundary of $\Omega$, $3Q$ may not be in $\Omega$.
Hence, we define
$\displaystyle Q^{*}=3Q\cap\Omega.$
Using a Vitali covering argument one can find the disjoint subset
$\\{\tilde{Q}_{1},\ldots,\tilde{Q}_{k}\\}$ of $\\{Q_{1},\ldots,Q_{n}\\}$
satisfying $K\subset\bigcup^{k}_{j=1}\tilde{Q_{j}}^{*}$ (see the proof of
Theorem 3.3.2 of [13]). To measure $|K|$ we compute the ratio
$\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}$. For
$Q_{j}=(t-\frac{r^{2}}{2},t+\frac{r^{2}}{2})\times(x^{1}-r,x^{1}+r)\times
B^{\prime}_{r}(x^{\prime})$ we have
$\displaystyle\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}=3^{d}\cdot\frac{\phi(x+3r)-\phi((x-3r)\vee
0)}{\phi(x+r)-\phi(x-r)},$
where $\phi(x)=x^{\theta-d+p+1}$ and $a\vee b:=\max\\{a,b\\}$. We note
$\displaystyle\frac{\phi(x+3r)-\phi((x-3r)\vee 0)}{\phi(x+r)-\phi(x-r)}$ (2.7)
$\displaystyle=$ $\displaystyle\frac{\phi(x-r)-\phi((x-3r)\vee
0)+\phi(x+r)-\phi(x-r)+\phi(x+3r)-\phi(x+r)}{\phi(x+r)-\phi(x-r)}$
$\displaystyle\leq$ $\displaystyle
2+\frac{\phi(x+3r)-\phi(x+r)}{\phi(x+r)-\phi(x-r)},$
where the last inequality is true since $\phi$ is increasing and convex. Now,
Lemma 2.11 with $x-r$, $2r$ instead of $x$, $r$ implies (2.7) is less than or
equal to $2+2^{\alpha+1}$. Hence, we have
$\displaystyle\frac{|\tilde{Q_{j}}^{*}|}{|\tilde{Q_{j}}|}\leq
3^{d}\cdot(2+2^{\alpha+1}),\quad|\tilde{Q_{j}}^{*}|\leq
3^{d}\cdot(2+2^{\alpha+1})|\tilde{Q_{j}}|.$
Thus,
$\displaystyle|K|$ $\displaystyle\leq$
$\displaystyle\sum^{k}_{j=1}|\tilde{Q_{j}}^{*}|\leq
3^{d}\cdot(2+2^{\alpha+1})\sum^{k}_{j=1}|\tilde{Q_{j}}|$ $\displaystyle\leq$
$\displaystyle
3^{d}\cdot(2+2^{\alpha+1})\lambda^{-1}\sum^{k}_{j=1}\int_{\tilde{Q_{j}}}g\,d\mu\leq
3^{d}\cdot(2+2^{\alpha+1})\lambda^{-1}\int_{\Omega}gI_{A(\lambda)}\,d\mu.$
The theorem is proved. ∎
## 3 A weighted $L_{p}$-theory for systems in a half space
Let $C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ denote the set of all
$\mathbb{R}^{d_{1}}$-valued infinitely differentiable functions with compact
support in $\mathbb{R}^{d}$. By $\mathcal{D}$ we denote the space of
$d$-dimensional distributions on
$C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$; precisely, for
$u\in\mathcal{D}$ and $\phi\in
C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ we define
$(u,\phi)\in\mathbb{R}^{d_{1}}$ with components
$(u,\phi)^{k}=(u^{k},\phi^{k})$, $k=1,\ldots,d_{1}$; each $u^{k}$ is a usual
scalar-valued distribution.
For $p\in(1,\infty)$ we define
$L_{p}=L_{p}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ as the space of all
$\mathbb{R}^{d_{1}}$-valued functions $u=(u^{1},\ldots,u^{d_{1}})$ satisfying
$\|u\|^{p}_{L_{p}}:=\sum^{d_{1}}_{k=1}\|u^{k}\|^{p}_{L_{p}}<\infty.$
Denote $x=(x^{1},\ldots,x^{d})$. In this paper we define
$\|u_{x}\|^{p}_{L_{p}}=\sum_{i=1}^{d}\|u_{x^{i}}\|^{p}_{L_{p}},\quad\|u_{xx}\|^{p}_{L_{p}}=\sum_{i,j=1}^{d}\|u_{x^{i}x^{j}}\|^{p}_{L_{p}},\quad\textrm{etc.}$
For any $\gamma\in\mathbb{R}$, define the space of Bessel potential
$H^{\gamma}_{p}=H^{\gamma}_{p}(\mathbb{R};\mathbb{R}^{d_{1}})$ as the space of
all distributions $u$ on $\mathbb{R}^{d}$ such that $(1-\Delta)^{\gamma/2}u\in
L_{p}$, where each component is defined by
$((1-\Delta)^{\gamma/2}u)^{k}=(1-\Delta)^{\gamma/2}u^{k}$
and the norm is given by
$\|u\|_{H^{\gamma}_{p}}:=\|(1-\Delta)^{\gamma/2}u\|_{L_{p}}.$
Then $H^{\gamma}_{p}$ is a Banach space with the given norm and
$C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ is dense in
$H^{\gamma}_{p}$ (see [19]). Note that $H^{\gamma}_{p}$ are usual Sobolev
spaces for $\gamma=0,1,2,\ldots$. It is well known that the first order
differentiation operator, $D:H^{\gamma}_{p}\to H^{\gamma-1}_{p}$, is bounded.
On the other hand, if $\text{supp}\,(u)\subset(a,b)$, where
$-\infty<a<b<\infty$, then
$\|u\|_{H^{\gamma}_{p}}\leq c(d,a,b)\|u_{x}\|_{H^{\gamma-1}_{p}}$ (3.1)
(see, for instance, Remark 1.13 in [10]).
Now we introduce the weighted Sobolev spaces taken from [10] and [17]. Take a
nonnegative real-valued function $\zeta(x)=\zeta(x^{1})\in
C^{\infty}_{0}(\mathbb{R}_{+})$ such that
$\sum_{n=-\infty}^{\infty}\zeta(e^{n+s})>c>0,\quad\forall s\in\mathbb{R},$
(3.2)
where $c$ is a constant. Note that any nonnegative function $\zeta$ with
$\zeta>0$ on $[1,e]$ satisfies (3.2). For $\theta\in\mathbb{R}$, let
$H^{\gamma}_{p,\theta}:=H^{\gamma}_{p,\theta}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$
denote the set of all $d$-dimensional distributions $u=(u^{1},u^{2},\cdots
u^{d_{1}})$ on $\mathbb{R}^{d}_{+}$ such that
$\|u\|^{p}_{H^{\gamma}_{p,\theta}}:=\sum_{n\in\mathbb{Z}}e^{n\theta}\|\zeta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}}<\infty.$
(3.3)
It is known that for different $\zeta$ satisfying (3.2), we get the same
spaces $H^{\gamma}_{p,\theta}$ with equivalent norms, and for any $\eta\in
C^{\infty}_{0}(\mathbb{R}_{+};\mathbb{R})$,
$\sum_{n=-\infty}^{\infty}e^{n\theta}\|\eta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}}\leq
N\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta(\cdot)u(e^{n}\cdot)\|^{p}_{H^{\gamma}_{p}},$
(3.4)
where $N$ depends only on $\gamma,\theta,p,d,d_{1},\eta,\zeta$. Furthermore,
if $\gamma$ is a nonnegative integer, then
$\displaystyle\|u\|^{p}_{H^{\gamma}_{p,\theta}}\sim\sum_{|\beta|\leq\gamma}\int_{\mathbb{R}_{+}^{d}}|(x^{1})^{|\beta|}D^{\beta}u(x)|^{p}(x^{1})^{\theta-d}\,dx.$
(3.5)
Let $M^{\alpha}$ be the operator of multiplying by $(x^{1})^{\alpha}$ and
$M:=M^{1}$. For $\nu\in(0,1]$, denote
$|u|_{C}=\sup_{x\in\mathbb{R}^{d}_{+}}|u(x)|,\quad[u]_{C^{\nu}}=\sup_{x\neq
y}\frac{|u(x)-u(y)|}{|x-y|^{\nu}}.$
Below we collect some other important properties of the spaces
$H^{\gamma}_{p,\theta}$.
###### Lemma 3.1.
$(\cite[cite]{[\@@bibref{}{kr99}{}{}]},\cite[cite]{[\@@bibref{}{kr99-1}{}{}]})$
Let $\gamma,\theta\in\mathbb{R}$ and $p\in(1,\infty)$.
* (i)
$C^{\infty}_{0}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ is dense in
$H^{\gamma}_{p,\theta}$.
* (ii)
Assume that $\gamma=m+\nu+d/p$ for some $m=0,1,\cdots$ and $\nu\in(0,1]$. Then
for any $u\in H^{\gamma}_{p,\theta}$ $i\in\\{0,1,\cdots,m\\}$, we have
$|M^{i+\theta/p}D^{i}u|_{C}+[M^{m+\nu+\theta/p}D^{m}u]_{C^{\nu}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$ (3.6)
* (iii)
Let $\alpha\in\mathbb{R}$. Then $M^{\alpha}H^{\gamma}_{p,\theta+\alpha
p}=H^{\gamma}_{p,\theta}$ and
$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|M^{-\alpha}u\|_{H^{\gamma}_{p,\theta+\alpha p}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$
* (iv)
For any $MD,DM:H^{\gamma}_{p,\theta}\to H^{\gamma-1}_{p,\theta}$ are bounded
linear operators, and
$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|_{H^{\gamma-1}_{p,\theta}}+N\|Mu_{x}\|_{H^{\gamma-1}_{p,\theta}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}},$ $\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|_{H^{\gamma-1}_{p,\theta}}+N\|(Mu)_{x}\|_{H^{\gamma-1}_{p,\theta}}\leq
N\|u\|_{H^{\gamma}_{p,\theta}}.$
Furthermore, if $\theta\neq d-1,d-1+p$, then
$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|Mu_{x}\|_{H^{\gamma-1}_{p,\theta}},\quad\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|(Mu)_{x}\|_{H^{\gamma-1}_{p,\theta}}.$ (3.7)
* (v)
For $i=0,1$ let
$\kappa\in[0,1],\;p_{i}\in(1,\infty),\;\gamma_{i},\;\theta_{i}\in\mathbb{R}$
and assume the relations
$\gamma=\kappa\gamma_{1}+(1-\kappa)\gamma_{0},\quad\frac{1}{p}=\frac{\kappa}{p_{1}}+\frac{1-\kappa}{p_{0}},\quad\frac{\theta}{p}=\frac{\theta_{1}\kappa}{p_{1}}+\frac{\theta_{0}(1-\kappa)}{p_{0}}.$
Then
$\|u\|_{H^{\gamma}_{p,\theta}}\leq
N\|u\|^{\kappa}_{H^{\gamma_{1}}_{p_{1},\theta_{1}}}\|u\|^{1-\kappa}_{H^{\gamma_{0}}_{p_{0},\theta_{0}}}.$
###### Remark 3.2.
Let $\theta\in(d-1,d-1+p)$ and $n$ be a nonnegative integer. By Lemma 3.1
$(iii),(iv)$
$\|M^{-n}v\|_{H^{\gamma}_{p,\theta}}\leq
N\|D^{n}v\|_{H^{\gamma-n}_{p,\theta}}$ (3.8)
for any $v\in C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$. Indeed,
since $\theta+mp\neq d-1,d-1+p$ for any integer $m$
$\displaystyle\|M^{-n}v\|_{H^{\gamma}_{p,\theta}}$ $\displaystyle\leq$
$\displaystyle N\|M^{-1}v\|_{H^{\gamma}_{p,\theta-(n-1)p}}\leq
N\|v_{x}\|_{H^{\gamma-1}_{p,\theta-(n-1)p}}$ $\displaystyle\leq$
$\displaystyle N\|M^{-1}v_{x}\|_{H^{\gamma-1}_{p,\theta-(n-2)p}}\leq
N\|D^{2}v\|_{H^{\gamma-2}_{p,\theta-(n-2)p}}\ldots.$
For $-\infty\leq S<T\leq\infty$, we define the Banach spaces:
$\mathbb{H}^{\gamma}_{p,\theta}(S,T):=L_{p}((S,T),H^{\gamma}_{p,\theta}),\;\mathbb{H}^{\gamma}_{p,\theta}(T):=\mathbb{H}^{\gamma}_{p,\theta}(0,T),\;\mathbb{L}_{p,\theta}(S,T):=H^{0}_{p,\theta}(S,T),\;\mathbb{L}^{\gamma}_{p,\theta}(T):=\mathbb{L}^{\gamma}_{p,\theta}(0,T)$
with norms given by
$\|u\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}=\int^{T}_{S}\|u(t)\|^{p}_{H^{\gamma}_{p,\theta}}dt.$
###### Lemma 3.3.
For $\phi,\psi\in C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})$, define
$(\phi,\psi)=\int_{S}^{T}\int_{\mathbb{R}^{d}_{+}}\phi(s,x)\psi(t,x)dtdx$. For
$p\in(1,\infty)$ and $\gamma,\theta\in\mathbb{R}$, define
$\gamma^{\prime},p^{\prime},\theta^{\prime}$ so that
$\gamma^{\prime}=-\gamma,\quad\frac{1}{p}+\frac{1}{p^{\prime}}=1,\quad\frac{\theta}{p}+\frac{\theta^{\prime}}{p^{\prime}}=d.$
Then for any $\phi\in C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})$
$\displaystyle\|\phi\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\leq
N\;\sup_{\psi\in
C^{\infty}_{0}((S,T)\times\mathbb{R}^{d}_{+})}\frac{(\phi,\psi)}{\|\psi\|_{\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)}}\leq
N\|\phi\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)},$
where $N$ is independent of $\phi$. Moreover the relation $(\phi,\psi)$ can be
extended by continuity on all $\phi\in\mathbb{H}^{\gamma}_{p,\theta}(S,T)$ and
$\psi\in\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)$, and
then it identifies the dual to $\mathbb{H}^{\gamma}_{p,\theta}(S,T)$ with
$\mathbb{H}^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}(S,T)$.
###### Proof.
See Theorem 2.5 of [11]; this actually proves the duality between
$H^{\gamma}_{p,\theta}$ and
$H^{\gamma^{\prime}}_{p^{\prime},\theta^{\prime}}$, but the proof of our claim
is essentially the same. The only difference is that one has to consider
integrations on the time variable, too. ∎
Finally, we set $U^{\gamma}_{p,\theta}:=M^{1-2/p}H^{\gamma-2/p}_{p,\theta}$,
meaning that any $u\in U^{\gamma}_{p,\theta}$ has the form $u=M^{1-2/p}\cdot
v$ with $v\in H^{\gamma-2/p}_{p,\theta}$ and
$\|u\|_{U^{\gamma}_{p,\theta}}:=\|M^{-1+2/p}u\|_{H^{\gamma-2/p}_{p,\theta}}=\|v\|_{H^{\gamma-2/p}_{p,\theta}}$.
Using these spaces, we define our solution spaces.
###### Definition 3.4.
We write $u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ if $u\in
M\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)$, $u(S,\cdot)\in
U^{\gamma+2}_{p,\theta}$ ($u(-\infty,\cdot):=0$ if $S=-\infty$), and for some
$\tilde{f}\in M^{-1}\mathbb{H}^{\gamma}_{p,\theta}(T)$ it holds
$u_{t}=\tilde{f}$ in the sense of distributions, that is for any $\phi\in
C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ the equality
$(u(t,\cdot),\phi)=(u(S,\cdot),\phi)+\int^{t}_{S}(\tilde{f}(s,\cdot),\phi)ds$
(3.9)
holds for all $t\in(S,T)$. In this case we write $u_{t}=\tilde{f}$. The norm
in $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ is defined by
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)}=\|M^{-1}u\|_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}+\|Mu_{t}\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}+\|u(S,\cdot)\|_{U^{\gamma+2}_{p,\theta}}.$
Define
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T):=\mathfrak{H}^{\gamma+2}_{p,\theta}(0,T)$
and
$\mathfrak{H}^{\gamma+2}_{p,\theta}:=\mathfrak{H}^{\gamma+2}_{p,\theta}(0,\infty)$.
###### Theorem 3.5.
(i) The space $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ is a Banach space.
(ii) Let $0<T<\infty$. Then for any
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$,
$\sup_{t\leq T}\|u(t)\|_{\mathbb{H}^{\gamma+1}_{p,\theta}}\leq
N(d,p,\theta,T)\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}.$
(iii) Let $0<T<\infty$. For any nonnegative integer $n\geq\gamma+2$, the set
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)\bigcap\bigcup_{k=1}^{\infty}C([0,T],C^{n}_{0}(G_{k}))$
where $G_{k}=(1/k,k)\times\\{|x^{\prime}|<k\\}$ is dense in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$.
###### Proof.
See Theorem 2.9 and Theorem 2.11 of [14]. ∎
Here are some interior Hölder estimates of functions in the space
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$.
###### Theorem 3.6.
Let $p>2$ and assume
$2/p<\alpha<\beta\leq 1,\quad\gamma+2-\beta-d/p=k+\varepsilon,$
where $k\in\\{0,1,2,\cdots\\}$ and $\varepsilon\in(0,1]$. Denote
$\sigma=\beta-1+\theta/p$. Then for any
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ and multi-indices $i,j$ such that
$|i|\leq j$ and $|j|=k$,
(i) the functions $D^{i}u(t,x)$ are continuous in
$[0,T]\times\mathbb{R}^{d}_{+}$ and
$M^{\sigma+|i|}D^{i}u(t,\cdot)-M^{\sigma+|i|}D^{i}u(0,\cdot)\in
C^{\alpha/2-1/p}([0,T],C(\mathbb{R}^{d}_{+}));$
(ii) there exists a constant $N=N(p,d,\alpha,\beta)$ so that
$\displaystyle\sup_{t,s\leq
T}\left(\frac{\big{|}M^{\sigma+|i|}D^{i}(u(t)-u(s))\big{|}_{C(\mathbb{R}^{d}_{+})}}{|t-s|^{\alpha/2-1/p}}+\frac{\big{[}M^{\sigma+|j|+\varepsilon}D^{j}(u(t)-u(s))\big{]}_{C^{\varepsilon}}}{|t-s|^{\alpha/2-1/p}}\right)$
(3.10) $\displaystyle\leq$ $\displaystyle
NT^{(\beta-\alpha)/2}\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}.$
###### Proof.
See Theorem 4.7 of [8]. ∎
###### Remark 3.7.
(see Remark 4.8 of [8] for details) For instance, if $\theta=d$,
$\gamma\geq-1$ and $\kappa_{0}=1-\frac{2}{p}-\frac{d}{p}>0$, then for any
$\kappa\in(0,\kappa_{0})$ and $u\in\mathfrak{H}^{1}_{p,\theta}(T)$ with
$u(0)=0$,
$\sup_{t\leq
T}\sup_{x,y\in\mathbb{R}^{d}_{+}}\frac{|u(t,x)-u(t,y)|}{|x-y|^{\kappa}}<\infty.$
(3.11) $\sup_{x\in\mathbb{R}^{d}_{+}}\sup_{s,t\leq
T}\frac{|u(t,x)-u(s,x)|}{|t-s|^{\kappa/2}}<\infty.$ (3.12)
Indeed, for (3.11) take $j=0,\beta=\kappa_{0}-\kappa+2/p$ and
$\varepsilon=1-\beta-d/p=\kappa=-\sigma$, then $\sigma+|j|+\varepsilon=0$ and
(3.10) yields (3.11). Also for (3.12), take
$i=0,\alpha=\kappa+2/p,\beta=1-d/p$ then $\sigma+|i|=0$, $2/p<\alpha<\beta<1$
and $\alpha/2-1/p=\kappa/2$.
For any $d_{1}\times d_{1}$ matrix $C=(c_{kr})$ we let
$|C|:=\sqrt{\sum_{k,r}(c_{kr})^{2}}.$
We set $A^{ij}=(a^{ij}_{kr})_{k,r=1,\ldots,d_{1}}$ for each $i,j=1,\ldots,d$.
Throughout the article we assume the followings.
###### Assumption 3.8.
For each $i$ and $j$, $A^{ij}$ depends only on $t$ and there exist finite
constants $\delta,K>0$ so that
$\delta|\xi|^{2}\leq\sum_{i,j=1}^{d}(\xi^{i})^{*}A^{ij}\;\xi^{i}$ (3.13)
for all (real valued) $d_{1}\times d$-matrix $\xi$, where $\xi^{i}$ denotes
the $i$-th column of $\xi$. Also, there exists a constant $K<\infty$ such that
$\left|A^{ij}\right|\leq K,\quad\forall\;\;i,j=1,\ldots,d,$ (3.14)
where $*$ means matrix transposition.
We recall (1.2) and write it as
$u^{k}_{t}=a^{ij}_{kr}(t)u^{r}_{x^{i}x^{j}}+f^{k},\quad
u^{k}(S)=u^{k}_{0},\quad\quad k=1,2,\cdots,d_{1},$ (3.15)
assuming the summation convention on indices $i,j,r$; such convention will be
used throughout the article. In short, we will write (3.15) as
$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f,\quad u(S)=u_{0},$ (3.16)
where we regard $u,u_{0},f$ as $d_{1}\times 1$ matrix-valued functions.
###### Definition 3.9.
A $d$-dimensional distribution-valued function $u$ defined on $(S,T)$ is a
solution of (3.16) in $\mathfrak{H}^{\gamma+2}_{p,\theta}(S,T)$ if $u\in
M\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)$, $u(S)\in U^{\gamma+2}_{p,\theta}$
($u(-\infty,\cdot):=0$ if $S=-\infty$) and (3.16) holds in the sense of
distributions, that is (3.9) holds with $\tilde{f}=A^{ij}u_{x^{i}x^{j}}+f$.
The following is our $L_{p}$-theory for the parabolic system (3.16). The proof
is given in section 6.
###### Theorem 3.10.
Let $p\in(1,\infty)$ and $\gamma\geq 0$. Assume $\theta\in(d+1-p,d+p-1)$ if
$p\in(1,2]$ and $\theta\in(d-1,d+1)$ if $p\in(2,\infty)$. Then for any $f\in
M^{-1}\mathbb{H}^{\gamma}_{p,\theta}(T)$ and $u_{0}\in
U^{\gamma+2}_{p,\theta}$ system (3.16) admits a unique solution
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$, and for this solution we have
$\|u\|_{\mathfrak{H}^{\gamma+2}_{p,\theta}(T)}\leq
N\left(\|Mf\|_{\mathbb{H}^{\gamma}_{p,\theta}(T)}+\|u_{0}\|_{U^{\gamma+2}_{p,\theta}}\right),$
(3.17)
where $N=N(\gamma,p,\theta,\delta,K)$.
###### Remark 3.11.
Various interior Hölder estimates of the solution in Theorem 3.10 can be
obtained according to Theorem 3.6. Also see Lemma 4.11 and Lemma 4.14.
###### Remark 3.12.
(i) The proof of Theorem 3.10 is based on a sharp function estimate (Lemma
5.7). If $d_{1}=1$, then Lemma 5.7 can be proved for any
$\theta\in(d-1,d-1+p)$ as long as $p>1$; we will prove this in a subsequent
article for parabolic equations with (weighted) BMO-coefficients.
(ii) It is known (see Remark 3.6 of [14]) that if $\theta\not\in(d-1,d-1+p)$,
then Theorem 3.10 is not true even for the heat equation $u_{t}=\Delta u+f$.
Now we present our $L_{p}$-theory for the elliptic system (1.1). The proof is
given in section 6.
###### Theorem 3.13.
Let $p\in(1,\infty)$, $\gamma\geq 0$ and $A^{ij}$ be independent of $t$.
Assume $\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$ and $\theta\in(d-1,d+1)$ if
$p\in(2,\infty)$. Then for any $f=(f^{1},f^{2},\cdots,f^{d_{1}})\in
M^{-1}H^{\gamma}_{p,\theta}$ the system (1.1) admits a unique solution $u\in
MH^{\gamma+2}_{p,\theta}$, and for this solution we have
$\|M^{-1}u\|_{H^{\gamma+2}_{p,\theta}}\leq N\|Mf\|_{H^{\gamma}_{p,\theta}},$
where $N=N(\gamma,p,\theta,\delta,K)$.
###### Remark 3.14.
Theorem 3.10 and Theorem 3.13 hold not only for $\gamma\geq 0$ but also for
any $\gamma<0$. This can be easily proved by using the results for $\gamma\geq
0$ and repeating the arguments used for single equations (see the proof of
Theorem 5.6 of [10]).
## 4 Preliminary estimates : Some local estimates of solutions
In this section we prove a version of Theorem 3.10 for $\theta=d$. This result
is used to derive some local estimates of $D^{\alpha}u$ for any multi-index
$\alpha$, where $u$ is a solution of (3.16).
First, we introduce some results for systems defined on the entire space. For
$-\infty\leq S<T\leq\infty$ we denote
$\mathbb{H}^{\gamma}_{p}(S,T):=L_{p}((S,T),H^{\gamma}_{p})$ and
$\mathbb{H}^{\gamma}_{p}(T):=\mathbb{H}^{\gamma}_{p}(0,T)$.
###### Theorem 4.1.
Let $\gamma\in\mathbb{R}$ and $-\infty\leq S<T\leq\infty$. Let
$f\in\mathbb{H}^{\gamma}_{p}(S,T)$ and $u\in\mathbb{H}^{\gamma+2}_{p}(S,T)$
satisfy
$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f,\quad t>S,x\in\mathbb{R}^{d}.$
Additionally assume $u(S,\cdot)=0$ if $S>-\infty$. Then
$\|u_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}\leq
N(d,p,\delta,K)\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$ (4.1)
Also if $-\infty<S<T<\infty$, then
$\|u\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S,T)}\leq
N(d,p,\delta,K,S,T)\|f\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$
###### Proof.
This is a classical result. See, for instance, Theorem 1.1 of [15]. Actually
in [15] the theorem is proved only when $\gamma=0$, but the general case
follows by the fact the operator
$(1-\Delta)^{\mu/2}:\mathbb{H}^{\gamma}_{p}(S,T)\to\mathbb{H}^{\gamma-\mu}_{p}(S,T)$
is an isometry. ∎
Theorem 4.1 yields the following result.
###### Corollary 4.2.
Let $u\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$. Then
$\|u_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(-\infty,\infty)}\leq
N(d,p,\delta,K)\;\|u_{t}-A^{ij}u_{x^{i}x^{j}}\|^{p}_{\mathbb{H}^{\gamma}_{p}(-\infty,\infty)}.$
(4.2)
###### Corollary 4.3.
Let $0<T<\infty$, $f^{i}\in\mathbb{L}_{p}(T)$, and $u\in\mathbb{H}^{1}_{p}(T)$
satisfies
$u_{t}=A^{ij}(t)u_{x^{i}x^{j}}+f^{i}_{x^{i}},\quad
t\in(0,T),x\in\mathbb{R}^{d}$
with zero initial condition $u(0)=0$. Then
$\|u_{x}\|^{p}_{\mathbb{L}_{p}(T)}\leq
N(d,p,\delta,K)\|f^{i}\|^{p}_{\mathbb{L}_{p}(T)}.$ (4.3)
$\|u\|^{p}_{\mathbb{H}^{1}_{p}(T)}\leq
N(d,p,\delta,K,T)\|f^{i}\|^{p}_{\mathbb{L}_{p}(T)}.$
###### Proof.
Remember
$\|f^{i}_{x}\|_{H^{-1}_{p}}\leq N\|f^{i}\|_{L_{p}},\quad\|u_{x}\|_{L_{p}}\leq
N(\|u_{xx}\|_{H^{-1}_{p}}+\|u\|_{L_{p}(T)}).$
By (4.1) with $\gamma=-1$,
$\|u_{x}\|_{\mathbb{L}(T)}\leq N(\|f^{i}\|_{\mathbb{L}(T)}+\|u\|_{L_{p}(T)}).$
(4.4)
Notice that, for any constant $c>0$, the function $u^{c}(t,x):=u(c^{2}t,cx)$
satisfies
$u^{c}_{t}=A^{ij}(c^{2}t)u^{c}_{x^{i}x^{j}}+(cf^{i}(c^{2}t,cx))_{x^{i}}.$
Thus for this function (4.4) with $c^{-2}T$ in place of $T$ becomes
$\|u_{x}\|_{\mathbb{L}(T)}\leq
N(\|f^{i}\|_{\mathbb{L}(T)}+c^{-1}\|u\|_{L_{p}}).$
Now we get (4.3) by taking $c\to\infty$. ∎
###### Corollary 4.4.
Let $u\in C^{\infty}_{0}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ and $A^{ij}$ be
independent of $t$. Then
$\|u_{xx}\|^{p}_{H^{\gamma}_{p}}\leq
N(d,p,\delta,K)\;\|A^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p}}.$ (4.5)
###### Proof.
Take a nonnegative smooth function $\eta(t)\in C^{\infty}_{0}(-1,1)$ so that
$\int_{\mathbb{R}}\eta^{p}(t)dt=1$. For each $n=1,2,\cdots$, define
$\eta_{n}(t)=n^{-1/p}\eta(t/n)$. Then applying (4.2) for
$v_{n}(t,x):=\eta_{n}(t)u(x)$,
$\|u_{xx}\|^{p}_{H^{\gamma}_{p}}\leq
N\|A^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p}}+N\|u\|^{p}_{H^{\gamma}_{p}}\int_{\mathbb{R}}|\eta^{\prime}_{n}|^{p}dt$
Now it is enough to let $n\to\infty$. The corollary is proved. ∎
Remember that for any $t\in\mathbb{R}$, $(x^{1},x^{\prime})\in\mathbb{R}^{d}$,
we defined
$B_{r}(x)=(x^{1}-r,x^{1}+r)\times B^{\prime}_{r}(x^{\prime}),\quad
Q_{r}(t,x)=(t,t+r^{2})\times B_{r}(x),$
where $B^{\prime}_{r}(x^{\prime})$ is the open ball in $\mathbb{R}^{d-1}$ of
radius $r$ with center $x^{\prime}$. By
$C^{\infty}_{loc}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$ we denote the set of
$\mathbb{R}^{d_{1}}$-valued functions $u$ defined on $\mathbb{R}^{d+1}$ and
such that $\zeta u\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$ for
any $\zeta\in C^{\infty}_{0}(\mathbb{R}^{d+1};\mathbb{R})$.
###### Theorem 4.5.
Let $q\in(1,\infty)$ and $(t,x)\in\mathbb{R}^{d+1}$. Then there exists a
constant $N$, depending only on $q,d,d_{1},\delta$ and $K$ so that for any
$\lambda\geq 4,r>0$ and $u\in
C^{\infty}_{loc}(\mathbb{R}^{d+1};\mathbb{R}^{d_{1}})$, we have
$\displaystyle-\int_{Q_{r}(t,x)}-\int_{Q_{r}(t,x)}|u_{xx}(s,y)-u_{xx}(r,z)|^{q}\,dsdydrdz$
$\displaystyle\leq$ $\displaystyle N\lambda^{-q}-\int_{Q_{\lambda
r}(t,x)}|u_{xx}|^{q}\,dsdy+N\lambda^{d+2}-\int_{Q_{\lambda
r}(t,x)}|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q}\,dsdy.$
###### Proof.
See Theorem 6.1.2 of [13]. Actually this theorem is proved when $d_{1}=1$, and
the proof is based on Theorem 4.1. Since Theorem 4.1 holds for any
$d_{1}=1,2,\cdots$, the theorem can be proved by repeating the proof of
Theorem 6.1.2 of [13] word for word. ∎
###### Corollary 4.6.
Let $u=u(x)\in C^{\infty}_{loc}(\mathbb{R}^{d};\mathbb{R}^{d_{1}})$ and
$A^{ij}$ be independent of $t$. Then for any $x\in\mathbb{R}^{d}$,
$\lambda\geq 4$ and $r>0$,
$\displaystyle-\int_{B_{r}(x)}-\int_{B_{r}(x)}|u_{xx}(y)-u_{xx}(z)|^{q}dydz$
$\displaystyle\leq$ $\displaystyle N\lambda^{-q}-\int_{B_{\lambda
r}(x)}|u_{xx}|^{q}dy+N\lambda^{d+2}-\int_{B_{\lambda
r}(x)}|A^{ij}u_{x^{i}x^{j}}|^{q}dy.$
From now on we consider systems defined on a half space. Remember
$\mathbb{H}^{\gamma}_{p,\theta}(S,T):=L_{p}((S,T),H^{\gamma}_{p,\theta}),\quad\quad\|u\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}:=\int^{T}_{S}\|u(t,\cdot)\|^{p}_{H^{\gamma}_{p,\theta}}\,dt.$
###### Lemma 4.7.
Let $\gamma,\theta\in\mathbb{R}$ and $p\in(1,\infty)$.
(i) Let $-\infty\leq S<T\leq\infty$ and suppose $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$
satisfies
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f,\quad(t,x)\in(S,T)\times\mathbb{R}^{d}_{+}$
and assume $u(T,\cdot)=0$ if $T<\infty$.
$\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}\leq
N(p,d,\theta,\delta,K)\left(\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+1}_{p,\theta}(S,T)}+\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\right).$
(4.6)
(ii) If $u(x)\in C^{\infty}_{0}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ and
$A^{ij}$ is independent of $t$, then
$\|M^{-1}u\|^{p}_{H^{\gamma+2}_{p,\theta}}\leq
N(p,d,\theta,\delta,K)\left(\|M^{-1}u\|^{p}_{H^{\gamma+1}_{p,\theta}}+\|MA^{ij}u_{x^{i}x^{j}}\|^{p}_{H^{\gamma}_{p,\theta}}\right).$
(4.7)
###### Proof.
(i). We proceed as in the proof of Lemma 5.8 of [10]. Denote $S_{n}=e^{-2n}S$
and $T_{n}=e^{-2n}T$. By Lemma 3.1(iii) and (3.3),
$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(\theta-p)}\|\zeta(x)u(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S,T)}$
(4.8) $\displaystyle=$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(2+\theta-p)}\|\zeta(x)u(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma+2}_{p}(S_{n},T_{n})}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(2+\theta-p)}\|(\zeta(x)u(e^{2n}t,e^{n}x))_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})},$
where the last inequality is due to (3.1). Denote
$v^{n}(t,x)=\zeta(x)u(e^{2n}t,e^{n}x)$, then it satisfies
$v^{n}_{t}+A^{ij}(e^{2n}t)v^{n}_{x^{i}x^{j}}=e^{2n}\zeta(x)f(e^{2n}t,e^{n}x)+2e^{n}A^{1j}(e^{2n}t)\zeta_{x}u_{x^{j}}(e^{2n}t,e^{n}x)+A^{11}(e^{2n}t)\zeta_{xx}u(e^{2n}t,e^{n}x)$
for $(t,x)\in(S_{n},T_{n})\times\mathbb{R}^{d}_{+}$. By (4.1),
$\displaystyle\|v^{n}_{xx}\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}$
$\displaystyle\leq$ $\displaystyle
Ne^{2np}\|\zeta(x)f(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}$
$\displaystyle+$ $\displaystyle
Ne^{np}\|\zeta_{x^{i}}u_{x^{j}}(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})}+N\|\zeta_{xx}u(e^{2n}t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S_{n},T_{n})},$
where $N$ is independent of $n$. Plugging this into (4.8) one gets
$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n(\theta+p)}\|\zeta(x)f(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}$
$\displaystyle+$ $\displaystyle
N\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta_{x}u_{x}(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}+N\sum_{n=-\infty}^{\infty}e^{n(\theta-p)}\|\zeta_{xx}u(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}.$
This, (3.4) and Lemma 3.1 easily lead us to (4.6). Indeed, for instance, by
(3.4)
$\sum_{n=-\infty}^{\infty}e^{n\theta}\|\zeta_{x}u_{x}(t,e^{n}x)\|^{p}_{\mathbb{H}^{\gamma}_{p}(S,T)}\leq
N\|u_{x}\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}$
and by Lemma 3.1(iv) applied to $M^{-1}u$ in place of $u$,
$\|u_{x}\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}=\|DM(M^{-1}u)\|_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}\leq
N\|M^{-1}u\|_{\mathbb{H}^{\gamma+1}_{p,\theta}(S,T)}.$
(ii) This is proved similarly based on (4.5). The lemma is proved. ∎
###### Remark 4.8.
Let $\gamma\geq 0$. By iterating (4.6), one gets
$\displaystyle\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\|M^{-1}u\|^{p}_{\mathbb{L}_{p,\theta}(S,T)}+N\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)}$
$\displaystyle\leq$ $\displaystyle
N\|Mu_{xx}\|^{p}_{\mathbb{L}_{p,\theta}(S,T)}+N\|Mf\|^{p}_{\mathbb{H}^{\gamma}_{p,\theta}(S,T)},$
where for the second inequality we use (3.7) twice. We use both inequalities
later to estimate $\|M^{-1}u\|^{p}_{\mathbb{H}^{\gamma+2}_{p,\theta}(S,T)}$.
Let $(w^{1}_{t},w^{2}_{t},\cdots,w^{d}_{t})$ be a $d$-dimensional Wiener
process defined on a probability space $(\Omega^{\prime},\mathcal{F},P)$.
Denote
$\xi_{t}=w^{1}_{t}\sqrt{2}+2t,\quad\eta_{t}=(\sqrt{2}\int^{t}_{0}e^{\xi_{s}}dw^{2}_{s},\cdots,\sqrt{2}\int^{t}_{0}e^{\xi_{s}}dw^{d}_{s})$
and define $d\times d$ matrix-valued process $\sigma_{t}$ so that
$(\sigma_{t}x)^{1}=e^{\xi_{t}}x^{1}$ and
$(\sigma_{t}x)^{\prime}=x^{\prime}+x^{1}\eta_{t}$. It is easy to check (see
[10], p.1628) that $x_{t}(x):=\sigma_{t}x$ is the unique solution of the
stochastic differential equation
$dx_{t}=\sqrt{2}x^{1}_{t}dw_{t}+3x^{1}_{t}e_{1}dt,\quad x_{0}(x)=x,$
where $e_{1}=(1,0,\cdots,0)$. For any $f\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ and $x\in\mathbb{R}^{d}$, define
$\mathcal{E}f(x)=\mathbb{E}\int^{\infty}_{0}f(\sigma_{t}x)\,dt:=\int_{\Omega^{\prime}}\int^{\infty}_{0}f(\sigma_{t}x)\,dtdP.$
(See below for the convergence of this integral). Note that if $x^{1}\leq 0$
then $(\sigma_{t}x)^{1}\leq 0$ and thus $\mathcal{E}f(x)=0$. Denote
$\mathcal{L}u:=M^{2}\Delta u+3MD_{1}u=\sum_{i=1}^{d}(MD_{i})^{2}+2MD_{1}.$
###### Lemma 4.9.
Let $f\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$.
(i) $\mathcal{E}f\in L_{p}(\mathbb{R}^{d})$ and $f=\mathcal{L}(\mathcal{E}f)$
in the sense of distributions on $\mathbb{R}^{d}$.
(ii) There exist $f^{1},f^{2},\cdots,f^{d}\in L_{p}(\mathbb{R}^{d})$ so that
$f=MD_{i}f^{i}$ in the sense of distributions on $\mathbb{R}^{d}$, and
$\sum_{i=1}^{d}\|f^{i}\|_{L_{p}(\mathbb{R}^{d})}\leq
N\|f\|_{L_{p}(\mathbb{R}^{d}_{+})}.$
###### Proof.
By Theorem 2.11 of [10] (with $\theta=d$ and $b=3$ there), the map
$\mathcal{L}$ is a bounded one-to-one operator from $H^{2}_{p,d}$ onto
$L_{p,d}$, and its inverse ($:=\mathcal{L}^{-1}$) is also bounded. Denote
$u:=\mathcal{L}^{-1}f\in H^{2}_{p,d}$. By Lemma 3.1(i), there exists a
sequence $u_{n}\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ so that $u_{n}\to u$ in
$H^{2}_{p,d}$. Denote $f_{n}(x):=\mathcal{L}u_{n}(x)$ for each
$x\in\mathbb{R}^{d}$. Then
$\mathcal{L}u_{n}\to\mathcal{L}u\,\,(=f)\quad\text{in}\quad
L_{p,d}\quad\quad\text{and}\quad\quad\|u_{n}-u_{m}\|_{H^{2}_{p,d}}\leq
N\|f_{n}-f_{m}\|_{L_{p,d}}.$ (4.9)
Obviously $u_{n}(x)=f_{n}(\sigma_{t}x)=0$ if $x^{1}\leq 0$. By Itô’s formula
(see (2.10) in [10] for details), we get
$u_{n}(x)=\mathbb{E}\int^{\infty}_{0}f_{n}(\sigma_{t}x)\,dt,\quad\quad\forall
x\in\mathbb{R}^{d}.$
The convergence of this improper integral is discussed in the proof of Theorem
2.11 of [10]. Actually there it is shown that for any $h\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ (here, $\theta=d$ and $b=3$ in our case),
$\mathbb{E}\int^{\infty}_{0}\|h(\sigma_{t}x)\|_{L_{p,d}}dt\leq
N\|h\|_{L_{p,d}}\int^{\infty}_{0}e^{-(\theta-d+1)(b-1)t+(\theta-d+1)^{2}t}dt=N\|h\|_{L_{p,d}},$
(4.10)
which also implies
$\|u_{n}-\mathcal{E}f\|_{L_{p,d}}=\|\mathbb{E}\int^{\infty}_{0}f_{n}(\sigma_{t}x)\,dt-\mathbb{E}\int^{\infty}_{0}f(\sigma_{t}x)\,dt\|_{L_{p,d}}\leq
N\|f_{n}-f\|_{L_{p,d}}\to 0\quad\text{as}\quad n\to\infty.$
Note $L_{p,d}=L_{p}(\mathbb{R}^{d}_{+})$. Since $u_{n}(x),f_{n}(x),f(x)$ and
$\mathcal{E}f$ vanish if $x^{1}\leq 0$, it follows that
$\|u_{n}-\mathcal{E}f\|_{L_{p}(\mathbb{R}^{d})}\to
0,\quad\quad\|f_{n}-f\|_{L_{p}(\mathbb{R}^{d})}\to 0$ (4.11)
as $n\to\infty$. Also (4.9) and fact
$\|u_{n}\|_{H^{2}_{p,d}}=\|\mathcal{L}^{-1}f_{n}\|_{H^{2}_{p,d}}\leq
N\|f_{n}\|_{L_{p,d}}$ show that $\\{MDu_{n}:n=1,2,\cdots\\}$ is a Cauchy
sequence in $L_{p}(\mathbb{R}^{d})$. Indeed, since each $u_{n}$ has compact
support in $\mathbb{R}^{d}_{+}$,
$\|MDu_{n}-MDu_{m}\|_{L_{p}(\mathbb{R}^{d})}=\|MDu_{n}-MDu_{m}\|_{L_{p,d}}\leq
N\|u_{n}-u_{m}\|_{H^{1}_{p,d}}\leq N\|f_{n}-f_{m}\|_{L_{p,d}}.$
Let $\mathcal{L}^{*}$ denote the adjoint operator of $\mathcal{L}$. For any
$\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, by (4.11),
$(f,\phi)=\lim_{n\to\infty}(f_{n},\phi)=\lim_{n\to\infty}(\mathcal{L}u_{n},\phi)=\lim_{n\to\infty}(u_{n},\mathcal{L}^{*}\phi)=(\mathcal{E}f,\mathcal{L}^{*}\phi)=(\mathcal{L}(\mathcal{E}f),\phi).$
Thus $f=\mathcal{L}(\mathcal{E}f)$ in the sense of distributions on
$\mathbb{R}^{d}$. Also since $u_{n}\to\mathcal{E}f$ in $L_{p}(\mathbb{R}^{d})$
and $\\{MDu_{n}\\}$ is a Cauchy sequence in $L_{p}(\mathbb{R}^{d})$, we have
$MD\mathcal{E}f\in L_{p}(\mathbb{R}^{d})$. Consequently,
$f=\mathcal{L}(\mathcal{E}f)=MD_{1}(MD_{1}\mathcal{E}f+2\mathcal{E}f)+\sum_{j=2}^{d}MD_{j}\mathcal{E}f=:\sum_{i=1}^{d}MD_{i}f^{i},$
and by (4.11),
$\sum_{i}\|f^{i}\|_{L_{p}(\mathbb{R}^{d})}=\lim_{n\to\infty}(\|u_{n}\|_{L_{p}}+\|MDu_{n}\|_{L_{p}})\leq\lim_{n\to\infty}\|u_{n}\|_{H^{2}_{p,d}}\leq
N\|f_{n}\|_{L_{p,d}}=N\|f\|_{L_{p,d}}.$
The lemma is proved. ∎
Now we prove a version of Theorem 3.10 for $\theta=d$.
###### Lemma 4.10.
Let $-\infty<S<T<\infty$, $p\in(1,\infty)$ and $n=0,1,2\cdots$. For any $f\in
M^{-1}\mathbb{H}^{n}_{p,d}(S,T)$, the equation
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f,\quad(t,x)\in(S,T)\times\mathbb{R}^{d}_{+}$
with the condition $u(T)=0$ has a unique solution
$u\in\mathfrak{H}^{n+2}_{p,d}(S,T)$, and for this solution
$\|M^{-1}u\|_{\mathbb{H}^{n+2}_{p,d}(S,T)}\leq
N(p,d,\delta,K)\|Mf\|_{\mathbb{H}^{n}_{p,d}(S,T)}.$ (4.12)
###### Proof.
As usual we only need to prove that the estimate (4.12) holds given that a
solution $u$ already exists. Furthermore we may assume $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Due to
Remark 4.8 and the inequality $\|M^{-1}u\|_{L_{p,d}}\leq
N(p,d)\|u_{x}\|_{L_{p,d}}$ (see Lemma 3.1(iv)), we only need to prove
$\|u_{x}\|_{\mathbb{L}_{p,d}(S,T)}\leq N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$
(4.13)
By Lemma 4.9, we can write $Mf=MD_{i}f^{i}$ on $\mathbb{R}^{d}$ (thus
$f=D_{i}f^{i}$), where $f^{i}=(f^{i1},\cdots,f^{id_{1}})$, so that
$f^{i}\in\mathbb{L}_{p}(S,T)$ (not only in $\mathbb{L}_{p,d}(S,T)$) and
$\sum_{i=1}^{d}\|f^{i}\|_{\mathbb{L}_{p}(S,T)}\leq
N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$
Thus by Corollary 4.3,
$\|u_{x}\|_{\mathbb{L}_{p,d}(S,T)}=\|u_{x}\|_{\mathbb{L}_{p}(S,T)}\leq
N\|f^{i}\|_{\mathbb{L}_{p}(S,T)}\leq N\|Mf\|_{\mathbb{L}_{p,d}(S,T)}.$
The lemma is proved. ∎
For $r,a>0$, denote
$Q_{r}(a)=Q_{r}(0,a,0)=(0,r^{2})\times(a-r,a+r)\times B^{\prime}_{r}(0),\quad
U_{r}=(-r^{2},r^{2})\times(-2r,2r)\times B^{\prime}_{r}(0).$
###### Lemma 4.11.
Let $0<s<r<\infty$, $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$ and
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0\quad\quad\text{for}\quad(t,x)\in Q_{r}(r).$
Then for any multi-index $\beta=(\beta^{1},\cdots,\beta^{d})$ there exists a
constant $N=N(p,|\beta|)$ so that the inequality
$\displaystyle\int_{Q_{s}(s)}\left(|M^{-1}D^{\beta}u|^{p}+|D^{\beta}u_{x}|^{p}+|MD^{\beta}u_{xx}|^{p}\right)(x^{1})^{\theta-d}dxdt$
(4.14) $\displaystyle\leq$ $\displaystyle
N(1+r)^{|\beta|p}\cdot(1+(r-s)^{-2})^{(|\beta|+1)p}\int_{Q_{r}(r)}|Mu(t,x)|^{p}(x^{1})^{\theta-d}dxdt$
holds for $\theta=d$.
###### Proof.
To prove (4.14) we use induction on $|\beta|$. Firstly, consider the case
$|\beta|=0$. We modify the proof of Lemma 2.4.4 of [13]. Denote $r_{0}=s$ and
$r_{m}=s+(r-s)\sum_{j=1}^{m}2^{-j}$ for $m=1,2,\cdots$. choose a smooth
function $\zeta_{m}$ so that $0\leq\zeta_{m}\leq 1$,
$\zeta_{m}=1\quad\text{on}\quad
U_{r_{m}},\quad\quad\zeta_{m}=0\quad\text{on}\quad\Omega\setminus
U_{r_{m+1}},$ $|\zeta_{mx}|\leq N(r-s)^{-1}2^{m},\quad|\zeta_{mxx}|\leq
N(r-s)^{-2}2^{2m},\quad|\zeta_{mt}|\leq N(r-s)^{-2}2^{2m}.$
Note that $(u\zeta_{m})(r^{2},x)=0$ on $\mathbb{R}^{d}_{+}$, and it satisfies
$(u\zeta_{m})_{t}+A^{ij}(u\zeta_{m})_{x^{i}x^{j}}=\zeta_{mt}u+2A^{ij}(u\zeta_{m+1})_{x^{i}}\zeta_{mx^{j}}+A^{ij}u\zeta_{mx^{i}x^{j}}=:f_{m},\quad\quad(t,x)\in(0,r^{2})\times\mathbb{R}^{d}_{+}.$
By Lemma 4.10 for $\gamma=0$,
$A_{m}:=\|M^{-1}u\zeta_{m}\|_{\mathbb{H}^{2}_{p,d}(r^{2})}\leq
N\|Mf_{m}\|_{\mathbb{L}_{p,d}(r^{2})}.$
Denote $B:=\left(\int_{Q_{r}(r)}|Mu|^{p}dxdt\right)^{1/p}$. Then
$\|\zeta_{mt}Mu+A^{ij}Mu\zeta_{mx^{i}x^{j}}\|_{\mathbb{L}_{p,d}(r^{2})}\leq
N(r-s)^{-2}2^{2m}(\int_{Q_{r}(r)}|Mu|^{p}dxdt)^{1/p}=N(r-s)^{-2}2^{2m}B,$
$\|A\zeta_{mx}M(u\zeta_{m+1})_{x}\|_{\L_{p,d}(r^{2})}\leq
N(r-s)^{-1}2^{m}\|M(u\zeta_{m+1})_{x}\|_{\mathbb{L}_{p,d}(r^{2})}\leq
N(r-s)^{-1}2^{m}\|u\zeta_{m+1}\|_{\mathbb{H}^{1}_{p,d}(r^{2})},$
and by Lemma 3.1 (v) (take
$p_{0}=p_{1}=p,\gamma=1,\gamma_{0}=0,\gamma_{1}=2,\theta=d,\theta_{0}=d+p,\theta_{1}=d-p$
and $\kappa=1/2$) for any $\varepsilon>0$
$(r-s)^{-1}2^{m}\|u\zeta_{m+1}\|_{\mathbb{H}^{1}_{p,d}(r^{2})}\leq\varepsilon
A_{m+1}+\varepsilon^{-1}(r-s)^{-2}2^{2m}B.$
It follows (with $\varepsilon$ different from the one above),
$A_{m}\leq\varepsilon A_{m+1}+N(1+\varepsilon^{-1})(r-s)^{-2}2^{2m}B.$
We take $\varepsilon=\frac{1}{16}$ and get
$\varepsilon^{m}A_{m}\leq\varepsilon^{m+1}A_{m+1}+N\varepsilon^{m}(1+\varepsilon^{-1})2^{2m}(r-s)^{-2}B,$
$A_{0}+\sum_{m=1}^{\infty}\varepsilon^{m}A_{m}\leq\sum_{m=1}^{\infty}\varepsilon^{m}A_{m}+N(r-s)^{-2}B.$
Note that the series $\sum_{m=1}\varepsilon^{m}A_{m}$ converges because
$A_{m}\leq N2^{2m}\|M^{-1}u\|_{\mathbb{H}^{2}_{p,d}(r^{2})}$. By Lemma
3.1(iii), for any $M^{-1}w\in H^{2}_{p,\theta}$,
$\|M^{-1}w\|_{H^{2}_{p,\theta}}\sim(\|M^{-1}w\|_{L_{p,\theta}}+\|w_{x}\|_{L_{p,\theta}}+\|Mw_{xx}\|_{L_{p,\theta}}).$
(4.15)
Therefore,
$\int_{Q_{s}(s)}\left(|M^{-1}u|^{p}+|u_{x}|^{p}+|Mu_{xx}|^{p}\right)dxdt\leq
NA^{p}_{0}\leq N(r-s)^{-2p}\int_{Q_{r}(r)}|u(t,x)|^{p}(x^{1})^{p}dxdt.$
Next assume that (4.14) holds whenever $s<r$ and $|\beta^{\prime}|=k$, that is
$\displaystyle\int_{Q_{s}(s)}\left(|M^{-1}D^{\beta^{\prime}}u|^{p}+|D^{\beta^{\prime}}u_{x}|^{p}+|MD^{\beta^{\prime}}u_{xx}|^{p}\right)(x^{1})^{\theta-d}dxdt$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{kp}\cdot(1+(r-s)^{-2})^{(k+1)p}\int_{Q_{r}(r)}|Mu(t,x)|^{p}(x^{1})^{\theta-d}dxdt$
Let $|\beta|=k+1$ and $D^{\beta}=D_{i}D^{\beta^{\prime}}$ for some $i$ and
$\beta^{\prime}$ with $|\beta^{\prime}|=k$. Fix a smooth function $\eta$ so
that $\eta=1$ on $U_{s}$, $\eta=0$ on $\Omega\setminus U_{(r+s)/2}$,
$|\eta_{x}|\leq N(r-s)^{-1},|\eta_{xx}|\leq N(r-s)^{-2}$ and $|\eta_{t}|\leq
N(r-s)^{-2}$. Note that $v:=\eta D^{\beta}u$ satisfies $v(r^{2},\cdot)=0$ and
$v_{t}+A^{ij}v_{x^{i}x^{j}}=f:=\eta_{t}D^{\beta}u+2A^{ij}\eta_{x^{i}}D^{\beta}u_{x^{j}}+A^{ij}\eta_{x^{i}x^{j}}D^{\beta}u,\quad\quad(t,x)\in(0,r^{2})\times\mathbb{R}^{d}_{+}.$
By Lemma 4.10 for $\gamma=0$ (also note that $x^{1}\leq r$ on the support of
$\eta$ and $(r-s)^{-1}\leq 1+(r-s)^{-2}$),
$\displaystyle\|M^{-1}v\|^{p}_{\mathbb{H}^{2}_{p,d}(r^{2})}$
$\displaystyle\leq$ $\displaystyle
N\|M\eta_{t}D^{\beta}u+2A\eta_{x}MD^{\beta}u_{x}+MA\eta_{xx}D^{\beta}u\|^{p}_{\mathbb{L}_{p,d}(r^{2})}$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{p}(1+(r-s)^{-2})^{p}\int_{Q_{(s+r)/2}((s+r)/2)}\left(|D^{\beta}u|^{p}+|MD^{\beta}u_{x}|^{p}\right)dxdt$
$\displaystyle\leq$ $\displaystyle
N(1+r)^{p}(1+(r-s)^{-2})^{p}\int_{Q_{(s+r)/2}((s+r)/2)}\left(|D^{\beta^{\prime}}u_{x}|^{p}+|MD^{\beta^{\prime}}u_{xx}|^{p}\right)dxdt.$
This and (4.15) show that the induction goes through, and hence the lemma is
proved. ∎
###### Remark 4.12.
The proof of Lemma 4.11 mainly depends on Lemma 4.10 and it can be easily
checked that the assertion of Lemma 4.11 holds for $\theta=\theta_{0}$
whenever Lemma 4.10 is true for $\theta=\theta_{0}$. Thus due to Theorem 3.10
(which will be proved in section 6), Lemma 4.11 holds for
$\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$ and $\theta\in(d-1,d+1)$ if
$p\in(2,\infty)$.
###### Lemma 4.13.
Let $u(t,x)\in
C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Then
for any $T>0$, $p>1$ and $n=0,1,2,\cdots$,
$\sup_{t\in[0,T]}\|u(t,\cdot)\|_{H^{n}_{p,\theta}}\leq
N(\|u\|_{\mathbb{H}^{n}_{p,\theta}(T)}+\|u_{t}\|_{\mathbb{H}^{n}_{p,\theta}(T)}).$
###### Proof.
See p. $66$ of [13]; actually in this book, weights are not used and hence we
give an outline of the proof. First of all, it is easy to check that for any
$\phi=\phi(t)\in W^{1}_{p}((0,T))$ (cf. p.32 of [13])
$\sup_{t\leq T}|\phi(t)|^{p}\leq
N\int^{T}_{0}(|\phi|^{p}+|\phi^{\prime}(t)|^{p})dt.$
Thus it suffices to prove
$\phi(t):=\|u(t,\cdot)\|_{H^{n}_{p,\theta}}\in
W^{1}_{p}((0,T)),\quad|\phi^{\prime}(t)|\leq\|u_{t}(t,\cdot)\|_{H^{n}_{p,\theta}}.$
(4.16)
One can prove (4.16) by repeating the proof of Exercise 2.4.8 on p.71 of [13].
It is enough to replace $H^{n}_{p}$ there by $H^{n}_{p,\theta}$. ∎
###### Lemma 4.14.
Let $\theta\leq d$, $p>1$, $s\in(0,r)$ and $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0$ for $(t,x)\in Q_{r}(r)$. Then for any multi-
index $\beta=(\beta^{1},\beta^{2},\cdots,\beta^{d})$,
$\displaystyle\max_{(t,x)\in
Q_{s}(s)}(|D^{\beta}u_{xx}|^{p}+|D^{\beta}u_{t}|^{p})\leq
N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{\theta-d+p}dxdt,$
where $N=N(s,r,\beta,p,\delta,K)$.
###### Proof.
Choose the smallest integer $n$ so that $np>d$. Let $v\in
C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ satisfy $v(x)=0$ for $x^{1}\geq 2r$. The
by Lemma 3.1 $(ii)$ with $\gamma=n$, $i=0$, $\theta=d$ and $u=M^{-n}v$,
$\sup_{x}|v(x)|\leq N(r)\sup_{x}|M^{d/p}M^{-n}v(x)|\leq
N\|M^{-n}v\|_{H^{n}_{p,d}}\leq N(r,p,n)\|D^{n}v\|_{L_{p,d}},$ (4.17)
where for the last inequality we use Remark 3.2.
Fix $\kappa\in(s,r)$. Let $\psi$ be a smooth function so that $\psi(x)=1$ for
$(t,x)\in Q_{s}(s)$ and $\psi=0$ for $(t,x)\not\in U_{\kappa}$. It follows
from (4.17) and Lemma 4.13 that
$\displaystyle\max_{Q_{s}(s)}\left(|D^{\beta}u_{xx}|+|D^{\beta}u_{t}|\right)$
$\displaystyle\leq$ $\displaystyle N\max_{(t,x)\in Q_{s}(s)}|(D^{\beta}\psi
u)_{xx}|$ $\displaystyle\leq$ $\displaystyle
N\max_{t\in[0,s^{2}]}\|D^{n}(D^{\beta}\psi u)_{xx}\|_{L_{p,d}}$
$\displaystyle\leq$ $\displaystyle N\left(\|D^{n}(D^{\beta}\psi
u)_{xx}\|_{\mathbb{L}_{p,d}(s^{2})}+\|D^{n}(D^{\beta}\psi
u_{t})_{xx}\|_{\mathbb{L}_{p,d}(s^{2})}\right)$ $\displaystyle\leq$
$\displaystyle N\sum_{|\alpha|\leq
n+|\beta|+4}\int_{Q_{\kappa}(\kappa)}|D^{\alpha}u|^{p}\,dxdt$
$\displaystyle\leq$ $\displaystyle N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{p}dxdt\leq
N\int_{Q_{r}(r)}|u|^{p}(x^{1})^{\theta-d+p}dxdt,$
where the last inequality is due to the fact that $1\leq
N(r)(x^{1})^{\theta-d}$ for $x^{1}\leq 2r$. The lemma is proved. ∎
###### Remark 4.15.
Actually by inspecting the proof of Lemma 4.14 it can be easily shown that if
Lemma 4.11 holds for some $\theta_{0}\in(d-1,d-1+p)$ then Lemma 4.14 holds for
any $\theta\in(d-1,\theta_{0}]$.
## 5 Main estimates : Sharp function estimations
Remember that we denote
$\nu_{\alpha}(dx)=\nu^{1}_{\alpha}(dx^{1})dx^{\prime}:=(x^{1})^{\alpha}dx^{1}dx^{\prime}.$
The following is a weighted version of Poincaré’s inequality.
###### Lemma 5.1.
Let $\alpha\geq 0$, $p\in[1,\infty)$, $D_{r}(a):=(a-r,a+r)\times
B^{\prime}_{r}(0)\subset\mathbb{R}^{d}_{+}$ , and $u\in
C^{\infty}_{loc}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$. Then
$\displaystyle\int_{D_{r}(a)}\int_{D_{r}(a)}|u(x)-u(y)|^{p}\nu_{\alpha}(dx)\;\nu_{\alpha}(dy)\leq
2^{\alpha+2}(2r)^{p}|D_{r}(a)|\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx),$
(5.1)
where $|D_{r}(a)|:=\nu_{\alpha}(D_{r}(a))$ and we define
$\displaystyle\int_{A}|f(x)|^{p}\nu_{\alpha}(dx)=\sum_{k=1}^{d_{1}}\int_{A}|f^{k}(x)|^{p}\nu_{\alpha}(dx)$
for $\mathbb{R}^{d_{1}}$-valued function $f$ and $A\subset\Omega$.
###### Proof.
We use the outline of the proof of Theorem 10.2.5 of [13]. Without loss of
generality we may assume $d_{1}=1$. For $x,y\in D_{r}(a)$ we have
$\displaystyle|u(x)-u(y)|^{p}\leq(2r)^{p}\int^{1}_{0}|u_{x}(tx+(1-t)y)|^{p}dt$
and the left-hand side of $(\ref{2010.03.17.1})$ is less than
$\displaystyle(2r)^{p}\int^{1}_{0}I(t)dt=2(2r)^{p}\int^{1}_{1/2}I(t)dt,$
where
$\displaystyle
I(t):=\int_{D_{r}(a)}\int_{D_{r}(a)}|u_{x}(tx+(1-t)y)|^{p}\nu_{\alpha}(dx)\;\nu_{\alpha}(dy)$
and $I$ satisfies $I(t)=I(1-t)$. For each $t\in[1/2,1]$ and $y$, substituting
$w=tx+(1-t)y$ and noticing $x^{1}=(w^{1}-(1-t)y^{1})/t\leq w^{1}/t$ since
$y^{1}\geq 0$, we get
$\displaystyle I(t)$ $\displaystyle\leq$ $\displaystyle
t^{-\alpha-1}\int_{D_{r}(a)}\left(\int_{tD_{r}(a)+(1-t)y}|u_{x}(w)|^{p}\nu_{\alpha}(dw)\right)\nu_{\alpha}(dy)$
$\displaystyle\leq$ $\displaystyle
2^{\alpha+1}\int_{D_{r}(a)}\left(\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx)\right)\nu_{\alpha}(dy)$
$\displaystyle=$ $\displaystyle
2^{\alpha+1}|D_{r}(a)|\int_{D_{r}(a)}|u_{x}(x)|^{p}\nu_{\alpha}(dx)$
with the observation $tD_{r}(a)+(1-t)y:=\\{tz+(1-t)y:z\in D_{r}(a)\\}\subset
D_{r}(a)$. Now, (5.1) follows. ∎
###### Lemma 5.2.
Let $\alpha>-1$. Recall $\nu_{\alpha}^{1}(dx^{1})=(x^{1})^{\alpha}dx^{1}$. For
any $B^{1}_{r}(a)\subset\mathbb{R}_{+}$ we have a non-negative function
$\zeta\in C^{\infty}_{0}(\mathbb{R}_{+};\mathbb{R})$ such that
$\displaystyle supp(\zeta)\in
B^{1}_{r/2}(a),\quad\int_{B^{1}_{r}(a)}\zeta(x^{1})\nu^{1}_{\alpha}(dx^{1})=1,\quad\sup\zeta\cdot|B^{1}_{r}(a)|\leq
N,\quad\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|\leq\frac{N}{r},$ (5.2)
where $N=N(\alpha)$ and $|B^{1}_{r}(a)|=\nu_{\alpha}^{1}(B^{1}_{r}(a))$.
###### Proof.
Choose a nonnegative smooth function $\psi=\psi(x^{1})\in
C^{\infty}_{0}(B^{1}_{1/2}(0))$ so that
$\int_{\mathbb{R}}\psi(x^{1})dx^{1}=1$. Define
$\zeta(x^{1})=\frac{(x^{1})^{-\alpha}}{r}\psi(\frac{x^{1}-a}{r}).$
Then the first and the second of (5.2) are obvious.
Case 1: Let $\alpha\geq 0$. Since $r\leq a$ and
$(a+r)^{\alpha+1}-(a-r)^{\alpha+1}\leq 2r(\alpha+1)(2a)^{\alpha}$, the third
follows:
$\displaystyle\sup|\zeta|\cdot|B^{1}_{r}(a)|\leq$ $\displaystyle N$
$\displaystyle\sup_{|x^{1}-a|\leq
r/2}\frac{(x^{1})^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})$
$\displaystyle\leq$ $\displaystyle
N\frac{(a/2)^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})\leq N.$
Similarly, the last inequality holds by
$\displaystyle\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|$ $\displaystyle\leq$
$\displaystyle N\sup_{|x^{1}-a|\leq
r/2}\left(\frac{(x^{1})^{-\alpha}}{r^{2}}+\frac{(x^{1})^{-\alpha-1}}{r}\right)\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})$
$\displaystyle\leq$
$\displaystyle\frac{N}{r}(1+\frac{(2a)^{\alpha+1}}{(a/2)^{\alpha+1}})\leq\frac{N}{r}.$
Case 2: Let $\alpha\in(-1,0)$. First assume $r\leq a/2$. Then by mean value
theorem $(a+r)^{\alpha+1}-(a-r)^{\alpha+1}\leq 2r(\alpha+1)(a/2)^{\alpha}$ and
thus the right term of (5) is bounded by a constant $N$. If $r\in[a/2,a]$,
then
$\sup_{|x^{1}-a|\leq
r/2}\frac{(x^{1})^{-\alpha}}{r}\cdot((a+r)^{\alpha+1}-(a-r)^{\alpha+1})\leq\frac{(2a)^{-\alpha}}{a/2}(2a)^{\alpha+1}\leq
N.$
One can handle $\sup|\zeta_{x^{1}}|\cdot|B^{1}_{r}(a)|$ similarly. The lemma
is proved. ∎
Now we consider the system
$\displaystyle
u_{t}+A^{ij}u_{x^{i}x^{j}}=f^{i}_{x^{i}}+g,\quad(t,x)\in\Omega=\mathbb{R}\times\mathbb{R}^{d}_{+};\quad
f^{i}=(f^{1i},\ldots,f^{d_{1}i}),$ (5.4)
i.e.,
$\displaystyle
u^{k}_{t}+a^{ij}_{kr}u^{r}_{x^{i}x^{j}}=f^{ki}_{x^{i}}+g^{k},\quad
k=1,2,\ldots,d_{1}.$
Recall that for $t\in\mathbb{R}$, $a\in\mathbb{R}_{+}$ and
$x^{\prime}\in\mathbb{R}^{d-1}$
$Q_{r}(t,a,x^{\prime}):=(t,t+r^{2})\times(a-r,a+r)\times
B^{\prime}_{r}(x^{\prime}),\quad Q_{r}(a):=Q_{r}(0,a,0).$
By $C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ we denote the set of
$\mathbb{R}^{d_{1}}$-valued functions $u$ defined on $\Omega$ and such that
$\zeta u\in C^{\infty}_{0}(\Omega;\mathbb{R}^{d_{1}})$ for any $\zeta\in
C^{\infty}_{0}(\Omega;\mathbb{R})$.
###### Lemma 5.3.
Let $\alpha\geq 0$, $p\in[1,\infty)$, $f^{i},g\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$. Assume that $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies (5.4) on
$Q_{r}(a)\subset\Omega$. Then
$\displaystyle\int_{Q_{r}(a)}\left|u(t,x)-u_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)\leq
Nr^{p}\int_{Q_{r}(a)}(|u_{x}(t,x)|^{p}+|f(t,x)|^{p}+r^{p}|g(t,x)|^{p})\mu_{\alpha}(dtdx),$
(5.5)
where $N=N(\theta,\alpha,p,d,d_{1},K)$.
###### Proof.
We follow the outline of the proof of Theorem 4.2.1 in [13]. We take the
scalar function $\zeta$ corresponding to $B^{1}_{r}(a)$ and $\alpha$ from
Lemma 5.2 and take a nonnegative function $\phi=\phi(x^{\prime})\in
C^{\infty}_{0}(B^{\prime}_{1}(0))$ with unit integral. Denote
$\eta(x^{\prime})=r^{-d+1}\phi(\frac{x^{\prime}}{r})$,
$D_{r}(a):=(a-r,a+r)\times B^{\prime}_{r}(0)$ as before, and for
$t\in(0,r^{2})$ set
$\displaystyle\bar{u}(t):=\int_{D_{r}(a)}\zeta(y^{1})\eta(y^{\prime})u(t,y)\nu_{\alpha}(dy).$
Then by Jensen’s inequality and the weighted version of Poincaré’s inequality
(Lemma 5.1),
$\displaystyle\int_{D_{r}(a)}|u(t,x)-\bar{u}(t)|^{p}\nu_{\alpha}(dx)$ (5.6)
$\displaystyle=$
$\displaystyle\int_{D_{r}(a)}\Big{|}\int_{D_{r}(a)}(u(t,x)-u(t,y))\zeta(y^{1})\eta(y^{\prime})\nu_{\alpha}(dy)\Big{|}^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$
$\displaystyle\int_{D_{r}(a)}\left(\int_{D_{r}(a)}|u(t,x)-u(t,y)|^{p}\zeta(y^{1})\eta(y^{\prime})\nu_{\alpha}(dy)\right)\nu_{\alpha}(dx)$
$\displaystyle\leq$
$\displaystyle|\sup\;\zeta|\cdot|\sup\,\eta|\,\int_{D_{r}(a)}\int_{D_{r}(a)}|u(t,x)-u(t,y)|^{p}\nu_{\alpha}(dx)\nu_{\alpha}(dy)$
$\displaystyle\leq$ $\displaystyle
Nr^{-d+1}|\sup\;\zeta|\cdot\nu_{\alpha}(D_{r}(a))\;r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$ $\displaystyle
Nr^{-d+1}|\sup\;\zeta|\cdot\nu_{\alpha}^{1}(B^{1}_{r}(a))\;r^{d-1}r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx)$
$\displaystyle\leq$ $\displaystyle
N\;r^{p}\int_{D_{r}(a)}|u_{x}(t,x)|^{p}\nu_{\alpha}(dx).$
We observe that for any constant vector $c\in\mathbb{R}^{d}$ the left-hand
side of (5.5) is less than $2\cdot 2^{p}$ times
$\displaystyle\int_{Q_{r}(a)}|u(t,x)-c|^{p}\mu_{\alpha}(dtdx)\leq
2^{p}\int_{Q_{r}(a)}|u(t,x)-\bar{u}(t)|^{p}\mu_{\alpha}(dtdx)+2^{p}\;\nu_{\alpha}(D_{r}(a))\int^{r^{2}}_{0}|\bar{u}(t)-c|^{p}dt.$
By (5.6) the first term is less than (5.5). To estimate the second term, we
take
$\displaystyle c=\frac{1}{r^{2}}\int^{r^{2}}_{0}\bar{u}(t)dt.$
Then by Poincaré’s inequality without a weight in variable $t$ we have
$\displaystyle\nu_{\alpha}(D_{r}(a))\int^{r^{2}}_{0}|\bar{u}(t)-c|^{p}dt$
(5.7) $\displaystyle\leq$ $\displaystyle
N\;\nu_{\alpha}(D_{r}(a))\;(r^{2})^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})u_{t}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt.$
Remember $u_{t}=-A^{ij}(t)u_{x^{i}x^{j}}+f^{i}_{x^{i}}+g$. We show that (5.7)
is less than (5.5). In fact, for handling the integral with $g$, using
Jensen’s inequality and taking the supremum out of the integral, we have
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})g(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\;|\sup\;\zeta|\,|\sup\eta|\,\int^{r^{2}}_{0}\int_{D_{r}(a)}|g(t,x)|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
N\nu_{\alpha}^{1}(B^{1}_{r}(a))r^{d-1}\;r^{2p}\;|\sup\;\zeta|\,r^{-d+1}\int^{r^{2}}_{0}\int_{D_{r}(a)}|g(t,x)|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
N(\theta,p,d)\;r^{2p}\int_{Q_{r}(a)}|g(t,x)|^{p}\mu_{\alpha}(dtdx),$
where we used $|\sup\zeta|\;\nu_{\alpha}^{1}(B^{1}_{r}(a))\leq N$ (Lemma 5.2).
Next, we handle the integral with $-A^{ij}u_{x^{i}x^{j}}$. Fix $i,j$. Firstly,
assume either $i$ or $j$ is $1$; say $j=1$. We use integration by parts and
observe
$\displaystyle\nu_{\alpha}(D_{r}(a))\;(r^{2})^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})A^{ij}(t)u_{x^{i}x^{j}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta_{x^{1}}(x^{1})\eta(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\quad+\nu_{\alpha}(D_{r}(a))\;r^{2p}|\alpha|^{p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\frac{1}{x}\zeta(x^{1})\eta(x^{\prime})A(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle=:$ $\displaystyle\quad I_{1}+I_{2}.$
For $I_{2}$ we use the fact $|A^{ij}u_{x^{i}}|\leq|A^{ij}||u_{x^{i}}|\leq
K|u_{x}|$ and $1/x\leq 2/r$ on the support of $\zeta$. The argument handling
the case of $g$ easily shows
$\displaystyle I_{2}\leq
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|u_{x}(t,x)|^{p}\mu_{\alpha}(dtdx).$
For $I_{1}$ we use Hölder’s inequality and get
$\displaystyle\nu_{\alpha}(D_{r}(a))\cdot|\int_{D_{r}(a)}\zeta_{x^{1}}\eta
A^{ij}u_{x^{i}}\;\nu_{\alpha}(dx)|^{p}$ $\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))^{p}\int_{D_{r}(a)}|\zeta_{x^{1}}\eta
A^{ij}u_{x^{i}}|^{p}\;\nu_{\alpha}(dx)$ $\displaystyle\leq$ $\displaystyle
N(\nu_{\alpha}^{1}(B^{1}_{r}(a))^{p}r^{(d-1)p}\cdot|\sup\zeta_{x^{1}}|^{p}r^{(-d+1)p}\int_{D_{r}(a)}|u_{x}|^{p}\nu_{\alpha}(dx).$
Since $\nu_{\alpha}^{1}(B^{1}_{r}(a))\cdot|\sup\zeta_{x}|\leq N/r$, it easily
follows that
$\displaystyle I_{1}\leq
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|u_{x}(t,x)|^{p}\mu_{\alpha}(dtdx).$
Secondly, if $i,j\neq 1$, by integration by parts, Hölder’s inequality and the
inequality $\sup|\eta_{x^{\prime}}|\leq Nr^{-d}$,
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})\left[-A^{ij}(t)u_{x^{i}x^{j}}(t,x)\right]\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle=$
$\displaystyle\nu_{\alpha}(D_{r}(a))\;r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta_{x^{j}}(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$
$\displaystyle\nu_{\alpha}(D_{r}(a))^{p}\;r^{2p}\int^{r^{2}}_{0}\int_{D_{r}(a)}\Big{|}\zeta(x^{1})\eta_{x^{j}}(x^{\prime})A^{ij}(t)u_{x^{i}}(t,x)\Big{|}^{p}\nu_{\alpha}(dx)\;dt$
$\displaystyle\leq$ $\displaystyle
N\nu_{\alpha}(D_{r}(a))^{p}\;r^{2p}\cdot\sup|\zeta|^{p}\cdot
r^{-dp}\int^{r^{2}}_{0}\int_{D_{r}(a)}|u_{x}|^{p}\nu_{\alpha}(dx)dt$
$\displaystyle\leq$ $\displaystyle
Nr^{p}\int_{Q_{r}(a)}|u_{x}|^{p}\mu_{\alpha}(dxdt).$
For the integral with $f^{i}_{x^{i}}$ we use a similar calculation to the one
of $-A^{ij}u_{x^{i}x^{j}}$ and get for each $i$
$\displaystyle\nu_{\alpha}(D_{r}(a))\,r^{2p}\int^{r^{2}}_{0}\Big{|}\int_{D_{r}(a)}\zeta(x^{1})\eta(x^{\prime})f_{x^{i}}(t,x)\nu_{\alpha}(dx)\Big{|}^{p}dt$
$\displaystyle\leq$ $\displaystyle
N(K,\theta,p,d)\;r^{p}\int_{Q_{r}(a)}|f(t,x)|^{p}\mu_{\alpha}(dtdx).$
The lemma is proved. ∎
###### Lemma 5.4.
Let $\alpha\geq 0,p\in[1,\infty)$, $0<r\leq a$ and $u\in
C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$.
(i) There is a constant $N=N(K,\theta,\alpha,p,d,d_{1})$ such that for any
$i=1,\cdots,d$ we have
$\displaystyle\int_{Q_{r}(a)}\left|u_{x^{i}}(t,x)-(u_{x^{i}})_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)\leq
Nr^{p}\int_{Q_{r}(a)}(|u_{xx}(t,x)|^{p}+|u_{t}(t,x)|^{p})\mu_{\alpha}(dtdx)$
(5.8)
(ii) Denote
$\kappa_{0}=\kappa_{0}(r,a):=(\nu^{1}_{\alpha}(B^{1}_{r}(a))^{-1}\cdot\int^{a+r}_{a-r}x^{1}\nu^{1}_{\alpha}(dx^{1})$.
Then
$\displaystyle\int_{Q_{r}(a)}\left|u(t,x)-u_{Q_{r}(a)}+\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i=1}^{d}x^{i}(u_{x^{i}})_{Q_{r}(a)}\right|^{p}\mu_{\alpha}(dtdx)$
(5.9) $\displaystyle\leq$ $\displaystyle
Nr^{p}\int_{Q_{r}(a)}(|u_{x}(t,x)-(u_{x})_{Q_{r}(a)}|^{p}+r^{p}|u_{t}(t,x)|^{p}+r^{p}|u_{xx}(t,x)|^{p})\mu_{\alpha}(dtdx)$
$\displaystyle\leq$ $\displaystyle
Nr^{2p}\int_{Q_{r}(a)}(|u_{xx}(t,x)|^{p}+|u_{t}(t,x)|^{p})\mu_{\alpha}(dtdx)$
###### Proof.
(i) For (5.8) we use the fact that for $v=u_{x^{i}}$,
$v_{t}-A^{jm}v_{x^{j}x^{m}}=(u_{t}-A^{jm}u_{x^{j}x^{m}})_{x^{i}}$ and apply
Lemma 5.3 with $f^{i}=u_{t}-A^{jm}u_{x^{j}x^{m}}$ for all $i$.
(ii) To prove (5.9), denote
$v(t,x):=u(t,x)-(u)_{Q_{r}(a)}+\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i}x^{i}(u_{x^{i}})_{Q_{r}(a)}$.
Then
$v_{Q_{r}(a)}=\kappa_{0}(u_{x^{1}})_{Q_{r}(a)}-\sum_{i}\frac{(u_{x^{i}})_{Q_{r}(a)}}{|Q_{r}(a)|}\int_{Q_{r}(a)}x^{i}\nu_{\alpha}(dx)dt=0,$
$v-v_{Q_{r}(a)}=v,\quad v_{x^{i}}=u_{x^{i}}-(u_{x^{i}})_{Q_{r}(a)},\quad
v_{t}-A^{ij}v_{x^{i}x^{j}}=g:=u_{t}-A^{ij}u_{x^{i}x^{j}}.$
Now it is enough to use Lemma 5.3 and (5.8). The lemma is proved. ∎
From this point on we fix $\alpha:=\theta-d+p$ (note $\alpha>0$) and denote
$\nu:=\nu_{\alpha},\quad\nu^{1}:=\nu_{\alpha}^{1},\quad\quad\mu(dxdt)=\nu(dx)dt=(x^{1})^{\theta-d+p}dxdt.$
###### Theorem 5.5.
Let $\theta\in(d-1,d]$, $0<r\leq a$ and $\lambda r/a\geq 2$.
(i) Assume that $u\in C^{\infty}_{loc}(\Omega;\mathbb{R}^{d_{1}})$ satisfies
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=0$ in $Q_{\lambda
r}(t_{0},a,x^{\prime}_{0})\cap\Omega$. Then there is a constant
$N=N(K,\delta,\theta,p,d,d_{1})$ so that
$\displaystyle-\int_{Q_{r}(t_{0},a,x^{\prime}_{0})}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(t_{0},a,x^{\prime}_{0})}|^{p}\mu(dtdx)$
(5.10) $\displaystyle\leq$ $\displaystyle\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{Q_{\lambda
r}(t_{0},a,x^{\prime}_{0})\cap\Omega}|u_{xx}(t,x)|^{p}\mu(dtdx).$
(ii) If $u\in C^{\infty}_{loc}(\mathbb{R}^{d}_{+};\mathbb{R}^{d_{1}})$,
$A^{ij}$ is independent of $t$ and $A^{ij}u_{x^{i}x^{j}}=0$ in $B_{\lambda
r}(a,x^{\prime}_{0})\cap\mathbb{R}^{d}_{+}$, then
$\displaystyle-\int_{B_{r}(a,x^{\prime}_{0})}|u_{xx}(x)-(u_{xx})_{B_{r}(a,x^{\prime}_{0})}|^{p}\nu(dx)$
(5.11) $\displaystyle\leq$ $\displaystyle\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{B_{\lambda
r}(a,x^{\prime}_{0})\cap\mathbb{R}^{d}_{+}}|u_{xx}(x)|^{p}\nu(dx).$
###### Proof.
(ii) is a consequence of (i). To prove (i), without loss of generality we may
assume $t_{0}=0$, $x^{\prime}_{0}=0$ and thus
$Q_{r}(t_{0},a,x^{\prime}_{0})=Q_{r}(a)$.
Step 1. First, we consider the case $a=1$. Note that
$r\leq 1,\quad 2\leq\lambda r,\quad\beta:=\frac{1+\lambda r}{2}\leq\lambda
r,\quad\frac{r}{\beta}\leq\frac{1}{\beta}\leq\frac{2}{3},\quad\quad
2\beta=1+\lambda r.$
Thus,
$Q_{\beta}(\beta)\subset Q_{\lambda r}(1)\cap\Omega,\quad
Q_{r/\beta}(\beta^{-1})\subset Q_{2/3}(2/3).$
Denote $w(t,x)=u(\beta^{2}t,\beta x)$, then obviously
$w_{t}+A^{ij}(\beta^{2}t)w_{x^{i}x^{j}}=0,\quad\quad\text{for}\quad(t,x)\in
Q_{1}(1)$
and
$\displaystyle-\int_{Q_{r}(1)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(1)}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle
N(d)\sup_{Q_{r}(1)}(|u_{xxx}|^{p}+|u_{xxt}|^{p})$ $\displaystyle\leq$
$\displaystyle
N(d)\beta^{-3p}\,\,\sup_{Q_{r/\beta}(\beta^{-1})}(|w_{xxx}|^{p}+|w_{xxt}|^{p})$
$\displaystyle\leq$ $\displaystyle
N(d)\beta^{-3p}\,\,\sup_{Q_{2/3}(2/3)}(|w_{xxx}|^{p}+|w_{xxt}|^{p}).$
Applying Lemma 4.14 to
$v(t,x)=w(t,x)-w_{Q_{1}(1)}+\kappa_{0}(w_{x^{1}})_{Q_{1}(1)}-\sum_{i=1}^{d}x^{i}(w_{x^{i}})_{Q_{1}(1)}$,
and then using Lemma 5.4
$\displaystyle\beta^{-3p}\,\,\sup_{Q_{2/3}(2/3)}(|w_{xxx}|^{p}+|w_{xxt}|^{p})$
$\displaystyle\leq$ $\displaystyle
N\beta^{-3p}\int_{Q_{1}(1)}|v|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle
N\beta^{-3p}\int_{Q_{1}(1)}|w_{xx}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle=$ $\displaystyle
N\beta^{-2p-2-\theta}\int_{Q_{\beta}(\beta)}|u_{xx}|^{p}(x^{1})^{\theta-d+p}dxdt.$
This leads to (5.10) since $|Q_{\lambda
r}(1)\cap\Omega|\sim\beta^{p+\theta+2}$.
Step 2. Let $a\neq 1$. Define $v(t,x):=u(a^{2}t,ax)$. Then
$v_{t}+A^{ij}(a^{2}t)v_{x^{i}x^{j}}=0$ in $Q_{\lambda r/a}(1)\cap\Omega$. As
easy to check,
$|Q_{r/a}(1)|=a^{-\theta-p-2}|Q_{r}(a)|,\quad(v_{xx})_{Q_{r/a}(1)}=a^{2}(u_{xx})_{Q_{r}(a)},\quad|Q_{\lambda
r/a}(1)\cap\Omega|=a^{-\theta-p-2}|Q_{\lambda r}(a)\cap\Omega|,$
and consequently
$-\int_{Q_{r/a}(1)}|v_{xx}(t,x)-(v_{xx})_{Q_{r/a}(1)}|^{p}(x^{1})^{\theta-d+p}dxdt=a^{2p}-\int_{Q_{r}(a)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(a)}|^{p}(x^{1})^{\theta-d+p}dxdt,$
$-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt=a^{2p}-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt.$
It follows
$\displaystyle-\int_{Q_{r}(a)}|u_{xx}(t,x)-(u_{xx})_{Q_{r}(a)}|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle=$ $\displaystyle a^{-2p}-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt$
$\displaystyle\leq$ $\displaystyle a^{-2p}\cdot\frac{N}{(1+\lambda
r/a)^{p}}\;-\int_{Q_{\lambda
r/a}(1)\cap\Omega}|v_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt$ $\displaystyle=$
$\displaystyle\frac{N}{(1+\lambda r/a)^{p}}\;-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{xx}(t,x)|^{p}(x^{1})^{\theta-d+p}dxdt.$
The theorem is proved. ∎
###### Remark 5.6.
Note that Theorem 5.5 is based on Lemma 4.14. It follows from Remark 4.12 and
Remark 4.15 that if $p\geq 2$ then Theorem 5.5 holds for any
$\theta\in(d-1,d+1)$ (not only for $\theta\in(d-1,d]$). Obviously we cannot
use this result yet since Remark 4.12 is valid only after we prove Theorem
3.10.
###### Lemma 5.7.
Assume $\theta\in(d-1,d]$ if $p\in(2,\infty)$ and $\theta\in(d-p+1,d]$ if
$p\in(1,2]$. Denote $q:=\theta-d+p$ which is in $(1,p]$.
(i) Let $u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$ and
$f:=u_{t}+A^{ij}(t)u_{x^{i}x^{j}}$. Suppose that $A^{ij}(t)$ is infinitely
differentiable and has bounded derivatives. Then for any $\varepsilon>0$,
$Q_{r}(t_{0},a,x^{\prime}_{0})\subset\Omega$ and $(t,x)\in
Q_{r}(t_{0},a,x^{\prime}_{0})$
$-\int_{Q_{r}(t_{0},a,x^{\prime}_{0})}|u_{xx}-(u_{xx})_{Q_{r}(t_{0},a,x^{\prime}_{0})}|^{q}\mu(dyds)\leq\varepsilon\mathbb{M}(|u_{xx}|^{q})(t,x)+N\mathbb{M}(|f|^{q})(t,x),$
(5.12)
where $N=N(\varepsilon,\theta,q,d,d_{1},\delta,K)$.
(ii) Furthermore, if $u\in C^{\infty}_{0}(\mathbb{R}^{d}_{+})$ and $A^{ij}$ is
independent of $t$, then for any $\varepsilon>0$,
$B_{r}(a,x^{\prime}_{0})\subset\mathbb{R}^{d}_{+}$ and $x\in
B_{r}(a,x^{\prime}_{0})$
$-\int_{B_{r}(a,x^{\prime}_{0})}|u_{xx}-(u_{xx})_{B_{r}(a,x^{\prime}_{0})}|^{q}\nu(dy)\leq\varepsilon\mathbb{M}(|u_{xx}|^{q})(x)+N\mathbb{M}(|A^{ij}u_{x^{i}x^{j}}|^{q})(x),$
(5.13)
where $N=N(\varepsilon,\theta,q,d,d_{1},\delta,K)$.
###### Proof.
(i) Without loss of generality we may take $t_{0}=0$ and $x^{\prime}_{0}=0$;
$Q_{r}(t_{0},a,x^{\prime}_{0})=Q_{r}(a)$. In fact, for other cases it is
enough to consider the function
$v(t,x):=u(t_{0}+t,x^{1},x^{\prime}_{0}+x^{\prime})$ in place of
$u(t,x^{1},x^{\prime})$.
Step 1. We prove that there exists $\kappa=\kappa(\varepsilon)\in(0,1)$ so
that (5.12) holds if $(r/a)\leq\kappa$.
Let $m$ denote the Lebesque measure on $\mathbb{R}^{d+1}$. Assume $\lambda\geq
4$ and $\lambda r\leq a/4$. Then $(3a/4)\leq x^{1}\leq(5a/4)$ if $x^{1}\in
B^{1}_{\lambda r}(a)$, and therefore
$(3/5)^{p+\theta-d}\frac{dtdx}{m(Q_{r}(a))}\leq\frac{\mu(dtdx)}{|Q_{r}(a)|}\leq(5/3)^{p+\theta-d}\frac{dtdx}{m(Q_{r}(a))}\quad\quad\text{on}\quad
Q_{r}(a),$ $(3/5)^{p+\theta-d}\frac{dtdx}{m(Q_{\lambda
r}(a))}\leq\frac{\mu(dtdx)}{|Q_{\lambda
r}(a)|}\leq(5/3)^{p+\theta-d}\frac{dtdx}{m(Q_{\lambda
r}(a))}\quad\quad\text{on}\quad Q_{\lambda r}(a).$
Denote $c_{0}:=(5/3)^{p+\theta-d}$. By Theorem 4.5,
$\displaystyle-\int_{Q_{r}(a)}|u_{xx}-(u_{xx})_{Q_{r}(a)}|^{q}\mu(dsdy)$
$\displaystyle\leq$
$\displaystyle\int_{Q_{r}(a)}\int_{Q_{r}(a)}|u_{xx}(s,y)-u_{xx}(\tau,\xi)|^{q}\frac{\mu(dsdy)}{|Q_{r}(a)|}\frac{\mu(d\tau
d\xi)}{|Q_{r}(a)|}$ $\displaystyle\leq$ $\displaystyle
c^{2}_{0}\int_{Q_{r}(a)}\int_{Q_{r}(a)}|u_{xx}(s,y)-u_{xx}(\tau,\xi)|^{q}\frac{dsdy}{m(Q_{r}(a))}\frac{d\tau
d\xi}{m(Q_{r}(a))}$ $\displaystyle\leq$ $\displaystyle
Nc^{2}_{0}\lambda^{d+2}\int_{Q_{\lambda r}(a)}|f|^{q}\frac{dyds}{m(Q_{\lambda
r}(a))}+Nc^{2}_{0}\lambda^{-q}\int_{Q_{\lambda
r}(a)}|u_{xx}|^{q}\frac{dyds}{m(Q_{\lambda r}(a))}$ $\displaystyle\leq$
$\displaystyle Nc^{3}_{0}\lambda^{d+2}-\int_{Q_{\lambda
r}(a)}|f|^{q}\mu(dyds)+Nc^{3}_{0}\lambda^{-q}-\int_{Q_{\lambda
r}(a)}|u_{xx}|^{q}\mu(dyds)$ $\displaystyle\leq$ $\displaystyle
N\lambda^{d+2}\mathbb{M}(|f|^{q})(t,x)+N\lambda^{-q}\mathbb{M}(|u_{xx}|^{q})(t,x),$
where $N$ depends only on $d,d_{1},p,\theta,\delta,K$. Note that the above
inequality holds as long as $r\lambda/a\leq 1/4$. Now we fix $\lambda$ so that
$N\lambda^{-q}=\varepsilon/2$, i.e. $\lambda=(2N/\varepsilon)^{1/q}$ and
define $\kappa=1/{(4\lambda)}=1/4\cdot(2N/\varepsilon)^{-1/q}$. Then whenever
$r/a\leq\kappa$ we have $(r/a)\lambda\leq 1/4$ and thus (5.12) follows.
Step 2. For given $\varepsilon$, take $\kappa=\kappa(\varepsilon)$ from Step
1. Assume $r/a\geq\kappa$. Choose $\lambda$, which will be specified later, so
that $r\lambda>4a$; this $\lambda$ is different from the one in step 1. Take a
$\zeta\in C^{\infty}_{0}(\mathbb{R}^{d+1})$ so that $\zeta(t,x)=1$ for
$(t,x)\in Q_{\lambda r/2}(a)\cap\Omega$ and $\zeta(t,x)=0$ if
$(t,x)\not\in(-\lambda^{2}r^{2},\lambda^{2}r^{2})\times(-a,a+\lambda r)\times
B^{\prime}_{\lambda r}$. Denote
$g=f\zeta,\quad h=f(1-\zeta).$
Take a large $T$ so that $u(t,x)=0$ if $t\geq T$. By Lemma 4.10 we can define
$v$ as the solution of
$v_{t}+A^{ij}v_{x^{i}x^{j}}=h,\quad t\in(S,T),\quad v(T,\cdot)=0$ (5.14)
so that $v\in\mathfrak{H}^{n}_{p,d}(S,T)$ for any $n$ and $S>-\infty$. Also
let $\bar{v}\in\mathfrak{H}^{n}_{p,d}(S,T+1)$ be the solution of
$\bar{v}_{t}+A^{ij}\bar{v}_{x^{i}x^{j}}=h,\quad
t\in(S,T+1),\quad\bar{v}(T+1,\cdot)=0.$
Then by considering the equation for $\bar{v}$ on $(T,T+1)$, since $h(t)=0$
for $t\geq T$, we conclude $\bar{v}(t)=0$ for $t\in[T,T+1]$. Thus $\bar{v}$
also satisfies (5.14) and $v=\bar{v}$. It follows from (3.6) that $v$ is
infinitely differentiable in $x$ (and hence in $t$) in $\Omega$. By applying
Theorem 5.5 with $\bar{p}=q,\bar{\theta}=d$ and $\lambda/2$ in places of
$p,\theta$ and $\lambda$ respectively,
$\displaystyle-\int_{Q_{r}(a)}|v_{xx}(t,x)-(v_{xx})_{Q_{r}(a)}|^{q}\bar{\mu}(dyds)$
$\displaystyle\leq$ $\displaystyle N\frac{1}{(1+\lambda
r/2a)^{q}}-\int_{Q_{\lambda
r/2}(a)\cap\Omega}|v_{xx}(t,x)|^{q}\bar{\mu}(dyds)$ (5.15) $\displaystyle\leq$
$\displaystyle N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|v_{xx}(t,x)|^{q}\bar{\mu}(dyds),$
where
$\bar{\mu}(dsdy):=(y^{1})^{\bar{\theta}-d+\bar{p}}dyds=(y^{1})^{q}dyds=\mu(dyds)$.
On the other hand, $w:=u-v$ satisfies $w(T,\cdot)=0$ and
$w_{t}+A^{ij}w_{x^{i}x^{j}}=g,\quad t\in(0,T).$
By Lemma 4.10,
$\int_{Q_{r}(a)}|w_{yy}|^{q}(y^{1})^{q}dyds\leq\int_{Q_{\lambda
r}(a)\cap\Omega}|w_{yy}|^{q}(y^{1})^{q}dyds\leq N\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}(y^{1})^{q}\;dyds,$
$\displaystyle-\int_{Q_{r}(a)}|w_{yy}|^{q}\mu(dyds)$ $\displaystyle\leq$
$\displaystyle N\frac{\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}}{(1+r/a)^{p+\theta-d+1}-(1-r/a)^{p+\theta-d+1}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}\mu(dyds)$ (5.16) $\displaystyle\leq$ $\displaystyle
N(\kappa)\lambda^{d+1}(1+\lambda r/a)^{p+\theta-d+1}-\int_{Q_{\lambda
r}(a)\cap\Omega}|f|^{q}\mu(dyds),$
where for the second inequality we use
$(1+r/a)^{p+\theta-d+1}-(1-r/a)^{p+\theta-d+1}\geq(1+\kappa)^{p+\theta-d+1}-1$.
Observing that $u=v+w$,
$\displaystyle I:$ $\displaystyle=$
$\displaystyle-\int_{Q_{r}(a)}|u_{yy}(t,x)-(u_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$
$\displaystyle\leq$ $\displaystyle
N(q)-\int_{Q_{r}(a)}|w_{yy}(t,x)-(w_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)+N(q)-\int_{Q_{r}(a)}|v_{yy}(t,x)-(v_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$
$\displaystyle\leq$ $\displaystyle
N(q)-\int_{Q_{r}(a)}|w_{yy}(t,x)|^{q}\mu(dyds)+N(q)-\int_{Q_{r}(a)}|v_{yy}(t,x)-(v_{yy})_{Q_{r}(a)}|^{q}\mu(dyds)$
and thus by (5.15) and (5.16),
$\displaystyle I$ $\displaystyle\leq$ $\displaystyle N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{Q_{\lambda r}(a)\cap\Omega}|f|^{q}\mu(dyds)$
$\displaystyle+N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|v_{yy}(t,x)|^{q}\mu(dyds)$ $\displaystyle\leq$ $\displaystyle
N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{(0,\lambda^{2}r^{2})\times(0,a+\lambda
r)}|f|^{q}\mu(dyds)$ $\displaystyle+N\frac{1}{(1+\lambda
r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}\left(|u_{yy}(t,x)|^{q}+|w_{yy}(t,x)|^{q}\right)\mu(dyds)$
$\displaystyle\leq$ $\displaystyle N\lambda^{d+1}(1+\lambda
r/a)^{p+\theta-d+1}-\int_{Q_{\lambda r}(a)\cap\Omega}|f|^{q}\mu(dyds)$
$\displaystyle+N\frac{1}{(1+\lambda r/a)^{q}}-\int_{Q_{\lambda
r}(a)\cap\Omega}|u_{yy}(t,x)|^{q}\mu(dyds).$
Now to prove the first assertion it is enough to choose $\lambda$ so large
that $N\frac{1}{(1+\lambda r/a)^{q}}\leq\varepsilon$. Also note that since
$r/a\geq\kappa$, we have
$N\lambda^{d+1}(1+\lambda r/a)^{p+\theta-d+1}\leq N(\lambda,\kappa).$
(ii) The second assertion is proved similarly based on Corollary 4.6 and
(5.11) in place Theorem 4.5 and (5.10). The lemma is proved.
∎
## 6 Proof of Theorem 3.10 and Theorem 3.13
Firstly, we give an $L_{p}$-theory for the following backward system defined
on $\mathbb{R}\times\mathbb{R}^{d}_{+}$.
###### Theorem 6.1.
Let $p\in(1,\infty)$. Assume $\theta\in(d-1,d+1)$ if $p\in(2,\infty)$, and
$\theta\in(d+1-p,d+p-1)$ if $p\in(1,2]$. Then for any
$f\in\mathbb{L}_{p,\theta}(-\infty,\infty)$ the system
$u_{t}+A^{ij}(t)u_{x^{i}x^{j}}=f$
has a unique solution $u$ in $M\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$ and
for this solution we have
$\|Mu_{t}\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}+\|M^{-1}u\|_{\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)}\leq
N\|Mf\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}.$ (6.1)
###### Proof.
If $A^{ij}u_{x^{i}x^{j}}=\Delta u=(\Delta u^{1},\ldots,\Delta u^{d_{1}})$,
then the theory of single equations is applied and the theorem is true for any
$\theta\in(d-1,d-1+p)$; see Theorem 5.6 in [10]. Actually the mentioned
theorem is proved for parabolic equations defined on
$(0,T)\times\mathbb{R}^{d}_{+}$, but one can easily check that the proofs in
[10] work for equations defined on $\mathbb{R}\times\mathbb{R}^{d}_{+}$.
For $\lambda\in[0,1]$ and $d_{1}\times d_{1}$ identity matrix $I$ we define
${A}^{ij}_{\lambda}=({a}^{ij}_{kr,\lambda}):=(1-\lambda)A^{ij}+\delta^{ij}\lambda\delta
I.$
Then for each $\lambda\in[0,1]$ the coefficient matrices
$\\{A^{ij}_{\lambda}:i,j=1,\ldots,d\\}$ satisfy Assumption 3.8 with the same
$\delta,K$. Thus due to the method of continuity, we only need to prove that a
priori estimate (6.1) holds given that a solution $u$ already exists.
Furthermore, since $C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$ is
dense in $M\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$, we may assume that
$u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$. By Remark 4.8, we
only need to prove the following:
$\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|u_{xx}(t,x)|^{p}\mu(dtdx)\leq
N\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|f(t,x)|^{p}\mu(dtdx).$ (6.2)
To prove this we certainly may assume that $A^{ij}$ are infinitely
differentiable and have bounded derivatives (remember that the constant $N$ in
(5.12) do not depend on the regularity of $A^{ij}$).
Case 1. Assume that either (i) $p\in(2,\infty)$ and $\theta\in(d-1,d]$ or (ii)
$p\in(1,2]$ and $\theta\in(d-p+1,d]$.
Define $q:=\theta-d+p$. Recall that the range of $q\in(1,p]$. By Lemma 5.7, if
$u\in C^{\infty}_{0}(\mathbb{R}\times\mathbb{R}^{d}_{+})$, then for any
$\varepsilon>0$
$(u_{xx})^{\sharp}(t,x)\leq\varepsilon\mathbb{M}^{1/q}(|u_{xx}|^{q})(t,x)+N(\varepsilon)\mathbb{M}^{1/q}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})(t,x).$
By Theorem 2.10 $($Fefferman-Stein$)$ and Theorem 2.12 $($Hardy-Littlewood$)$,
$\displaystyle\|Mu_{xx}\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}$
$\displaystyle=$ $\displaystyle\|u_{xx}\|_{L_{p}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle N\|(u_{xx})^{\sharp}\|_{L_{p}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle
N\varepsilon\|\mathbb{M}^{1/q}(|u_{xx}|^{q})\|_{L_{p}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|\mathbb{M}^{1/q}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})\|_{L_{p}(\Omega,\mu)}$
$\displaystyle=$ $\displaystyle
N\varepsilon\|\mathbb{M}(|u_{xx}|^{q})\|^{1/q}_{L_{p/q}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|\mathbb{M}(|u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q})\|^{1/q}_{L_{p/q}(\Omega,\mu)}$
$\displaystyle\leq$ $\displaystyle
N\varepsilon\||u_{xx}|^{q}\|^{1/q}_{L_{p/q}(\Omega,\mu)}+N\cdot
N(\varepsilon)\||u_{t}+A^{ij}u_{x^{i}x^{j}}|^{q}\|^{1/q}_{L_{p/q}(\Omega,\mu)}$
$\displaystyle=$ $\displaystyle
N\varepsilon\|u_{xx}\|_{L_{p}(\Omega,\mu)}+N\cdot
N(\varepsilon)\|u_{t}+A^{ij}u_{x^{i}x^{j}}\|_{L_{p}(\Omega,\mu)}.$
This obviously yields (6.2).
Case 2. Assume that either (i) $p\in(2,\infty)$ and $\theta\in[d,d+1)$ or (ii)
$p\in(1,2]$ and $\theta\in[d,d+p-1)$. By Remark 4.8 we only need to prove the
following:
$\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|M^{-1}u(t,x)|^{p}(x^{1})^{\theta-d}dxdt\leq
N\int^{\infty}_{-\infty}\int_{\mathbb{R}^{d}_{+}}|Mf(t,x)|^{p}(x^{1})^{\theta-d}dxdt.$
(6.3)
To prove this, we use a duality (Lemma 3.3). Denote $p^{\prime}=p/(p-1)$ and
choose $\bar{\theta}$ so that $\theta/p+\bar{\theta}/p^{\prime}=d$. Then
$\bar{\theta}\in(d-1,d]$ if $p^{\prime}\in(2,\infty)$ and
$\bar{\theta}\in(d-p^{\prime}+1,d]$ if $p^{\prime}\in(1,2]$.
Changing the variable $t\to-t$ shows that the result of case 1 is applicable
to the operator $u_{t}-A^{ij}u_{x^{i}x^{j}}$ in place of
$u_{t}+A^{ij}u_{x^{i}x^{j}}$. Therefore for any $v\in
M\mathbb{H}^{2}_{p^{\prime},\bar{\theta}}(-\infty,\infty)$, by integration by
parts,
$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}M^{-1}uM(v_{t}-A^{ij}v_{x^{i}x^{j}})dxdt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}u(v_{t}-A^{ij}v_{x^{i}x^{j}})dxdt$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d+1}_{+}}M(-u_{t}-A^{ij}u_{x^{i}x^{j}})M^{-1}vdxdt$
$\displaystyle\leq$
$\displaystyle\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\|M^{-1}v\|_{\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)}$
$\displaystyle\leq$ $\displaystyle
N\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\|M(v_{t}-A^{ij}v_{x^{i}x^{j}})\|_{\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)}.$
Since, by Case 1, $\\{v_{t}-A^{ij}v_{x^{i}x^{j}}:v\in
M\mathbb{H}^{2}_{p^{\prime},\bar{\theta}}(-\infty,\infty)\\}$ is dense in
$M^{-1}\mathbb{L}_{p^{\prime},\bar{\theta}}(-\infty,\infty)$, it follows that
$\|M^{-1}u\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}\leq
N\|M(u_{t}+A^{ij}u_{x^{i}x^{j}})\|_{\mathbb{L}_{p,\theta}(-\infty,\infty)}.$
The theorem is proved. ∎
Proof of Theorem 3.10 As usual, we assume $u_{0}=0$. For details see the proof
of Theorem 5.1 in [9].
Case 1. Let $T=\infty$. As before we only prove the a priori estimate. Suppose
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\infty)$ satisfies
$u_{t}=A^{ij}u_{x^{i}x^{j}}+f,\quad t\in(0,T)\,;\quad u(0,\cdot)=0.$ (6.4)
Define $v(t,x)=u(t,x)I_{t>0}$ and $\bar{f}=fI_{t>0}$, then $v\in
M^{-1}\mathbb{H}^{2}_{p,\theta}(-\infty,\infty)$ and $v$ satisfies (see
Definition 3.9)
$v_{t}=A^{ij}u_{x^{i}x^{j}}+\bar{f},\quad(t,x)\in\mathbb{R}^{d+1}_{+}.$
By Theorem 6.1,
$\|Mu_{xx}\|_{\mathbb{L}_{p,\theta}(\infty)}\leq
N\|Mf\|_{\mathbb{L}_{p,\theta}(\infty)}.$
By Remark 4.8, this certainly proves (3.17).
Case 2. Let $T<\infty$. The existence of the solution in
$\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ is obvious. Now suppose that
$u\in\mathfrak{H}^{\gamma+2}_{p,\theta}(T)$ is a solution of (6.4). By the
result of Case 1, the system
$v_{t}=\Delta v+(A^{ij}u_{x^{i}x^{j}}+f-\Delta u)I_{t\leq T},\quad t>0\,;\quad
v(0,\cdot)=0$ (6.5)
has a unique solution $v\in\mathfrak{H}^{\gamma+2}_{p,\theta}(0,\infty)$. Then
$v-u$ satisfies
$(v-u)_{t}=\Delta(v-u),\quad t\in(0,T)\,;\quad(v-u)(0,\cdot)=0.$
If follows from the theory of single equations (see, for instance, Theorem 5.6
in [10]), $u=v$ for $t\in[0,T]$. For $t\geq 0$, define
$A^{ij}_{T}=(a^{ij}_{T,kr}),\quad a^{ij}_{T,kr}=a^{ij}_{kr}I_{t\leq
T}+\delta^{ij}\delta^{kr}I_{t>T}.$
Then (6.5) and the fact $u=v$ for $t\in[0,T]$ show that $v$ satisfies (replace
$u$ by $v$ for $t\leq T$ in (6.5))
$v_{t}=A^{ij}_{T}v_{x^{i}x^{j}}+fI_{t<T},\quad t>0\,;\,\,v(0,\cdot)=0.$ (6.6)
By Case 1, $v\in\mathfrak{H}^{\gamma+2}_{p,\theta}(\infty)$ is the unique
solution of (6.6), and $u=v$ on $[0,T]$ whenever $u$ is a solution of (6.4) on
$[0,T]$. This obviously yields the uniqueness. The theorem is proved. $\Box$
Proof of Theorem 3.13 The proof is very similar to that of the proof of
Theorem 3.10 and is based on (5.13). We leave the details to the readers as an
exercise.
## References
* [1] M. Bramanti and M.C. Ceruti, $W^{1,2}_{p}$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO doeffieients, Comm. Partial Differential Equations, 18 (1993), no. 9-10, 1735-1763.
* [2] Sun-Sig Bun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations, 209 (2005), no. 2, 229-265.
* [3] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), no. 2, 841-853.
* [4] D. Kim, Elliptic equations with nonzero boundary condtions in weighted Sobolev spaces, J. Math. Anal. Appl. 337 (2008), 1465-1479.
* [5] P. Grisvard, Elliptic Problems in nonsmooth domains, Monographs and Studies in Mathematics 24 (1985), Pittman, Boston-London-Melbourn.
* [6] R. Haller-Dintelmann, H. Heck and M. Hieber, $L^{p}-L^{q}$-estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc. (2) 74 (2006), no. 3, 717-736.
* [7] Doyoon Kim and N.V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345-361.
* [8] N.V. Krylov, Some properties of traces for stochastic and determistic parabolic weighted Sobolev spaces, Journal of Functional Analysis 183 (2001), 1-41.
* [9] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, 64 (1999), AMS, Providence, RI.
* [10] N.V. Krylov, Weighted Sobolev spaces and Laplace equations and the heat equations in a half space, Comm. in PDEs, 23 (1999), no. 9-10, 1611-1653.
* [11] N.V. Krylov, Some properties of weighted Sobolev spaces in $\mathbb{R}^{d}_{+}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (1999), no. 4, 675-693.
* [12] N.V. Krylov, A $W^{n}_{2}$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab.Theory Relat.Fields 98(1994), 389-421.
* [13] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, American Mathematical Society, Prividence, RI, 2008.
* [14] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. on Math. Anal., 31 (1999), no. 1, 19-33.
* [15] Kijung Lee, On a Deterministic Linear Partial Differential System, Journal of Mathematical Analysis and Applications, 353, (2009), no. 1, 24-42.
* [16] J.-L. Lions and E. Magenes, Problèmes aux limites non homogénes et applications $1$, Dunod, Paris, 1968.
* [17] S.V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods and Applications of Analysis, 1 (2000), no. 1, 195-204.
* [18] V.A. Solonnikov, Solvability of the classical initial-boundary-value problems for the heat-conduction equations in a dihedral angle, Zapiski Nauchnykh Seminarov LOMI 138 (1984), 146-180 in Russian; English translation in Journal of Soviet Math. 32 (1986), no. 5, 526-546.
* [19] H. Triebel, Theory of function spaces, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983.
|
arxiv-papers
| 2012-04-11T03:15:32 |
2024-09-04T02:49:29.583751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kyeong-Hun Kim and Kijung Lee",
"submitter": "Kyeong-Hun Kim",
"url": "https://arxiv.org/abs/1204.2325"
}
|
1204.2401
|
# Controlling complex networks: How much energy is needed?
Gang Yan Temasek Laboratories, National University of Singapore, 117411,
Singapore Jie Ren Department of Physics and Centre for Computational Science
and Engineering, National University of Singapore, 117542, Singapore Ying-
Cheng Lai School of Electrical, Computer and Energy Engineering, Department
of Physics, Arizona State University, Tempe, AZ 85287, USA Choy-Heng Lai
Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, 117542, Singapore Baowen Li Department of
Physics and Centre for Computational Science and Engineering, National
University of Singapore, 117542, Singapore Center for Phononics and Thermal
Energy Science, Department of Physics, Tongji University, 200092, Shanghai,
China
###### Abstract
The outstanding problem of controlling complex networks is relevant to many
areas of science and engineering, and has the potential to generate
technological breakthroughs as well. We address the physically important issue
of the energy required for achieving control by deriving and validating
scaling laws for the lower and upper energy bounds. These bounds represent a
reasonable estimate of the energy cost associated with control, and provide a
step forward from the current research on controllability toward ultimate
control of complex networked dynamical systems.
###### pacs:
89.75.-k, 89.75.Fb
Complex networks are ubiquitous in natural, social, and man-made systems, such
as gene regulatory networks, social networks, mobile sensor networks and so on
Barabasi:review . A network is composed of nodes and edges. The nodes
represent individual units (e.g., genes, persons, sensors) and the edges
represent connections or interactions between the nodes. The state of a node
(e.g., protein being expressed, opinion of a person, position of a sensor)
normally evolves over time. And the evolution depends not only on the node’s
intrinsic dynamics but also on the couplings with its nearest neighbors
Newman:book .
On one hand, the couplings between nodes increase the complexity of collective
behaviors, which stimulates much interest of modeling, analyzing, and
predicting dynamical processes on complex networks DynamicalBook . On the
other hand, one may utilize the couplings to control a whole network, i.e.,
steering a network from any initial state (vector) to a desired final state,
by driving only a few suitable nodes with external signals. In this direction
there are good attempts recently from physics PRL1997 ; controlRMP ; LH:2007 ;
PhysicaD2010 ; LSB:2011 ; CowanarXiv ; WangarXiv , biology biologycontrolbook1
; RajaBiology and engineering networkcontrolbook2 ; Network_Control ;
LCWX:2008 ; RJME:2009 research communities. Among others, Liu _et al._
studied the controllability of various real-world networks, i.e., the ability
to steer a complex network as measured by the minimum number of driver nodes.
A main result was that the number of driver nodes required for full control is
determined by the network’s degree distribution LSB:2011 . Issues such as
achieving control by using only one controller RJME:2009 ; CowanarXiv and
making structural perturbations to the network to minimize the number of
control inputs WangarXiv have also been addressed.
When control a complex network, an important and unavoidable issue is the cost
of control. For instance, in order to control a social network some efforts
has to be devoted to change a few individuals’ opinions, while to control an
electronic or a mechanical network, some energy has to be consumed to drive a
few elements. Even if a network is controllable in principle, it may not be
controllable in practice if it costs an infinite amount of energy or if it
requires too much time to achieve the control. In this Letter, we address this
outstanding issue of _energy cost_ , i.e., the amount of efforts or energy
that are necessary to produce external signals for steering a complex network,
and focus on its lower and upper bounds. Suppose a complex network is deemed
to be controlled to a desired state in finite time $T_{f}$, our main results
[see Eqs. (7) and (8)] show the scaling laws of the energy cost bounds with
the control time $T_{f}$ in two different regimes separated by the
characteristic time. The results give faithful estimates for the required
energy and thus can provide significant insights into bridging network
controllability with actual control.
To be able to analyze the energy cost, we study linear networked systems
subject to control inputs. This is the currently standard framework, upon
which the network controllability analysis is built LH:2007 ; RJME:2009 ;
LSB:2011 ; CowanarXiv ; WangarXiv . A typical system of $N$ nodes and $M$
controllers can be written as
$\dot{\mathbf{x}}_{t}=\mathbf{A}\mathbf{x}_{t}+\mathbf{B}\mathbf{u}_{t},$ (1)
where $\mathbf{x}_{t}=[x_{1}(t),x_{2}(t),\ldots,x_{N}(t)]^{\text{T}}$ is the
state vector of nodes,
$\mathbf{u}_{t}=[u_{1}(t),u_{2}(t),\ldots,u_{M}(t)]^{\text{T}}$ is the input
vector of external signals, $\mathbf{B}=\\{b_{im}\\}$ is the $N\times M$ input
matrix with $b_{im}=1$ if controller $m$ connects to node $i$ and $b_{im}=0$
otherwise, $\mathbf{A}=\\{a_{ij}\\}$ is the weighted network’s adjacency
matrix including linear nodal dynamics $\\{a_{ii}\\}$.
The typical situation of controlling a complex dynamical network can be
characterized as using external signals $\mathbf{u}_{t}$ to direct the system
Eq. (1) from an arbitrary initial state $\mathbf{x}_{0}$ toward an arbitrary
desired state $\mathbf{x}_{T_{f}}$ in the time interval $t\in[0,T_{f}]$.
Assuming that the networked system is controllable LSB:2011 ; supplement , our
goal is to obtain analytic estimate of the energy cost required for achieving
control, which is defined as controlbook1
$\mathcal{E}(T_{f})\equiv\int_{0}^{T_{f}}\|\mathbf{u}_{t}\|^{2}dt$. Generally,
an infinite number of possibilities exist for choosing the control input
$\mathbf{u}_{t}$ to steer the system Eq. (1) from $\mathbf{x}_{0}$ to
$\mathbf{x}_{T_{f}}$. Of all the possible inputs, the optimal control input is
given by
$\mathbf{u}_{t}=\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}(T_{f}-t)}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}}$,
which minimizes the energy cost controlbook1 ; systembiology . The
corresponding minimized energy cost is then
$\mathcal{E}(T_{f})=\mathbf{v}^{\text{T}}_{T_{f}}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}}$,
where
$\mathbf{W}_{T_{f}}\equiv\int_{0}^{T_{f}}e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}t}dt$
and
$\mathbf{v}_{T_{f}}\equiv\mathbf{x}_{T_{f}}-e^{\mathbf{A}T_{f}}\mathbf{x}_{0}$
denotes the difference vector between the desired state under control and the
final state during free evolution. For convenience, we set the origin as the
desired state $\mathbf{x}_{T_{f}}=\mathbf{0}$ and rewrite the energy cost as
$\mathcal{E}(T_{f})=\mathbf{x}^{\text{T}}_{0}\mathbf{H}^{-1}\mathbf{x}_{0},$
(2)
where $\mathbf{H}(T_{f})\equiv
e^{-\mathbf{A}T_{f}}\mathbf{W}_{T_{f}}e^{-\mathbf{A}^{\text{T}}T_{f}}$ is the
symmetric Gramian matrix controlbook1 . When the system is controllable,
$\mathbf{H}$ is positive-definite (PD), otherwise it is non-invertible. In the
following we focus on the normalized energy cost
$E(T_{f})={\mathcal{E}(T_{f})}/{\|\mathbf{x}_{0}\|^{2}}=\frac{\mathbf{x}^{\text{T}}_{0}\mathbf{H}^{-1}\mathbf{x}_{0}}{\mathbf{x}^{\text{T}}_{0}\mathbf{x}_{0}}.$
(3)
When $\mathbf{x}_{0}$ is parallel to the direction of one of $\mathbf{H}$’s
eigenvectors, the corresponding inverse of the eigenvalue has the physical
meaning of normalized energy cost associated with controlling the system along
the particular eigendirection. Using the Rayleigh-Ritz theorem matrixbook , we
can bound the normalized energy cost as
$\frac{1}{\eta_{\text{max}}}\equiv E_{\text{min}}\leq E(T_{f})\leq
E_{\text{max}}\equiv\frac{1}{\eta_{\text{min}}},$ (4)
where $\eta_{\text{max}}$ and $\eta_{\text{min}}$ are the maximal and minimal
eigenvalues of the PD matrix $\mathbf{H}$, respectively.
To proceed, we focus on the lower and upper bounds of normalized energy cost
for the case of single-node control. To analytically calculate the quantities
$1/\eta_{\text{max}}$ and $1/\eta_{\text{min}}$, for weighted undirected
networks, we decompose the matrix $\mathbf{A}$ in terms of its eigenvectors as
$\mathbf{A=VSV}^{\text{T}}$, where $\mathbf{V}$ is the orthonormal eigenvector
matrix that satisfies
$\mathbf{VV}^{\text{T}}=\mathbf{V}^{\text{T}}\mathbf{V}=\mathbf{I}$,
$\mathbf{S}=\mathrm{diag}\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\\}$
with descending order $\lambda_{1}>\lambda_{2}>\ldots>\lambda_{N}$. We thus
have
$e^{\mathbf{A}t}=e^{\mathbf{A}^{\text{T}}t}=\mathbf{V}e^{\mathbf{S}t}\mathbf{V}^{\text{T}}$.
Substituting these expressions into the Gramian matrix and noting that
$\mathbf{V}$ is time-independent, we have
$\mathbf{H}=\mathbf{V}e^{-\mathbf{S}T_{f}}(\int_{0}^{T_{f}}e^{\mathbf{S}t}\mathbf{V}^{\text{T}}\mathbf{B}\mathbf{B}^{\text{T}}\mathbf{V}e^{\mathbf{S}t}dt)e^{-\mathbf{S}T_{f}}\mathbf{V}^{\text{T}}.$
(5)
Denoting the only node under direct control as $c$, we have that $\mathbf{B}$
is an $N\times 1$ matrix, of which all elements are zeros except the $c$th
element, which is one. After some amount of algebra, we obtain
$H_{ij}=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{j\beta}}{\lambda_{\alpha}+\lambda_{\beta}}\left(1-e^{-\left(\lambda_{\alpha}+\lambda_{\beta}\right)T_{f}}\right),$
(6)
where the Roman letters $i,j,c$ are node indices in the real space while the
Greek letters $\alpha,\beta$ are running indices in the eigenspace.
To carry the analysis further, we note that there are two distinct regimes in
terms of the control time $T_{f}$. In the small $T_{f}$ regime where $T_{f}\ll
1/|\lambda_{\alpha}+\lambda_{\beta}|$, we can expand
$e^{-(\lambda_{\alpha}+\lambda_{\beta})T_{f}}\approx
1-(\lambda_{\alpha}+\lambda_{\beta})T_{f}$ and obtain $H_{ij}\approx
T_{f}\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}V_{i\alpha}V_{c\alpha}V_{c\beta}V_{j\beta}=T_{c}\delta_{ic}\delta_{cj}$.
In this case, we have $H_{ij}\approx 0$ for all $i$ and $j$ except
$H_{cc}\approx T_{f}$ so that the maximal eigenvalue of matrix $\mathbf{H}$
can be approximated as $T_{f}$. Consequently, for the small $T_{c}$ regime, we
have $E_{\text{min}}\equiv 1/\eta_{\text{max}}\approx 1/T_{f}$, regardless of
the form of the matrix $\mathbf{A}$ and of the value of $c$. In contrast, in
the large $T_{f}$ regime characterized by $T_{f}\gg
1/|\lambda_{\alpha}+\lambda_{\beta}|$, we can approximate the maximal
eigenvalue of $\mathbf{H}$ by its trace, which has been numerically verified:
$\eta_{\text{max}}\approx\sum^{N}_{\alpha=1}\eta_{\alpha}\equiv\mathrm{Tr}[\mathbf{H}]=\sum_{i}^{N}\sum^{N}_{\alpha}\sum^{N}_{\beta}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{i\beta}}{\lambda_{\alpha}+\lambda_{\beta}}\left(1-e^{-(\lambda_{\alpha}+\lambda_{\beta})T_{f}}\right)=\sum_{\alpha=1}^{N}\frac{V_{c\alpha}^{2}}{2\lambda_{\alpha}}\left(1-e^{-2\lambda_{\alpha}T_{f}}\right)$.
If $\mathbf{A}$ is PD, the term $e^{-2\lambda_{\alpha}T_{f}}$ vanishes for
large $T_{f}$. We thus have $E_{\text{min}}\equiv 1/\eta_{\text{max}}\approx
1/\sum_{\alpha=1}^{N}\frac{V_{c\alpha}^{2}}{2\lambda_{\alpha}}\left(1-e^{-2\lambda_{\alpha}T_{f}}\right)\approx
1/\sum_{\alpha=1}^{N}\frac{V_{c\alpha}^{2}}{2\lambda_{\alpha}}=1/[(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}]_{cc}$.
Note that, since the matrix $\mathbf{A}$ is independent of $T_{f}$, the factor
$1/[(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}]_{cc}$ is time-independent too.
This means that, when $\mathbf{A}$ is PD, the lower bound of the energy cost
converges to a constant value for large $T_{f}$. If $\mathbf{A}$ is not PD,
i.e., at least one of $\mathbf{A}$’s eigenvalues is negative, the most
negative eigenvalue $\lambda_{N}$ will dominate the behavior of $\mathbf{H}$:
$H_{ij}\approx\frac{V_{iN}V_{cN}^{2}V_{jN}}{2\lambda_{N}}\left(1-e^{-2\lambda_{N}T_{f}}\right)\sim
e^{-2\lambda_{N}T_{f}}$. As a result, the maximal eigenvalue of $\mathbf{H}$
grows exponentially with $T_{f}$: $\eta_{\text{max}}\sim
e^{-2\lambda_{N}T_{f}}$ so that $E_{\text{min}}\sim e^{2\lambda_{N}T_{f}}$.
Since $\lambda_{N}<0$, the lower bound of the energy cost vanishes
exponentially with the control time $T_{f}$. In the borderline case where
$\mathbf{A}$ is semi PD, i.e., $\lambda_{\alpha}>0$ for
$\alpha=1,2,\ldots,N-1$ and $\lambda_{N}=0$, the behavior of $\mathbf{H}$ can
be characterized as: $H_{ij}\approx\lim_{\lambda_{N}\rightarrow
0}\frac{V_{iN}V_{cN}^{2}V_{jN}}{2\lambda_{N}}\left(1-e^{-2\lambda_{N}T_{f}}\right)\sim
T_{f}^{-1}$.
Our theoretical estimates for the _lower bound_ $E_{\text{min}}$ of the energy
cost can be summarized as
$E_{\text{min}}\begin{cases}\approx T_{f}^{-1}&\text{small $T_{f}$}\\\
\approx\frac{1}{[(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}]_{cc}}&\text{large
$T_{f}$, $\mathbf{A}$ is PD}\\\
\xrightarrow[\sim\exp{\left(2\lambda_{N}T_{f}\right)}]{\sim~{}T_{f}^{-1}}0&\text{large
$T_{f}$, $\mathbf{A}$ is }\frac{\text{semi PD}}{\text{not PD}}\end{cases}.$
(7)
Numerical support for Eq. (7) is shown in Fig. 1. We use scale-free networks
generated by the Barabási-Albert (BA) model BAmodel and Erdös-Réyni (ER) type
of random networks ERmodel . The link weights are randomly generated from the
uniform interval $[0.5,1.5]$. The linear nodal dynamics are set as
${a_{ii}}=-(a+s_{i})$ where $s_{i}=\sum^{N}_{j=1,j\neq i}a_{ij}$ is the
strength of node $i$, and $a$ is such a tunable parameter that one can
conveniently change $\mathbf{A}$ between positive and negative definite. We
note that other node-dependent settings of $a_{ii}$ will not affect our
results. Use the method proposed in LSB:2011 one can find the weighted
network is controllable, except some pathological link-weights sets of measure
zero, by any single driver node. We numerically compute the lower bound
according to Eqs. (4) and (5). From Figs. 1(a) and inset of 1(b), we see that,
for the small $T_{f}$ regime, $E_{\text{min}}$ decays as a power law
$T_{f}^{-1}$, regardless of $\mathbf{A}$ and $c$, agreeing with our
theoretical result. In the large $T_{f}$ regime, the behavior of
$E_{\text{min}}$ is determined by the signs of the eigenvalues of
$\mathbf{A}$. In particular, if the eigenvalues are all positive, the dynamics
in the absence of control, i.e.,
$\dot{\mathbf{x}}_{t}=\mathbf{A}\mathbf{x}_{t}$, will force the nodal states
to depart away from the zero state. Thus, even given sufficiently large time,
one has to consume some amount of energy to steer the nodes back. As shown in
Fig. 1(a), $E_{\text{min}}$ converges to a constant value as $T_{f}$ is
increased, which agrees with our predicted value
$1/[(\mathbf{A}+\mathbf{A}^{T})^{-1}]_{cc}$. In contrast, if $\mathbf{A}$ is
not PD, $E_{\text{min}}$ vanishes exponentially, as shown in Fig. 1(b). The
corresponding exponent is $2\lambda_{N}$, which is consistent with our
theoretical estimate in Eq. (7) as well.
Figure 1: (color online). Lower bound of the energy cost $E_{\text{min}}\equiv
1/\eta_{\text{max}}$ versus the control time $T_{f}$. All networks are
weighted BA scale-free networks except one weighted ER random network in (a),
with the same size $N=500$ and $\langle s\rangle=20$. The $s_{c}$ denotes the
strength of directly controlled node. In (a) $a=-150$ which makes $\mathbf{A}$
PD. In (b) $a=-50$ thus $\mathbf{A}$ is not PD. The dashed line in the semi-
log plot in (b) has a slope $2\lambda_{N}$. The symbols represent the same
quantities calculated numerically and the solid lines represent the results
from the estimation $\eta_{\text{max}}\approx\text{Tr}[\mathbf{H}]$.
Figure 2: (color online). Upper bound of the control energy cost
$E_{\text{max}}\equiv 1/\eta_{\text{min}}$ for a weighted BA network with 20
nodes. In (a), $a=2$ thus $\mathbf{A}$ is ND. In (b), $a=-5$. In (c), $a=-20$
so that $\mathbf{A}$ is PD. In (a-c), ( $\bullet$) represent the upper bound
$E_{\text{max}}$ while ( $\blacktriangleleft$) represent the corresponding
lower bound $E_{\text{min}}$ (included for comparison). In (d) the decaying
behavior of $E_{\text{max}}$ is shown for different $s_{c}$ and $a$ values.
The dash line has a slope $-36$. In (e) the exponential decay of
$E_{\text{max}}$ for large $T_{f}$ is plotted for different values of $a$. The
slopes of dashed lines are $2\lambda_{1}$ respectively. In (f) the constant
values of the energy cost in (c) are shown as a function of $|a+s_{c}|$. The
slope of the dashed line is $2$.
We now turn to the _upper bound_ of the energy cost $E_{\text{max}}\equiv
1/\eta_{\text{min}}$. As indicated by Eq. (6), most elements of the matrix
$\mathbf{H}$ are small, especially for the small $T_{f}$ regime. Consequently,
$\mathbf{H}$ is generally ill-conditioned matrixbook and its minimal
eigenvalue is typically very small (though positive). Thus, to control a
large-size network, $E_{\text{max}}$ can be very large. The underlying
physical reason is that, when only one node is subject to control, the effect
on other nodes will not be direct but instead will be indirect through various
paths on the network. The end result is that we need to steer the whole system
in the state space by following highly circuitous, though smooth, routes
supplement , a process that requires a large amount of energy.
Typical results computed from Eqs. (4) and (5) are shown in Figs. 2(a-c). For
small $T_{f}$, the upper bound $E_{\text{max}}$ exhibits power-law decay,
similar to the behavior of the lower bound, but the decay exponent for
$E_{\text{max}}$ assumes a much larger value that is independent of $a$ and
$c$ [see Fig. 2(d)]. For large $T_{f}$, $E_{\text{max}}$ will converge to a
constant value if $\mathbf{A}$ is not negative definite (ND), or will vanish
exponentially if $\mathbf{A}$ is ND. The corresponding exponent is given by
$2\lambda_{1}$, where $\lambda_{1}$ is the least negative eigenvalue of
$\mathbf{A}$, as shown in Fig. 2(e). This is due to the fact that, in the
large $T_{f}$ limit, the behavior of $H_{ij}^{-1}$ is dominated by the mode
with the least negative eigenvalue $\lambda_{1}$, which contributes the
slowest increase to $H_{ij}$. As a result, we have
$E_{\mathrm{max}}\sim[\mathbf{H}^{-1}]_{ij}\sim
H_{ij}^{-1}\sim\frac{2\lambda_{1}}{\left(1-\exp{(-2\lambda_{1}T_{f})}\right)}\sim
e^{2\lambda_{1}T_{f}}$. In the borderline case, i.e., $\mathbf{A}$ is semi ND,
the upper bound decays according to $T_{f}^{-1}$:
$E_{\mathrm{max}}\sim\lim_{\lambda_{1}\rightarrow
0}\frac{2\lambda_{1}}{\left(1-\exp{(-2\lambda_{1}T_{f})}\right)}\sim
T_{f}^{-1}$. Such a behavior in both $E_{\text{max}}$ and $E_{\text{min}}$ has
been numerically verified supplement .
The results for the _upper bound_ can be summarized as:
$E_{\text{max}}\begin{cases}\approx T_{f}^{-\theta}\;(\theta\gg 1)&\text{small
$T_{f}$}\\\ =\varepsilon(\mathbf{A},c)&\text{large $T_{f}$, $\mathbf{A}$ is
not ND}\\\
\xrightarrow[\sim\exp{\left(2\lambda_{1}T_{f}\right)}]{\sim~{}T_{f}^{-1}}0&\text{large
$T_{f}$, $\mathbf{A}$ is }\frac{\text{semi ND}}{\text{ND}}\end{cases},$ (8)
where $\varepsilon(\mathbf{A},c)$ denotes a positive value that depends on the
matrix $\mathbf{A}$ and the controlled node $c$. For the constant value of the
lower bound as described in Eq. (7), one may approximate
$1/[(\mathbf{A}+\mathbf{A}^{T})^{-1}]_{cc}\approx 2a_{cc}$ so that
$E_{\mathrm{min}}$ is proportional to $|a+s_{c}|$. However, as shown in Fig.
2(f), there appears no proportional relationship between the constant value
$\varepsilon(\mathbf{A},c)$ of $E_{\mathrm{max}}$ and $a_{cc}$ of the
controlled node. This indicates that directly controlling a node with larger
degree does not generally result in less energy cost.
Actually, when the system matrix $\mathbf{A}$ is PD and the control time
$T_{f}\rightarrow\infty$, Eq. 6 reduces to
$H^{\infty}_{ij}=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{j\beta}}{\lambda_{\alpha}+\lambda_{\beta}}$
which is the solution of
$\mathbf{AH}^{\infty}+\mathbf{H}^{\infty}\mathbf{A}^{\text{T}}=\mathbf{BB}^{\text{T}}$
and can be naturally interpreted as dynamical correlation RWLL:PRL10 , between
nodes $i$ and $j$ with respect to controlled (driver) node $c$. So
$\varepsilon(\mathbf{A},c)$ is the inverse of the smallest eigenvalue of the
correlation matrix $\mathbf{H}^{\infty}$. From this point of view, two
indications come out immediately: Firstly, to find optimal driver node in a
network, one should consider the node viewing from which the rest nodes are
most dissimilar. The reason is that, controlling a central hub node, though
may transmit external signals fast, can induce star-like structure which makes
the rest nodes more similar to each other. When nodes are more structurally
similar, they tend to have more similar dynamical correlations with other
nodes so that the corresponding rows in $\mathbf{H}^{\infty}$ become more
similar. As a consequence, the smallest eigenvalue of $\mathbf{H}^{\infty}$
will be less. In other words, we have to consume more energy to independently
steer similar nodes in order to fully control the network. Secondly, for
randomized networks, the more heterogeneous the node-degrees, the higher the
energy cost of control, on the average (see Section III of supplement ). Take
randomized BA and ER networks for example, we compare the values of
$\varepsilon(\mathbf{A},c)$, i.e., $\varepsilon_{\text{BA}}$ and
$\varepsilon_{\text{ER}}$ in Fig. 3(a). It shows that the upper bound of
energy cost for controlling BA networks is much larger than that for
controlling ER networks.
Figure 3: (color online). (a) The ratio $\varepsilon_{BA}/\varepsilon_{ER}$
for different network size $N$. In order to eliminate the effects of nodal
dynamics and strength, we fix the values of $a_{ii}$ and $\langle s\rangle$.
The results include the ratio for optimal driver node ( $\blacktriangleleft$)
and the ratio of averaging over different driver nodes ( $\bullet$). The error
bars are caused by different generations of network topology and link weights.
(b) $E_{\text{min}}$ ( $\blacktriangleleft$, left) and $E_{\text{max}}$ (
$\bullet$, right) versus $n_{c}$, the number of directly controlled nodes. The
dot pointed by the arrow corresponds to the node with largest degree in the
network.
We have also studied the energy cost associated with the control scheme
proposed in a recent work CowanarXiv , i.e., controlling more than one node by
a common controller. Fig. 3(b) shows the effect of $n_{c}$, the number of
directly controlled nodes, on the energy cost, which reveals that controlling
more nodes will induce smaller value of the lower energy bound. This, however,
does not hold for the upper bound. In fact, adding a node with large degree
into the directly controlled node-set may drastically increase the energy
cost. This result is consistent, to a certain degree, with that found in Ref.
LSB:2011 which shows the driver nodes tend to avoid the high-degree nodes.
It is noteworthy that our results can be easily generalized to weighted
_directed_ networks. If a network is controllable by one driver node, the
eigenvalues of the corresponding system matrix $\mathbf{A}$ are non-degenerate
RJME:2009 though may be not all real. Thus we have
$\mathbf{A}=\mathbf{VSV}^{-1}$ where
$\mathbf{S}=\text{diag}\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\\}$ with
descending order of the real part
$\text{Re}\lambda_{1}\geq\text{Re}\lambda_{2}\geq\ldots\geq\text{Re}\lambda_{N}$.
Similarly, $e^{\mathbf{A}t}=\mathbf{V}e^{\mathbf{S}t}\mathbf{V}^{-1}$ and
$e^{\mathbf{A}^{\text{T}}t}=(\mathbf{V}^{-1})^{\text{T}}e^{\mathbf{S}t}\mathbf{V}^{\text{T}}$.
As a consequence, Eq. 6 is replaced by
$H_{ij}=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{i\alpha}(V^{-1})_{\alpha
c}(V^{-1})_{\beta
c}V_{j\beta}}{\lambda_{\alpha}+\lambda_{\beta}}\left(1-e^{-\left(\lambda_{\alpha}+\lambda_{\beta}\right)T_{f}}\right)$.
Therefore, the scaling laws in Eqs. 7 and 8 keep unchanged while the decaying
exponents are replaced by $2\text{Re}\lambda_{N}$ and $2\text{Re}\lambda_{1}$
respectively. Moreover, for large $T_{f}$ and PD $\mathbf{A}$, the constant in
Eq. 7 is still proportional to $2a_{cc}$ by using first-order approximation in
RWLL:PRL10 .
In conclusion, we have reduced the complexity of the fundamental problem of
control cost from the complicated and intractable Gramian matrix to the simple
system matrix which is directly related to the network structure. Our results
have revealed that energy cost of controlling complex networks has different
scaling behaviors with control time in two time scales, separated by the
characteristic time, $\frac{1}{2|\text{Re}\lambda_{N}|}$ and
$\frac{1}{2|\text{Re}\lambda_{1}|}$ for the lower and the upper bound
respectively. In the small-time regime, setting a relatively longer time for
control always leads to less energy cost. While, in the large-time regime,
there exists the situation where we cannot reduce the energy cost even given
much more time. Furthermore, our results indicate that the lower (upper) bound
of energy cost is less when controlling a randomized network with
heterogeneous (homogeneous) node-degrees. These implications are important
when considering the trade-off between the energy cost and the control time,
which may find applications not only for classical LSB:2011 ; controlRMP but
also for biological biologycontrolbook1 ; systembiology ; RajaBiology and
quantum quantum_network networks. Although we have given some heuristics, a
method to choose an optimal control node-set for minimizing the energy cost is
lack, which is a promising future work.
We thank Drs. Maho Nakata and W.-X. Wang for helpful discussions. YCL thanks
the National University of Singapore for great hospitality, and he is
supported by AFOSR under Grant No. FA9550-10-1-0083. GY and CHL are supported
by DSTA of Singapore under Grant No. POD0613356.
## References
* (1) A.-L. Barabási, Science 325, 412 (2009); Nature Physics 8, 14 (2012).
* (2) M. E. J. Newman, Networks: An Introduction (Oxford University Press, Oxford, UK, 2010).
* (3) A. Barrat, M. Barthelémy, and A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, NY, USA, 2008).
* (4) R. O. Grigoriev, M. C. Cross, and H. G. Schuster, Phys. Rev. Lett. 79, 2795 (1997).
* (5) J. Bechhoefer, Rev. Mod. Phys. 77, 783 (2005).
* (6) A. Lombardi and M. Hörnquist, Phys. Rev. E 75, 056110 (2007).
* (7) M. Porfiri and F. Fiorilli, Physica D 239, 454 (2010).
* (8) Y.-Y. Liu, J.-J. Slotine and A.-L. Barabási, Nature (London), 473, 167 (2011).
* (9) N. J. Cowan, E. J. Chastain, D. A. Vilhena, J. S. Freudenberg, and C. T. Bergstrom, arXiv:1106.2573 (2011).
* (10) W.-X. Wang, X. Ni, Y.-C. Lai, and C. Grebogi, Phys. Rev. E 85, 026115 (2012).
* (11) I. Shmulevich, E. R. Dougherty, _Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks_ (SIAM, PA, USA, 2009).
* (12) I. Rajapakse, M. Groudine, and M. Mesbahi, Proc. Natl. Acad. Sci. U.S.A. 108, 17257 (2011).
* (13) M. Mesbahi and M. Egerstedt, _Graph Theoretic Methods in Multiagent Networks_ (Princeton University Press, NJ, USA, 2010).
* (14) X. Li, X. F. Wang and G. Chen, IEEE Trans. Circ. Syst.-I: 51, 2074 (2004).
* (15) B. Liu, T. Chu, L. Wang and G. Xie, IEEE Trans. Automat. Contr. 53, 1009 (2008).
* (16) A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, SIAM J. Contr. Optim. 48, 162 (2009).
* (17) Supplemental Materials associated with this paper.
* (18) W. J. Rugh, _Linear System Theory (2nd ed.)_ (Prentice-Hall, NJ, USA, 1996).
* (19) E. Klipp, W. Liebermeister, C. Wierling, A. Kowald, H. Lehrach, and R. Herwig, _Systems Biology: A Textbook_ (Wiley-VCH, Weinheim, Germany, 2009).
* (20) R. A. Horn and C. R. Johnson, _Matrix Analysis_ (Cambridge University Press, NY, USA, 1985).
* (21) A.-L. Barabási and R. Albert, Science 286, 509 (1999).
* (22) P. Erdös and A. Rényi, Publicationes Mathematicae 6, 290 (1959).
* (23) J. Ren, W.-X. Wang, B. Li, and Y.-C. Lai, Phys. Rev. Lett. 104, 058701 (2010).
* (24) See, for example, M. Yanagisawa, Phys. Rev. A 73, 022342 (2006); S. G. Schirmer, I. C. H. Pullen, and P. J. Pemberton-Ross, Phys. Rev. A 78, 062339 (2008); D. Burgarth, D. D’Alessandro, L. Hogben, S. Severini, and M. Young, arXiv:1111.1475.
Supplemental Materials
for “Controlling complex networks: How much energy is needed?”
## I Decay behaviors of $E_{\text{min}}$ and $E_{\text{max}}$ in borderline
cases
In the main text we argue that, for large control time $T_{f}$, if
$\mathbf{A}$ is semi positive definite (PD), the lower bound of the energy
cost $E_{\text{min}}$ will decay as $T_{f}^{-1}$ and, if $\mathbf{A}$ is semi
negative definite (ND) the upper bound of the energy cost $E_{\text{max}}$
will also decay as $T_{f}^{-1}$. To provide numerical confirmation for these
theoretical results, we consider the situation of controlling an undirected
network of 20 nodes which can be controllable by one single driver node. Just
as in the main text, let nodal dynamics be $a_{ii}=-(a+s_{i})$ where $s_{i}$
is the strength of node $i$. Setting $a=0$ so that $\mathbf{A}$ is semi ND, we
obtain the decay behavior of $E_{\text{min}}$, as depicted in Fig. S1(a).
However, if $a=-14.148$, $\mathbf{A}$ becomes semi PD. In this case, we
observe the decay of $E_{\text{max}}$ as shown in Fig. S1(b). The dashed lines
in both figures have the same slope $-1$. Thus, in these borderline
situations, $E_{\text{min}}$ and $E_{\text{max}}$ decay as $T_{f}^{-1}$ for
relatively large $T_{f}$, which is consistent with our theoretical
predictions.
Figure S1: Power-law decays of $E_{\text{min}}$ and $E_{\text{max}}$ for the
borderline situations. The dashed lines in (a) and (b) have the same slope
$-1$.
## II Optimal control route
In the main text, we argue that the upper bound of the energy cost associated
with controlling a complex network can be very large, because even the optimal
control route of steering the whole network from some initial state to the
origin (zero state) is in general highly circuitous, though smooth. Here we
provide numerical result of the optimal route for a simple directed network
used in MotterComment :
$\mathbf{A}=\begin{pmatrix}1&0\\\ 1&0\end{pmatrix},\hskip
14.22636pt\mathbf{B}=\begin{pmatrix}1\\\ 0\end{pmatrix}.$
Figure S2 shows that, when steering the network from
$\mathbf{x}_{0}=(1.0,0.5)^{\text{T}}$ to the desired state
$\mathbf{x}_{T_{f}}=(0,0)^{\text{T}}$ in the allowed time range $[0,1]$, the
optimal route is indeed not direct (dashed line) but smooth and circuitous (
$\bullet$). For undirected networks, we obtain similar highly circuitous
optimal routes (not shown here).
Figure S2: (color online). Optimal control routes to steer a simple directed
network from the initial state $(1.0,0.5)^{\text{T}}$ to the desired state
$(0,0)^{\text{T}}$ in the allowed time range [0,1].
## III Notes on structural equivalence of randomized networks
As we stated in the main text, the constant value $\varepsilon(\mathbf{A},c)$
of energy cost in Eq. (8) is the inverse of smallest eigenvalue of the
correlation matrix $\mathbf{H}^{\infty}$ with elements
$H^{\infty}_{ij}=\sum^{N}_{\alpha=1}\sum^{N}_{\beta}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{j\beta}}{\lambda_{\alpha}+\lambda_{\beta}}$.
Although it is difficult, if not impossible, to estimate the smallest
eigenvalue of $\mathbf{H}^{\infty}$, we can obtain some indications from the
viewpoint of correlations between nodes. When nodes are more structurally
similar, they tend to have more similar dynamical correlations with other
nodes so that corresponding columns and rows in $\mathbf{H}^{\infty}$ become
more similar. As a consequence, the smallest eigenvalue will be less and
$\mathbf{H}^{\infty}$ tend to be ill-conditioned so that the upper bound of
control energy cost increases drastically.
Because the energy cost of controlling a network depends not only on the
network topology but also on the link weights and nodal dynamics, for the
convenience of comparing the impacts of network structures on the control
cost, one may set the networks with equal size, edge numbers, but different
degree distributions, and set all nodal dynamics as a function of node-
degrees. At this situation, when two nodes are _structural equivalent_ , the
network cannot be controllable by any other driver node. Here _structural
equivalent_ is defined for any two nodes $a$ and $b$ with the same degree as
follows: nodes $a$ and $b$ are structural equivalent if and only if there
exists a nontrivial automorphism $f$ of the network graphtheory such that
$f(a)=b$. More apparently, take unweighted undirected networks for example,
denote the neighbor-set of any node $i$ by $\mathcal{S}(i)$, nodes $a$ and $b$
are structural equivalent if any of the following conditions are satisfied:
1\. $\mathcal{S}(a)=\mathcal{S}(b)$, or
2\. $a$ and $b$ are connected with each other and
$(\mathcal{S}(a)-\\{b\\})=(\mathcal{S}(b)-a)$, or
3\. $\forall v_{1}\in\mathcal{S}(a)$, $\exists v_{2}\in\mathcal{S}(b)$ such
that $(\mathcal{S}(v_{1})-a)=(\mathcal{S}(v2)-b)$ or $v_{1}$ and $v_{2}$ are
connected with each other and
$(\mathcal{S}(v_{1})-a-v_{2})=(\mathcal{S}(v_{2})-b-v_{1})$, or
4\. higher-order neighbors of $a$ and $b$ meet condition 3.
Therefore, for randomized networks, the probability of that any two nodes are
structural equivalent can provide hints for the energy cost of controlling
them: the larger the probability, the more the energy needed for controlling.
We here focus on the probabilities of the most possible conditions 1 and 2,
and ignore the high-order probabilities of conditions 3 and 4.
We will derive the probability of that two nodes are structural equivalent for
a randomized network with degree distribution $p(k)$. For a randomized
network, two nodes $i$ and $j$ are connected with the probability
$p_{ij}=\frac{k_{i}k_{j}}{\langle k\rangle N}$, where $k_{i}$ and $k_{j}$ are
the degrees of nodes $i$ and $j$ respectively, $\langle k\rangle$ is the
average degree and $N$ is the network size. Thus the probability of two nodes
$a$ and $b$ with the same degree $k_{0}$ have the same neighbors is
$\displaystyle p_{\text{eq}}$
$\displaystyle=\binom{k_{0}}{N-2}\prod^{k_{0}}_{i=1}p_{ai}p_{bi}\prod^{N-2}_{i=k_{0}+1}(1-p_{ai})(1-p_{bi})$
$\displaystyle=\binom{k_{0}}{N-2}\prod^{k_{0}}_{i=1}\bigg{(}\frac{k_{0}k_{i}}{\langle
k\rangle
N}\bigg{)}^{2}\prod^{N-2}_{i=k_{0}+1}\bigg{(}1-\frac{k_{0}k_{i}}{\langle
k\rangle N}\bigg{)}^{2}.$
As the degree $k_{i}$ of node $i$ is generated from the distribution $p(k)$
independently, the expectation value of $p_{\text{eq}}$ can be obtained as
$\langle
p_{\text{eq}}\rangle=\int_{k_{1}}\int_{k_{2}}\ldots\int_{k_{N-2}}[p_{\text{eq}}p(k_{1})p(k_{2})\ldots
p(k_{N-2})]dk_{1}dk_{2}\ldots dk_{N-2}$ where the integrations begin at the
smallest degree $k_{\text{min}}$ and end at the largest degree
$k_{\text{max}}$. Therefore, we have
$\displaystyle\langle p_{\text{eq}}\rangle=$ $\displaystyle
Z\prod^{k_{0}}_{i=1}\int_{k_{i}}\bigg{(}\frac{k_{i}}{\langle
k\rangle}\bigg{)}^{2}p(k_{i})dk_{i}\prod_{i=k_{0}+1}^{N-2}\int_{k_{i}}\bigg{(}\frac{N}{k_{0}}-$
$\displaystyle\frac{k_{i}}{\langle k\rangle}\bigg{)}^{2}p(k_{i})dk_{i}$
$\displaystyle=$ $\displaystyle Z\bigg{(}\frac{\langle k^{2}\rangle}{\langle
k\rangle^{2}}\bigg{)}^{k_{0}}\bigg{(}(\frac{N}{k_{0}})^{2}-2(\frac{N}{k_{0}})+\frac{\langle
k^{2}\rangle}{\langle k\rangle^{2}}\bigg{)}^{N-2-k_{0}},$
where $Z=\binom{k_{0}}{N-2}(k_{0}/N)^{2(N-2)}$. For the condition 2 mentioned
above, one can obtain similar result. Thus, when $k_{0}<N/2$, the larger the
factor $\frac{\langle k^{2}\rangle}{\langle k\rangle^{2}}$, the larger the
expectation value $\langle p_{\text{eq}}\rangle$.
From the result one can expect that controlling randomized Barabási-Albert
networks need much more energy than controlling Erdös-Rényi networks, as shown
in Fig. 3(a) in the main text. Moreover, for randomized scale-free networks
with degree distribution $p(k)\propto k^{-\gamma}$ ($\gamma>2$), the smaller
the value of $\gamma$, the more the energy needed for controlling. Especially,
if ignore the nodal dynamics, when $\gamma\rightarrow 2$ the value of
$p_{\text{eq}}\rightarrow 1$ for most of degrees $k_{0}$ because
$\frac{\langle k^{2}\rangle}{\langle k\rangle^{2}}$ is very large at this
situation. The ending result is that controlling a scale-free network with
$\gamma=2$ needs an infinite amount of energy. In other words, the network is
uncontrollable at $\gamma=2$ unless most of the nodes are set as driver nodes.
This indication is qualitatively consistent with the result found in
LSB:2011nature .
## IV Reachability: from initial state $\mathbf{x}_{0}=0$ to desired state
$\mathbf{x}_{T_{f}}\neq 0$
In the main text, we consider the case of controlling a networked system from
an arbitrary state $\mathbf{x}_{0}\neq 0$ to the origin
$\mathbf{x}_{T_{f}}=0$, which defines _controllability_. Here, we consider the
case of steering the system described in Eq. (1) from $\mathbf{x}_{0}=0$ to
$\mathbf{x}_{T_{f}}\neq 0$. This situation is referred to as _reachability_
linearcontrol .
To analyze the energy cost associated with reachability, we write down the
energy expression in the complete form as
$\mathcal{E}(T_{f})=(\mathbf{x}_{T_{f}}^{\text{T}}-\mathbf{x}_{0}^{\text{T}}e^{\mathbf{A}^{\text{T}}T_{f}})\mathbf{W}_{T_{f}}^{-1}(\mathbf{x}_{T_{f}}-e^{\mathbf{A}T_{f}}\mathbf{x}_{0})$,
where
$\mathbf{W}_{T_{f}}=\int_{0}^{T_{f}}e^{\mathbf{A}t}\mathbf{BB}^{\text{T}}e^{\mathbf{A}^{\text{T}}t}dt$.
Since $\mathbf{x}_{0}=0$, we have
$\mathcal{E}(T_{f})=\mathbf{x}_{T_{f}}^{\text{T}}\mathbf{W}_{T_{f}}^{-1}\mathbf{x}_{T_{f}}$.
For undirected networks, we factorize $\mathbf{A}$ as
$\mathbf{A}=\mathbf{VSV}^{\text{T}}$, where $\mathbf{V}$ is the orthonormal
eigenvector matrix with
$\mathbf{VV}^{\text{T}}=\mathbf{V}^{\text{T}}\mathbf{V}=\mathbf{I}$ and
$\mathbf{S}=\text{diag}\\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\\}$ with
descending order: $\lambda_{1}>\lambda_{2}>\ldots>\lambda_{N}$. Assume that
only the $c$-th node is controlled, we have
$W_{ij}=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{j\beta}}{\lambda_{\alpha}+\lambda_{\beta}}(e^{(\lambda_{\alpha}+\lambda_{\beta})T_{f}}-1).$
(S1)
Note that the term in the parenthesis is different from that of Eq. (6) in the
main text.
Recall the normalized energy cost
$E(T_{f})=\mathcal{E}(T_{f})/\parallel\mathbf{x}_{T_{f}}\parallel^{2}$, which
satisfies
$1/\xi_{\text{max}}=E_{\text{min}}\leq E(T_{f})\leq
E_{\text{max}}=1/\xi_{\text{min}},$ (S2)
where $\xi_{\text{min}}$ and $\xi_{\text{max}}$ are the minimal and maximal
eigenvalues of the matrix $\mathbf{W}_{T_{f}}$, respectively. Following a
similar analysis in the main text, we approximate $\xi_{\text{max}}$ by the
trace of $\mathbf{W}_{T_{f}}$ (verified again numerically, as shown in Fig.
S3) and get that
$\displaystyle\xi_{\text{max}}$ $\displaystyle\approx\sum_{i=1}^{N}W_{ii}\ \ \
[\equiv\text{Tr}(\mathbf{W}_{T_{f}})]$ (S3)
$\displaystyle=\sum_{i}^{N}\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{i\alpha}V_{c\alpha}V_{c\beta}V_{i\beta}}{\lambda_{\alpha}+\lambda_{\beta}}(e^{(\lambda_{\alpha}+\lambda_{\beta})T_{f}}-1)$
$\displaystyle=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{c\alpha}V_{c\beta}}{\lambda_{\alpha}+\lambda_{\beta}}(e^{(\lambda_{\alpha}+\lambda_{\beta})T_{f}}-1)(\sum_{i=1}^{N}V_{i\alpha}V_{i\beta})$
$\displaystyle=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\frac{V_{c\alpha}V_{c\beta}}{\lambda_{\alpha}+\lambda_{\beta}}(e^{(\lambda_{\alpha}+\lambda_{\beta})T_{f}}-1)\delta_{\alpha\beta}$
$\displaystyle=\sum_{\alpha=1}^{N}\frac{V_{c\alpha}V_{c\alpha}}{2\lambda_{\alpha}}(e^{2\lambda_{\alpha}T_{f}}-1).$
Figure S3: (color online). In the context of reachability where a network is
controlled from $\mathbf{x}_{0}=0$ to the desired state
$\mathbf{x}_{T_{f}}\neq 0$, the lower bound of the energy cost
$E_{\text{min}}=1/\xi_{\text{max}}$ versus the control time $T_{f}$. All
networks considered are scale-free except the one noted as ER Net in (a),
which is an Erdös-Réyni random network. The networks have the same size
$N=500$ and the same average degree $\langle k\rangle=6$. In all figures,
$k_{c}$ is the degree of the directly controlled node, and the symbols
represent the results of our numerical computation and the corresponding solid
lines are the results of our estimation
$\xi_{\text{max}}\approx\text{Tr}[\mathbf{W}]$. In (a), the parameter is
$a=2$, which makes the matrix $\mathbf{A}$ ND. In (b-d), $a=-2$ so that
$\mathbf{A}$ is not ND. The dashed lines in the log-log plots in (a) and (c)
have the same slope $-1$, which agrees with our theoretical result
$T_{f}^{-1}$ for small $T_{f}$. The dashed line in the semi-log plot in (d)
has the slope $-3.99\pm 0.01$, which corresponds to our theoretical estimate
$e^{-2\lambda_{1}T_{f}}$, as $-2\lambda_{1}=2a=-4.0$ in this case (see the
text for details).
Figure S4: (color online). (a,b) For the context of reachability, the lower
bound of the energy cost $E_{\text{min}}=1/\xi_{\text{max}}$ versus the degree
$k_{c}$ of the directly controlled node. The scale-free networks have the same
size $N=200$ and average degree $\langle k\rangle=6$, and the largest degree
is $k_{m}=54$. For comparison, (c) and (d) show the corresponding results for
the case of _controllability_ as discussed in the main text, i.e., steering
the network from $\mathbf{x}_{0}\neq 0$ to the desired state
$\mathbf{x}_{T_{f}}=0$. In (a) and (b), we set $a=2$ so that $\mathbf{A}$ is
ND and, hence, following the analysis in Sec. IV, we find that
$E_{\text{min}}$ converges to a constant value. In (c) and (d), we set $a=-80$
so that $\mathbf{A}$ becomes PD and, hence, following the analysis in the main
text, $E_{\text{min}}$ converges to a constant value. While (b) and (d)
display the relationship between the lower bound of the energy cost and the
degree $k_{c}$ of the controlled single node for different values of the
allowed control time $T_{f}$, (a) and (c) show the behaviors for large time
$T_{f}=10$, where the node size represents its degree, and the node color
represents $E_{\text{min}}$ when controlling the corresponding node only. Note
that the bottom curves in both (b) and (d) show the behavior
$E_{\text{min}}\propto|a+k_{c}|$, which are consistent with our theoretical
result
$E_{\text{min}}\approx|1/(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}|\approx
2|a+k_{c}|$.
From Eq. (S3), we obtain the behavior of the lower bound
$E_{\text{min}}=1/\xi_{\text{max}}$:
_Small $T_{f}$ regime_: $1/\xi_{\text{max}}\approx
1/\sum_{\alpha=1}^{N}\frac{V_{c\alpha}V_{c\alpha}}{2\lambda_{\alpha}}(e^{2\lambda_{\alpha}T_{f}}-1)\approx
1/\sum_{\alpha=1}^{N}V_{c\alpha}V_{c\alpha}T_{f}\approx 1/T_{f}$.
_Large $T_{c}$ regime_: if $\mathbf{A}$ is ND, i.e., all of $\mathbf{A}$’s
eigenvalues are negative, we have
$1/\xi_{\text{max}}\approx-1/\sum_{\alpha=1}^{N}\frac{V_{c\alpha}V_{c\alpha}}{2\lambda_{\alpha}}=-1/[(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}]_{cc}$.
If $A$ is not ND, i.e., at least one of $A$’s eigenvalues is positive,
$E_{\text{min}}$ will vanish exponentially as $1/\xi_{\text{max}}\sim
e^{-2\lambda_{1}T_{f}}$, where $\lambda_{1}$ is the most positive eigenvalue
of $\mathbf{A}$. For the borderline situation that $\mathbf{A}$ is semi-ND, we
can obtain the large-time decay behavior by setting $\lambda_{1}\rightarrow 0$
in Eq. (S3), which gives $E_{\text{min}}\sim 1/T_{f}$.
The behaviors of the lower bound of the energy cost associated with the
_reachability_ can then be summarized, as follows.
$E_{\text{min}}\begin{cases}\approx T_{f}^{-1}&\text{small $T_{f}$}\\\
\approx-\frac{1}{[(\mathbf{A}+\mathbf{A}^{\text{T}})^{-1}]_{cc}}&\text{large
$T_{f}$, $\mathbf{A}$ is ND}\\\
\xrightarrow[\sim\exp{\left(-2\lambda_{1}T_{f}\right)}]{\sim~{}T_{f}^{-1}}0&\text{large
$T_{f}$, $\mathbf{A}$ is }\frac{\text{semi ND}}{\text{not ND}}\end{cases}$
(S4)
We take unweighted networks for example, and show the numerical results on the
behaviors of $E_{\text{min}}$ versus the allowed control time $T_{f}$ in Fig.
S3. We have also studied the relationship between $E_{\text{min}}$ and the
degree $k_{c}$ of the directly controlled node, as shown in Fig. S4. We see
that, if the matrix $\mathbf{A}$ is ND, $E_{\text{min}}$ associated with
_reachability_ will converge to a constant value. In contrast, if $\mathbf{A}$
is PD, $E_{\text{min}}$ associated with _controllability_ will converge to a
constant value. Furthermore, we recall that, in the context of controllability
in the main text, driving a node with higher degree will induce smaller
$E_{\text{min}}$. However, in the context of reachability treated here,
driving a node with a higher degree will induce larger $E_{\text{min}}$, as
shown in Figs. S4(a) and S4(c).
Following a similar analysis in the main text, we obtain the behaviors of the
energy upper bound $E_{\text{max}}=1/\xi_{min}$ associated with _reachability_
, as follows (numerical results not shown here).
$E_{\text{max}}\begin{cases}\approx T_{f}^{-\nu}(\nu\gg 1)&\text{small
$T_{f}$}\\\ \approx\varsigma(\mathbf{A},c)&\text{large $T_{f}$, $\mathbf{A}$
is not PD}\\\
\xrightarrow[\sim\exp{\left(-2\lambda_{N}T_{f}\right)}]{\sim~{}T_{f}^{-1}}0&\text{large
$T_{f}$, $\mathbf{A}$ is }\frac{\text{semi PD}}{\text{PD}},\end{cases}$ (S5)
where $\varsigma(\mathbf{A},c)$ is a positive value depending on $\mathbf{A}$
and $c$, and $\lambda_{N}$ is the eigenvalue of $\mathbf{A}$ with the least
positive real part.
## V Derivation of optimal control $\mathbf{u}_{t}$
In this paper we have studied the behaviors of the energy cost
$\mathcal{E}(T_{f})=\int_{0}^{T_{f}}\mathbf{u}_{t}^{\text{T}}\mathbf{u}_{t}dt$
associated with controlling complex networks while choosing
$\mathbf{u}_{t}=\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}(T_{f}-t)}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}}$,
where
$\mathbf{W}_{T_{f}}=\int_{0}^{T_{f}}e^{\mathbf{A}t}\mathbf{B}\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}t}dt$,
and $\mathbf{v}_{T_{f}}=\mathbf{x}_{T_{f}}-e^{\mathbf{A}T_{f}}\mathbf{x}_{0}$.
We state in the main text without proof that this form of $\mathbf{u}_{t}$
minimizes the energy cost $\mathcal{E}(T_{f})$. Although the derivation of
this statement can be found in books on _optimal control_ (e.g., Soptimal ,
among others) and is not the subject of our present paper, we would like to
include it here for readers’ convenience.
_System_ : $\mathbf{\dot{x}}_{t}=\mathbf{Ax}_{t}+\mathbf{B\tilde{u}}_{t}$,
$\mathbf{x}(0)=\mathbf{x}_{0}$, $\mathbf{x}(T_{f})=\mathbf{x}_{T_{f}}$.
_Problem_ : Choose an optimal $\mathbf{u}_{t}$ out of
$\tilde{\mathbf{u}}_{t}$: $[0,T_{f}]\rightarrow\mathbf{R}^{N}$ to minimize
$J=\int_{0}^{T_{f}}\mathbf{\tilde{u}}_{t}^{\text{T}}\mathbf{\tilde{u}}_{t}dt$.
The optimal problem can be solved by using Pontryagin’s Maximum Principle
(PMP). Firstly, define the Hamiltonian
$M_{t}=\mathbf{\tilde{u}}_{t}^{\text{T}}\mathbf{\tilde{u}}_{t}+\mathbf{\lambda}_{t}^{\text{T}}(\mathbf{Ax}_{t}+\mathbf{B{\tilde{u}}}_{t})$
(S6)
where $\mathbf{\lambda}_{t}$ is the vector of Lagrange multipliers. Then,
according to the PMP, the optimal control signal $\mathbf{u}_{t}$ obeys
$\displaystyle\mathbf{\dot{x}}_{t}=\Big{(}\frac{\partial
M_{t}}{\partial\mathbf{\lambda}_{t}}\Big{)}^{\text{T}}\Big{|}_{\mathbf{u}_{t}}$
$\displaystyle=$ $\displaystyle\mathbf{Ax}_{t}+\mathbf{Bu}_{t},$ (S7a)
$\displaystyle-\mathbf{\dot{\lambda}}_{t}=\Big{(}\frac{\partial
M_{t}}{\partial\mathbf{x}_{t}}\Big{)}^{\text{T}}\Big{|}_{\mathbf{u}_{t}}$
$\displaystyle=$ $\displaystyle\mathbf{A}^{\text{T}}\mathbf{\lambda}_{t},$
(S7b) $\displaystyle 0=\Big{(}\frac{\partial
M_{t}}{\partial\mathbf{\tilde{u}}_{t}}\Big{)}^{\text{T}}\Big{|}_{\mathbf{u}_{t}}$
$\displaystyle=$
$\displaystyle\mathbf{u}_{t}+\mathbf{B}^{\text{T}}\mathbf{\lambda}_{t}.$ (S7c)
The solution of Eq. (S7b) is
$\lambda_{t}=e^{-\mathbf{A}^{\text{T}}t}\mathbf{c}$, where $\mathbf{c}$ is a
vector independent of time $t$. Substituting it into Eq. (S7c), we get
$\mathbf{u}_{t}=-\mathbf{B}^{\text{T}}e^{-\mathbf{A}^{\text{T}}t}\mathbf{c}.$
(S8)
The solution of Eq. (S7a) under the boundary condition
$\mathbf{x}_{t=0}=\mathbf{x}_{0}$ is
$\mathbf{x}_{t}=e^{\mathbf{A}t}(\mathbf{x}_{0}+\int_{0}^{t}e^{-\mathbf{A}s}\mathbf{Bu}_{s}ds).$
(S9)
Substituting Eq. (S8) into Eq. (S9) gives
$\mathbf{x}_{t}=e^{\mathbf{A}t}\mathbf{x}_{0}-(\int_{0}^{t}e^{\mathbf{A}(t-s)}\mathbf{B}\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}(t-s)}ds)e^{-\mathbf{A}^{\text{T}}t}\mathbf{c}$.
By a change of variables $t-s=\tau$ and thus $d\tau=-ds$ in the integral, we
can rewrite the equation as
$\mathbf{x}_{t}=e^{\mathbf{A}t}\mathbf{x}_{0}-(\int_{0}^{t}e^{\mathbf{A}\tau}\mathbf{B}\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}\tau}d\tau)e^{-\mathbf{A}^{\text{T}}t}\mathbf{c}$.
Denoting
$\mathbf{W}_{t}=\int_{0}^{t}e^{\mathbf{A}\tau}\mathbf{B}\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}\tau}d\tau$
(S10)
and using the boundary condition $\mathbf{x}_{t=T_{f}}=\mathbf{x}_{T_{f}}$, we
have
$\mathbf{x}_{T_{f}}=e^{\mathbf{A}T_{f}}\mathbf{x}_{0}-\mathbf{W}_{T_{f}}e^{-\mathbf{A}^{\text{T}}T_{f}}\mathbf{c},$
(S11)
where $\mathbf{W}_{T_{f}}$ is symmetric as well as PD if the system is
controllable. It is also referred to as the Gramian matrix of controllability.
Thus
$\mathbf{c}=-e^{\mathbf{A}^{\text{T}}T_{f}}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}},$
(S12)
where
$\mathbf{v}_{T_{f}}=\mathbf{x}_{T_{f}}-e^{\mathbf{A}T_{f}}\mathbf{x}_{0}.$
(S13)
Substituting Eq. (S12) into Eqs. (S8) and (S9), we obtain
$\displaystyle\mathbf{u}_{t}$ $\displaystyle=$
$\displaystyle\mathbf{B}^{\text{T}}e^{\mathbf{A}^{\text{T}}(T_{f}-t)}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}},$
(S14a) $\displaystyle\mathbf{x}_{t}$ $\displaystyle=$ $\displaystyle
e^{\mathbf{A}t}\mathbf{x}_{0}+\mathbf{W}_{t}e^{\mathbf{A}^{\text{T}}(T_{f}-t)}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}},$
(S14b) $\displaystyle\mathcal{E}(T_{f})$ $\displaystyle=$ $\displaystyle
J_{\text{optimal}}=\mathbf{v}_{T_{f}}^{\text{T}}\mathbf{W}_{T_{f}}^{-1}\mathbf{v}_{T_{f}}.$
(S14c)
## References
* (1) J. Sun, S. P. Cornelius, W. L. Kath, A. E. Motter, arXiv:1108.5739v1 (2011).
* (2) B. Bollobás, _Random Graphs (2nd ed.)_ (Cambridge University Press, Cambridge, UK, 2001).
* (3) Y.-Y. Liu, J.-J. Slotine and A.-L. Barabási, Nature (London), 473, 167 (2011).
* (4) W. J. Rugh, _Linear System Theory (2nd ed.)_ (Prentice-Hall, NJ, USA, 1996).
* (5) F. L. Lewis and V. L. Syrmos, _Optimal Control (2nd ed.)_ (Wiley, NY, USA, 1995).
|
arxiv-papers
| 2012-04-11T10:28:38 |
2024-09-04T02:49:29.594426
|
{
"license": "Public Domain",
"authors": "Gang Yan, Jie Ren, Ying-Cheng Lai, Choy-Heng Lai, Baowen Li",
"submitter": "Gang Yan",
"url": "https://arxiv.org/abs/1204.2401"
}
|
1204.2420
|
# Variational Principle underlying Scale Invariant Social Systems
A. Hernando1, A. Plastino2, 3 1 Laboratoire Collisions, Agrégats, Réactivité,
IRSAMC, Université Paul Sabatier 118 Route de Narbonne 31062 - Toulouse CEDEX
09, France
2National University La Plata, IFLP-CCT-CONICET, C.C. 727, 1900 La Plata,
Argentina
3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,
Spain
###### Abstract
MaxEnt’s variational principle, in conjunction with Shannon’s logarithmic
information measure, yields only exponential functional forms in
straightforward fashion. In this communication we show how to overcome this
limitation via the incorporation, into the variational process, of suitable
dynamical information. As a consequence, we are able to formulate a somewhat
generalized Shannonian Maximum Entropy approach which provides a unifying
“thermodynamic-like” explanation for the scale-invariant phenomena observed in
social contexts, as city-population distributions. We confirm the MaxEnt
predictions by means of numerical experiments with random walkers, and compare
them with some empirical data.
###### pacs:
89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k
## I Introduction
Scale-invariant phenomena are plentiful in Nature and in artificial systems.
Illustrations can be encountered that range from physical and biological to
technological and social sciences uno . Among the latter, one may speak of
empirical data from scientific collaboration networks cites , cites of physics
journals nosotrosZ , the Internet traffic net1 , Linux packages links linux ,
popularity of chess openings chess , as well as electoral results elec1 ; ccg
, urban agglomerations ciudad ; ciudad2 and firm sizes all over the world
firms . These systems lack a typical size, length or frequency for observables
under scrutiny, a fact that usually leads to power law distributions
$p_{X}(x)\sim 1/x^{1+\lambda},$ (1)
with $\lambda\geq 0$. The class of universality defined by $\lambda=1$,
corresponding to the so-called Zipf’s law (ZL) in the cumulative distribution
or the rank-size distribution received much attention zipf ; nosotrosZ ; net1
; linux ; chess ; ciudad ; ciudad2 ; firms ; chin ; upf ; citis . Maillart et
al. linux have found that links’ distributions follow ZL as a consequence of
stochastic proportional growth. Such kind of growth assumes that an element of
the system becomes enlarged proportionally to its size $k$, being governed by
a Wiener process. The $\lambda=1-$class emerges from a condition of
stationarity (dynamic equilibrium) citis . ZL also applies for processes
involving either self-similarity chess or fractal hierarchy chin , all of
them mere examples amongst very general stochastic ones upf . The instance
$\lambda=0$ is representative of a second universality-class following Costa
Filho et al. elec1 , who studied vote-distributions in Brazil’s electoral
results. Therefrom emerge multiplicative processes in complex networks ccg .
$\lambda=0-$behavior ensues as well in i) city-population rank distributions
nosotros , ii) Spanish electoral results nosotros , and iii) the degree
distribution of social networks nosotros2 . As shown in Ref. benford2 , this
universality class encompasses Benford’s Law benford1 . In the present vein,
still another kind of idiosyncratic distribution is often reported: the log-
normal one lnwiki , that has been observed in biology (length and sizes of
living tissue bio ), finance (in particular the Black Scholes model black ),
and firms-sizes. The latter instance obeys Gibrat’s rule of proportionate
growth gibrat , that also applies to cities’ sizes. _Remarkably enough, all
these variegated and sometimes quite complex systems share a scale-free growth
behavior_.
Together with geometric Brownian motion, there is a variety of models arising
in different fields that yield Zipf’s law and other power laws on a case-by-
case basis ciudad ; ciudad2 ; citis ; mod1 ; exp ; renorm , as preferential
attachment net1 and competitive cluster growth ccg in complex networks, used
to explain many of the scale-free properties of social, technological and
biological networks. For instance we may mention urban dynamics ud , opinion
dynamics oppi , and electoral results elec1 ; elec2 , that develop detailed
realistic approaches. Ref. otros is highly recommendable as a primer on urban
modelling. Of course, the renormalization group is intimately related to scale
invariance and associated techniques have been fruitfully exploited in these
matters (as a small sample see renorm ; voteGalam ).
Can the celebrated maximum entropy principle of Jaynes’ (MaxEnt) jaynes ; katz
explain the disparate phenomena recounted above? At first sight, one faces a
seemingly unsurmountable difficulty. MaxEnt’s variational principle, in
conjunction with Shannon’s logarithmic information measure, can not in
straightforward fashion yield power laws (nor more involved combinations that
include power laws), but only exponential distributions katz . This fact
constituted one major motivation pla for introducing Tsallis’ information
measure tbook (and its associated Tsallis-MaxEnt treatment pp93 ). One
immediately gets power laws thereby as a result of varying the measure. We
have shown in Ref. ourRef how to overcome the above Shannonian limitation via
suitable incorporation, into the MaxEnt principle, of dynamical information.
As a consequence, we are now able to formulate a Shannonian Maximum Entropy
approach jaynes ; katz common to all these systems. As stated in ourRef ,
this technique exhibits the peculiarity of including “equations of motion” as
a part of the required a priori knowledge which always needs to be
incorporated into the accompanying Jaynes’ variational problem. The desired
goal can in this way be successfully achieved, which provides a unifying
“thermodynamic-like” explanation for the above mentioned disparate phenomena.
The approach explicitly reconciles two apparent different viewpoints, those
attached to i) growth models and ii) information-treatments X2 .
## II Theoretical procedure
### II.1 General approach
Accordingly, consider a rather general stochastic dynamical equation for the
observable $x$, namely,
$\dot{x}(t)=k\,g[x(t)];\,\,g\,\,{\rm arbitrary}.$ (2)
where $k$ represents the derivative of a Wiener process (characterized by
stationary independent increments) that frequently occurs in economic and
social systems. We assume that there exists an appropriate transformation of
$x$ that makes (2) invariant and a hallmark of the _symmetry_ characterizing
the system. Thermodynamics and many areas of physics have been shown by
Frieden and Soffer FS to be typified by _translational_ symmetry if they are
theoretically described à la Fisher FS . This entails $g(x)=1$, which makes
Eq. (2) the emblematic equation for linear Brownian motion. Our considerations
will revolve around a new variable $u=u(x)$ such that $dx/du=g(x)$, a variable
that
1. 1.
should linearize the dynamic equation,
2. 2.
make the original $x-$symmetry a translational one $\dot{u}(t)=k$, and
3. 3.
according to the “central hypothesis” of Ref. ourRef , constitutes the tool
for introducing dynamical information into MaxEnt by working with the
$u-$entropy of the system.
One has ourRef
$S[p_{U}(u)]=\int_{\Omega}dup_{U}(u)\log[p_{U}(u)],$ (3)
with $\Omega$ an appropriate “volume” in $u-$space. The equilibrium
probability density $p_{U}(u)$ is derived from Jaynes’ MaxEnt principlejaynes
$\delta_{p}\left\\{S[p_{U}(u)]-\sum_{i}\lambda_{i}\langle
f_{i}(u)\rangle\right\\}=0,$ (4)
where the mean values of the functions $f_{i}(u)$ describe the a-priori known
constraints on the system, while $\lambda_{i}$ are the associated Lagrange
multipliers. Our constraints represent conservation rules, operating on our
system, that strongly condition the configuration of the equilibrium state.
The general solution of the Jaynes problem is jaynes
$p_{U}(u)du=\exp{\left[-\sum_{i}\lambda_{i}f_{i}(u)\right]}du$ (5)
which, changing back to $x$ lead us to a $p_{X}(x)$ of the form
$p_{X}(x)dx=\exp{\left[-\sum_{i}\lambda_{i}f_{i}(u(x))\right]}\frac{dx}{g(x)}.$
(6)
The main difference with the usual MaxEnt (ME) solution is the Jacobian
$du/dx=1/g(x)$. This looks trivial enough, but the Jacobian contains dynamical
information, otherwise absent from the treatment. In a manner of speaking, we
are thereby “extending” the exponential-like form of the Jaynes’ ME-solutions
to other analytical possibilities.
### II.2 Scale-invariant systems
To illustrate this procedure we consider the emblematic equation used in many
models of mathematical finance (e.g., the Black-Scholes model black ) and
cities’ and firms’ sizes (Gibrat’s law lognorm ), and also many of the
“social” examples listed above. We speak then of the so-called geometrical
Brownian motion:
$\dot{x}(t)=kx(t),$ (7)
which is symmetric under scale transformations $x^{\prime}=cx$ with $c$ an
arbitrary constant. We are here defining a proportional growth or
multiplicative process. A systems following such dynamics can be described by
our approach. We set $u=\log(x/x_{0})$, where $x_{0}$ is the minimum allowed
value for $x$ (or a “reference”-one). The Jacobian of the transformation is
$du/dx=1/x$, and the volume in $u$ space is defined as $\Omega$: $[0\leq u\leq
u_{M}]$. Here $u_{M}=\log(x_{M}/x_{0})$ corresponds to some maximum allowable
value (which can be infinity). When no constraints are included and an
infinite volume $\Omega$ is considered, the constituents of system diffuse as
random walkers in $u$. Accordingly, the density distribution (DD) $p_{U}(u)du$
is a gaussian distribution that becomes a log-normal one in $x$. This no-
constraint case has been widely studied, and most city-population
distributions follow it lognorm .
Let us discuss now just how adding constraints affect the equilibrium DD by
considering the simple form $f_{i}(u)=u^{i}$. The case $i=0$ corresponds to
normalization of the DD, while $i=1$ refers to having $\langle u\rangle$ as a
constraint. We tackle the variational solution subjected to these two
constraints, setting $\mu=\lambda_{0}$ and $\lambda=\lambda_{1}$. Accordingly,
the equilibrium DD $p_{U}(u)$ extremizes the functional
$F=S-\mu-\lambda\langle u\rangle,$ so that
$p_{U}(u)du=e^{-\mu-\lambda u}du.$ (8)
The values for $\mu$ and $\lambda$ are obtained from the conservation rules
$1=\int_{\Omega}du~{}e^{-\mu-\lambda u}$ and $\langle
u\rangle=\int_{\Omega}du~{}e^{-\mu-\lambda u}u$ which yield $\langle
u\rangle=e^{\mu}=1/\lambda$ if $u_{M}\rightarrow\infty$. The DD, as a function
of $x$, becomes
$p_{X}(x)dx=e^{-\mu}x_{0}^{\lambda}\frac{dx}{x^{\lambda+1}},$ (9)
i.e., a power law whose exponent is characterized by $\lambda$, as that
presented in Eq. (1). Look first at the particular solution $\lambda=0$ (no
constraint on $\langle u\rangle$). One is led to $e^{\mu}=u_{M}$ and $\langle
u\rangle=u_{M}/2$. In terms of $x$ the DD is
$p_{X}(x)dx=\frac{1}{u_{M}}\frac{dx}{x}.$ (10)
As remarked above, such DD is related to Benford’s Law. Using a thermodynamic
analogy, the law describes the simple scenario of a non-interacting system
confined to a finite volume of $u$-space, with a Gaussian distribution for
$\dot{u}$, i.e., a scale-free ideal gas (SFIG)nosotros . Even if the
normalization seems to diverge here for $u_{M}\rightarrow\infty$, it can be
kept finite in going to the thermodynamic limit (see nosotros ): if $N$ is the
total number of system’s “elements”, its density $\rho(x)=Np_{X}(x)$ is
normalized in the limit ($N,\,u_{M}\rightarrow\infty$) if
$N/u_{M}\rightarrow\rho_{0}$, where $\rho_{0}$ is a constant. Let us pass now
to the particular solution $\mu=0$ (relaxing the normalization constraint). If
$u_{M}\rightarrow\infty$, we obtain from the conservation rules $\lambda=1$
and $\langle u\rangle=1$. Accordingly, the DD for the observable $x$ is
$p_{X}(x)dx=x_{0}\frac{dx}{x^{2}},$ (11)
i.e., Zipf’s law (ZL). In a thermal context, the absence of normalization can
be understood as an inability of the system to reach the thermodynamic limit.
One may then speak of a Zipf regime nosotros . The lack of normalization
constraint is discussed also in Refs. nosotrosZ ; upf . An explanation may be
concocted: these elements might be distributed on the surface of an
appropriate volume nosotrosZ ; nosotros . Accordingly, ZL usually holds for
the upper tail and not for the “bulk” of the associated distribution. There is
a free interchange of elements between the two regions with no “energetic
cost” since the “chemical potential” $\mu$ is zero. Accordingly, the number of
elements does not remain constant (photon statistics). This number-fluctuation
has no effect in the bulk region where the thermodynamic limit is reached, but
it totally determines how the density behaves on our putative “surface”.
## III Present results: numerical experiments and empirical observations
So as to confirm the preceding MaxEnt predictions we have performed numerical
experiments with random walkers and compared them with empirical city-
population data. We solve numerically the equation $\dot{x}(t)=kx(t)$,
discretizing the time in intervals of $\Delta t$. $k$ is randomly generated at
each iteration from a Gaussian distribution with zero mean and variance $K$.
Figure 1: Top panel: Evolution of a random walkers’ distribution without
constraints from an initial delta-one (purple solid line), passing through
intermediate stages (red dotted, green dashed, and solid blue lines). Inset:
logarithmic size $u$ vs. relative growth $\dot{u}$ (linear fit in black).
Bottom panel: Florida’s cities-population-distribution for 1990, 2000, and
2010 (same color-code as TP) compared with a normal distribution (smooth solid
blue line). Inset: same as in top panel’s inset.
### III.1 Free evolution
We first study evolution without constraints (as a control case), starting
with $N=10000$ walkers located at the same position $x=100$ ($u=2\log 10$),
that evolve with $K=10$ and a time interval $\Delta t=10^{-5}$ in $x$. We also
include a drift, as in Ref. gibrat . As expected, we obtain the diffusion
process depicted in upper panel of Fig. 1, which is a log-normal DD in $x$
(and a Gaussian in $u$).
We now compare such evolution with that of the cities-population of Florida
State using Census Bureau’s data for the years 1990, 2000, and 2010 usa , also
finding a log-normal in $x$ with growing width (1.64, 1.65 and 1.72,
respectively). The proportional growth condition is checked by computing the
correlation between the logarithm of the population of each place $u$ with its
relative change $\dot{u}$, using two-points formulas for the time derivative.
For $\dot{u}=k$ both observables are independent for scale-free systems. The
correlation value is 0.027, small enough to confirm that geometrical motion is
taking place (see inset of Fig. 1, bottom panel). Even if the populations of
Florida-cities do not seem to match the non-interacting assumption of our
simulation, we expect a short-range nature for the correlation between the
population of each pair of cities Gmodel , relative to the State’s size.
According to our thermodynamic interpretation, we can think of a dilute scale-
free gas at zero pressure and expanding freely.
### III.2 Evolution constrained by normalization and finite volume
For an example with normalization constraint and finite volume ($\lambda=0$),
we arbitrarily define $x_{0}=1$, $x_{M}=10^{4}$, ($u_{M}=4\log 10$) and use
the same initial conditions as in the precedent case. Now, a “move” is not
accepted if the new position falls outside the appropriate region. After some
iterations the system reaches equilibrium, as shown in Fig. 2, top panel. We
find a nice fitting of this equilibrium distribution to that predicted by
MaxEnt [Eq. (10)], confirming the validity of our approach.
A system in this thermodynamical condition
1. 1.
obeys the proportional growth dynamics,
2. 2.
exhibits low correlation between its elements,
3. 3.
conserves the particle-number, and
4. 4.
is characterized by an objective, measurable size-constraint.
This last condition, for city-populations, has the form of a geographical
constraint. Such is the case of, e.g., the Marshall Islands: this particular
geographical area covers 181.3 km2 distributed into 29 atolls and 5 islands.
Traversing the sea may reduce correlations between cities as compared to terra
firma. The migration pattern concentrates in the two main population-centers,
Rita and Ebeye, so that only indirect correlations are expected between the
rest of the cities. We verified this issue by recourse to data from 1980,
1988, and 1999 mar . The islands’ logarithmic population $u$, exhibits a
correlation coefficient of $9\times 10^{-5}$ with a relative increment
$\dot{u}$, confirming the dynamics’ nature (inset of Fig. 1b, bottom panel).
We fitted the raw data for all low-correlated centers (154 populations) to the
MaxEnt prediction and also to a log-normal with the same log-mean and log-
variance that characterize the data (Fig. 1b, bottom panel). We found a
correlation of $0.991$ in the former case and of $0.979$ in the latter one.
One thus may visualize the Marshall Islands as a closed-volume scale-free
ideal ga (SFIG) in thermal equilibrium, again empirically confirming the
validity of our approach.
Figure 2: Top panel: walkers’ distribution for the first text-example
$\lambda=0$ (origin of Benford Law). The blue solid line indicates convergence
towards the MaxEnt-prediction (black solid line). Inset: equilibrium rank-
distribution. Bottom Panel: rank-distribution of Marshall Islands’ city-
population versus i) MaxEnt-prediction for $\lambda=0$ (black line) and ii) a
log-normal (red line). Inset: logarithmic size $u$ vs. relative growth
$\dot{u}$ (linear fit in black).
### III.3 Evolution constrained by a mean value-condition
The second example presented here ($\mu=0$) uses a simple algorithm that
reproduces Zipf’s Law. $N$ walkers “unfold” while guaranteeing the
conservation laws’ proper working: $\langle\log(x/x_{0})\rangle=\langle
u\rangle=1$ (or $\sum_{i=1}^{N}\log(x_{i}/x_{0})=\sum_{i=1}^{N}u_{i}=N$). We
start all the walkers at $x=e\times x_{0}$ (any distribution with $\langle
u\rangle=1$ is adequate) and proceed iterating as follows:
1. 1.
select an arbitrary walker and change its position in proportional fashion
according to $x(t+\Delta t)=x(t)\times(1+k\Delta t)$, with a random value $k$;
2. 2.
assume that we have a bulk reservoir so that the exchange of elements has no
’energetic cost’. We randomly select a second walker and remove him from his
position $x^{\prime}$;
3. 3.
add a new walker at a position that preserves the operating conservation rule,
that is $x^{\prime}(t+\Delta t)=x^{\prime}(t)\times(1-k\Delta t)$.
We iterate these steps till convergence is achieved. After some iterations, we
do converge to the predicted MaxEnt distribution Eq. (11) (Fig. 3, top panel,
with the same values for $x_{0}$, $N$, $K$, and $\Delta t$ used in the first
example). thusly Once more, we reconfirm that the approach presented here
works in reasonable fashion. Remarkably enough, a similar algorithm is also
able to reproduce any arbitrary power law with exponent $\lambda+1$ by
changing the value of the conservation rule in the fashion $\langle
u\rangle=1/\lambda$.
For city-populations, elements-exchange occurs in a continuous fashion.
However the data are recorded at intervals of years, so that the ensuing
exchange-effects are similar to those of our simulation. Areas belonging to
the most-populated sites eventually change with time. Thus, we deal with an
open system with no fixed number of elements, like our photon-gas above. As a
well known example consider the most populated metropolitan USA-areas citis .
We have confirmed the proportional growth hypothesis using data from years
1990, 2000, and 2010 usa to find a small correlation of 0.016 between $u$ and
$\dot{u}$ (inset of Fig. 1c, bottom panel). Our first $N=150$ areas are in
Zipf’s regime (Fig. 3, bottom panel) and constitute the surface of the
statistical system at hand. We have verified the prevailing conservation rule,
finding $\sum_{i=1}^{150}\log(x_{i}/x_{150})=145.7$, 150.8, and 154.9 for each
year, respectively, close to the MaxEnt prediction of 150. We have also
verified that 10 of these areas (a 6.7%) that in 1990 pertained to the Zipf
regime are not characterized by it in 2010.
Figure 3: Top panel: walkers distribution for the second text-example $\mu=0$
(Zipf’s Law). Inset: equilibrium rank-distribution (slope$=1$ in black).
Bottom panel: rank-distribution of USA’s most-populated metropolitan areas for
1990, 2000, and 2010; Inset: logarithmic size $u$ vs. relative growth
$\dot{u}$ (linear fit in black).
## IV Conclusions
Different phenomena involving scale-invariance can be unified via Jaynes’
MaxEnt principle, provided that adequate dynamical information is suitably
incorporated into the variational process, in the manner here prescribed. This
allows one to perform a thermodynamic-like description of social systems,
related to that of an ideal gas.
ACKNOWLEDGMENT: This work was partially supported by ANR DYNHELIUM
(ANR-08-BLAN-0146-01) Toulouse, project PIP1177 of CONICET, (Argentina),
project FIS2008-00781/FIS (MICINN), and project FEDER (EU) (Spain, EU).
## References
* (1) M. Schroeder, Fractals, chaos and power laws (Freeman, NY, 1990).
* (2) M. E. J. Newman, Phys. Rev. E 64, (2001) 016131.
* (3) A. Hernando et al., A. Plastino, _Phys. Lett. A_ 374, 18 (2009).
* (4) A.-L. Barabasi, R. Albert, Rev. Mod. Phys. 74, 47 (2002).
* (5) T. Maillart, et al., Phys. Rev. Lett. 101, 218701 (2008).
* (6) B. Blasius, R. Tönjes, Phys. Rev. Lett. 103, 218701 (2009).
* (7) R. N. Costa Filho, et al., Phys. Rev. E 60, 1067 (1999).
* (8) A. A. Moreira, et al., Phys. Rev. E, 73, 065101(R) (2006).
* (9) L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, Phys. Rev. E 65, 017106 (2001).
* (10) M. Marsili, Y. C. Zhang, Phys. Rev. Lett. 80, 2741 (1998).
* (11) R. L. Axtell, Science 293, 1818 (2001).
* (12) G. K. Zipf, _Human Behavior and the Principle of Least Effort_ (Addison-Wesley, Cambridge, Mass., 1949).
* (13) Y. Chen, _Chaos Solitons Fract._ 45, 63 (2012).
* (14) B. C. Murtra, R.V. Solé, _Phys. Rev. E_ 82, 011102 (2010).
* (15) X. Gabaix, Quart. J. Econom. 114, 739 (1999).
* (16) A. Hernando, C. Vesperinas, A. Plastino, _Physica A_ 389, 490 (2010).
* (17) A. Hernando et al., _Eur. Phys. J. B_ 76, 87 (2010).
* (18) Weisstein, Eric W., _Benford’s Law_ from MathWorld.
* (19) F. Benford, _Proceed. Am. Phil. Soc._ 78, 551572 (1938).
* (20) Wikipedia http://en.wikipedia.org/wiki/Log-normal.
* (21) J. S. Huxley, (1932) Problems of relative growth (Methuen & Co. Lmtd., London, 1932).
* (22) S. M. Ross, Introduction to Probability Models, 9th edition (Academis Press, NY, 2007).
* (23) J. Eeckhout, _Ame. Eco. Rev._ 94, 1429 (2004).
* (24) H. Rozenfeld, et al., _Proc. Nat. Acad. Sci._ 105, 18702 (2008).
* (25) S. Ree, Phys. Rev. E 73 (2006) 026115.
* (26) W. J. Reed, B. D. Hughes, Phys. Rev. E 66 (2002) 067103.
* (27) S. Galam, J. Stat. Phys. 61, 943 (1990).
* (28) UrbanSim: http://www.urbansim.org.
* (29) C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys., 81, 591 (2009).
* (30) S. Fortunato, C. Castellano, Phys. Rev. Lett. 99, 138701 (2007).
* (31) M. Batty, _Cities and Complexity_ (MIT Press, Cambridge, MA, 2005).
* (32) S. Galam, Physica A 285, 66 (2000).
* (33) A. Katz, Principles of stat. mechanics (Freeman, San Francisco, 1967).
* (34) A. Katz, Principles of statistical mechanics (Freeman, San Francisco, 1967).
* (35) A. Plastino, _Physica A_ 344, 608 (2004).
* (36) C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, NY, 2009).
* (37) A. R. Plastino, A. Plastino, _Phys. Lett. A_ 177, 177 (1993).
* (38) A. Hernando, A. Plastino, A.R. Plastino, arXiv:1201.0889 and _Eur. Phys. J. B_ (2012) in Press.
* (39) H. Simon, _Biometrika_ 42, 425 (1955).
* (40) B.R. Frieden, B.H. Soffer, _Phys. Rev. E_ 52, 2274 (1995).
* (41) Census bureau, Government of USA (web).
* (42) G. Krings et al., _J. Stat. Mech_ , L07003 (2009).
* (43) Econom. Pol. Plann. $\&$ Stat. Office, Rep. of the Marshall Islands (web).
|
arxiv-papers
| 2012-04-11T11:39:32 |
2024-09-04T02:49:29.601559
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Hernando, A. Plastino",
"submitter": "Alberto Hernando",
"url": "https://arxiv.org/abs/1204.2420"
}
|
1204.2422
|
# Scale-invariance underlying the logistic equation and its social
applications
A. Hernando1, A. Plastino2, 3 1 Laboratoire Collisions, Agrégats, Réactivité,
IRSAMC, Université Paul Sabatier 118 Route de Narbonne 31062 - Toulouse CEDEX
09, France
2National University La Plata, IFLP-CCT-CONICET, C.C. 727, 1900 La Plata,
Argentina
3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,
Spain
###### Abstract
On the basis of dynamical principles we derive the Logistic Equation (LE),
widely employed (among multiple applications) in the simulation of population
growth, and demonstrate that scale-invariance and a mean-value constraint are
sufficient and necessary conditions for obtaining it. We also generalize the
LE to multi-component systems and show that the above dynamical mechanisms
underlie large number of scale-free processes. Examples are presented
regarding city-populations, diffusion in complex networks, and popularity of
technological products, all of them obeying the multi-component logistic
equation in an either stochastic or deterministic way. So as to assess the
predictability-power of our present formalism, we advance a prediction,
regarding the next 60 months, for the number of users of the three main web
browsers (Explorer, Firefox and Chrome) popularly referred as “Browser Wars”.
###### pacs:
89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k
## I Introduction
It is well-known that the logistic equation (LE) (sometimes called the
Verhulst model or logistic growth curve) is a phenomenological model of
population growth first published by Pierre Verhulst in the 1840’s. The model
is continuous in time, but a modification of the pertinent equation to a
discrete quadratic recurrence equation known as the logistic map is also
widely used. The continuous version of the logistic model for the evolution of
the population $x(t)$ is described by the differential equation
$\dot{x}(t)=kx(t)\left(1-\frac{x(t)}{N}\right),$ (1)
where $k$ is the Malthusian parameter (rate of maximum population growth) and
$N$ is the so-called carrying capacity (i.e., the maximum sustainable
population). The LE has as a solution
$x(t)=\frac{N}{1+(N/x(0)-1)e^{-kt}},$ (2)
i.e., the sigmoid function. The discrete version of the LE is the celebrated
logistic map. A typical application of the logistic equation refers to a
1838-model of population growth, originally due to Verhulst, in which the rate
of reproduction is proportional to both the existing population and the amount
of available resources, all else being equal. The Verhulst equation was
published after he had read Thomas Malthus’ An Essay on the Principle of
Population. Verhulst derived his logistic equation to describe the self-
limiting growth of a biological population. Today, proper referencing to the
logistic equation’s variegated applications to multiple fields of endeavor
would require pages and pages of citations. Of this immense body we just
mention, as a tiny sample, 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 .
Some ad-hoc LE-deductions have been previously published in a case-by-case
basis. One such demonstration is that provided by A. D. Zimm for companies or
firms zimm , that grow according to their commercial successfulness with a
classic linear Marshallian price-volume relationship. Other analytical
derivations are also found with some ad-hoc assumptions6 ; toston . Out
present goal is to describe an universal and generic dynamical mechanism that
leads in natural fashion to the logistic equation. The above cited derivations
do not have a dynamical character, as ours does. Our procedure is based on
* •
_scale symmetry_ and
* •
a _mean-value constraint_.
We will show that these are necessary an sufficient conditions for a LE-
derivation. The two items above are empirically known to be related to the LE
zimm ; 6 ; toston but they are used here for the first time as its pure
mathematical basis.
## II Derivation from dynamical principles
Consider an $n-$components system, each of them characterized by a population
$x_{i}$. Let us further assume that a multiplicative, time-evolution of
population takes place via free proportional growth, i.e.,
$\dot{x}_{i}(t)=k_{i}(t)x_{i}(t),$
where $k_{i}$ is the growth-ratio per unit-time for the $i$-th component.
Scale-symmetry is here apparent so that it proves convenient to transform
coordinates to $u_{i}=\log(x_{i})$ as in Refs. prl, ; epjb, . Thereby one is
led to the linear equation
$\dot{u}_{i}(t)=k_{i}(t),$
where the scale-invariance of $x$ is now a translational invariance in $u$. We
assume that the total population is finite, namely
$\sum_{i=1}^{n}x_{i}(t)=\sum_{i=1}^{n}e^{u_{i}(t)}=N,$
and that also $n$ remains constant, so that $\langle x\rangle=N/n$. For each
arbitrarily small time-interval $\Delta t$, the $u-$population grows freely
via
$u^{\prime}_{i}(t+\Delta t)=u(t)+\Delta tk_{i}(t),$
but conservation of the mean value $\langle e^{u}\rangle$ demands a global
“self-correction” process that should respect the original symmetry of the
system (translational for $u$). Accordingly,
$u_{i}(t+\Delta t)=u_{i}^{\prime}(t+\Delta t)+A,$
where $A$ is a value that guarantees fulfillment of
$\sum_{i=1}^{n}e^{u_{i}(t+\Delta t)}=N$. This is achieved if
$A=-\ln\left[\frac{1}{N}\sum_{j=1}^{n}\,e^{u_{j}^{\prime}(t+\Delta
t)}\right]=-\frac{\Delta t}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)},$
where a Taylor-expansion to first order is justified since $\Delta t$ is
arbitrarily small. One is then led to
$u_{i}(t+\Delta t)=u_{i}(t)+\Delta
t\left[k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)}\right].$
In the continuum-limit one finds
$\dot{u}_{i}(t)=k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)},$
that written in $x-$parlance leads to what we call the _multi-component
logistic equation_ (MCLE)
$\dot{x}_{i}(t)=x_{i}(t)\left(k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)x_{j}(t)\right).$
(3)
A matrix version of this equation is presented in the Appendix, together with
with some applicability perspectives. It is easy to check that the MCLE
retains the original scale-symmetry of $x$, and also exhibits translational
symmetry in $k$. The latter allows for some arbitrariness in the definition of
the $k_{i}$ rates in the fashion $k^{\prime}_{i}=k_{i}-k_{0}$. The same
results obtain for primed or unprimed $k$’s. If the $k_{i}$ are constant, or
exhibit just a slow dependence on $t$ (quasi-statics), the solution to the
MCLE is
$x_{i}(t)=\frac{Nx_{i}(0)e^{k_{i}t}}{\sum_{j=1}^{n}x_{j}(0)e^{k_{j}t}},$ (4)
where $x_{i}(0)$ are the initial conditions of the evolutive-process. The
logistic equation is directly derived from the MCLE Eq. (3) in a
straightforward fashion. We recover Eq. (1) by i) considering a bi-component
system ($n=2$), ii) defining $x(t)\equiv x_{1}(t)$, $k\equiv k_{1}-k_{2}$, and
iii) taking into account that $x_{2}(t)=N-x_{1}(t)$. Similarly, the sigmoid
function Eq. (2) is recovered from Eq. (4) with the same assumptions. The
second component acts here as a population-reservoir and the first component
becomes the only evolutive degree of freedom.
## III Possible physical regimes
According to the nature of the growth-ratios $k_{i}$, variegated kinds of
processes can be described by the MCLE.
1. 1.
Constant values or deterministic time-dependencies lead to mechanical systems
exhibiting deterministic evolution while
2. 2.
adding noise to the pertinent mean values gives rise to stochastic systems,
with interesting behaviors and applications.
Without aiming to be exhaustive, we consider here three different tableaus for
MCLE-applicability, according to the amount of ‘noise’ in the system: totally
stochastic (or thermodynamic regime), an intermediate level of randomness
(involving diffusive processes), and totally deterministic dynamics.
### III.1 Thermodynamic regime
Consider a multi-component case with dozens or hundreds of elements, and a
very high level of noise. Assume that each $k_{i}$ describes the derivative of
a Wiener process. We write $k_{i}(t)=\overline{k}_{i}+\sigma_{i}\xi(t)$, where
$\overline{k}_{i}$ is the time-average of $k_{i}$, $\sigma_{i}$ the standard
deviation measured in a certain interval $\Delta t$, and $\xi(t)$ an
independent normal-distributed random number. Defining
$\langle\sigma^{2}\rangle=\sum_{i=1}^{n}\sigma_{i}^{2}/n$, one asserts that we
thermalize the system if i)
$|\sigma_{i}-\sigma_{j}|^{2}/\langle\sigma^{2}\rangle\ll 1$ and ii)
$|\overline{k}_{i}-\overline{k}_{j}|^{2}/(\langle\sigma^{2}\rangle\Delta t)\ll
1$, $\forall i,j$, i.e., if all elements exhibit similar deviations and the
differences between mean values are much smaller than the noise. If $n$ is
large enough (as stated above, in the hundreds), dynamical equilibrium is
encountered after some finite time, meaning that the system is well-described
by the MaxEnt approach. The MaxEnt solution for scale-free systems describes
an equilibrium density $p_{X}(x)$ that follows the general form prl
$p_{X}(x)dx=\exp\left[-\sum_{a}f_{a}(x)\right]\frac{dx}{x},$
where $f_{a}(x)$ is the $a$-th constraint of the system. For a constraint in
the normalization ($n$ is invariant) we have $f_{\langle 1\rangle}(x)=\mu$,
and for one in the mean value we write $f_{\langle x\rangle}(x)=\lambda x$,
where $\mu$ and $\lambda$ are constants, univocally determined by the
fulfillment of each constraint. The density-distribution obeys in this case
the relation epjb
$p_{X}(x)dx=\frac{1}{\Gamma(0,\lambda x_{0})}\frac{e^{-\lambda
x}}{x}dx;\leavevmode\nobreak\ \leavevmode\nobreak\ x_{0}\leq x<\infty,$
where $\Gamma$ is the incomplete gamma function and $x_{0}$ is the smallest
allowed population for the elements (that can be 1). The rank-distribution is
then written as
$x=\frac{1}{\lambda}\Gamma^{-1}\left[0,\Gamma(0,\lambda x_{0})r/n\right],$ (5)
where $r$ is the (continuous) rank from 0 to $n$, and $\Gamma^{-1}(z)$ denotes
the inverse function of $\Gamma:$ $\Gamma(\Gamma^{-1}(z))=z$. The value of
$\lambda$ is obtained from the mean value
$\frac{e^{-\lambda x_{0}}}{\lambda\Gamma(0,\lambda x_{0})}=\frac{N}{n}.$ (6)
In order to test the above relationships we have carried out a calculation
that simulates human population dynamics. We also compare the result with
empirical data regarding city and place-populations, using as example data
from Ohio State (United States) ohio . We consider $n=1000$ random walkers
that mimic the population of cities, and set the total population to
$N=6000000$. We fix the minimum allowed size at the Dunbar’s number epjb
$x_{0}=150$ people (empirically known to be the usual size of small human
communities, and related to the maximum of social relationships/links that a
human can comfortably handle). We make the walkers to stochastically obey the
MCLE Eq. (3) so as to simulate migration patters between cities, with a
constant total population. We evolve the walkers in small intervals $\Delta
t=0.03$ and generate gaussian-distributed random numbers at each interval for
each $k_{i}$, using $\sigma_{i}=1$ for all $i$. Due to the translational
symmetry in $k$, we can set $\overline{k}_{i}=0$ for all $i$. To respect the
minimum size, a walker’ ‘move’ is not accepted if it leads to a value lower
than the Dunbar’s one. We start all walkers at $x_{i}=N/n$. After some
iterations we get the equilibrium distribution of Fig. 1, which perfectly fits
the MaxEnt prediction ($\lambda=0.00533$ humans-1). The available empirical
data covers years 2010, 2000 and 1990 with $n=1204$, $1015$, and $928$ cities
and places, respectively. We discard places with populations of under 150
people (67, 48 and 39 centers respectively). We also discard very large cities
(their potential economic correlation with the rest of the country compromises
the hypothesis of isolated systems). Excluding the four largest cities, the
total population is $N=6318170$, $6019960$, and $5477830$, respectively. We
have checked the proportional growth condition comparing $\log(x_{i})$ vs.
$|\dot{x}_{i}/x_{i}|^{2}$ (or equivalently, $u_{i}$ vs. $|\dot{u}_{i}|^{2}$).
No correlation between these two observables is expected for scale-invariance.
We have found a correlation coefficient of $0.0018$, as shown in Fig. 1, thus
confirming the proportional growth hypothesis (the same correlation
coefficient in the precedent simulation is $0.0027$). According to Eq. (6),
the predicted values of $\lambda$ are $0.00585$, $0.00507$, and $0.00513$,
respectively. A direct fit of the data to the form Eq. (5) yields
$\lambda=0.00636(2)$, $0.00502(3)$, and $0.00522(3)$, respectively, close
enough to the former values so as to confirm the MaxEnt prediction.
Figure 1: Top panel: behavior of random walkers obeying the stochastic MCLE
(blue dots) compared with the associated MaxEnt prediction (black line).
Inset: walkers’ squared-relative-growth $|\dot{u}_{i}|^{2}$ vs. logarithmic
size $u$. There is no correlation in view of the linear regression (red line).
Bottom panel: Rank-distributions for Ohio (red: year 2010; green: year 2000;
blue: year 1990) and rank-distribution of random walkers obeying the
stochastic MCLE (yellow) compared with the corresponding MaxEnt predictions
(black lines). Inset: same as upper panel with the empirical data. In view of
the regression line, no correlation is detected.
### III.2 Intermediate regime
Let us pass now to consideration of an intermediate stochastic regime in the
bi-component case. One party acts as a population-reservoir while the other
obeys Eq. (1). Defining the new variable
$y(t)=-\log(N/x(t)-1),$ (7)
the LE linearizes itself for $\dot{y}(t)=k$. The rate $k$ is here again the
derivative of a Wiener process $k(t)=\overline{k}+\sigma\xi$, but we now
include a drift obeying $|\overline{k}|^{2}>\sigma^{2}\Delta t$. Working with
an ensemble of independent walkers following this equation is equivalent to
handling Brownian motion in $y$-space. Consequently, the usual diffusion
equation for the density of walkers for $p_{Y}(y,t)$ ensues BM
$\partial_{t}p_{Y}(y,t)=-\overline{k}\partial_{y}p_{Y}(y,t)+D\partial_{y}^{2}p_{Y}(y,t),$
where $D$ is the diffusion-coefficient, related to $\sigma_{k}$ via
$\sigma_{k}=\sqrt{2D/\Delta t}$, and $\Delta t$ stands for the time-interval
used in the random-walk numerical simulation. The diffusion-equation’s kernel
is a Gaussian
$p_{Y}(y,t)dy=\frac{1}{\sqrt{4\pi
Dt}}e^{-\frac{(y-y_{0}-\overline{k}t)^{2}}{4Dt}}dy,$
with $y_{0}$ a reference-value. If at $t=0$ all walkers are located in
$x$-space at $x_{0}=N/(1+e^{-y_{0}})$, they will later evolve via
$p_{X}(x,t)dx=\frac{1}{\sqrt{4\pi
Dt}x(1-x/N)}e^{-\frac{(\log(N/x-1)-y_{0}+\overline{k}t)^{2}}{4Dt}}dx,$ (8)
since $p_{X}(x,t)dx=p_{Y}(y,t)dy$.
We have verified this prediction with a diffusion process taking place inside
a scale-free ideal network (SFIN) ournets , a random network with a degree-
distribution following the scale-free ideal gas one $p(c)\sim c^{-1}$, where
$c\leq c_{M}$. We have generated a SFIN of $N=20000$ nodes with a maximum
degree of $c_{M}=100$-connections and carried out a multitude of cluster-
growth processes ournets ; clusters . Diffusion in networks generally starts
i) by using a randomly chosen node as a seed, ii) with its first neighbors
being added to the cluster in the first iteration, iii) and the neighbors of
those “first" neighbors, afterwards (and so on). The process ends when all
nodes of the network belong to the cluster. The size of the cluster at each
iteration depends on the particular node selected as seed, via its position
inside the network. We depict in Fig. 2 the result of a large number of these
processes, indicating the size of the cluster at each iteration. All of them
start with $x=1$ and end up with $x=N$, but processes exhibit deviations at
intermediate steps. The associated median clearly follows a logistic
evolution, as shown in Fig. 2. Changing to the variable $y$ of Eq. (7) we find
a straight line with slope $\overline{k}=3.09$. The deviations can be
described by $y$-random walkers with $\sigma=0.7$ ($\Delta t=1$). The
statistics of the processes can be nicely described with Eq. (8) via
$x_{0}=1$, $D=0.245$, and with the above value of $\overline{k}$, as
illustrated by the comparison depicted in Fig. 2.
Figure 2: Diffusion inside a scale-free ideal network (dots) compared with the
analytical one provided by Eq. (8), derived from the logistic equation.
### III.3 Deterministic regime
We study now the deterministic evolution of a multi-component system with few
elements and very low level of (or without) noise. Assume that the growth-
ratios $k_{i}$ are now constants or represent a quasi-static evolution in
time. Assume further that we have data about the temporal population-evolution
but do not known the explicit values (or the tendency) of the $k_{i}$ rates.
These can be easily obtained from the solution of MCLE Eq. (3) taking
advantage of its property of translational symmetry in $k$. By arbitrarily
setting $k_{1}=0$, all the remaining values are obtained from the population
data thanks to the functions
$h_{i}(t)=\log\left[\frac{x_{i}(0)}{x_{i}(t)}\frac{x_{1}(t)}{x_{1}(0)}\right].$
(9)
If the growth-ratios are constants, $h_{i}(t)=k_{i}t$, the entire evolution-
path can be predicted (for any arbitrary time). If our functions $h_{i}(t)$
exhibit small time-fluctuations we can parameterize them, via a fitting
procedure, to any given analytical form. The desired solution is obtained by
substituting the arguments in the exponentials of Eq. (3) by these functions
$k_{i}t\rightarrow h_{i}(t)$. We have tested this last statement using data
regarding web-browsers’ statistics so as to study the past and future of the
(popularly called) _Browser Wars_ wikiBW . We considered the $n=3$ system
composed of Microsoft Explorer (E), Mozilla Firefox (F), and Google Chrome
(C). Our analysis of the popularity of each uses data from _w3schools_ w3 and
_statcounter_ sc (depicted in Fig. 3).
We take $N=100$% and choose the origin $t=0$ at March 2012 (as this
communication was being written). Setting $k_{E}=0$ we have applied Eq. (9) to
the data, finding a small dependence on time in both $k_{F}$ and $k_{C}$. We
show in Fig. 3 that the functions $h_{i}(t)$ can be nicely fitted to a simple
exponential form $h(t)=ae^{-bt}t+c$ (that can be regarded as a pure
exponential time-dependence of $k$ plus a correction on the initial value
$x(0)$ via $e^{c}$). Note that the small fluctuations of the data become more
apparent as we approach the reference point at $t=0$. However, our accuracy
remains sufficiently high for our purposes. We obtain
$\begin{array}[]{rl}h_{F}(t)=&0.0074(5)\exp[-0.021(1)t]t-0.008(14)\leavevmode\nobreak\
\mathrm{and}\\\ h_{C}(t)=&0.0579(24)\exp[-0.0097(10)t]t+0.104(26),\end{array}$
for _w3schools_ , and
$\begin{array}[]{rl}h_{F}(t)=&0.0022(5)\exp[-0.043(6)t]t+0.026(13)\leavevmode\nobreak\
\mathrm{and}\\\ h_{C}(t)=&0.055(2)\exp[-0.015(1)t]t+0.015(27),\end{array}$
for _statcounter_ , that are compared in the top panels of Fig. 3 to empirical
data. Our predictions for the popularity evolution are evaluated using Eq. (4)
and substituting $k_{E}t$, $k_{F}t$, and $k_{C}t$ by the above described
extrapolations of $h_{E}$, $h_{F}$, and $h_{C}$. A proper correction is
finally added in the later case to improve the fitting by using
$N^{\prime}=1.03N$. We depict in Fig 3 our monthly prediction for the next 5
years regarding browsers’ usage. In the two reported instances, Google Chrome
grows till coming ahead in the competition, saturating effects being
noticeable at 80% and 60%, respectively. Thus, according to our prediction,
Google Chrome will win the competition but it will not acquire such a dominant
position as the MS Explorer attained in the past.
Figure 3: Left panels: data from _w3schools_. Right panels: data from
_statcounter_. Top panels: increment rate of M Firefox (red squares) and G
Chrome (yellow triangles) defined as $(h(t)-c)/t$ relative to MS Explorer (see
text), compared with the analytical fit (solid lines). Bottom panels: relative
users of MS Explorer (blue circles), M Firefox (red squares) and G Chrome
(yellow triangles), compared with our prediction (lines).
## IV Conclusions
We have been able here to demonstrated that scale-invariance and a mean-value
constraint are sufficient and necessary conditions for obtaining the LE from
first dynamical principles. Then, the LE was generalized to multi-component
systems. This allowed us to show that these dynamical mechanisms underlie
interesting scale-free processes, which was illustrated with reference to
city-populations phenomena, diffusion in complex networks, and popularity of
Net Browsers.
## Appendix A Generalization of the multi-component logistic equation to a
matrix equation
We generalize here the formalism discussed above. If we define a new set of
variable $\chi_{i}=\sqrt{x_{i}/N}$, the total-population’s constraint can be
recast in the fashion $\sum_{i=1}^{n}\chi_{i}^{2}=1$. This condition becomes
formally equivalent to the conservation of the modulus of a vector
$\mathbf{\chi}=\\{\chi_{i}\\}_{i=1}^{n}$ in a $n$-dimensional space. We can
also generalize the definition of the growth-ratios $k_{i}$, promoting them to
a matrix $K_{ij}=k_{i}\delta_{ij}$, and write the MCLE as a matrix equation.
Using bra-ket notation $\mathbf{\chi}=|\chi\rangle$ one has
$\partial_{t}|\chi\rangle=\frac{1}{2}\left\\{K-\frac{\langle\chi|K|\chi\rangle}{\langle\chi|\chi\rangle}\right\\}|\chi\rangle.$
This equation is formally identical to that used in quantum physics to find
the ground state wave-function of a Hamiltonian (here, $K$). We are speaking
of the Imaginary Time Method (ITM) widely used in the literature ITM . The
mean-value term is also used to guarantee the conservation of the
normalization of $|\chi\rangle$ during the process. All our examples can be
regarded as particular applications of this formalism, calling attention to
the “functional" definition of our effective ‘Hamiltonians’ $K$. In our
examples $K$ has a diagonal form, which only indicates that we were working in
the eigenbasis of the dynamics defined by $K$. A generalized definition can
include off-diagonal terms as well, indicative of some kind of interaction
between populations. Density functional theories (DFT) also use the above
equation for many-body quantum systems DFT . A phenomenological Hamiltonian is
defined by means of a parametric functional form, than can also be a
functional of the own state-vector $\chi$. The associated parameters are
chosen so as to reproduce well-known empirical facts regarding the system of
interest. We expect that the bridge we have here built up between the MCLE and
the DFT can open new vistas with respect to the possibility of studying scale-
free systems. Such approach would take advantage of the huge experience
accumulated regarding DFT methods.
## References
* (1) S. Jannedy, R. Bod, J. Hay, Probabilistic Linguistics (MIT Press, Cambridge, Massachusetts, 2003).
* (2) N. A. Gershenfeld, The Nature of Mathematical Modeling (Cambridge University Press, Cambridge, UK, 1999).
* (3) S. E. Kingsland, Modeling nature: episodes in the history of population ecology (University of Chicago Press, Chicago, 1995).
* (4) E. W. Weisstein, Logistic Equation, from Wolfram Research Mathworld, Repository hosted at UIUC.
* (5) M. Fuentes, H. Larrondo, M. T. Martin, A. Plastino, O. Rosso, Phys. Rev. Lett. 99 (2007) 154102.
* (6) E. O. Wilson, W. H. Bossert. A Primer of Population Biology (Sinauer Associates, Sunderland, MA 01375, 1971).
* (7) J.P. Gabriel, F. Saucy, L. F. Bersier, Ecological Modelling 185 (2005) 147.
* (8) A. D. Zimm, Comp. & Math. Org. Theo., 11 (2005) 37.
* (9) T. Royama, Analytical Population Dynamics (Chapman and Hall, London, 1992).
* (10) E. T. Jaynes, (1957). Phys. Rev. 106, 620 (1957); 108, 171 (1957); IEEE Trans. Syst. Sci. & Cyb. 4, 227 (1968).
* (11) A. Katz, Principles of statistical mechanics: the information theory approach (W. H. Freeman, San Francisco, 1967).
* (12) A. Hernando, A. Plastino, _Variational Principle underlying Scale Invariant Social Systems_. Pre-print (2012).
* (13) A. Hernando, A.R. Plastino, A. Plastino, Eur. Phys. J. B., accepted for publication (2012).
* (14) Census bureau website, Government of USA, www.census.gov.
* (15) B. H. Lavenda, Nonequilibrium Statistical Thermodynamics (John Wiley & Sons Inc., 1985).
* (16) A. Hernando, D. Villuendas, C. Vesperinas, M. Abad, A. Plastino, Eur. Phys. J. B 76, 87 (2010).
* (17) C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod. Phys., 81, 591 (2009).
* (18) Wikipedia, _Browser wars_. http://en.wikipedia.org/wiki/Browser_wars.
* (19) http://www.w3schools.com/browsers/browsers_stats.asp
* (20) http://gs.statcounter.com/
* (21) V. S. Popov, Phys. of Atom. Nuclei, 68 (2008) 686-708.
* (22) Parr, R. G.; Yang, W.Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York. 1989)
|
arxiv-papers
| 2012-04-11T11:52:26 |
2024-09-04T02:49:29.606366
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Hernando, A. Plastino",
"submitter": "Alberto Hernando",
"url": "https://arxiv.org/abs/1204.2422"
}
|
1204.2472
|
# Stacked Central Configurations for Newtonian N+4-Body Problems 111This work
is supported by NSF of China and Youth found of Mianyang Normal University.
Furong Zhao1,2 and Shiqing Zhang1
1Department of Mathematics, Sichuan University, Chengdu, 610064,P.R.China
2Department of Mathematics and Computer Science, Mianyang Normal University,
Mianyang, Sichuan,621000,P.R.China
Abstract: In this paper,we study spatial central configurations where N bodies
are at the vertices of a regular N-gon $T$ and the other $4$ bodies are
symmetrically located on the straight line that is perpendicular to the plane
that contains $T$ and passes through the center of $T$.We study the necessary
conditions about masses for the bodies which can form a central configuration
and show the existence of central configurations for Newtonian N+4-body
problems.
Keywords : N+4-body problems, central configurations ,stacked central
configurations.
MSC: 34C15,34C25.
## 1 Introduction and Main Results
The Newtonian n-body problems([1],[23]) concern with the motions of n
particles with masses $m_{j}\in R^{+}$ and positions $q_{j}\in
R^{3}$$(j=1,2,...,n)$ , the motion is governed by Newton’s second law and the
Universal law:
$m_{j}\ddot{q}_{j}=\frac{\partial U(q)}{\partial{q}_{j}},$ (1.1)
where $q=(q_{1},q_{2},\cdots,q_{n})$ and $U(q)$ is Newtonian potential:
$U(q)=\sum_{1\leqslant j<k\leqslant n}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|},$ (1.2)
Consider the space
$X=\\{q=(q_{1},q_{2},\cdots,q_{n})\in R^{3n}:\sum_{j=1}^{n}m_{j}q_{j}=0\\},$
(1.3)
i.e,suppose that the center of mass is fixed at the origin of the space.
Because the potential is singular when two particles have same position, it is
natural to assume that the configuration avoids the collision set
$\triangle=\\{q=(q_{1},\cdots,q_{n}):q_{j}=q_{k}$ for some $k\neq j\\}$.The
set $X\backslash\triangle$ is called the configuration space.
Definition 1.1([20,24]):A configuration $q=(q_{1},q_{2},\cdots,q_{n})\in
X\backslash\triangle$ is called a central configuration if there exists a
constant $\lambda$ such that
$\sum_{j=1,j\neq
k}^{n}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant n.$ (1.4)
The value of constant $\lambda$ in (1.4) is uniquely determined by
$\lambda=\frac{U}{I},$ (1.5)
Where
$I=\sum_{k=1}^{n}m_{k}|q_{k}|^{2}.$ (1.6)
Since the general solution of the n-body problem can’t be given, great
importance has been attached to search for particular solutions from the very
beginning. A homographic solution is a configuration which is preserved for
all time. Central configurations and homographic solutions are linked by the
Laplace theorem (see [24]).Collaps orbits and parabolic orbits have relations
with the central configurations([17,19,20]).So finding central configurations
becomes very important. The main general open problem for the cental
configurations is due to Winter[24]and Smale[22]:Is the number of central
configurations finite for any choice of positive masses
$m_{1},...,m_{n}$?Hampton and Moeckel([6]) have proved this conjecture for
four any given positive masses.
For 5-body problem ,Hampton ([5])provided a new family of planar central
configurations,called stacked central configurations which has some proper
subset of three or more points forming a central configuration.
Ouyang ,Xie and Zhang([15]) studied pyramidal central configurations for
Newtonian N+1-body problems; Zhang and Zhou([25]) studied double pyramidal
central configurations for Newtonian N+2-body problems; Mello and
Fernandes([11]) studied new classes of spatial central configurations for the
N+3-body problem.
Based the above works,we study stacked central configuration for Newtonian
N+4-body problems. in N+4-body problems, for which $N$ bodies are at the
vertices the vertices of a regular polygon , the other $4$ bodies are
symmetrically located on the straight line that is perpendicular to the plane
that contains $T$ and passes through the center of $T$,the vertical line
passes the geometrical center of the regular polygon.(see Fig 1 for $N=4$).
Related assumptions will be interpreted more precisely in the following.
Without loss of generality we can take a coordinate system such that
$q_{j}=(\cos(\frac{(j-1)}{N}2\pi),\sin(\frac{(j-1)}{N}2\pi),0)$ where
$j=1,\cdots,N$;
$q_{N+1}=(0,0,r_{1})$, $q_{N+2}=(0,0,-r_{1})$, $q_{N+3}=(0,0,r_{2})$,
$q_{N+4}=(0,0,-r_{2})$.
$m_{6}$$m_{8}$$z$$y$$m_{1}$$x$$m_{2}$$m_{3}$$m_{4}$$m_{5}$$m_{7}$Fig.1
We have :
Theorem1.1:If $m_{N}+1=m_{N}+2$(or $m_{N}+3=m_{N}+4$) and
$m_{1},\cdots,m_{N+4}$ form a central configuration,then
(1):$\Sigma_{j=1}^{N}m_{j}q_{j}=0$
(2):$m_{N}+3=m_{N}+4$( $m_{N}+1=m_{N}+2$).
(3):$m_{1},\cdots,m_{N}$ also form a central configuration.
(4):$m_{1}=\cdots=m_{N}$.
Theorem1.2:Assume that
$m_{1}=\cdots=m_{N}=1$,$m_{N+1}=m_{N+2}=M_{1}$,$m_{N+3}=m_{N+4}=M_{1}$,then
there exist $\epsilon(r_{1},r_{2})>0$ , $\delta>0$ such that $\forall$
$(r_{1},r_{2})$$\in$
$\\{(r_{1},r_{2})|r_{2}>r_{1}>\delta,r_{2}-r_{1}<\epsilon(r_{1},r_{2})\\}$, we
have positive masses $M_{1}=M_{1}(r_{1},r_{2})$,$M_{2}=M_{2}(r_{1},r_{2})$ and
all the $N+4$ bodies form a central configuration.
Remark 1:$M_{1}=\frac{b_{1}a_{22}-b_{2}a_{12}}{a_{11}a_{22}-a_{12}a_{21}}$,
$M_{2}=\frac{b_{2}a_{11}-b_{1}a_{21}}{a_{11}a_{22}-a_{12}a_{21}}$.
Where :
$a_{11}=\frac{1}{4r_{1}^{3}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}$,$a_{12}=\frac{1}{|r_{1}+r_{2}|^{2}r_{1}}-\frac{1}{|r_{1}-r_{2}|^{2}r_{1}}-\frac{2}{|1+r_{2}^{2}|^{3/2}}$,
$a_{22}=\frac{1}{4r_{2}^{3}}-\frac{2}{|1+r_{2}^{2}|^{3/2}}$,
$a_{21}=\frac{1}{|r_{1}+r_{2}|^{2}r_{2}}+\frac{1}{|r_{1}-r_{2}|^{2}r_{2}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}$.
$b_{1}=\lambda^{*}-\frac{N}{|1+r_{1}^{2}|^{3/2}}$
$b_{2}=\lambda^{*}-\frac{N}{|1+r_{2}^{2}|^{3/2}}$
Remark 2:When $N=2$ ,the Theorem1.2 is related to the Theorem 1.3 in [8].
## 2 The Proofs of Theorems
### 2.1 Some Lemmas
We need some Lemmas.
If $n\times n$ matrix $A=(a_{ij})$ satisfies
$a_{i,j}=a_{i-1,j-1},1\leq i,j\leq n,$ (2.1)
where we assume $a_{i,0}=a_{i,n},a_{0,j}=a_{n,j}$,then A is called a circulant
matrix.
Lemma2.1(see [10]).Let $A=(a_{ij})$ be a circulant matrix,then the eigenvalues
$\lambda_{k}$ and eigenvectors $\overrightarrow{v}_{k}$ of A are
$\lambda_{k}(A)=\sum_{j=1}^{n}a_{1,j}\rho_{k-1}^{j-1}$ (2.2)
and
$\overrightarrow{v}_{k}=(\rho_{k-1},\rho_{k-1}^{2},\cdots,\rho_{k-1}^{n})^{T}$
(2.3)
where $\rho_{k}=e^{\sqrt{-1}\frac{2k\pi}{n}}$.
Lemma2.2([24]):For $n\geq 3$,and $m_{1}=m_{2}=\cdots=m_{n}$, if
$(m_{1},m_{2},\cdots,m_{n})$ locate at vertices of a regular polygon ,then
they form a central configuration.
From (1.4)and (1.3),notice that we have
$\begin{split}\sum_{j=1,j\neq
k}^{n}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k}=-\lambda m_{k}(q_{k}-q_{0})\\\ =-\lambda
m_{k}(q_{k}-\frac{\sum_{j=1}^{n}m_{j}q_{j}}{M})=-m_{k}\frac{\lambda}{M}\sum_{j=1}^{n}m_{j}(q_{k}-q_{j})\end{split}$
(2.4)
where $M=\sum_{j=1}^{n}m_{j}$,$q_{0}=\frac{\sum_{j=1}^{n}m_{j}q_{j}}{M}$,
So (1.4) is also equivalent to
$\sum_{j=1,j\neq
k}^{n}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})=0,k=1,2,3,\cdots,n.$
(2.5)
### 2.2 The Proofs of Theorem 1.1 and Theorem 1.2
#### 2.2.1 The Proof of Theorem 1.1
If $m_{1},\cdots,m_{N+4}$ form a central configuration,we have
$\sum_{j=1,j\neq
k}^{N+4}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})=0,k=1,\cdots,N+4.$
(2.6)
Notice that (2.6) can be also written as :
$\begin{split}\sum_{j=1,j\neq
k}^{N}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})+\\\
\sum_{j=1}^{4}m_{N+j}(\frac{1}{|q_{N+j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{N+j}-q_{k})=0,\\\
k=1,\cdots,N.\end{split}$ (2.7)
and
$\begin{split}\sum_{j=1}^{N}m_{j}(\frac{1}{|q_{j}-q_{N+l}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{N+l})+\\\
\sum_{j=1,j\neq
l}^{4}m_{N+j}(\frac{1}{|q_{N+j}-q_{N+l}|^{3}}-\frac{\lambda}{M})(q_{N+j}-q_{N+l})=0,\\\
l=1,2,3,4.\end{split}$ (2.8)
Now (2.8) is taken inner product with vectors $\overrightarrow{e}_{1}=(1,0,0)$
and $\overrightarrow{e}_{2}=(0,1,0)$,then we get:
$\begin{split}(\frac{1}{|q_{j}-q_{N+l}|^{3}}-\frac{\lambda}{M})\sum_{j=1}^{N}m_{j}\cos(\frac{(j-1)}{N}2\pi)=0\\\
(\frac{1}{|q_{j}-q_{N+l}|^{3}}-\frac{\lambda}{M})\sum_{j=1}^{N}m_{j}\sin(\frac{(j-1)}{N}2\pi)=0\\\
j=1,\cdots,N.\end{split}$ (2.9)
(2.9) can be also written as
$(\frac{1}{|q_{j}-q_{N+l}|^{3}}-\frac{\lambda}{M})\sum_{j=1}^{N}m_{j}q_{j}=0,j=1,\cdots,N.\\\
$ (2.10)
It is obvious that
$(\frac{1}{|q_{j}-q_{N+l}|^{3}}-\frac{\lambda}{M})=(\frac{1}{|q_{k}-q_{N+l}|^{3}}-\frac{\lambda}{M}),1\leq
k,j\leq N,$ (2.11)
we get
$\sum_{j=1}^{N}m_{j}q_{j}=0$ (2.12)
(2.8) is taken inner product with vector $\overrightarrow{e}_{3}=(0,0,1)$,then
we have:
$\begin{split}\sum_{j=1}^{N}m_{j}(\frac{1}{|1+r_{1}^{2}|^{3/2}}-\frac{\lambda}{M})r_{1}+0m_{N+1}+2r_{1}(\frac{1}{|2r_{1}|^{3}}-\frac{\lambda}{M})m_{N+2}+\\\
(r_{1}-r_{2})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})m_{N+3}+(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})m_{N+4}=0\end{split}$
(2.13)
$\begin{split}\sum_{j=1}^{N}m_{j}(\frac{1}{|1+r_{1}^{2}|^{3/2}}-\frac{\lambda}{M})r_{1}+2r_{1}(\frac{1}{|2r_{1}|^{3}}-\frac{\lambda}{M})m_{N+1}+0m_{N+2}+\\\
(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})m_{N+3}+(r_{1}-r_{2})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})m_{N+4}=0\end{split}$
(2.14)
$\begin{split}\sum_{j=1}^{N}m_{j}(\frac{1}{|1+r_{2}^{2}|^{3/2}}-\frac{\lambda}{M})r_{2}+(r_{2}-r_{1})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})m_{N+1}+\\\
(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})m_{N+2}+0m_{N+3}+2r_{2}(\frac{1}{|2r_{2}|^{3}}-\frac{\lambda}{M})m_{N+4}=0\end{split}$
(2.15)
$\begin{split}\sum_{j=1}^{N}m_{j}(\frac{1}{|1+r_{2}^{2}|^{3/2}}-\frac{\lambda}{M})r_{2}+(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})m_{N+1}+\\\
(r_{2}-r_{1})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})m_{N+2}+2r_{2}(\frac{1}{|2r_{2}|^{3}}-\frac{\lambda}{M})m_{N+3}+0m_{N+4}=0\end{split}$
(2.16)
By(2.13)and(2.14),we have:
$\begin{split}2r_{1}(\frac{1}{|2r_{1}|^{3}}-\frac{\lambda}{M}))(m_{N+1}-m_{N+2})+\\\
[(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})-(r_{1}-r_{2})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})](m_{N+3}-m_{N+4})=0\end{split}$
(2.17)
By(2.15)and(2.16),we have:
$\begin{split}[(r_{2}-r_{1})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})-(r_{1}+r_{2})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})])(m_{N+1}-m_{N+2})+\\\
2r_{2}(\frac{1}{|2r_{2}|^{3}}-\frac{\lambda}{M}))(m_{N+4}-m_{N+3})=0\end{split}$
(2.18)
We
define:$f(x)=x(\frac{1}{x^{3}}-\frac{\lambda}{M}))$,$\frac{df(x)}{dx}=-\frac{2}{x^{3}}-\frac{\lambda}{M}<0,$so
$f(r_{2}-r_{1})=(r_{2}-r_{1})(\frac{1}{|r_{1}-r_{2}|^{3}}-\frac{\lambda}{M})\neq(r_{2}+r_{1})(\frac{1}{|r_{1}+r_{2}|^{3}}-\frac{\lambda}{M})=f(r_{2}+r_{1})$
(2.19)
If $m_{N+1}=m_{N+2}$,by (2.17) and (2.18), we have $m_{N+3}=m_{N+4}$.
If $m_{N+3}=m_{N+4}$,by (2.17) and (2.18), we have $m_{N+1}=m_{N+2}$.
By $m_{N+1}=m_{N+2}$,$m_{N+3}=m_{N+4}$ and (2.7), we have
$\sum_{j=1,j\neq
k}^{N}m_{j}(\frac{1}{|q_{j}-q_{k}|^{3}}-\frac{\lambda}{M})(q_{j}-q_{k})=0,k=1,\cdots,N.$
(2.20)
Since $q_{j}$ locates on a unit circle,let
$q_{k}=\exp(\frac{2(k-1)\pi i}{N})$,$i=\sqrt{-1}$.By (2.20) we have
$\sum_{j=1,j\neq
k}^{N}m_{j}(\frac{1}{|q_{j-k}-1|^{3}}-\frac{\lambda}{M})(q_{j-k}-1)=0,k=1,\cdots,N.$
(2.21)
We define the $N\times N$ matrix $C=(c_{k,j})$,where
$c_{k,j}=0$,for
$j=k$;$c_{k,j}=(\frac{1}{|q_{j-k}-1|^{3}}-\frac{\lambda}{M})(q_{j-k}-1)$,for
$j\neq k$.
$C$ is circulant matrix since
$c_{k-1,j-1}=c_{k,j}=0$,for $j=k$;
$c_{k-1,j-1}=(\frac{1}{|q_{(j-1)-(k-1)}-1|^{3}}-\frac{\lambda}{M})(q_{(j-1)-(k-1)}-1)$
=$(\frac{1}{|q_{j-k}-1|^{3}}-\frac{\lambda}{M})(q_{j-k}-1)=c_{k,j}$for $j\neq
k$.
Then (2.21) can be written as
$CM^{*}=0$ (2.22)
where $M^{*}=(m_{1},\cdots,m_{N})^{T}$.
By Lemma2.1 and (2.22) we have
$m_{1}=m_{2}=\cdots=m_{N}.$ (2.23)
By Lemma2.2 and (2.23) we know that
$m_{1},\cdots,m_{N}$ also form a central configuration.
The proof of Theorem1.1 is completed.
#### 2.2.2 The Proof of Theorem 1.2
Notice that $(q_{1},\cdots,q_{N+4})$ is a central configuration if and only if
$\sum_{j=1,j\neq
k}^{N+4}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leqslant k\leqslant N+4.$ (2.24)
Since the symmetries,(2.24) is equivalent to
$\sum_{j=1,j\neq
k}^{N+4}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},k=1,N+1,N+3.$ (2.25)
That is
$\begin{split}-\lambda(1,0,0)=-\lambda^{*}(1,0,0)+\frac{(-1,0,r_{1})}{|1+r_{1}^{2}|^{3/2}}M_{1}+\frac{(-1,0,-r_{1})}{|1+r_{1}^{2}|^{3/2}}M_{1}\\\
+\frac{(-1,0,r_{2})}{|1+r_{2}^{2}|^{3/2}}M_{2}+\frac{(-1,0,-r_{2})}{|1+r_{2}^{2}|^{3/2}}M_{2}\end{split}$
(2.26)
$\begin{split}-\lambda(0,0,-r_{1})=\frac{N(0,0,r_{1})}{|1+r_{1}^{2}|^{3/2}}+\frac{(0,0,2r_{1})}{|2r_{1}|^{3}}M_{1}+\frac{(0,0,r_{1}+r_{2})}{|r_{1}+r_{2}|^{3}}M_{2}\\\
+\frac{(0,0,r_{1}-r_{2})}{|r_{1}-r_{2}|^{3}}M_{2}\end{split}$ (2.27)
$\begin{split}-\lambda(0,0,-r_{2})=\frac{N(0,0,r_{2})}{|1+r_{2}^{2}|^{3/2}}+\frac{(0,0,r_{1}+r_{2})}{|r_{1}+r_{2}|^{3}}M_{1}+\frac{(0,0,-r_{1}+r_{2})}{|r_{1}-r_{2}|^{3}}M_{1}\\\
+\frac{(0,0,2r_{2})}{|2r_{2}|^{3}}M_{2}\end{split}$ (2.28)
where $\lambda^{*}$ such that
$\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda^{*}m_{k}q_{k},k=1,\cdots,N.$
(2.26),(2.27)and (2.28) are equivalent to
$\lambda=\lambda^{*}+\frac{2}{|1+r_{1}^{2}|^{3/2}}M_{1}+\frac{2}{|1+r_{2}^{2}|^{3/2}}M_{2}$
(2.29)
$\begin{split}\lambda=\frac{N}{|1+r_{1}^{2}|^{3/2}}+\frac{1}{4r_{1}^{3}}M_{1}+(\frac{1}{|r_{1}+r_{2}|^{2}r_{1}}-\frac{1}{|r_{1}-r_{2}|^{2}r_{1}})M_{2}\end{split}$
(2.30)
$\begin{split}\lambda=\frac{N}{|1+r_{2}^{2}|^{3/2}}+(\frac{1}{|r_{1}+r_{2}|^{2}r_{2}}+\frac{1}{|r_{1}-r_{2}|^{2}r_{2}})M_{1}+\frac{1}{4r_{2}^{3}}M_{2}\end{split}$
(2.31)
(2.29),(2.30)and (2.31) are equivalent to
$\begin{split}(\frac{1}{4r_{1}^{3}}-\frac{2}{|1+r_{1}^{2}|^{3/2}})M_{1}+(\frac{1}{|r_{1}+r_{2}|^{2}r_{1}}-\frac{1}{|r_{1}-r_{2}|^{2}r_{1}}-\frac{2}{|1+r_{2}^{2}|^{3/2}})M_{2}\\\
=\lambda^{*}-\frac{N}{|1+r_{1}^{2}|^{3/2}}\end{split}$ (2.32)
$\begin{split}(\frac{1}{|r_{1}+r_{2}|^{2}r_{2}}+\frac{1}{|r_{1}-r_{2}|^{2}r_{2}}-\frac{2}{|1+r_{1}^{2}|^{3/2}})M_{1}+(\frac{1}{4r_{2}^{3}}-\frac{2}{|1+r_{2}^{2}|^{3/2}})M_{2}\\\
=\lambda^{*}-\frac{N}{|1+r_{2}^{2}|^{3/2}}\end{split}$ (2.33)
when $a_{11}a_{22}-a_{12}a_{21}\neq 0$,we have
$M_{1}=\frac{b_{1}a_{22}-b_{2}a_{12}}{a_{11}a_{22}-a_{12}a_{21}}$ (2.34)
$M_{2}=\frac{b_{2}a_{11}-b_{1}a_{21}}{a_{11}a_{22}-a_{12}a_{21}}$ (2.35)
Where :
$a_{11}=\frac{1}{4r_{1}^{3}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}$,$a_{12}=\frac{1}{|r_{1}+r_{2}|^{2}r_{1}}-\frac{1}{|r_{1}-r_{2}|^{2}r_{1}}-\frac{2}{|1+r_{2}^{2}|^{3/2}}$,
$a_{22}=\frac{1}{4r_{2}^{3}}-\frac{2}{|1+r_{2}^{2}|^{3/2}}$,
$a_{21}=\frac{1}{|r_{1}+r_{2}|^{2}r_{2}}+\frac{1}{|r_{1}-r_{2}|^{2}r_{2}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}$.
$b_{1}=\lambda^{*}-\frac{N}{|1+r_{1}^{2}|^{3/2}}$
$b_{2}=\lambda^{*}-\frac{N}{|1+r_{2}^{2}|^{3/2}}$
If
$a_{11}a_{22}-a_{12}a_{21}<0,b_{1}a_{22}-b_{2}a_{12}<0,b_{2}a_{11}-b_{1}a_{21}<0,$
(2.36)
then
$M_{1}>0,M_{2}>0$ (2.37)
Notice that (2.36) is equivalent to
$\frac{a_{11}}{a_{21}}<\frac{b_{1}}{b_{2}}<\frac{a_{12}}{a_{22}}$ (2.38)
Notice that
$\begin{split}\frac{a_{11}}{a_{21}}=\frac{\frac{1}{4r_{1}^{3}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}}{\frac{1}{|r_{1}+r_{2}|^{2}r_{2}}+\frac{1}{|r_{1}-r_{2}|^{2}r_{2}}-\frac{2}{|1+r_{1}^{2}|^{3/2}}}\\\
=\frac{|1+r_{1}^{2}|^{3/2}-8r_{1}^{3}}{4r_{1}^{3}\times|1+r_{1}^{2}|^{3/2}}\times\frac{(r_{2}^{2}-r_{1}^{2})^{2}r_{2}(1+r_{1}^{2})^{3/2}}{2(r_{1}^{2}+r_{2}^{2})(1+r_{1}^{2})^{3/2}-2(r_{2}^{2}-r_{1}^{2})^{2}}\\\
=\frac{|1+r_{1}^{2}|^{3/2}-8r_{1}^{3}}{4r_{1}^{3}}\times\frac{(r_{2}^{2}-r_{1}^{2})^{2}r_{2}}{2(r_{1}^{2}+r_{2}^{2})(1+r_{1}^{2})^{3/2}-2(r_{2}^{2}-r_{1}^{2})^{2}r_{2}}\\\
\end{split}$ (2.39)
Since
$\lim_{r_{1}\rightarrow+\infty}\frac{|1+r_{1}^{2}|^{3/2}-8r_{1}^{3}}{4r_{1}^{3}}=-\infty,$
and for $r_{2}=r_{1}$,$2(r_{2}^{2}-r_{1}^{2})^{2}r_{2}=0$. Then there exists
$\delta_{1}>0$,$\epsilon(r_{1},r_{2})>0$,such that for $r_{1}>\delta_{1},$ we
have
$\frac{|1+r_{1}^{2}|^{3/2}-8r_{1}^{3}}{4r_{1}^{3}}<0,$
and for $r_{2}-r_{1}<\epsilon(r_{1},r_{2}),$ we have
$2(r_{1}^{2}+r_{2}^{2})(1+r_{1}^{2})^{3/2}-2(r_{2}^{2}-r_{1}^{2})^{2}r_{2}>0$
So
$\begin{split}\frac{a_{11}}{a_{21}}<0,\forall(r_{1},r_{2})\in\\{(r_{1},r_{2})|r_{2}>r_{1}>\delta,r_{2}-r_{1}<\epsilon(r_{1},r_{2})\\}\end{split}$
(2.40)
We also notice that
$\lim_{r_{1}\rightarrow+\infty}\frac{b_{1}}{b_{2}}=1$ (2.41)
$\begin{split}\frac{a_{12}}{a_{22}}=\frac{4r_{2}(1+r_{2}^{2})^{3/2}+2(r_{2}^{2}-r_{1}^{2})^{2}}{(r_{2}^{2}-r_{1}^{2})^{2}(1+r_{2}^{2})^{3/2}}\times\frac{4r_{2}^{3}(1+r_{2}^{2})^{3/2}}{8r_{2}^{3}-(1+r_{2}^{3})^{3/2}}\\\
=\frac{16(1+\frac{1}{r_{2}^{2}})^{3/2}+8(1-(\frac{r_{1}}{r_{2}})^{2})}{(1-(\frac{r_{1}}{r_{2}})^{2})(8-(1+\frac{1}{r_{2}^{2}})^{3/2})}\end{split}$
There exists $\delta_{2}>0$ ,such that for $r_{2}>\delta_{2}>0,$ we have
$\frac{a_{12}}{a_{22}}>2$ (2.42)
By (2.40),(2.41)and(2.42), there exists
$\delta\geq\max{\\{\delta_{1},\delta_{2}\\}}$,such that for
$\delta<r_{1}<r_{2}$ and $r_{2}-r_{1}<\epsilon(r_{1},r_{2}),$ we have
$\frac{a_{11}}{a_{21}}<\frac{b_{1}}{b_{2}}<\frac{a_{12}}{a_{22}}$
The proof of Theorem1.2 is completed.
## References
* [1] Abraham R.and Marsden J.E.,Foundation of Mechanics,2nd edn,Benjamin,New York,1978.
* [2] Albouy A.,The symmetric central configurations of four equal masses, Amer.Math.Soc,Providence,RI,1996,131-135.
* [3] Albouy A., Fu Y. and Sun S.Z.,Symmetry of planar four-body convex central configurations,Proc.R.Soc.A 464(2008),1355-1365.
* [4] Diacu F.,The masses in a symmetric centered solution of the n-body problem,Proc.AMS 109(1990),1079-1085.
* [5] Hampton M.,Stacked central configurations:new examples in the planar five-body problem,Nonlinearity 18(2005),2299-2304.
* [6] Hampton M.,Moeckel R.,Finiteness of relative equilibria of the four-body problem.Invent.Math,163(2)(2006)289-312.
* [7] Lei J.and Santoprete M.,Rosette central configurations, degenerate central configurations and bifurcations,Celetial Mechanics and Dynamical Astronomy 94(2006),271-287.
* [8] Llibre J.,Mello L.F.,Triple and quadruple nested central configurations for the planar n-body problem,Physica D238(2009)563-571.
* [9] Long Y.,Admissible shapes of 4-body non-collinear relative equilibria,Adv.Nonlinear Stud.1(2003),495-509.
* [10] Marcus M.and Minc H.,A survey of matrix theory and matrix inqualities,Allyn and Bacon,Boston,1964.
* [11] Mello L.F.and Fernandes A.C.,New classes of spatial central configurations for the n+3-body problem ,Nonlinear Analysis:Real World Applications 12(2011)723-730.
* [12] Moeckel R.,On central configurations,Math.Z.205(1990)499-517.
* [13] Moeckel R.,Simo C.,Bifurcation of spatial central configurations from planar ones,SIAM J.Math.Anal.26(1995):978-998.
* [14] Moulton,F.R.,The straight line solutions of the n-body problem,Annals of Math, 12(1910),1-17.
* [15] Ouyang T.C.,Xie Z.F.and Zhang S.,Pyramidal central configurations and perverse solution,Electronic Journal of Differential Equations,106(2004),1-9.
* [16] Perko L.M.and Walter E.L.,Regular polygon solutions of N-body problem,Pro.AMS,94(1985),301-309.
* [17] Saari,D.G.,Singularities and collions of Newtonian gravitational systems,Arch.Rational Mech.49(1973),311-320.
* [18] Saari,D.G.,On the role and properties of N body central configurations,Celestial Mechanics and Dynamical Astronomy 21(1980),9-20.
* [19] Saari,D.G.,On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem,J.Diff.Eqs. 41(1981), 27-43.
* [20] Saari D.G.,Collisions,Rings and Other Newtonian N-body Problems,AMS Providence,Rhode Island.2005.
* [21] Shi J.,Xie Z.,Classification of four-body central configurations with three equal masses,J.Math.Anal.Appl.363(2010),512-524.
* [22] Smale S.,Mathematical problems for the next century,Math. Intelligenceer 20(1998)141-145.
* [23] Smale S.,Topology and mechanics II,Inv.Math,11(1970),45-64.
* [24] WintnerA.,The analytical foundations of celestial mechanics, Princeton Univ. Press, 1941.
* [25] Zhang S.Q.and Zhou Q.,Double pyramidal central configurations,Physics Letters A,281(2001),240-248.
|
arxiv-papers
| 2012-04-11T15:20:09 |
2024-09-04T02:49:29.612008
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Furong Zhao and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1204.2472"
}
|
1204.2513
|
# The $\\{-3\\}$-reconstruction and the $\\{-3\\}$-self duality of tournaments
###### Abstract
Let $T=(V,A)$ be a (finite) tournament and $k$ be a non negative integer. For
every subset $X$ of $V$ is associated the subtournament $T[X]=(X,A\cap(X\times
X))$ of $T$, induced by $X$. The dual tournament of $T$, denoted by
$T^{\ast}$, is the tournament obtained from $T$ by reversing all its arcs. The
tournament $T$ is self dual if it is isomorphic to its dual. $T$ is
$\\{-k\\}$-self dual if for each set $X$ of $k$ vertices, $T[V\setminus X]$ is
self dual. $T$ is strongly self dual if each of its induced subtournaments is
self dual. A subset $I$ of $V$ is an interval of $T$ if for $a,b\in I$ and for
$x\in V\setminus I$, $(a,x)\in A$ if and only if $(b,x)\in A$. For instance,
$\emptyset$, $V$ and $\\{x\\}$, where $x\in V$, are intervals of $T$ called
trivial intervals. $T$ is indecomposable if all its intervals are trivial;
otherwise, it is decomposable. A tournament $T^{\prime}$, on the set $V$, is
$\\{-k\\}$-hypomorphic to $T$ if for each set $X$ on $k$ vertices,
$T[V\setminus X]$ and $T^{\prime}[V\setminus X]$ are isomorphic. The
tournament $T$ is $\\{-k\\}$-reconstructible if each tournament
$\\{-k\\}$-hypomorphic to $T$ is isomorphic to it.
Suppose that $T$ is decomposable and $\mid V\mid\geq 9$. In this paper, we
begin by proving the equivalence between the $\\{-3\\}$-self duality and the
strong self duality of $T$. Then we characterize each tournament
$\\{-3\\}$-hypomorphic to $T$. As a consequence of this characterization, we
prove that if there is no interval $X$ of $T$ such that $T[X]$ is
indecomposable and $\mid V\setminus X\mid\leq 2$, then $T$ is
$\\{-3\\}$-reconstructible. Finally, we conclude by reducing the
$\\{-3\\}$-reconstruction problem to the indecomposable case (between a
tournament and its dual). In particular, we find and improve, in a less
complicated way, the results of [6] found by Y. Boudabbous and A. Boussaïri.
Mouna Achour(a), Youssef Boudabbous(a), Abderrahim Boussaïri(b)
(a) Département de Mathématiques, Faculté des Sciences de Sfax, Université de
Sfax, BP $1171$, $3038$ Sfax, Tunisie. Fax: (00216) 74.27.44.37
E-mail: mouna achour@yahoo.fr, youssef boudabbous@yahoo.fr (b) Faculté des
Sciences Aïn Chock, Département de Mathématiques et Informatique, Km 8 route
d’El Jadida, BP 5366 Maarif, Casablanca, Maroc. E-mail: aboussairi@hotmail.com
## 1 Introduction
### 1.1 Preliminaries on tournaments
A (finite) _tournament_ $T$ consists of a finite set $V$ of _vertices_ with a
prescribed collection $A$ of ordered pairs of distinct vertices, called the
set of _arcs_ of $T$, which satisfies: for $x,y\in V$ with $x\neq y$,
$(x,y)\in A$ if and only if $(y,x)\not\in A$. Such a tournament is denoted by
$(V,A)$. If $(x,y)$ is an arc of $T$, then we say that $x$ _dominates_ $y$
(symbolically $x\rightarrow y$). The _dual_ of the tournament $T$ is the
tournament $T^{\ast}=(V,A^{\ast})$ defined by: for all $x$, $y\in V$,
$(y,x)\in A^{\ast}$ if and only if $(x,y)\in A$. The tournament $T$ is
_transitive_ or a _linear order_ provided that for any $x,y,z\in V$, if
$(x,y)\in A$ and $(y,z)\in A$, then $(x,z)\in A$. For example, a total order
on a finite set $E$ can be identified to a transitive tournament with a vertex
set $E$ in the following way: for $x,y\in E$ with $x\neq y$, $x\rightarrow y$
if and only if $x<y$. The tournament corresponding to the usual order on
$\\{1,\ldots,n\\}$ (where $n\in{\mathbb{N}}^{\ast}$) is denoted by $O_{n}$. An
_almost transitive tournament_ is a tournament obtained from a transitive
tournament with at least three vertices by reversing the arc formed by its two
extremal vertices.
For every finite sets $E$ and $F$, we denote $E\subset F$ when $E$ is a subset
of $F$ and $\mid E\mid$ the cardinality of $E$.
Given a tournament $T=(V,A)$, for each subset $X$ of $V$ we associate the
_subtournament_ of $T$ induced by $X$, that is the tournament
$T[X]=(X,A\cap\\\ (X\times X))$. For convenience, the subtournament
$T[V\backslash X]$ is denoted by $T-X$, and by $T-x$ whenever $X=\\{x\\}$.
Let $T=(V,A)$ be a tournament, a subset $I$ of $V$ is an _interval_ of $T$ if
for every $x\in V\setminus I$, $x$ dominates or is dominated by all elements
of $I$. For instance, $\emptyset$, $V$ and $\\{x\\}$, where $x\in V$, are
intervals of $T$ called _trivial_ intervals. A tournament is _indecomposable_
if all its intervals are trivial; otherwise, it is _decomposable_. For
example, the tournament $C_{3}=(\\{1,2,3\\},\\{(1,2),(2,3),(3,1)\\})$ is
indecomposable, whereas, the tournaments
$C_{4}=(\\{1,2,3,4\\},\\{(1,2),(2,3),(3,4),(4,1),(3,1),(2,4)\\})$,
$\delta^{+}=(\\{1,2,3,4\\},\\\ \\{(1,2),(2,3),(3,1),(1,4),(2,4),(3,4)\\})$ and
$\delta^{-}=(\delta^{+})^{\ast}$ are decomposable.
Given two tournaments $T=(V,A)$ and $T^{\prime}=(V^{\prime},A^{\prime})$, an
_isomorphism_ from $T$ onto $T^{\prime}$ is a bijection $f$ from $V$ onto
$V^{\prime}$ satisfying: for any $x,y\in V$, $(x,y)\in A$ if and only if
$(f(x),f(y))\in A^{\prime}$. The tournaments $T$ and $T^{\prime}$ are
_isomorphic_ if there exists an isomorphism from one onto the other. This is
denoted by $T\sim T^{\prime}$. A tournament $T^{\prime}$ _embeds_ into a
tournament $T$ (or $T$ _embeds_ $T^{\prime}$ ), if $T^{\prime}$ is isomorphic
to a subtournament of $T$. A _$3$ -cycle_ (resp. _$4$ -cycle_) is a tournament
which is isomorphic to $C_{3}$ (resp. $C_{4}$). Moreover, a _positive diamond_
(resp. _negative diamond_) is a tournament that is isomorphic to $\delta^{+}$
(resp. $\delta^{-}$). A _diamond_ is a positive or a negative diamond. For
convenience, a set $X$ of vertices of a tournament $T$ is called diamond of
$T$ if $T[X]$ is a diamond.
### 1.2 Self duality and reconstruction
A tournament $T$ on a set $V$ is _self dual_ if $T$ and $T^{\ast}$ are
isomorphic, it’s _strongly self dual_ if for every subset $X$ of $V$, $T[X]$
and $T^{\ast}[X]$ are isomorphic. For each non negative integer $k$, the
tournament $T$ is _$(\leq k)$ -self dual_ whenever for every set $X$ of at
most $k$ vertices, the subtournament $T[X]$ is self dual. It is easy to see
that a transitive tournament or an almost transitive tournament is strongly
self dual. Conversely, Reid and Thomassen [23] was proved that a strongly self
dual tournament with at least $8$ vertices is transitive or almost transitive.
This result was used by K. B. Reid and C. Thomassen [23] in order to
characterize the pair of _hereditarily isomorphic_ tournaments, that is, the
pair of tournaments $T$ , $T^{\prime}$ on a set $V$ such that for every subset
$X$ of $V$, the subtournaments $T[X]$ and $T^{\prime}[X]$ are isomorphic. A
relaxed version of this notion is the following. Consider two tournaments $T$
and $T^{\prime}$ on the same vertex set $V$, with $\mid V\mid=n\geq 2$. Let
$k$ be an non negative integer $k$ with $k\leq n$. The tournaments $T$ and
$T^{\prime}$ are _$\\{k\\}$ -hypomorphic_, whenever for every set $X$ of $k$
vertices, the subtournaments $T[X]$ and $T^{\prime}[X]$ are isomorphic. If $T$
and $T^{\prime}$ are $\\{n-k\\}$-hypomorphic, we say that $T$ and $T^{\prime}$
are _$\\{-k\\}$ -hypomorphic_. Let $F$ be a set of integers. The tournaments
$T$ and $T^{\prime}$ are _$F$ -hypomorphic_, if for every $p\in F$, $T$ and
$T^{\prime}$ are $\\{p\\}$-hypomorphic, in particular, if
$F=\\{0,\ldots,k\\}$, we say that $T$ and $T^{\prime}$ are _$(\leq k)$
-hypomorphic_. For example, every two tournaments on the same vertex set with
at least $2$ elements are $(\leq 2)$-hypomorphic. The tournament $T$ is _$F$
-reconstructible_ provided that every tournament $F$-hypomorphic to $T$ is
isomorphic to $T$. This notion was introduced by R. Fraïssé [11] in 1970.
In 1972, G. Lopez ([15],[16]) showed that a tournament, with at least $6$
vertices, is $(\leq 6)$-reconstructible (see also [17]). It follows from a
”Combinatorial Lemma” of M. Pouzet (see Section $2$) that a tournament, with
at least $12$ vertices, is $\\{-6\\}$-reconstructible.
On the other hand, P. K. Stockmeyer [25] showed that the tournaments are not,
in general, $\\{-1\\}$-reconstructible, invalidating so the conjecture of Ulam
[26] for tournaments. Then, M. Pouzet ([1],[2]) proposed the
$\\{-k\\}$-reconstruction problem of tournaments. P. Ille [14] established
that a tournament with at least $11$ vertices is $\\{-5\\}$-reconstructible.
G. Lopez and C. Rauzy ([18],[19]) showed that a tournament with at least $10$
vertices is $\\{-4\\}$-reconstructible. The $\\{-k\\}$-reconstruction problem
of tournaments is still open for $k\in\\{2,3\\}$.
In 1995, Y. Boudabbous and A. Boussaïri [6] studied the $\\{-3\\}$ -recons-
truction of decomposable tournaments, for which they give a partial positive
answer.
In this paper, we find and improve, in a less complicated way, the results of
this study. In order to present our results, we need the Gallai’s
decomposition theorem [12] .
### 1.3 Gallai’s decomposition
Given a tournament $T=(V,A)$, we define on $V$ a binary relation $\mathcal{R}$
as follows: for all $x\in V$, $x\mathcal{R}x$ and for $x\neq y\in V$,
$x\mathcal{R}y$ if there exist two integers $n,\,m\geq 1$ and two sequences
$x_{0}=x,\ldots,x_{n}=y$ and $y_{0}=y,\ldots,y_{m}=x$ of vertices of $T$ such
that $x_{i}\longrightarrow x_{i+1}$ for $i=0,\ldots,n-1$ and
$y_{j}\longrightarrow y_{j+1}$ for $j=0,\ldots,m-1$. Clearly, $\mathcal{R}$ is
an equivalence relation on $V$. The equivalence classes of $\mathcal{R}$ are
called the _strongly connected components_ of $T$. A tournament is then
_strongly connected_ if it has at most one strongly connected component,
otherwise, it is _non-strongly connected_.
The next result is due to J. W. Moon [20].
###### Lemma 1
[20] Given a strongly connected tournament $T=(V,A)$ with $n\geq 3$ vertices,
for every integer $k\in\\{3,\ldots,n\\}$ and for every $x\in V$, there exists
a subset $X$ of $V$ such that $x\in X$, $\mid X\mid=k$ and the subtournament
$T[X]$ is strongly connected.
Let $T$ be a tournament on a set $V$. A partition $\mathcal{P}$ of $V$ is an
_interval partition_ of $T$ if all the elements of $\mathcal{P}$ are intervals
of $T$. It ensues that the elements of $\mathcal{P}$ may be considered as the
vertices of a new tournament, the _quotient_
$T/\mathcal{P}=(\mathcal{P},A/\mathcal{P})$ _of $T$ by $\mathcal{P}$_, defined
in the following way: for any $X\neq Y\in\mathcal{P}$, $(X,Y)\in
A/\mathcal{P}$ if $(x,y)\in A$, for $x\in X$ and $y\in Y$. On another hand, a
subset $X$ of $V$ is a _strong interval_ of $T$ provided that $X$ is an
interval of $T$ and for every interval $Y$ of $T$, if $X\cap Y\neq\emptyset$,
then $X\subset Y$ or $Y\subset X$. Here, for each tournament $T=(V,A)$ with
$\mid V\mid\geq 2$, $\mathcal{P}(T)$ denotes the family of maximal, strong
intervals of $T$, under the inclusion, amongst the strong intervals of $T$
distinct from $V$. Clearly, $\mathcal{P}(T)$ realizes an interval partition of
$T$.
Consider a tournament $H=(V,A)$. For every $x\in V$ is associated the
tournament $T_{x}=(V_{x},A_{x})$ such that the $V_{x}$’s are mutually
disjoint. The lexicographical sum of $T_{x}$’s over $H$ is the tournament $T$
denoted by $H(T_{x};x\in V)$ and defined on the union of $V_{x}$’s as follows:
given $u\in V_{x}$ and $v\in V_{y}$, where $x$, $y\in V$, $(u,v)$ is an arc of
$T$ if either $x=y$ and $(u,v)\in A_{x}$ or $x\neq y$ and $(x,y)\in A$. This
operation consists in fact to replace every vertex $x$ of $V$ by $T_{x}$ so
that $V_{x}$ becomes an interval; we say that the vertex $x$ is _dilated_ by
$T_{x}$. For example, an almost transitive tournament is obtained from a
$3$-cycle by dilating one of its vertices by a transitive tournament.
The Gallai’s decomposition theorem [12] consists in the following examination
of the quotient $T/\mathcal{P}(T)$.
###### Theorem 2
([9],[12]) Let $T$ be a tournament with at least two vertices.
1. 1.
The tournament $T$ is non-strongly connected if and only if $T/\mathcal{P}(T)$
is transitive. In addition, if $T$ is non-strongly connected, then
$\mathcal{P}(T)$ is the family of the strongly connected components of $T$.
2. 2.
The tournament $T$ is strongly connected if and only if $T/\mathcal{P}(T)$ is
indecomposable and $\mid\mathcal{P}(T)\mid\geq 3$.
We complete this subsection by the following notation.
###### Notation 3
For every tournament $T$ defined on a vertex set $V$ with at least two
elements, we associate the partition $\widetilde{\mathcal{P}}(T)$ of $V$
defined from $\mathcal{P}(T)$ as follows:
* •
If $T$ is strongly connected, $\widetilde{\mathcal{P}}(T)=\mathcal{P}(T)$.
* •
If $T$ is non-strongly connected, a subset $A$ of $V$ belongs to
$\widetilde{\mathcal{P}}(T)$ if and only if either $A\in\mathcal{P}(T)$ and
$\mid A\mid\geq 2$, or $A$ is a maximal union of consecutive vertices of the
transitive tournament $T/\mathcal{P}(T)$ which are singletons.
### 1.4 Statement of the results
In this paper, we begin by proving the following theorem (Section $3$). That
improves the result obtained by K. B. Reid and C. Thomassen [23]. (Note that,
the study of [23] is easly reduced to the decomposable case).
###### Theorem 4
A decomposable tournament which has at least $9$ vertices is $\\{-3\\}$-self
dual if and only if it is either transitive or almost transitive.
Then, we characterize each tournament $\\{-3\\}$-hypomorphic to a decomposable
tournament with at least $9$ vertices (Section $4$). For the statement, we
need additional notation.
###### Notation 5
Consider a set $P$ of non zero integers, an integer $n\geq 6$ and
$q\in\\{2,3\\}$. We denote by:
* •
$I_{n,P}$, the class of tournaments with $n$ vertices which are
indecomposable, not self dual and $\\{p\\}$-self dual for every $p\in P$;
these tournaments being considered up to an isomorphism.
* •
$C_{3}(I_{n,P})$ (resp. $O_{q}(I_{n,P})$), the class of tournaments with $n+2$
(resp. $n+q-1$) vertices obtained, from the $3$-cycle $C_{3}$ (resp. from the
transitive tournament $O_{q}$), by dilating one of its vertices by a
tournament belonging to the class $I_{n,P}$; these tournaments being
considered up to an isomorphism.
* •
For each integer $m\geq 8$, we denote by $\Omega_{m}$ the union
$C_{3}(I_{m-2,\\{-1,-2,-3\\}})\\\ \cup O_{3}(I_{m-2,\\{-1,-2,-3\\}})\cup
O_{2}(I_{m-1,\\{-2,-3\\}})$.
Here is the characterization.
###### Theorem 6
Consider a decomposable tournament $T$ with $n\geq 9$ vertices and let
$T^{\prime}$ be a tournament $\\{-3\\}$-hypomorphic to $T$. Then, we have:
1. 1.
$\widetilde{\mathcal{P}}(T^{\prime})=\widetilde{\mathcal{P}}(T)$ and one of
the following situations is achieved.
$(a)$
$T$ is almost transitive and $T^{\prime}\sim T$.
$(b)$
$T$ is not almost transitive,
$T/\widetilde{\mathcal{P}}(T)=T^{\prime}/\widetilde{\mathcal{P}}(T)$ and one
of the following situations is achieved.
$(i)$
$T\not\in\Omega_{n}$, for every $X\in\widetilde{\mathcal{P}}(T)$,
$T^{\prime}[X]\sim T[X]$ and $T^{\prime}\sim T$.
$(ii)$
$T\in\Omega_{n}$ and $T^{\prime}\not\sim T$.
2. 2.
$T$ is not $\\{-3\\}$-reconstructible if and only if $T\in\Omega_{n}$.
We deduce the following results.
###### Corollary 7
Let $T$ be a decomposable tournament on a set $V$, with $\mid V\mid\geq 9$. If
there is no interval $X$ of $T$ such that $T[X]$ is indecomposable and $\mid
V\setminus X\mid\leq 2$, then $T$ is $\\{-3\\}$-reconstructible.
###### Corollary 8
If there exists an integer $n_{0}\geq 7$ for which the indecomposable
tournaments with at least $n_{0}$ vertices are $\\{-3\\}$-reconstructible,
then the tournaments with at least $n_{0}+2$ vertices are
$\\{-3\\}$-reconstructible.
By Theorem 17 (Section $2$) and Corollary 8, we reduce the
$\\{-3\\}$-reconstruction problem of tournaments to the indecomposable case
between a tournament and its dual.
## 2 Preliminary results
In this section, we recall and prove some results which will be used in next
sections.
First, concerning the decomposability, we recall the following notation, lemma
and corollary.
###### Notation 9
Given a tournament $T=(V,A)$, for each subset $X$ of $V$, such that
$\mid\\!X\\!\mid\geq 3$ and $T[X]$ is indecomposable, we associate the
following subsets of $V\setminus X$.
* •
$Ext(X)$ is the set of $x\in V\setminus X$ such that $T[X\cup\\{x\\}]$ is
indecomposable.
* •
$[X]$ is the set of $\ x\in V\setminus X\,$ such that $X$ is an interval of
$T[X\cup\\{x\\}]$.
* •
For every $u\in X$, $X(u)$ is the set of $x\in V\setminus X\ $such that
$\\{u,x\\}$ is an interval of $T[X\cup\\{x\\}]$.
###### Lemma 10
[10] Let $T=(V,A)$ be a tournament and let $X$ be a subset of $V$ such that
$\mid\\!X\\!\mid\geq 3$ and $T[X]$ is indecomposable. The family $Ext(X),[X]$
and $X(u)$ where $u\in X$ constitutes a partition of $V\setminus X$. (Some
elements of this family can be empty).
The next result follows from Lemma 10.
###### Corollary 11
[10] Let $T=(V,A)$ be an indecomposable tournament. If $X$ is a subset of $V$,
such that $\mid\\!X\\!\mid\geq 3$, $\mid\\!V\setminus X\\!\mid\geq 2$ and
$T[X]$ is indecomposable, then there are distinct $x,y\in V\setminus X$ such
that $T[X\cup\\{x,y\\}]$ is indecomposable.
Second, we consider the following remark and notation.
###### Remark 12
i)
Up to an isomorphism, there exist four tournaments with four vertices:
$O_{4}$, $C_{4}$, $\delta^{+}$ and $\delta^{-}$. In addition, both of them is
decomposable.
ii)
Given two $\\{3\\}$-hypomorphic tournaments $T$ and $T^{\prime}$ with the same
vertex set $V$ with $\mid V\mid\geq 4$, $T$ and $T^{\prime}$ are $(\leq
4)$-hypomorphic if and only if for every subset $X$ of $V$, if $T[X]$ or
$T^{\prime}[X]$ is a diamond, then $T^{\prime}[X]\sim T[X]$.
iii)
Consider two $\\{3\\}$-hypomorphic tournaments $T$ and $T^{\prime}$. If $T$ is
without diamonds then $T^{\prime}$ and $T$ are $\\{4\\}$-hypomorphic.
###### Notation 13
Given a tournament $T=(V,A)$, a subset $F$ of $V$ and a tournament $H$, we
denote by $S(T,H;F)=\\{X\subset V;\,F\subset X$ and $T[X]\sim H\\}$ and
$n(T,H;F)=\mid S(T,H;F)\mid$.
Then, we recall the following ”Combinatorial Lemma” of M. Pouzet [21].
###### Lemma 14
[21] Let $p$, $r$ be two positive integers, $E$ be a set of at least $p+r$
elements and $U$, $U^{\prime}$ be two sets of subsets of $p$ elements of $E$.
If for each subset $Q$ of $E$ with $\mid Q\mid=p+r$, the number of the
elements of $U$ which are contained in $Q$ is equal to the number of the
elements of $U^{\prime}$ which are contained in $Q$, then for every finite
subsets $P^{\prime}$ and $Q^{\prime}$ of $E$, such that $P^{\prime}$ is
contained in $Q^{\prime}$ and $Q^{\prime}\setminus P^{\prime}$ has at least
$p+r$ elements, the number of elements of $U$ containing $P^{\prime}$ and
included in $Q^{\prime}$ is equal to the number of elements of $U^{\prime}$
containing $P^{\prime}$ and included in $Q^{\prime}$. In particular, if $E$
has at least $2p+r$ elements, then $U$ and $U^{\prime}$ are equal.
From the Combinatorial Lemma, we have the next corollaries.
###### Corollary 15
[22] Consider positive integers $n$, $p$, $h$ such that $p<n$ and $h\leq n-p$,
a tournament $H$ with $h$ vertices and two tournaments $T$ and $T^{\prime}$
defined on the same vertex set $V$ with $\mid V\mid=n$. If $T$ and
$T^{\prime}$ are $\\{-p\\}$-hypomorphic then for each subset $X$ of $V$ of at
most $p$ elements, $n(T^{\prime},H;X)=n(T,H;X)$.
###### Corollary 16
[21] Consider two tournaments $T=(V,A)$ and $T^{\prime}=(V,A^{\prime})$ and an
integer $p$ such that $0<p<\mid V\mid$. If $T$ and $T^{\prime}$ are
$\\{p\\}$-hypomorphic, then $T$ and $T^{\prime}$ are $\\{q\\}$-hypomorphic for
each $q\in\\{1,\ldots,min(p,\mid V\mid-p)\\}$.
In particular, if $\mid V\mid\geq 6$ and $T$ and $T^{\prime}$ are
$\\{-3\\}$-hypomorphic, then $T$ and $T^{\prime}$ are $(\leq 3)$-hypomorphic.
Now, recall the following theorem, called the ”Inversion Theorem”, which was
obtained by A. Boussaïri, P. Ille, G. Lopez and S. Thomassé ([8],[9]).
###### Theorem 17
([8],[9]) Given an indecomposable tournament $T$ with at least $3$ vertices,
the only tournaments which are $\\{3\\}$-hypomorphic to $T$ are $T$ and
$T^{\ast}$.
The following corollary is a consequence of Theorem 17.
###### Corollary 18
[9] Let $T$ and $T^{\prime}$ be two $\\{3\\}$-hypomorphic tournaments with at
least $3$ verices.
i)
$\mathcal{P}(T)=\mathcal{P}(T^{\prime})$.
ii)
$T$ is strongly connected (resp. indecomposable) if and only if $T^{\prime}$
is strongly connected (resp. indecomposable).
iii)
If $T$ is strongly connected, then the quotients $T^{\prime}/\mathcal{P}(T)$
and $T/\mathcal{P}(T)$ are either equal or dual.
From this corollary, we obtain the following remark.
###### Remark 19
Let $T$ and $T^{\prime}$ be two $\\{3\\}$-hypomorphic tournaments on a set $V$
with $\mid V\mid\geq 3$, and $I$ be a subset of $V$ such that
$T\left[I\right]$ is strongly connected. If $I$ is an interval of $T$, then
$I$ is an interval of $T^{\prime}$.
Given a tournament $T$ on a set $V$ and a subset $I$ of $V$, we denote by
$I^{+}_{T}$ (resp. $I^{-}_{T}$ ) the set of vertices $x\in V\setminus I$ such
that $I\longrightarrow x$ (resp. $x\longrightarrow I$).
We complete this section by the following result.
###### Proposition 20
Let $T$ and $T^{\prime}$ be two tournaments on a set $V$ with
$\left|V\right|\geq 6$, and $I$ be an interval of $T$ such that $\mid
I\mid\geq 3$ and $T\left[I\right]$ is indecomposable.
i)
If $T$ and $T^{\prime}$ are $\\{3,-2\\}$-hypomorphic (resp.
$\\{-3\\}$-hypomorphic) and $\left|V\setminus I\right|\geq 2$ (resp.
$\left|V\setminus I\right|\geq 3$), then $T\left[I\right]\backsim
T^{\prime}\left[I\right]$.
ii)
If $T$ and $T^{\prime}$ are $\\{3,-2\\}$-hypomorphic (resp.
$\\{-3\\}$-hypomorphic) and $\left|V\setminus I\right|\geq 3$ (resp.
$\left|V\setminus I\right|\geq 4$), then
$\left|I^{+}_{T}\right|=\left|I^{+}_{T^{\prime}}\right|$ and
$\left|I^{-}_{T}\right|=\left|I^{-}_{T^{\prime}}\right|$.
iii)
If $T$ and $T^{\prime}$ are $\\{-3\\}$-hypomorphic and $\left|V\setminus
I\right|\geq 4$, then $T\left[I\right]\backsim T^{\prime}\left[I\right]$,
$I^{+}_{T}=I^{+}_{T^{\prime}}$ and $I^{-}_{T}=I^{-}_{T^{\prime}}$.
Proof.
First, note that if $T$ and $T^{\prime}$ are $\\{-3\\}$-hypomorphic, then $T$
and $T^{\prime}$ are $\\{3\\}$-hypomorphic (by Corollary 16). Moreover, as
$T[I]$ is indecomposable (in particular, it is strongly connected because
$\mid I\mid\geq 3$) and $T[I]$ and $T^{\prime}[I]$ are $\\{3\\}$-hypomorphic,
then by Corollary 18, $T^{\prime}[I]$ is indecomposable and by Remark 19, $I$
is an interval of $T^{\prime}$.
i)
Let $a\neq b\in I$ and $J$ be a subset of $V$ containing $\\{a,b\\}$ such that
$T[J]$ is indecomposable and $\mid I\mid=\mid J\mid$ and denote by $H$ the
subtournament $T[I]$. As $I$ is an interval of $T$, then $I\cap J$ is an
interval of $T[J]$. However, $\\{a,b\\}\subset I\cap J$ and $T[J]$ is
indecomposable, then $I\cap J=J$ and hence $I=J$ (because $\mid I\mid=\mid
J\mid$). Thus, $I$ is the only subset $J$ of $V$ containing $\\{a,b\\}$ such
that $T[J]$ is indecomposable and $\mid J\mid=\mid I\mid$. In particular
$S(T,H;\\{a,b\\})=\\{I\\}$ and then $n(T,H;\\{a,b\\})=1$.
By interchanging $T$ and $T^{\prime}$ in the previous result, $I$ is the only
subset $J$ of $V$ containing $\\{a,b\\}$ such that $T^{\prime}[J]$ is
indecomposable and $\mid I\mid=$ $\mid J\mid$. In particular,
$S(T^{\prime},H;\\{a,b\\})\subset\\{I\\}$. Lastly, as $T$ and $T^{\prime}$ are
$\\{-2\\}$-hypomorphic (resp. $\\{-3\\}$-hypomorphic), $\mid V\mid\geq 6$ and
$\mid I\mid\leq$
$\mid V\mid-2$ (resp. $\mid I\mid\leq\mid V\mid-3$), then by Corollary 15,
$n(T,H;\\{a,b\\})=n(T^{\prime},H;\\{a,b\\})$. As $n(T,H;\\{a,b\\})=1$, then
$S(T^{\prime},H;\\{a,b\\})\neq\emptyset$ and so
$S(T^{\prime},H;\\{a,b\\})=\\{I\\}$. Consequently, $T^{\prime}[I]\sim T[I]$.
ii)
Let $a\neq b\in I$ and denote by $H$ a tournament with vertex set
$I\cup\\{u\\}$ (where $u\notin I$) such that $H[I]=T[I]$ and $I\longrightarrow
u$. Clearly, if $I^{+}_{T}\neq\emptyset$, then for each $x\in I^{+}_{T}$,
$I\cup\\{x\\}\in S(T,H;\\{a,b\\})$. Conversely, assume that
$S(T,H;\\{a,b\\})\neq\emptyset$ and consider an element $J$ of
$S(T,H;\\{a,b\\})$. Let $f$ be an isomorphism from $H$ to $T[J]$ and let
$\alpha=f(u)$. As $I$ is the unique non trivial interval of $H$ and
$f(I)=J\setminus\\{\alpha\\}$, then $J\setminus\\{\alpha\\}$ is the unique non
trivial interval of $T[J]$. However, $I\cap J$ is an interval of $T[J]$ and
$\\{a,b\\}\subset I\cap J$, then $I\cap J=J\setminus\\{\alpha\\}$ and hence
$J\setminus\\{\alpha\\}\subset I$. So, $J\setminus\\{\alpha\\}=I$ (because
$\mid I\mid=\mid J\setminus\\{\alpha\\}\mid$). Thus, $J=I\cup\\{\alpha\\}$ and
$\alpha\in I^{+}_{T}$.
We conclude that $S(T,H;\\{a,b\\})=\\{I\cup\\{x\\};\,x\in I^{+}_{T}\\}$ and
hence,
$n(T,H;\\{a,b\\})=\mid I^{+}_{T}\mid$.
As $T^{\prime}[I]\sim T[I]$ (by i)), then by interchanging $T$ and
$T^{\prime}$ in the previous result, we deduce that
$n(T^{\prime},H;\\{a,b\\})=$ $\mid I^{+}_{T^{\prime}}\mid$. Lastly, as $T$ and
$T^{\prime}$ are $\\{-2\\}$-hypomorphic (resp. $\\{-3\\}$-hypomorphic) and
$\mid I\cup\\{u\\}\mid\leq\mid V\mid-2$ (resp. $\mid I\cup\\{u\\}\mid\leq\mid
V\mid-3$), then by Corollary 15, $n(T,H;\\{a,b\\})=n(T^{\prime},H;\\{a,b\\})$.
Therefore, $\mid I^{+}_{T}\mid=\mid I^{+}_{T^{\prime}}\mid$ and hence, $\mid
I^{-}_{T}\mid=\mid I^{-}_{T^{\prime}}\mid$.
iii)
By ii), we have $\left|I^{+}_{T}\right|=\left|I^{+}_{T^{\prime}}\right|$.
Assume now that $I^{+}_{T}\setminus I^{+}_{T^{\prime}}\neq\emptyset$ and let
$x\in I^{+}_{T}\setminus I^{+}_{T^{\prime}}$. As $I$ is an interval of
$T^{\prime}$, then $x\in I^{-}_{T^{\prime}}$. The tournaments $T-x$ and
$T^{\prime}-x$ are $\\{3,-2\\}$-hypomorphic, $\mid V\setminus\\{x\\}\mid\geq
6$, and $\left|(V\setminus\left\\{x\right\\})\setminus I\right|\geq 3$, then
by ii), $\left|I^{+}_{T-x}\right|=\left|I^{+}_{T^{\prime}-x}\right|$. However,
$\left|I^{+}_{T-x}\right|=\left|I^{+}_{T}\right|-1\text{ and
}\left|I^{+}_{T^{\prime}-x}\right|=\left|I^{+}_{T^{\prime}}\right|$;
contradiction. It follows that $I^{+}_{T}\subset I^{+}_{T^{\prime}}$ and then
$I^{+}_{T}=I^{+}_{T^{\prime}}$. By duality, we obtain
$I^{-}_{T}=I^{-}_{T^{\prime}}$.
$\Box$
## 3 Proof of Theorem 4
For the tournaments without diamonds, H. Bouchaala and Y. Boudabbous [5]
established the following result.
###### Proposition 21
[5] Given a tournament $T$ without diamonds and which has at least $9$
vertices, $T$ is $\\{-3\\}$-self dual if and only if it is strongly self dual.
We present now some results concerning tournaments embedding a diamond.
###### Remark 22
A diamond $\delta$ has a unique non trivial interval $I$. Moreover,
$\delta[I]$ is a $3$-cycle.
###### Lemma 23
[4] If $T=(V,A)$ is a tournament embedding a diamond, then each vertex of $T$
is contained in at least one diamond of $T$.
The following proposition was obtained by M. Sghiar in 2004. This result plays
an important role in the proof of Theorem 4.
###### Proposition 24
[24] Let $T$ be a tournament, with at least $8$ vertices, embedding a diamond.
If $T$ has an interval of cardinality $2$, then $T$ is not $\\{-3\\}$-self
dual.
For the proof of this proposition, we need some definitions and notations.
Given a tournament $T=(V,A)$, if $X=\\{a,b,c,d\\}$ is a subset of $V$ such
that $T[X]$ is a diamond and $T[\\{a,b,c\\}]$ is a $3$-cycle, we say that $X$
is a diamond of $T$ of _center_ $d$ and cycle $\\{a,b,c\\}$. Let $x\neq y\in
V$, we denote:
* •
$\delta_{T,\\{x,y\\}}^{+}$ (resp. $\delta_{T,\\{x,y\\}}^{-}$), the number of
positive (resp. negative) diamonds of $T$ whose cycle contains $\\{x,y\\}$.
* •
$C_{T,\\{x,y\\}}$, the set of elements $w$ of $V$ such that $T[\\{x,y,w\\}]$
is a $3$-cycle.
* •
$\delta_{T,\\{x,y,w\\}}^{+}$ (resp. $\delta_{T,\\{x,y,w\\}}^{-}$), the number
of positive (resp. negative) diamonds of $T$ whose cycle is $\\{x,y,w\\}$,
where $w$ is an element of $C_{T,\\{x,y\\}}$.
* •
$D_{T,\\{x\\}}^{+}(y)$ (resp. $D_{T,\\{x\\}}^{-}(y)$), the number of positive
(resp. negative) diamonds of $T$ passing by $x$ and whose center is $y$.
* •
$D_{T,\\{x,y\\}}^{+}$ (resp. $D_{T,\\{x,y\\}}^{-}$), the number of positive
(resp. negative) diamonds of $T$ passing by $x$ and $y$.
* •
$\delta_{T}^{+}(x)$ (resp. $\delta_{T}^{-}(x)$), the number of positive (resp.
negative) diamonds of $T$ whose center is $x$.
###### Lemma 25
[24] Let $T=(V,A)$ be a $\\{-3\\}$-self dual tournament with at least $7$
vertices. If $T$ embeds a diamond, then every vertex of $T$ is the center of
at least one diamond of $T$.
Proof. Suppose for a contradiction that there exists a vertex $x$ of $T$ such
that $\delta^{+}_{T}(x)=\delta^{-}_{T}(x)=0$. From Lemma 23, there exists a
diamond $\sigma$ of $T$ containing $x$. By interchanging $T$ and $T^{\ast}$,
we can assume that $\sigma$ is a negative diamond. Let $y$ be the center of
$\sigma$. So, $D^{-}_{T,\\{x\\}}(y)\neq 0$ and $D^{+}_{T,\\{y\\}}(x)=0$. If
$C_{T,\\{x,y\\}}=\emptyset$, then
$\delta^{+}_{T,\\{x,y\\}}=\delta^{-}_{T,\\{x,y\\}}=0$. If
$C_{T,\\{x,y\\}}\neq\emptyset$, then pick $w\in C_{T,\\{x,y\\}}$ and let
$X=\\{x,y,w\\}$. As $T$ and $T^{\ast}$ are $\\{-3\\}$-hypomorphic and $\mid
V\mid\geq 7$, then from Corollary 15,
$n(T,\delta^{+};X)=n(T^{\ast},\delta^{+};X)$. So,
$n(T,\delta^{+};X)=n(T,\delta^{-};X)$ and hence
$\delta^{+}_{T,X}=\delta^{-}_{T,X}$. However,
$\delta^{+}_{T,\\{x,y\\}}=\sum\limits_{w\in
C_{T,\\{x,y\\}}}\delta_{T,\\{x,y,w\\}}^{+}$ and
$\delta_{T,\\{x,y\\}}^{-}=\sum\limits_{w\in
C_{T,\\{x,y\\}}}\delta_{T,\\{x,y,w\\}}^{-}$. Thus,
$\delta^{+}_{T,\\{x,y\\}}=\delta^{-}_{T,\\{x,y\\}}$. On the other hand, we
have
$D^{-}_{T,\\{x,y\\}}=n(T,\delta^{-};\\{x,y\\})=n(T^{\ast},\delta^{+};\\{x,y\\})$,
$D^{+}_{T,\\{x,y\\}}=n(T,\delta^{+};\\{x,y\\})$ and from Corollary 15,
$n(T,\delta^{+};\\{x,y\\})=n(T^{\ast},\delta^{+};\\{x,y\\})$, hence
$D^{+}_{T,\\{x,y\\}}=D^{-}_{T,\\{x,y\\}}$. However,
$D^{+}_{T,\\{x,y\\}}=D^{+}_{T,\\{y\\}}(x)+\delta^{+}_{T,\\{x,y\\}}$,
$D^{-}_{T,\\{x,y\\}}=D^{-}_{T,\\{x\\}}(y)+\delta^{-}_{T,\\{x,y\\}}$ and
$\delta^{+}_{T,\\{x,y\\}}=\delta^{-}_{T,\\{x,y\\}}$, thus,
$D^{+}_{T,\\{y\\}}(x)=D^{-}_{T,\\{x\\}}(y)$; which contradicts the fact that
$D^{-}_{T,\\{x\\}}(y)\neq 0$ and $D^{+}_{T,\\{y\\}}(x)=0$. $\Box$
###### Lemma 26
[24] Consider a $\\{-2\\}$-self dual (resp. $\\{-3\\}$-self dual) tournament
$T=(V,A)$ with at least $7$ (resp. $8$) vertices and two distinct vertices
$a,b$ of $T$. If $\\{a,b\\}$ is an interval of $T$ then
$\delta^{+}_{T}(a)=\delta^{-}_{T}(a)$.
Proof. Let $H$ be the tournament obtained from one positive diamond by
dilating its center by a tournament with $2$ vertices. Let
$\Delta^{+}_{T}(a)=\\{X\subset V;\,T[X]$ is a positive diamond with center
$a\\}$. Let $X$ be an element of $\Delta^{+}_{T}(a)$. As $\\{a,b\\}$ is an
interval of $T$, so $\\{a,b\\}\cap X$ is an interval of $T[X]$. Then, by
Remark 22, $b\not\in X$ and $\\{a,b\\}$ is an interval of $T[X\cup\\{b\\}]$.
Hence, $X\cup\\{b\\}\in S(T,H;\\{a,b\\})$. Let’s consider the map
$f\,:\,\Delta^{+}_{T}(a)\longrightarrow S(T,H;\\{a,b\\})$ defined by: for each
$X\in\Delta^{+}_{T}(a)$, $f(X)=X\cup\\{b\\}$. Clearly, $f$ is bijective and so
$\delta^{+}_{T}(a)=n(T,H;\\{a,b\\})$. By interchanging $T$ and $T^{\ast}$, we
deduce that
$\delta^{-}_{T}(a)=\delta^{+}_{T^{\ast}}(a)=n(T^{\ast},H;\\{a,b\\})$. On the
other hand, as $T$ and $T^{\ast}$ are $\\{-2\\}$-hypomorphic (resp.
$\\{-3\\}$-hypomorphic) and $\mid V\mid\geq 7$ (resp. $\mid V\mid\geq 8$),
then from Corollary 15, $n(T,H;\\{a,b\\})=n(T^{\ast},H;\\{a,b\\})$. Thus,
$\delta^{+}_{T}(a)=\delta^{-}_{T}(a)$. $\Box$
Proof of Proposition $24$. Assume by contradiction, that $T$ is
$\\{-3\\}$-self dual and has an interval $\\{a,b\\}$ with $a\neq b$. By Lemma
25, $T$ has a diamond $T\left[X\right]$ with center $a$. Assume for example
that $T\left[X\right]$ is a positive diamond. Clearly, $b\notin X$. Consider a
vertex $x\in X\setminus\\{a\\}$. As $T-x$ (resp. $T$) is $\\{-2\\}$-self dual
(resp. $\\{-3\\}$-self dual) and $\\{a,b\\}$ is an interval of $T-x$ (resp.
$T$), then, by Lemma 26, $\delta_{T-x}^{+}(a)=\delta_{T-x}^{-}(a)$ (resp.
$\delta_{T}^{+}(a)=\delta_{T}^{-}(a)$). So,
$0=\delta_{T}^{+}(a)-\delta_{T}^{-}(a)=\delta_{T-x}^{+}(a)+D_{T,\\{x\\}}^{+}(a)-\delta_{T-x}^{-}(a)=D_{T,\\{x\\}}^{+}(a)$;
contradiction.$\Box$
Theorem 4 is an immediate consequence of Proposition 21 and the below
proposition.
###### Proposition 27
Every decomposable tournament with at least $8$ vertices embedding a diamond
is not $\\{-3\\}$-self dual.
Proof. Let $T=(V,A)$ be a decomposable tournament with at least $8$ vertices
embedding a diamond. Assume by contradiction that $T$ is $\\{-3\\}$-self dual.
By Proposition 24, $T$ has no interval of size $2$. Let $I$ be a minimal
(under the inclusion) non trivial interval of $T$. Clearly, the subtournament
$T\left[I\right]$ is indecomposable. And then it is strongly connected.
Firstly, assume that $\left|V\setminus I\right|\geq 4$. Let $z\in V\setminus
I$ and suppose, for example that $z\longrightarrow I$ in $T$. As $T$ and
$T^{\ast}$ are $\\{-3\\}$-hypomorphic, then from Proposition 20,
$z\longrightarrow I$ in $T^{\ast}$; contradiction. Secondly, assume that
$\left|V\setminus I\right|\leq 3$. Let $k=\mid V\setminus I\mid$. We have so
$k\in\\{1,2,3\\}$ and $\mid I\mid\geq 8-k$. As the subtournament $T[I]$ is
strongly connected and $8-k\in\\{5,6,7\\}$, then from Lemma 1, there exists a
subset $X$ of $I$ such that $\mid X\mid=4-k$ and $T[I]-X$ is strongly
connected. Let $Y$ be a subset of $V\setminus I$ such that $\mid Y\mid=k-1$.
Clearly, the subtournament $T-(X\cup Y)$ is not self dual; contradicts the
fact that $\mid X\cup Y\mid=3$. $\Box$
## 4 Proof of Theorem 6
The proof of Theorem 6 is based on the next result.
###### Proposition 28
Let $T$ be a strongly connected and decomposable tournament on a set $V$ with
$\mid V\mid=n\geq 9$, which is not almost transitive. If $T^{\prime}$ is a
tournament $\\{-3\\}$-hypomorphic to $T$, then the following assertions hold.
1. 1.
$\mathcal{P}(T^{\prime})=\mathcal{P}(T)$ and
$T^{\prime}/\mathcal{P}(T)=T/\mathcal{P}(T)$.
2. 2.
If there exists $X\in\mathcal{P}(T)$ such that $T^{\prime}[X]\not\sim T[X]$,
then $\mid\mathcal{P}(T)\mid=3$ and $\mid X\mid=n-2$.
3. 3.
If for each $X\in\mathcal{P}(T)$, $\mid X\mid\leq n-3$, then for each
$X\in\mathcal{P}(T)$, $T^{\prime}[X]\sim T[X]$ and in particular
$T^{\prime}\sim T$.
For the proof of the last proposition, we use the following remark and lemma.
###### Remark 29
Let $T$ and $T^{\prime}$ be two tournaments on a set $V$ with $\mid V\mid\geq
3$ and $\Gamma$ be a common interval partition of $T$ and $T^{\prime}$ such
that $T/\Gamma=T^{\prime}/\Gamma$. Given a non empty subset $A$ of $V$ and let
$\Gamma_{A}=\\{X\cap A;\,X\in\Gamma$ and $X\cap A\neq\emptyset\\}$. Then
$\Gamma_{A}$ is a common interval partition of $T[A]$ and $T^{\prime}[A]$ and
$T[A]/\Gamma_{A}=T^{\prime}[A]/\Gamma_{A}$. Suppose that for each
$Y\in\Gamma_{A}$, there exists an isomorphism $\varphi_{Y}$ from $T[Y]$ onto
$T^{\prime}[Y]$ and consider the map $f\,:\,A\longrightarrow A$ defined by:
for each $x\in A$, $f(x)=\varphi_{Y}(x)$ where $Y$ is the unique element of
$\Gamma_{A}$ such that $x\in Y$. Then, $f$ is an isomorphism from $T[A]$ onto
$T^{\prime}[A]$. In particular, if for each $X\in\Gamma$, $T[X]$ and
$T^{\prime}[X]$ are hereditarily isomorphic, then $T$ and $T^{\prime}$ are
hereditarily isomorphic.
###### Lemma 30
[3] Let $T=(V,A)$ and $T^{\prime}=(V^{\prime},A^{\prime})$ be two isomorphic
tournaments, $f$ be an isomorphism from $T$ onto $T^{\prime}$, $i\in V$ and
$R_{i}$ (resp. $R_{i}^{\prime}$) be a tournament defined on a vertex set
$I_{i}$ (resp. $I_{i}^{\prime}$) disjoint from $V$ (resp. $V^{\prime}$). Let
$R$ (resp. $R^{\prime}$) be the tournament obtained from $T$ (resp.
$T^{\prime}$) by dilating the vertex $i$ (resp. $f(i)$) by $R_{i}$ (resp.
$R_{i}^{\prime}$). Then $R\sim R^{\prime}$ if and only if $R_{i}\sim
R_{i}^{\prime}$.
Note that this lemma is a simple generalization of a result communicated by A.
Boussaïri, and on which $V^{\prime}=V$ and $f=id_{V}$.
Proof of Proposition 28.
1. 1.
By Corollary 16, $T$ and $T^{\prime}$ are $(\leq 3)$-hypomorphic. So, from
Corollary 18, $\mathcal{P}(T)=\mathcal{P}(T^{\prime})$ and
$T^{\prime}/\mathcal{P}(T)=T/\mathcal{P}(T)$ or
$T^{\prime}/\mathcal{P}(T)=T^{\ast}/\mathcal{P}(T)$.
Assume by contradiction that
$T^{\prime}/\mathcal{P}(T)=T^{\ast}/\mathcal{P}(T)$. In this case, we are
going to show that for every $X\in\mathcal{P}(T)$, $T[X]$ is transitive, and
thus by Remark 29, $T^{\prime}$ is hereditarily isomorphic to $T^{\ast}$.
Since $T^{\prime}$ is $\\{-3\\}$-hypomorphic to $T$, then $T$ is
$\\{-3\\}$-self dual. By Theorem 4, the tournament $T$ is almost transitive;
which contradicts the hypothesis. For that, proceed by contradiction and
consider an element $X$ of $\mathcal{P}(T)$ and a subset
$\\{\alpha,\beta,\gamma\\}$ of $X$ such that $T[\\{\alpha,\beta,\gamma\\}]$ is
a $3$-cycle. As $T$ is strongly connected, there is $a\in V\setminus X$ such
that $X\longrightarrow a$. From Corollary 15,
$n(T,\delta^{+};\\{\alpha,\beta,a\\})=n(T^{\prime},\delta^{+};\\{\alpha,\beta,a\\})$.
Moreover, $n(T,\delta^{+};\\{\alpha,\beta,a\\})\neq 0$, because
$T[\\{\alpha,\beta,\gamma,a\\}]\sim\delta^{+}$, then there exists a subset $K$
of $V$ such that $\\{\alpha,\beta,a\\}\subset K$ and $T^{\prime}[K]$
$\sim\delta^{+}$. Hence, $\mid K\cap X\mid=3$, because otherwise $K\cap X$ is
an interval with two elements of $T^{\prime}[K]$; which contradicts Remark 22.
So, $T^{\prime}[K]$ is written: $(K\cap X)\longleftarrow a$; which contradicts
the fact that $T^{\prime}[K]\sim\delta^{+}$.
2. 2.
By 1, we have $\mathcal{P}(T^{\prime})=\mathcal{P}(T)$ and
$T^{\prime}/\mathcal{P}(T)=T/\mathcal{P}(T)$. We distinguish the following two
cases.
* •
If for every $X\in\mathcal{P}(T)$, $\mid X\mid\leq
n-\left|\mathcal{P}(T)\right|-2$.
Let $X\in\mathcal{P}(T)$ and let $H$ be the tournament obtained from
$T/\mathcal{P}(T)$ by dilating the vertex $X$ by $T[X]$. Assume that $\mid
X\mid\geq 2$ and consider a subset $A$ of $X$ with $2$ elements. Consider a
subset $B$ of $V$ containing $X$ such that:
$\forall\,Y\in\mathcal{P}(T)\setminus\\{X\\}$, $\mid B\cap Y\mid=1$. Clearly,
$B\in S(T,H;A)$ and hence $n(T,H;A)\neq\emptyset$. From Corollary 15,
$n(T^{\prime},H;A)=n(T,H;A)$, then $n(T^{\prime},H;A)\neq 0$. So, there exists
a subset $K$ of $V$ such that: $A\subset K$ and $T^{\prime}[K]\sim H$. Then
$\mathcal{P}(T^{\prime}[K])$ have a unique element $J$ non reduced to a
singleton. Clearly, $T^{\prime}[J]\sim T[X]$, in particular, $\mid J\mid=\mid
X\mid$.
Let $P_{1}=\\{Y\in\mathcal{P}(T);\,\mid Y\cap K\mid\geq 2\\}$ and
$P_{2}=\\{Y\in\mathcal{P}(T);\\\ \mid Y\cap K\mid=1\\}$. The set $K$ is the
union of the two disjoint sets $K_{1}=\bigcup\limits_{Y\in P_{1}}K\cap Y$ and
$K_{2}=\bigcup\limits_{Y\in P_{2}}K\cap Y$. As $A\subset X\cap K$, then $X\in
P_{1}$ and so, $P_{2}\subset(\mathcal{P}(T)\setminus\\{X\\})$, in particular,
$\mid P_{2}\mid\leq\mid\mathcal{P}(T)\mid-1$. For all $Y\in P_{1}$, $Y\cap K$
is a non trivial interval of $T^{\prime}[K]$, then $Y\cap K\subset J$, so
$K_{1}\subset J$ and thus $(K\setminus J)\subset K_{2}$.
Thus, $\mid\mathcal{P}(T)\mid-1=\mid K\setminus J\mid\leq\sum\limits_{Y\in
P_{2}}\mid K\cap Y\mid=\mid P_{2}\mid\leq\mid\mathcal{P}(T)\mid-1$. So, $\mid
P_{2}\mid=\mid\mathcal{P}(T)\mid-1$ and then
$P_{2}=\mathcal{P}(T)\setminus\\{X\\}$ and $P_{1}=\\{X\\}$. Thus, $K=(X\cap
K)\cup K_{2}$. So, $\mid X\cap K\mid=\mid K\mid-\mid K_{2}\mid=$ $\mid
K\mid-\mid P_{2}\mid=\mid K\mid-\mid\mathcal{P}(T)\mid+1=\mid J\mid$. As in
addition $X\cap K\subset J$, then, $X\cap K=J$. So, $J\subset X$ and then
$J=X$ because $\mid J\mid=\mid X\mid$. Consequently, $T^{\prime}[X]\sim T[X]$.
* •
If there exists an element $X$ of $\mathcal{P}(T)$ such that $\mid
X\mid>n-\left|\mathcal{P}(T)\right|-2$.
In this case, it is clear that for every $Y\in\mathcal{P}(T)\setminus\\{X\\}$,
$\mid Y\mid\leq 3$, so $T^{\prime}[Y]$ and $T[Y]$ are hereditarily isomorphic.
Suppose that $\mid V\setminus X\mid\geq 3$. Let $x$ be an element of $X$ and
$B$ be a subset of $V\setminus X$ such that $\mid B\mid=3$. Denote by
$V_{(x,B)}$ the set $(V\setminus(X\cup B))\cup\\{x\\}$ and by $T_{(x,B)}$
(resp. $T_{(x,B)}^{\prime}$) the subtournament $T[V_{(x,B)}]$ (resp.
$T^{\prime}[V_{(x,B)}]$). The tournament $T-B$ (resp. $T^{\prime}-B$) is
obtained from the tournament $T_{(x,B)}$ (resp. $T_{(x,B)}^{\prime}$) by
dilating the vertex $x$ by $T[X]$ (resp. $T^{\prime}[X]$). Moreover, by Remark
29, there exists an isomorphism $g$ from $T_{(x,B)}$ onto $T_{(x,B)}^{\prime}$
such that $g(x)=x$. As in addition, $T-B$ and $T^{\prime}-B$ are isomorphic,
then, from Lemma 30, $T^{\prime}[X]$ is isomorphic to $T[X]$; which permits to
conclude.
3. 3.
Is a direct consequence of 2.
$\square$
###### Lemma 31
Every strongly connected and decomposable tournament, which has $8$ vertices,
is $\\{-2,-3\\}$-reconstructible.
Proof. Let $H$ be a strongly connected and decomposable tournament defined on
a vertex set $X$ with $\mid X\mid=8$ and let $H^{\prime}$ be a tournament
$\\{-2,-3\\}$-hypomorphic to $H$. The tournaments $H$ and $H^{\prime}$ are
$\\{3\\}$-hypomorphic, by Corollary 16, so they are $\\{3,5,6\\}$-hypomorphic.
Besides, by Corollary 18, $\mathcal{P}(H)=\mathcal{P}(H^{\prime})$, and
$H^{\prime}/\mathcal{P}(H)=H/\mathcal{P}(H)$ or
$H^{\prime}/\mathcal{P}(H)=H^{\ast}/\mathcal{P}(H)$. Let’s put
$Q=\mathcal{P}(H)$, and discuss according to its cardinal.
* •
If $\mid Q\mid>3$.
If for every $Z\in Q$, $\mid Z\mid\leq 3$. By $(\leq 3)$-hypomorphy,
$H^{\prime}[Z]\sim H[Z];\,\forall Z\in Q$. If $H^{\prime}/Q=H/Q$, clearly
$H^{\prime}\sim H$. Suppose hence that $H^{\prime}/Q=H^{\ast}/Q$. In this
case, $H^{\prime}$ is hereditarily isomorphic to $H^{\ast}$. Then, $H$ is
$\\{-3\\}$-self dual. From Proposition 27, $H$ is without diamonds. By Remark
12, iii), $H$ is $\\{4\\}$-self dual. Thus, $H$ is $\\{4,5,6\\}$-self dual.
So, $H$ is $(\leq 6)$-self dual and it is thus self dual, by the $(\leq
6)$-reconstruction of tournaments with at least $6$ vertices [17]. It follows
that $H^{\prime}\sim H$.
If there exists $Y\in Q$ such that $\mid Y\mid\geq 4$. In this case, as $\mid
X\mid=8$, $\mid Q\mid>3$ and every tournament with $4$ vertices is
decomposable, then $\mid Q\mid=5$, $\mid Y\mid=4$ and for every $Z\in
Q\setminus\\{Y\\}$, $\mid Z\mid=1$.
* –
If $H^{\prime}/Q=H/Q$. For $z\in X\setminus Y$, as $\mid Y\cup\\{z\\}\mid=\mid
X\mid-3$, then, $H[Y\cup\\{z\\}]\sim H^{\prime}[Y\cup\\{z\\}]$. It follows
that $H^{\prime}[Y]\sim H[Y]$, and then $H^{\prime}\sim H$.
* –
If $H^{\prime}/Q=H^{\ast}/Q$.
* *
If $H[Y]$ is not a diamond. In this case, clearly $H^{\prime}[Y]$ is
hereditarily isomorphic to $H^{\ast}[Y]$ and hence $H^{\prime}$ and $H^{\ast}$
are hereditarily isomorphic. As $H^{\prime}$ is $\\{-3\\}$-hypomorphic to $H$,
then $H$ is $\\{-3\\}$-self dual. From Proposition 27, $H$ is then without
diamonds, and it is clearly self dual and thus, $H^{\prime}\sim H$.
* *
If $H[Y]$ is a diamond. In this case, if $H^{\prime}[Y]$ is not isomorphic to
$H[Y]$, then $H^{\prime}$ is hereditarily isomorphic to $H^{\ast}$ and so, $H$
is $\\{-3\\}$-self dual; which contradicts Proposition 27. So,
$H^{\prime}[Y]\sim H[Y]$. For $z\in X\setminus Y$, it is easy to verify that
$H^{\prime}[Y\cup\\{z\\}]$ is not isomorphic to $H[Y\cup\\{z\\}]$; which
contradicts the $\\{-3\\}$-hypomorphy between $H^{\prime}$ and $H$.
* •
If $\mid Q\mid=3$. We distinguish the following two sub-cases.
* –
If there exists an unique $Y\in Q$ such that $\mid Y\mid>1$.
In this case, $Q$ is written: $Q=\\{\\{a\\},\\{b\\},Y\\}$ where $\mid Y\mid=6$
and $a\longrightarrow Y\longrightarrow b$ in $H$. As $\mid Y\mid=6$, we have:
$H^{\prime}[Y]\sim H[Y]$ by $\\{-2\\}$-hypomorphy. Thus, $H^{\prime}\sim H$.
* –
If there are $Y\neq Z\in Q$ such that $\min(\mid Y\mid,\mid Z\mid)>1$.
In this case, we can write: $Q=\\{Y_{1},Y_{2},Y_{3}\\}$ with: $\mid
Y_{3}\mid\leq\mid Y_{2}\mid\leq$ $\mid Y_{1}\mid$, $\mid Y_{2}\mid\geq 2$ and
$\mid Y_{1}\mid\geq 3$. By considering a subset $A_{1}$ (resp. $A_{2}$) with
$3$ (resp. $2$) elements of $Y_{1}$ (resp. $Y_{2}$) and an element $y_{3}$ of
$Y_{3}$, we see that the isomorphy between $H[A_{1}\cup A_{2}\cup\\{y_{3}\\}]$
and $H^{\prime}[A_{1}\cup A_{2}\cup\\{y_{3}\\}]$ requires that:
$H^{\prime}/Q=H/Q$. So, if $\mid Y_{1}\mid=3$, then for every
$i\in\\{1,2,3\\}$, $H^{\prime}[Y_{i}]\sim H[Y_{i}]$ and then $H^{\prime}\sim
H$. Suppose thus that $\mid Y_{1}\mid\geq 4$. As $\mid X\mid=8$, then $\mid
Y_{1}\mid\in\\{4,5\\}$ and $\mid Y_{2}\mid\leq 3$. By considering an element
$y_{2}$ of $Y_{2}$, we see that the isomorphy between
$H[Y_{1}\cup\\{y_{2}\\}]$ and $H^{\prime}[Y_{1}\cup\\{y_{2}\\}]$ requires the
isomorphy between $H^{\prime}[Y_{1}]$ and $H[Y_{1}]$. It follows that
$H^{\prime}\sim H$.
$\Box$
Proof of Theorem 6. Consider a decomposable tournament $T$ defined on a vertex
set $V$ with $\mid V\mid=n\geq 9$, a tournament $T^{\prime}$
$\\{-3\\}$-hypomorphic to $T$. By Corollary 16, $T^{\prime}$ is $(\leq
3)$-hypomorphic to $T$. So, from Corollary 18,
$\mathcal{P}(T)=\mathcal{P}(T^{\prime})$. In particular, if $T$ is strongly
connected, then $\widetilde{P}(T)=\widetilde{P}(T^{\prime})$.
If $T$ is almost transitive, then clearly $T^{\prime}\sim T$. Let’s suppose
that $T$ is not almost transitive. For the proof, we distinguish the following
two cases.
$\ast$
$T$ is strongly connected.
In this case, from Proposition 28,
$T^{\prime}/\mathcal{P}(T)=T/\mathcal{P}(T)$ and we can suppose that
$\mid\mathcal{P}(T)\mid=3$ and there exists $X\in\mathcal{P}(T)$ such that
$\mid X\mid=n-2$. Let $\mathcal{P}(T)=\\{X,\\{a\\},\\{b\\}\\}$ where
$X\longrightarrow a\longrightarrow b\longrightarrow X$ in $T$. We verify
easily that $T^{\prime}[X]$ and $T[X]$ are $\\{-1,-2,-3\\}$-hypomorphic
(because $T^{\prime}$ and $T$ are $\\{-3\\}$-hypomorphic). We consider the
following three cases.
* •
$T[X]$ is non-strongly connected.
As the tournaments $T^{\prime}[X]$ and $T[X]$ are $\\{-1\\}$-hypomorphic, then
they are isomorphic, since the non-strongly connected tournaments with at
least $5$ vertices are $\\{-1\\}$-reconstructible [13].
* •
$T[X]$ is strongly connected and decomposable.
Let $Q=\mathcal{P}(T[X])$. Assume that there exists $Y\in Q$ such that $\mid
Y\mid=$
$\mid X\mid-2$. In this case, we have $\mid Q\mid=3$ and $T[X]/Q$ is a
$3$-cycle. Moreover, as $T^{\prime}[X]$ and $T[X]$ are $(\leq 3)$-hypomorphic,
then by Corollary 18, $\mathcal{P}(T^{\prime}[X])=\mathcal{P}(T[X])$ and
$T^{\prime}[X]/\mathcal{P}(T[X])=T[X]/\mathcal{P}(T[X])$ or
$T^{\prime}[X]/\mathcal{P}(T[X])=T^{\ast}[X]/\mathcal{P}(T[X])$. As in
addition, $T^{\prime}[Y]\sim T[Y]$ (because $T^{\prime}[X]$ and $T[X]$ are
$\\{-2\\}$-hypomorphic), then $T^{\prime}[X]\sim T[X]$. Thus, clearly
$T^{\prime}\sim T$.
Now, suppose that for every $Y\in Q$, $\mid Y\mid<\mid X\mid-2$. In this case,
$T[X]$ is not almost transitive. If $\mid X\mid\geq 9$, as $T^{\prime}[X]$ and
$T[X]$ are $\\{-3\\}$-hypomorphic, then by Proposition 28, $T^{\prime}[X]\sim
T[X]$ and hence, clearly $T^{\prime}\sim T$. If $\mid X\mid=7$. As
$T^{\prime}[X]$ and $T[X]$ are $\\{-1,-2,-3\\}$-hypomorphic, then they are
$(\leq 6)$-hypomorphic, and thus $T^{\prime}[X]\sim T[X]$ (by [17]). If $\mid
X\mid=8$. As $T^{\prime}[X]$ and $T[X]$ are $\\{-2,-3\\}$-hypomorphic, then,
by Lemma 31, $T^{\prime}[X]\sim T[X]$.
* •
If $T[X]$ is indecomposable.
In this case, from Theorem 17, $T^{\prime}[X]=T[X]$ or
$T^{\prime}[X]=T^{\ast}[X]$ (because $T^{\prime}[X]$ and $T[X]$ are $(\leq
3)$-hypomorphic). If $T^{\prime}[X]=T[X]$, then clearly $T^{\prime}\sim T$. If
$T^{\prime}[X]=T^{\ast}[X]$, then $T[X]$ is $\\{-1,-2,-3\\}$-self dual
(because $T^{\prime}[X]$ and $T[X]$ are $\\{-1,-2,-3\\}$-hypomorphic). We
obtain: If $T[X]$ is self dual, then $T^{\prime}[X]\sim T[X]$ and clearly
$T^{\prime}\sim T$. If $T[X]$ is not self dual, then $T\in
C_{3}(I_{n-2,\\{-1,-2,-3\\}})\subset\Omega_{n}$ and clearly
$T^{\prime}\not\sim T$.
$\ast$
$T$ is non-strongly connected.
Observe that if $T$ is a transitive tournament, then $T^{\prime}$ is also
transitive (by $(\leq 3)$-hypomorphy) and the result is obvious. Suppose then
that $T$ is a non-strongly connected tournament which is not transitive. The
result follows from the following five facts.
Fact 1. Let $X$ be an element of $\widetilde{\mathcal{P}}(T)$ such that $T[X]$
is strongly connected with $\mid X\mid\geq 3$ and let $a$ be an element of
$V\setminus X$. As $\mathcal{P}(T^{\prime})=\mathcal{P}(T)$, then
$T^{\prime}[X]$ is strongly connected, $X\in\widetilde{P}(T^{\prime})$ and we
have:
$a\longrightarrow X$ in $T$ if and only if $a\longrightarrow X$ in
$T^{\prime}$.
Indeed :
Let $\\{\alpha,\beta,\gamma\\}$ be a subset of $X$ such that
$T[\\{\alpha,\beta,\gamma\\}]$ is a $3$-cycle and suppose, for example, that
$X\longrightarrow a$ in $T$. From Corollary 15,
$n(T,\delta^{+};\\{\alpha,\beta,a\\})$
$=n(T^{\prime},\delta^{+};\\{\alpha,\beta,a\\})$. As
$n(T,\delta^{+};\\{\alpha,\beta,a\\})\neq 0$, thus there exists a subset $K$
of $V$ such that $\\{\alpha,\beta,a\\}\subset K$ and
$T^{\prime}[K]\sim\delta^{+}$. Hence, $\mid K\cap X\mid=3$, because otherwise
$K\cap X$ is an interval with two elements of $T^{\prime}[K]$; which
contradicts Remark 22. As $a\in V\setminus X$ and $K\setminus\\{a\\}\subset
X$, then $K\setminus\\{a\\}$ is an interval of $T^{\prime}[K]$ and thus
$T^{\prime}[K]$ is a diamond of center $a$. So, $X\longrightarrow a$ in
$T^{\prime}$.
Fact 2. $\widetilde{\mathcal{P}}(T^{\prime})=\widetilde{\mathcal{P}}(T)$.
Indeed :
Consider an element $Y$ of $\widetilde{\mathcal{P}}(T)$. We distinguish the
following two cases.
* •
If $T[Y]$ is strongly connected and $\mid Y\mid\geq 3$. In this case,
$Y\in\mathcal{P}(T)$. Then $Y\in\mathcal{P}(T^{\prime})$ and hence,
$Y\in\widetilde{\mathcal{P}}(T^{\prime})$.
* •
If $T[Y]$ is a transitive. In this case, there exists an element $Z$ of
$\widetilde{\mathcal{P}}(T^{\prime})$ such that $Y\subset Z$, because
otherwise, $Y$ admits a partition $\\{Y_{1},Y_{2}\\}$ such that there is
$K\in\mathcal{P}(T^{\prime})$ with $\mid K\mid\geq 3$, $T^{\prime}[K]$ is
strongly connected and in $T^{\prime}$ we have $Y_{1}\longrightarrow
K\longrightarrow Y_{2}$; which contradicts the Fact 1. While exchanging the
roles of $T$ and $T^{\prime}$, we can hence deduce that $Y=Z$ and then,
$Y\in\widetilde{\mathcal{P}}(T^{\prime})$ .
Fact 3. $T^{\prime}/\widetilde{\mathcal{P}}(T)=T/\widetilde{\mathcal{P}}(T)$.
Indeed:
Proceed by the absurd and suppose that there exist two distinct elements $X$
and $Y$ of $\widetilde{\mathcal{P}}(T)$ such that $X\longrightarrow Y$ in $T$
and $Y\longrightarrow X$ in $T^{\prime}$. From the Fact 1, $T[X]$ and $T[Y]$
are transitive. So, $X$ and $Y$ are not consecutive in
$T/\widetilde{\mathcal{P}}(T)$. Thus, there exists an element $Z$ of
$\widetilde{\mathcal{P}}(T)$ such that $T[Z]$ is strongly connected, $\mid
Z\mid\geq 3$ and $X\longrightarrow Z\longrightarrow Y$ in $T$. So, by Fact 1,
$X\longrightarrow Z\longrightarrow Y$ in $T^{\prime}$ and then
$X\longrightarrow Y$ in $T^{\prime}$; which is absurd.
Fact 4. If $T\not\in\Omega_{n}$, then for all
$X\in\widetilde{\mathcal{P}}(T)$, $T^{\prime}[X]\sim T[X]$.
Indeed:
Suppose that $T\not\in\Omega_{n}$ and consider an element $X$ of
$\widetilde{\mathcal{P}}(T)$. As $T$ and $T^{\prime}$ are $(\leq
3)$-hypomorphic, we can assume that $\mid X\mid\geq 4$ and $T[X]$ is strongly
connected.
We distinguish the following cases.
* •
If $\mid X\mid\leq n-3$.
Consider $H=T[X]$ and $A\subset X$ such that $\mid A\mid=2$. From Corollary
15, $n(T,H;A)=n(T^{\prime},H;A)$. As $n(T,H;A)\neq 0$, then
$n(T^{\prime},H;A)\neq 0$. So, there exists a subset $K$ of $V$ such that
$A\subset K$ and $T^{\prime}[K]\sim H$. We have then $K=X$, because otherwise,
there exists $Y\in\widetilde{\mathcal{P}}(T)\setminus\\{X\\}$ such that $K\cap
Y\neq\emptyset$ and hence $T^{\prime}[K]$ is non-strongly connected (because
$K\cap X$ is also non empty); which is absurd. So, $T^{\prime}[X]\sim H=T[X]$.
* •
If $\mid X\mid=n-1$.
Let $\\{a\\}=V\setminus X$ and suppose, for example, that $X\longrightarrow a$
in $T$. As $\mid X\mid-2=n-3$, then $T^{\prime}[X]$ and $T[X]$ are
$\\{-2\\}$-hypomorphic. Now we shall prove that these two tournaments are
$\\{-3\\}$-hypomorphic. For that, consider a subset $A$ of $X$ such that $\mid
A\mid=3$. It is clear that $T^{\prime}-A$ and $T-A$ are isomorphic and in
these two tournaments, we have $(X\setminus A)\longrightarrow a$. So,
$T^{\prime}[X\setminus A]\sim T[X\setminus A]$. Hence, $T^{\prime}[X]$ and
$T[X]$ are $\\{-3\\}$-hypomorphic. So, $T^{\prime}[X]$ and $T[X]$ are
$\\{-2,-3\\}$-hypomorphic.
* –
If $T[X]$ is decomposable. Let $Q=\mathcal{P}(T[X])$. If there exists $Y\in Q$
such that $\mid Y\mid=\mid X\mid-2$. In this case, $\mid Q\mid=3$ and $T[X]/Q$
is a $3$-cycle. As $\mid Y\mid=n-3$, then $T^{\prime}[Y]\sim T[Y]$. As in
addition, $T^{\prime}[X]$ and $T[X]$ are $(\leq 3)$-hypomorphic, then, by
Corollary 18, $\mathcal{P}(T^{\prime}[X])=Q$ and $T^{\prime}[X]/Q$ $=T[X]/Q$
or $T^{\prime}[X]/Q$ $=T^{\ast}[X]/Q$. So we deduce immediately that
$T^{\prime}[X]\sim T[X]$. Now suppose that for every $Y\in Q$, $\mid
Y\mid<\mid X\mid-2$. In this case, $T[X]$ is not almost transitive.
Distinguish the following two cases. If $\mid X\mid\geq 9$. In this case, by
applying Proposition 28 for the tournaments $T[X]$ and $T^{\prime}[X]$, we
obtain $T^{\prime}[X]\sim T[X]$. If $\mid X\mid=8$. As $T^{\prime}[X]$ and
$T[X]$ are $\\{-2,-3\\}$-hypomorphic, then, by Lemma 31, $T^{\prime}[X]\sim
T[X]$.
* –
If $T[X]$ is indecomposable. We have $T^{\prime}[X]$ and $T[X]$ are $(\leq
3)$-hypomorphic. So, by Theorem 17, $T^{\prime}[X]=T[X]$ or
$T^{\prime}[X]=T^{\ast}[X]$. Consider then the case where
$T^{\prime}[X]=T^{\ast}[X]$. If $T[X]$ is self dual, then $T^{\prime}[X]\sim
T[X]$. Suppose hence that $T[X]$ is not self dual. As $T[X]$ is
$\\{-2,-3\\}$-self dual (because $T^{\prime}[X]$ and $T[X]$ are
$\\{-2,-3\\}$-hypomorphic), then $T\in
O_{2}(I_{n-1,\\{-2,-3\\}})\subset\Omega_{n}$; which is absurd.
* •
If $\mid X\mid=n-2$.
Let $\\{a,b\\}=V\setminus X$ and suppose, for example, that $X\longrightarrow
a$ in $T$. As $\mid X\mid-1=n-3$, then $T^{\prime}[X]$ and $T[X]$ are
$\\{-1\\}$-hypomorphic. Now, we shall prove that $T^{\prime}[X]$ and $T[X]$
are $\\{-2,-3\\}$-hypomorphic. For that, for every $i\in\\{2,3\\}$, let
$A_{i}$ be a subset of $X$ such that $\mid A_{i}\mid=i$. We have
$T-(A_{2}\cup\\{a\\})\sim T^{\prime}-(A_{2}\cup\\{a\\})$. From Fact 3,
$(X\setminus A_{2})\longrightarrow b$ in $T$ if and only if $(X\setminus
A_{2})\longrightarrow b$ in $T^{\prime}$. So, $T^{\prime}[X\setminus
A_{2}]\sim T[X\setminus A_{2}]$ and thus, $T^{\prime}[X]$ and $T[X]$ are
$\\{-2\\}$-hypomorphic. Furthermore, as $T^{\prime}-A_{3}\sim T-A_{3}$, then
by Fact 3, we can see that $T^{\prime}[X\setminus A_{3}]\sim T[X\setminus
A_{3}]$. So, $T^{\prime}[X]$ and $T[X]$ are $\\{-1,-2,-3\\}$-hypomorphic.
* –
If $T[X]$ is decomposable. Pose $Q=\mathcal{P}(T[X])$. If there exists $Y\in
Q$ such that $\mid Y\mid=\mid X\mid-2$. In this case, $\mid Q\mid=3$ and
$T[X]/Q$ is a $3$-cycle. Besides, by $\\{-3\\}$-hypomorphy, the two
subtournaments $T^{\prime}[Y\cup\\{a\\}]$ and $T[Y\cup\\{a\\}]$ are isomorphic
(because $\mid Y\cup\\{a\\}\mid=n-3$). As in addition, $Y\longrightarrow a$ in
both these tournaments, then $T^{\prime}[Y]\sim T[Y]$. As in addition,
$T^{\prime}[X]$ and $T[X]$ are $(\leq 3)$-hypomorphic, then, by Corollary 18,
we can see that $T^{\prime}[X]\sim T[X]$. Now, suppose that for every $Y\in
Q$, $\mid Y\mid<\mid X\mid-2$. In this case, $T[X]$ is not almost transitive.
We distinguish the following three cases.
If $\mid X\mid\geq 9$, we conclude by Proposition 28.
If $\mid X\mid=8$, we conclude by Lemma 31.
If $\mid X\mid=7$, as $T^{\prime}[X]$ and $T[X]$ are
$\\{-1,-2,-3\\}$-hypomorphic, then they are $(\leq 6)$-hypomorphic and hence
$T^{\prime}[X]\sim T[X]$.
* –
If $T[X]$ is indecomposable. As $T^{\prime}[X]$ and $T[X]$ are $(\leq
3)$-hypomorphic, then from Theorem 17, $T^{\prime}[X]=T[X]$ or
$T^{\prime}[X]=T^{\ast}[X]$. Consider then the case where
$T^{\prime}[X]=T^{\ast}[X]$. If $T[X]$ is self dual, then $T^{\prime}[X]\sim
T[X]$. Suppose now that $T[X]$ is not self dual. As in addition $T[X]$ is
$\\{-1,-2,-3\\}$-self dual (because $T^{\prime}[X]$ and $T[X]$ are
$\\{-1,-2,-3\\}$-hypomorphic), hence the tournament $T$ belongs to
$O_{3}(I_{n-2,\\{-1,-2,-3\\}})$ and then to $\Omega_{n}$; which is absurd.
Fact 5. If $T\in\Omega_{n}$, then $T^{\prime}\not\sim T$.
This fact is an immediate consequence of facts 2 and 3.
$\square$
We shall now prove Corollary 8. Before that let us observe that Corollary 7 is
an immediate consequence of Theorem 6 because each element $T$ of $\Omega_{n}$
(where $n\geq 9$) has an interval $X$ such that $T[X]$ is indecomposable and
$\mid X\mid\in\\{n-1,n-2\\}$.
Proof of Corollary 8. Suppose that there exists an integer $n_{0}\geq 7$ such
that the indecomposable tournaments with at least $n_{0}$ vertices are
$\\{-3\\}$-reconstructible and consider a tournament $T$ with $n\geq n_{0}+2$
vertices. Then, the classes $I_{n-2,\\{-3\\}}$ and $I_{n-1,\\{-3\\}}$ are
empty. So, the classes $I_{n-2,\\{-1,-2,-3\\}}$ and $I_{n-1,\\{-2,-3\\}}$ are
empty. Thus, $\Omega_{n}$ is empty and then $T\notin\Omega_{n}$. By Theorem 6,
the tournament $T$ is then $\\{-3\\}$-reconstructible.
$\square$
_AKNOWLEDGEMENT_
We would like to thank the anonymous referee for his insightful comments.
## References
* [1] J. A. Bondy, A graph reconstructor’s manual, in Surveys in Combinatorics, (1991), Keendwell éd., London. Math. Soc. Lecture Note Ser., Cambridge Univ. Press, 221 - 252.
* [2] J. A. Bondy and R. L. Hemminger, Graph reconstruction, a survey, J. Graph theory, 1, (1977), 227 - 268.
* [3] M. Bouaziz and Y. Boudabbous, La demi-isomorphie et les tournois fortement connexes finis, C. R. Acad. Sci. Paris Série I Math. 335 (2002), 411-416.
* [4] H. Bouchaala, Sur la répartition des diamants dans un tournoi, C. R. Acad. Sci. Paris, Serie. I 338 (2004) 109-112.
* [5] H. Bouchaala and Y. Boudabbous, La $\\{-k\\}$-autodualité des sommes lexicographiques finies de tournois suivant un $3$-cycle ou un tournoi critique, Ars Combin. 81 (2006). 33-64.
* [6] Y. Boudabbous and A. Boussaïri, Reconstruction des tournois et dualité, C.R. Acad. Sci. Paris, t. 320, Série I, (1995), 397 - 400.
* [7] Y. Boudabbous, J. Dammak and P. Ille, Indecomposability and duality of tournaments, Discrete Mathematics 223 (2000) 55-82.
* [8] A. Boussaïri, P. Ille, G. Lopez and S. Thomassé, Hypomorphie et inversion locale entre graphes, C. R. Acad. Sci. Paris, Série I 137 (1993), 125-128.
* [9] A. Boussaïri, P. Ille, G. Lopez and S. Thomassé, The $C_{3}$-structure of tournaments, Discrete Math., 277 (2004), 29-43.
* [10] A. Ehrenfeucht and G. Rozenberg, Primitivity is hereditary for $2$-structures, Theoret. Comput. Sci. 3 (70) (1990) 343-358.
* [11] R. Fraïssé, Abritement entre relations et spécialemnt entre chaînes, Symposi. Math. Instituto Nazionale di Alta. Mathematica, 5, (1970) 203-251.
* [12] T. Gallai, Transitiv orientierbare Graphen, Acta. Math. Acad. Sci. Hungar., 18, (1967), 25-66.
* [13] F. Harary and E. Palmer, On the problem of reconstructing a tournament from subtournaments, Monatsch. Math. 71, (1967), 14-23.
* [14] P. Ille, La reconstruction des relations binaires, C. R. Acad. Sci. Paris, t. 306. Série I, (1988), 635-638.
* [15] G. Lopez, Deux résultats concernant la détermination d’une relation par les types d’isomorphie de ses restrictions, C. R. Acad. Sci. Paris, t. 274, Série A, (1972), 1525 - 1528.
* [16] G. Lopez, Sur la détermination d’une relation par les types d’isomorphie de ses restrictions. C. R. Acad. Sci Paris (A) t. 275 (1972) 951-953.
* [17] G. Lopez, L’indéformabilité des relations et multirelations binaires, Z. Math. Logik Grundlag. Math., 24, (1978), 303-317.
* [18] G. Lopez and C. Rauzy, Reconstruction of binary relations from their restrictions of cardinality 2, 3, 4 and (n-1), I, Z. Math. Logik Grundlag. Math., 38, (1992), 27-37.
* [19] G. Lopez and C. Rauzy, Reconstruction of binary relations from their restrictions of cardinality 2, 3, 4 and (n-1), II, Z. Math. Logik Grundlag. Math., 38, (1992), 157-168.
* [20] J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York (1968).
* [21] M. Pouzet, Application d’une propriété combinatoire des parties d’un ensemble aux groupes et aux relations, Math. Z. 150 (1976) 117-134.
* [22] M. Pouzet, Relations non reconstructibles par leurs restrictions. Journal of Combin. Theory B. vol. 1.26, (1979), 22-34.
* [23] K. B. Reid and C. Thomassen, Strongly Self-Complementary and Hereditarily Isomorphic Tournaments, Monatshefte Math. 81, (1976), 291-304.
* [24] M. Sghiar, Private communication.
* [25] P. K. Stockmeyer, The falsity of the reconstruction conjecture for tournaments, J. Graph Theory 1 (1977), 19-25.
* [26] S. M. Ulam, A collection of Mathematical Problems, Intersciences Publishers, New York (1960).
|
arxiv-papers
| 2012-04-11T18:26:56 |
2024-09-04T02:49:29.617307
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mouna Achour, Youssef Boudabbous and Abderrahim Boussairi",
"submitter": "Youssef Boudabbous",
"url": "https://arxiv.org/abs/1204.2513"
}
|
1204.2542
|
# The spacetime structure of MOND with Tully-Fisher relation and Lorentz
invariance violation
Xin Li lixin@ihep.ac.cn Zhe Chang changz@ihep.ac.cn Institute of High Energy
Physics
and
Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences, 100049 Beijing, China
###### Abstract
It is believed that the modification of Newtonian dynamics (MOND) is possible
alternate for dark matter hypothesis. Although Bekenstein’s TeVeS supplies a
relativistic version of MOND, one may still wish a more concise covariant
formulism of MOND. In this paper, within covariant geometrical framwork, we
present another version of MOND. We show the spacetime structure of MOND with
properties of Tully-Fisher relation and Lorentz invariance violation.
###### pacs:
95.35.+d,02.40.-k, 04.25.Nx
## I Introduction
In 1932, Physicists Oort ; Zwicky found from the observations of galaxies and
galaxy clusters that the Newton’s gravity could not provide enough force to
attract the matters of galaxies. The recent astronomical observations show
that the rotational velocity curves of all spiral galaxies tend to some
constant valuesTrimble . These include the Oort discrepancy in the disk of the
Milky WayBahcall , the velocity dispersions of dwarf Spheroidal galaxiesVogt ,
and the flat rotation curves of spiral galaxiesRubin . These facts violate
sharply the prediction of Newton’s gravity. The most widely adopted way to
resolve these difficulties is the dark matter hypothesis. It is assumed that
all visible stars are surrounded by massive nonluminous matters.
The dark matter hypothesis has dominated astronomy and cosmology for almost 80
years. However, up to now, no direct observations has been substantially
tested. Some models have been built for alternative of the dark matter
hypothesis. Their main ideas are to assume that the Newtonian gravity or
Newton’s dynamics is invalid in galactic scale. The most successful and famous
model is MOND Milgrom . It assumed that the Newtonian dynamics does not hold
in galactic scale. The particular form of MOND is given as
$\begin{array}[]{l}m\mu\left(\displaystyle\frac{a}{a_{0}}\right)\mathbf{a}=\mathbf{F},\\\\[11.38092pt]
\displaystyle\lim_{x\gg 1}\mu(x)=1,~{}~{}~{}\lim_{x\ll 1}\mu(x)=x,\end{array}$
(1)
where $a_{0}$ is at the order of $10^{-8}$ cm/s2. At beginning, as a
phenomenological model, MOND explains well the flat rotation curves with a
simple formula and a new parameter. In particular, it deduce naturally a well-
known global scaling relation for spiral galaxies, the Tully-Fisher relationTF
. The Tully-Fisher relation is an empirical relation between the total
luminosity of a galaxy and the maximum rotational speed. The relation is of
the form $L\propto v^{a}_{max}$, where $a\approx 4$ if luminosity is measured
in the near-infrared. Tully and Pierce Tully showed that the Tully-Fisher
relation appears to be convergence in the near-infrared. McGaugh McGaugh
investigated the Tully-Fisher relation for a large sample of galaxies, and
concluded that the Tully-Fisher relation is fundamental relation between the
total baryonic mass and the rotational speed. MOND Milgrom predicts that the
rotational speed of galaxy has an asymptotic value
$\lim_{r\rightarrow\infty}v^{4}=GMa_{0}$, which explains the Tully-Fisher
relation.
By introducing several scalar, vector and tensor fields, BekensteinBekenstein
rewrote the MOND into a covariant formulation (TeVeS). He showed that the MOND
satisfies all four classical tests on Einstein’s general relativity in Solar
system. Beside Bekenstein’s theory, there are other MOND theories(for example,
Einstein-aether theoryZlosnik ). These MOND theories modify gravity with
additional scalar/vector/tensor fields. Bekenstein’s theory and Einstein-
aether theory both admit a preferred reference frame and break local Lorentz
invariance. It means that local Lorentz symmetry violation is a feature of
MOND.
The local Lorentz symmetry violation implies that the space structure of
galaxy is not Minkowskian at large scale, and the relationship between the
Tully-Fisher relation and MOND implies that the space structure of galaxy
depends on rotational speed. Finsler gravity based on Finsler geometry
involves the above features. It is natural to assume that the Finsler gravity
is a covariant formulism of MOND. Finsler geometry Book by Bao as a nature
extension of Riemann geometry involves Riemann geometry as its special case.
The length element of Finsler geometry depends not only on the positions but
also the velocities. Finsler gravity naturally preserves fundamental
principles and results of general relativity. A new geometry (Finsler
geometry) involves new spacetime symmetry. The Lorentz invariance violation is
intimately linked to Finsler geometry. Kostelecky Kostelecky has studied
effective field theories with explicit Lorentz invariance violation in Finsler
spacetime. In this paper, within the covariant geometric framework, we try to
present a covariant formulism of MOND with explicit Tully-Fisher relation and
Lorentz invariance violation.
## II Vacuum field equation in Finsler spacetime
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given as Book by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (2)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(3)
The Finsler structure represents the length element of Finsler space.
The parallel transport has been studied in the framework of Cartan connection
Matsumoto ; Antonelli ; Szabo . The notation of parallel transport in Finsler
manifold means that the length $F\left(\frac{dx}{d\tau}\right)$ is constant.
The geodesic equation for Finsler manifold is given as Book by Bao
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+2G^{\mu}=0,$ (4)
where
$G^{\mu}=\frac{1}{4}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (5)
is called geodesic spray coefficient. Obviously, if $F$ is Riemannian metric,
then
$G^{\mu}=\frac{1}{2}\tilde{\gamma}^{\mu}_{\nu\lambda}y^{\nu}y^{\lambda},$ (6)
where $\tilde{\gamma}^{\mu}_{\nu\lambda}$ is the Riemannian Christoffel
symbol. Since the geodesic equation (4) is directly derived from the integral
length
$L=\int F\left(\frac{dx}{d\tau}\right)d\tau,$ (7)
the inner product
$\left(\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}=F\left(\frac{dx}{d\tau}\right)\right)$
of two parallel transported vectors is preserved.
In Finsler manifold, there exists a linear connection - the Chern connection
Chern . It is torsion freeness and almost metric-compatibility,
$\Gamma^{\alpha}_{\mu\nu}=\gamma^{\alpha}_{\mu\nu}-g^{\alpha\lambda}\left(A_{\lambda\mu\beta}\frac{N^{\beta}_{\nu}}{F}-A_{\mu\nu\beta}\frac{N^{\beta}_{\lambda}}{F}+A_{\nu\lambda\beta}\frac{N^{\beta}_{\mu}}{F}\right),$
(8)
where $\gamma^{\alpha}_{\mu\nu}$ is the formal Christoffel symbols of the
second kind with the same form of Riemannian connection, $N^{\mu}_{\nu}$ is
defined as
$N^{\mu}_{\nu}\equiv\gamma^{\mu}_{\nu\alpha}y^{\alpha}-A^{\mu}_{\nu\lambda}\gamma^{\lambda}_{\alpha\beta}y^{\alpha}y^{\beta}$
and $A_{\lambda\mu\nu}\equiv\frac{F}{4}\frac{\partial}{\partial
y^{\lambda}}\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}(F^{2})$ is the Cartan tensor (regarded as a measurement of deviation
from the Riemannian Manifold). In terms of Chern connection, the curvature of
Finsler space is given as
$R^{~{}\lambda}_{\kappa~{}\mu\nu}=\frac{\delta\Gamma^{\lambda}_{\kappa\nu}}{\delta
x^{\mu}}-\frac{\delta\Gamma^{\lambda}_{\kappa\mu}}{\delta
x^{\nu}}+\Gamma^{\lambda}_{\alpha\mu}\Gamma^{\alpha}_{\kappa\nu}-\Gamma^{\lambda}_{\alpha\nu}\Gamma^{\alpha}_{\kappa\mu},$
(9)
where $\frac{\delta}{\delta x^{\mu}}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}_{\mu}\frac{\partial}{\partial y^{\nu}}$.
The gravity in Finsler spacetime has been investigated for a long time Takano
; Ikeda ; Tavakol1 ; Bogoslovsky1 . In this paper, we introduce vacuum field
equation by the way discussed first by Pirani Pirani ; Rutz . In Newton’s
theory of gravity, the equation of motion of a test particle is given as
$\frac{d^{2}x^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial x^{i}},$
(10)
where $\phi=\phi(x)$ is the gravitational potential and $\eta^{ij}$ is
Euclidean metric. For an infinitesimal transformation $x^{i}\rightarrow
x^{i}+\epsilon\xi^{i}$($|\epsilon|\ll 1$), the equation (10) becomes, up to
first order in $\epsilon$,
$\frac{d^{2}x^{i}}{dt^{2}}+\epsilon\frac{d^{2}\xi^{i}}{dt^{2}}=-\eta^{ij}\frac{\partial\phi}{\partial
x^{i}}-\epsilon\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial x^{j}\partial
x^{k}}.$ (11)
Combining the above equations(10) and (11), we obtain
$\frac{d^{2}\xi^{i}}{dt^{2}}=\eta^{ij}\xi^{k}\frac{\partial^{2}\phi}{\partial
x^{j}\partial x^{k}}\equiv\xi^{k}H^{i}_{k}.$ (12)
In Newton’s theory of gravity, the vacuum field equation is given as
$H^{i}_{i}=\bigtriangledown^{2}\phi=0$. It means that the tensor $H^{i}_{k}$
is traceless in Newton’s vacuum.
In general relativity, the geodesic deviation gives similar equation
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}\tilde{R}^{\mu}_{~{}\nu},$ (13)
where
$\tilde{R}^{\mu}_{~{}\nu}=\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $\tilde{R}^{~{}\mu}_{\lambda~{}\nu\rho}$ is Riemannian curvature tensor,
$D$ denotes the covariant derivative alone the curve $x^{\mu}(\tau)$. The
vacuum field equation in general relativity gives
$\tilde{R}^{~{}\lambda}_{\mu~{}\lambda\nu}=0$Weinberg . It implies that the
tensor $\tilde{R}^{\mu}_{~{}\nu}$ is also traceless,
$\tilde{R}\equiv\tilde{R}^{\mu}_{~{}\mu}=0$.
In Finsler spacetime, the geodesic deviation gives Book by Bao
$\frac{D^{2}\xi^{\mu}}{D\tau^{2}}=\xi^{\nu}R^{\mu}_{~{}\nu},$ (14)
where
$R^{\mu}_{~{}\nu}=R^{~{}\mu}_{\lambda~{}\nu\rho}\frac{dx^{\lambda}}{d\tau}\frac{dx^{\rho}}{d\tau}$.
Here, $R^{~{}\mu}_{\lambda~{}\nu\rho}$ is Finsler curvature tensor defined in
(9), $D$ denotes covariant derivative
$\frac{D\xi^{\mu}}{D\tau}=\frac{d\xi^{\mu}}{d\tau}+\xi^{\nu}\frac{dx^{\lambda}}{d\tau}\Gamma^{\mu}_{\nu\lambda}(x,\frac{dx}{d\tau})$.
Since the vacuum field equations of Newton’s gravity and general relativity
have similar form, we may assume that vacuum field equation in Finsler
spacetime hold similar requirement as the case of Netwon’s gravity and general
relativity. It implies that the tensor $R^{\mu}_{~{}\nu}$ in Finsler geodesic
deviation equation should be traceless, $R\equiv R^{\mu}_{~{}\mu}=0$. In fact,
we have proved that the analogy from the geodesic deviation equation is valid
at least in Finsler spacetime of Berwald type Finsler DM . We suppose that
this analogy is still valid in general Finsler spacetime.
It should be noticed that $H$ is called the Ricci scaler, which is a
geometrical invariant. For a tangent plane $\Pi\subset T_{x}M$ and a non-zero
vector $y\in T_{x}M$, the flag curvature is defined as
$K(\Pi,y)\equiv\frac{g_{\lambda\mu}R^{\mu}_{~{}\nu}u^{\nu}u^{\lambda}}{F^{2}g_{\rho\theta}u^{\rho}u^{\theta}-(g_{\sigma\kappa}y^{\sigma}u^{\kappa})^{2}},$
(15)
where $u\in\Pi$. The flag curvature is a geometrical invariant and a
generalization of the sectional curvature in Riemannian geometry. It is clear
that the Ricci scaler $R$ is the trace of $R^{\mu}_{~{}\nu}$, which is the
predecessor of flag curvature. Therefore, the value of Ricci scaler $R$ is
invariant under the coordinate transformation. Furthermore, the predecessor of
flag curvature could be written in terms of the geodesic spray coefficient
$R^{\mu}_{~{}\nu}=2\frac{\partial G^{\mu}}{\partial
x^{\nu}}-y^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial x^{\lambda}\partial
y^{\nu}}+2G^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial y^{\lambda}\partial
y^{\nu}}-\frac{\partial G^{\mu}}{\partial y^{\lambda}}\frac{\partial
G^{\lambda}}{\partial y^{\nu}}.$ (16)
Thus, the Ricci scaler $R$ is insensitive to specific connection form. It only
depends on the length element $F$. The gravitational vacuum field equation
$R=0$ is universal in any types of theories of Finsler gravity. Pfeifer et al.
Pfeifer1 have constructed gravitational dynamics for Finsler spacetimes in
terms of an action integral on the unit tangent bundle. Their results also
show that the gravitational vacuum field equation in Finsler spacetime is
$R=0$.
## III The Newtonian limit in Finsler spacetime
It is well known that the Minkowski spacetime is a trivial solution of
Einstein’s vacuum field equation. In the Finsler spacetime, the trivial
solution of Finslerian vacuum field equation is called locally Minkowski
spacetime. A Finsler spacetime is called a locally Minkowshi spacetime if
there is a local coordinate system $(x^{\mu})$, with induced tangent space
coordinates $y^{\mu}$, such that $F$ depends only on $y$ and not on $x$. Using
the formula (16), one knows obvious that locally Minkowski spacetime is a
solution of Finslerian vacuum field equation.
In Ref. Finsler GW , we supposed that the metric is close to the locally
Minkowski metric $\eta_{\mu\nu}(y)$,
$g_{\mu\nu}=\eta_{\mu\nu}(y)+h_{\mu\nu}(x,y),~{}~{}|h_{\mu\nu}|\ll 1$ (17)
and found that the gravitational vacuum field equation is of the form
$\eta^{\mu\nu}\frac{\partial^{2}h_{\alpha\beta}}{\partial x^{\mu}\partial
x^{\nu}}-\frac{1}{2}\frac{\partial\eta^{\mu\nu}}{\partial
y^{\mu}}\frac{\partial^{2}h_{\alpha\beta}}{\partial x^{\nu}\partial
x^{\lambda}}y^{\lambda}=0,$ (18)
where the lowering and raising of indices are carried out by $\eta_{\mu\nu}$
and its matrix inverse $\eta^{\mu\nu}$. To make contact with Newton’s gravity,
we consider the gravitational field $h_{\mu\nu}$ is stationary and particle
moving very slowly. In general relativity, gravitational field $h_{00}$
corresponds to the Newton’s gravitational potential. Here, we consider that
only $h_{00}$ is the non vanish components of $h_{\mu\nu}$. Then, we obtain
from (18) that
$\eta^{ij}\frac{\partial^{2}h_{00}}{\partial x^{i}\partial x^{j}}=0.$ (19)
The solution of (19) is given as
$h_{00}=\frac{C}{R}\eta_{00},~{}~{}(R\neq 0),$ (20)
where $C$ is a constant and $R^{2}=\eta_{ij}x^{i}x^{j}$. The gravitational
theory should reduce to general relativity, if the Finsler metric $g_{\mu\nu}$
reduces to Riemann metric. Thus, for a giving gravitational source with mass
$M$ at $R=0$, the gravitational field equation should be of the form
$\eta^{ij}\frac{\partial^{2}h_{00}}{\partial x^{i}\partial
x^{j}}=8\pi_{F}G\rho\eta_{00},$ (21)
where $\rho$ is the energy density of gravitational source. In Finsler
spacetime, the space volume of $\eta_{\mu\nu}(y)$ Book by Bao is different
the one in Euclidean space. We used $\pi_{F}$ in (21) to represent the
difference, where
$\pi_{F}\equiv\frac{3}{4}\int_{R=1}\sqrt{g}dx^{1}\wedge dx^{2}\wedge dx^{3}$
(22)
and $g$ is the determinant of $\eta_{ij}$. The solution of (21) is given as
$h^{0}_{0}=\frac{2GM}{R}.$ (23)
Here, we have used $h^{0}_{0}$ to denote the gravitational field instead of
$h_{00}$. It is due to the fact that $\eta_{00}\neq 1$ and we want to obtain
the formula (23) which insensitive to the spacetime index.
In the approximation of Newton’s limit, the geodesic equation (4) reduces to
$\displaystyle\frac{d^{2}x^{0}}{d\tau^{2}}-\frac{\eta^{0i}}{2}\frac{\partial
h_{00}}{\partial x^{i}}\frac{dx^{0}}{d\tau}\frac{dx^{0}}{d\tau}$
$\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\frac{d^{2}x^{i}}{d\tau^{2}}-\frac{\eta^{ij}}{2}\frac{\partial
h_{00}}{\partial x^{j}}\frac{dx^{0}}{d\tau}\frac{dx^{0}}{d\tau}$
$\displaystyle=$ $\displaystyle 0.$ (25)
The equation (24) implies that $\frac{dx^{0}}{d\tau}$ is a function of
$h_{00}$. Since $|h_{00}|\ll 1$, $\frac{dx^{0}}{d\tau}$ could be treated as a
constant in equation (25). Then, we find from (25) that
$\frac{d^{2}x^{i}}{{dx^{0}}^{2}}=-\frac{GM}{R^{2}}\frac{x^{i}}{R},$ (26)
where ${dx^{0}}^{2}=\eta_{00}dx^{0}dx^{0}$. The formula (26) means that the
law of gravity in Finsler spacetime is similar to the Newton’s gravity. The
difference is that space length is Finslerian. It is what we expect from
Finslerian gravity, because the length difference is the major attribute of
Finsler geometry.
The Newton’s gravity or general relativity is compatible with the astronomical
tests in solar system. It means that the Minkowski spacetime well describes
the physics in solar system. It may be expected that the Finsler gravity (26)
domains in large scale. The empirical Tully-Fisher relation TF between galaxy
luminosity and rotational speed implies that the rotational speed of galaxy
has a limitation. It hints that we may consider the space length of galaxies
to be Finslerian
$\eta_{ij}=\delta_{ij}\sqrt{1-\left(\frac{GMa_{0}{y^{0}}^{4}}{(\delta_{mn}y^{m}y^{n})^{2}}\right)^{2}},$
(27)
where $a_{0}$ is the constant of MOND. In Finsler spacetime, the speed of
particle is given as $v^{i}\equiv\frac{dx^{i}}{dx^{0}}=\frac{y^{i}}{y^{0}}$.
The space length of galaxies (27) could be write into a concise form
$R=r\sqrt{1-\left(\frac{GMa_{0}}{v^{4}}\right)^{2}},$ (28)
where $r^{2}\equiv\delta_{ij}x^{i}x^{j}$ and
$v^{2}\equiv\delta_{ij}v^{i}v^{j}$. By making use of formula (26), we obtain
the approximate dynamical equation in galaxy system. It is given as
$\frac{GM}{R^{2}}=\frac{v^{2}}{R}.$ (29)
Substituting formula (28) into equation (29), we obtain that
$\frac{GM}{r\sqrt{1-\left(GMa_{0}/v^{4}\right)^{2}}}=v^{2}.$ (30)
One could find from (30) that the rotational speed of galaxies has an
asymptotic value
$\lim_{r\rightarrow\infty}v^{4}=GMa_{0}.$ (31)
The asymptotic speed stems from the Tully-Fisher relation. The formula (30)
could be written into a familiar form
$\frac{GM}{r^{2}}=\frac{v^{2}}{r}\mu\left(\frac{v^{2}}{ra_{0}}\right),$ (32)
where $\mu(x)=x/\sqrt{x^{2}+1}$ is interpolating function of MOND. It
indicates that the law of gravity (30) in Finsler spacetime is MOND and the
spacetime structure of MOND is “Tully-Fisher” like (28). It should be noticed
that the formula (30) is given in natural units ($c=1$). The term $L_{g}\equiv
GM/c^{2}$ corresponds to the typical galaxy scale where the gravity of galaxy
is dominated. The term $L_{0}\equiv c^{2}/a_{0}\approx 2\pi L_{H}\approx
10^{29}{\rm cm}$MOND , where $L_{H}$ is the Hubble radius. Thus, the constant
term $GMa_{0}$ in (30) equals ratio of galaxy scale $L_{g}$ to cosmological
scale $L_{0}$. The law of gravity (30) in Finsler spacetime hints that there
is connection between MOND and cosmology.
## IV Conclusions
In this paper, we presented a Finsler geometry origin of MOND. We showed that
the spacetime structure of galaxies may be Finslerian (28). The “Tully-Fisher”
like length (28) could be the reason of the empirical Tully-Fisher relation.
The law of gravity in galaxies was shown as formula (30). It hints that there
is connection between MOND and cosmology.
The strong and weak gravitational lensing observations of Bullet Cluster
1E0657-558Bullet could not be explained well by MOND and Bekenstein’s
relativistic version of MONDAngus . The surface density $\Sigma$-map and the
convergence $\kappa$-map of Bullet Cluster 1E0657-558 show that the center of
baryonic matters separates from the center of gravitational force, and the
distribution of gravitational force do not possess spherical symmetry. One
should notice that Finsler metric (27) satisfies
$\eta_{ij}(-y)=\eta_{ij}(y).$ (33)
Most of galaies possess spherical symmetry, so length element should satisfy
(33). There is a class of Finsler spacetime that do not satisfy (33). For
example, the Randers spacetime Randers
$F(x,y)=\sqrt{a_{\mu\nu}y^{\mu}y^{\nu}}+b_{\mu}y^{\mu}~{}.$ (34)
The space structure of Bullet Cluster may be one of this class, and its
Finslerian gravitational behavior may account for the observations of Bullet
Cluster.
###### Acknowledgements.
We would like to thank M. H. Li, S. Wang, Y. G. Jiang and H. N. Lin for useful
discussions. The work was supported by the NSF of China under Grant No.
11075166 and 11147176.
## References
* (1) J. Oort, Bull. Astron. Inst. Netherlands 6, 249 (1932).
* (2) F. Zwicky, Helv. Phys. Acta 6, 110 (1933).
* (3) V. T. Trimble, Ann. Rev. Astron. Astrophys. 25, 425 (1987).
* (4) J. N. Bahcall, C. Flynn, and A. Gould, Astrophys. J. 389, 234 (1992).
* (5) S. S. Vogt, M. Mateo, E. W. Olszewski, and M. J. Keane, Astron. J. 109, 151 (1995).
* (6) V. C. Rubin, W. K. Ford, and N. Thonnard, Astrophys. J. 238, 471 (1980).
* (7) M. Milgrom, Astrophys. J. 270, 365 (1983).
* (8) R. B. Tully and J. R. Fisher, Astr. Ap. 54, 661 (1977).
* (9) R. B. Tully and M. J. Pierce, Astrophys. J. 533, 744 (2000).
* (10) S. S. McGaugh, Astrophys. J. 632, 859 (2005).
* (11) J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004).
* (12) T. G Zlosnik, P. G Ferreira and G. D. Starkman, Phys. Rev. D 75, 044017 (2007).
* (13) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics 200, Springer, New York, 2000.
* (14) V. A. Kostelecky, Phys. Rev. D 69, 105009 (2004); Phys. Lett. B 701, 137 (2011).
* (15) M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa Shigaken, Japan 1986.
* (16) P. L. Antonelli and S. F. Rutz, Finsler Geometry, Advanced studies in Pure Mathematics 48, Sapporo (2005) p. 210 -In memory of M.Matsumoto.
* (17) Z. Szabo, Ann. Glob. Anal. Geom 34, 381 (2008).
* (18) S. S. Chern, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 95 (1948); or Selected Papers, vol. II, 194, Springer 1989.
* (19) Y. Takano, Lett. Nuovo Cimento, 10, 747 (1974).
* (20) S. Ikeda, Ann. der Phys., 44, 558 (1987).
* (21) R. Tavakol, N. van den Bergh, Phys. Lett. A 112, 23 (1985).
* (22) G. Yu. Bogoslovsky, Phys. Part. Nucl., 24, 354 (1993).
* (23) F. A. E. Pirani, Lectures on General Relativity, Brandeis Summer Institute in Theoretical Physics, Vol. 1, 1964.
* (24) S. F. Rutz, Computer Physcis Communications 115, 300 (1998).
* (25) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.
* (26) Z. Chang and X. Li, Phys. Lett. B 668, 453 (2008); X. Li and Z. Chang, arXiv:1108.3443v1 [gr-qc].
* (27) C. Pfeifer and M. N. R. Wohlfarth, arXiv:1112.5641[gr-qc].
* (28) Xin Li and Zhe Chang, arXiv: 1111.1383[gr-qc].
* (29) M. Milgrom, arXiv:0801.3133[astro-ph].
* (30) D. Clowe, S.W. Randall, and M. Markevitch, http://flamingos.astro.ufl.edu/1e0657/index.html; Nucl. Phys. B, Proc. Suppl. 173, 28 (2007).
* (31) G. W. Angus, B. Famaey and H. S. Zhao, Mon.Not.Roy.Astron. Soc. 371, 138 (2006); G. W. Angus, H. Y. Shan, H. S. Zhao and B. Famaey, Astrophys. J. Lett. 654, L13 (2007).
* (32) G. Randers, Phys. Rev. 59, 195 (1941).
|
arxiv-papers
| 2012-04-11T08:46:20 |
2024-09-04T02:49:29.624416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1204.2542"
}
|
1204.2548
|
# Superconducting Phase Transistor in Diffusive Four-terminal Ferromagnetic
Josephson Junctions
Mohammad Alidoust phymalidoust@gmail.com Department of Physics, Norwegian
University of Science and Technology, N-7491 Trondheim, Norway Granville
Sewell sewell@utep.edu Mathematics Department, University of Texas El Paso,
El Paso, TX 79968, USA Jacob Linder jacob.linder@ntnu.no Department of
Physics, Norwegian University of Science and Technology, N-7491 Trondheim,
Norway
###### Abstract
We study diffusive magnetic Josephson junctions with four superconducting
terminals in the weak proximity limit where the leads are arranged in cross
form. Employing the linearized Keldysh-Usadel technique, the anomalous Green’s
function and Josephson current are analytically obtained based on a
quasiclassical theory using the Fourier series method. The derived results may
be reduced to non-magnetic junctions by setting the exchange field equal to
zero. We find that increments of the magnetic barrier thickness may cause a
reversal of the supercurrent direction flowing into some of the leads, whereas
the direction of current-flow remains invariant at the others. The reversal
direction can be switched by tuning the perpendicular superconducting phases.
In the non-magnetic case, we find that the supercurrent flowing between the
leads in one direction can be tuned by changing the superconducting phase
difference in the perpendicular direction. These findings suggest the
possibility of constructing a nano-scale superconducting phase transistor
whose core element consists of the proposed four-terminal Josephson junction
with rich switching aspects.
###### pacs:
74.50.+r, 74.45.+c, 74.78.Na
## I introduction
When a weak link is established between two superconductors, a gradient in the
superconducting phases can drive a supercurrent through the system. This
Josephson effect cite:josephson ; cite:yason ; cite:shapiro and the
associated current-phase relation in weak links has been investigated
extensively in previous literature, see for example the comprehensive reviews
Refs. cite:likharev, and cite:golubov, (see also Refs. cite:buzdin2, and
cite:bergeret, for magnetic Josephson junctions).
The proximity effect between superconductors and normal diffusive metals was
first studied by W.L. McMillan in 1965 cite:mcmillam . It is known that the
electronic properties of a normal metal become altered when placed in
proximity to a host superconductor. For instance, the electronic spectrum of
the normal metal connected to a superconductor exhibits a minigap
cite:mcmillam ; cite:belzig ; cite:hammer1 ; cite:hammer2 ; cite:sueur_prl_10
; cite:alidoust_prb_10 . Very recently, the key properties of density of
states (DOS) of a sandwiched normal metal between superconductors were
employed in an experiment for producing a superconducting quantum interference
proximity transistor (SQUIPT) cite:giazotto . Moreover, superconductor-normal
metal-superconductor (S/N/S) Josephson junctions have been studied under non-
equilibrium conditions where two additional normal leads are connected to the
sandwiched normal layer. It has been demonstrated that this type of S/N/S
Josephson junctions is able to produce a $\pi$-junction depending on the
applied voltage to the normal sandwiched layer cite:crosser ; cite:Baselmans .
Such $\pi$-junctions may also be observed in three terminal junctions
cite:crosser ; cite:Huang .
The proximity-induced interplay between superconductivity and ferromagnetism
in hybrid structures is also known to establish intriguing physical phenomena.
The wavefunction describing the leakage of Cooper pairs inside a ferromagnet
oscillates in a damped fashion. One of the most interesting phenomena in the
proximity of ferromagnetism and superconductivity is 0-$\pi$ transition which
may occur in superconductor-ferromagnet-superconductor (S/F/S) junctions
cite:buzdin1 ; cite:buzdin2 ; cite:Garifullin ; cite:Sidorenko ; cite:Jiang .
The transition usually occurs over a narrow length $\xi_{F}=\sqrt{D_{F}/h}$ in
which $D_{F}$ and $h$ represent the diffusion constant and the exchange field
of the sandwiched ferromagnetic layer, respectively. At this crossover point,
the minimum energy of junction is switched between zero and
$\pi$-superconducting phase difference by changing the energy scales of the
system such as Thouless energy, exchange field and temperature. Also it has
been demonstrated that the spin-flip scattering may render the junction energy
minimum from 0 to $\pi$ cite:ryazanov1 ; cite:buzdin2 ; cite:buzdin4 ;
cite:linder_prb_08 and that the supercurrent itself may become spin-polarized
if the magnetization texture is inhomogeneous cite:spin_josephson .
So far in the literature, the main emphasis has mostly been on one-dimensional
systems where two superconductors are coupled via e.g. a constriction or
diffusive metal. On the other hand, the interplay between multiple
superconducting terminals cite:crosser in a Josephson junction would require
an extension to higher dimensions cite:malek1 ; cite:amin2 . This in turn
complicates the analytical treatment of the system, and one is usually forced
to resort to numerical means within the diffusive regime cite:bergeret2 . It
would therefore be of interest to clarify how the transport characteristics of
a diffusive ferromagnetic Josephson junction is influenced by the presence of
multiple superconducting phase differences, and also to provide an analytical
framework for studying such phenomena. Multi-terminal Josephson point contacts
had intensively been investigated (both AC and DC characteristics) using the
Ginzburg-Landau theory cite:omel1 ; cite:omel2 ; cite:omel3 and was followed
by studying the four-terminal S/N/S Josephson junctions in the clean limit via
the Eilenberger equations cite:malek1 ; cite:malek2 ; cite:amin . Interesting
phenomena such as phase dragging (the production of phase difference between
two terminals by means of phase variation between other terminals), magnetic
flux transfer and bistable states were found due to non-local coupling and
additional degrees of freedom in such classes of Josephson junctions
cite:malek1 ; cite:malek2 ; cite:omel1 . Such point contacts also have been
fabricated and intensively studied in experiments cite:omel2 .
Motivated by this, we consider in this paper a diffusive Josephson junction
with four superconducting leads where are arranged in a cruciate form and
study the supercurrent flowing in this junction. The superconducting leads are
separated by a metal that may or may not be ferromagnetic. We use the
quasiclassical Usadel equations in the diffusive regime and formulate the
current-phase relation as a function of all the available parameters in the
system such as superconducting phases in the magnetic junction. We recover the
results of Refs. cite:malek1, and cite:malek2, obtained in the clean S/N/S
junctions: namely, when the dimensions $L$ (length) and $W$ (width) of the
sandwiched metal are comparable to each other, i.e. $L\simeq W$, the standard
sinusoidal supercurrent is strongly modified by all the condensate phases. We
also use a phenomenological Ginzburg-Landau theory to confirm our analytical
expressions obtained via the quasiclassical framework. In particular, we
demonstrate that the Josephson current flowing between leads along one axis
may be tuned via the superconducting phase gradient in the perpendicular
direction.
Moreover, we find that increments of the magnetic barrier thickness may cause
a reversal of the supercurrent direction flowing into some of the leads,
whereas the direction of current-flow remains invariant at the others. These
findings are suggestive in terms of designing a nano-scale superconducting
phase transistor where current switching effects in one direction is possible
by variation of macroscopic superconducting phase in the perpendicular
direction as has also been pointed out in Ref. cite:amin, and cite:amin2,
for ballistic contacts.
The paper is organized as follows. In Sec. II we present our main analytical
findings. In Subsect. II.1 the basic equations of the quasiclassical method
are presented and in Subsect. II.2 the cruciate Josephson junction is studied
analytically via the Green’s function method. We formulate the current-phase
relation as a function of the four superconducting phases for a magnetic
Josephson junction. In Subsect. II.3 we confirm our results and findings via a
macroscopic Ginzburg-Landau theory. In Sec. IV we employ a ’Jacobi’ numerical
method cite:alidoust (which shall be explained in detail) and investigate the
behavior of the supercurrent which confirms our analytical derived expressions
in Subsect. II.2 and their dependencies on the superconducting U(1) phases,
also the behavior of junction is analyzed in more detail. Sec. IV is devoted
to the study of the supercurrent behavior in S/F/S four-terminal junctions as
a function of ferromagnetic barrier thickness. Concluding remarks are finally
given in Sec. V.
## II Theory and analytical discussions
We consider four superconducting leads coupled via a ferromagnetic or normal
diffusive metal. As in Fig. 1, the nano-scale diffusive metal is assumed to be
located in the $xy$ plane, where $x\in[0,L]$ and $y\in[0,W]$. The four
superconducting terminals are assumed to have equal magnitudes for the gap
$\Delta$ and are connected to each edge of the diffusive strip. The
suppression of the pair potential is neglected near interfaces due to a low
interface transparency and the superconducting phases are assumed to be
different in each of the four terminals:
$\theta_{\text{up}},\theta_{\text{down}},\theta_{\text{left}}$ and
$\theta_{\text{right}}$. One may expect that superconducting correlations
inside the system interfere, resulting in a quite complicated coherent system.
The S/F/S system is studied in the diffusive limit and current-phase
relationship is obtained at each terminal similar to clean S/N/S four-terminal
junctions cite:malek1 ; cite:malek2 . In our approach, we start with a
magnetic four-terminal Josephson junction and derive our analytical results
for the magnetic system. We then may achieve the non-magnetic Josephson
junction characteristics by setting the magnetic exchange field $h$ equal to
zero.
Figure 1: Experimental schematic setup of the cruciate Josephson junction.
The junction is assumed to lie in the $xy$ plane with interfaces located at
$x=0,L$ and $y=0,W$. The four spin-singlet superconductors have different
superconducting phases:
$\theta_{\text{up}},\theta_{\text{down}},\theta_{\text{left}}$ and
$\theta_{\text{right}}$. Exchange field h, is assumed to be oriented in the
$z$ direction perpendicular to the sandwiched layer plane.
### II.1 Microscopic Green’s function approach
In this subsection, we present basic equations of the quasiclassical Keldysh-
Usadel method. In order to study the transport properties of the proposed
four-terminal device, we employ the quasiclassical method. In the diffusive
regime, due to the existence of strong scattering sources, quasiparticles’
momentums are integrated over all directions in space. In this case, the
Eilenberger equations reduce to the Usadel equations cite:usadel . Under
equilibrium conditions, the system under consideration can be described by a
$4\times 4$ matrix propagator in Nambu space: the retarded Green’s function
$G^{R}$. The total Green’s function describing the system compactly reads
cite:mortenthesis :
$\hat{G}(R,\varepsilon,T)=\left(\begin{array}[]{cc}G^{A}&G^{K}\\\
\mathbf{0}&G^{R}\end{array}\right),\;G^{R}=\left(\begin{array}[]{cc}g^{R}&f^{R}\\\
-\tilde{f}&-\tilde{g}\end{array}\right),$ (1)
where the meaning of the $\tilde{...}$-operation depends on the notation
adopted. In our notation, it denotes complex conjugation and a change in sign
for the energy argument. The advanced and Keldysh blocks are made from
retarded block by $G^{A}=-(\tau_{3}G^{R}\tau_{3})^{{\dagger}}$ and
$G^{K}=\tanh(\beta\varepsilon)(G^{R}-G^{A})$ in which $\tau_{3}$ is the Pauli
matrix and $\beta=k_{B}T/2$. In the presence of exchange energy
$\textbf{h}=(h_{x},h_{y},h_{z})$ inside the ferromagnetic layer, the Usadel
equation can be give by;
$\displaystyle
D[\hat{\partial},\hat{G}[\hat{\partial},\hat{G}]]+i[\varepsilon\hat{\rho}_{3}+\text{diag}[\textbf{h}\cdot\underline{\sigma},(\textbf{h}\cdot\underline{\sigma})^{\tau}],\hat{G}]=0,$
(2)
where $\hat{\rho}_{3}$ and $\underline{\sigma}$ are $4\times 4$ and $2\times
2$ Pauli matrixes, respectively. Here $D$ is diffusive constant of the
sandwiched medium. Also, $\varepsilon$ is the quasiparticles’ energy which is
measured from Fermi surface.
The so-called weak proximity regime occurs in the case of very low transparent
interfaces or for temperatures near to the critical temperature of the
superconducting leads. The superconducting correlations leak into the
ferromagnetic region weakly and so the normal and anomalous Green’s functions
can be approximated by $\underline{\text{g}}\simeq\underline{1}$ and
$\underline{f}\ll\underline{1}$, respectively. In this limit one can linearize
the Usadel equation which yields a set of uncoupled complex boundary value
partial differential equations. The energy representation is used in this
paper, however, one may reach the Matsubara representation by replacing
$\varepsilon\rightarrow i\omega_{n}$, where $\omega_{n}=(2n+1)\pi k_{B}T$ are
Matsubara frequencies. For the sake of simplicity, a uniform exchange field
for the ferromagnetic layer is considered throughout the paper i.e.
$\textbf{h}=(0,0,h_{z}=h)$. In the weak proximity regime that mentioned above,
the Green’s function read cite:linder_prb_08
$\displaystyle\hat{G}^{R}\approx\begin{pmatrix}\underline{1}&\underline{f}^{R}\\\
-\underline{\tilde{f}}^{R}&-\underline{1}\\\ \end{pmatrix},$ (3)
in fact, we have expanded the Green’s function around the bulk solution
$\hat{G}_{0}$ as $\hat{G}\simeq\hat{G}_{0}+\hat{f}$, where
$\hat{G}_{0}=\text{diag(\text@underline{1},-\text@underline{1})}$
cite:bergeret . The retarded Green’s function now can be given by;
$\displaystyle\hat{G}^{R}=\begin{pmatrix}1&0&0&f^{R}_{+}(\varepsilon)\\\
0&1&f^{R}_{-}(\varepsilon)&0\\\ 0&[-f^{R}_{+}(-\varepsilon)]^{\ast}&-1&0\\\
[-f^{R}_{-}(-\varepsilon)]^{\ast}&0&0&-1\\\ \end{pmatrix}.$ (4)
If we assume that the exchange field is uniform throughout the sample and is
oriented in the $z$ direction, so the Usadel equations reduce to two
dimensional form as belows:
$\displaystyle\partial_{x}^{2}f^{R}_{\pm}(-\varepsilon)+\partial_{y}^{2}f^{R}_{\pm}(-\varepsilon)-\frac{2i(\varepsilon\mp
h)}{D}f^{R}_{\pm}(-\varepsilon)=0,$ (5)
$\displaystyle\partial_{x}^{2}[f^{R}_{\pm}(\varepsilon)]^{*}+\partial_{y}^{2}[f^{R}_{\pm}(\varepsilon)]^{*}-\frac{2i(\varepsilon\pm
h)}{D}[f^{R}_{\pm}(\varepsilon)]^{*}=0.$ (6)
We employ the Kupriyanov-Lukichev boundary conditions at F/S interfaces
cite:zaitsev and control their opacities using a parameter $\zeta$ that
depends on the resistance of the interface and the diffusive normal region;
$\zeta(\hat{G}\hat{\partial}\hat{G})\cdot\hat{\boldsymbol{n}}=[\hat{G}_{\text{BCS}}(\theta),\hat{G}],$
(7)
where $\hat{\boldsymbol{n}}$ is a unit vector denoting the perpendicular
direction to an interface. The bulk solution , $\hat{G}_{\text{BCS}}$ for a
$s$-wave superconductor is cite:mortenthesis ;
$\displaystyle\hat{G}^{R}_{\text{BCS}}(\theta)=\left(\begin{array}[]{cc}\mathbf{1}\cosh(\vartheta(\varepsilon))&i\tau_{2}\sinh(\vartheta(\varepsilon))e^{i\theta}\\\
i\tau_{2}\sinh(\vartheta(\varepsilon))e^{-i\theta}&-\mathbf{1}\cosh(\vartheta(\varepsilon))\\\
\end{array}\right),$ (10)
$\vartheta(\varepsilon)=\text{arctanh}(\frac{\mid\Delta\mid}{\varepsilon}),$
$\displaystyle s(\varepsilon)\equiv\sinh(\vartheta(\varepsilon))e^{i\theta}=$
$\displaystyle-\Delta\left\\{\frac{\text{sgn}(\varepsilon)}{\sqrt{\varepsilon^{2}-\Delta^{2}}}\Theta(\varepsilon^{2}-\Delta^{2})-\frac{i}{\sqrt{\Delta^{2}-\varepsilon^{2}}}\Theta(\Delta^{2}-\varepsilon^{2})\right\\}$
$\displaystyle c(\varepsilon)\equiv\cosh(\vartheta(\varepsilon))=$
$\displaystyle\frac{\mid\varepsilon\mid}{\sqrt{\varepsilon^{2}-\Delta^{2}}}\Theta(\varepsilon^{2}-\Delta^{2})-\frac{i\varepsilon}{\sqrt{\Delta^{2}-\varepsilon^{2}}}\Theta(\Delta^{2}-\varepsilon^{2}).$
$\Delta$ is superconducting gap in the $s$-wave superconductors and the
Heaviside step-function is denoted by $\Theta(\varepsilon)$. In this paper, we
have defined $\theta_{u}$, $\theta_{d}$, $\theta_{l}$, $\theta_{r}$ as the
condensate phases in the up, down, left and right superconductor leads,
respectively. If we now open up the compacted boundary conditions Eq. (7) at
left F/S interface for instance, $x=0$, we reach at;
$\displaystyle(\zeta\partial_{x}-c^{*}(\varepsilon))f^{R}_{\pm}(-\varepsilon)=\pm
s^{*}(\varepsilon)e^{i\theta_{l}}$
$\displaystyle(\zeta\partial_{x}-c^{*}(\varepsilon))[f^{R}_{\pm}(\varepsilon)]^{*}=\mp
s^{*}(\varepsilon)e^{-i\theta_{l}},$ (11)
and at $x=L$
$\displaystyle(\zeta\partial_{x}+c^{*}(\varepsilon))f^{R}_{\pm}(-\varepsilon)=\mp
s^{*}(\varepsilon)e^{i\theta_{r}}$
$\displaystyle(\zeta\partial_{x}+c^{*}(\varepsilon))[f^{R}_{\pm}(\varepsilon)]^{*}=\pm
s^{*}(\varepsilon)e^{-i\theta_{r}}.$ (12)
Also at $y=0$
$\displaystyle(\zeta\partial_{y}-c^{*}(\varepsilon))f^{R}_{\pm}(-\varepsilon)=\pm
s^{*}(\varepsilon)e^{i\theta_{d}}$
$\displaystyle(\zeta\partial_{y}-c^{*}(\varepsilon))[f^{R}_{\pm}(\varepsilon)]^{*}=\mp
s^{*}(\varepsilon)e^{-i\theta_{d}},$ (13)
and at $y=W$ the boundary condition takes the below form
$\displaystyle(\zeta\partial_{y}+c^{*}(\varepsilon))f^{R}_{\pm}(-\varepsilon)=\mp
s^{*}(\varepsilon)e^{i\theta_{u}}$
$\displaystyle(\zeta\partial_{y}+c^{*}(\varepsilon))[f^{R}_{\pm}(\varepsilon)]^{*}=\pm
s^{*}(\varepsilon)e^{-i\theta_{u}}.$ (14)
In the equilibrium conditions, the current density vector is given by Keldysh
block as
${\mathbf{J}}\text{(}\mathbf{R}\text{)}=J_{0}\int
d\varepsilon\text{Tr}\\{\rho_{3}(\hat{G}[\hat{\partial},\hat{G}])^{K}\\}$ (15)
here $J_{0}$ is a normalization constant. The current density vector
determines the direction and amplitude of current density inside the
sandwiched layer as a function of coordinates. If we substitute the total
Green’s function Eq. (1) into the current density relation namely, Eq. (15) we
arrive at:
$\displaystyle\mathbf{J}(\mathbf{R})=J_{0}\int_{-\infty}^{\infty}d\varepsilon\tanh(\varepsilon\beta)\left\\{f^{R}_{-}(-\varepsilon)\vec{\nabla}[f^{R}_{+}(\varepsilon)]^{*}\right.$
$\displaystyle+f^{R}_{+}(-\varepsilon)\vec{\nabla}[f^{R}_{-}(\varepsilon)]^{*}-f^{R}_{+}(\varepsilon)\vec{\nabla}[f^{R}_{-}(-\varepsilon)]^{*}-f^{R}_{-}(\varepsilon)$
$\displaystyle\vec{\nabla}[f^{R}_{+}(-\varepsilon)]^{*}+[f^{R}_{-}(-\varepsilon)]^{*}\vec{\nabla}f^{R}_{+}(\varepsilon)+[f^{R}_{+}(-\varepsilon)]^{*}\vec{\nabla}f^{R}_{-}(\varepsilon)$
$\displaystyle\left.-[f^{R}_{+}(\varepsilon)]^{*}\vec{\nabla}f^{R}_{-}(-\varepsilon)-[f^{R}_{-}(\varepsilon)]^{*}\vec{\nabla}f^{R}_{+}(-\varepsilon)\right\\}.$
(16)
To obtain total supercurrent flowing through the junction, for example at
right superconducting gate, one needs to perform an integration of Eq. (15)
over the $y$ coordinate , $I\text{(}\phi\text{)}=I_{0}\int\int
dyd\varepsilon\text{Tr}\\{\rho_{3}(\check{\text{g}}[\hat{\partial},\check{\text{g}}])^{K}\\}$.
At this point it suffices that Eqs. (5) be solved together with appropriate
boundary conditions (i.e. Eqs. (II.1), (II.1), (II.1) and (II.1)) in order to
capture the transport characteristics of the present class of Josephson
junctions in the diffusive limit.
### II.2 Analytical microscopic discussions
In this subsection we derive explicit analytical expressions describing the
supercurrent at each superconducting terminal. To this end, we consider the
weak proximity limit of diffusive regime where the Keldysh-Usadel method
yields a set of uncoupled complex elliptic partial differential equations. The
simplified Usadel equations and corresponding boundary conditions are give by
Eqs. (5), (6), (II.1), (II.1), (II.1) and (II.1). For simplicity in our
analytical calculations we exclude first-order terms of the anomalous Green’s
function in the Kupryianov-Lukichev boundary conditions, Eq. (7). We use the
Fourier series method in the presence of non-homogenous boundary conditions
and obtain analytical solutions for the Usadel equations. The method leads a
somewhat lengthy solutions, for instance one of the anomalous components of
Green’s function namely, $f_{+}^{R}(\varepsilon)$ after long calculations is
given by Eq. (17);
$\displaystyle f_{+}^{R}(\varepsilon)$
$\displaystyle=-\left\\{\frac{\Delta\text{sgn}(\varepsilon)}{\sqrt{\varepsilon^{2}-\Delta^{2}}}\Theta(\varepsilon^{2}-\Delta^{2})-\frac{i\Delta}{\sqrt{\Delta^{2}-\varepsilon^{2}}}\Theta(\Delta^{2}-\varepsilon^{2})\right\\}\left\\{\frac{e^{i\theta_{l}}}{L\zeta}(x-\frac{x^{2}}{2L}+\frac{D}{2iL(\varepsilon+h)}-\frac{L}{3}-\right.$
(17) $\displaystyle\sum_{k=1}^{\infty}\frac{4iL(\varepsilon+h)\cos(\frac{k\pi
x}{L})}{k^{2}\pi^{2}(Dk^{2}\pi^{2}/L^{2}-2i(\varepsilon+h))})-\frac{e^{i\theta_{r}}}{L\zeta}(\frac{x^{2}}{2L}-\frac{D}{2iL(\varepsilon+h)}-\frac{L}{6}+\sum_{k=1}^{\infty}\frac{4iL(\varepsilon+h)(-1)^{k}\cos(\frac{k\pi
x}{L})}{k^{2}\pi^{2}(Dk^{2}\pi^{2}/L^{2}-2i(\varepsilon+h))})$
$\displaystyle+\frac{e^{i\theta_{d}}}{W\zeta}(y-\frac{y^{2}}{2W}+\frac{D}{2iW(\varepsilon+h)}-\frac{W}{3}-\sum_{l=1}^{\infty}\frac{4iW(\varepsilon+h)\cos(\frac{l\pi
y}{W})}{l^{2}\pi^{2}(Dl^{2}\pi^{2}/W^{2})-2i(\varepsilon+h)})-\frac{e^{i\theta_{u}}}{W\zeta}(\frac{y^{2}}{2W}-\frac{D}{2iW(\varepsilon+h)}$
$\displaystyle\left.-\frac{W}{6}+\sum_{l=1}^{\infty}\frac{4iW(\varepsilon+h)(-1)^{l}\cos(\frac{l\pi
y}{W})}{l^{2}\pi^{2}(Dl^{2}\pi^{2}/W^{2}-2i(\varepsilon+h))})\right\\}.$
The length and width of the ferromagnetic region sandwiched between the
superconductors are denoted by $L$ and $W$. As can be seen, the anomalous
component of the retarded Green’s function depends on all four condensation
phases, which in turn leads to an interference between these superconducting
phases in the Josephson current. In Eq. (II.1) there are 8 different terms of
anomalous component of Green’s function involved the supercurrent relation.
Therefore, one must find 8 similar solutions as Eq. (17) for other terms and
substitute them into the supercurrent relation Eq. (II.1) in order to obtain
the supercurrent at one terminal. To obtain analytical solutions for the total
supercurrent flowing at the other superconducting terminals, one must repeat
the latter described process. We have done so and arrived at the analytical
expressions describing the supercurrent in the system as follows. Supercurrent
at $x=0,L$ terminals are obtained as
$\displaystyle\frac{I_{x}(x=0)}{I_{0}}=\int_{-\infty}^{\infty}\frac{d\varepsilon}{\Delta_{0}}\frac{\Delta^{2}\tanh(\beta\varepsilon)}{\Delta^{2}-\varepsilon^{2}}\sum_{\sigma=\pm}\left\\{(\frac{WD}{L^{3}\zeta^{2}(\varepsilon+\sigma
h)}\right.$
$\displaystyle\left.+\frac{8WD}{L^{3}\zeta}\sum_{k=1}^{\infty}\frac{(-1)^{k}(\varepsilon+\sigma
h)}{D^{2}k^{4}\pi^{4}/L^{4}+4(\varepsilon+\sigma
h)^{2}})\sin(\theta_{l}-\theta_{r})+\right.$
$\displaystyle\left.\frac{D\sin(\theta_{l}-\theta_{u})}{LW\zeta^{2}(\varepsilon+\sigma
h)}+\frac{D\sin(\theta_{l}-\theta_{d})}{LW\zeta^{2}(\varepsilon+\sigma
h)}\right\\}$ (18)
$\displaystyle\frac{I_{x}(x=L)}{I_{0}}=\int_{-\infty}^{\infty}\frac{d\varepsilon}{\Delta_{0}}\frac{\Delta^{2}\tanh(\beta\varepsilon)}{\Delta^{2}-\varepsilon^{2}}\sum_{\sigma=\pm}\left\\{(\frac{WD}{L^{3}\zeta^{2}(\varepsilon+\sigma
h)}\right.$
$\displaystyle\left.+\frac{8WD}{L^{3}\zeta}\sum_{k=1}^{\infty}\frac{(-1)^{k}(\varepsilon+\sigma
h)}{D^{2}k^{4}\pi^{4}/L^{4}+4(\varepsilon+\sigma
h)^{2}})\sin(\theta_{l}-\theta_{r})+\right.$
$\displaystyle\left.\frac{D\sin(\theta_{d}-\theta_{r})}{LW\zeta^{2}(\varepsilon+\sigma
h)}+\frac{D\sin(\theta_{u}-\theta_{r})}{LW\zeta^{2}(\varepsilon+\sigma
h)}\right\\}$ (19)
and also at the $W=0,L$ terminals:
$\displaystyle\frac{I_{y}(y=0)}{I_{0}}=\int_{-\infty}^{\infty}\frac{d\varepsilon}{\Delta_{0}}\frac{\Delta^{2}\tanh(\beta\varepsilon)}{\Delta^{2}-\varepsilon^{2}}\sum_{\sigma=\pm}\left\\{(\frac{LD}{W^{3}\zeta^{2}(\varepsilon+\sigma
h)}\right.$
$\displaystyle\left.+\frac{8LD}{W^{3}\zeta}\sum_{l=1}^{\infty}\frac{(-1)^{l}(\varepsilon+\sigma
h)}{D^{2}l^{4}\pi^{4}/W^{4}+4(\varepsilon+\sigma
h)^{2}})\sin(\theta_{d}-\theta_{u})+\right.$
$\displaystyle\left.\frac{D\sin(\theta_{d}-\theta_{r})}{LW\zeta^{2}(\varepsilon+\sigma
h)}+\frac{D\sin(\theta_{d}-\theta_{l})}{LW\zeta^{2}(\varepsilon+\sigma
h)}\right\\}$ (20)
$\displaystyle\frac{I_{y}(y=W)}{I_{0}}=\int_{-\infty}^{\infty}\frac{d\varepsilon}{\Delta_{0}}\frac{\Delta^{2}\tanh(\beta\varepsilon)}{\Delta^{2}-\varepsilon^{2}}\sum_{\sigma=\pm}\left\\{(\frac{LD}{W^{3}\zeta^{2}(\varepsilon+\sigma
h)}\right.$
$\displaystyle\left.+\frac{8LD}{W^{3}\zeta}\sum_{l=1}^{\infty}\frac{(-1)^{l}(\varepsilon+\sigma
h)}{D^{2}l^{4}\pi^{4}/W^{4}+4(\varepsilon+\sigma
h)^{2}})\sin(\theta_{d}-\theta_{u})+\right.$
$\displaystyle\left.\frac{D\sin(\theta_{l}-\theta_{u})}{LW\zeta^{2}(\varepsilon+\sigma
h)}+\frac{D\sin(\theta_{r}-\theta_{u})}{LW\zeta^{2}(\varepsilon+\sigma
h)}\right\\}$ (21)
$\sigma=\pm$ comes from the spin-dependent nature of the ferromagnetic
material which is sandwiched between the four superconducting terminals. To be
more specific, $I_{x}(x=0)$, $I_{x}(x=L)$, $I_{y}(y=0)$ and $I_{y}(y=W)$
represent the Josephson current in the $x$ direction at $x=0,L$ and $y$
direction at $y=0,W$, respectively. The above currents involve three
sinusoidal terms whose arguments include phase differences of the lead which
supercurrent is being calculated at and the three other terminals. As
expected, the obtained supercurrents show explicitly that this interfering
terms in the $x$ and $y$ directions vanish for large $L$ and $W$,
respectively. This fact is also found in ballistic junctions cite:malek1 ;
cite:malek2 . In these two limits, either large $L$ or $W$, the system takes
on quasi-one dimensional features and we recover the well-known standard
sinusoidal Josephson relation for the supercurrent. However, in the opposite
regime where $L\simeq W$, the proximity-induced order parameters from the
superconducting terminals overlap substantially and additional terms compared
to the one dimensional case appear in the expressions for the supercurrent. As
we shall see, the supercurrent can behave strongly different from one
dimensional junctions as a function of the phase in one superconducting
terminal due to this overlap. In fact, the supercurrent is a function of a
superposition of sinusoidal phase differences between the different
superconducting leads and one may express the supercurrent relations as
$I(x_{i})=\sum_{j}I_{j}\sin(\theta_{i}-\theta{j})$ in weakly coupled systems
cite:omel1 ; cite:omel2 ; cite:omel3 ; cite:malek1 ; cite:malek2 . The
conservation of charge current is also satisfied by the current relationships
namely, Eqs. (II.2), (II.2), (II.2) and (II.2). It can be verified explicitly
that:
$\displaystyle I_{x}(x=0)+I_{y}(y=0)=I_{x}(x=L)+I_{y}(y=W).$ (22)
which constitutes the Kirchhoff law of electricity. We will proceed to
investigate and justify the obtained analytical supercurrent numerically and
study how they depend on the superconducting phases of the terminals. First,
we compare our analytical expressions for the supercurrent with the results
obtained via a macroscopic Ginzburg-Landau theory in the next subsection.
### II.3 Ginzburg-Landau approach: analytical macroscopic discussions
In this subsection, we make a complementary discussion and examine
qualitatively the quasiclassical findings of the previous subsection by
comparison with a phenomenological Ginzburg-Landau (GL) theory cite:GL . The
phenomenological approach is a macroscopic theory which is unable to explain
the microscopic mechanism underlying superconductivity, but instead describes
the macroscopic properties near a phase transition of the system by writing
the free energy as an expansion in the order parameter. We note that the
smallness of the superconducting order parameter may be compared directly with
the weak proximity effect regime in the quasiclassical theory for temperatures
near $T_{c}$. We assume here that the normal regions characteristic length
scale ($d$) satisfies $\xi\gg d$ where $\xi$ is the coherence length. In this
case the condensation wavefunctions overlap effectively via the proximity
effect. It is instructive to briefly consider first the one dimensional case,
where one may write an ansatz for the wavefunction as follows cite:likharev ;
cite:larkin :
$\displaystyle\psi=\psi_{1}\mathrm{e}^{\mathrm{i}\theta_{1}}\mathcal{X}+\psi_{2}\mathrm{e}^{\mathrm{i}\theta_{2}}(1-\mathcal{X}).$
(23)
Here, $\psi_{j}$ is the amplitude of the condensate wavefunction in region
$j=1,2$ while $\theta_{j}$ is the corresponding superconducting phase. The
function $\mathcal{X}$ is unknown, but assumed to satisfy $\mathcal{X}\to 1$
inside region 1 while $\mathcal{X}\to 0$ inside region 2. We now generalize
this ansatz to the present four-terminal two dimensional case. Assume that
deep inside the superconducting banks the order parameter is given as
$\psi=\psi_{u}e^{i\theta_{u}},\;\psi_{d}e^{i\theta_{d}},\;\psi_{l}e^{i\theta_{l}},\;\psi_{r}e^{i\theta_{r}}.$
(24)
Inside the contact region, the four condensation’s wavefunctions overlap and
consequently we expect a solution as
$\displaystyle\psi=\psi_{r}e^{i\theta_{r}}\mathcal{X}\mathcal{Y}(1-\mathcal{Y})+\psi_{l}e^{i\theta_{l}}(1-\mathcal{X})\mathcal{Y}(1-\mathcal{Y})+$
$\displaystyle\psi_{u}e^{i\theta_{u}}\mathcal{Y}\mathcal{X}(1-\mathcal{X})+\psi_{d}e^{i\theta_{d}}(1-\mathcal{Y})\mathcal{X}(1-\mathcal{X}),$
(25)
here we have generalized the mentioned one dimensional ansatz for the four-
terminal junction. The functions $\mathcal{X}$ and $\mathcal{Y}$ satisfy the
following asymptotic behavior: $\mathcal{X}\rightarrow 0$ in the left,
$\mathcal{X}\rightarrow 1$ in the right, $\mathcal{Y}\rightarrow 0$ in the
bottom and $\mathcal{Y}\rightarrow 1$ in the top superconductors. The
supercurrent density can now be defined by the second GL equation cite:larkin
; cite:likharev
$\mathbf{j}_{s}=\frac{\alpha\hbar e}{\beta
m}\text{Im}\left\\{\psi^{\ast}\nabla\psi\right\\},$ (26)
where $\alpha$ and $\beta$ are phenomenological coefficients in the GL theory.
After some calculations, we find the following expressions for
$\mathfrak{j}_{x}$ and $\mathfrak{j}_{y}$, the supercurrent components in the
$x$ and $y$ directions,
$\displaystyle\mathfrak{j}_{x}$
$\displaystyle=\mathcal{X}^{\prime}(1-\mathcal{Y})\mathcal{Y}\left\\{-\mathcal{Y}(1-\mathcal{Y})\psi_{l}\psi_{r}\sin(\theta_{l}-\theta_{r})-\right.$
(27)
$\displaystyle\left.\mathcal{X}^{2}(1-\mathcal{Y})\psi_{d}\psi_{r}\sin(\theta_{d}-\theta_{r})-\mathcal{X}^{2}\mathcal{Y}\psi_{u}\psi_{r}\sin(\theta_{u}-\theta_{r})\right.$
$\displaystyle\left.+(1-\mathcal{X})^{2}(1-\mathcal{Y})\psi_{d}\psi_{l}\sin(\theta_{d}-\theta_{l})+\right.$
$\displaystyle\left.\mathcal{Y}(1-\mathcal{X})^{2}\psi_{u}\psi_{l}\sin(\theta_{u}-\theta_{l})\right\\}$
$\displaystyle\mathfrak{j}_{y}$
$\displaystyle=\mathcal{Y}^{\prime}(1-\mathcal{X})\mathcal{X}\left\\{-\mathcal{X}(1-\mathcal{X})\psi_{u}\psi_{d}\sin(\theta_{u}-\theta_{d})-\right.$
(28)
$\displaystyle\left.\mathcal{Y}^{2}(1-\mathcal{X})\psi_{l}\psi_{u}\sin(\theta_{l}-\theta_{u})-\mathcal{Y}^{2}\mathcal{X}\psi_{r}\psi_{u}\sin(\theta_{r}-\theta_{u})\right.$
$\displaystyle\left.+(1-\mathcal{Y})^{2}(1-\mathcal{X})\psi_{l}\psi_{d}\sin(\theta_{l}-\theta_{d})+\right.$
$\displaystyle\left.\mathcal{X}(1-\mathcal{Y})^{2}\psi_{d}\psi_{r}\sin(\theta_{r}-\theta_{d})\right\\}$
in which the prime sign denotes derivation. The obtained results illustrate
that, for instance in $\mathfrak{j}_{x}$, the terms coupling the top and
bottom superconducting terminals vanish. In this way, we see that the
phenomenological GL approach produces identical dependencies on the
superconducting phase differences as the microscopic approach using
quasiclassical theory. Direct comparison with e.g. Eqs. (II.2) and (II.2) in
the appropriate limits for $\mathcal{X}$ shows consistency with Eq. (27).
## III Four terminal non-magnetic Josephson junction
In this section, we first set $h=0$ (the exchange field of ferromagnetic
layer) and consider an S/N/S junction. Basically, there are two methods for
inducing a supercurrent into our Josephson system: 1) via an external flux
where the external magnetic field penetrates the junction through a SQUID-like
geometry and 2) via a current-bias where the supercurrent is injected into the
system. A combination of these two methods is also possible by utilizing
different configurations of a multi-terminal system (for a comprehensive
investigation of such possibilities, see Refs. cite:omel1, , cite:omel2, ,
cite:omel3, , cite:malek1, , cite:malek2, ). The supercurrent at each terminal
can be generally expressed as
$I_{i}=\sum_{i,j}I_{i,j}\sin(\theta_{i}-\theta_{j})$. Thus if one is able to
tune the superconducting phases independently, the supercurrent will be a
$2\pi$-periodic function of one of the superconducting phases.
### III.1 Numerical justification of current phase relationships
In this subsection, we discuss the analytical findings obtained in the
previous section and present numerical results using a real energy
representation. In the actual plots, we consider a temperature $T=0.05T_{c}$
and also set the normal region’s length and width to $L=W\simeq 2.5\xi_{S}$.
In this representation, we normalize lengths against $\xi_{S}$ and introduce
the Thouless energy $\varepsilon_{T}=(\hbar D/L^{2})$. Also, we have
normalized the quasiparticles’ energy by the superconducting gap at zero
temperature $\Delta_{0}$ and consider units so that $\hbar=k_{B}=1$. Moreover,
we add a small imaginary number $\eta/\Delta_{0}=0.1$ to the quasiparticle
energy to account for inelastic scattering which leads to a finite lifetime
for quasiparticle excitations. Setting $\zeta=7$ ensures the validity of weak
proximity in numerical calculations. Solving numerically the resultant complex
boundary value partial differential equations, the approximate solution
components of the Usadel equation are assumed to be linear combinations of
bicubic Hermite basis functions, and required to satisfy the Usadel equations
(5) and (6) exactly at 4 collocation points in each subrectangle of a grid,
and to satisfy the boundary conditions exactly at certain boundary collocation
points. We mention in passing that we include first-order terms of the
anomalous Green’s function in the Kupryianov-Lukichev boundary conditions, as
done in Ref. cite:iver, , in contrast to the usual approximation in the
literature where such terms are discarded. By doing so, we improve the
accuracy of the analytical solution in our numerical investigations. Finally,
the linear algebraic equations resulting from the collocation method, which
are highly nonsymmetric and thus difficult to solve using iterative and sparse
direct solvers, are solved using a “Jacobi” conjugate-gradient method, which
means that the conjugate gradient method (Section 4.8 of Ref. cite:sewell2a, )
is applied to the preconditioned equations $D^{-1}A^{T}A{\bf
x}=D^{-1}A^{T}{\bf b}$, where D is the diagonal part of $A^{T}A$. For a
generalized discussion see Ref. cite:sewell2b, . The same framework was very
recently used in Ref. cite:alidoust, to study the anomalous Fraunhofer
pattern appearing in an inhomogeneous S/F/S structure.
Figure 2: Top left: Supercurrent in the $x$ direction as a function of left
condensation phase, $\theta_{\text{left}}$, at left superconductor gate i.e.
$x=0$. Top right: Supercurrent in the $x$ direction vs left superconducting
phase ,$\theta_{\text{left}}$, at right superconductor gate i.e. $x=L$. Bottom
left: Supercurrent in the $y$ direction as a function of left condensation
phase at down superconductor gate i.e. $y=0$. Bottom right: Supercurrent in
the $y$ direction vs left superconducting phase at up gate i.e. $y=W$. Here
other superconductor phases namely, $\theta_{\text{up}}$ and
$\theta_{\text{down}}$ are assumed to be zero.
In order to clarify the behavior of the supercurrent in the present four-
terminal Josephson junction with respect to condensate phases of the four
superconductors, we use the following strategy. We focus on the behavior of
the supercurrent with respect to one superconductor’s phase (the left one) and
set two phases equal to zero: $\theta_{\text{down}}=\theta_{\text{right}}=0$,
while varying $\theta_{\text{up}}$. The motivation for this is to see if the
supercurrent flowing in one direction can be tuned explicitly by the
superconducting phase difference in the transverse direction, which would
correspond to a superconducting phase transistor-like device.
In general, the supercurrent inside the normal diffusive region is described
by a vector field and depends on the position. The total flowing current is
conserved, as we have proven analytically. We focus here on the supercurrent
flowing into and out of the terminals, i.e. at the positions $x=0$, $y=0$,
$x=L$ and $y=W$ gates. The results are shown in Fig. 2 where we plot the
supercurrent at the four gates as a function of left superconducting phase
where $\theta_{u}$ is varied while $\theta_{d}=\theta_{r}=0$. The top left
frame shows the supercurrent at $x=0$ as a function of the left
superconducting phase, top right is the supercurrent at $x=L$, bottom left
frame displays the supercurrent at $y=0$, and finally the bottom right frame
shows the supercurrent at $y=W$. The standard sinusoidal current-phase
relation appears at all gates in the special case where $\theta_{u}$ is equal
to zero. This behavior can be understood by considering Eqs. (II.2), (II.2),
(II.2) and (II.2). In this case, only terms with $\sin(\theta_{l})$ survive
and the supercurrent exhibits a pure sinusoidal relation vs $\theta_{l}$. When
$\theta_{u}$ increases, the phase shift effectively adds a constant which can
be either positive or negative. In particular, the currents at $x=L$ and $y=0$
shift either upwards or downwards depending on the value of $\theta_{u}$, as
can be understood by looking at Eqs. (II.2) and (II.2): a change in
$\theta_{u}$ only varies constant terms involving $\sin(\theta_{u})$.
In contrast, variation in $\theta_{u}$ influences the currents at $x=0$ and
$y=W$ in a more complicated manner. In this case, there is an explicit
dependence on the phase difference $\theta_{l}-\theta_{u}$, which induces a
strongly non-sinusoidal behavior in the current-phase relation. Interestingly,
we see that it is possible to cancel out the current even for a finite value
of $\theta_{l}$ by choosing $\theta_{u}$ appropriately. This observation
suggests that the present four-terminal device can act as a superconducting
phase transistor where the phase difference in one direction controls the
supercurrent flowing in the perpendicular direction. The underlying mechanism
behind this is the interference between the condensate wavefunctions in the
diffusive normal region, which results in an intricate phase-dependence of the
supercurrent as shown in the analytical results.
## IV Four-terminal magnetic Josephson junction
In this section, we consider a four-terminal Josephson junction with a
ferromagnetic barrier where the exchange field of the magnetic layer is
oriented along the $z$ direction. In the usual two-terminal magnetic Josephson
junctions, an increment of the ferromagnetic barrier thickness not only
reverses the current direction at particular thicknesses but also renders the
minimum of junction energy to change from $0$ superconducting phase difference
to a $\pi$ phase. The phenomenon is so called 0-$\pi$ transition. As has been
discussed in Ref. cite:malek1, the junction energy where there are several
superconducting leads can be expressed as
$E_{J}=\sum_{j<i}\gamma_{j,i}(1-\cos(\theta_{j}-\theta_{i}))$. Here, the $i$
and $j$ indices stand for the $i$th and $j$th superconducting leads. Below, we
demonstrate that an increment in the thickness of the ferromagnet can reverse
the flow of supercurrent into a pair of the superconducting terminals (along
the direction of increment), whereas the current direction in the other
terminal pair remains unaltered.
### IV.1 The behavior of critical supercurrent as a function of magnetic
barrier thickness
Figure 3: Critical supercurrent as a function of the normalized junction
length $L/\xi_{S}$ at different superconducting gates and for various values
of $\theta_{up}$, the superconducting phase of the up terminal. Top left: at
the left superconductor gate i.e. $x=0$. Top right: at the right
superconductor gate i.e. $x=L$. Bottom left: at the down superconductor gate
i.e. $y=0$. Bottom right: at the up gate i.e. $y=W$. The other superconductor
phases are fixed at zero.
We here present a numerical study of the transport properties of four-terminal
ferromagnetic Josephson junctions. Although the numerical results are
confirmed by the analytical expressions presented in Sec. III, we include
first-order terms of the anomalous Green’s function in the Kupryianov-Lukichev
boundary conditions in contrast to the approximation used for deriving the
analytical expressions for supercurrent where such terms are dropped. We now
consider a non-zero value of the ferromagnetic exchange field $h$. For a weak,
diffusive ferromagnetic alloy such as $\text{Pd}_{x}\text{Ni}_{1-x}$, the
exchange field $h/\Delta_{0}$ is tunable by means of the doping level $x$ to
take values in the range meV to tens of meV. Here, we will fix
$h=10\Delta_{0}$, which typically places the exchange field $h$ in the range
$10$-$20$ meV. In order to investigate the effects of magnetic barrier
thickness on the supercurrent at each terminal and the influence of the
various superconducting phases, we follow a similar strategy as in the
previous section. $\theta_{l}$ is varied from $0$ to $2\pi$ where magnetic
barrier length, $L$, is being varied from $L=2\xi_{S}$ to $L=5\xi_{S}$. The
other superconducting phases are fixed at zero except $\theta_{u}$ which is
changed in order to demonstrate the possible influence of the other
superconducting phases. The critical value of the supercurrent at each
terminal is calculated separately for each value of $\theta_{u}$.
Fig. 3 indicates the behavior of critical supercurrent at each superconductor
lead as a function of normalized junction length $L/\xi_{S}$ for various
values of $\theta_{u}$. The top left frame exhibits the critical current at
left terminal. Except for $\theta_{u}=\pi$ which shows two points changing the
supercurrent direction, the other values give rise to one sign-change in the
critical current. Identical qualitative behavior appears for the current at
the right terminal except when $\theta_{u}=0$, as shown in the bottom left
frame. Top and bottom right frames exhibit the critical supercurrent vs
$L/\xi_{S}$ at the down and up terminals, respectively. The critical
supercurrent at the two terminals show a smooth function of $L/\xi_{S}$ which
is in stark contrast with the behavior of the critical supercurrent at the
left and right terminal. Thus, the increment of the junction length primarily
affects the critical supercurrent flowing into leads along the same direction
of the increment. Moreover, the direction of the current can be drastically
switched by tuning the superconducting phase of up terminal. In contrast, the
current flowing into the superconducting banks perpendicular direction to
junction length increment is left unchanged. This class of multi-terminal
ferromagnet Josephson junction then offers an interesting synthesis between 0
and $\pi$-states, and possibly $\phi$-states, due to the fact that the
coefficients $I_{j}$ can change sign depending on the junction parameters such
as $L$ and $W$.
## V Conclusions
In conclusion, we have studied a four-terminal Josephson junction where a
diffusive normal or ferromagnetic metal with sides $L$ and $W$ is sandwiched
among four $s$-wave superconductor leads. We have obtained explicit analytical
results using the quasiclassical Keldysh-Usadel method for the supercurrent in
the system. We find that the wavefunctions of the four superconductors
interfere efficiently when $L\simeq W$ and modifies the standard sinusoidal
current-phase relation which confirm previous findings in ballistic junctions.
These findings are confirmed qualitatively by using a macroscopic Ginzburg-
Landau theory. We have presented numerical results for the behavior of the
supercurrent, and demonstrated that the current flowing along one axis may be
tuned by the superconducting phase-difference along the perpendicular
direction. It is demonstrated that such four-terminal junctions can provide a
rich switching circuit element (due to additional degrees of freedom in
comparison with one-dimensional two-terminal Josephson junctions) where the
various superconducting phases influence considerably the current behavior at
the terminals. In particular, we show that a reversal in critical current
direction as a function of junction length can be strongly switched by means
of variation of superconducting phase of perpendicular terminals. The present
investigations of diffusive cruciate magnetic Josephson junction may provide
new perspectives for the design of a superconducting phase switches where can
be used in quantum circuits as switching elements.
Acknowledgments: We would like to thank K. Halterman for his generosity
regarding compiler source and also F. S. Bergeret for fruitful discussions.
## References
* (1) B. D. Josephson, Phys. Lett. 1 251 (1962).
* (2) I. K. Yason, V. M. Svistunov, I. M. Dmitrenko, Zh. Eksp. Teor. Fiz. 48, 976 (1965) [English transl. Sov. Phys. JETP 21, 650 (1965).]
* (3) S. Shapiro, Phys. Rev. Lett. 11 80 (1965).
* (4) K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979).
* (5) A. A. Golubov, M. Yu. Kupriyanov, and E. Ilichev, Rev. Mod. Phys. 76, 411 (2004).
* (6) A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).
* (7) F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005).
* (8) W. L. McMillan, Phys. Rev. 175, 537 (1968).
* (9) W. Belzig, C. Bruder and G. Sch n, Phys. Rev. B54,9443 (1996).
* (10) J. C. Hammer, J. C. Cuevas, F. S. Bergeret, and W. Belzig, Phys. Rev. B76, 064514 (2007).
* (11) J. C. Cuevas, J. Hammer, J. Kopu, J. K. Viljas, and M. Eschrig, Phys. Rev. B73, 184505 (2006).
* (12) H. le Sueur, P. Joyez, H. Pothier, C. Urbina, and D. Esteve, Phys. Rev. Lett. 100, 197002 (2008).
* (13) M. Alidoust, G. Rashedi, J. Linder, and A. Sudbø, Phys. Rev. B82, 014532 (2010).
* (14) F. Giazotto, J. T. Peltonen, M. Meschke, and J. P. Pekola, Nature Physics 6, 254 (2010).
* (15) M. S. Crosser, J. Huang, F. Pierre, P. Virtanen, T. T. Heikkil , F. K. Wilhelm, and N. O. Birge, Phys. Rev. B77, 014528 (2008).
* (16) F. S. Bergeret and J. C. Cuevas, J Low Temp Phys 153, 304 (2008).
* (17) K. D. Usadel,Phys. Rev. Lett. 25, 507 (1970);A. I. Larkin and Y. N. Ovchinnikov, in Nonequilibrium Superconductivity, edited by D. Langenberg and A. Larkin (Elsevier, Amesterdam, 1986), P. 493.
* (18) J. P. Morten, M.Sc. thesis, Norwegian University of Science and Technology (2003).
* (19) A. V. Zaitsev, Zh. Eksp. Teor. Fiz. 86, 1742 (1984) (Sov. Phys. JETP 59, 1015 (1984);M. Y. Kuprianov _et al._ , Sov. Phys. JETP 67, 1163 (1988).
* (20) V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
* (21) L.G. Aslamazov and A.I. Larkin: Pis’ma Zh. Eksp. Teor. Fiz. 48, 976 (1965).
* (22) I. B. Sperstad, J. Linder, and A. Sudbø, Phys. Rev. B78, 104509 (2008).
* (23) G. Sewell, Advances in Engineering Software41, Iss. 5, 748-753 (2010).
* (24) G. Sewell, The Numerical Solution of Ordinary and Partial Differential Equations, second edition, John Wiley $\&$ Sons, (2005).
* (25) A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, Pis ma Zh. Eksp. Teor. Fiz. 35, 147 (1982) [JETP Lett. 35, 178 (1982)]; A. I. Buzdin and M. Y. Kuprianov, Pis ma Zh. Eksp. Teor. Fiz. 53, 308 (1991) [JETP Lett. 53, 321 (1991)].
* (26) J. S. Jiang, D. Davidovic, D. H. Reich, and C. L. Chien, Phys. Rev. Lett. 74, 314 (1995).
* (27) A. S. Sidorenko, V. I. Zdravkov, A. A. Prepelitsa, C. Helbig, Y. Luo, S. Gsell, M. Schreck, S. Klimm, S. Horn, L. R. Tagirov, and R. Tidecks, Ann. Phys. 12, 37 (2003).
* (28) I. A. Garifullin, D. A. Tikhonov, N. N. Garifyanov, L. Lazar, Yu. V. Goryunov, S. Ya. Khleb- nikov, L. R. Tagirov, K. Westerholt, and H. Zabel, Phys. Rev. B66, 020505(R) (2002).
* (29) V. A. Oboznov, V. V. Bolginov, A. K. Feofanov, V. V. Ryazanov, and A. I. Buzdin, Phys. Rev. Lett. 96, 197003 (2006).
* (30) M. Faure, A. I. Buzdin, A. A. Golubov, and M. Yu. Kupriyanov, Phys. Rev. B73, 064505 (2006).
* (31) J. Linder, T. Yokoyama, and A. Sudbø, Phys. Rev. B77, 174514 (2008).
* (32) M. Alidoust, J. Linder, G. Rashedi, T. Yokoyama, and A. Sudbø, Phys. Rev. B81, 014512 (2010).
* (33) R. de Bruyn Ouboter, A. N. Omelyanchouk, and E. D. Vol, Physica B 205, 153 (1995).
* (34) B. j. Vleeming, A. V. Zakarian, A. N. Omelyanchouk and R. de Bruyn Ouboter, Physica B 226, 253 (1996); B. j. Vleeming, F. J. C. van Bemmelen, M. R. Berends, R. de Bruyn Ouboter, and A. N. Omelyanchouk, Physica B 262 , 296 (1999)
* (35) R. de Bruyn Ouboter, and A. N. Omelyanchouk, Superlattice and Mircrostructures vol. 25, No. 5/6, (1999).
* (36) M. Zareyan, and A. N. Omelyanchouk, Low Temp. Phys. 25, 175 (1999).
* (37) A. N. Omelyanchouk, and M. Zareyan, Physica C B 81, 291 (2000).
* (38) M. H. S. Amin, A. N. Omelyanchouk, and M. Zagoskin, Low Temp. Phys. 27, 616 (2001).
* (39) M. H. S. Amin, A. N. Omelyanchouk, and M. Zagoskin, Physica C 372, 178 (2002); M. H. S. Amin, A. N. Omelyanchouk, A. Blais, Alec Maassen van den Brink, G. Rose, T. Duty, and A. M. Zagoskin, Physica C 368, 310 (2002).
* (40) R. de Bruyn Ouboter, A.N. Omelyanchouk, Physica C 254, 134 (1998).
* (41) J. J. A. Baselmans, A. F. Morpurgo, B. J. van Wees, and T. M. Klapwijk, Nature 397, 43 (1999).
* (42) M. Alidoust, G. Sewell, and J. Linder, Phys. Rev. Lett. 108, 037001 (2012).
* (43) Jian Huang, F. Pierre, Tero T. Heikkila, Frank K. Wilhelm, and Norman O. Birge, Phys. Rev. B66, 020507(R) (2007).
|
arxiv-papers
| 2012-04-11T20:00:01 |
2024-09-04T02:49:29.629464
|
{
"license": "Public Domain",
"authors": "Mohammad Alidoust, Granville Sewell and Jacob Linder",
"submitter": "Mohammad Alidoust",
"url": "https://arxiv.org/abs/1204.2548"
}
|
1204.2692
|
# Asynchronous Physical-layer Network Coding Scheme for Two-way OFDM Relay
Xiaochen Xia, Kui Xu, Youyun Xu Institute of Communication Engineering,
PLAUST
Nanjing 210007, P. R. China
Email: Xia1382084@gmail.com
###### Abstract
In two-way OFDM relay, carrier frequency offsets (CFOs) between relay and
terminal nodes introduce severe inter-carrier interference (ICI) which
degrades the performance of traditional physical-layer network coding (PLNC).
Moreover, traditional algorithm to compute the _posteriori_ probability in the
presence of ICI would incur prohibitive computational complexity at the relay
node. In this paper, we proposed a two-step asynchronous PLNC scheme at the
relay to mitigate the effect of CFOs. In the first step, we intend to
reconstruct the ICI component, in which space-alternating generalized
expectation-maximization (SAGE) algorithm is used to jointly estimate the
needed parameters. In the second step, a channel-decoding and network-coding
scheme is proposed to transform the received signal into the XOR of two
terminals’ transmitted information using the reconstructed ICI. It is shown
that the proposed scheme greatly mitigates the impact of CFOs with a
relatively lower computational complexity in two-way OFDM relay.
## I Introduction
Nowadays, there is increasing interest in employing the idea of network coding
[1] in wireless communication to improve the system throughput [2]-[5]. The
simplest scenario in which network coding can be applied is the two-way relay
channel (TWRC), as illustrated in Fig. 1. In TWRC, two terminal nodes $T_{1}$
and $T_{2}$ exchange statistically independent information with the help of a
relay node $R$. Traditionally, this process can be achieved within four time
slots, that is, $T_{1}\to R$, $R\to T_{2}$, $T_{2}\to R$ and $R\to T_{1}$, as
illustrated in Fig. 1(a). To enhance the system throughput of TWRC, physical-
layer network coding (PLNC) has been introduced in [6]. PLNC reduces the
required time slots for one round of information exchange from four to two
comparing with the traditional protocol, as shown in Fig. 1(b).
In this paper, we consider the OFDM modulated TWRC or two-way OFDM relay
(TWOR). A key issue in practical application of PLNC in TWOR is how to deal
with the frequency asynchrony between the signals transmitted by the two
terminal nodes. That is, symbols transmitted by different terminals may arrive
at the relay node with different CFOs. Due to the impact of CFOs, traditional
channel-decoding and network-coding mapping method in [6] suffers from
severely performance degradation. Moreover, traditional algorithm [7] to
compute the _posteriori_ probability at the relay node may introduce
prohibitively expensive computation for practical implementation due to
correlations among the received samples caused by ICI. On the other hand, the
OFDM modulated PLNC assigns the same subcarrier to both terminals which is
very different from OFDMA where the subcarriers of different users are
orthogonal. That is to say, the received signal in each subcarrier is the
composition of symbols transmitted by $T_{1}$ and $T_{2}$. Due to this
observation, traditional CFO compensation methods developed for OFDMA [8][9]
are difficult to be utilized in the PLNC system. In [10], Lu investigates the
frequency asynchronous PLNC for OFDM system and proposes a method to
compensate the CFOs with the mean of two terminals’ estimated CFOs at the
relay node. Unfortunately, this scheme will not perform well when the relative
CFO between the two terminals becomes larger.
In this paper, we develop a two-step asynchronous PLNC scheme at the relay
node. Comparing with the previous work: 1) The proposed method can effectively
mitigate the effect of frequency offsets in TWOR system; 2) It can cope with
the situation that the relative CFO is larger without incurring severe
performance degradation with respect to perfectly synchronized system; 3) The
proposed scheme has a relatively lower computational complexity.
Figure 1: A system model for two-way relay channel: (a) Traditional four-slot
protocol; (b) PLNC.
_Notation_ : Lower and upper case bold symbols denote column vectors and
matrices, respectively. $(\cdot)^{T}$, $(\cdot)^{*}$, and $(\cdot)^{H}$ denote
transpose, complex conjugate, and Hermitian transpose, respectively.
$E\left(\cdot\right)$ stands expectation operation.
$\textrm{diag}\left(\cdot\right)$ denote a diagonal matrix. Let
$\hat{\vartheta}$ denote the estimate of ${\vartheta}$. Let $|\cdot|$ and
$\|\cdot\|$ denote the magnitude and the Euclidean norm, respectively. We use
$\boldsymbol{I}_{N}$ and $\boldsymbol{0}_{N\times M}$ for the $N\times N$
identity matrix and $N\times M$ matrix with all zero entries, respectively.
## II System Model
We consider the TWOR network as shown in Fig. 1, where $T_{1}$ and $T_{2}$
exchange statistically independent information with the help of node $R$. It
is assumed that all nodes are half-duplex, that is, a node cannot transmit and
receive simultaneously. It is also assumed that each node is equipped with a
single antenna and no direct link is existed between $T_{1}$ and $T_{2}$.
We consider the two-phase transmission scheme which consists of a multiple-
access (MAC) phase and a broadcasting (BC) phase as illustrated in Fig. 1(b).
During the MAC phase, terminals $T1$ and $T2$ send OFDM modulated signals to
the relay node $R$ simultaneously. Let $\boldsymbol{b}_{i}$,
$\boldsymbol{c}_{i}$ and $\boldsymbol{u}_{i}$ denote the uncoded source
vector, channel coded vector and the modulated vector of terminal $T_{i}$,
respectively. Let $N$ denote the total number of subcarriers. Let
${\boldsymbol{x}}_{i}(n)$ denote the $n$th frequency domain OFDM block of node
$T_{i}$, where
${\boldsymbol{x}}_{i}(n)=[x_{i,0}(n),x_{i,1}(n),\cdots,x_{i,N-1}(n)]^{T}$,
$i\in\\{1,2\\}$. We define $\boldsymbol{A}_{i}$ as the subcarrier allocation
matrix,
$\left[\boldsymbol{A}_{i}\right]_{q,k}=\left\\{\begin{array}[]{ll}1&\textrm{if
the $q$th subcarrier is allocated to the $k$th}\\\ &\textrm{element of
$\boldsymbol{u}_{i}(n)$}\\\ 0&\textrm{if the $q$th subcarrier is not allocated
to $T_{i}$}\end{array}\right..$ (1)
Then we have
${\boldsymbol{x}_{i}}\left(n\right)={\boldsymbol{A}_{i}}{\boldsymbol{u}_{i}}\left(n\right)$.
Notably, during the MAC phase, $T_{1}$ and $T_{2}$ are allocated a same subset
of $K$ subcarriers due to the application of PLNC, so we can obtain that
$\boldsymbol{A}_{1}=\boldsymbol{A}_{2}=\boldsymbol{A}$. Let
${\boldsymbol{h}}_{i}=[h_{i}(0),h_{i}(1),\cdots,h_{i}(L_{i}-1),{\boldsymbol{0}}_{\left(N-L_{i}\right)\times
1}^{T}]^{T}$ denote the channel impulsive response (CIR) between $T_{i}$ and
the relay node. Here we assume that the length of cyclic prefix (CP)
$N_{g}\geq\max\\{L_{1},L_{2}\\}$ to avoid the inter-block interference (IBI).
Therefore, we concentrate only on the $n$th OFDM block and omit the index $n$
in the rest of this work. Then the received signal samples at node $R$ in the
end of the MAC phase can be expressed as
${\boldsymbol{y}}_{R}={\boldsymbol{E}}(\varepsilon_{1}){\boldsymbol{F}}{\boldsymbol{X}}_{1}{\boldsymbol{D}}{\boldsymbol{h}}_{1}+{\boldsymbol{E}}(\varepsilon_{2}){\boldsymbol{F}}{\boldsymbol{X}}_{2}{\boldsymbol{D}}{\boldsymbol{h}}_{2}+\boldsymbol{w},$
(2)
in which
$\bullet$
$\boldsymbol{E}(\varepsilon_{i})=\textrm{diag}\\{1,e^{j2\pi\varepsilon_{i}/N},\cdots,e^{j2\pi(N-1)\varepsilon_{i}/N}\\}$
and $\varepsilon_{i}$ is the normalized CFO for node $T_{i}$;
$\bullet$ $\boldsymbol{F}$ is an $N\times N$ matrix with elements
$[\boldsymbol{F}]_{p,q}=\sqrt{1/N}e^{j2{\pi}pq/N}$ for $0\leq p,q\leq N-1$;
$\bullet$
${\boldsymbol{X}}_{i}=\textrm{diag}\\{x_{i,0},x_{i,1},\cdots,x_{i,N-1}\\}^{T}$
is a diagonal matrix;
$\bullet$ $\boldsymbol{D}$ is a Fourier matrix with elements
$[\boldsymbol{D}]_{p,q}=e^{-j2{\pi}pq/N}$ for $0\leq p,q\leq N-1$;
$\bullet$ ${\boldsymbol{w}}=[w(0),w(1),\cdots,w(N-1)]^{T}$ is the additive
white Gaussian noise vector with zero mean and covariance matrix
$\sigma^{2}_{w}\boldsymbol{I}_{N}.$
By multiplying both sides of (2) by matrix $\boldsymbol{F}^{H}$, we obtain the
frequency domain received samples which can be expressed as
$\displaystyle\boldsymbol{Y}_{R}$
$\displaystyle=\boldsymbol{A}^{T}\left(\boldsymbol{\Pi}_{1}\boldsymbol{S}_{1}+\boldsymbol{\Pi}_{2}\boldsymbol{S}_{2}\right)+\boldsymbol{W}$
(3)
$\displaystyle{}=\underbrace{\boldsymbol{A}^{T}\left(\boldsymbol{\Lambda}_{1}\boldsymbol{S}_{1}+\boldsymbol{\Lambda}_{2}\boldsymbol{S}_{2}\right)}_{\textrm{Desired
signal}}$
$\displaystyle{}+\underbrace{\boldsymbol{A}^{T}\left(\left(\boldsymbol{\Pi}_{1}-\boldsymbol{\Lambda}_{1}\right)\boldsymbol{S}_{1}+\left(\boldsymbol{\Pi}_{2}-\boldsymbol{\Lambda}_{2}\right)\boldsymbol{S}_{2}\right)}_{\textrm{ICI}}+\boldsymbol{W},$
in which
$\boldsymbol{\Pi}_{i}=\boldsymbol{F}^{H}\boldsymbol{E}(\varepsilon_{i})\boldsymbol{F}$
is defined as the interference matrix,
$\boldsymbol{\Lambda}_{i}=\textrm{diag}\left[\boldsymbol{\Pi}_{i}\right]$ and
$\boldsymbol{S}_{i}={\boldsymbol{X}}_{i}{\boldsymbol{D}}{\boldsymbol{h}}_{i}$.
Obviously, $\boldsymbol{\Lambda}_{i}=\boldsymbol{I}_{N}$ and
$\left(\boldsymbol{\Pi}_{i}-\boldsymbol{\Lambda}_{i}\right)=\boldsymbol{0}_{N\times
N}$ for synchronous case. In (3), we can see that, with non-zero CFOs, each
output symbol is affected by ICI from all other subcarriers due to the loss of
orthogonality among subcarriers. This results in poor performance for
traditional channel-decoding and network coding mapping method [6].
Using the proposed mapping scheme for asynchronous PLNC detailed in the next
section, relay node $R$ transforms the received superimposed signal in the
presence of ICI into the XORed massages
${\boldsymbol{b}}_{1}\oplus{\boldsymbol{b}}_{2}={\rm
T}\left(\boldsymbol{Y}_{R}\right)$. After that, in the BC phase, relay then
broadcasts ${\boldsymbol{b}}_{1}\oplus{\boldsymbol{b}}_{2}$. Both $T_{1}$ and
$T_{2}$ try to decode ${\boldsymbol{b}}_{1}\oplus{\boldsymbol{b}}_{2}$ from
their corresponding received signals. Since $T_{1}\left(T_{2}\right)$ knows
its own bits, after decoding ${\boldsymbol{b}}_{1}\oplus{\boldsymbol{b}}_{2}$,
it can extract the bits transmitted by $T_{2}\left(T_{1}\right)$ from the
XORed massages by subtracting its own information.
## III Proposed Scheme
For frequency asynchronous PLNC, a critical challenge is how to map the
received signal at the relay node into the XOR of two terminals’ transmitted
information. In this section, we present a two-step asynchronous PLNC scheme
to deal with this problem. In the first step, we intend to reconstruct the ICI
component in (3), in which SAGE algorithm is employed to jointly
estimate111The reason why we update the CFOs during the payload is two-fold:
i) It is necessary to estimate the residual CFO due to the estimation error in
the preamble; ii) For scenarios with time-varying CFOs, reconstructing the ICI
with estimates from the preamble may results in poor performance.
$\boldsymbol{\varepsilon}=\left[\varepsilon_{1},\varepsilon_{2}\right]^{T}$,
$\boldsymbol{h}=\big{[}\boldsymbol{h}_{1}^{T},\boldsymbol{h}_{2}^{T}\big{]}^{T}$
and
$\boldsymbol{X}=\left[\boldsymbol{X}_{1}^{T},\boldsymbol{X}_{2}^{T}\right]$.
Here we suppose a coarse CFO compensation has been operated before the uplink
frame, as a result, we need to concentrate only on the situation that CFOs are
less than half of the subcarrier spacing, i.e., $-1/2<\varepsilon_{i}<1/2$,
for $i=1,2$. Secondly, using the reconstructed ICI, an channel-decoding and
network-coding scheme for asynchronous PLNC is performed to map the received
signal into the XOR of two terminals’ transmitted information.
### III-A SAGE Based ICI Reconstruction
Let
$\boldsymbol{\theta}=\left[\boldsymbol{\varepsilon}^{T},\boldsymbol{h}^{T},\boldsymbol{X}^{T}\right]^{T}$
denote a set of parameters to be estimated from the observed data
$\boldsymbol{y}_{R}$ with conditional probability density function
$p(\boldsymbol{y}_{R}|\boldsymbol{\theta})$. Obviously, the maximization
problem of $p(\boldsymbol{y}_{R}|\boldsymbol{\theta})$ with respect to the
unknown parameters $\boldsymbol{\theta}$ is equivalent to the maximization of
the log-likelihood function which is given by
$\displaystyle L(\boldsymbol{\theta})=$
$\displaystyle-\frac{1}{{\sigma_{w}^{2}}}{\left\|\boldsymbol{y_{R}}-\boldsymbol{E}\left({{{\boldsymbol{\varepsilon}}_{1}}}\right)\boldsymbol{F}{{{\boldsymbol{X}}}_{1}}\boldsymbol{D}{{{\boldsymbol{h}}}_{1}}-\boldsymbol{E}\left({{{{\boldsymbol{\varepsilon}}}_{2}}}\right)\boldsymbol{F}{{{\boldsymbol{X}}}_{2}}\boldsymbol{D}{{{\boldsymbol{h}}}_{2}}\right\|^{2}}$
(4) $\displaystyle{}+const.$
Then we should consider parameter estimation from the viewpoint of maximizing
$L(\boldsymbol{\theta})$, that is
$\hat{\boldsymbol{\theta}}=\arg\mathop{\max}\limits_{\boldsymbol{\theta}}L(\boldsymbol{\theta}).$
(5)
However, direct computation of the maximization problem would require an
exhaustive search over multiple-dimensional space spanned by
$\boldsymbol{\varepsilon}=\left[\varepsilon_{1},\varepsilon_{2}\right]^{T}$,
$\boldsymbol{h}=\big{[}\boldsymbol{h}_{1}^{T},\boldsymbol{h}_{2}^{T}\big{]}^{T}$
and
$\boldsymbol{X}=\left[\boldsymbol{X}_{1}^{T},\boldsymbol{X}_{2}^{T}\right]$,
which may incur prohibitively expensive computation for practical
implementation. To reduce the computational complexity, we propose a SAGE
based scheme to estimate the multiple-dimensional parameters iteratively.
To operate the SAGE algorithm for asynchronous PLNC, we should divide the
parameters to be estimated into two groups of
$\boldsymbol{\theta}_{i}=\left[\varepsilon_{i},\boldsymbol{h}_{i}^{T},\boldsymbol{X}_{i}^{T}\right]^{T}$,
for $i=1,2$. A hidden space [11] must be chosen for each group so that the
update process of one group can be taken place while the other is kept fixed
at its latest value. Here we define the hidden space as
${\boldsymbol{y}}_{i}={\boldsymbol{E}}\left(\varepsilon_{i}\right){\boldsymbol{F}}{\boldsymbol{X}}_{i}{\boldsymbol{D}}{\boldsymbol{h}}_{i}+\boldsymbol{w}.$
(6)
In (6), we include all the noise to the hidden space of
$\boldsymbol{\theta}_{i}$. [11] has shown that such a choice is optimal to
reduce the Fisher information and increase the convergence rate.
The update process of $\boldsymbol{\theta}_{i}$, $\forall
i\in\left\\{{1,2}\right\\}$, at $(m+1)$th iteration can be described as
follow:
1) Expectation–step: In this step, we define the conditional log-likelihood
function [11] or $Q$ function of $\boldsymbol{\theta}_{i}$, that is
$Q\left(\boldsymbol{\theta}_{i}\big{|}\hat{\boldsymbol{\theta}}^{[m]}\right)\buildrel\Delta\over{=}E\left\\{\log
p\left(\boldsymbol{y}_{i}\big{|}\boldsymbol{\theta}_{i},\hat{\boldsymbol{\theta}}_{\bar{i}}^{[m]}\right)|\boldsymbol{y}_{R},\hat{\boldsymbol{\theta}}^{[m]}\right\\},$
(7)
in which
$p\left(\boldsymbol{y}_{i}\big{|}\boldsymbol{\theta}_{i},\hat{\boldsymbol{\theta}}_{\bar{i}}^{[m]}\right)$
is the conditional probability density function of $\boldsymbol{y}_{i}$,
$\displaystyle
p\left(\boldsymbol{y}_{i}\big{|}\boldsymbol{\theta}_{i},\hat{\boldsymbol{\theta}}_{\bar{i}}^{[m]}\right)=p\left(\boldsymbol{y}_{i}\big{|}\boldsymbol{\theta}_{i}\right)$
(8)
$\displaystyle{}=\frac{1}{(\pi\sigma_{w})^{N}}\exp\left\\{-\frac{1}{\sigma_{w}^{2}}\left\|\boldsymbol{y}_{i}-\boldsymbol{E}\left(\varepsilon_{i}\right)\boldsymbol{F}\boldsymbol{X}_{i}\boldsymbol{D}\boldsymbol{h}_{i}\right\|^{2}\right\\}.$
where $\bar{i}\buildrel\Delta\over{=}\left\\{1,2\right\\}/\left\\{i\right\\}$.
Substitute (8) into (7) and remove the terms that do not relate to
$\boldsymbol{\theta}_{i}$, we can rewrite (7) as
$Q\left(\boldsymbol{\theta}_{i}\big{|}\hat{\boldsymbol{\theta}}^{[m]}\right)=\textrm{Re}\left\\{\left(\hat{\boldsymbol{y}}_{i}^{[m]}\right)^{H}\boldsymbol{E}\left(\varepsilon_{i}\right)\boldsymbol{F}\boldsymbol{X}_{i}\boldsymbol{D}\boldsymbol{h}_{i}\right\\},$
(9)
in which $\hat{\boldsymbol{y}}_{i}^{[m]}$ is the estimate of
$\boldsymbol{y}_{i}$ at the $(m+1)$th iteration and
$\hat{\boldsymbol{y}}_{i}^{[m]}=\boldsymbol{y}_{R}-\boldsymbol{E}\left(\hat{\varepsilon}_{\bar{i}}^{[m]}\right)\boldsymbol{F}\hat{\boldsymbol{X}}_{\bar{i}}^{[m]}\boldsymbol{D}\hat{\boldsymbol{h}}_{\bar{i}}^{[m]}.$
(10)
2) Maximization–step: In this step, we update the value of $\varepsilon_{i}$,
$\boldsymbol{h}_{i}$ and $\boldsymbol{X}_{i}$ sequentially. The channel
estimation at the $(m+1)$th iteration can be obtained by maximizing (9) with
respect to $\boldsymbol{h}_{i}$ while fixing $\hat{\varepsilon}_{i}$ and
$\hat{\boldsymbol{X}}_{i}$ to their latest estimates, i.e.,
$\displaystyle\hat{\boldsymbol{h}}_{i}^{[m+1]}$
$\displaystyle=\arg\mathop{\max}\limits_{{\boldsymbol{h}_{i}}}\textrm{Re}\left\\{\left(\boldsymbol{y}_{i}^{[m]}\right)^{H}\boldsymbol{E}\left(\hat{\varepsilon}_{i}^{[m]}\right)\boldsymbol{F}\hat{\boldsymbol{X}}_{i}^{[m]}\boldsymbol{D}\boldsymbol{h}_{i}\right\\}$
(11)
$\displaystyle{}=\boldsymbol{K}\boldsymbol{D}^{H}\left(\hat{\boldsymbol{X}}_{i}^{[m]}\right)^{H}\boldsymbol{F}^{H}\boldsymbol{E}^{H}\left(\hat{\varepsilon}_{i}^{[m]}\right)\boldsymbol{y}_{i}^{[m]},$
in which
$\boldsymbol{K}=\left(\sigma_{w}^{2}\boldsymbol{R}_{i}^{-1}+\boldsymbol{D}^{H}\left(\hat{\boldsymbol{X}}_{i}^{[m]}\right)^{H}\hat{\boldsymbol{X}}_{i}^{[m]}\boldsymbol{D}\right)^{-1}$
(12)
and $\boldsymbol{R}_{i}$ is the covariance matrix of $\boldsymbol{h}_{i}$,
$\boldsymbol{R}_{i}=E\left\\{\boldsymbol{h}_{i}\boldsymbol{h}_{i}^{H}\right\\}$.
It is seen that (11) is equivalent to the MMSE estimation [12] obtained with
the latest estimates of $\hat{\varepsilon}_{i}$ and
$\hat{\boldsymbol{X}}_{i}$.
The frequency offset estimation at the $(m+1)$th iteration can be obtained by
maximizing (9) with respect to $\varepsilon_{i}$ while keeping
$\hat{\boldsymbol{h}}_{i}$ and $\hat{\boldsymbol{X}}_{i}$ fixed at their
latest value, i.e.,
$\hat{\varepsilon}_{i}^{[m+1]}=\arg\mathop{\max}\limits_{{\varepsilon_{i}}}\textrm{Re}\left\\{\left(\hat{\boldsymbol{y}}_{i}^{[m]}\right)^{H}\boldsymbol{E}\left(\varepsilon_{i}\right)\boldsymbol{F}\hat{\boldsymbol{X}}_{i}^{[m]}\boldsymbol{D}\hat{\boldsymbol{h}}_{i}^{[m+1]}\right\\}.$
(13)
To cope with the nonlinear problem in (13), we assume
$\left|\varepsilon_{i}-\hat{\varepsilon}_{i}^{[m]}\right|$ is sufficient small
such that we can replace $e^{j\frac{2\pi\varepsilon_{i}n}{N}}$ with its
Taylor’s series expansion around $\hat{\varepsilon}_{i}^{[m]}$ to the second
order term, i.e.,
$\displaystyle
e^{j\frac{2\pi\Delta\varepsilon_{i}^{[m]}n}{N}}{\approx}1+j\frac{2\pi
n}{N}\Delta\varepsilon_{i}^{[m]}+\frac{1}{2}\left(j\frac{2\pi
n}{N}\right)^{2}\left(\Delta\varepsilon_{i}^{[m]}\right)^{2},$ (14)
in which
$\Delta\varepsilon_{i}^{[m]}=\varepsilon_{i}-\hat{\varepsilon}_{i}^{[m]}$.
Substitute (14) into (13) and remove the terms that do not relate to
$\varepsilon_{i}$, we have
$\displaystyle\hat{\varepsilon}_{i}^{[m+1]}=\hat{\varepsilon}_{i}^{[m]}$ (15)
$\displaystyle{}-\frac{N}{2\pi}\frac{\sum\limits_{n=0}\limits^{N-1}n\textrm{Im}\left\\{\left(\hat{\boldsymbol{y}}_{i}^{[m]}\left(n\right)\right)^{\ast}\boldsymbol{\Omega}_{i}^{[m]}\left(n\right)\exp\left\\{\frac{j2\pi\hat{\varepsilon}_{i}^{[m]}n}{N}\right\\}\right\\}}{\sum\limits_{n=0}\limits^{N-1}n^{2}\textrm{Re}\left\\{\left(\boldsymbol{y}_{i}^{[m]}\left(n\right)\right)^{\ast}\boldsymbol{\Omega}_{i}^{[m]}\left(n\right)\exp\left\\{\frac{j2\pi\hat{\varepsilon}_{i}^{[m]}n}{N}\right\\}\right\\}}.$
in which we let
$\boldsymbol{\Omega}_{i}^{[m]}=\boldsymbol{F}\hat{\boldsymbol{X}}_{i}^{[m]}\boldsymbol{D}\hat{\boldsymbol{h}}_{i}^{[m+1]}$.
In order to update the value of $\boldsymbol{X}_{i}$, we replace equation (10)
with
$\boldsymbol{y}_{i}^{[m]}=\boldsymbol{E}\left(\varepsilon_{i}\right)\boldsymbol{F}\boldsymbol{X}_{i}\boldsymbol{D}\boldsymbol{h}_{i}+\boldsymbol{I}_{\bar{i}}^{[m]}+\boldsymbol{w},$
(16)
in which
$\boldsymbol{I}_{\bar{i}}^{[m]}=\boldsymbol{E}\left(\varepsilon_{\bar{i}}\right)\boldsymbol{F}\boldsymbol{X}_{\bar{i}}\boldsymbol{D}\boldsymbol{h}_{\bar{i}}-\boldsymbol{E}\left(\hat{\varepsilon}_{\bar{i}}^{[m]}\right)\boldsymbol{F}\hat{\boldsymbol{X}}_{\bar{i}}^{[m]}\boldsymbol{D}\hat{\boldsymbol{h}}_{\bar{i}}^{[m]}$
is the residual interference from $T_{\bar{i}}$ after the $m$th iteration.
Note that $\boldsymbol{I}_{\bar{i}}^{[m]}\left(k\right)$ is a linear function
of all symbols transmitted by $T_{\bar{i}}$. Therefore, it is rational to
assume that $\boldsymbol{I}_{\bar{i}}^{[m]}$ is nearly Gaussian distributed
with zero mean and covariance matrix $\sigma_{I}^{2}\boldsymbol{I}_{N}$
following the central limit theorem. Then we obtain
$\hat{\boldsymbol{X}}_{i}^{[m+1]}\left(k,k\right)=\arg\min\limits_{\boldsymbol{X}_{i}\left(k,k\right)}\left|\hat{\boldsymbol{Y}}_{i}\left(k\right)-\boldsymbol{X}_{i}\left(k,k\right)\hat{\boldsymbol{H}}_{i}\left(k\right)\right|^{2},$
(17)
in which
$\hat{\boldsymbol{Y}}_{i}=\boldsymbol{A}^{T}\boldsymbol{F}^{H}\boldsymbol{E}^{H}\left(\hat{\varepsilon}_{i}^{[m+1]}\right)\boldsymbol{y}_{i}^{[m]}$
and $\hat{\boldsymbol{H}}_{i}=\boldsymbol{D}\hat{\boldsymbol{h}}_{i}^{[m+1]}$.
Let
$\hat{\boldsymbol{\theta}}=\left[\hat{\boldsymbol{\varepsilon}}^{T},\hat{\boldsymbol{h}}^{T},\hat{\boldsymbol{X}}^{T}\right]^{T}$
denote the final estimate of
$\boldsymbol{\theta}=\left[{\boldsymbol{\varepsilon}}^{T},{\boldsymbol{h}}^{T},{\boldsymbol{X}}^{T}\right]^{T}$
after $M$ iterations. Then the ICI component in (3) can be reconstructed by
$\hat{\boldsymbol{I}}_{R}=\left(\hat{\boldsymbol{\Pi}}_{1}-\hat{\boldsymbol{\Lambda}}_{1}\right)\hat{\boldsymbol{S}}_{1}+\left(\hat{\boldsymbol{\Pi}}_{2}-\hat{\boldsymbol{\Lambda}}_{2}\right)\hat{\boldsymbol{S}}_{2},$
(18)
in which
$\hat{\boldsymbol{\Pi}}_{i}=\boldsymbol{F}^{H}\boldsymbol{E}\left(\hat{\varepsilon}_{i}\right)\boldsymbol{F}$,
$\hat{\boldsymbol{\Lambda}}_{i}=\textrm{diag}\left(\hat{\boldsymbol{\Pi}}_{i}\right)$
and
$\hat{\boldsymbol{S}}_{i}=\hat{\boldsymbol{X}}_{i}\boldsymbol{D}\hat{\boldsymbol{h}}_{i}$.
### III-B Channel-Decoding and Network-Coding Scheme
In this subsection, we investigate the channel-decoding and network-coding
scheme for frequency asynchronous PLNC. Notably, for synchronous PLNC, the
channel-decoding and network-coding at the relay node consists of the
following two steps [7].
Step-1: In the first step, the relay maps received samples
$\boldsymbol{Y}_{R}$ into the XORed massages
$\boldsymbol{b}_{1}\oplus\boldsymbol{b}_{2}$ by function $T$. Specifically,
the relay firstly computes the _posteriori_ probability
$p\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)|\boldsymbol{Y}_{R}\left(k\right)\right)$
from the received samples. Then the log-likelihood ratios (LLRs) of network-
coded information can be obtained by
$\displaystyle\Phi\left(\boldsymbol{c}_{1}\left(l\right)\oplus\boldsymbol{c}_{2}\left(l\right)\right)=$
(19)
$\displaystyle{}\log\left({\frac{{\sum\limits_{\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)\right):\boldsymbol{c}_{1}\left(l\right)\oplus\boldsymbol{c}_{2}\left(l\right)=1}p\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)|\boldsymbol{Y}_{R}\left(k\right)\right)}}{{\sum\limits_{\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)\right):\boldsymbol{c}_{1}\left(l\right)\oplus\boldsymbol{c}_{2}\left(l\right)=0}p\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)|\boldsymbol{Y}_{R}\left(k\right)\right)}}}\right).$
Since with the same linear channel code at both source nodes, the XOR of two
codewords $\boldsymbol{c}_{1}\oplus\boldsymbol{c}_{2}$ is also a valid
codeword. Therefore, the relay can directly perform channel decoding over
$\Phi\left(\boldsymbol{c}_{1}\left(l\right)\oplus\boldsymbol{c}_{2}\left(l\right)\right)$
to obtain $\boldsymbol{b}_{1}\oplus\boldsymbol{b}_{2}$.
Step-2: In the second step, the relay re-channel encodes
$\boldsymbol{b}_{1}\oplus\boldsymbol{b}_{2}$ and broadcasts the coded
information in the BC phase.
Figure 2: BER performance versus number of SAGE iterations for different
normalized CFOs: (a) SNR=10dB; (b) SNR=15dB.
For asynchronous case, we need to compute the _posteriori_ probability
$p\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)|\boldsymbol{Y}_{R},\boldsymbol{\varepsilon},\boldsymbol{h}\right)$
in order to apply (18). However, the major difficulty occurs here is that it
takes an exhaustive search over $2(N-1)$ dimensional space to compute this
probability even for BPSK modulation with perfect knowledge of CIRs and CFOs.
This is caused by correlations in the received samples. That is, due to the
loss of orthogonality among subcarriers, each received symbol is affected by
the interference from all other subcarriers as shown in (3). Consequently,
each sample is correlated with all other samples.
To circumvent this obstacle, we present a three-step process to perform the
channel-decoding and network-coding mapping at the node $R$.
Step-1: The first step is referred to as the interference cancellation step.
In this step, we intend to remove the ICI component in (3) using the
reconstructed ICI presented in (18). By removing the ICI component from the
frequency domain received samples, we obtain
${\boldsymbol{Y}_{R}^{\prime}}=\boldsymbol{A}^{T}\left(\boldsymbol{\Lambda}_{1}\boldsymbol{S}_{1}+\boldsymbol{\Lambda}_{2}\boldsymbol{S}_{2}\right)+\boldsymbol{A}^{T}\left(\boldsymbol{I}_{R}-\hat{\boldsymbol{I}}_{R}\right)+\boldsymbol{W},$
(20)
in which
$\boldsymbol{I}_{R}=\left(\boldsymbol{\Pi}_{1}-\boldsymbol{\Lambda}_{1}\right)\boldsymbol{S}_{1}+\left(\boldsymbol{\Pi}_{2}-\boldsymbol{\Lambda}_{2}\right)\boldsymbol{S}_{2}$.
Step-2: From (18), it is seen that $\hat{\boldsymbol{I}}_{R}$ is the estimate
of ${\boldsymbol{I}}_{R}$. Here we suppose the elements of
$\boldsymbol{I}_{R}-\hat{\boldsymbol{I}}_{R}$ or the estimation errors are
sufficient small so that we can compute
$p\left(\boldsymbol{u}_{1}\left(k\right),\boldsymbol{u}_{2}\left(k\right)|\boldsymbol{Y}_{R},\boldsymbol{\varepsilon},\boldsymbol{h}\right)$
approximately by
$\displaystyle
p\left(\boldsymbol{u}_{1}\left(k\right)=a,\boldsymbol{u}_{2}\left(k\right)=b|\boldsymbol{Y}_{R},\boldsymbol{\varepsilon},\boldsymbol{h}\right)\approx$
(21)
$\displaystyle{}C\exp\left\\{{-\frac{1}{{\sigma_{w}^{2}}}{{\left|{\boldsymbol{Y}_{R}^{\prime}\left(k\right)-a{\boldsymbol{\Gamma}_{1}}\left(k\right)-b{\boldsymbol{\Gamma}_{2}}\left(k\right)}\right|}^{2}}}\right\\}$
in which
$\boldsymbol{\Gamma}_{i}=\boldsymbol{A}^{T}\hat{\boldsymbol{\Lambda}}_{i}\boldsymbol{D}\hat{\boldsymbol{h}}_{i}$
and $C$ is a constant independent of $\boldsymbol{u}_{1}$ and
$\boldsymbol{u}_{2}$. Then LLRs of the network-coded codewords could be
computed by (18). After that, channel decoder is employed to map the LLRs into
$\boldsymbol{b}_{1}\oplus\boldsymbol{b}_{2}$.
Step-3: This step is identical with the Step-2 for synchronous PLNC.
### III-C Complexity Analysis
In this subsection, we study the computational complexity of the proposed
scheme. Note that multiplications by matrices $\boldsymbol{F}$ and
$\boldsymbol{D}$ are equivalent to DFT(IDFT) operations, which could be
efficiently computed by FFT with $N{\log_{2}}N$ complex additions and
${N\mathord{\left/{\vphantom{N2}}\right.\kern-1.2pt}2}{\log_{2}}N$ complex
multiplications, respectively. Multiplications by matrices
$\boldsymbol{E}\left(\hat{\varepsilon}_{i}\right)$ and
$\hat{\boldsymbol{X}}_{i}$ require $N$ and $K$ complex multiplications,
respectively. Therefore, it is shown that the total computational complexity
for each SAGE iteration is $6N{\log_{2}}N+2N+K$ complex additions and
$3N{\log_{2}}N+5N+K$ complex multiplications. Also, we can obtain that the
computational load for (18)-(20) is $6N{\log_{2}}N+7K$ complex additions and
$3N{\log_{2}}N+28K$ complex multiplications. According to the analysis above,
it is seen that the overall complexity involved in the proposed two-step
asynchronous PLNC scheme is approximately
$\left(12M+6\right)N{\log_{2}}N+4MN+\left(2M+6\right)K$ complex additions and
$\left(6M+3\right)N{\log_{2}}N+10MN+\left(2M+28\right)K$ complex
multiplications, in which $M$ is the maximum number of SAGE iterations.
Notably, for scenarios that the CFOs are nearly constant, (15) can be computed
only in the first block after the preamble to further reduce the computational
complexity.
## IV Numerical Results
Figure 3: BER performance versus normalized CFO, SNR is set to 15dB. Figure 4:
BER performance versus SNR with constant CFOs in one uplink frame, where
$\boldsymbol{\varepsilon}$ is set to
$\boldsymbol{\varepsilon}=\left[0.1,-0.1\right]^{T}$.
In this section, simulation results of the proposed scheme are presented. For
the simulation setup, we consider a TWOR system with $N=128$ subcarriers. For
simplicity, we allocate all the subcarriers to each terminal. BPSK modulation
is assumed. Quasi-cyclic LDPC code [13] with codewords of length $1270$ and
code rate $1/2$ is chosen and all nodes are assumed to use the same channel
code. Channels between terminal nodes and relay node are modeled as six-tap
frequency-selective fading and the power delay profile of CIR is presented as
$E\left|h_{i}\left(l\right)\right|^{2}\propto e^{-l/2}$ for
$l=0,1,\cdots,L_{i}-1$ and
$\sum\limits_{l=0}^{{L_{i}-1}}E{{{\left|{{h_{i}}\left(l\right)}\right|}^{2}}=1}$.
It is assumed that the uplink frame of each terminal consists of 10 OFDM
blocks. At the beginning of each frame, a preamble [10] is employed to
estimate the CIRs which will be utilized to initial the iteration at the first
OFDM block. The initial CFOs are set to
$\hat{\boldsymbol{\varepsilon}}^{[0]}=\left[0,0\right]^{T}$. The CIRs are
supposed to be constant in one uplink frame and the final estimates of
$\boldsymbol{\varepsilon}$ and $\boldsymbol{h}$ at the last OFDM block are
utilized to initial the next block.
Figure 5: BER performance versus SNR with time-varying CFOs in one uplink
frame, where $\xi$ is set to 0.1.
In Fig. 2, the BER performance versus number of SAGE iterations for different
normalized CFOs is presented, where the SNRs are set to 10dB and 15dB. We set
the CFOs as a function of $\xi$, that is,
$\boldsymbol{\varepsilon}=\xi\cdot\left[-1,1\right]^{T}$ in which $\xi$ is
modeled as a deterministic scalar belonging to interval $\left[0,0.5\right]$.
The synchronous system with perfect knowledge of CIR is also considered to
provide a benchmark. As shown in the figure, two iterations are sufficient for
convergence of the proposed scheme. Hence, we fix the maximum number of SAGE
iterations $M$ to $2$ in the rest of this section.
In Fig. 3, we present the BER performance of the proposed scheme as a function
of normalized CFO. The normalized CFOs are also set as
$\boldsymbol{\varepsilon}=\xi\cdot\left[-1,1\right]^{T}$ in which $\xi$ varies
between -0.15 and 0.15. The compensation scheme proposed in [10] (This scheme
is referred to as the mean operation.) and the synchronous system are also
presented for comparison. As shown in the figure, the BER performance of both
the proposed scheme and the mean operation degrades as the increase of
normalized CFO. However, we can see that the proposed scheme remarkably
outperforms the mean operation as well as the curve without CFO compensation.
In Fig. 4 and Fig. 5, the BER performance versus SNR for the proposed scheme
is depicted. The CFOs are assumed to be constant during each uplink frame in
Fig. 4. So the CFO estimation in (15) is operated only in the first block
after the preamble. It is seen from the figure that the proposed scheme
effectively mitigates the effect of frequency offsets in OFDM modulated PLNC.
Particularly, the SNR loss is approximately 0.5dB at a BER of $10^{-3}$ for
the case $\boldsymbol{\varepsilon}=\left[-0.1,0.1\right]^{T}$. In Fig. 5, we
assume the CFO varies as a sinusoidal function of block index $n$ with an
amplitude of $\%5$ of the intercarrier spacing [14], i.e.,
$\varepsilon_{i}\left(n\right)=(-1)^{i}\xi+0.05\sin\left(\frac{2}{5}\pi
n\right)$, $n=1,2,\cdots,10$. Also, it can be observed that the proposed
scheme remarkably outperforms the mean operation and the scheme without CFO
compensation. The SNR loss is approximately 1.5dB at a BER of $10^{-3}$ for
$\xi=0.1$.
In Fig. 6, we compare the BER performance of the proposed scheme with the mean
operation for different relative CFOs. Here we define the relative CFO as
$\left|\varepsilon_{1}-\varepsilon_{2}\right|$. Without loss of generality, we
set $\varepsilon_{1}=0.05$ and let $\varepsilon_{2}$ vary between $-0.15$ and
$0.15$. It is seen from the figure that BER performance of the mean operation
deteriorates greatly as the relative CFO increases. However, our proposed
scheme remarkably mitigates the performance degradation at the whole
observation interval.
## V Conclusions
In this paper, we propose a two-step scheme to cope with the frequency
asynchrony in TWOR. In the proposed scheme, SAGE algorithm is applied to
reconstruct the ICI component from received signal at the relay. Then a
channel-decoding and network-coding scheme is employed to map the received
samples into the XOR of two terminals’ information. It can be shown that the
proposed scheme greatly mitigates the degradation due to CFOs with a
relatively lower complexity and is robust to larger relative CFO comparing
with the existing strategy.
## Acknowledgment
This work is supported by the Jiangsu Province Natural Science Foundation
under Grant BK2011002, Major Special Project of China (2010ZX03003-003-01) and
National Natural Science Foundation of China (No. 60972050).
Figure 6: BER performance versus normalized CFO, where $\varepsilon_{1}=0.05$
and SNR=15dB.
## References
* [1] R. Ahlswede, N. Cai, S-Y. R. Li, and R. W. Yeung, “Network Information Flow,” in _IEEE Trans. on Information Theory_ , 46(4):1204-1216, July 2000.
* [2] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, J. Crowcroft, “XORs in The Air: Practical Wireless Network Coding,” in _Proc. ACM Sigcomm Conference_ , Sep. 2006.
* [3] C. Hausl and J. Hagenauer, “Iterative network and channel decoding for the two-way relay channel,” in _Proc. IEEE International Conference on Communication (ICC 2006)_ , Istanbul, Turkey, Jun. 2006.
* [4] J.-S. Park, D.S. Lun, M. Gerla,and et al., “Performance of Network Coding in Ad Hoc Networks,” in _Proc. 25th Military Communica-tions Conf (MILCOM 2006)_ , Washington D C, 2006, 1 6.
* [5] Y. Yan, B. Zhang, H.T. Mouftah, and et al., “Practical Coding-Aware Mechanism for Opportunistic Routing in Wireless Mesh Networks,” in _Proc. IEEE International Conference on Communications (ICC 2008)_ , Beijing, 19-23 May 2008, 2871 2876
* [6] S. Zhang, S. C. Liew, and P. P. Lam, “Hot topic: physical layer network coding,” in _Proc. 12th MobiCom_ , pp. 358-365, 2006.
* [7] L. Lu and S. Liew, “Asynchronous Physical-layer Network Coding,“ _IEEE Trans. Wireless Communication_ , vol. 11, issue. 2, pp. 819-831, Dec. 2011.
* [8] Z. Cao, U. O. Tureli, Y. Yao and et al., “Frequency Synchronization for Generalized OFDMA Uplink,” _Proc. IEEE Global Telecommunications Conference (GLOBECOM 2004)_ , USA, Dec. 2004, pp. 1071-1075.
* [9] T. Yucek, and H. Arslan, “Carrier Frequency Offset Compensation with Successive Cancellation in Uplink OFDMA Systems,” _IEEE Trans. Wireless Communication_ , vol. 6, no. 10, pp. 3546-3551, Oct. 2007.
* [10] L. Lu, T. Wang, S. C. Liu and S. Zhang, “Implementation of Physical-Layer Network Coding,,” _Physical Communication_ , available at http://arxiv.org/abs/1105.3416, May 2011.
* [11] J. A. Fessler, and A. O. Hero, “Space-Alternating Generalized Expectation-Maximization Algorithm,” _IEEE Trans. Signal Processing_ , vol. 42, no. 10, pp. 2664-2672, Oct. 1994.
* [12] M. Morelliand and U. Mengali, “A comparison of pilot-aided channel estimation methods for OFDM systems,” _IEEE Trans. Signal Processing_ , vol. 49, no. 12, pp. 3065 C3073, Dec. 2001.
* [13] S. Myung,; K. Yang, J. Kim, “Quasi-cyclic LDPC codes for fast encoding ,” _IEEE Trans. Communication_ , vol. 51, no. 8, pp. 2894-2901 , Aug. 2005.
* [14] J. Beek, P. O. Borjesson, M. Boucheret and et al., “A Time and Frequency Synchronization Scheme for Multiuser OFDM,” _IEEE Jour. Select. Areas in Comm._ , vol. 17, no. 11, pp. 1900-1914, Nov. 1999.
|
arxiv-papers
| 2012-04-12T11:48:30 |
2024-09-04T02:49:29.637697
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaochen Xia, Kui Xu, Youyun Xu",
"submitter": "Xiaochen Xia",
"url": "https://arxiv.org/abs/1204.2692"
}
|
1204.2693
|
# A nice acyclic matching on the nerve of the partition lattice
Ralf Donau Fachbereich Mathematik, Universität Bremen, Bibliothekstraße 1,
28359 Bremen, Germany ruelle@math.uni-bremen.de
###### Abstract
The author has already proven that the space $\Delta(\Pi_{n})/G$ is homotopy
equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers
$n\geq 3$ and all subgroups $G\subset S_{1}\times S_{n-1}$. We wish to
construct an acyclic matching on $\Delta(\Pi_{n})/G$ that allows us to give a
basis of its cohomology. This is also a more elementary approach to
determining the number of spheres. Furthermore we give a description of the
group action by an action on the spheres. We also obtain another result that
we call Equivariant Patchwork Theorem.
###### keywords:
Discrete Morse Theory, Regular trisp, Acyclic matching, Equivariant homotopy
††journal: Topology and its Applications††journal: arXiv.org
## 1 Introduction
Let $n\geq 3$ and let $\Pi_{n}$ denote the poset consisting of all partitions
of $[n]:=\\{1,\dots,n\\}$ ordered by refinement, such that the finer partition
is the smaller partition. Let $\overline{\Pi}_{n}$ denote the poset obtained
from $\Pi_{n}$ by removing both the smallest and greatest element, which are
$\\{\\{1\\},\dots,\\{n\\}\\}$ and $\\{[n]\\}$, respectively. We consider
$\overline{\Pi}_{n}$ as a category, which is acyclic, and define
$\Delta(\overline{\Pi}_{n})$ to the nerve of the acyclic category
$\overline{\Pi}_{n}$, which is a regular trisp, see [8, Chapter 10]. The
symmetric group $S_{n}$ operates on $\Delta(\overline{\Pi}_{n})$ in a natural
way.
It is well-known that $\Delta(\overline{\Pi}_{n})$ is homotopy equivalent to a
wedge of spheres of dimension $n-3$. The following two theorems are the first
results concerning the topology of the quotient
$\Delta(\overline{\Pi}_{n})/G$, where $G$ is a non-trivial subgroup of
$S_{n}$.
###### Theorem 1.1 (Kozlov, [7]).
For any $n\geq 3$, the topological space $\Delta(\overline{\Pi}_{n})/S_{n}$ is
contractible.
We set $S_{1}\times S_{n-1}:=\\{\sigma\in S_{n}\mid\sigma(1)=1\\}$.
###### Theorem 1.2 (Donau, [2]).
Let $n\geq 3$ and $G\subset S_{1}\times S_{n-1}$ be a subgroup, then the
topological space $\Delta(\overline{\Pi}_{n})/G$ is homotopy equivalent to a
wedge of $k$ spheres of dimension $n-3$, where $k$ is the index of $G$ in
$S_{1}\times S_{n-1}$.
This leads to a general question of determining the homotopy type of
$\Delta(\overline{\Pi}_{n})/G$ for an arbitrary subgroup $G\subset S_{n}$. One
might conjecture that $\Delta(\overline{\Pi}_{n})/G$ is homotopy equivalent to
a wedge of spheres for any $n\geq 3$ and any subgroup $G\subset S_{n}$. But
unfortunately this statement is not true as the following example will show:
Let $p\geq 5$ be a prime number and let $C_{p}$ denote the subgroup of $S_{p}$
that is generated by the cycle $(1,2,\dots,p)$. Then the fundamental group of
$\Delta(\overline{\Pi}_{p})/C_{p}$ is isomorphic to
$\mathbbm{Z}/p\mathbbm{Z}$. In particular $\Delta(\overline{\Pi}_{p})/C_{p}$
cannot be homotopy equivalent to a wedge of spheres. A proof, which uses facts
about covering spaces111See [6, Chapter 1.3], can be found in [3].
In this paper we construct an $(S_{1}\times S_{n-1})$-equivariant acyclic
matching on the face poset ${\cal F}(\Delta(\overline{\Pi}_{n}))$ of
$\Delta(\overline{\Pi}_{n})$ for $n\geq 3$ such that we have a description of
the critical simplices. This induces an acyclic matching on
$\Delta(\overline{\Pi}_{n})/G$ for any subgroup $G\subset S_{1}\times
S_{n-1}$. Another benefit of having a description of the critical simplices is
that we can easily give cocycle representatives of the generators of the
cohomology, which can be useful for further analysis of the cohomology.
Equivariant acyclic matchings are also useful to find equivariant homotopies
between spaces, since there exists an equivariant version of the Main Theorem
of Discrete Morse Theory, see [5]. For the construction of an equivariant
acyclic matching we have similar tools as in Discrete Morse Theory. An
equivariant closure operator induces an equivariant trisp closure map which
induces an equivariant acyclic matching. A detailed description of the non-
equivariant versions of these tools can be found in [7].
## 2 Discrete Morse Theory
The definitions of regular trisps, partial matchings, acyclic matchings and
foundations of Discrete Morse Theory can be found in [4, 7, 8]. The following
two theorems of Discrete Morse Theory are frequently used in our proofs.
###### Theorem 2.1 (Patchwork Theorem).
Let $\varphi:P\longrightarrow Q$ be an order-preserving map and assume we have
acyclic matchings on the subposets $\varphi^{-1}(q)$ for all $q\in Q$. Then
the union of these matchings is an acyclic matching on $P$.
###### Theorem 2.2.
Let $\Delta$ be a finite regular trisp and let $M$ be an acyclic matching on
the poset ${\cal F}(\Delta)\setminus\\{\hat{0}\\}$. Let $c_{i}$ denote the
number of critical $i$-dimensional simplices of $\Delta$. Then $\Delta$ is
homotopy equivalent to a CW complex with $c_{i}$ cells of dimension $i$.
The proofs of Theorems 2.1 and 2.2 as well as further facts on Discrete Morse
Theory can be found in [8, Chapter 11].
## 3 An Equivariant Patchwork Theorem
We wish to construct an equivariant acyclic matching on a poset by gluing
together smaller equivariant acyclic matchings on parts of the poset. This is
similar to Theorem 2.1 with the difference that we also create copies of these
matchings in our construction, see Figure 1.
###### Definition 3.1.
Let $P$ be a poset and let $G$ be a group acting on $P$. Let $M$ be an acyclic
matching on $P$. We call $M$ an _$G$ -equivariant acyclic matching_ if
$(a,b)\in M$ implies $(ga,gb)\in M$ for all $g\in G$ and $a,b\in P$.
Let $G$ be a group acting on some posets $P$ and $Q$. For an element $q\in Q$
we set $G_{q}:=\\{g\in G\mid gq=q\\}$, known as the stabilizer subgroup of
$q$.
###### Proposition 3.2.
Let $\varphi:P\longrightarrow Q$ be an order-preserving $G$-map and let
$R\subset Q$ be a subset such that $R$ contains exactly one representative for
each orbit in $Q$. Assume for each $r\in R$ we have an $G_{r}$-equivariant
acyclic matching $M_{r}$ on $\varphi^{-1}(r)$. For $r\in R$, let $C_{r}$
denote the set of critical elements of $M_{r}$. Then we have an
$G$-equivariant acyclic matching on $P$ such that
$\bigcup_{g\in G,r\in R}gC_{r}$
is the set of critical elements.
Let $r\in R$ and assume $G_{r}$ acts transitively on $C_{r}$. Then $G$ acts
transitively on
$\bigcup_{g\in G}gC_{r}$
###### Proof.
We define acyclic matchings on the fibers of $\varphi$ as follows. For each
$q\in Q$ we choose $r\in R$ and $g\in G$ with $gr=q$. The map
$g:\varphi^{-1}(r)\longrightarrow\varphi^{-1}(q)$, which is an isomorphism of
posets, induces an acyclic matching on $\varphi^{-1}(q)$. If we choose another
$h\in G$ with $hr=q$, then we obtain the same matching. By Theorem 2.1 the
union of these acyclic matchings is an acyclic matching which is
$G$-equivariant by construction. The second statement is easy to see. ∎
Figure 1: A simple example: An $\mathbbm{Z}_{2}$-equivariant acyclic matching
composed of acyclic matchings on the fibers of $0$ and $1$. The matching pair
in the fiber of $1$ is copied to the fiber of $2$. $\mathbbm{Z}_{2}$ acts on
both posets by reflection across the vertical line.
###### Remark 3.3.
Let $G$ be a group acting on a regular trisp $\Delta$. Assume we have an
$G$-equivariant acyclic matching $M$ on ${\cal
F}(\Delta)\setminus\\{\hat{0}\\}$. Let $C$ be the set of critical simplices.
Clearly we have an action of $G$ on $C$. Let $H\subset G$ be a subgroup. Then
$M/H$ is an acyclic matching on ${\cal F}(\Delta/H)\setminus\\{\hat{0}\\}$,
where $C/H$ is the set of critical simplices. In particular, if $\Delta$ is
$G$-collapsible, then $\Delta/H$ is collapsible. Furthermore if $H$ is a
normal subgroup, then the acyclic matching $M/H$ is $(G/H)$-equivariant.
We also have an equivariant version of the Main Theorem of Discrete Morse
Theory.
###### Theorem 3.4 (Freij, [5]).
Let $G$ be a finite group. Let $\Delta$ be a finite regular $G$-trisp and let
$M$ be a $G$-equivariant acyclic matching on the poset ${\cal
F}(\Delta)\setminus\\{\hat{0}\\}$. Let $c_{i}$ denote the number of critical
$i$-dimensional simplices of $\Delta$. Then $\Delta$ is $G$-homotopy
equivalent to a $G$-CW complex where the cells correspond to the critical
simplices of $M$ and the action of $G$ is the same as the action on $\Delta$
restricted to the critical simplices of $M$.
## 4 The main result
Let $n\geq 3$ be a fixed natural number.
###### Definition 4.1.
Let $A$ be the set of all vertices of $\Delta(\overline{\Pi}_{n})$ where all
blocks not containing $1$ are singleton. We define the following set of
simplices of $\Delta(\overline{\Pi}_{n})$, see Figure 2.
$C_{n}:=\\{\sigma\in{\cal
F}(\Delta(\overline{\Pi}_{n}))\mid\text{$V(\sigma)\subset A$ and
$\dim\sigma=n-3$}\\}$
$V(\sigma)$ denotes the set of vertices of $\sigma$ and ${\cal
F}(\Delta(\overline{\Pi}_{n}))$ denotes the face poset of
$\Delta(\overline{\Pi}_{n})$. Furthermore we set $\alpha_{n}$ to the vertex
$\\{\\{1\\},\\{2,\dots,n\\}\\}$.
Figure 2: A simplex in $C_{5}$ which has dimension $2$.
###### Remark 4.2.
The cardinality of $C_{n}$ is $(n-1)!$.
###### Proposition 4.3.
There exists an $(S_{1}\times S_{n-1})$-equivariant acyclic matching on ${\cal
F}(\Delta(\overline{\Pi}_{n}))$, such that $C_{n}\cup\\{\alpha_{n}\\}$ is the
set of critical simplices.
Let $V$ be the set of all vertices where the block containing $1$ has exactly
two elements and any other block is singleton. Such a vertex can be written as
$v_{k}:=\\{\\{1,k\\},\\{2\\},\dots,\widehat{\\{k\\}},\dots,\\{n\\}\\}$
with $k\in\\{2,\ldots,n\\}$. The element with the hat above is omitted.
We define a poset $P:=V\cup\\{0\\}$ such that $0$ is the smallest element of
$P$ and the only element that is comparable with some other element. That
means $x,y\in P$, $x<y$ implies $x=0$. We define an order-preserving map
$\varphi:{\cal F}(\Delta(\overline{\Pi}_{n}))\longrightarrow P$ as follows.
Let $\sigma\in{\cal F}(\Delta(\overline{\Pi}_{n}))$, then we map $\sigma$ to
$0$ if $V(\sigma)\cap V=\emptyset$. Otherwise we map $\sigma$ to the special
vertex of $V$ that belongs to $\sigma$, which is unique. Notice that
$S_{1}\times S_{n-1}$ acts on $P$ in a natural way and $\varphi$ is a
$(S_{1}\times S_{n-1})$-map. $P$ has two orbits where one consists of one
element which is $0$. The other orbit may be represented by $v_{n}$.
###### Lemma 4.4.
There exists an $(S_{1}\times S_{n-1})$-equivariant acyclic matching on
$\varphi^{-1}(0)$, such that $\alpha_{n}$ is the only critical simplex.
###### Proof.
The proof of Lemma 4.4 is the same as the proof of Lemma 3.2 in [2] for the
case $G=\\{\operatorname{id}_{[n]}\\}$. It is easy to see that the acyclic
matching, that is constructed in this proof, is $(S_{1}\times
S_{n-1})$-equivariant. ∎
###### Proof of Proposition 4.3.
It is easy to see, that the statement is true for $n=3$. Now we assume $n>3$
and proceed by induction.
We define acyclic matchings on $\varphi^{-1}(0)$ and $\varphi^{-1}(v_{n})$ as
follows. By Lemma 4.4 there exists an $(S_{1}\times S_{n-1})$-equivariant
acyclic matching on $\varphi^{-1}(0)$, where $\alpha_{n}$ is the only critical
simplex.
We define a map
$\psi:{\cal
F}(\Delta(\overline{\Pi}_{n-1}))\longrightarrow\varphi^{-1}(v_{n})\setminus\\{v_{n}\\}$
as follows. We add $n$ to the block that contains $1$ in each partition and
append $v_{n}$ to the bottom of the chain, see Figure 3. The map $\psi$ is an
isomorphism of posets. A more general definition of $\psi$ as well as a
detailed description of its inverse can be found in the proof of Lemma 4.1 in
[2].
Figure 3: Example for the map $\psi$, where $n=5$.
Via $\psi$ we get an acyclic matching $M$ on $\varphi^{-1}(v_{n})$, where the
set of critical simplices consists of the simplices in $\psi[C_{n-1}]$, one
critical simplex $s_{n}$ consisting of the two vertices $v_{n}$ and
$\\{\\{1,n\\},\\{2,\dots,(n-1)\\}\\}$, which has dimension $1$. Additionally
we have the critical simplex that has only the vertex $v_{n}$. Finally we
match $v_{n}$ with $s_{n}$.
We have to show that $\sigma(v_{n})=v_{n}$ and $(a,b)\in M$ implies $(\sigma
a,\sigma b)\in M$ for all $\sigma\in S_{1}\times S_{n-1}$ and all
$a,b\in\varphi^{-1}(v_{n})$. Let $\sigma\in S_{1}\times S_{n-1}$.
$\sigma(v_{n})=v_{n}$ implies $\sigma(1)=1$ and $\sigma(n)=n$. We define a
$\widetilde{\sigma}\in S_{1}\times S_{n-2}$ by setting
$\widetilde{\sigma}(x):=\sigma(x)$ for $1\leq x\leq n-1$. Notice
$\sigma\psi=\psi\widetilde{\sigma}$ which implies
$\psi^{-1}\sigma=\widetilde{\sigma}\psi^{-1}$. Let $(a,b)\in M$. Clearly we
have $(v_{n},s_{n})=(\sigma v_{n},\sigma s_{n})$, hence we assume
$a\not=v_{n}$ and $b\not=s_{n}$. By the induction hypothesis, we have an
acyclic matching $\widetilde{M}$ on ${\cal F}(\Delta(\overline{\Pi}_{n-1}))$
which is $(S_{1}\times S_{n-2})$-equivariant. By the construction of $M$ we
have $(\psi(a)^{-1},\psi(b)^{-1})\in\widetilde{M}$. This implies
$(\widetilde{\sigma}\psi(a)^{-1},\widetilde{\sigma}\psi(b)^{-1})\in\widetilde{M}$,
hence $(\sigma a,\sigma b)\in M$.
By Proposition 3.2 there exists an $(S_{1}\times S_{n-1})$-equivariant acyclic
matching on ${\cal F}(\Delta(\overline{\Pi}_{n}))$ such that
$\left(\bigcup_{g\in S_{1}\times
S_{n-1}}g\psi[C_{n-1}]\right)\cup\\{\alpha_{n}\\}$
is the set of critical elements. It is easy to see that this set equals
$C_{n}\cup\\{\alpha_{n}\\}$. ∎
###### Corollary 4.5.
Let $G\subset S_{1}\times S_{n-1}$ be a subgroup. Then there exists an acyclic
matching on ${\cal F}(\Delta(\overline{\Pi}_{n}))/G$, such that the set of
critical simplices consists of the simplices in $C_{n}/G$ and $\alpha_{n}/G$.
###### Proof.
Apply Remark 3.3. ∎
###### Example 4.6.
Assume $G=S_{1}\times S_{n-1}$. The vertices of
$\Delta(\overline{\Pi}_{n})/S_{1}\times S_{n-1}$ can be indexed with number
partitions of $n$, which we may write as $v_{0}\oplus v_{1}+\dots+v_{r}$, that
distinguish the first number, i.e. $\oplus$ is non-commutative. The number on
the left side of $\oplus$, that is $v_{0}$, corresponds to the block that
contains $1$. There exists an acyclic matching on the poset ${\cal
F}(\Delta(\overline{\Pi}_{n})/S_{1}\times S_{n-1})$, where the set of critical
simplices consists of the vertex $1\oplus(n-1)$ and the unique simplex
$\sigma$ whose vertices are $v_{0}\oplus 1^{n-v_{0}}$ with
$v_{0}=2,\dots,n-1$, which has dimension $n-3$.
A slightly different proof of the result in Example 4.6, as well as a detailed
description of $\Delta(\overline{\Pi}_{n})/S_{1}\times S_{n-1}$, can be found
in [1].
## 5 Applications
Let $n\geq 3$ be a fixed natural number.
###### Corollary 5.1.
The topological space $\Delta(\overline{\Pi}_{n})$ is $(S_{1}\times
S_{n-1})$-homotopy equivalent to a wedge of $(n-1)!$ spheres of dimension
$n-3$. The spheres are indexed with the simplices in $C_{n}$, which induces an
action of $S_{1}\times S_{n-1}$ on the $(n-1)!$ spheres.
###### Proof.
Apply Theorem 3.4. ∎
###### Lemma 5.2.
$S_{1}\times S_{n-1}$ acts freely and transitively on $C_{n}$.
###### Proof.
Since $C_{n}=\bigcup_{g\in S_{1}\times S_{n-1}}g\psi[C_{n-1}]$, the action is
transitive, which follows inductively by the second statement of Proposition
3.2.
By Remark 4.2 the cardinality of $C_{n}$ is $(n-1)!$ which equals the
cardinality of $S_{1}\times S_{n-1}$. Hence the action is free. ∎
Let $G\subset S_{1}\times S_{n-1}$ be an arbitrary subgroup.
###### Remark 5.3.
The cardinality of $C_{n}/G$ is the index of $G$ in $S_{1}\times S_{n-1}$.
###### Proof.
Apply Lemma 5.2. ∎
Now Theorem 1.2 follows as a corollary. We can either apply Corollary 4.5 or
Corollary 5.1.
###### Corollary 5.4.
The topological space $\Delta(\overline{\Pi}_{n})/G$ is homotopy equivalent to
a wedge of spheres of dimension $n-3$. The number of spheres is the index of
$G$ in $S_{1}\times S_{n-1}$.
## Acknowledgments
The author would like to thank Dmitry N. Kozlov for this interesting problem,
Ragnar Freij and Giacomo d’Antonio for the helpful discussions.
## References
* [1] R. Donau, On a quotient topology of the partition lattice with forbidden block sizes, Topology and its Applications 159 (8) (2012), pp. 2052-2057.
* [2] R. Donau, Quotients of the order complex $\Delta(\overline{\Pi}_{n})$ by subgroups of the Young subgroup $S_{1}\times S_{n-1}$, Topology and its Applications 157 (16) (2010), pp. 2476-2479.
* [3] R. Donau, Quotients of the topology of the partition lattice which are not homotopy equivalent to wedges of spheres, arXiv:1202.4368v2 [math.AT] (2012).
* [4] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1) (1998), pp. 90-145.
* [5] R. Freij, Equivariant discrete Morse theory, Discrete Mathematics 309 (12) (2009), pp. 3821-3829.
* [6] A. Hatcher, Algebraic Topology, Cambridge University Press, 2008.
* [7] D.N. Kozlov, Closure maps on regular trisps, Topology and its Applications 156 (15) (2009), pp. 2491-2495.
* [8] D.N. Kozlov, Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21, Springer-Verlag Berlin Heidelberg, 2008.
|
arxiv-papers
| 2012-04-12T11:49:55 |
2024-09-04T02:49:29.643269
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Ralf Donau",
"submitter": "Ralf Donau",
"url": "https://arxiv.org/abs/1204.2693"
}
|
1204.2813
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-092 LHCb-PAPER-2012-004 June 15, 2012
Measurement of the polarization amplitudes and triple product asymmetries in
the $B^{0}_{s}\rightarrow\phi\phi$ decay
The LHCb collaboration 111Authors are listed on the following pages.
Using $1.0~{}\mbox{\,fb}^{-1}$ of $pp$ collision data collected at a centre-
of-mass energy of $\sqrt{\mathrm{s}}=7~{}{\mathrm{TeV}}$ with the LHCb
detector, measurements of the polarization amplitudes, strong phase difference
and triple product asymmetries in the $B^{0}_{s}\rightarrow\phi\phi$ decay
mode are presented. The measured values are
$\displaystyle|A_{0}|^{2}$ $\displaystyle=$ $\displaystyle 0.365\pm
0.022\,({\rm stat})\pm 0.012\,({\rm syst})\,,$ $\displaystyle|A_{\perp}|^{2}$
$\displaystyle=$ $\displaystyle 0.291\pm 0.024\,({\rm stat})\pm 0.010\,({\rm
syst})\,,$ $\displaystyle\cos(\delta_{\parallel})$ $\displaystyle=$
$\displaystyle-0.844\pm 0.068\,({\rm stat})\pm 0.029\,({\rm syst})\,,$
$\displaystyle A_{U}$ $\displaystyle=$ $\displaystyle-0.055\pm 0.036\,({\rm
stat})\pm 0.018\,({\rm syst})\,,$ $\displaystyle A_{V}$ $\displaystyle=$
$\displaystyle 0.010\pm 0.036\,({\rm stat})\pm 0.018\,({\rm syst})\,.$
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, S.
Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y.
Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F.
Archilli18,35, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G.
Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W. Baldini16, R.J.
Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10, Th.
Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28, E.
Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P. David8, P.N.Y.
David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De Cian37, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35, O. Deschamps5,
F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1, F. Domingo
Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D. Dossett45, A.
Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, S. Easo46, U.
Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38, F. Eisele11, S.
Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch. Elsasser37, D. Elsby42, D.
Esperante Pereira34, A. Falabella16,e,14, C. Färber11, G. Fardell47, C.
Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4,
V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35,
A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A.
Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S.
Gregson44, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
R. Harji50, N. Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P.
Hunt52, T. Huse49, R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41,
P. Ilten12, J. Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E.
Jans38, F. Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D.
Johnson52, C.R. Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A.
Keune36, B. Khanji6, Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38,
M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G.
Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j,
V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35,
C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33,
N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12,
C. Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35,
G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P.
Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35,
A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T.
Nikodem11, A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V.
Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M.
Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A.
Palano13,b, M. Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C.
Parkes51, C.J. Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, S.K.
Paterson50, G.N. Patrick46, C. Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos
Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe Altarelli35, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-Calero Yzquierdo33,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrolini19,i, A. Phan53, E.
Picatoste Olloqui33, B. Pie Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R.
Plackett48, S. Playfer47, M. Plo Casasus34, G. Polok23, A. Poluektov45,31, E.
Polycarpo2, D. Popov10, B. Popovici26, C. Potterat33, A. Powell52, J.
Prisciandaro36, V. Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43,
B. Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52,
M.M. Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A.
Roa Romero5, P. Robbe7, E. Rodrigues48,51, F. Rodrigues2, P. Rodriguez
Perez34, G.J. Rogers44, S. Roiser35, V. Romanovsky32, M. Rosello33,n, J.
Rouvinet36, T. Ruf35, H. Ruiz33, G. Sabatino21,k, J.J. Saborido Silva34, N.
Sagidova27, P. Sail48, B. Saitta15,d, C. Salzmann37, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, K. Sobczak5,
F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B. Souza De Paula2, B. Spaan9,
A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O. Steinkamp37, S.
Stoica26, S. Stone53,35, B. Storaci38, M. Straticiuc26, U. Straumann37, V.K.
Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36, T. Szumlak24, S.
T’Jampens4, E. Teodorescu26, F. Teubert35, C. Thomas52, E. Thomas35, J. van
Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39, S. Topp-Joergensen52, N.
Torr52, E. Tournefier4,50, S. Tourneur36, M.T. Tran36, A. Tsaregorodtsev6, N.
Tuning38, M. Ubeda Garcia35, A. Ukleja25, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34, S. Vecchi16, J.J.
Velthuis43, M. Veltri17,g, B. Viaud7, I. Videau7, D. Vieira2, X. Vilasis-
Cardona33,n, J. Visniakov34, A. Vollhardt37, D. Volyanskyy10, D. Voong43, A.
Vorobyev27, V. Vorobyev31, H. Voss10, R. Waldi55, S. Wandernoth11, J. Wang53,
D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, K.
Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47, O.
Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
In the Standard Model, the flavour-changing neutral current decay
$B^{0}_{s}\rightarrow\phi\phi$ proceeds via a $b\rightarrow s\bar{s}s$ penguin
process. Studies of the polarization amplitudes and triple product asymmetries
in this decay provide powerful tests for the presence of contributions from
processes beyond the Standard Model [1, 2, 3, 4, 5].
The $B^{0}_{s}\rightarrow\phi\phi$ decay is a pseudoscalar to vector-vector
transition. As a result, there are three possible spin configurations of the
vector meson pair allowed by angular momentum conservation. These manifest
themselves as three helicity states, with amplitudes denoted $H_{+1},H_{-1}$
and $H_{0}$. It is convenient to define linear polarization amplitudes, which
are related to the helicity amplitudes through the following transformations
$\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle H_{0}\,,$ $\displaystyle
A_{\perp}$ $\displaystyle=$ $\displaystyle\frac{H_{+1}-H_{-1}}{\sqrt{2}}\,,$
$\displaystyle A_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{H_{+1}+H_{-1}}{\sqrt{2}}\,.$ (1)
The $\phi\phi$ final state can be a mixture of $C\\!P$-even and $C\\!P$-odd
eigenstates. The longitudinal ($A_{0}$) and parallel ($A_{\parallel}$)
components are $C\\!P$-even and the perpendicular component ($A_{\perp}$) is
$C\\!P$-odd. From the V–A structure of the weak interaction, the longitudinal
component,
$f_{L}=|A_{0}|^{2}/(|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2})$, is
expected to be dominant [6, 7, 8]. However, roughly equal longitudinal and
transverse components are found in measurements of $B^{+}\rightarrow\phi
K^{*+}$, $B^{0}\rightarrow\phi K^{*0}$, $B^{+}\rightarrow\rho^{0}K^{*+}$ and
$B^{0}\rightarrow\rho^{0}K^{*0}$ decays at the B-factories [9, 10, 11, 12, 13,
14]. To explain this, large contributions from either penguin annihilation
effects [15] or final state interactions [16] have been proposed. Recent
calculations where phenomenological parameters are adjusted to account for the
data allow $f_{L}$ in the range $0.4-0.7$ [6, 7]. Another pseudoscalar to
vector-vector penguin decay is $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. A recent measurement by the LHCb
Collaboration in this decay mode has found a value of $f_{L}=0.31\pm 0.12\pm
0.04$ [17].
$\hat{n}_{1}$$\hat{n}_{2}$$B^{0}_{s}$$\Phi$$\theta_{2}$$\theta_{1}$$K^{-}$$K^{+}$$K^{-}$$K^{+}$$\phi_{1}$$\phi_{2}$
Figure 1: Decay angles for the $B^{0}_{s}\rightarrow\phi\phi$ decay, where
the $K^{+}$ momentum in the $\phi_{1,2}$ rest frame, and the parent
$\phi_{1,2}$ momentum in the rest frame of the $B^{0}_{s}$ meson span the two
$\phi$ meson decay planes, $\theta_{1,2}$ is the angle between the $K^{+}$
track momentum in the $\phi_{1,2}$ meson rest frame and the parent
$\phi_{1,2}$ momentum in the $B^{0}_{s}$ rest frame, $\Phi$ is the angle
between the two $\phi$ meson decay planes and $\hat{n}_{1,2}$ is the unit
vector normal to the decay plane of the $\phi_{1,2}$ meson.
The time-dependent differential decay rate for the
$B^{0}_{s}\rightarrow\phi\phi$ mode can be written as
$\frac{d^{4}\Gamma}{d\cos\theta_{1}d\cos\theta_{2}d\Phi
dt}\propto\sum^{6}_{i=1}K_{i}(t)f_{i}(\theta_{1},\theta_{2},\Phi)\,,$ (2)
where the helicity angles $\Omega=(\theta_{1},\theta_{2},\Phi)$ are defined in
Fig. 1. The angular functions $f_{i}(\Omega)$ are [18]
$\displaystyle f_{1}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle 4\cos^{2}\theta_{1}\cos^{2}\theta_{2}\,,$ $\displaystyle
f_{2}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1+\cos 2\Phi)\,,$
$\displaystyle f_{3}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{1}\sin^{2}\theta_{2}(1-\cos 2\Phi)\,,$
$\displaystyle f_{4}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle-2\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin 2\Phi\,,$
$\displaystyle f_{5}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\cos\Phi\,,$
$\displaystyle f_{6}(\theta_{1},\theta_{2},\Phi)$ $\displaystyle=$
$\displaystyle-\sqrt{2}\sin 2\theta_{1}\sin 2\theta_{2}\sin\Phi\,.$ (3)
The time-dependent functions $K_{i}(t)$ are given by [19]
$\displaystyle K_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}A_{0}^{2}[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{2}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}A_{\parallel}^{2}[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{3}(t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}A_{\perp}^{2}[(1-\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1+\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\mp
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{4}(t)$
$\displaystyle=$ $\displaystyle|A_{\parallel}||A_{\perp}|[\pm
e^{-\Gamma_{s}t}\\{\sin\delta_{1}\cos(\Delta m_{s}t)-\cos\delta_{1}\sin(\Delta
m_{s}t)\cos\phi_{s}\\}$
$\displaystyle-\frac{1}{2}(e^{-\Gamma_{\rm{H}}t}-e^{-\Gamma_{\rm{L}}t})\cos\delta_{1}\sin\phi_{s}]\,,$
$\displaystyle K_{5}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}|A_{0}||A_{\parallel}|\cos(\delta_{2}-\delta_{1})$
$\displaystyle[(1+\cos\phi_{s})e^{-\Gamma_{\rm{L}}t}+(1-\cos\phi_{s})e^{-\Gamma_{\rm{H}}t}\pm
2e^{-\Gamma_{s}t}\sin(\Delta m_{s}t)\sin\phi_{s}]\,,$ $\displaystyle K_{6}(t)$
$\displaystyle=$ $\displaystyle|A_{0}||A_{\perp}|[\pm
e^{-\Gamma_{s}t}\\{\sin\delta_{2}\cos(\Delta m_{s}t)-\cos\delta_{2}\sin(\Delta
m_{s}t)\cos\phi_{s}\\}$ (4)
$\displaystyle-\frac{1}{2}(e^{-\Gamma_{\rm{H}}t}-e^{-\Gamma_{\rm{L}}t})\cos\delta_{2}\sin\phi_{s}]\,,$
where the upper of the $\pm$ or $\mp$ signs refers to the $B^{0}_{s}$ meson
and the lower refers to a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson. Here, $\Gamma_{\rm L}$
and $\Gamma_{\rm H}$ are the decay widths of the light and heavy $B^{0}_{s}$
mass eigenstates,222Units are adopted such that $\hbar=1$. $\Delta m_{s}$ is
the $B^{0}_{s}$ oscillation frequency,
$\delta_{1}=\arg(A_{\perp}/A_{\parallel})$ and
$\delta_{2}=\arg(A_{\perp}/A_{0})$ are $C\\!P$-conserving strong phases and
$\phi_{s}$ is the weak $C\\!P$-violating phase. It is assumed that the weak
phases of the three polarization amplitudes are equal. The quantities
$\Gamma_{\rm H}$ and $\Gamma_{\rm L}$ correspond to the observables
$\Delta\Gamma_{s}=\Gamma_{\rm L}-\Gamma_{\rm H}$ and $\Gamma_{s}=(\Gamma_{\rm
L}+\Gamma_{\rm H})/2$. In the Standard Model, the value of $\phi_{s}$ for this
mode is expected to be very close to zero due to a cancellation between the
phases arising from mixing and decay [20].333The convention used in this
Letter is that the symbol $\phi_{s}$ refers solely to the weak phase
difference measured in the $B^{0}_{s}\rightarrow\phi\phi$ decay. A calculation
based on QCD factorization provides an upper limit of 0.02 rad for $\phi_{s}$
[21, 6]. This is different to the situation in the $B^{0}_{s}\rightarrow
J/\psi\phi$ decay, where the Standard Model predicts
$\phi_{s}(J/\psi\phi)=-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.036\pm
0.002$ rad [22]. The magnitude of both weak phase differences can be enhanced
in the presence of new physics in $B^{0}_{s}$ mixing, where recent results
from LHCb have placed stringent constraints [23]. For the
$B^{0}_{s}\rightarrow\phi\phi$ decay, new particles could also contribute in
$b\rightarrow s$ penguin loops.
To measure the polarization amplitudes, a time-integrated untagged analysis is
performed, assuming that an equal number of $B^{0}_{s}$ and $\bar{B}_{s}^{0}$
mesons are produced and that the $C\\!P$-violating phase is zero as predicted
in the Standard Model.444In the case of non-zero $\phi_{s}$ deviations from
these formulas are suppressed by a factor of $\Delta\Gamma_{s}/\Gamma_{s}$ and
hence only small variations would be observed on the fitted parameters. In
this case, the functions $K_{i}(t)$ integrate to
$\displaystyle K_{1}$ $\displaystyle=$ $\displaystyle|A_{0}|^{2}/\Gamma_{\rm
L}\,,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle|A_{\parallel}|^{2}/\Gamma_{\rm L}\,,$ $\displaystyle K_{3}$
$\displaystyle=$ $\displaystyle|A_{\perp}|^{2}/\Gamma_{\rm H}\,,$
$\displaystyle K_{4}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
K_{5}$ $\displaystyle=$
$\displaystyle|A_{0}||A_{\parallel}|\cos(\delta_{\parallel})/\Gamma_{\rm
L}\,,$ $\displaystyle K_{6}$ $\displaystyle=$ $\displaystyle 0\,,$ (5)
where the strong phase difference is defined by
$\delta_{\parallel}\equiv\delta_{2}-\delta_{1}=\arg(A_{\parallel}/A_{0})$ and
the time integration assumes uniform time acceptance.
In addition, a search for physics beyond the Standard Model is performed by
studying the triple product asymmetries [1, 2, 3] in the
$B^{0}_{s}\rightarrow\phi\phi$ decay. Non-zero values of these quantities can
be either due to $T$-violation or final-state interactions. Assuming $C\\!PT$
conservation, the former case implies that $C\\!P$ is violated.
Experimentally, the extraction of the triple product asymmetries is
straightforward and provides a measure of $C\\!P$ violation that does not
require flavour tagging or a time-dependent analysis.
There are two observable triple products denoted $U=\sin(2\Phi)/2$ and
$V=\pm\sin(\Phi)$, where the positive sign is taken if the $T$-even quantity
$\cos\theta_{1}\cos\theta_{2}\geq 0$ and the negative sign otherwise. These
variables correspond to the $T$-odd triple products
$\displaystyle\sin\Phi$ $\displaystyle=$
$\displaystyle(\hat{n}_{1}\times\hat{n}_{2})\cdot\hat{p}_{1}\,,$
$\displaystyle\sin(2\Phi)/2$ $\displaystyle=$
$\displaystyle(\hat{n}_{1}\cdot\hat{n}_{2})(\hat{n}_{1}\times\hat{n}_{2})\cdot\hat{p}_{1}\,,$
(6)
where $\hat{n}_{i}$ ($i=1,2$) is a unit vector perpendicular to the $\phi_{i}$
decay plane and $\hat{p}_{1}$ is a unit vector in the direction of the
$\phi_{1}$ momentum in the $B^{0}_{s}$ rest frame. The triple products, $U$
and $V$, are proportional to the $f_{4}$ and $f_{6}$ angular functions which,
for $\phi_{s}=0$, vanish in the untagged decay rate for any value of $t$. The
$f_{4}$ and $f_{6}$ angular functions would not vanish in the presence of new
physics processes that cause the polarization amplitudes to have different
weak phases [1]. Therefore, a measurement of significant asymmetries would be
an unambiguous signal for the effects of new physics [1, 3].
The asymmetry, $A_{U}$, is defined as
$A_{U}=\frac{N_{+}-N_{-}}{N_{+}+N_{-}}\,,$ (7)
where $N_{+}$ ($N_{-}$) is the number of events with $U>0$ ($U<0$). Similarly
$A_{V}$ is defined as
$\displaystyle A_{V}=\frac{M_{+}-M_{-}}{M_{+}+M_{-}}\,,$ (8)
where $M+$ ($M_{-}$) is the number of events with $V>0$ ($V<0$). The triple
product asymmetries, $A_{U}$ and $A_{V}$ are proportional to the interference
terms $\mathcal{I}m(A_{\perp}A_{\parallel}^{*})$ and
$\mathcal{I}m(A_{\perp}A_{0}^{*})$ in the decay rate.
The $B^{0}_{s}\rightarrow\phi\phi$ decay mode was first observed by the CDF
Collaboration [24]. More recently, CDF has reported measurements of the
polarization amplitudes and triple product asymmetries in this mode based on a
sample of 295 events [25]. In this Letter, measurements of the polarization
amplitudes, $|A_{0}|^{2}$ and $|A_{\perp}|^{2}$, the strong phase difference,
$\delta_{\parallel}$, and the triple product asymmetries, $A_{U}$ and $A_{V}$,
are presented. The dataset consists of $801\pm 29$ candidates collected in
$1.0~{}\mbox{\,fb}^{-1}$ of $pp$ collisions at the LHC. The Monte Carlo (MC)
simulation samples used are based on the Pythia 6.4 generator [26] configured
with the parameters detailed in Ref. [27]. The EvtGen [28] and Geant4 [29]
packages are used to generate hadron decays and simulate interactions in the
detector, respectively.
## 2 Detector description
The LHCb detector [30] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has a momentum resolution $\Delta p/p$ that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter resolution
of 20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors. Photon, electron
and hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a muon system composed of
alternating layers of iron and detector stations. The trigger consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage which applies a full event reconstruction.
The software trigger used in this analysis requires a two-, three- or four-
track secondary vertex with a high sum of the transverse momentum, $p_{\rm
T}$, of the tracks, significant displacement from the primary interaction, and
at least one track with $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; impact parameter $\chi^{2}$
with respect to the primary interaction greater than 16; and a track fit
$\chi^{2}/\rm{ndf}<2$ where ndf is the number of degrees of freedom in the
track fit. A multivariate algorithm is used for the identification of the
secondary vertices [31]. The $B^{0}_{s}\rightarrow\phi\phi$ candidates are
selected with high efficiency either by identifying events containing a $\phi$
meson or using topological information to select hadronic $b$ decays. Events
passing the software trigger are stored for subsequent offline processing.
## 3 Event selection
The $B^{0}_{s}\rightarrow\phi\phi$ channel is reconstructed using events where
both $\phi$ mesons decay into a $K^{+}K^{-}$ pair. The
$B^{0}_{s}\rightarrow\phi\phi$ selection criteria were optimized using a data-
driven approach based on the $\phantom{}{}_{s}\mathcal{P}lot$ technique
employing the four-kaon mass as the unfolding variable [32] to separate signal
($S$) and background ($B$) with the aim of maximizing $S/\sqrt{S+B}$. The
resulting cuts are summarized in Table 1. Good quality track reconstruction is
ensured by a cut on the transverse momentum ($p_{\rm T}$) of the daughter
particles and a cut on the $\chi^{2}/{\rm ndf}$ of the track fit.
Combinatorial background is reduced by cuts on the minimum impact parameter
significance of the tracks with respect to all reconstructed $pp$ interaction
vertices and also by imposing a requirement on the vertex separation
$\chi^{2}$ of the $B^{0}_{s}$ candidate. Well-identified $\phi$ meson
candidates are selected by requiring that the two particles involved are
identified as kaons by the ring-imaging Cherenkov detectors using a cut on the
difference in the global likelihood between the kaon and pion hypotheses
($\Delta\ln{\cal L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}>0$) and by requiring
that the reconstructed mass of each $K^{+}K^{-}$ pair is within
$12~{}\rm\,MeV\\!/\\!{\it c}^{2}$ of the nominal mass of the $\phi$ meson
[33]. Further signal purity is achieved by cuts on the transverse momentum of
the $\phi$ candidates.
Table 1: Selection criteria for the $B^{0}_{s}\rightarrow\phi\phi$ decay. The abbreviation IP stands for impact parameter and $\mbox{$p_{\rm T}$}^{\phi 1}$ and $\mbox{$p_{\rm T}$}^{\phi 2}$ refer to the transverse momentum of the two $\phi$ candidates. Variable | Value
---|---
Track $\chi^{2}/{\rm ndf}$ | $<~{}5$
Track $p_{\rm T}$ | $>~{}500~{}\rm\,MeV\\!/\\!{\it c}$
Track IP $\chi^{2}$ | $>~{}21$
$\Delta\ln{\cal L}_{{\rm\mathit{K}}{\rm\mathit{\pi}}}$ | $>~{}0$
$|M_{\phi}-M_{\phi}^{\rm{PDG}}|$ | $<~{}12~{}\rm\,MeV\\!/\\!{\it c}^{2}$
$\mbox{$p_{\rm T}$}^{\phi 1},\,\mbox{$p_{\rm T}$}^{\phi 2}$ | $>~{}900~{}\rm\,MeV\\!/\\!{\it c}$
$\mbox{$p_{\rm T}$}^{\phi 1}\cdot\mbox{$p_{\rm T}$}^{\phi 2}$ | $>~{}2$ GeV${}^{2}/c^{2}$
$\phi$ vertex $\chi^{2}/{\rm ndf}$ | $<~{}24$
$B^{0}_{s}$ vertex $\chi^{2}/{\rm ndf}$ | $<~{}7.5$
$B^{0}_{s}$ vertex separation $\chi^{2}$ | $>~{}270$
$B^{0}_{s}$ IP $\chi^{2}$ | $<~{}15$
Figure 2 shows the four-kaon invariant mass distribution for selected events.
To determine the signal yield an unbinned maximum likelihood fit is performed.
The $B^{0}_{s}\rightarrow\phi\phi$ signal component is modelled by two
Gaussian functions with a common mean. The resolution of the first Gaussian is
measured from data to be $13.9\pm 0.6$ $\rm\,MeV\\!/\\!{\it c}^{2}$. The
relative fraction and resolution of the second Gaussian are fixed to 0.785 and
29.5 $\rm\,MeV\\!/\\!{\it c}^{2}$ respectively, where values have been
obtained from simulation. Combinatorial background is modelled using an
exponential function. Background from $B^{0}\rightarrow\phi K^{*0}$ and
$B^{0}_{s}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays is found to be negligible both in simulation and data driven studies.
Fitting the probability density function (PDF) described above to the data, a
signal yield of $801\pm 29$ events is found.
In addition to the dominant P-wave $\phi\rightarrow K^{+}K^{-}$ component
described in Section 1, other contributions, either from $f_{0}\rightarrow
K^{+}K^{-}$ or non-resonant $K^{+}K^{-}$, are possible. The size of these
contributions, neglecting interference effects, is studied by relaxing the
$\phi$ mass cut to be within $25$ $\rm\,MeV\\!/\\!{\it c}^{2}$ of the nominal
value555This is a larger window than the $\pm 12$ $\rm\,MeV\\!/\\!{\it c}^{2}$
window used in the polarization amplitude and strong phase difference
measurements. and using the $\phantom{}{}_{s}\mathcal{P}lot$ technique in
conjunction with the $\phi$ mass to subtract the combinatorial background.
Figure 2: Invariant $K^{+}K^{-}K^{+}K^{-}$ mass distribution for selected
$B^{0}_{s}\rightarrow\phi\phi$ candidates. A fit of a double Gaussian signal
component together with an exponential background (dotted line) is
superimposed.
The resulting $\phi$ mass distribution is shown in Fig. 3. A fit of a
relativistic P-wave Breit-Wigner function together with a two body phase space
component to model the S-wave contribution is superimposed. In a $\pm
25~{}\rm\,MeV\\!/\\!{\it c}^{2}$ mass window, the size of the S-wave component
is found to be $(1.3\pm 1.2)\%$. Since the S-wave yield is consistent with
zero, it will be neglected in the following section. A systematic uncertainty
arising from this assumption will be assigned.
Figure 3: Invariant mass distribution of $K^{+}K^{-}$ pairs for the
$B^{0}_{s}\rightarrow\phi\phi$ data without a $\phi$ mass cut. The background
has been removed using the ${}_{s}\mathcal{P}lot$ technique in conjunction
with the $K^{+}K^{-}$ invariant mass. There are two entries per $B^{0}_{s}$
candidate. The solid line shows the result of the fit model described in the
text. The fitted S-wave component is shown by the dotted line.
## 4 Results
The polarization amplitudes ($|A_{0}|^{2}$, $|A_{\perp}|^{2}$,
$|A_{\parallel}|^{2}$), are determined by performing an unbinned maximum
likelihood fit to the reconstructed mass and helicity angle distributions. For
each event, the $\phi$ meson used to define $\theta_{1}$ is chosen at random.
Both the signal and background PDFs are the products of a mass component
described in Section 3 together with an angular component. The angular
component of the signal is given by Eq. 3 multiplied by the angular acceptance
of the detector. The acceptance is determined using the simulation and is
calculated separately according to trigger type, i.e. whether the event was
triggered by the signal candidate or other particles in the event. In total
the fit for the polarization amplitudes has eight free parameters: the signal
angular parameters $|A_{0}|^{2}$, $|A_{\perp}|^{2}$ and
$\cos(\delta_{\parallel})$ defined in Section 1, the fractions of signal for
each trigger type, the resolution of the core Gaussian, the $B^{0}_{s}$ mass
and the slope of the mass background. The sum of squared amplitudes is
constrained such that $|A_{0}|^{2}+|A_{\perp}|^{2}+|A_{\parallel}|^{2}=1$. The
angular distributions for the background have been studied using the mass
sidebands in the data, where mass sidebands are defined to be between $60$ and
$300$ $\rm\,MeV\\!/\\!{\it c}^{2}$ either side of the nominal $B^{0}_{s}$ mass
[33]. With the current sample size these distributions are consistent with
being flat in ($\cos\theta_{1},\cos\theta_{2},\Phi$). Therefore, a uniform
angular PDF is assumed and more complicated shapes are considered as part of
the systematic studies. The values of $\Gamma_{s}=0.657\pm 0.009\pm
0.008{\rm\,ps}^{-1}$ and $\Delta\Gamma_{s}=0.123\pm 0.029\pm
0.011{\rm\,ps}^{-1}$ together with their correlation coefficient of $-0.3$
quoted in [23] are used as a Gaussian constraint. The validity of the fit
model has been extensively tested using simulated data samples. The results
are given in Table 2 and the angular projections are shown in Fig. 4.
Table 2: Measured polarization amplitudes and strong phase difference. The uncertainties are statistical only. The sum of the squared amplitudes is constrained to unity. The correlation coefficient between ${|A_{0}|}^{2}$ and ${|A_{\perp}}|^{2}$ is $-0.47$. Parameter | Measurement
---|---
${|A_{0}|}^{2}$ | 0.365$\pm$ | 0.022
${|A_{\perp}|}^{2}$ | 0.291$\pm$ | 0.024
${|A_{\parallel}|}^{2}=1-(|A_{0}|^{2}+|A_{\perp}|^{2})$ | 0.344$\pm$ | 0.024
$\cos(\delta_{\parallel})$ | $-$0.844$\pm$ | 0.068
Figure 4: Angular distributions for (a) $\Phi$, (b) $\cos\theta_{1}$ and (c)
$\cos\theta_{2}$ of $B_{s}^{0}\rightarrow\phi\phi$ events with the fit
projections for signal and background superimposed for the total fitted PDF
(solid line) and background component (dotted line).
Several sources of systematic uncertainty on the determination of the
polarization amplitudes are considered and summarized in Table 3. With the
present size of the dataset, the S-wave component is consistent with zero.
From the studies described in Section 3 and fits to the data including the
S-wave terms in the PDF [34], we consider a maximum S-wave component of $2\%$.
Simulation studies have been performed to investigate the effect of neglecting
an S-wave component of this size. As discussed in Section 1, the integration
that leads to Eq. 5 assumes uniform time acceptance. This is not the case due
to lifetime biasing cuts in the trigger and offline selections. The functional
form of the decay time acceptance is obtained through the use of Monte Carlo
events. The difference between using this functional form in simulation
studies and using uniform time acceptance is taken as a systematic
uncertainty. The uncertainty on the angular acceptance for the signal is
propagated to the observables also using Monte Carlo studies. The analysis was
repeated with an alternative background angular distribution, taken from a
coarsely binned histogram in $(\cos\theta_{1},\cos\theta_{2},\Phi)$ of the
mass sidebands, and the difference taken as a systematic uncertainty. An
additional uncertainty arises from angular acceptance dependencies on trigger
type. This dependency is corrected for using Monte Carlo events, with half of
the effect on fitted parameters assigned as systematic uncertainties. The
total systematic uncertainty is obtained from the sum in quadrature of the
individual uncertainties.
Table 3: Systematic uncertainties on the measured polarization amplitudes and the strong phase difference. Source | $|A_{0}|^{2}$ | $|A_{\perp}|^{2}$ | $|A_{\parallel}|^{2}$ | $\cos\delta_{\parallel}$
---|---|---|---|---
S-wave component | 0.007 | 0.005 | 0.012 | 0.001
Decay time acceptance | 0.006 | 0.006 | 0.002 | 0.007
Angular acceptance | 0.007 | 0.006 | 0.006 | 0.028
Trigger category | 0.003 | 0.002 | 0.001 | 0.004
Background model | 0.001 | – | 0.001 | 0.003
Total | 0.012 | 0.010 | 0.014 | 0.029
Figure 5: Distributions of the $U$ (left) and $V$ (right) observables for the
$B_{s}^{0}\rightarrow\phi\phi$ data in the mass range
$5286.6<M(K^{+}K^{-}K^{+}K^{-})<5446.6~{}\rm\,MeV\\!/\\!{\it c}^{2}$. The
distribution for the background is taken from the mass sidebands, normalized
to the same mass range and is shown by the solid histogram.
The distributions of the $U$ and $V$ triple product observables are shown in
Fig. 5 for the mass range
$5286.6<M(K^{+}K^{-}K^{+}K^{-})<5446.6~{}\rm\,MeV\\!/\\!{\it c}^{2}$. To
determine the triple product asymmetries, the dataset is partitioned according
to whether $U$ ($V$) is less than or greater than zero. Simultaneous fits are
performed to the mass distributions for each of the two partitions
corresponding to each observable individually. In these fits, the mean and
resolution of the Gaussian signal component together with the slope of the
exponential background component are common parameters. The asymmetries are
left as free parameters and are fitted for directly in the simultaneous fit.
The measured values are
$A_{U}$ $~{}=~{}$ | $-$0.055$\,\,\pm\,$ | 0.036 $\,,$
---|---|---
$A_{V}$ $~{}=~{}$ | 0.010$\,\,\pm\,$ | 0.036 $\,.$
Systematic uncertainties due to the residual effect of the decay time,
geometrical acceptance and the signal and background fit models have been
evaluated and are summarized in Table 4. The effect of the decay time
acceptance has been found using the same method as for the polarization
amplitudes. The impact of angular acceptance on the measured values has been
obtained from simplified simulation studies. The total systematic uncertainty
is conservatively estimated by choosing the larger of the two individual
systematic uncertainties on $A_{U}$ and $A_{V}$. The contributions are
combined in quadrature to determine the total systematic error. Various cross-
checks of the stability of the result have been performed. For example,
dividing the data according to how the event was triggered or by magnet
polarity. No significant bias is observed in these studies.
Table 4: Systematic uncertainties on the triple product asymmetries $A_{U}$ and $A_{V}$. The total uncertainty is the quadratic sum of the larger of the two components. Source | $A_{U}$ | $A_{V}$ | Final uncertainty
---|---|---|---
Angular acceptance | 0.009 | 0.006 | 0.009
Decay time acceptance | 0.006 | 0.014 | 0.014
Fit model | 0.004 | 0.005 | 0.005
Total | | $0.018$
## 5 Summary
The polarization amplitudes and strong phase difference in the
$B^{0}_{s}\rightarrow\phi\phi$ decay mode are measured to be
$|A_{0}|^{2}$ $~{}=~{}$ | 0.365$\,\pm\,$ | 0.022 (stat) $\pm\,$ | 0.012 (syst) | $\,,$
---|---|---|---|---
$|A_{\perp}|^{2}$ $~{}=~{}$ | 0.291$\,\pm\,$ | 0.024 (stat) $\pm\,$ | 0.010 (syst) | $\,,$
$|A_{\parallel}|^{2}$ $~{}=~{}$ | 0.344$\,\pm\,$ | 0.024 (stat) $\pm\,$ | 0.014 (syst) | $\,,$
$\cos(\delta_{\parallel})$ $~{}=~{}$ | $-$0.844$\,\pm\,$ | 0.068 (stat) $\pm\,$ | 0.029 (syst) | $\,,$
where the sum of the squared amplitudes is constrained to be unity. These
values agree well with the CDF measurements [25]. Measurements in other
$B\rightarrow VV$ penguin transitions at the B factories generally give higher
values of $f_{L}$ [9, 10, 11, 12, 13, 14]. It is interesting to note that the
value of $f_{L}$ found in the $B^{0}_{s}\rightarrow\phi\phi$ channel is almost
equal to that in the $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay [17]. The results are in
agreement with QCD factorization predictions [6, 7], but disfavour the pQCD
estimate given in [8].
The triple product asymmetries in this mode are measured to be
$A_{U}$ $~{}=~{}$ | $-$0.055$\,\pm\,$ | 0.036 (stat) $\pm\,$ | 0.018 (syst) | $\,,$
---|---|---|---|---
$A_{V}$ $~{}=~{}$ | 0.010$\,\pm\,$ | 0.036 (stat) $\pm\,$ | 0.018 (syst) | $\,.$
Both values are in good agreement with those reported by the CDF Collaboration
[25] and consistent with the hypothesis of $C\\!P$ conservation.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] M. Gronau and J. L. Rosner, Triple product asymmetries in $K$, $D_{(s)}$ and $B_{(s)}$ decays, Phys. Rev. D84 (2011) 096013, arXiv:1107.1232
* [2] W. Bensalem and D. London, $T$-violating triple product correlations in hadronic $b$ decays, Phys. Rev. D64 (2001) 116003, arXiv:hep-ph/0005018
* [3] A. Datta and D. London, Triple-product correlations in $B\rightarrow V_{1}V_{2}$ decays and new physics, Int. J. Mod. Phys. A19 (2004) 2505, arXiv:hep-ph/0303159
* [4] S. Nandi and A. Kundu, New physics in $b\rightarrow s\bar{s}s$: study of $B\rightarrow V_{1}V_{2}$ modes, J. Phys. G32 (2006) 835, arXiv:hep-ph/0510245
* [5] A. Datta, M. Duraisamy, and D. London, Searching for new physics with B-Decay fake triple products, Phys. Lett. B701 (2011) 357, arXiv:1103.2442
* [6] M. Beneke, J. Rohrer, and D. Yang, Branching fractions, polarisation and asymmetries of ${B}\rightarrow{VV}$ decays, Nucl. Phys. B774 (2007) 64, arXiv:hep-ph/0612290
* [7] H.-Y. Cheng and C.-K. Chua, QCD factorization for charmless hadronic $B_{s}$ decays revisited, Phys. Rev. D80 (2009) 114026, arXiv:0910.5237
* [8] A. Ali et al., Charmless nonleptonic ${B}_{s}$ decays to $PP$, $PV$, and $VV$ final states in the perturbative QCD approach, Phys. Rev. D76 (2007) 074018, arXiv:hep-ph/0703162
* [9] Belle collaboration, K.-F. Chen et al., Measurement of polarization and triple-product correlations in $B\rightarrow\phi K^{*}$ decays, Phys. Rev. Lett. 94 (2005) 221804, arXiv:hep-ex/0503013
* [10] BaBar collaboration, B. Aubert et al., Vector-tensor and vector-vector decay amplitude analysis of ${B}^{0}\rightarrow\phi{K}^{*0}$, Phys. Rev. Lett. 98 (2007) 051801, arXiv:hep-ex/0610073
* [11] BaBar collaboration, B. Aubert et al., Time-dependent and time-integrated angular analysis of ${B}\rightarrow\phi{K}_{S}^{0}\pi^{0}$ and $\phi{K}^{\pm}\pi^{\mp}$, Phys. Rev. D78 (2008) 092008, arXiv:0808.3586
* [12] BaBar collaboration, P. del Amo Sanchez et al., Measurements of branching fractions, polarizations, and direct CP-violation asymmetries in $B^{+}\rightarrow\rho^{0}K^{*+}$ and $B^{+}\rightarrow f_{0}(980)K^{*+}$ decays, Phys. Rev. D83 (2011) 051101, arXiv:1012.4044
* [13] Belle collaboration, J. Zhang et al., Measurements of branching fractions and polarization in $B^{+}\rightarrow\rho^{+}K^{*0}$ decays, Phys. Rev. Lett. 95 (2005) 141801, arXiv:hep-ex/0408102
* [14] BaBar collaboration, B. Aubert et al., Measurements of branching fractions, polarizations, and direct CP-violation asymmetries in $B\rightarrow\rho K^{*}$ and $B\rightarrow f_{0}(980)K^{*}$ decays, Phys. Rev. Lett. 97 (2006) 201801, arXiv:hep-ex/0607057
* [15] A. L. Kagan, Polarization in ${B}\rightarrow{VV}$ decays, Phys. Lett. B601 (2004) 151, arXiv:hep-ph/0405134
* [16] A. Datta et al., Testing explanations of the ${B}\rightarrow\phi{K}^{*}$ polarization puzzle, Phys. Rev. D76 (2007) 034015, arXiv:0705.3915
* [17] LHCb collaboration, R. Aaij et al., First observation of the decay ${B}^{0}_{s}\rightarrow{K}^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, Phys. Lett. B709 (2012) 50, arXiv:1111.4183
* [18] C.-W. Chiang and L. Wolfenstein, Observables in the decays of B to two vector mesons, Phys. Rev. D61 (2000) 074031, arXiv:hep-ph/9911338
* [19] C.-W. Chiang, Angular distribution of charming ${B}\rightarrow\mathrm{VV}$ decays and time evolution effects, Phys. Rev. D62 (2000) 014017, arXiv:hep-ph/0002243
* [20] M. Raidal, CP asymmetry in $B\rightarrow\phi K_{S}$ decays in left-right models and its implications for $B_{s}$ decays, Phys. Rev. Lett. 89 (2002) 231803, arXiv:hep-ph/0208091
* [21] M. Bartsch, G. Buchalla, and C. Kraus, $B\rightarrow V_{L}V_{L}$ decays at next-to-leading order in QCD, arXiv:0810.0249
* [22] J. Charles et al., Predictions of selected flavour observables within the Standard Model, Phys. Rev. D84 (2011) 033005, arXiv:1106.4041
* [23] LHCb collaboration, R. Aaij et al., Measurement of the $CP$-violating phase $\phi_{s}$ in the decay ${B}_{s}^{0}\rightarrow J/\psi\phi$, Phys. Rev. Lett. 108 (2012) 101803, arXiv:1112.3183
* [24] CDF collaboration, D. E. Acosta et al., First evidence for $B_{s}\rightarrow\phi\phi$ decay and measurements of branching ratio and $A_{CP}$ for $B^{+}\rightarrow\phi K^{+}$, Phys. Rev. Lett. 95 (2005) 031801, arXiv:hep-ex/0502044
* [25] CDF collaboration, T. Aaltonen et al., Measurement of polarization and search for $CP$ violation in $B_{s}^{0}\rightarrow\phi\phi$ decays, Phys. Rev. Lett. 107 (2011) 261802, arXiv:1107.4999
* [26] T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175
* [27] I. Belyaev et al., Handling of the generation of primary events in Gauss, the LHCb simulation framework, Nuclear Science Symposium Conference Record (NSS/MIC) IEEE (2010) 1155
* [28] D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A462 (2001) 152
* [29] S. Agostinelli et al., GEANT4 – a simulation toolkit, Nucl. Instrum. Meth. A506 (2003) 250
* [30] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005
* [31] V. Gligorov, C. Thomas and M. Williams, LHCb collaboration, The HLT inclusive $B$ triggers, LHCb-PUB-2011-016
* [32] M. Pivk and F. R. Le Diberder, sPlot: A statistical tool to unfold data distributions, Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
* [33] Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys. G37 (2010) 075021
* [34] Y. Xie, P. Clarke, G. Cowan, and F. Muheim, Determination of $2\beta_{s}$ in $B^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$ decays in the presence of a $K^{+}K^{-}$ S-wave contribution, JHEP 09 (2009) 074, arXiv:0908.3627
|
arxiv-papers
| 2012-04-12T19:00:34 |
2024-09-04T02:49:29.653132
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis,\n J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov,\n M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, V. Balagura,\n W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W. Barter, A. Bates, C.\n Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov,\n V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V.\n Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I.\n Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo\n Gomez, A. Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A.\n Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M.\n Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, G.\n Corti, B. Couturier, G. A. Cowan, R. Currie, C. D'Ambrosio, P. David, P. N.\n Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J. M. De\n Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi,\n L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens,\n H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D.\n Esperante Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez,\n V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H.\n Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E.\n Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S. C.\n Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison,\n P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A.\n Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E.\n Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J.\n Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim,\n M. Knecht, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi,\n D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti,\n G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T.\n Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu,\n G. Liu, J. von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu,\n J. Luisier, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O.\n Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, S.\n Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan,\n B. Muryn, B. Muster, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, B. K. Pal, J.\n Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M. Patel, S. K. Paterson,\n G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina,\n A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P.\n Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci,\n M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B.\n Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, H. Voss, R. Waldi, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams,\n F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Sean Benson",
"url": "https://arxiv.org/abs/1204.2813"
}
|
1204.2902
|
# Lepton-flavor violating decays of neutral Higgs to muon and tauon in
supersymmetric economical 3-3-1 model
P.T.Giang L.T.Hue D.T.Huong H.N.Long Institute of Physics, VAST, 10 Dao Tan,
Ba Dinh, Hanoi, Vietnam
###### Abstract
We investigate Lepton-Flavor Violating (LFV) decays of Higgs to muon-tau in
the Supersymmetric Economical 3-3-1 (SUSYE331) model. In the presence of
flavor mixing in sleptons $\\{\tilde{\mu},\tilde{\tau}\\}$ and large values of
$v/v^{\prime}$, the ratio of
$Br(H\rightarrow\tau^{+}\mu^{-})/Br(H\rightarrow\tau^{+}\tau^{-})$ can reach
non-negligible values $\mathcal{O}(10^{-3})$, as in many known SUSY models. We
predict that for the Standard Model Higgs boson, the LHC may detect its decay
to muon and tauon. We also investigate the asymmetry between left and right
LFV values of corrections and prove that the LFV effects are dominated by the
left FLV term, which is $\mathcal{O}(10^{3})$ times larger than the right LFV
term in the limit of small values of $|\mu_{\rho}|/m_{SUSY}$. The
contributions of Higgs-mediated effects to the decay
$\tau\rightarrow\mu\mu\mu$ are also discussed.
###### keywords:
Supersymmetric models, Decays of taus, Supersymmetric Higgs bosons
###### PACS:
12.60.Jv , 13.35.Dx, 14.80.Da
††thanks: Email: ptgiang@grad.iop.vast.ac.vn††thanks: Email:
lthue@iop.vast.ac.vn††thanks: Email: dthuong@iop.vast.ac.vn††thanks: Email:
hnlong@iop.vast.ac.vn
## 1 Introduction
The experimental evidences of non-zero neutrino masses and mixing [1] have
shown that the Standard Model (SM) of fundamental particles and interactions
must be extended. Among many extensions of the SM known today, the models
based on gauge symmetry
$\mathrm{SU}(3)_{C}\otimes\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ (called
3-3-1 models) [2, 3] have interesting features. The model requires that the
number of fermion families $N$ be a multiple of the quark color in order to
cancel anomalies, which suggests an interesting connection between the number
of flavors and the strong color group. If further one uses the condition of
QCD asymptotic freedom, which is valid only if the number of families of
quarks is to be less than five, it follows that $N$ is equal to 3. In
addition, the third quark generation has to be different from the first two,
so this leads to the possible explanation of why top quark is
uncharacteristically heavy (see, for example, [4]). The 3-3-1 models can also
provide a solution of electric charge quantization observed in the nature [5].
In one of the 3-3-1 models [3] three $\mathrm{SU}(3)_{L}$ lepton triplets are
of the form $(\nu_{l},l,\nu_{l}^{c})_{L}$, where $\nu_{l}^{c}$ is related to
the right-handed component of the neutrino field $\nu_{l}$ (a model with
right-handed neutrinos). The scalar sector of this model requires three Higgs
triplets, and it is interesting to note that two Higgs triplets has the same
$\mathrm{U}(1)_{X}$ charge with two neutral components at their top and
bottom. Giving all neutral Higgs fields a vacuum expectation value (VEV), we
can remove one Higgs triplet. Hence the Higgs sector of the obtained model
becomes minimal, and it has been called the economical 3-3-1 model [6].
The lepton-flavor is absolutely conserved in the SM. Recently, experiments on
neutrino oscillations have proved that lepton flavor is not conserved. It
leads to motivation on a search for the signals of lepton flavor violations
(LFV) beyond the SM. Many versions of the extension of Minimal Supersymmetric
Standard Model (MSSM) with large $\tan\beta$ have been investigated in Higgs
LFV decay. The interesting here is there exists parameter space that predicts
the branching ratio of these types of decays are very sizable, enough to be
detected by present colliders such as CERN Large Hadron Collider (LHC) [7] or
International Linear Collider (ILC) [8]. For example, the SM [9] predicted
that branching ratio of $H\rightarrow\mu\tau$ is very suppressed. However, in
the beyond SM, this ratio can reach large values, more than $10^{-4}$. In
particular, Refs. [10, 11] showed that, in the MSSM,
$BR(H\rightarrow\mu^{+}\tau^{-})\sim 10^{-4}$ if $m_{H}/M_{SUSY}\sim 10^{-1}$
. The Minimal Supersymmetric Neutrino Seesaw Models ($\nu$MSSM) [12] predicted
the branching ratio of heavy Higgs LFV decay is of order $10^{-4}$ while that
of light Higgs LFV decay is of order $10^{-8}$. For more discussions in
details about the Higgs LFV decay, readers are referred to [13] for general
LFV framework, to [14] for two Higgs doublet models, to [13, 14, 15, 16, 17,
18, 19] for MSSM, and $\nu$MSSM and to [20] for little Higgs models (LTH).
The MSSM has shown that in the limit of large $\tan\beta$, the radiative
corrections become non-negligible in many Higgs LFV decay processes. For
example, refs. [15, 17] showed that the ratio $R_{b/t}\equiv
BR(H\rightarrow\bar{b}b)/BR(H\rightarrow\bar{t}t)$ can be distinguished
between the MSSM and non-supersymmetric models. The main reason is that the
Higgs boson couplings to down-type fermions receive a large corrections
enhanced by $\tan\beta$. It leads to many interesting decay processes in quark
sector such as $b\rightarrow s\gamma$ [21, 22]. The large value of $\tan\beta$
also leads to many interesting effects in the lepton sector, especially when
the LFV source in sleptons is included.
Recently, the Supersymmetric Economical 3-3-1 model (SUSYE331) has been
constructed [23]. Apart from interesting features that mentioned in refs. [23,
24, 25, 26], the scalar sector is minimal, and therefore it has been called
the economical. In a series of works [23, 24, 25, 26], we have developed and
proved that the non-supersymmetric version [6] and supersymmetric version are
consistent, realistic and very rich in physics. In the previous work [24], we
skip the LFV source in the soft sector. However, the model predicts more
interesting phenomenology if there exists LFV source in the soft breaking
terms. In this paper, we will concentrate on LFV Higgs decays to $\mu\tau$
with the presence of misalignment of sleptons $\\{\tilde{\mu},\tilde{\tau}\\}$
and their sneutrinos contained in soft breaking terms. In SUSYE331 model, for
generating fermion masses as well as canceling anomaly, one needs four Higgs
triplets. In particular, the ”up” $\rho^{0}$ Higgs gives mass for neutrinos
and the remain, ”down” $\rho^{\prime 0}$, gives mass for charged leptons [23,
25] and other Higgs give mass for quarks.The ratio of VEVs, namely
$\frac{<\rho^{0}>}{<\rho^{\prime 0}>}$, is denoted by $\tan\gamma$ which is
similar to $\tan\beta$ in MSMS. Hence,the $\rho^{0}$ and $\rho^{\prime 0}$
Higgs play very important roles if we consider effects of radiative correction
in lepton sector in the limit of large $\tan\gamma$. The corrections may cause
many non-negligible effects, such as the correction of lepton mass, branching
ratio of LFV Higgs decay…On the other hand, the model 331 is the extension of
SM based on extended gauge symmetry. Therefore, comparing to MSSM, the
SUSYE331 model contains new gauge bosons and new Higgses as well as their
superpartners. Because of appearing of new particles, the number of diagrams
contributing to LFV Higgs decay in SUSYE331 model is predicted more than that
in MSSM. It leads to LFV in Higgs decay effected in SUSYE331 may be larger
than in MSSM. Hence, in this work, we investigate the flavor violating Higgs
coupling in SUSYE331 model, specially we focus on the $\\{\mu,\tau\\}$
generations.
Our work is arranged as follows: In Section 2, we review the particle content
in SUSYE331 model. The analytic expressions of the Higgs effective couplings
are studied in Section 3. In Section 4, we study a numerical estimation on
decay $H\rightarrow\mu\tau$ at colliders and compare contribution from the
left and right LFV radiative corrections into the mentioned decay. In this
section, we also consider the contributions of Higgs exchange to branching
ratio of $\tau\rightarrow 3\mu$ decay. In the last section, we summarize our
main results.
## 2 Particle content
Let us give brief report on the particle content in SUSYE331 model [23]. The
superfields in the anomaly-free model are given by
$\widehat{L}_{aL}=\left(\widehat{\nu}_{a},\widehat{l}_{a},\widehat{\nu}^{c}_{a}\right)^{T}_{L}\sim(1,3,-1/3),\hskip
14.22636pt\widehat{l}^{c}_{aL}\sim(1,1,1),\hskip 14.22636pta=1,2,3$ (1)
$\widehat{Q}_{1L}=\left(\widehat{u}_{1},\ \widehat{d}_{1},\
\widehat{u}^{\prime}\right)^{T}_{L}\sim(3,3,1/3),$ $\widehat{u}^{c}_{1L},\
\widehat{u}^{\prime
c}_{L}\sim(3^{*},1,-2/3),\widehat{d}^{c}_{1L}\sim(3^{*},1,1/3),$ (2)
$\begin{array}[]{ccc}\widehat{Q}_{\alpha
L}=\left(\widehat{d}_{\alpha},-\widehat{u}_{\alpha},\widehat{d^{\prime}}_{\alpha}\right)^{T}_{L}\sim(3,3^{*},0),\hskip
14.22636pt\alpha=2,3,\end{array}$ (3) $\widehat{u}^{c}_{\alpha
L}\sim\left(3^{*},1,-2/3\right),\hskip 14.22636pt\widehat{d}^{c}_{\alpha L},\
\widehat{d}^{\prime c}_{\alpha L}\sim\left(3^{*},1,1/3\right),$ (4)
$\widehat{\chi}=\left(\widehat{\chi}^{0}_{1},\widehat{\chi}^{-},\widehat{\chi}^{0}_{2}\right)^{T}\sim(1,3,-1/3),\hskip
14.22636pt\widehat{\rho}=\left(\widehat{\rho}^{+}_{1},\widehat{\rho}^{0},\widehat{\rho}^{+}_{2}\right)^{T}\sim(1,3,2/3),$
(5) $\widehat{\chi}^{\prime}=\left(\widehat{\chi}^{\prime
o}_{1},\widehat{\chi}^{\prime+},\widehat{\chi}^{\prime
o}_{2}\right)^{T}\sim(1,3^{*},1/3),\,\widehat{\rho}^{\prime}=\left(\widehat{\rho}^{\prime-}_{1},\widehat{\rho}^{\prime
o},\widehat{\rho}^{\prime-}_{2}\right)^{T}\sim(1,3^{*},-2/3).$ (6)
Here we use some new notations as
$\widehat{\psi}^{c}_{L}=(\widehat{\psi}_{R})^{c}\equiv\widehat{\psi}_{R}^{\dagger}$
and exotic quarks are denoted by usual quarks with prime-superscripts
($u^{\prime}$ with the electric charge $q_{u^{\prime}}=2/3$ and $d^{\prime}$
with $q_{d^{\prime}}=-1/3$). The values in each parenthesis show corresponding
quantum numbers of the
$(\mathrm{SU}(3)_{c},\mathrm{SU}(3)_{L},\mathrm{U}(1)_{X})$ symmetry. In this
model, the $\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}$ gauge group is broken
via two steps:
$\mathrm{SU}(3)_{L}\otimes\mathrm{U}(1)_{X}\stackrel{{\scriptstyle
w,w^{\prime}}}{{\longrightarrow}}\
\mathrm{SU}(2)_{L}\otimes\mathrm{U}(1)_{Y}\stackrel{{\scriptstyle
v,v^{\prime},u,u^{\prime}}}{{\longrightarrow}}\mathrm{U}(1)_{Q},$ (7)
where the VEVs are defined by
$\sqrt{2}\langle\chi\rangle^{T}=\left(u,0,w\right),\hskip 14.22636pt\hskip
14.22636pt\sqrt{2}\langle\chi^{\prime}\rangle^{T}=\left(u^{\prime},0,w^{\prime}\right)$
(8) $\sqrt{2}\langle\rho\rangle^{T}=\left(0,v,0\right),\hskip 14.22636pt\hskip
14.22636pt\sqrt{2}\langle\rho^{\prime}\rangle^{T}=\left(0,v^{\prime},0\right).$
(9)
The vector superfields $\widehat{V}_{c}$, $\widehat{V}$ and
$\widehat{V}^{\prime}$ containing the usual gauge bosons are given in [23,
25]. The supersymmetric model possessing a general Lagrangian is studied in
[25]. In the following, only terms relevant to our calculations are displayed.
## 3 Higgs-muon-tauon effective interactions
In the SUSYE331 model [23, 25], at the tree level, the down-type leptons
($e,\mu,\tau$) only couple to the neutral Higgs $(\rho^{\prime 0})$ through
the Yukawa interaction given by
$\displaystyle\mathcal{L}_{llH}=-\frac{\lambda_{1ab}}{3}(L_{aL}l^{c}_{bL}\rho^{\prime
0}+\rm H.c.).$ (10)
In general case, $\lambda_{1ab}\neq 0$, the Lagrangian given in (10) not only
provides mass for the charged leptons but also gives the source of the lepton
flavor mixing at the tree level. It means that if the couplings
$\lambda_{1ab}\neq 0$ with $(a\neq b)$, the LFV processes, such as
Higgs$\rightarrow\mu\tau$, must be existed. In this case, our theory predicts
very large branching ratios of LFV processes which exceed to experimental
results discussed in [27]. Hence, in the following calculation, we skip the
$\lambda_{1ab}$ with $(a\neq b$) in (10) .
Let us consider another source of LFV which is caused by slepton mixing. More
details of slepton mixing, one can find in Appendix C. Because of slepton
mixing, the leading effective interactions of leptons with
$\rho^{o},\rho^{\prime o}$ Higgs can appeare at the one-loop order. In this
paper, we will concentrate only on the couplings of Higgs with
$\\{\mu,\tau\\}$ leptons.
In order to consider the $\mu,\tau$ flavor mixing at the one loop level, first
we rewrite the original Lagrangian (10) in terms of two component spinor
notations which are familiar to those in literature, namely
$\displaystyle-\mathcal{L}_{0\mu\tau}=\left(Y_{\mu}\mu^{c}_{L}\mu_{L}+Y_{\tau}\tau^{c}_{L}\tau_{L}\right)\rho^{\prime
0}+\mathrm{H.c.},$ (11)
where $Y_{\mu}\equiv\lambda_{122}/3,Y_{\tau}\equiv\lambda_{133}/3$.
At the one-loop level, if we skip all of the terms which are proportional to
$Y_{\mu}$ except terms contributing to mass of muon, then Yukawa interactions
containing Higgs-lepton-lepton couplings can be divided into two parts:
* •
The lepton-flavor conversing (LFC) part given by
$\displaystyle-\Delta\mathcal{L}_{FC}$ $\displaystyle=$
$\displaystyle\left(Y_{\mu}\Delta^{1\rho}_{\mu}+Y_{\tau}\Delta^{2\rho}_{\mu}\right)\mu^{c}_{L}\mu_{L}\rho^{0*}+Y_{\tau}\Delta^{\rho}_{\tau}\tau^{c}_{L}\tau_{L}\rho^{0*}$
(12) $\displaystyle+$
$\displaystyle\left(Y_{\mu}\Delta^{1\rho^{\prime}}_{\mu}+Y_{\tau}\Delta^{2\rho^{\prime}}_{\mu}\right)\mu^{c}_{L}\mu_{L}\rho^{\prime
0}+Y_{\tau}\Delta^{\rho^{\prime}}_{\tau}\tau^{c}_{L}\tau_{L}\rho^{\prime
0}+\mathrm{H.c.},$
* •
The lepton-flavor violating (LFV) part given as
$\displaystyle-\Delta\mathcal{L}_{FV}$ $\displaystyle=$ $\displaystyle
Y_{\tau}\left(\Delta^{\rho}_{L}\tau^{c}_{L}\mu_{L}+\Delta^{\rho}_{R}\mu^{c}_{L}\tau_{L}\right)\rho^{0*}$
(13) $\displaystyle+$ $\displaystyle
Y_{\tau}\left(\Delta^{\rho^{\prime}}_{L}\tau^{c}_{L}\mu_{L}+\Delta^{\rho^{\prime}}_{R}\mu^{c}_{L}\tau_{L}\right)\rho^{\prime
0}+\rm H.c.,$
where all of
$\Delta^{1\rho}_{\mu},\Delta^{2\rho}_{\mu},\Delta^{1\rho^{\prime}}_{\mu},\Delta^{2\rho^{\prime}}_{\mu},\Delta^{\rho}_{\tau},\Delta^{\rho^{\prime}}_{\tau},\Delta^{\rho}_{L},\Delta^{\rho^{\prime}}_{L},\Delta^{\rho}_{R}$
and $\Delta^{\rho^{\prime}}_{R}$ are the leading effective couplings.
From now on, for convenience, we use notation $\Delta$ to imply any radiative
correction of couplings appearing in (12) and (13). Note that $\Delta$ is a
dimensionless function of mass parameters and $\Delta^{\rho}_{\mu}$,
$\Delta^{\rho}_{\tau}$ are non-zero value even if we assume that there is no
flavor mixing in slepton sector. We emphasize that $\Delta^{\rho}_{\tau}$ is
one of quantities affecting on many observable quantities such as the ratio of
branching ratios $Br(H\rightarrow b\bar{b})/BR(H\rightarrow\tau\bar{\tau})$.
The contribution of $\Delta^{\rho}_{\tau}$ to that of branching ratios in the
SUSY model is studied in [15, 17]. The diagrams which contribute to all of
$\Delta$s are drawn in Appendix A.
Now let us construct the total effective Lagrangian for Higgs, muon and tauon
couplings in terms of physical eigenstates. First we write down the whole
Lagrangian coming from all of Eqs. (11), (12) and (13) in the matrix form
$\displaystyle-\mathcal{L}$ $\displaystyle=$ $\displaystyle
Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{1}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)~{}\rho^{\prime
0}+Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{2}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)\rho^{0*}+\mathrm{H.c.},$ (20)
where $\mathcal{Y}_{l_{1}}$ and $\mathcal{Y}_{l_{2}}$ are matrices defined by
the following formulas:
$\displaystyle\mathcal{Y}_{l_{1}}=\left(\begin{array}[]{cc}\Delta^{o\rho^{\prime}}_{\mu}&\Delta^{\rho^{\prime}}_{R}\\\
\Delta^{\rho^{\prime}}_{L}&1+\Delta^{\rho^{\prime}}_{\tau}\\\
\end{array}\right);\hskip 14.22636pt\hskip
14.22636pt\mathcal{Y}_{l_{2}}=\left(\begin{array}[]{cc}\Delta^{o\rho}_{\mu}&\Delta^{\rho}_{R}\\\
\Delta^{\rho}_{L}&\Delta^{\rho}_{\tau}\\\ \end{array}\right),$ (25)
with $y\equiv Y_{\mu}/Y_{\tau}$ , $\Delta^{o\rho}_{\mu}\equiv
y\Delta^{1\rho}_{\mu}+\Delta^{2\rho}_{\mu}$ and
$\Delta^{o\rho^{\prime}}_{\mu}\equiv
y+y\Delta^{1\rho^{\prime}}_{\mu}+\Delta^{2\rho^{\prime}}_{\mu}$.
Because of loop corrections, the mass matrix of the $\mu,\tau$ in (20) is no
longer diagonal. In order to find the physical eigenstates of muon and tauon,
we expand the Higgs $\rho$ and $\rho^{\prime}$ around the vacuum expectation
values. As a consequence, the mixing mass matrix for the muon and tauon are
$\displaystyle-\mathcal{L}_{mass}$ $\displaystyle=$ $\displaystyle
Y_{\tau}v^{\prime}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)+\mathrm{H.c.},$ (29)
where
$\displaystyle\mathcal{Y}_{l}$ $\displaystyle\equiv$
$\displaystyle\mathcal{Y}_{l_{1}}+t_{\gamma}\mathcal{Y}_{l_{2}}=(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{R}\\\
\epsilon_{L}&1\\\ \end{array}\right)$ (32) $\displaystyle=$
$\displaystyle(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})Y_{\epsilon},$
(33)
with
$\displaystyle t_{\gamma}$ $\displaystyle\equiv$
$\displaystyle\tan\gamma=\frac{v}{v^{\prime}}=\frac{\langle\rho^{0}\rangle}{\langle\rho^{{}^{\prime}0}\rangle},\hskip
14.22636pt\epsilon_{\mu}\equiv\frac{\Delta^{o\rho^{\prime}}_{\mu}+\Delta^{o\rho}_{\mu}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}},$
$\displaystyle\epsilon_{L}$ $\displaystyle\equiv$
$\displaystyle\frac{\Delta^{\rho^{\prime}}_{L}+\Delta^{\rho}_{L}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}},\hskip
14.22636pt\epsilon_{R}\equiv\frac{\Delta^{\rho^{\prime}}_{R}+\Delta^{\rho}_{R}t_{\gamma}}{1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma}}$
(34)
and
$\displaystyle
Y_{\epsilon}=\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{R}\\\
\epsilon_{L}&1\\\ \end{array}\right)$ (37)
It is easy to see that the mixing mass matrix of muon and tauon given in (37)
is a general matrix. Finding the mass eigenvalues of left-right leptons is
equivalent to finding a matrix $C$ satisfying:
$\displaystyle
C^{\dagger}\mathcal{Y}_{\epsilon}^{\dagger}\mathcal{Y}_{\epsilon}C$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}y_{\mu}^{2}&0\\\
0&y^{2}_{\tau}\\\ \end{array}\right)\equiv\mathcal{Y}^{2}_{d}.$ (40)
In our theory, the matrix $C$ can be found in a form
$\displaystyle C=\left(\begin{array}[]{cc}c_{\Lambda}&s_{\Lambda}\\\
-s_{\Lambda}&c_{\Lambda}\\\ \end{array}\right),$ (43)
where $c_{\Lambda}\equiv\cos\Lambda,s_{\Lambda}\equiv\sin\Lambda$ and
$\Lambda$ is the rotation angle given by
$\displaystyle
t_{2\Lambda}\equiv\tan(2\Lambda)=\frac{2(\epsilon_{\mu}\epsilon_{R}+\epsilon_{L})}{1+\epsilon^{2}_{R}-(\epsilon^{2}_{\mu}+\epsilon^{2}_{L})}.$
(44)
In addition, $\mathcal{Y}_{d}=\mathrm{diag}(y_{\mu},~{}y_{\tau})$ in which
$(y_{\mu},~{}y_{\tau})$ are defined as follows
$\displaystyle y^{2}_{\mu}$ $\displaystyle=$ $\displaystyle
r^{\prime}-rs^{2}_{\Lambda},\hskip
14.22636pty^{2}_{\tau}=r^{\prime}+rc^{2}_{\Lambda},$ (45)
where
$\displaystyle r^{2}$ $\displaystyle\equiv$ $\displaystyle
4(\epsilon_{\mu}\epsilon_{R}+\epsilon_{L})^{2}+\left[1+\epsilon^{2}_{R}-(\epsilon^{2}_{\mu}+\epsilon^{2}_{L})\right]^{2},\hskip
14.22636ptr^{\prime}\equiv\epsilon^{2}_{\mu}+\epsilon^{2}_{L}~{}.$ (46)
Note that the mass eigenvalues of muon and tauon are proportional to
$(y_{\mu},y_{\tau})$, namely
$\displaystyle
m_{\mu}=y_{\mu}Y_{\tau}v^{\prime}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}),\hskip
14.22636ptm_{\tau}=y_{\tau}Y_{\tau}v^{\prime}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}).$
(47)
On the other hand, the mass eigenstates of leptons $(\mu,\tau)$ and
$(\mu^{c},\tau^{c})$ are determined from two transformations
$\displaystyle L^{c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\mu^{c}\\\ \tau^{c}\\\
\end{array}\right)=(U_{l})^{T}\left(\begin{array}[]{c}\mu^{c}_{L}\\\
\tau^{c}_{L}\\\ \end{array}\right)=U^{T}_{l}L^{c}_{L},$ (52) $\displaystyle L$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}\mu\\\ \tau\\\
\end{array}\right)=V_{l}\left(\begin{array}[]{c}\mu_{L}\\\ \tau_{L}\\\
\end{array}\right)=V_{l}L_{L},$ (57)
where $U_{l}$ and $V_{l}$ have come from (40), namely
$\displaystyle U^{\dagger}_{l}$ $\displaystyle=$
$\displaystyle\mathcal{Y}^{-1}_{d}C^{\dagger}\mathcal{Y}^{\dagger}_{\epsilon}=\left(\begin{array}[]{cc}\frac{1}{y_{\mu}}&0\\\
0&\frac{1}{y_{\tau}}\\\
\end{array}\right)\left(\begin{array}[]{cc}c_{\Lambda}&-s_{\Lambda}\\\
s_{\Lambda}&c_{\Lambda}\\\
\end{array}\right)\left(\begin{array}[]{cc}\epsilon_{\mu}&\epsilon_{L}\\\
\epsilon_{R}&1\\\ \end{array}\right),$ (64) $\displaystyle V_{l}^{\dagger}$
$\displaystyle=$ $\displaystyle
C=\left(\begin{array}[]{cc}c_{\Lambda}&s_{\Lambda}\\\
-s_{\Lambda}&c_{\Lambda}\\\ \end{array}\right).$ (67)
Next, we replace $\mathcal{Y}_{l_{1}}$ in Eq. (20) by a new form deduced from
Eq.(33)
$\mathcal{Y}_{l_{1}}=(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})Y_{\epsilon}-\mathcal{Y}_{l_{2}}t_{\gamma}$
Now we have obtained a new expression of (20) as follows
$\displaystyle-\mathcal{L}$ $\displaystyle=$ $\displaystyle
Y_{\tau}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{\epsilon}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)~{}\rho^{\prime o}$ (71) $\displaystyle+$
$\displaystyle Y_{\tau}\left(\begin{array}[]{cc}\mu^{c}_{L}&\tau^{c}_{L}\\\
\end{array}\right)\mathcal{Y}_{l_{2}}\left(\begin{array}[]{c}\mu_{L}\\\
\tau_{L}\\\ \end{array}\right)(\rho^{o*}-t_{\gamma}\rho^{\prime
o})+\mathrm{H.c.},$ (75)
In the basis of mass eigenstates of the muon and tauon given in Eq. (57), the
Lagrangian (75) transforms into
$\displaystyle-\mathcal{L}_{d}$ $\displaystyle=$ $\displaystyle
Y_{\tau}(1+\Delta^{\rho^{\prime}}_{\tau}+\Delta^{\rho}_{\tau}t_{\gamma})L^{cT}~{}\mathcal{Y}_{d}~{}L\rho^{\prime
o}$ (76) $\displaystyle+$ $\displaystyle
Y_{\tau}L^{cT}(U^{\dagger}_{l}\mathcal{Y}_{2}V_{l}^{\dagger})L(\rho^{o*}-t_{\gamma}\rho^{\prime
o})+\mathrm{H.c.}$
It is needed to emphasize that the first term in Eq. (76) generates only
masses for muon and tauon while the second creates masses as well as give rise
to the lepton flavor mixing. Sources of flavor mixing are two off-diagonal
elements of the matrix $(U^{\dagger}_{l}\mathcal{Y}_{2}V_{l}^{\dagger})$ :
$\displaystyle\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{12}$
$\displaystyle=$
$\displaystyle\frac{c^{2}_{\Lambda}\Delta^{\rho}_{R}\epsilon_{\mu}}{y_{\mu}}+\frac{(c^{2}_{\Lambda}\epsilon_{L}-c_{\Lambda}s_{\Lambda})\Delta^{\rho}_{\tau}}{y_{\mu}}$
(77) $\displaystyle+$
$\displaystyle\frac{c_{\Lambda}s_{\Lambda}(\Delta^{\rho}_{L}\epsilon_{L}+\Delta^{o\rho}_{\mu}\epsilon_{\mu}-\Delta^{\rho}_{R}\epsilon_{R})-s^{2}_{\Lambda}(\Delta^{\rho}_{L}+\epsilon_{R}\Delta^{\rho}_{R})}{y_{\mu}},$
$\displaystyle\left(U_{l}^{\dagger}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{21}$
$\displaystyle=$
$\displaystyle\frac{c^{2}_{\Lambda}\Delta^{\rho}_{L}}{y_{\tau}}+\frac{c^{2}_{\Lambda}\Delta^{o\rho}_{\mu}\epsilon_{R}-c_{\Lambda}s_{\Lambda}\Delta^{\rho}_{\tau}}{y_{\tau}}$
(78) $\displaystyle+$
$\displaystyle\frac{c_{\Lambda}s_{\Lambda}(\Delta^{\rho}_{L}\epsilon_{L}+\Delta^{o\rho}_{\mu}\epsilon_{\mu}-\Delta^{\rho}_{R}\epsilon_{R})-s^{2}_{\Lambda}(\Delta^{\rho}_{\tau}\epsilon_{L}+\epsilon_{\mu}\Delta^{\rho}_{R})}{y_{\tau}}$
In the further calculations, we consider a case of $(t_{\gamma}\Delta)\ll 1$
but large enough (as investigated in MSSM) to cause many interesting effects,
and we will comment more details after some numerical calculations. On the
other hand, the rotation angle given in Eq. (44) is very small, so we can set
$c_{\Lambda}\simeq 1,s_{\Lambda}\simeq\Lambda$. As a result, Eqs. (45), (77)
and (78) can be presented as very simple formulas:
$\displaystyle y_{\mu}$ $\displaystyle\simeq$
$\displaystyle\epsilon_{\mu},\hskip 14.22636pty_{\tau}\simeq 1,$
$\displaystyle\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{12}$
$\displaystyle\simeq$ $\displaystyle\Delta^{\rho}_{R},\hskip
14.22636pt\left(U^{\dagger}_{l}\mathcal{Y}_{l_{2}}V^{\dagger}\right)_{21}\simeq\Delta^{\rho}_{L},$
(79)
and the above LFV Lagrangian also appears in a simple form:
$\displaystyle-\mathcal{L}_{FV}$ $\displaystyle\simeq$ $\displaystyle
Y_{\tau}(\Delta^{\rho}_{R}\mu^{c}\tau+\Delta^{\rho}_{L}\tau^{c}\mu)(\rho^{0*}-t_{\gamma}\rho^{\prime
0})+\mathrm{H.c.}.$ (80)
Finally, in the mass-eigenstate basis for both lepton and Higgs, we obtain the
effective LFV Lagrangian:
$\displaystyle-\mathcal{L}_{FV}$ $\displaystyle\simeq$
$\displaystyle\sqrt{2}Y_{\tau}(\Delta^{\rho}_{R}\mu^{c}\tau+\Delta^{\rho}_{L}\tau^{c}\mu)\left(s_{\alpha}s_{\gamma}\phi_{S_{a36}}-c_{\alpha}s_{\gamma}\varphi_{S_{a36}}\right)+\mathrm{H.c.},$
(81)
where $\varphi_{S_{a36}}$ and $\phi_{S_{a36}}$ are the Higgs mass eigenstates
generated from the mixing of two original Higgs bosons $\rho^{0}$ and
$\rho^{\prime 0}$. The expressions of the Higgs mass eigenstates were
introduced in [25]. They are summarized in the Appendix C. The emphasis here
is that in the general supersymmetric model there exist both the leading
interactions of the muon, tauon with neutral scalar and pseudo scalar Higgs.
However, the SUSYE331 model contains only interactions among muon, tauon and
scalar Higgs.
The effective couplings given in (81) are widely investigated for many LFV
low-energy processes, specially in the MSSM [10, 18, 28]. In this paper we
first concentrate on some simple aspects of LFV in the SUSYE331 model. In
particular, we are going to consider the LFV in decays of the scalar Higgs,
i.e. $\Phi^{0}\rightarrow\tau^{\pm}\mu^{\mp}$, where
$\Phi^{0}=\varphi_{S_{a36}}$ or $\phi_{S_{a36}}$. First, we start with
studying the branching ratios of neutral Higgs decay into muon and tauon. The
SUSYE331 model predicts that the formula of these branching ratios is
$\displaystyle BR(\Phi^{0}\rightarrow\tau^{+}\mu^{-})$ $\displaystyle=$
$\displaystyle BR(\Phi^{0}\rightarrow\tau^{-}\mu^{+})$ $\displaystyle=$
$\displaystyle
2(1+\tan^{2}\gamma)\left(\mid\Delta^{\rho}_{L}\mid^{2}+\mid\Delta^{\rho}_{R}\mid^{2}\right)~{}BR(\Phi^{0}\rightarrow\tau^{+}\tau^{-}).$
This result is similar to that one given in [10], except the absence of angle
of mixing among Higgses. In the limit of appropriately large $\tan\gamma$, the
effects of LFV in the Higgs decay processes is not to be ignored. Hence, our
theoretical prediction is not much different from that of previous results
given in [10, 11, 15, 16, 28]. For details, we will study some numerical
calculations for the branching ratios indicated by Eq.(LABEL:Br1). In our
paper, we use the assumption for slepton mixing presented in Appendix C. The
diagrams giving contributions to $\Delta^{\rho}_{R}$ and $\Delta^{\rho}_{L}$
are shown in Fig.1. The relevant vertices to our calculation are presented in
Appendix B.
$\mu$$\tau^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\small\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(a)$
$\mu$$\tau^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda^{3}_{A}$$\lambda^{8}_{A}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(b)$
$\mu$$\tau^{c}$$\tilde{\rho}_{1}^{\prime-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho^{0*}$$\tilde{\nu}_{L\alpha}$$(c)$
$\mu$$\tau^{c}$$\tilde{\rho}_{2}^{\prime-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho^{0*}$$\tilde{\nu}_{R\alpha}$$(d)$
$\mu$$\tau^{c}$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{0*}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(e)$
$\mu$$\tau^{c}$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{0*}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(f)$
$\tau$$\mu^{c}$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{R_{\alpha}}$$(i)$
$\mu$$\tau^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(k)$
$\tau$$\mu^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(l)$
Figure 1: Diagrams contributing to $\Delta^{\rho}_{L}$
[$(a),(b),(c),(d),(e),(f),(k)$] and $\Delta^{\rho}_{R}$ [$(i),(l)$].
Using Feynman rules, we can obtain the expression $\Delta_{L}^{\rho}$ from the
diagrams in Fig.1, namely:
$\displaystyle\Delta^{\rho}_{L}$ $\displaystyle=$
$\displaystyle\Delta^{\rho}_{La}+\Delta^{\rho}_{Lb}+\Delta^{\rho}_{Lc}+\Delta^{\rho}_{Ld}+\Delta^{\rho}_{Le}+\Delta^{\rho}_{Lf}+\Delta^{\rho}_{Lk},$
(83)
where
$\displaystyle\Delta^{\rho}_{La}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{216\pi^{2}}\mu_{\rho}m^{\prime}c_{L}s_{L}\left[I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})-I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Lb}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}c_{L}s_{L}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Lc}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}c_{\nu_{L}}s_{\nu_{L}}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right],$
$\displaystyle\Delta^{\rho}_{Ld}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}c_{\nu_{R}}s_{\nu_{R}}\left[I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})-I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right],$
$\displaystyle\Delta^{\rho}_{Le}$ $\displaystyle=$
$\displaystyle\frac{(Y_{\nu_{\mu\tau}})h_{\mu\tau}-h_{\tau\mu}\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s_{\nu_{(L-R)}}\left(s_{\nu_{L}}s_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+c_{\nu_{L}}c_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c_{\nu_{(L-R)}}\left(s_{\nu_{R}}c_{\nu_{L}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})-s_{\nu_{L}}c_{\nu_{R}}I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right],$
$\displaystyle\Delta^{\rho}_{Lf}$ $\displaystyle=$
$\displaystyle-\Delta^{\rho}_{Le},$ $\displaystyle\Delta^{\rho}_{Lk}$
$\displaystyle=$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{L}c_{L}\left[s^{2}_{R}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right)\right.$ (84)
$\displaystyle+$ $\displaystyle\left.c^{2}_{R}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$
Also, the $\Delta^{\rho}_{R}$ receives contributions from two diagrams ($i$)
and ($l$) of Fig.1 too,
$\displaystyle\Delta^{\rho}_{R}$ $\displaystyle=$
$\displaystyle\Delta^{\rho}_{Ri}+\Delta^{\rho}_{Rl},$ (85)
where
$\displaystyle\Delta^{\rho}_{Ri}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{72\pi^{2}}\mu_{\rho}m^{\prime}c_{R}s_{R}\left[I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{R_{3}})\right],$
$\displaystyle\Delta^{\rho}_{Rl}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{R}c_{R}\left[s^{2}_{L}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$ (86)
$\displaystyle+$ $\displaystyle\left.c^{2}_{L}\left(I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right]$
Here we have used some new notations
$\displaystyle s_{\nu_{(L-R)}}\equiv
s_{\nu_{L}}c_{\nu_{R}}-s_{\nu_{R}}c_{\nu_{L}},\hskip
14.22636ptc_{\nu_{(L-R)}}\equiv
s_{\nu_{L}}s_{\nu_{R}}+c_{\nu_{L}}c_{\nu_{R}},$ (87)
where $s_{L}$, $c_{L}$ and $s_{R}$, $c_{R}$ are deduced from mixing angles for
left and right handed sleptons (for details, see Appendix C.2). The same
relations hold for sneutrino sector, with corresponding notations for mixing
angles $s_{\nu_{L}},s_{\nu_{R}},c_{\nu_{L}}$ and $c_{\nu_{R}}$. The function
$I_{3}(x,y,z)$ is similar to that mentioned in literature [11],
$\displaystyle
I_{3}(x,y,z)=\frac{xy\log(x/y)+yz\log(y/z)+zx\log(z/x)}{(x-y)(y-z)(z-x)}.$
(88)
The analytical results appearing in (84) show that contributions from two
diagrams (e) and (f) to the $\Delta^{\rho}_{L}$ always are the same magnitude
but opposite in sign. Therefore the total contribution of these two diagrams
to $\Delta^{\rho}_{L}$ vanishes. On the other hand, results obtained from the
Eq.(84) show that if we neglect the terms of the slepton mixing, namely
$s_{L}=s_{R}=s_{\nu_{L}}=s_{\nu_{R}}=0$, the amounts collected from the
$\Delta^{\rho}_{L,R}$ class diagrams given in Fig. 1 are all zero. This
corresponds to the case of lepton flavor conservation:
$\Delta^{\rho}_{L}=\Delta^{\rho}_{R}=0.$
We also remind that analytical expressions of other $\Delta$ functions can be
found in Appendix A. These results demonstrate that the values
$\Delta^{1\rho}_{\mu},\Delta^{2\rho}_{\mu}$ and $\Delta^{\rho}_{\tau}$ given
by expressions (94), (95) and (96) obtained at one loop approximation, do not
vanish even if we assume no mixing of the sleptons. These quantities create
non-negligible effects of the lepton masses. They are widely discussed in many
previous papers. Another feature of the SUSYE331 model that we would like to
remind here: there are two independent sources (Yukawa coupling at tree level)
to create masses of slepton and neutrino sectors. Hence, contributions to LFV
corrections come from two independent sources: mixing of lepton and sneutrino
sectors. We assume that the model contains both LFV sources.
Before coming to numerical computation section, it is necessary to note that
the formulas of LFV corrections, such as $\Delta$s in this case, have not been
established for the SUSYE331 model before. So let us give some general
comments on the formulas of $\Delta$s which discriminate against those in MSSM
versions:
* •
At the one loop approximation, the effective couplings $\Delta$s are obtained
from the diagrams such as those listed in Figs. (7) and (8). We can
distinguish two types of diagram which give contribution to $\Delta$s. The
first type of diagram does not include any Higgsino propagators, for example
Fig. 1 (k) and (l), and they are known as pure gaugino-mediated diagrams. The
second, containing at least one Higgsino propagator like remaining diagrams,
is Higgsino-mediated type. In general case, each of these kinds of diagram may
give main contribution to the $\Delta$s depending on regions of mass parameter
space. If each Higgsino-mediated diagram gives the dominated contribution to
$\Delta$s that reach single maximum value. In contrast, each $\Delta$ that
gains values from pure gaugino-mediated diagrams, $\Delta^{\rho}_{Lk}$ for
example, is proportional to $|\mu_{\rho}|$. Additionally, we can see the
analytic expressions of $\Delta$s given in (84), (86) and Appendix A. It is
well known in beyond MSSM theories [10], all of effective couplings $\Delta$s
are obtained from both types of diagrams, except $\Delta^{\prime}_{\mu}$. In
the limit of large values of $|\mu_{\rho}|$, the dominated contributions of
$\Delta$s are caused by pure gaugino-mediated diagrams. This conclusion also
is happened in the SUSYE331. However, in the SUSYE331, there are the
additional $SU(3)_{L}$ gaugino-mediated diagrams. Hence the values of
$\Delta^{\rho}_{L}$ can be changed in comparison with other models. Details of
this difference are discussed in section 4.
* •
The difference between the predictions of the model under consideration and
other ones due to hypercharge structure of particle content. For example, let
us compare our expressions of $\Delta$s with those of $\Delta$s in MSMS [10].
All contributions to the $\Delta$s obtained from Fig.1, are proportional to
$I_{3}$ functions. Rate coefficients in both models are the same level for
diagrams of Higgsino-mediated type whereas the rate coefficients in the model
under consideration are smaller than that in the MSSM model for diagrams of
pure gaugino-mediated type. As a consequence, the large contribution to the
$\Delta$s from the pure gaugino-mediated type will happen if mass parameters
are large. Furthermore, in this limit of mass parameters, the pure gaugino-
mediated diagrams are the only source giving contribution to radiative
corrections $\Delta^{2\rho}_{\mu}$ of muon mass. It is nature to keep the
ratio $Y_{\tau}/Y_{\mu}$, at one loop correction, to be the same as those at
tree level. This leads to the limit of the mass parameters, which does not
exceed $10$ TeV .
In the next section we will investigate some numerical results. On that basis,
we will compare the effects of the LFV origin in the left- and right-slepton
sectors as well as sneutrino sectors. In order to investigate numerically, we
are going to use results from [25] such as:
$g^{\prime}/g=\frac{3\sqrt{2}s_{W}}{\sqrt{4c^{2}_{W}-1}}$, $s^{2}_{W}=0.2312$
and $\alpha^{-1}_{em}=128$ at the weak scale.
## 4 Numerical results
|
---|---
Figure 2: $|\Delta^{\rho}_{R}|^{2}$ as a function of $|\mu_{\rho}|/\tilde{m}_{R}$ with four different choices of masses ratios: 1) blue curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green curve–$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve- $m^{\prime}=\tilde{m}_{R}=3\tilde{m}_{L}$; 4) red curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}/3$. Two black lines correspond to two values $10^{-5}$ and $10^{-3}$ of $|50\Delta^{\rho}_{R}|^{2}$ . |
---|---
Figure 3: $|\Delta^{\rho}_{L}|^{2}$ as a function of
$|\mu_{\rho}|/\tilde{m}_{L}$ with four different choices of masses ratios: 1)
blue curve–$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green
curve–$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve–
$m^{\prime}=\tilde{m}_{L}=3\tilde{m}_{R}$; 4) red
curve–$m^{\prime}=\tilde{m}_{L}=\tilde{m}_{R}/3$. A black line corresponds to
value $10^{-3}$ of $|50\Delta^{\rho}_{L}|^{2}$.
In this section we firstly discuss some numerical results that relate to any
signals of LFV decays $H\rightarrow\mu\tau$. Let us start with the maximum LFV
in both left and right sectors, especially
$s_{L}c_{L}=s_{R}c_{R}=s_{\nu_{L}}c_{\nu_{L}}=s_{\nu_{R}}c_{\nu_{R}}=0.5$. It
means that we can assign values of mass parameters like:
$\displaystyle\tilde{m}^{2}_{(\tau_{L},\tau_{R},\nu_{\tau_{L}},\nu_{\tau_{R}})}$
$\displaystyle=$
$\displaystyle\tilde{m}^{2}_{(\mu_{L},\mu_{R},\nu_{\mu_{L}},\nu_{\mu_{R}})}$
$\displaystyle=$ $\displaystyle\tilde{m}^{2}_{L,R,\nu_{L},\nu_{R}},$
$\displaystyle\tilde{m}^{2}_{(L_{2},R_{2},\nu_{L_{2}},\nu_{R_{2}})}$
$\displaystyle=$ $\displaystyle 0.2\,\tilde{m}^{2}_{L,R,\nu_{L},\nu_{L}}$
and
$\tilde{m}^{2}_{(L_{3},R_{3},\nu_{L_{3}},\nu_{R_{3}})}=1.8~{}\tilde{m}^{2}_{(L,R,\nu_{L},\nu_{R})},$
where $\tilde{m}^{2}_{(L,R,\nu_{L},\nu_{R})}$ are mass parameters used to
compare with SUSY mass scale $m_{SUSY}$. We would like to emphasize that
branching ratios of Higgs decays to muon and tauon are sizable if $\tan\gamma$
is large enough. Therefore, in the following calculations, we take
$\tan\gamma\sim 50$.
The Fig.2 displays the quantity $|50\Delta^{\rho}_{R}|^{2}$ as a function of
$|\mu_{\rho}|/\tilde{m}_{R}$ while Fig.3 displays the
$|50\Delta^{\rho}_{L}|^{2}$ as a function of $|\mu_{\rho}|/\tilde{m}_{L}$
where all other relevant parameters are fixed. Each curve presented in Figs.2,
3 contains a single peak. All the peaks of the curves are obtained at mass
parameters at which the contribution of the Higgs-mediated diagrams to
$\Delta$s are dominated. Corresponding to each curve, there are two regions of
mass parameter space separated by deep wells. Deep wells, which divide the
parameter space into two parts. The first part, the mass parameters are
located in the right hand side of deep wells. In this region of parameter
space,the pure gaugino-mediated type can give main contribution to $\Delta$s.
The second part, the mass parameters are located in the left handed side of
deep wells at which the dominated contribution to $\Delta$s is obtained by the
Higgs-mediated. All of the maximum points of the curves in the Fig.2 and Fig.3
are reached at $|\mu_{\rho}|/\tilde{m}_{L,R}\sim\mathcal{O}(10^{-1})$ and
these values depend weekly on the changes of values of $\tilde{m}_{L}$ and
$\tilde{m}_{R}$. On the other hand, the maximum values of the
$\Delta^{\rho}_{L}$ is $\mathcal{O}(10^{-3})$, as concerned in the MSSM [10,
12, 18, 29] while the maximum values of the $\Delta^{\rho}_{R}$ are much
smaller than those of $\Delta^{\rho}_{L}$, specifically
max$(|\Delta^{\rho}_{R}|)^{2}/\mathrm{max}(|\Delta^{\rho}_{L}|)^{2}\sim
10^{-3}$. This large difference comes from the symmetry of
$SU(3)_{L}\times\mathrm{U}(1)_{X}$ model. In particular, in the left side of
wells the main contributions to $\Delta^{\rho}_{L}$ of SUSYE331 model come
from the $SU(3)_{L}$ gaugino-mediated diagrams, namely diagrams ((b), (c),
(d)) in Fig.1. In contrast, the main contributions to $\Delta^{\rho}_{R}$ come
from only $\mathrm{U}(1)$ gaugino-mediated diagram. Figs.2 and 3 also show
that both $\Delta^{\rho}_{L}$ and $\Delta^{\rho}_{R}$ are very sensitive with
the changes of $\tilde{m}_{L}$ and $\tilde{m}_{R}$. More details, Fig.4 draws
the dependence of $|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ on
$|\mu_{\rho}|/\tilde{m}_{L}$, where
$m^{\prime}=m_{\lambda}=\tilde{m}_{L}\equiv m_{SUSY}$ and four different fixed
values of $\tilde{m}_{R}$. The maximal and minimal values of the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ on all the curves in Fig.4
have the same value at different values of $|\mu_{\rho}|/\tilde{m}_{SUSY}$. In
the parameter region where the Higgs-mediated diagrams give dominated
contribution to $\Delta$s, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ is very small $(<10^{-3})$.
But in the remaining parameters, that ratio is increased. In the limit
$|\mu_{\rho}|/\tilde{m}_{SUSY}\geq 30$, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ reaches a constant value.
More general, we can investigate the influence of
$\tilde{m}_{R}/\tilde{m}_{L}$ on the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ through contour plots drawn
in Fig.5. On the drawing results showed that
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}\leq\mathcal{O}(10^{-2})$
whenever $|\mu_{\rho}|/m_{SUSY}\leq 5$ and that ratio does not depend too much
on the ratio $\tilde{m}_{R}/\tilde{m}_{L}$. However, in the limit
$|\mu_{\rho}|/m_{SUSY}\geq 7$ and $\tilde{m}_{R}<0.5\tilde{m}_{L}$, the ratio
$|\Delta^{\rho}_{R}|^{2}/|\Delta^{\rho}_{L}|^{2}$ changes very rapidly if
small changes $\tilde{m}_{L}$ and $\tilde{m}_{R}$. It means that chirality
effects of phenomena relating with $\Delta^{\rho}_{L}$ and $\Delta^{\rho}_{R}$
are sensitive with the change of ratio $\tilde{m}_{R}/\tilde{m}_{L}$ at large
values of $\mu_{\rho}$. On the other hand, the left picture in Fig.5 indicates
that when the ratio $|\mu_{\rho}|/m_{SUSY}\geq 7$, it will exist in some
regions of parameter space of $\tilde{m}_{R},\tilde{m}_{L}$ at which the
contributions of left- and right-lepton sectors into the $H\rightarrow\mu\tau$
decay process are of the same order. In this case, the pure gaugino-mediated
diagrams give the dominated contribution to both $\Delta^{\rho}_{L}$ and
$\Delta^{\rho}_{R}$, and also $\Delta^{2\rho}_{\mu}$ ( see (95) in Appendix
A). Recalling that large values of $\Delta^{2\rho}_{\mu}$ can strongly affect
directly on the ratio $Y_{\mu}/Y_{\tau}$. The results presented in Fig.5 again
confirm that whenever $|\mu_{\rho}|/m_{SUSY}\geq 7$ and
$\tilde{m}_{R}<0.5\tilde{m}_{L}$, the right-lepton sector gives dominated
contribution to the branching ratio of $H\rightarrow\mu\tau$ decay process.
|
---|---
Figure 4: $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ as a function of $|\mu_{\rho}|/\tilde{m}_{L}$ with four different choices of masses ratios: 1) blue curve-$m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 2) green curve-$3m^{\prime}=\tilde{m}_{R}=\tilde{m}_{L}$ ; 3) yellow curve- $m^{\prime}=\tilde{m}_{L}=3\tilde{m}_{R}$; 4) red curve-$m^{\prime}=\tilde{m}_{L}=\tilde{m}_{R}/3$. A black line in the left side of figure corresponding to the value $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ equals 1. Both black lines in the right side of figure presenting $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$ are $2\times 10^{-3}$ and $0.1$. |
---|---
Figure 5: Contour plot of $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}$, $\tilde{m}_{R}/\tilde{m}_{L}$ vs $|\mu_{\rho}|/m_{SUSY}$, where $\tilde{m}_{R}=\tilde{m}_{\nu_{R}}$, $m^{\prime}=m_{\lambda}=\tilde{m}_{L}=\tilde{m}_{\nu_{L}}=m_{SUSY}$. The red region corresponds to the values of $\frac{|\Delta^{\rho}_{R}|^{2}}{|\Delta^{\rho}_{L}|^{2}}\geq 0.5$. |
---|---
Figure 6: Contour plot of $BR(H\rightarrow\mu\tau)/BR(H\rightarrow\tau\tau)$,
$\tilde{m}_{g}$ vs $|\mu_{\rho}|/m_{SUSY}$, where
$m^{\prime}=m_{\lambda}=\tilde{m}_{g}$ and
$\tilde{m}_{R}=\tilde{m}_{\nu_{R}}=\tilde{m}_{L}=\tilde{m}_{\nu_{L}}=m_{SUSY}$.
In the left picture, the green and yellow regions correspond to the values of
$BR(H\rightarrow\mu\tau)/BR(H\rightarrow\tau\tau)\geq\mathcal{O}(10^{-3})$ .
Now we investigate more details the region of parameter space where Higgsino-
mediated diagrams give a dominated contribution. In this region of parameter
space, both $\Delta^{2\rho}_{\mu}$ and $\Delta^{\rho}_{R}$ are much smaller
than $\Delta^{\rho}_{L}$ so we just focus on $\Delta^{\rho}_{L}$. From Fig.6,
we can estimate the ratio of
$Br(H\rightarrow\mu\tau)/Br(H\rightarrow\tau\tau)$ that can reach the order of
$10^{-3}$ in the limit $0.1\leq|\mu_{\rho}|/M_{SUSY}\leq 6$ and
$0.1\leq|\tilde{m}_{g}|/M_{SUSY}\leq 7$ where $\tilde{m}_{g}$ is mass of
gauginos. Let us briefly review the decay properties of neutral Higgs bosons
in the SUSYE331 model. At the tree level, the couplings of neutral Higgs
bosons to up-fermions, down-fermion are modified with respect to the SM
coupling by factors which are given in Table 1.
Table 1: Coupling of neutral Higgs bosons to fermion. Particles | Up-fermion | Down-fermion | Exotic up-quark | Exotic down-quark
---|---|---|---|---
SM Higgs | 1 | 1 | 0 | 0
$\varphi_{Sa36}$ | $c_{\alpha}$ | $c_{\alpha}$ | $s_{\alpha}/s_{\gamma}$ | $c_{\alpha}/s_{\gamma}$
$\phi_{Sa36}$ | $s_{\alpha}$ | $s_{\alpha}$ | $c_{\alpha}$ | $s_{\alpha}$
We assume that all exotic quarks have masses heavier than that of all neutral
Higgses. It means that the neutral Higgs cannot decay into the exotic quarks.
The neutral Higgs bosons may decay mainly into the pairs of fermions. This
prediction depends on the mass of the neutral Higgs. For neutral Higgs
$\varphi_{Sa36}$, its mass depends on the vacuum expectation values
$v,v^{\prime}$. So it should be predicted SM Higgs with mass smaller than
about $130$ GeV. Decay of $\varphi_{Sa36}$ to $b\overline{b}$ and
$\tau\overline{\tau}$ are dominated, the branching ratios of $90$ percent and
$8$ percent, respectively. Combined with the results in Fig.6, the branching
ratio $Br(\varphi_{Sa36}\rightarrow\mu\tau)$ is $8\times 10^{-3}$ percent.
This may be a good signification of new physics in the present limits of
colliders. For neutral Higgs $\phi_{Sa36}$, it is heavy Higgs, the main
productions of decay are the the gauge bosons such as $W^{+}W^{-}$,
$ZZ$,…Hence, the branching ratio $\phi\rightarrow\mu\tau$ is very suppressed.
We would like to note that the effective interactions of the muon, tauon and
Higgs given in (81) not only leads to the LFV of Higgs decay process, but also
affects the other physical processes with lepton-flavor violations. Some of
these processes which are looked seriously by present experiments are, for
instance, $\tau\rightarrow\mu\mu\mu$ and $\tau\rightarrow\mu\gamma$. Let us
apply the effective couplings given in (81) to the $\tau\rightarrow\mu\mu\mu$
decay process. In a general way-regardless of the model, the general effective
Lagrangian describing decay of $\tau\rightarrow\mu\mu\mu$ was studied in [30].
However, in this work we focus on the effect of the Higgs-mediated LFV
interactions on the $\tau\rightarrow\mu\mu\mu$ decay process. Hence, the four
dimensional effective Lagrangian which is built through Higgs exchange is
formulated by
$\displaystyle\mathcal{L}^{eff}_{\tau\mu\mu\mu}$ $\displaystyle=$
$\displaystyle-2\sqrt{2}G_{F}m_{\mu}m_{\tau}\tan\gamma\left(\frac{s^{2}_{\alpha}}{m^{2}_{\phi_{Sa36}}}+\frac{c^{2}_{\alpha}}{m^{2}_{\varphi_{Sa36}}}\right)(\mu^{c}\mu+\bar{\mu}\bar{\mu}^{c})$
(89) $\displaystyle\times$
$\displaystyle(\Delta^{\rho}_{L}\tau^{c}\mu+\Delta^{\rho}_{R}\mu^{c}\tau)+\mathrm{H.c.}.$
We would like to remind that the decay process of $\tau\rightarrow\mu\mu\mu$
were investigated, by [11, 30] for examples, in a general model-independent
way. The predicted results show that when Higgs exchange effects are much
smaller than other ones, the ratio $Br(\tau\rightarrow
3\mu)/BR(\tau\rightarrow\mu\gamma)$ becomes constant with a value
$\sim\mathcal{O}(10^{-3})$. Now, we will discuss in more details whether
Higgs-mediated effects can make any significations to the ratio
$Br(\tau\rightarrow 3\mu)/BR(\tau\rightarrow\mu\gamma)$ in the SUSYE331 model.
We can divide our results into two cases, namely $\phi^{*}_{S_{a36}}$ and
$\varphi^{*}_{S_{a36}}$ Higgs-mediated effects. The results can be written in
two respective forms:
$\displaystyle
BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{\phi^{*}_{S_{a36}}}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tan^{2}\gamma~{}\frac{m^{2}_{\mu}m^{2}_{\tau}}{m^{4}_{\phi_{S_{a36}}}}s^{4}_{\alpha}\left(|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}\right)$
$\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\phi_{S_{a36}}}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right)s^{4}_{\alpha}$
and
$\displaystyle
BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{\varphi^{*}_{S_{a36}}}$
$\displaystyle=$
$\displaystyle\frac{1}{8}\tan^{2}\gamma~{}\frac{m^{2}_{\mu}m^{2}_{\tau}}{m^{4}_{\varphi_{S_{a36}}}}c^{4}_{\alpha}\left(|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}\right)$
(91) $\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\varphi_{S_{a36}}}}\right)^{4}$
$\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right)c^{4}_{\alpha}.$
These results immediately lead to a consequence: the maximum contribution of
Higgs exchange processes can be estimated through the formula:
$\displaystyle BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})_{H^{*}}$
$\displaystyle\simeq$ $\displaystyle 7\times
10^{-11}\left(\frac{\tan\gamma}{40}\right)^{2}\left(\frac{100\mathrm{GeV}}{m_{\mathrm{H}}}\right)^{4}$
(92) $\displaystyle\times$
$\displaystyle\left(\frac{|\Delta^{\rho}_{L}|^{2}+|\Delta^{\rho}_{R}|^{2}}{10^{-3}}\right).$
The values of the branching ratios decrease rapidly corresponding to the
enhancement of Higgs masses . We stress that in the model under consideration,
the $\varphi_{S_{a36}}$ is identified with the SM Higgs boson and the remain,
$\phi_{S_{a36}}$, is heavy one. Overall, in our model, the SM Higgs-mediated
gives larger contribution to the branching ratio of the
$\tau\rightarrow\mu\mu\mu$ decay process than that of heavy Higgs. That kind
of branching increases if the $\tan\gamma$ increases. The branching ratio
estimated in (92) is $\simeq 10^{-11}$ in the limit of $\tan\gamma\simeq 50$
and the Higgs mass is of the order of $100$ GeV. However, the branching ratio
of $\tau\rightarrow\mu\mu\mu$ can be reached at the present limits of
experiment $BR(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})\leq 3.2\times
10^{-8}$ [31]. It means that the contribution to $\tau\rightarrow\mu\mu\mu$ is
suppressed in the limit of $\tan\gamma\simeq 50$. This result is different
from that predicted in the MSSM model [16, 10, 11]. In particular, for the
MSSM, the dominant contributions to $BR(\tau\rightarrow\mu\mu\mu)$ are induced
by the dipole term and the Higgs-mediated term at the limit
$\tan\beta=50,m_{A}=100$ GeV.
Because of sub-dominated contribution of Higgs-mediated to
$BR(\tau\rightarrow\mu\mu\mu)$, the dominated contribution to that branching
ratio is still obtained from the photon-penguin couplings. This result leads
to the values of well-known ratios such as
$\displaystyle\frac{Br(\tau\rightarrow
3\mu)}{BR(\tau\rightarrow\mu\gamma)}\simeq\mathcal{O}(10^{-3})$ (93)
The predicted result is the concerned result given in [16, 11, 30, 32].
We emphasize that in the SUSYE331 model, in order to get the dominated
contribution to the $B(\tau\rightarrow\mu\mu\mu)$, the values of $\tan\gamma$
must be $10^{2}$. In this limit of $\tan\gamma$, the result given in (93) is
not holden.
## 5 Conclusions
In this paper, we have studied the LFV interactions of Higgs bosons in the
SUSYE331 model. We have the unique existence of the lepton-number violation in
the slepton sector at the tree level. On the basis of this assumption we have
examined the lepton-number violating interactions of Higgs bosons at the one-
loop level. Specially we have concentrated our study on the LFV couplings of
Higgs bosons with muon and tauon. The analytical expressions of the effective
Higgs-muon-tauon couplings are established at the one-loop level. One of the
features is that the model does not contain the LFV interactions of neutral
pseudo-scalar Higgs bosons. For the neutral Higgs scalars, the model contains
two types of radiative interactions that violate lepton number, namely,
$\phi_{Sa36}\mu\tau$ and $\varphi_{Sa36}\mu\tau$. These effective couplings
depend on ratios of SUSY mass parameters and $\tan\gamma$. There is an exactly
similar to the other SUSY models, all LFV couplings are built from two types
of diagrams as Higgs-mediated diagrams and pure gaugino-mediated ones.
Depending on the SUSY parameters, each type of diagram gives the main
contribution to LFV couplings. In this work, we have also studied the
branching ratio of the neutral Higgs decay into muon and tauon. In the limit
$|\mu_{\rho}|/m_{SUSY}\leq 7$, the ratio of
$BR(H\rightarrow\tau\mu)/BR(H\rightarrow\tau\tau)$ in this region can reach
values that can be observed by near future experiments and the contributions
from both left and right LFV sectors to $Br(H\rightarrow\mu\tau)$ are of the
same order. Outside this region the effects of left and right LFV terms mix in
different ways in different regions of mass parameter space. We predicted that
for the SM Higgs boson, LHC may detect the decay of SM Higgs boson to muon and
tauon. For heavy Higgs bosons, the branching of LFV decay is very suppressed.
We have also studied the contribution of Higgs exchange to decay $H\rightarrow
3\mu$. In the limit $\tan\gamma=50$, the $Br(H\rightarrow 3\mu)$ is very
small, out of direct detection of present searching of experiment and it leads
to predicted results such as the ratio of $BR(\tau\rightarrow
3\mu)/BR(\tau\rightarrow\mu\gamma)$.
## Acknowledgments
L.T.H would like to thank the Organizers of KEK-Vietnam Visiting Program 2011
(Exchange Program for East Asia Young Researchers, JSPS), especially Prof. Y.
Kurihara, for the support for his initial work at KEK. This work was supported
in part by the National Foundation for Science and Technology Development
(NAFOSTED) of Vietnam under Grant No. 103.01-2011.63.
## Appendix A Analytic formulas and diagrams contributing to $\Delta$s
Let us write down all expressions of $\Delta$s given as following
$\displaystyle\Delta^{1\rho}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{108\pi^{2}}\mu_{\rho}m^{\prime}\left[c^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+s^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$ (94) $\displaystyle-$
$\displaystyle\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+s^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})+s^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[c^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})+s^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right]$
$\displaystyle+$
$\displaystyle\frac{Y_{\nu_{\mu\tau}}(h_{\mu\tau}-h_{\tau\mu})\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}\left[c^{2}_{L}\left(c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})+s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$
$\displaystyle+$ $\displaystyle\left.s^{2}_{L}\left(c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$
$\displaystyle\Delta^{2\rho}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}s_{L}c_{L}s_{R}c_{R}\left[I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})-I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right.$ (95) $\displaystyle-$
$\displaystyle\left.I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right].$
$\displaystyle\Delta^{\rho}_{\tau}$ $\displaystyle=$
$\displaystyle-\frac{g^{\prime
2}}{108\pi^{2}}\mu_{\rho}m^{\prime}\left[s^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+c^{2}_{L}I_{3}(m^{\prime
2},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$ (96) $\displaystyle-$
$\displaystyle\frac{g^{2}}{24\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{2}})+c^{2}_{L}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{L_{3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}})+c^{2}_{\nu_{L}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}})\right]$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\mu_{\rho}m_{\lambda}\left[s^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda},\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R2}})+c^{2}_{\nu_{R}}I_{3}(m^{2}_{\lambda}\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{R3}})\right]$
$\displaystyle+$
$\displaystyle\frac{Y_{\nu_{\mu\tau}}(h_{\mu\tau}-h_{\tau\mu})\mu_{\rho}}{8\pi^{2}}$
$\displaystyle\times$
$\displaystyle\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
$\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{3L}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})\right)\right]$
$\displaystyle+$ $\displaystyle\frac{g^{\prime
2}}{288\pi^{2}}\mu_{\rho}m^{\prime}\left[s^{2}_{L}\left(s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})+c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right)\right.$
$\displaystyle+$ $\displaystyle\left.c^{2}_{L}\left(s^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})+c^{2}_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right)\right].$
$\displaystyle\Delta^{1\rho^{\prime}}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{Y^{2}_{\nu_{\mu\tau}}}{4\pi^{2}}\mu^{2}_{\rho}\left[s^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R2}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R3}})\right)\right.$
(97) $\displaystyle+$
$\displaystyle\left.c^{2}_{\nu_{(L-R)}}\left(I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L2}},\tilde{m}^{2}_{\nu_{R3}})+I_{3}(\mu^{2}_{\rho},\tilde{m}^{2}_{\nu_{L3}},\tilde{m}^{2}_{\nu_{R2}})\right)\right].$
$\displaystyle\Delta^{2\rho^{\prime}}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right].$
$\displaystyle\Delta^{\rho^{\prime}}_{\tau}$ $\displaystyle=$
$\displaystyle\Delta^{1\rho^{\prime}}_{\mu}+\frac{g^{\prime
2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right]$
In the case of $\Delta^{\rho^{\prime}}_{L}$, it also receives contributions
from two diagrams (similar to diagrams (e) and (f) in Fig.1 ) which cancel
each other. Therefore we do not repeat them in Figs.and 9 and 10 . Formula of
$\Delta^{\rho^{\prime}}_{L}$ then is
$\displaystyle\Delta^{\rho^{\prime}}_{L}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{\tilde{L}_{2}},\tilde{m}^{2}_{\tilde{R}_{3}})\right].$
$\displaystyle\Delta^{\rho^{\prime}}_{R}$ $\displaystyle=$
$\displaystyle\frac{g^{\prime 2}m^{\prime}}{144\pi^{2}}$ $\displaystyle\times$
$\displaystyle\left[\frac{h^{\prime}_{\mu}c_{L}c_{R}+h^{\prime}_{\tau}s_{L}s_{R}+h^{\prime}_{\mu\tau}c_{L}s_{R}+h^{\prime}_{\tau\mu}s_{L}c_{R}}{Y_{\tau}}s_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{h^{\prime}_{\mu}s_{L}s_{R}+h^{\prime}_{\tau}c_{L}c_{R}-h^{\prime}_{\mu\tau}s_{L}c_{R}-h^{\prime}_{\tau\mu}c_{L}s_{R}}{Y_{\tau}}c_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{3}})\right.$ $\displaystyle+$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}s_{L}c_{R}+h^{\prime}_{\tau}c_{L}s_{R}-h^{\prime}_{\mu\tau}s_{L}s_{R}+h^{\prime}_{\tau\mu}c_{L}c_{R}}{Y_{\tau}}c_{L}c_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{3}},\tilde{m}^{2}_{R_{2}})\right.$ $\displaystyle-$
$\displaystyle\left.\frac{-h^{\prime}_{\mu}c_{L}s_{R}+h^{\prime}_{\tau}s_{L}c_{R}+h^{\prime}_{\mu\tau}c_{L}c_{R}-h^{\prime}_{\tau\mu}s_{L}s_{R}}{Y_{\tau}}s_{L}s_{R}I_{3}(m^{\prime
2},\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{R_{3}})\right].$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(a)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda^{3}_{A}$$\lambda^{8}_{A}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$(b)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}_{1}^{\prime-}$$\tilde{\rho}_{1}^{+}$$\tilde{W}^{-}$$\tilde{W}^{+}$$\rho^{0*}$$\tilde{\nu}_{L\alpha}$$(c)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}_{2}^{\prime-}$$\tilde{\rho}_{2}^{+}$$\tilde{Y}^{-}$$\tilde{Y}^{+}$$\rho^{0*}$$\tilde{\nu}_{R\alpha}$$(d)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{0*}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(e)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{0*}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(f)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime
0}$$\tilde{\rho}^{0}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{R_{\alpha}}$$(i)$
$Y_{\mu(\tau)}$$\mu$$\mu^{c}$$\lambda_{B}$$\rho^{*0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(k)$$(\tau)$$(\tau^{c})$
Figure 7: Diagrams contributing to $\Delta^{1\rho}_{\mu}$ ( or
$\Delta^{\rho}_{\tau})$.
$Y_{\tau}$$\mu$$\mu^{c}$$\lambda_{B}$$\rho^{0*}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$
Figure 8: Diagram contributing to $\Delta^{2\rho}_{\mu}$.
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{2}$$\tilde{\rho}^{+}_{2}$$\rho^{\prime
0}$$\tilde{\nu}_{L_{\alpha}}$$\tilde{\nu}_{R_{\beta}}$$(a)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\tilde{\rho}^{\prime-}_{1}$$\tilde{\rho}^{+}_{1}$$\rho^{\prime
0}$$\tilde{\nu}_{R_{\alpha}}$$\tilde{\nu}_{L_{\beta}}$$(b)$
$\mu$$\mu^{c}$$(\tau)$$(\tau^{c})$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(c)$
Figure 9: Diagrams contributing to $\Delta^{1\rho^{\prime}}_{\mu}$
$[(a),(b)]$, $\Delta^{2\rho^{\prime}}_{\mu}$ $[(c)]$ ( or
$\Delta^{\rho^{\prime}}_{\tau}$ $[(a),(b),(c)]$).
$\mu$$\tau^{c}$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{l}_{L_{\alpha}}$$\tilde{l}_{R_{\beta}}$$(a)$
$\tau$$\mu^{c}$$\lambda_{B}$$\rho^{\prime
0}$$\tilde{e}_{L_{\alpha}}$$\tilde{e}_{R_{\beta}}$$(b)$
Figure 10: Diagrams contributing to $\Delta^{\rho^{\prime}}_{L}$ [(a)] and
$\Delta^{\rho^{\prime}}_{R}$ [(b)].
## Appendix B Lagrangian
We have denoted by $f_{L,R}(\bar{f}_{L,R}^{c})$ two component spinor of the
generic matter left-handed and right-handed fermion, respectively. The
$\tilde{f}_{L}(\tilde{f}_{L}^{c})$ are their superpartners which satisfy
$(\bar{f}_{L}^{c}=(f_{R})^{\dagger T},\tilde{f}_{L}^{c}=\tilde{f}^{*}_{R}$).
The four-component Dirac spinor can be represented through two-component
spinor such as: $\mu\equiv(\mu_{L},\;\mu_{R})=(\mu_{L},\;\bar{\mu}_{L}^{c})$
and
$\widetilde{{\mu}}\equiv(\tilde{\mu}_{L},\;\tilde{\mu}_{R})=(\widetilde{\mu}_{L},\;\widetilde{\mu}_{L}^{c*})$.
We also emphasize that three-left handed leptons contained in the triplet
$L_{aL}$ of $\mathrm{SU}(3)_{L}$ are $(f_{a1L},f_{a2L},f_{a3L})$:
$\displaystyle f_{a1L}$ $\displaystyle\in$ $\displaystyle\\{\nu_{eL},\nu_{\mu
L},\nu_{\tau L}\\}\equiv\\{\nu_{1L},\nu_{2L},\nu_{3L}\\},$ $\displaystyle
f_{a2L}$ $\displaystyle\in$ $\displaystyle\\{e_{L},\mu_{L},\tau_{L}\\},$
$\displaystyle f_{a3L}$ $\displaystyle\in$
$\displaystyle\\{\nu^{c}_{eL},\nu^{c}_{\mu L},\nu^{c}_{\tau
L}\\}\equiv\\{\nu^{c}_{1L},\nu^{c}_{2L},\nu^{c}_{3L}\\}\equiv\\{(\nu^{c}_{1})_{L},(\nu^{c}_{2})_{L},(\nu^{c}_{3})_{L}\\}$
(102)
while $f_{aL}^{c}$ is singlet under $\mathrm{SU}(3)_{L}$. Conventions for two
component spinors used in our paper are the same as those given in [11] except
the lower index $L$, which is used to distinguish between Dirac spinors $f$
and left-handed Weyl spinors $f_{L}$.
Next, let us find the interactional vertices relating with our calculation. We
start to collect related terms from the Lagrangian given in [23, 33] into the
following ones:
1. 1.
Gaugino and Higgsino mass terms:
$\displaystyle\mathcal{L}_{gh}$ $\displaystyle=$
$\displaystyle-\left[\frac{1}{2}m_{\lambda}\sum_{b=1}^{8}\left(\lambda^{b}_{A}\lambda^{b}_{A}\right)+\frac{1}{2}m^{\prime}\lambda_{B}\lambda_{B}+\mu_{\chi}\widetilde{\chi}\widetilde{\chi}^{\prime}+\mu_{\rho}\widetilde{\rho}\widetilde{\rho}^{\prime}\right]+\mathrm{H.c.}$
(103) $\displaystyle=$
$\displaystyle-\frac{1}{2}m^{\prime}\lambda_{B}\lambda_{B}-m_{\lambda}\tilde{W}^{+}\tilde{W}^{-}-\frac{1}{2}m_{\lambda}\lambda_{A}^{3}\lambda_{A}^{3}$
$\displaystyle-$ $\displaystyle
m_{\lambda}\tilde{Y}^{\prime+}\tilde{Y}^{\prime-}-m_{\lambda}\tilde{X^{0*}}\tilde{X^{0}}-\frac{1}{2}m_{\lambda}\lambda_{A}^{8}\lambda_{A}^{8}-\mu_{\chi}\left(\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{\prime
0}_{1}+\widetilde{\chi}^{-}\widetilde{\chi}^{\prime+}+\widetilde{\chi}^{0}_{2}\widetilde{\chi}^{\prime
0}_{2}\right)$ $\displaystyle-$
$\displaystyle\mu_{\rho}\left(\widetilde{\rho}^{+}_{1}\widetilde{\rho}^{\prime-}_{1}+\widetilde{\rho}^{0}\widetilde{\rho}^{\prime
0}+\widetilde{\rho}^{+}_{2}\widetilde{\rho}^{\prime-}_{2}\right)+\mathrm{H.c.}.$
where
$\displaystyle\tilde{W}^{\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(\lambda_{A}^{1}\mp i\lambda_{A}^{2}),\hskip
14.22636pt\tilde{Y}^{\pm}\equiv\frac{1}{\sqrt{2}}(\lambda_{A}^{6}\pm
i\lambda_{A}^{7}),$ $\displaystyle\tilde{X}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(\lambda_{A}^{4}+i\lambda_{A}^{5}),\hskip
14.22636pt\tilde{X}^{*}\equiv\frac{1}{\sqrt{2}}(\lambda_{A}^{4}-i\lambda_{A}^{5})$
(104)
where we have used the results of mass eigenstate states for Higgsinos and
gauginos given in [24] and [26].
2. 2.
Fermion-sfermion-gaugino interaction terms:
$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{V}}$ $\displaystyle=$
$\displaystyle-\frac{ig^{\prime}}{\sqrt{3}}\left[-\frac{1}{3}(\bar{L}\tilde{L}\bar{\lambda}_{B}-\tilde{L}^{\dagger}L\lambda_{B})+(\bar{l}^{c}\tilde{l}^{c}\bar{\lambda}_{B}-\tilde{l}^{c*}l^{c}\lambda_{B})\right]$
(105) $\displaystyle-$
$\displaystyle\frac{ig}{\sqrt{2}}(\bar{L}\lambda^{i}\tilde{L}\bar{\lambda}^{i}_{A}-\tilde{L}^{\dagger}\lambda^{i}L\lambda^{i}_{A}),$
where $i=1,2,...,8$ is a color index and
$L\equiv(L_{1L},\;L_{2L},\;L_{3L})^{T}$,
$\tilde{L}\equiv(\tilde{L}_{1L},\;\tilde{L}_{2L},\;\tilde{L}_{3L})^{T}$,
$\bar{l}^{c}=(l_{1R},\;l_{2R},\;l_{3R})^{T}\equiv(\bar{l}^{c}_{1L},\;\bar{l}^{c}_{2L},\;\bar{l}^{c}_{3L})^{T}\equiv(e_{R},\mu_{R},\tau_{R})^{T}\equiv(\bar{e}^{c}_{L},\;\bar{\mu}^{c}_{L},\;\bar{\tau}^{c}_{L})^{T}$,
and
$\tilde{l}^{c}=(\tilde{l}^{*}_{1R},\;\tilde{l}^{*}_{2R},\;\tilde{l}^{*}_{3R})^{T}\equiv(\tilde{e}^{*}_{R},\;\tilde{\mu}^{*}_{R},\;\tilde{\tau}^{*}_{R},)^{T}$.
In this paper we just focus on interactions relating with two fermions, namely
$l_{aL}=\\{\mu_{L},\tau_{L}\\}$. All interested terms are given as
$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{V}}$ $\displaystyle=$
$\displaystyle\left[\bar{\mu}_{L}\left(\frac{ig^{\prime}}{3\sqrt{3}}\bar{\lambda}_{B}+\frac{ig}{\sqrt{2}}(\bar{\lambda}^{3}_{A}-\frac{1}{\sqrt{3}}\bar{\lambda}^{8}_{A})\right)\tilde{\mu}_{L}\right.$
(106) $\displaystyle-$
$\displaystyle\left.\mu_{L}\left(\frac{ig^{\prime}}{3\sqrt{3}}\lambda_{B}+\frac{ig}{\sqrt{2}}(\lambda^{3}_{A}-\frac{1}{\sqrt{3}}\lambda^{8}_{A})\right)\tilde{\mu}^{*}_{L}\right.$
$\displaystyle-$
$\displaystyle\left.\frac{ig^{\prime}}{\sqrt{3}}(\bar{\mu}^{c}_{L}\tilde{\mu}^{c}_{L}\bar{\lambda}_{B}-\mu^{c}_{L}\tilde{\mu}^{c*}_{L}\lambda_{B})+\frac{ig}{\sqrt{2}}\left(\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{3}_{A}-\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{3}_{A}\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{ig}{\sqrt{2}}\left(\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{8}_{A}-\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{8}_{A}\right)\right.$
$\displaystyle-$
$\displaystyle\left.ig\left((\bar{\mu}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\mu
L}-\mu_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\mu
L})+(\bar{\mu}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\mu
L}-\mu_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\mu L})\right)\right]$
$\displaystyle+$ $\displaystyle[\mu\rightarrow\tau].$
From this, we list the related vertices in Table 2
Vertex | Factor | Vertex | Factor
---|---|---|---
$\bar{\mu}_{L}\bar{\lambda}_{B}\tilde{\mu}_{L}$ | $-\frac{g^{\prime}}{3\sqrt{3}}$ | $\mu_{L}\lambda_{B}\tilde{\mu}^{*}_{L}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\mu}^{c}_{L}\tilde{\mu}^{c}_{L}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{\sqrt{3}}$ | $\mu^{c}_{L}\tilde{\mu}^{c*}_{L}\lambda_{B}$ | $\frac{-g^{\prime}}{\sqrt{3}}$
$\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\mu}_{L}\tilde{\mu}_{L}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\mu}^{*}_{L}\mu_{L}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\mu}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\mu L}$ | $g$ | $\mu_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\mu L}$ | $-g$
$\bar{\mu}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\mu L}$ | $g$ | $\mu_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\mu L}$ | $-g$
$\bar{\tau}_{L}\bar{\lambda}_{B}\tilde{\tau}_{L}$ | $-\frac{g^{\prime}}{3\sqrt{3}}$ | $\tau_{L}\lambda_{B}\tilde{\tau}^{*}_{L}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tau}^{c}_{L}\tilde{\tau}^{c}_{L}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{\sqrt{3}}$ | $\tau^{c}_{L}\tilde{\tau}^{c*}_{L}\lambda_{B}$ | $\frac{-g^{\prime}}{\sqrt{3}}$
$\bar{\tau}_{L}\tilde{\tau}_{L}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\tau}^{*}_{L}\tau_{L}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tau}_{L}\tilde{\tau}_{L}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\tau}^{*}_{L}\tau_{L}\lambda^{8}_{A}$ | $\frac{g}{-\sqrt{6}}$
$\bar{\tau}_{L}\overline{\tilde{W}}^{+}\tilde{\nu}_{\tau L}$ | $g$ | $\tau_{L}\tilde{W}^{+}\tilde{\nu}^{*}_{\tau L}$ | $-g$
$\bar{\tau}_{L}\overline{\tilde{Y}}^{+}\tilde{\nu}^{c}_{\tau L}$ | $g$ | $\tau_{L}\tilde{Y}^{+}\tilde{\nu}^{c*}_{\tau L}$ | $-g$
Table 2: Vertices of lepton-slepton-gaugino interaction at tree level.
3. 3.
Higgs-Higgsino-gaugino:
$\displaystyle\mathcal{L}_{H\tilde{H}\tilde{V}}$ $\displaystyle=$
$\displaystyle-\frac{ig^{\prime}}{\sqrt{3}}\left[-\frac{1}{3}(\bar{\tilde{\chi}}\chi\bar{\lambda}_{B}-\chi^{\dagger}\tilde{\chi}\lambda_{B})+\frac{1}{3}(\bar{\tilde{\chi}}^{\prime}\chi^{\prime}\bar{\lambda}_{B}-\chi^{\prime\dagger}\tilde{\chi}^{\prime}\lambda_{B})\right.$
(107) $\displaystyle+$
$\displaystyle\left.\frac{2}{3}(\bar{\tilde{\rho}}\rho\bar{\lambda}_{B}-\rho^{\dagger}\tilde{\rho}\lambda_{B})-\frac{2}{3}(\bar{\tilde{\rho}}^{\prime}\rho^{\prime}\bar{\lambda}_{B}-\rho^{\prime\dagger}\tilde{\rho}^{\prime}\lambda_{B})\right]$
$\displaystyle-$
$\displaystyle\frac{ig}{\sqrt{2}}\left[\bar{\tilde{\rho}}\lambda^{a}\rho\bar{\lambda}^{a}_{A}-\rho^{\dagger}\lambda^{a}\tilde{\rho}\lambda^{a}_{A}+\bar{\tilde{\chi}}\lambda^{a}\chi\bar{\lambda}^{a}_{A}-\chi^{\dagger}\lambda^{a}\tilde{\chi}\lambda^{a}_{A}\right.$
$\displaystyle-$
$\displaystyle\left.\bar{\tilde{\rho}}^{\prime}\lambda^{*a}\rho^{\prime}\bar{\lambda}^{a}_{A}+\rho^{\prime\dagger}\lambda^{*a}\tilde{\rho}^{\prime}\lambda^{a}_{A}-\bar{\tilde{\chi}}^{\prime}\lambda^{*a}\chi^{\prime}\bar{\lambda}^{a}_{A}+\chi^{\prime\dagger}\lambda^{*a}\tilde{\chi}^{\prime}\lambda^{a}_{A}\right]$
Vertices of neutral Higgs-Higgsino-gaugino interactions are shown in table 3
Table 3: Vertices of the neutral Higgs-Higgsino-gaugino interactions. Vertex | Factor | Vertex | Factor
---|---|---|---
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{0}_{2}\chi^{0}_{2}\bar{\lambda}_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{0}_{2}\chi^{0\dagger}_{2}\lambda_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}^{3}_{A}$ | $\frac{g}{\sqrt{2}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$
$\bar{\tilde{\chi}}^{0}_{1}\chi^{0}_{1}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\chi}^{0}_{1}\chi^{0\dagger}_{1}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{0}_{2}\chi^{0}_{2}\bar{\lambda}^{8}_{A}$ | $\frac{-g\sqrt{2}}{\sqrt{3}}$ | $\tilde{\chi}^{0}_{2}\chi^{0\dagger}_{2}\lambda^{8}_{A}$ | $\frac{g\sqrt{2}}{\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\chi^{\prime 0}_{2}\bar{\lambda}_{B}$ | $\frac{g^{\prime}}{3\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{2}\chi^{\prime 0\dagger}_{2}\lambda_{B}$ | $\frac{-g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\chi^{\prime 0}_{1}\bar{\lambda}^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$ | $\tilde{\chi}^{\prime 0}_{1}\chi^{\prime 0\dagger}_{1}\lambda^{8}_{A}$ | $\frac{g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\chi^{\prime 0}_{2}\bar{\lambda}^{8}_{A}$ | $\frac{g\sqrt{2}}{\sqrt{3}}$ | $\tilde{\chi}^{\prime 0}_{2}\chi^{\prime 0\dagger}_{2}\lambda^{8}_{A}$ | $\frac{-g\sqrt{2}}{\sqrt{3}}$
$\bar{\tilde{\rho}}^{0}\rho^{0}\bar{\lambda}_{B}$ | $\frac{2g^{\prime}}{3\sqrt{3}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda_{B}$ | $\frac{-2g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\rho^{0}}}\rho^{0}\bar{\lambda}^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda^{3}_{A}$ | $\frac{g}{\sqrt{2}}$
$\bar{\tilde{\rho}}^{0}\rho^{0}\bar{\lambda}^{8}_{A}$ | $\frac{g}{\sqrt{6}}$ | $\tilde{\rho}^{0}\rho^{0\dagger}\lambda^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$
$\bar{\tilde{\rho}}^{\prime 0}\rho^{\prime 0}\bar{\lambda}_{B}$ | $\frac{-2g^{\prime}}{3\sqrt{3}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda_{B}$ | $\frac{2g^{\prime}}{3\sqrt{3}}$
$\bar{\tilde{\rho^{\prime 0}}}\rho^{\prime 0}\bar{\lambda}^{3}_{A}$ | $\frac{g}{\sqrt{2}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda^{3}_{A}$ | $\frac{-g}{\sqrt{2}}$
$\bar{\tilde{\rho}}^{\prime 0}\rho^{\prime 0}\bar{\lambda}^{8}_{A}$ | $\frac{-g}{\sqrt{6}}$ | $\tilde{\rho}^{\prime 0}\rho^{\prime 0\dagger}\lambda^{8}_{A}$ | $\frac{g}{\sqrt{6}}$
$\bar{\tilde{\chi}}^{-}\bar{\tilde{W}}^{+}\chi^{0}_{1}$ | $g$ | $\chi^{0*}\tilde{W}^{+}\tilde{\chi}^{-}$ | -$g$
$\bar{\tilde{\chi}}^{-}\bar{\tilde{Y}}^{+}\chi^{0}_{2}$ | $g$ | $\chi^{0*}_{2}\tilde{Y}^{+}\tilde{\chi}^{-}$ | $-g$
$\bar{\tilde{\chi}}^{0}_{2}\bar{\tilde{X}}\chi^{0}_{1}$ | $g$ | $\chi^{0*}_{1}\tilde{X}\tilde{\chi}^{0}_{2}$ | $-g$
$\bar{\tilde{\chi}}^{0}_{1}\bar{\tilde{X}}^{*}\chi^{0}_{2}$ | $g$ | $\chi^{0*}_{2}\tilde{X}^{*}\tilde{\chi}^{0}_{1}$ | $-g$
$\bar{\tilde{\rho}}^{+}_{1}\bar{\tilde{W}}^{-}\rho^{0}$ | $g$ | $\rho^{0*}\tilde{W}^{-}\tilde{\rho}^{+}_{1}$ | -$g$
$\bar{\tilde{\rho}}^{+}_{2}\bar{\tilde{Y}}^{-}\rho^{0}$ | $g$ | $\rho^{0*}\tilde{Y}^{-}\tilde{\rho}^{+}_{2}$ | -$g$
$\bar{\tilde{\chi}}^{\prime+}\bar{\tilde{W}}^{-}\chi^{\prime 0}_{1}$ | $-g$ | $\chi^{0*}_{1}\tilde{W}^{-}\tilde{\chi}^{\prime+}$ | $g$
$\bar{\tilde{\chi}}^{\prime+}\bar{\tilde{Y}}^{-}\chi^{\prime 0}_{2}$ | $-g$ | $\chi^{\prime 0*}_{2}\tilde{Y}^{-}\tilde{\chi}^{\prime+}$ | $g$
$\bar{\tilde{\chi}}^{\prime 0}_{2}\bar{\tilde{X}}^{*}\chi^{\prime 0}_{1}$ | $-g$ | $\chi^{\prime 0*}_{1}\tilde{X}\tilde{\chi}^{\prime 0}_{2}$ | $g$
$\bar{\tilde{\chi}}^{\prime 0}_{1}\bar{\tilde{X}}\chi^{\prime 0}_{2}$ | $-g$ | $\chi^{\prime 0*}_{2}\tilde{X}\tilde{\chi}^{\prime 0}_{1}$ | $g$
$\bar{\tilde{\rho}}^{\prime-}_{1}\bar{\tilde{W}}^{+}\rho^{\prime 0}$ | $-g$ | $\rho^{\prime 0*}\tilde{W}^{+}\tilde{\rho}^{\prime-}_{1}$ | $g$
$\bar{\tilde{\rho}}^{\prime-}_{2}\bar{\tilde{Y}}^{+}\rho^{\prime 0}$ | $-g$ | $\rho^{\prime 0*}\tilde{Y}^{+}\tilde{\rho}^{\prime-}_{2}$ | $g$
4. 4.
Yukawa interaction terms:
$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}\left(L_{aL}\tilde{\rho}^{\prime}\tilde{l}^{c}_{bL}+\tilde{L}_{aL}\tilde{\rho}^{\prime}l^{c}_{bL}\right)-\frac{\lambda_{3ab}}{3}\left(L_{aL}\tilde{\rho}\tilde{L}_{bL}+\tilde{L}_{aL}\tilde{\rho}L_{bL}\right),$
(108) $\displaystyle\mathcal{L}_{llH}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}L_{aL}l^{c}_{bL}\rho^{\prime}-\frac{\lambda_{3ab}}{3}\epsilon^{\alpha\beta\gamma}(L_{aL})_{\alpha}(L_{bL})_{\beta}(\rho)_{\gamma}+\mathrm{H.c.}.$
(109)
Our work needs only terms which include leptons $\mu$ or $\tau$, such as:
$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{1ab}}{3}\left[l_{aL}\tilde{\rho}^{\prime
0}\tilde{l}^{c}_{bL}+(\tilde{\nu}_{aL}\tilde{\rho}^{\prime-}_{1}+\tilde{l}_{aL}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{aL}\tilde{\rho}^{\prime-}_{2})l^{c}_{bL}\right]$ (110)
$\displaystyle-$
$\displaystyle\frac{\lambda_{3ab}}{3}\left[l_{aL}\tilde{\rho}^{+}_{2}\widetilde{\nu}_{bL}-l_{aL}\widetilde{\rho}^{+}_{1}\widetilde{\nu}^{c}_{bL}+\widetilde{\nu}^{c}_{aL}\widetilde{\rho}^{+}_{1}l_{bL}-\widetilde{\nu}_{aL}\widetilde{\rho}^{+}_{2}l_{bL}\right],$
From now we just note that because the conversation of lepton flavor in the
lepton sector at tree level then $\lambda_{1ab}=0$ with $a\neq b$ and
$\lambda_{3cd}=0$ with $c=d$. For simplicity, we use new notations:
$Y_{e}\equiv\lambda_{111}/3,Y_{\mu}\equiv\lambda_{122}/3,Y_{\tau}\equiv\lambda_{133}/3$
and
$Y_{\nu_{e\mu}}\equiv\lambda_{312}/3,Y_{\nu_{\mu\tau}}\equiv\lambda_{323}/3,Y_{\nu_{e\tau}}\equiv\lambda_{313}/3$.
Eq.(110) now can be written in the common form:
$\displaystyle\mathcal{L}_{l\tilde{l}\tilde{H}}$ $\displaystyle=$
$\displaystyle-Y_{\mu}\left[\mu_{L}\tilde{\mu}^{c}_{L}\tilde{\rho}^{\prime
0}+(\tilde{\nu}_{\mu
L}\tilde{\rho}^{\prime-}_{1}+\tilde{\mu}_{L}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{\mu L}\tilde{\rho}^{\prime-}_{2})\mu^{c}_{L}\right]$ (111)
$\displaystyle-$ $\displaystyle
Y_{\tau}\left[\tau_{L}\tilde{\tau}^{c}_{L}\tilde{\rho}^{\prime
0}+(\tilde{\nu}_{\tau
L}\tilde{\rho}^{\prime-}_{1}+\tilde{\tau}_{L}\tilde{\rho}^{\prime
0}+\tilde{\nu}^{c}_{\tau L}\tilde{\rho}^{\prime-}_{2})\tau^{c}_{L}\right]$
$\displaystyle-$ $\displaystyle
Y_{\nu_{ab}}[l_{aL}\tilde{\rho}^{+}_{2}\tilde{\nu}_{bL}-l_{aL}\tilde{\rho}^{+}_{1}\tilde{\nu}^{c}_{bL}+\tilde{\nu}^{c}_{aL}\tilde{\rho}^{+}_{1}l_{bL}-\tilde{\nu}_{aL}\tilde{\rho}^{+}_{2}l_{bL}]$
Corresponding vertices are shown in Fig. 4
Vertex | Factor | vertex | Factor
---|---|---|---
$\mu_{L}\tilde{\mu}^{c}_{L}\tilde{\rho}^{\prime 0}$ | $-iY_{\mu}$ | $\tilde{\nu}_{\mu L}\tilde{\rho}^{\prime-}_{1}\mu^{c}_{L}$ | $-iY_{\mu}$
$\tilde{\mu}_{L}\tilde{\rho}^{\prime 0}\mu^{c}_{L}$ | $-iY_{\mu}$ | $\tilde{\nu}^{c}_{\mu L}\tilde{\rho}^{\prime-}_{2}\mu^{c}_{L}$ | $-iY_{\mu}$
$\tau_{L}\tilde{\tau}^{c}_{L}\tilde{\rho}^{\prime 0}$ | $-iY_{\tau}$ | $\tilde{\nu}_{\tau L}\tilde{\rho}^{\prime-}_{1}\tau^{c}_{L}$ | $-iY_{\tau}$
$\tilde{\tau}_{L}\tilde{\rho}^{\prime 0}\tau^{c}_{L}$ | $-iY_{\tau}$ | $\tilde{\nu}^{c}_{\tau L}\tilde{\rho}^{\prime-}_{2}\tau^{c}_{L}$ | $-iY_{\tau}$
$\mu_{L}~{}\tilde{\rho}^{+}_{2}~{}\tilde{\nu}_{\tau_{L}}$ | $-2iY_{\nu_{\mu\tau}}$ | $\mu_{L}~{}\tilde{\rho}^{+}_{1}~{}\tilde{\nu}^{c}_{\tau}$ | $2iY_{\nu_{\mu\tau}}$
$\tau_{L}~{}\tilde{\rho}^{+}_{2}~{}\tilde{\nu}_{\mu_{L}}$ | $-2iY_{\nu_{\tau\mu}}$ | $\tau_{L}~{}\tilde{\rho}^{+}_{1}~{}\tilde{\nu}^{c}_{\mu_{L}}$ | $2iY_{\nu_{\tau\mu}}$
Table 4: Higgsino-lepton-slepton interacions
5. 5.
In the soft Lagrangian, the mass term of sleptons is given by
$\displaystyle\mathcal{L}_{\tilde{f}mass}$ $\displaystyle=$
$\displaystyle-\tilde{m}^{2}_{Lab}\tilde{L}_{aL}^{\dagger}\tilde{L}_{bL}-\tilde{m}^{2}_{Rab}\tilde{l}_{aL}^{c*}\widetilde{l}_{bL}^{c}-\left[h^{\prime}_{ab}\tilde{L}^{T}_{aL}\rho^{\prime}\tilde{l}^{c}_{bL}\right.$
(112) $\displaystyle+$
$\displaystyle\left.h_{ab}\varepsilon^{\alpha\beta\gamma}(\tilde{L}_{aL})_{\alpha}(\tilde{L}_{bL})_{\beta}(\rho)_{\gamma}+\frac{\lambda_{1ab}}{3}\mu_{\rho}\rho^{*}\tilde{L}_{aL}\tilde{l}^{c}_{bL}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\lambda_{3ab}}{3}\mu_{\rho}\varepsilon^{\alpha\beta\gamma}(\rho^{\prime*})_{\alpha}(\tilde{L}_{aL})_{\beta}(\tilde{L}_{bL})_{\gamma}+\mathrm{H.c.}\right],$
here $a,b$ are flavor indices $\\{a,b=1,2,3\\}$ or $a,b=\\{e,\mu,\tau\\}$ and
$\alpha,\beta,\gamma$ are component indices of $SU(3)_{L}$. The
$\varepsilon^{\alpha\beta\gamma}$ is the antisymmetric tensor. In this paper
we focus on the mixing of slepton $\widetilde{\mu}$ and $\widetilde{\tau}$.
This mixing makes mass-eigenstate basis of slepton is different from [24]. For
more detail, please see in Appendix C. The Lagrangian relating with Higgs-
lepton-slepton interactions has the form
$\displaystyle\mathcal{L}_{\tilde{\mu}\tilde{\tau}H^{0}}$ $\displaystyle=$
$\displaystyle-\left[(h^{\prime}_{\mu\tau}\tilde{\mu}_{L}\rho^{\prime
0}\tilde{\tau}^{c}_{L}+h^{\prime}_{\tau\mu}\tilde{\tau}_{L}\rho^{\prime
0}\tilde{\mu}^{c}_{L}+h^{\prime}_{\tau}\tilde{\tau}^{c}_{L}\rho^{\prime
0}\tilde{\tau}_{L}+h^{\prime}_{\mu}\tilde{\mu}^{c}_{L}\rho^{\prime
0}\tilde{\mu}_{L})\right.$ (113) $\displaystyle+$
$\displaystyle\left.\rho^{0}(h_{\mu\tau}-h_{\tau\mu})(\tilde{\nu}^{c}_{\mu
L}\tilde{\nu}_{\tau L}-\tilde{\nu}_{\mu L}\tilde{\nu}^{c}_{\tau
L})+\frac{1}{2}Y_{\tau}\mu_{\rho}\rho^{0*}\tilde{\tau}_{L}\tilde{\tau}^{c}_{L}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2}Y_{\mu}\mu_{\rho}\rho^{0*}\tilde{\mu}_{L}\tilde{\mu}^{c}_{L}+Y_{\nu_{\mu\tau}}\mu_{\rho}\rho^{\prime
0*}\left(\tilde{\nu}^{c}_{\tau L}\tilde{\nu}_{\mu L}-\tilde{\nu}^{c}_{\mu
L}\tilde{\nu}_{\tau L}\right)+\mathrm{H.c.}\right]$
Vertices of Higgs- slepton-slepton interactions are listed in Table 5.
Vertex | Factor | Vertex | Factor
---|---|---|---
$\tilde{\mu}^{c}_{L}\tilde{\mu}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\mu}$ | $\tilde{\tau}^{c}_{L}\tilde{\tau}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\tau}$
$\tilde{\mu}_{L}\tilde{\tau}^{c}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\mu\tau}$ | $\tilde{\tau}_{L}\tilde{\mu}^{c}_{L}\rho^{\prime 0}$ | $-ih^{\prime}_{\tau\mu}$
$\rho^{0}\tilde{\nu}^{c}_{\mu L}\tilde{\nu}_{\tau L}$ | $-i(h_{\mu\tau}-h_{\tau\mu})$ | $\rho^{0}\tilde{\nu}_{\mu L}\tilde{\nu}^{c}_{\tau L}$ | $i(h_{\mu\tau}-h_{\tau\mu})$
$\rho^{0}\tilde{\tau}^{*}_{L}\tilde{\tau}^{c*}_{L}$ | $-\frac{i}{2}Y_{\tau}\mu_{\rho}$ | $\rho^{0}\tilde{\mu}^{*}_{L}\tilde{\mu}^{c*}_{L}$ | $-\frac{i}{2}Y_{\mu}\mu_{\rho}$
$\rho^{\prime 0}\tilde{\nu}^{*}_{\tau_{L}}\tilde{\nu}^{c*}_{\mu_{L}}$ | $-iY_{\nu_{\mu\tau}}\mu_{\rho}$ | $\rho^{\prime 0}\tilde{\nu}^{*}_{\mu_{L}}\tilde{\nu}^{c*}_{\tau_{L}}$ | $iY_{\nu_{\mu\tau}}\mu_{\rho}$
Table 5: Slepton-slepton-Higgs vertices.
.
## Appendix C Mass eigenstates of particles in the SUSYE331 model
### C.1 Neutral Higgs
The physical states of Higgs (mass eigenstates) have been studied in [24]. For
convenience we review the main results in this appendix. First we expand the
neutral Higgs components around the VEVs by
$\displaystyle\chi^{T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{u+S_{1}+iA_{1}}{\sqrt{2}},&\chi^{-},&\frac{w+S_{2}+iA_{2}}{\sqrt{2}}\\\
\end{array}\right),\hskip
14.22636pt\rho^{T}=\left(\begin{array}[]{ccc}\rho^{+}_{1},&\frac{v+S_{5}+iA_{5}}{\sqrt{2}},&\rho^{+}_{2}\\\
\end{array}\right)$ (116) $\displaystyle\chi^{\prime T}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{u^{\prime}+S_{3}+iA_{3}}{\sqrt{2}},&\chi^{\prime+},&\frac{w+S_{4}+iA_{4}}{\sqrt{2}}\\\
\end{array}\right),\hskip 14.22636pt\rho^{\prime
T}=\left(\begin{array}[]{ccc}\rho^{\prime-}_{1},&\frac{v^{\prime}+S_{6}+iA_{6}}{\sqrt{2}},&\rho^{\prime-}_{2}\\\
\end{array}\right)$ (119)
where $\\{u,w,u^{\prime},w^{\prime},v,v^{\prime}\\}$,
$\\{S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}\\}$ and
$\\{A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}\\}$ are VEV, scalar, and pseudo scalar
parts of neutral Higgs, respectively. The Higgs mass spectrum and the Higgs
mass eigenstates given in [24] showed that:
Scalar Higgs: Mass eigenstates of six original scalar Higgs
$\\{S_{1},S_{2},S_{3},S_{4},S_{5},S_{6}\\}$ are defined as three massless
eigenstates $\\{S^{\prime}_{1a},S^{\prime}_{5},\varphi_{S_{24}}\\}$ and three
massive ones $\\{\phi_{S_{24}},\varphi_{S_{a36}},\phi_{S_{a36}}\\}$. The
relations between the original and the mass-eigenstate base are 111 There are
some different definitions for $\gamma$ in [23, 24, 25]. In this paper we use
notations identifying with those of [23].:
$\displaystyle\left(\begin{array}[]{c}S_{1}\\\ S_{2}\\\ S_{3}\\\ S_{4}\\\
S_{5}\\\ S_{6}\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}c_{\beta}s_{\theta}&-s_{\beta}c_{\theta}&-c_{\beta}c_{\theta}&-s_{\alpha}s_{\beta}s_{\theta}&-c_{\alpha}s_{\beta}s_{\theta}&0\\\
c_{\beta}c_{\theta}&s_{\beta}s_{\theta}&c_{\beta}s_{\theta}&-s_{\alpha}s_{\beta}c_{\theta}&-c_{\alpha}s_{\beta}c_{\theta}&0\\\
s_{\beta}s_{\theta}&-c_{\beta}c_{\theta}&s_{\beta}c_{\theta}&s_{\alpha}c_{\beta}s_{\theta}&c_{\alpha}c_{\beta}s_{\theta}&0\\\
s_{\beta}c_{\theta}&c_{\beta}s_{\theta}&-s_{\beta}s_{\theta}&s_{\alpha}c_{\beta}c_{\theta}&c_{\alpha}c_{\beta}c_{\theta}&0\\\
0&0&0&-c_{\alpha}s_{\gamma}&s_{\alpha}s_{\gamma}&c_{\gamma}\\\
0&0&0&c_{\alpha}c_{\gamma}&-s_{\alpha}c_{\gamma}&s_{\gamma}\\\
\end{array}\right)\left(\begin{array}[]{c}S^{\prime}_{1a}\\\
\varphi_{S_{24}}\\\ \phi_{S_{24}}\\\ \varphi_{S_{a36}}\\\ \phi_{S_{a36}}\\\
S^{\prime}_{5}\\\ \end{array}\right)$ (138)
where some new notations are defined as follows:
$t_{\theta}\equiv\tan\theta\equiv\frac{u}{w}=\frac{u^{\prime}}{w^{\prime}},\;c_{\theta}\equiv\cos\theta,\;s_{\theta}\equiv\sin\theta,$
$t_{\beta}\equiv\tan\beta\equiv\frac{w}{w^{\prime}},\;c_{\beta}\equiv\cos\beta,\;s_{\beta}\equiv\sin\beta,$
and
$t_{\gamma}\equiv\tan\gamma\equiv\frac{v}{v^{\prime}},\;s_{\gamma}\equiv\sin\gamma,\;c_{\gamma}\equiv\cos\gamma,$
and $\alpha$ is determined through relations:
$\tan{2\alpha}\equiv\frac{-2m^{2}_{36a}}{m^{2}_{66a}-m^{2}_{33a}},\;c_{\alpha}\equiv\cos\alpha,\;s_{\alpha}\equiv\sin\alpha$
$m^{2}_{33a}=\frac{18g^{2}+g^{\prime 2}}{54c^{2}_{\theta}}(w^{2}+w^{\prime
2})=\frac{(18g^{2}+g^{\prime 2})w^{2}}{54c^{2}_{\theta}s^{2}_{\beta}}$
$m^{2}_{66a}=\frac{9g^{2}+2g^{\prime 2}}{27}(v^{2}+v^{\prime
2})=\frac{(9g^{2}+2g^{\prime 2})v^{2}}{27s^{2}_{\gamma}}$
$m^{2}_{36a}=\frac{9g^{2}+2g^{\prime 2}}{54}\sqrt{\frac{(v^{2}+v^{\prime
2})(w^{2}+w^{\prime 2})}{c^{2}_{\theta}}}=\frac{9g^{2}+2g^{\prime
2}}{54}\frac{vw}{|c_{\theta}s_{\gamma}s_{\beta}|}$
The mass eigenvalues of three physical Higgses
$\phi_{S_{24}},\varphi_{S_{a36}}$ and $\phi_{S_{a36}}$ are:
$\displaystyle m^{2}_{\phi_{S_{24}}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}(1+t^{2}_{\theta})(w^{2}+w^{\prime
2})=\frac{g^{2}w^{2}}{c^{2}_{\theta}s^{2}_{\beta}}$ (139) $\displaystyle
m^{2}_{\varphi_{S_{a36}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{33a}+m^{2}_{66a}-\sqrt{(m^{2}_{33a}-m^{2}_{66a})^{2}+4m^{4}_{36a}}\right]$
(140) $\displaystyle m^{2}_{\phi_{S_{a36}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{33a}+m^{2}_{66a}+\sqrt{(m^{2}_{33a}-m^{2}_{66a})^{2}+4m^{4}_{36a}}\right]$
(141)
Pseudo-scalar Higgs: There are five Goldstone bosons
$\\{A_{5},A_{6},A^{\prime}_{1},A^{\prime}_{2},\varphi_{A}\\}$ and one massive
physical Higgs $\varphi_{A}$. Because the $\varphi_{A}$ does not receive any
contributions from the $A_{5},A_{6}$ and sleptons as well as their sneutrinos
only couple to $\rho,\rho^{\prime}$, hence there is no coupling of pseudo
scalar $\varphi_{A}$ to muon and tauon at the one loop approximation. This is
the difference between MSMS and our model.
### C.2 Mass eigenstates of sleptons
The masses of sleptons in SUSYE331 models were studied in details in [24]. In
the work, they assumed that lepton numbers are conserved even in the slepton
sector. This assumption leaded to the absence of mixing terms in slepton
sector. Our work is interested in studying the source of LFV caused by the
mixing between slepton $\tilde{\mu}$ and $\tilde{\tau}$ and ignore all other
sources of FLV . So with two assumptions of R-parity conversation and the
small left-right mixing in slepton sector, we can base on [24] to write the
mass terms of charged sleptons in the form:
$\displaystyle-\mathcal{L}_{\tilde{l}\tilde{l}^{*}}$ $\displaystyle=$
$\displaystyle\sum_{\tilde{l}_{L_{a}}}\tilde{m}^{2}_{\tilde{l}_{L_{a}}}\tilde{l}^{*}_{L_{a}}\tilde{l}_{L_{a}}+\left(\tilde{m}^{2}_{L_{\mu\tau}}\tilde{\mu}^{*}_{L}\tilde{\tau}_{L}+\mathrm{H.c.}\right)$
(142) $\displaystyle+$
$\displaystyle\sum_{\tilde{l}_{R_{a}}}\tilde{m}^{2}_{\tilde{l}_{R_{a}}}\tilde{l}^{*}_{R_{a}}\tilde{l}_{R_{a}}+\left(\tilde{m}^{2}_{R_{\mu\tau}}\tilde{\mu}^{*}_{R}\tilde{\tau}_{R}+\mathrm{H.c.}\right)$
where
$\tilde{l}_{L_{a}}=\\{\tilde{e}_{L},\tilde{\mu}_{L},\tilde{\tau}_{L}\\}$,
$\tilde{l}_{R_{a}}=\\{\tilde{e}_{R},\tilde{\mu}_{R},\tilde{\tau}_{R}\\}$ and
$\displaystyle\tilde{m}^{2}_{\tilde{l}_{L_{a}}}\equiv B_{aa}$ $\displaystyle=$
$\displaystyle M^{2}_{aa}+\frac{1}{4}\mu_{0a}^{2}+\frac{v^{\prime
2}}{18}\lambda^{2}_{1aa}+\frac{1}{18}\lambda^{2}_{2a}(u^{2}+w^{2})$
$\displaystyle-$
$\displaystyle\frac{g^{2}}{2}\left(H_{3}-\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right),$
$\displaystyle\tilde{m}^{2}_{\tilde{l}_{R_{a}}}\equiv C_{aa}$ $\displaystyle=$
$\displaystyle m^{2}_{aa}+\frac{v^{\prime
2}}{18}\lambda^{2}_{1aa}+g^{2}t^{2}H_{1},$
$\displaystyle\tilde{m}^{2}_{L_{\mu\tau}}\equiv B_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(u^{2}+w^{2}),$
$\displaystyle\tilde{m}^{2}_{R_{\mu\tau}}\equiv C_{23}$ $\displaystyle=$
$\displaystyle m^{2}_{23},$ $\displaystyle H_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{6}\left[(u^{2}+w^{2})\frac{\cos
2\beta}{s^{2}_{\beta}}-2v^{2}\frac{\cos 2\gamma}{s^{2}_{\gamma}}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left[(u^{2}+w^{2})\left(\cot^{2}\beta-1\right)-2v^{2}\left(\cot^{2}\gamma-1\right)\right]$
$\displaystyle H_{3}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4}\left[u^{2}\frac{\cos
2\beta}{s^{2}_{\beta}}-2v^{2}\frac{\cos 2\gamma}{s^{2}_{\gamma}}\right]$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\left[u^{2}\left(\cot^{2}\beta-1\right)-2v^{2}\left(\cot^{2}\gamma-1\right)\right]$
$\displaystyle H_{8}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4\sqrt{3}}\left[v^{2}\frac{\cos
2\gamma}{s^{2}_{\gamma}}+(u^{2}-2w^{2})\frac{\cos
2\beta}{s^{2}_{\beta}}\right],$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{3}}\left[v^{2}\left(\cot^{2}\gamma-1\right)+(u^{2}-2w^{2})\left(\cot^{2}\beta-1\right)\right],$
$\displaystyle t^{2}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{g^{\prime}}{g}\right)^{2}=\frac{6s^{2}_{W}}{3-4s^{2}_{W}}$
(143)
Note that here we use notations $M^{2}_{aa}$ and $m^{2}_{aa}$ in stead of
$m^{2}_{aL}$ and $m^{2}_{la}$ in the soft breaking term of [23]. We assume
that the mixing matrix of the $\tilde{\mu}_{L,R}$ and $\tilde{\tau}_{L,R}$
slepton masses is given by
$\displaystyle-\mathcal{L}_{\tilde{\mu}\tilde{\tau}}$ $\displaystyle=$
$\displaystyle\left(\tilde{\mu}_{L}^{*},\;\tilde{\tau}_{L}^{*}\right)\left(\begin{array}[]{cc}\tilde{m}^{2}_{\mu_{L}}&\tilde{m}^{2}_{L_{\mu\tau}}\\\
\tilde{m}^{*2}_{L_{\mu\tau}}&\tilde{m}^{2}_{\tau_{L}}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\mu_{L}}\\\
\tilde{\tau_{L}}\\\
\end{array}\right)+\left(\tilde{\mu}^{c*}_{L},\;\tilde{\tau}^{c*}_{L}\right)\left(\begin{array}[]{cc}\tilde{m}^{2}_{\mu_{R}}&\tilde{m}^{2}_{R_{\mu\tau}}\\\
\tilde{m}^{*2}_{R_{\mu\tau}}&\tilde{m}^{2}_{\tau_{R}}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\mu}^{c}_{L}\\\
\tilde{\tau}^{c}_{L}\\\ \end{array}\right)$ (152)
This form is the same as that given in [10, 11] ( for detail, see [10],
Appendix A.1). Hence, the mass eigenstates and eigenvalues of sleptons in our
model are similar to that in the MSSM [10, 11]. In particular, the mass mixing
matrix of the left handed and right handed of sleptons given in (113) produce
the mass eigenstates such as $\\{\tilde{l}_{L_{2}},\tilde{l}_{L_{3}}\\}$ and
$\\{\tilde{l}_{R_{2}},\tilde{l}_{R_{3}}\\}$. The corresponding mass
eigenvalues are $\\{\tilde{m}^{2}_{L_{2}},\tilde{m}^{2}_{L_{3}}\\}$ and
$\\{\tilde{m}^{2}_{R_{2}},\tilde{m}^{2}_{R_{3}}\\}$.
From now we adopt conventions the flavor states of sleptons are
$\tilde{\mu}_{L},\tilde{\tau}_{L}$ and
$\tilde{\mu}^{c}_{L},\tilde{\tau}^{c}_{L}$ while the mass eigenstates are
$\tilde{l}_{L_{2}},\tilde{l}_{L_{3}}$ and
$\widetilde{l}_{R_{2}},\widetilde{l}_{R_{3}}$, respectively. The relations
between these two kinds of basics are:
$\tilde{\mu}_{L}=c_{L}\tilde{l}_{L_{2}}-s_{L}\tilde{l}_{L_{3}}$,
$\tilde{\tau}_{L}=s_{L}\tilde{l}_{L_{2}}+c_{L}\tilde{l}_{L_{3}}$, with
$c_{L}=\cos\theta_{L}$, $s_{L}=\sin\theta_{L}$;
$\widetilde{\mu}^{c}_{L}=c_{R}\widetilde{l}_{R_{2}}-s_{R}\widetilde{l}_{R_{3}}$,
$\widetilde{\tau}^{c}_{L}=s_{R}\widetilde{l}_{R_{2}}+c_{R}\widetilde{l}_{R_{3}}$,
with $c_{R}=\cos\theta_{R}$, $s_{R}=\sin\theta_{R}$. The mixing parameters
satisfy the following relations:
$\displaystyle
s_{L}c_{L}=\frac{\tilde{m}^{2}_{L\mu\tau}}{\tilde{m}^{2}_{L_{3}}-\tilde{m}^{2}_{L_{2}}},\hskip
14.22636pts_{R}c_{R}=\frac{\tilde{m}^{2}_{R\mu\tau}}{\tilde{m}^{2}_{R_{3}}-\tilde{m}^{2}_{R_{2}}}.$
(154)
### C.3 Sneutrinos
The general Lagrangian which gains masses for sneutrinos is given in [24] as
follows
$\displaystyle\mathcal{L}_{\tilde{\nu}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\tilde{\nu}^{*}_{aL},&\tilde{\nu}^{*}_{aR}\\\
\end{array}\right)\left(\begin{array}[]{cc}A_{ab}&E_{ab}\\\ E_{ab}&G_{ab}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{\nu}_{bL}\\\
\tilde{\nu}_{bR}\\\ \end{array}\right),$ (160)
where
$\displaystyle\nu_{aL}\equiv(\nu_{1L},~{}\nu_{2L},~{}\nu_{3L})^{T},\hskip
14.22636pt\nu_{aR}\equiv\nu^{c*}_{aL}=(\nu^{c*}_{1L},\nu^{c*}_{2L},\nu^{c*}_{3L}),$
(161)
and
$\displaystyle A_{ab}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\delta_{ab}\left(H_{3}+\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right)+M^{2}_{ab}+\frac{1}{4}\mu_{0a}\mu_{0b},$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda_{2a}\lambda_{2b}(v^{2}+w^{2})+\frac{2}{9}\lambda_{3ca}\lambda_{3cb}v^{2},$
$\displaystyle G_{ab}$ $\displaystyle=$
$\displaystyle-g^{2}\delta_{ab}\left(\frac{1}{\sqrt{3}}H_{8}+\frac{t^{2}}{3}H_{1}\right)+M^{2}_{ab}+\frac{1}{4}\mu_{0a}\mu_{0b},$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda_{2a}\lambda_{2b}(v^{2}+u^{2})+\frac{2}{9}\lambda_{3ca}\lambda_{3cb}v^{2},$
$\displaystyle E_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\left[(\varepsilon_{ab}-\varepsilon_{ba})v+\frac{1}{6}\mu_{\rho}v^{\prime}(\lambda_{3ab}-\lambda_{3ba})\right].$
(162)
If the LFV happens only in the $\\{\tilde{\nu}_{\mu},\tilde{\nu}_{\tau}\\}$
sector, we can rewrite the non-vanishing terms given in (162) in more explicit
formulas:
$\displaystyle m^{2}_{\tilde{\nu}_{aL}}\equiv A_{aa}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}\left(H_{3}+\frac{1}{\sqrt{3}}H_{8}-\frac{2t^{2}}{3}H_{1}\right)+M^{2}_{aa}+\frac{1}{4}\mu^{2}_{0a}$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda^{2}_{2a}(v^{2}+w^{2})+\frac{2}{9}v^{2}\sum_{c}\lambda^{2}_{3ca},$
$\displaystyle m^{2}_{\tilde{\nu}_{aR}}\equiv G_{aa}$ $\displaystyle=$
$\displaystyle-g^{2}\left(\frac{1}{\sqrt{3}}H_{8}+\frac{t^{2}}{3}H_{1}\right)+M^{2}_{aa}+\frac{1}{4}\mu^{2}_{0a}$
$\displaystyle+$
$\displaystyle\frac{1}{18}\lambda^{2}_{2a}(v^{2}+u^{2})+\frac{2}{9}v^{2}\sum_{c}\lambda^{2}_{3ca},$
$\displaystyle m^{2}_{\tilde{\nu}_{L\mu\tau}}\equiv A_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(v^{2}+w^{2})+\frac{2}{9}v^{2}\lambda^{2}_{3c2}\lambda_{3c3},$
$\displaystyle m^{2}_{\tilde{\nu}_{R\mu\tau}}\equiv A_{23}$ $\displaystyle=$
$\displaystyle
M^{2}_{23}+\frac{1}{4}\mu_{02}\mu_{03}+\frac{1}{18}\lambda_{22}\lambda_{23}(v^{2}+u^{2})+\frac{2}{9}v^{2}\lambda_{3c2}\lambda_{3c3}.$
(163)
Similar to the charged sleptons sector, we denote by
$\\{\tilde{\nu}_{\mu_{L}},~{}\tilde{\nu}_{\tau_{L}},~{}\tilde{\nu}_{\mu_{R}},~{}\tilde{\nu}_{\tau_{R}}\\}$
the flavor eigenstates while by
$\\{\tilde{\nu}_{L2},~{}\tilde{\nu}_{L3},~{}\tilde{\nu}_{R2},~{}\tilde{\nu}_{R3}\\}$
the mass eigenstates. Also, notations
$\\{\tilde{m}^{2}_{\nu_{L2}},~{}\tilde{m}^{2}_{\nu_{L3}},$
$~{}\tilde{m}^{2}_{\nu_{R2}},\tilde{m}^{2}_{\nu_{R3}}\\}$ denote the mass
eigenstates of sneutrinos. Here the relations between two bases are:
$\displaystyle\widetilde{\nu}_{\mu_{L}}$ $\displaystyle=$ $\displaystyle
c_{\nu_{L}}\tilde{\nu}_{L2}-s_{\nu_{L}}\tilde{\nu}_{L3},\hskip
14.22636pt\widetilde{\nu}_{\tau_{L}}=s_{\nu_{L}}\tilde{\nu}_{L2}+c_{\nu_{L}}\tilde{\nu}_{L3},$
$\displaystyle\widetilde{\nu}_{\mu_{R}}$ $\displaystyle=$ $\displaystyle
c_{\nu_{R}}\tilde{\nu}_{R2}-s_{\nu_{R}}\tilde{\nu}_{R3},\hskip
14.22636pt\widetilde{\nu}_{\tau_{R}}=s_{\nu_{R}}\tilde{\nu}_{R2}+c_{\nu_{R}}\tilde{\nu}_{R3},$
$\displaystyle s_{\nu_{L}}c_{\nu_{L}}$ $\displaystyle=$
$\displaystyle\frac{\tilde{m}^{2}_{\nu_{L\mu\tau}}}{\tilde{m}^{2}_{\nu_{L3}}-\tilde{m}^{2}_{\nu_{L2}}},\hskip
14.22636pts_{\nu_{R}}c_{\nu_{R}}=\frac{\tilde{m}^{2}_{\nu_{R\mu\tau}}}{\tilde{m}^{2}_{\nu_{R3}}-\tilde{m}^{2}_{\nu_{R2}}}.$
(164)
## References
* [1] K. Nakamura et al. [Particle Data Group], J. Phys. G 37, 075021 (2010).
* [2] F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992); P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992); R. Foot, O. F. Hernandez, F. Pisano and V. Pleitez, Phys. Rev. D 47, 4158 (1993).
* [3] M. Singer, J. W. F. Valle and J. Schechter, Phys. Rev. D 22, 738 (1980); R. Foot, H. N. Long and Tuan A. Tran, Phys. Rev. D 50, 34(R) (1994) [arXiv: hep-ph/9402243] ; J. C. Montero, F. Pisano and V. Pleitez, Phys. Rev. D 47, 2918 (1993); H. N. Long, Phys. Rev. D 54, 4691 (1996); Phys. Rev. D 53, 437 (1996); H. N. Long, Mod. Phys. Lett. A13, (1998) 1865.
* [4] H. N. Long and V. T. Van, J. Phys. G 25, 2319 (1999).
* [5] F. Pisano, Mod. Phys. Lett A 11, 2639 (1996); A. Doff and F. Pisano, Mod. Phys. Lett. A 14, 1133 (1999); C. A. de S. Pires and O. P. Ravinez, Phys. Rev. D 58, 035008 (1998); C. A. de S. Pires, Phys. Rev. D 60, 075013 (1999); P. V. Dong and H. N. Long, Int. J. Mod. Phys. A 21, 6677 (2006).
* [6] P. V. Dong, H. N. Long, D. T. Nhung, D. V. Soa, Phys. Rev. D 73, 035004 (2006), P. V. Dong, H. N. Long, Adv. High Energy Phys., 2008, 739492 (2008); arXiv: 0804.3239; W. Ponce, Y. Giraldo, L. A. Sanchez, Phys. Rev. D 67, 075001 (2003).
* [7] ATLAS Collaboration (G. Aad et al.), January 2009\.
* [8] ILC Collaboration (Gerald Aarons (SLAC) et al.). September 2007.
* [9] E. Arganda, A. M. Curiel, M. J. Herrero, D. Temes, Phys. Rev. D 71 (2005) 035011, arxiv: hep-ph/0407302.
* [10] A. Brignoble, A. Rossi, Phys. Lett. B 66 (2003) 217, arXiv:hep-ph/0304081.
* [11] A. Brignole, A. Rossi, Nucl. Phys. B 701 (2004),3-53, arXiv:hep-ph/0404211.
* [12] J. L. Diaz-Cruz, D. K. Gosh and S. Morreti, Phys. Lett. B 679 (2009) 376, arXiv:0809.5158.
* [13] G. Blankenburg, J. Ellis and G. Isidori, Flavour-Changing Decays of a 125 GeV Higgs-like Particle, e-Print: arXiv:1202.5704 [hep-ph]
* [14] S. Kanemura, K. Matsuda, T. Ota, T. Shindou, E. Takasugi and K.Tsumura Phys. Lett. B599(2004)83, hep-ph/0406316; S. Kanemura, T. Ota, T. Shindou and K.Tsumura, Phys. Lett. D73 (2006) 016006, hep-ph/0505191.
* [15] K. S. Babu and C. Kolda, Phys. Lett B 451 (1999) 77, arxiv: 9811308.
* [16] K. S. Babu and C. Kolda, Phys. Rev. Lett. 89 (2002) 241802, arXiv: 0206310; K. S. Babu and C. F. Kolda, Phys. Rev. Lett. 84 (2000) 228, e-Print: hep-ph/9909476.
* [17] J. Guasch, W. Hollik and S. Penaranda, Phys. Lett. B 515 (2001) 367, arxiv: 0106027.
* [18] M. Cannoni and O. Panella, Phys. Rev. D 79 (2009) 056001, Arxiv: 0812.2875; Phys. Rev. D 81 (2010) 036009, arXiv:0910.3316; Nuovo Cim. C 33 (2010) 183, arXiv:1002.3697.
* [19] M. S. Carena, D. Garcia, U. Nierste and C. E. M. Wagner, Nucl. Phys. B 577 (2000) 88, arXiv: 9912516.
* [20] T. Goto, Y. Okada and Y. Yamamota, Phys. Rev. D 83 (2011) 053011, arXiv:1012.4385.
* [21] T. Blazek and S. Raby, Phys. Rev. D 59 (1999) 095002, Arxiv: 9712257.
* [22] G. Isidori and A. Retico, JHEP 0111 (2001) 001, Arxiv: 0110121.
* [23] P. V. Dong, D. T. Huong, M. C. Rodriguez, H. N. Long, Nucl. Phys. B 772 (2007) 150; e-Print: hep-ph/0701137.
* [24] P. V. Dong, Tr. T. Huong, N. T. Thuy and H. N. Long, JHEP 0711 (2007)073, arXiv:0708.3155 [hep-ph].
* [25] P. V. Dong, D. T. Huong, N. T. Thuy and H. N. Long, Nucl. Phys. B 795 (2008) 361; arXiv:0707.3712
* [26] D. T. Huong and H. N. Long, JHEP 0807 (2008) 049, arXiv:0804.3875.
* [27] D. Atwood, L. Reina and A. Soni, Phys. Rev D 55, 3156 (1997)
* [28] P. Paradisi, JHEP 0608 (2006) 047, e-Print: hep-ph/0601100.
* [29] J. L. Diaz-Cruz, JHEP 0305 (2003) 036, e-Print: hep-ph/0207030.
* [30] Y. Okada, K. Okumura, Y. Shimizu, Phys.Rev. D 61 (2000) 094001, e-Print: hep-ph/9906446.
* [31] Y. Miyazaki _et al_. [Belle Collaboration], Phys. Lett. B 660 (2008) 154 ,arXiv:0711.2189.
* [32] J.Hisano and D. Nomura, Phys. Rev. D 59, 116005 (1999); R. Bernstein, in: 4th International Workshop on Nuclear and Particle Physics at J-PARC (NP08),Mito, Ibaraki, Japan, March 2008,http://j-parc.jp/researcher/Hadron/NP08/index.html.
* [33] J. C. Montero, V. Pleitez, M. C. Rodriguez, Phys. Rev. D 70 (2004) 075004.
|
arxiv-papers
| 2012-04-13T07:27:01 |
2024-09-04T02:49:29.661936
|
{
"license": "Public Domain",
"authors": "P. T. Giang, L. T. Hue, D. T. Huong, H. N. Long",
"submitter": "Tho Hue Le",
"url": "https://arxiv.org/abs/1204.2902"
}
|
1204.2980
|
# Realizable Rate Distortion Function and Bayesian Filtering Theory
Photios A. Stavrou, Charalambos D. Charalambous and Christos K. Kourtellaris
ECE Department, University of Cyprus, Green Park, Aglantzias 91,
P.O. Box 20537, 1687, Nicosia, Cyprus
e-mail:{stavrou.fotios, chadcha, kourtellaris.christos}@ucy.ac.cy
###### Abstract
The relation between rate distortion function (RDF) and Bayesian filtering
theory is discussed. The relation is established by imposing a causal or
realizability constraint on the reconstruction conditional distribution of the
RDF, leading to the definition of a causal RDF. Existence of the optimal
reconstruction distribution of the causal RDF is shown using the topology of
weak convergence of probability measures. The optimal non-stationary causal
reproduction conditional distribution of the causal RDF is derived in closed
form; it is given by a set of recursive equations which are computed backward
in time. The realization of causal RDF is described via the source-channel
matching approach, while an example is briefly discussed to illustrate the
concepts.
## I INTRODUCTION
Shannon’s information theory for reliable communication evolved over the years
without much emphasis on real-time realizability or causality imposed on the
communication sub-systems. In particular, the classical rate distortion
function (RDF) for source data compression deals with the characterization of
the optimal reconstruction conditional distribution subject to a fidelity
criterion [1, 2], without regard for realizability. Hence, coding schemes
which achieve the RDF are not realizable.
On the other hand, filtering theory is developed by imposing real-time
realizability on estimators with respect to measurement data. Specifically,
least-squares filtering theory deals with the characterization of the
conditional distribution of the unobserved process given the measurement data,
via a stochastic differential equation which causally depends on the
observation data.
Although, both reliable communication and filtering (state estimation for
control) are concerned with the reconstruction of processes, the main
underlying assumptions characterizing them are different. There are, however,
examples in which the gap between the two disciplines in both the underlying
assumption and the form of reconstruction is bridged [1, 3, 4, 5, 6]. In
information theory, the real-time realizability or causality of a
communication system is addressed via joint source-channel coding [7] (for
memoryless channels and sources).
Historically, the work of R. Bucy [8] appears to be the first to consider the
direct relation between distortion rate function and filtering, by carrying
out the computation of a realizable distortion rate function with square
criteria for two samples of the Ornstein-Uhlenbeck process. The earlier work
of A. K. Gorbunov and M. S. Pinsker [9] on $\epsilon$-entropy defined via a
causal constraint on the reproduction distribution of the RDF, although not
directly related to the realizability question pursued by Bucy, computes the
causal RDF for stationary Gaussian processes via power spectral densities. The
realizability constraints imposed on the reproduction conditional distribution
in [8] and [9] are different, the actual computation of the distortion rate or
RDF in these works is based on the Gaussianity of the process, while no
general theory is developed to handle arbitrary processes.
The objective of this paper is to develop the general theory by further
investigating the connection between realizable rate distortion theory and
filtering theory for general distortion functions and random processes on
abstract Polish spaces. The connection is established via optimization on the
spaces of conditional distributions which satisfy a causality constraint and
an average distortion constraint.
The main results obtained are the following.
1. a)
Existence of optimal reconstruction distribution minimizing the causal RDF
using the topology of weak convergence of probability measures on Polish
spaces.
2. b)
Closed form expression of the optimal reconstruction conditional distribution
for non-stationary processes, via recursive equations computed backward in
time.
3. c)
Realization procedure of the filter based on the causal RDF.
4. d)
Example to demonstrate the realization of the filter.
Although, the operational meaning of the causal RDF in terms of causal and
sequential codes is not pursued, it is pointed out that by utilizing the
assumptions and coding theorem derived in [10], the causal RDF derived is the
optimal performance theoretically achievable (OPTA) for sequential codes,
while it is related to the OPTA for causal codes [11].
Next, we give a high level discussion on RDF and filtering theory, and discuss
their connection.
Consider a discrete-time process
$X^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{0},X_{1},\ldots,X_{n}\\}\in{\cal
X}_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{n}{\cal X}_{i}$,
and its reconstruction
$Y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{0},Y_{1},\ldots,Y_{n}\\}\in{\cal
Y}_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{n}{\cal Y}_{i}$
where ${\cal X}_{i}$ and ${\cal Y}_{i}$ are Polish spaces.
Bayesian Estimation Theory. In classical filtering, one is given a
mathematical model that generates the process $X^{n}$,
$\\{P_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1}):i=0,1,\ldots,n\\}$, often induced via
discrete-time recursive dynamics, a mathematical model that generates observed
data obtained from sensors, say, $Z^{n}$, $\\{P_{Z_{i}|Z^{i-1},X^{i}}$
$(dz_{i}|z^{i-1},x^{i}):i=0,1,\ldots,n\\}$, while $Y^{n}$ are the causal
estimates of some function of the process $X^{n}$ based on the observed data
$Z^{n}$. The classical Kalman Filter is a well-known example [12], where
$\widehat{X}_{i}=\mathbb{E}[X_{i}|Z^{i-1}],~{}i=0,1,\ldots,n$, is the
conditional mean which minimizes the average least-squares estimation error.
Thus, in classical filtering theory both models which generate the unobserved
and observed processes, $X^{n}$ and $Z^{n}$, respectively, are given á priori.
Fig. 1 is the block diagram of the filtering problem.
Figure 1: Block Diagram of Filtering Problem
Causal Rate Distortion Theory and Estimation. In causal rate distortion theory
one is given a distribution for the process $X^{n}$, which induces
$\\{P_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1}):~{}i=0,1,\ldots,n\\}$, and determines
the causal reconstruction conditional distribution
$\\{P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i}):~{}i=0,1,\ldots,n\\}$ which
minimizes the mutual information between $X^{n}$ and $Y^{n}$ subject to
distortion fidelity constraint, via a causal (realizability) constraint. The
filter $\\{Y_{i}:~{}i=0,1,\ldots,n\\}$ of $\\{X_{i}:~{}i=0,1,\ldots,n\\}$ is
found by realizing the reconstruction distribution
$\\{P_{Y_{i}|X^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i}):~{}i=0,1,\ldots,n\\}$ via a
cascade of sub-systems as shown in Fig. 2.
Figure 2: Block Diagram of Filtering via Causal Rate Distortion Function
The precise problem formulation necessitates the definitions of distortion
function or fidelity, and mutual information.
The distortion function or fidelity between $x^{n}$ and its reconstruction
$y^{n}$, is a measurable function defined by
$\displaystyle d_{0,n}:{\cal X}_{0,n}\times{\cal
Y}_{0,n}\rightarrow[0,\infty],\>\>d_{0,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\sum^{n}_{i=0}\rho_{0,i}(x^{i},y^{i})$
The mutual information between $X^{n}$ and $Y^{n}$, for a given distribution
${P}_{X^{n}}(dx^{n})$, and conditional distribution
$P_{Y^{n}|X^{n}}(dy^{n}|x^{n})$, is defined by [2]
$\displaystyle I(X^{n};Y^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\int\log\Big{(}\frac{P_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{{P}_{Y^{n}}(dy^{n})}\Big{)}$
$\displaystyle P_{Y^{n}|X^{n}}(dy^{n}|x^{n})\otimes{P}_{X^{n}}(dx^{n})$ (I.1)
The realizability constraint is introduced next. Define the causal
$(n+1)-$fold convolution measure
${\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes^{n}_{i=0}P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})-a.s.$
(I.2)
The realizability constraint for a causal filter is defined by
$\displaystyle{\overrightarrow{Q}}_{ad}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\Big{\\{}P_{Y^{n}|X^{n}}(dy^{n}|x^{n}):$ $\displaystyle
P_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})-a.s.\Big{\\}}$
(I.3)
The realizability condition (I.3) is necessary, otherwise the connection
between filtering and realizable rate distortion theory cannot be established.
This is due to the fact that
$P_{Y^{n}|X^{n}}(dy^{n}|x^{n})=\otimes_{i=0}^{n}{P}_{Y_{i}|Y^{i-1},X^{n}}(dy_{i}|y^{i-1},x^{n})-a.s.$,
and hence in general, for each $i=0,1,\ldots,n$, the conditional distribution
of $Y_{i}$ depends on future symbols $\\{X_{i+1},X_{i+2},\ldots,X_{n}\\}$ in
addition to the past and present symbols $\\{Y^{i-1},X^{i}\\}$.
Causal RDF. The causal RDF is defined by
${R}^{c}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\in\overrightarrow{Q}_{ad}:\mathbb{E}\big{\\{}d_{0,n}(X^{n},Y^{n})\leq{D}\big{\\}}}I(X^{n};Y^{n})$
(I.4)
Note that realizability condition (I.3) is different from the realizability
condition in [8], which is defined under the assumption that $Y_{i}$ is
independent of
$X_{j|i}^{*}\stackrel{{\scriptstyle\triangle}}{{=}}X_{j}-\mathbb{E}\Big{(}X_{j}|X^{i}\Big{)},j=i+1,i+2,\ldots,$.
The claim here is that realizability condition (I.3) is more natural and
applies to processes which are not necessarily Gaussian having square error
distortion function. Realizability condition (I.3) is weaker that the
causality condition found in [9] defined by $X_{n+1}^{\infty}\leftrightarrow
X^{n}\leftrightarrow Y^{n}$.
The point to be made regarding (I.4) is that the realizability constraint
${P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})-a.s.,$
is equivalent to the following (see also Lemma II.1):
$\displaystyle{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})-a.s.{\Longleftrightarrow}$
$\displaystyle
I(X^{n};Y^{n})=\int\log\Big{(}\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{{P}_{Y^{n}}(dy^{n})}\Big{)}$
$\displaystyle{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n}){P}_{X^{n}}(dx^{n})\equiv{\mathbb{I}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$
(I.5)
where ${\mathbb{I}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$ indicates
the functional dependence of $I(X^{n};{Y^{n}})$ on
$\\{P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}}\\}$.
Therefore, by finding the solution of (I.4), then one can realize it via a
channel from which one can construct an optimal filter causally as in Fig. 2.
This paper is organized as follows. Section II discusses the formulation on
abstract spaces. Section III establishes existence of optimal minimizing
distribution, and Section IV derives the non-stationary solution recursively.
Section V describes the realization of causal RDF, while Section VI provides
an example. Lengthy derivations are omitted due to space limitation.
## II CAUSAL RDF ON ABSTRACT SPACES
The source and reconstruction alphabets are sequences of Polish spaces [13] as
defined in the previous section. Probability distributions on any measurable
space $({\cal Z},{\cal B}({\cal Z}))$ are denoted by ${\cal M}_{1}({\cal Z})$.
It is assumed that the $\sigma$-algebras
$\sigma\\{X^{-1}\\}=\sigma\\{Y^{-1}\\}=\\{\emptyset,\Omega\\}$. For $({\cal
X},{\cal B}({\cal X})),({\cal Y},{\cal B}({\cal Y}))$ measurable spaces, the
set of conditional distributions $P_{Y|X}(\cdot|X=x)$ is denoted by ${\cal
Q}({\cal Y};{\cal X})$ and it is equivalent to stochastic kernels.
Mutual information is defined via the Kullback-Leibler distance:
$\displaystyle
I(X^{n};Y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{D}(P_{X^{n},Y^{n}}||P_{X^{n}}\times{P_{Y^{n}}})$
$\displaystyle=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{P_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{P_{Y^{n}}(dy^{n})}\Big{)}P_{Y^{n}|X^{n}}(dy^{n}|x^{n})$
$\displaystyle\otimes{P}_{X^{n}}(dx^{n})=\int_{{\cal
X}_{0,n}}\mathbb{D}(P_{Y^{n}|X^{n}}(\cdot|x^{n})||P_{Y^{n}}(\cdot))P_{X^{n}}(dx^{n})$
$\displaystyle\equiv\mathbb{I}(P_{X^{n}},P_{Y^{n}|X^{n}})$ (II.1)
Note that (II.1) states that mutual information is expressed as a functional
of $\\{P_{X^{n}},P_{Y^{n}|X^{n}}\\}$.
The next lemma (stated without prove) relates causal product conditional
distributions and conditional independence.
###### Lemma II.1
The following are equivalent.
1. 1.
$P_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})-a.s.$.
2. 2.
For each $i=0,1,\ldots,n-1$,
$Y_{i}\leftrightarrow(X^{i},Y^{i-1})\leftrightarrow(X_{i+1},X_{i+2},\ldots,X_{n})$
forms a Markov chain.
3. 3.
For each $i=0,1,\ldots,n-1$, $Y^{i}\leftrightarrow X^{i}\leftrightarrow
X_{i+1}$ forms a Markov chain.
According to Lemma II.1, mutual information subject to causality reduces to
$\displaystyle I(X^{n};Y^{n})=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{P_{Y^{n}}(dy^{n})}\Big{)}$
$\displaystyle{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|dx^{n})\otimes
P_{X^{n}}(dx^{n})\equiv{\mathbb{I}}(P_{X^{n}},\overrightarrow{P}_{Y^{n}|X^{n}})$
(II.2)
where
$P_{Y^{n}}(dy^{n})=\int{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|dx^{n})\otimes
P_{X^{n}}(dx^{n})$, and (II.2) states that $I(X^{n};Y^{n})$ is a functional of
$\\{P_{X^{n}},\overrightarrow{P}_{Y^{n}|X^{n}}\\}$. Hence, causal RDF is
defined by optimizing ${\mathbb{I}}(P_{X^{n}},{P}_{Y^{n}|X^{n}})$ over
${P}_{Y^{n}|X^{n}}$ subject to the realizability constraint
${P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})-a.s.,$
which satisfies a distortion constraint.
###### Definition II.2
$($Causal Rate Distortion Function$)$ Suppose
$d_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\sum^{n}_{i=0}\rho_{0,i}(x^{i},y^{i})$,
where $\rho_{0,i}:{\cal X}_{0,i}\times{\cal Y}_{0,i}\rightarrow[0,\infty)$, is
a sequence of ${\cal B}({\cal X}_{0,i})\times{\cal B}({\cal
Y}_{0,i})$-measurable distortion functions, and let
$\overrightarrow{Q}_{0,n}(D)$ (assuming is non-empty) denotes the average
distortion or fidelity constraint defined by
$\displaystyle{\overrightarrow{Q}_{{0,n}}(D)}\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}{P}_{Y^{n}|X^{n}}\in{\cal
Q}({\cal Y}_{0,n};{\cal X}_{0,n}):$
$\displaystyle\ell_{d_{0,n}}({P}_{Y^{n}|X^{n}})\stackrel{{\scriptstyle\triangle}}{{=}}\int_{{\cal
X}_{0,n}\times{\cal
Y}_{0,n}}d_{0,n}(x^{n},y^{n}){P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$
$\displaystyle\otimes{P}_{X^{n}}(dx^{n})\leq
D\Big{\\}}\bigcap{\overrightarrow{Q}}_{ad},~{}D\geq 0$ (II.3)
where ${\overrightarrow{Q}_{ad}}$ is the realizability constraint (I.3). The
causal RDF is defined by
$\displaystyle{R}^{c}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{{{P}_{Y^{n}|X^{n}}\in\overrightarrow{Q}_{0,n}(D)}}{\mathbb{I}}({P}_{X^{n}},{P}_{Y^{n}|X^{n}})$
(II.4)
Clearly, ${R}^{c}_{0,n}(D)$ is characterized by minimizing mutual information
or equivalently $\mathbb{I}({P}_{X^{n}},{P}_{Y^{n}|X^{n}})$ over
$\overrightarrow{Q}_{0,n}(D)$.
## III EXISTENCE OF OPTIMAL CAUSAL RECONSTRUCTION
In this section, the existence of the minimizing causal product kernel in
(II.4) is established by using the topology of weak convergence of probability
measures on Polish spaces. Let $BC({\cal Y}_{0,n})$ denotes the set of bounded
continuous real-valued functions on ${\cal Y}_{0,n}$. The assumptions required
are the following.
1. 1)
${\cal Y}_{0,n}$ is a compact Polish space, ${\cal X}_{0,n}$ is a Polish
space;
2. 2)
for all $h(\cdot){\in}BC({\cal Y}_{0,n})$, the function
$(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal Y}_{0,n-1}\mapsto\int_{{\cal
Y}_{n}}h(y)P_{Y|Y^{n-1},X^{n}}(dy|y^{n-1},x^{n})\in\mathbb{R}$ is continuous
jointly in the variables $(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}$;
3. 3)
$d_{0,n}(x^{n},\cdot)$ is continuous on ${\cal Y}_{0,n}$;
4. 4)
the distortion level $D$ is such that there exist sequence
$(x^{n},y^{n})\in{\cal X}_{0,n}\times{\cal Y}_{0,n}$ satisfying
$d_{0,n}(x^{n},y^{n})<D$.
Note that since it is assumed that ${\cal Y}_{0,n}$ is a compact Polish space,
then ${\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$ is weakly compact.
###### Lemma III.1
Assume that conditions 1), 2) hold.
Then
* 1)
The realizability constraint set ${\overrightarrow{Q}}_{ad}$ is a closed
subset of a weakly compact set ${\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$
(hence compact).
* 2)
Under the additional conditions 3), 4) the set ${\overrightarrow{Q}}_{0,n}(D)$
is a closed subset of ${\overrightarrow{Q}}_{ad}$ (hence compact).
The previous results follow from Prohorov’s theorem that relates tighness and
weak compactness.
The next theorem establishes existence of the minimizing reconstruction kernel
for (II.4); it follows from Lemma III.1 and the lower semicontinuity of
$\mathbb{I}(P_{X^{n}},\cdot)$ with respect to $P_{Y^{n}|X^{n}}$.
###### Theorem III.2
Suppose the conditions of Lemma III.1 hold. Then ${R}^{c}_{0,n}(D)$ has a
minimum.
## IV NON-STATIONARY OPTIMAL RECONSTRUCTION
In this section the form of the optimal causal product reconstruction kernels
is derived under non-stationarity assumption. The Gateaux differential of the
$(n+1)-$fold convolution product
${\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$ should be varied in each
direction of ${P}_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i}),i=0,1,\ldots,n$.
###### Theorem IV.1
Suppose
${\mathbb{I}}_{{P}_{X^{n}}}(P_{Y_{i}|Y^{i-1},X^{i}}:i=0,1,\ldots,n)\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{I}}({P}_{X^{n}},\overrightarrow{P}_{Y^{n}|X^{n}})$
is well defined for every $\overrightarrow{P}_{Y^{n}|X^{n}}\in{\cal Q}({\cal
Y}_{0,n};{\cal X}_{0,n})$ possibly taking values from the set $[0,\infty].$
Then
$\\{P_{Y_{i}|Y^{i-1},X^{i}}:i=0,1,\ldots,n\\}\rightarrow{\mathbb{I}}_{{P}_{X^{n}}}(P_{Y_{i}|Y^{i-1},X^{i}}:i=0,1,\ldots,n)$
is Gateaux differentiable at every point in ${\cal Q}({\cal Y}_{0,n};{\cal
X}_{0,n})$, and the Gateaux derivative at the points
${P}_{Y_{i}|Y^{i-1},X^{i}}^{0}$ in each direction
$\delta{P_{Y_{i}|Y^{i-1},X^{i}}}=P_{Y_{i}|Y^{i-1},X^{i}}-{P}_{Y_{i}|Y^{i-1},X^{i}}^{0}$,
$i=0,\ldots,n$, is
$\displaystyle\delta{\mathbb{I}}_{{P}_{X^{n}}}({P}_{Y_{i}|Y^{i-1},X^{i}}^{0},{P}_{Y_{i}|Y^{i-1},X^{i}}-{P}_{Y_{i}|Y^{i-1},X^{i}}^{0}:i=0,\ldots,n)$
$\displaystyle=\sum_{i=0}^{n}\int_{{\cal X}_{0,i}\times{\cal
Y}_{0,i}}\log\Bigg{(}\frac{{P}_{Y_{i}|Y^{i-1},X^{i}}^{0}}{{P}_{Y_{i}|Y^{i-1}}^{0}}\Bigg{)}\frac{d}{d\epsilon}\overrightarrow{P}_{Y^{i}|X^{i}}^{\epsilon}\Big{|}_{\epsilon=0}{P}_{X^{i}}(dx^{i})$
where
$\overrightarrow{P}_{Y^{i}|X^{i}}^{\epsilon}\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{j=0}^{i}P_{Y_{j}|Y^{j-1},X^{j}}^{\epsilon}$,
$P_{Y_{j}|Y^{j-1},X^{j}}^{\epsilon}=P_{Y_{j}|Y^{j-1},X^{j}}^{0}+\epsilon\Big{(}P_{Y_{j}|Y^{j-1},X^{j}}-P_{Y_{j}|Y^{j-1},X^{j}}^{0}\Big{)}$,
$j=0,1,\ldots,i,~{}~{}i=0,1,\ldots,n$,
$\displaystyle\frac{d}{d\epsilon}{P}_{Y_{0}|X^{0}}^{\epsilon}\Big{|}_{\epsilon=0}=\delta{P}_{{Y_{0}|X^{0}}}$
$\displaystyle\frac{d}{d\epsilon}\overrightarrow{P}_{Y^{1}|X^{1}}^{\epsilon}\Big{|}_{\epsilon=0}=\delta{P}_{{Y_{0}|X^{0}}}\otimes{P}^{0}_{Y_{1}|Y^{0},X^{1}}+{P}^{0}_{{Y_{0}|X^{0}}}\otimes\delta{P}_{{Y_{1}|Y^{0},X^{1}}}$
$\displaystyle\ldots$
$\displaystyle\frac{d}{d\epsilon}\overrightarrow{P}_{Y^{i}|X^{i}}^{\epsilon}\Big{|}_{\epsilon=0}=\delta{P}_{{Y_{0}|X^{0}}}\otimes_{j=1}^{i}{P}^{0}_{Y_{j}|Y^{j-1},X^{j}}+$
$\displaystyle{P}^{0}_{{Y_{0}|X^{0}}}\delta{P}_{{Y_{1}|Y^{0},X^{1}}}\otimes_{j=2}^{i}{P}^{0}_{{Y_{j}|Y^{j-1},X^{j}}}+\ldots+$
$\displaystyle\otimes_{j=0}^{i-1}{P}^{0}_{{Y_{j}|Y^{j-1},X^{j}}}\otimes\delta{P}_{{Y_{i}|Y^{i-1},X^{i}}},~{}i=0,1,\ldots,n.$
The constrained problem defined by (II.4) can be reformulated using Lagrange
multipliers as follows (equivalence of constrained and unconstrained problems
follows from [14]).
$\displaystyle{R}_{0,n}^{c}(D)$
$\displaystyle=\inf_{\overrightarrow{P}_{Y^{n}|X^{n}}={\otimes_{i=0}^{n}{P}_{Y_{i}|Y^{i-1},X^{i}}}}\Big{\\{}{{\mathbb{I}}}({P}_{X^{n}},\overrightarrow{P}_{Y^{n}|X^{n}})-$
$\displaystyle
s(\ell_{{d}_{0,n}}(\overrightarrow{P}_{Y^{n}|X^{n}})-D)\Big{\\}}$ (IV.1)
and $s\in(-\infty,0]$ is the Lagrange multiplier.
Note that ${P}_{Y_{i}|Y^{i-1},X^{i}}\in{\cal Q}({\cal Y}_{i};{\cal
Y}_{0,i-1}\times{\cal X}_{0,i})$, therefore, one should introduce another set
of Lagrange multipliers to obtain an optimization problem without constraints.
This process is involved, hence we state the main results.
General Recursions for Non-Stationary Optimal Reconstruction
For $k=0,\ldots,n$
$\displaystyle
g_{n,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}0,\>\>g_{n-k,n}(x^{n-k},y^{n-k})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}-\int_{{\cal
X}_{n-k+1}}P_{X_{n-k+1}|X^{n-k}}(dx_{n-k+1}|x^{n-k})$
$\displaystyle\log\int_{{\cal
Y}_{n-k+1}}e^{s\rho_{0,n-k+1}-g_{n-k+1,n}}P_{Y_{n-k+1}|Y^{n-k}}^{*}(dy_{n-k+1}|y^{n-k})$
the optimal reconstruction is given by
$\displaystyle P_{Y_{n-k}|Y^{n-k-1},X^{n-k}}^{*}(dy_{n-k}|y^{n-k-1},x^{n-k})=$
$\displaystyle\frac{e^{s\rho_{0,n-k}-g_{n-k,n}}P_{Y_{n-k+1}|Y^{n-k}}^{*}(dy_{n-k}|y^{n-k-1})}{\int_{{\cal
Y}_{n}}e^{s\rho_{0,n-k}-g_{n-k,n}}P_{Y_{n-k}|Y^{n-k-1}}^{*}(dy_{n-k}|y^{n-k-1})}$
The causal RDF is given by
$\displaystyle R_{0,n}^{c}(D)=sD+\sum_{i=0}^{n}\int_{{\cal
X}_{0,i-1}\times{\cal Y}_{0,i-1}}$
$\displaystyle\bigg{(}\otimes_{j=0}^{i-1}P_{X_{j}|X^{j-1}}(dx_{j}|x^{j-1})\otimes{P}_{Y_{j}|Y^{j-1},X^{j}}^{*}(dy_{j}|y^{j-1},x^{j})\bigg{)}$
$\displaystyle\int_{{\cal
X}_{i}}P_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1})\bigg{(}-\int_{{\cal
Y}_{i}}g_{i,n}P_{Y_{i}|Y^{i-1},X^{i}}^{*}(dy_{i}|y^{i-1},x^{i})$
$\displaystyle-\log\int_{{\cal
Y}_{i}}e^{s\rho_{0,i}-g_{i,n}}P_{Y_{i}|Y^{i-1}}^{*}(dy_{i}|y^{i-1})\bigg{)}$
The above recursions illustrate the causality, since
$g_{n-k,n}(x^{n-k},y^{n-k})$ appearing in the exponent of the reconstruction
distribution integrate out future reconstruction distributions. Note also that
for the stationary case all reconstruction conditional distributions are the
same and hence, $g_{n-k,n}(\cdot,\cdot)=0,k=0,1,\ldots,n$. The above
recursions are general, while depending on the application they can be
simplified considerably.
## V REALIZATION OF CAUSAL RDF
The realization of the causal RDF (optimal reconstruction kernel) is
equivalent to identifying a communication channel, an encoder and a decoder
such that the reconstruction from the sequence $X^{n}$ to the sequence $Y^{n}$
matches the causal rate distortion minimizing reconstruction kernel. Fig. 3
illustrates the cascade sub-systems that realize the causal RDF. This is
called source-channel matching in information theory [7]. It is also described
in [6] and [10]; this technique allows one to design encoding/decoding schemes
without encoding and decoding delays. The realization of the optimal
reconstruction kernel is given below.
###### Definition V.1
Given a source
$\\{P_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1}):i=0,\ldots,n\\}$, a
channel $\\{P_{B_{i}|B^{i-1},A^{i}}(db_{i}|b^{i-1},a^{i}):i=0,\ldots,n\\}$ is
a realization of the optimal reconstruction distribution if there exists a
pre-channel encoder
$\\{P_{A_{i}|A^{i-1},B^{i-1},X^{i}}(da_{i}|a^{i-1},b^{i-1},x^{i}):i=0,\ldots,n\\}$
and a post-channel decoder
$\\{P_{Y_{i}|Y^{i-1},B^{i}}(dy_{i}|y^{i-1},b^{i}):i=0,\ldots,n\\}$ such that
$\displaystyle{\overrightarrow{P}}_{Y^{n}|X^{n}}^{*}(dy^{n}|x^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{i=0}^{n}P^{*}_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})-a.s.$
where the joint distribution is
$\displaystyle P_{X^{n},A^{n},B^{n},Y^{n}}(dx^{n},da^{n},db^{n},dy^{n})$
$\displaystyle=\otimes_{i=0}^{n}P_{Y_{i}|Y^{i-1},B^{i}}(dy_{i}|y^{i-1},b^{i})\otimes
P_{B_{i}|B^{i-1},A^{i}}(db_{i}|b^{i-1},a^{i})$ $\displaystyle\otimes
P_{A_{i}|A^{i-1},B^{i-1},X^{i}}(da_{i}|a^{i-1},b^{i-1},x^{i})$
$\displaystyle\otimes P_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1})$
Figure 3: Block Diagram of Realizable Causal Rate Distortion Function
The filter is given by $\\{P_{X_{i}|B^{i-1}}(dx_{i}|b^{i-1}):i=0,\ldots,n\\}$.
Thus, if $\\{P_{B_{i}|B^{i-1},A^{i}}(db_{i}|b^{i-1},a^{i}):i=0,\ldots,n\\}$ is
a realization of the causal RDF minimizing distribution then the channel
connecting the source, encoder, channel, decoder achieves the causal RDF, and
the filter is obtained.
## VI EXAMPLE: BINARY MARKOV SOURCE
Consider a binary Markov source, while the objective is to detect consecutive
sequences of $\\{1\\}$’s subject to a specific, pre-defined distortion or
error criterion. The Markov source has the following transition probability
matrix.
$\displaystyle P(x_{i}=0|x_{i-1}=0)=1-p,\ P(x_{i}=1|x_{i-1}=0)=p$
$\displaystyle P(x_{i}=0;|x_{i-1}=1)=q,\ P(x_{i}=1|x_{i-1}=1)=1-q$
The steady state joint probabilities $P(x_{i},x_{i-1})$ are given by
$\displaystyle P(x_{i}=0,x_{i-1}=0)=\frac{(1-p)q}{p+q}$ $\displaystyle
P(x_{i}=0,x_{i-1}=1)=\frac{pq}{p+q}=P(x_{i}=1,x_{i-1}=0)$ $\displaystyle
P(x_{i}=1,x_{i-1}=1)=\frac{p(1-q)}{p+q}$
The distortion function is described in Table I.
$(x_{i},x_{i-1})$
| 00 | 01 | 10 | 11
---|---|---|---|---
$y_{i}=0$ | 0 | 0 | 0 | 1
$y_{i}=1$ | 1 | 1 | 1 | 0
TABLE I: Distortion: $d(x_{i},x_{i-1},y_{i})$
For the given distortion measure the optimal reconstruction kernel has the
following form
${P}^{*}(y_{i}|x_{i},x_{i-1})=\frac{e^{sd(x_{i},x_{i-1},y_{i})}P^{*}({y}_{i})}{\int_{{\cal
Y}_{i}}e^{sd(x_{i},x_{i-1},y_{i})}P^{*}({y}_{i})}$
in which ${P}^{*}(y_{i}|y_{i-1})=P^{*}({y}_{i}).$ The Lagrange parameter $s$
is the slope of the causal RDF. Then
$\displaystyle P^{*}(1|0,0)=P^{*}(1|0,1)=P^{*}(1|1,0)=1-\alpha$ $\displaystyle
P^{*}(0|0,0)=P^{*}(0|0,1)=P^{*}(0|1,0)=\alpha$ $\displaystyle
P^{*}(0|1,1)=1-P^{*}(1|1,1)=1-\beta$ $\displaystyle
P^{*}(y_{i}=0)=1-P^{*}(y_{i}=1)=\gamma$
where $\alpha=\frac{(1-D)(q-Dp-Dq+pq)}{q(1-2D)(1+p)}$,
$\beta=\frac{(1-D)(Dp-p+Dq+pq)}{p(1-2D)(1+q)}$, $\gamma=\frac{q-Dp-
Dq+pq}{(1-2D)(p+q)}$. The causal RDF is
$R^{c}(D)=\left\\{\begin{array}[]{ll}H\Big{(}\frac{q(1+p)}{p+q}\Big{)}-H(D)&\mbox{if
$D\leq D_{max}$}\\\ 0&\mbox{if $D>D_{max}$}\end{array}\right.$ $\displaystyle
D_{max}$
$\displaystyle=\min_{y_{i}}\sum_{x_{i},x_{i-1}}P(d{x}_{i},d{x}_{i-1}){d}(x_{i},x_{i-1},y_{i})$
$\displaystyle=\min\Big{(}\frac{q(1+p)}{p+q},\frac{p(1-q)}{p+q}\Big{)}$
Figure 4: $R^{c}$(D) for p=0.55 and q=0.45
The filter which realizes the optimal reproduction kernel
$P^{*}(\cdot|\cdot,\cdot)$ via the specification of an encoder, channel and
decoder which achieves the causal RDF, $R^{c}(D)$, is described in [7].
Special Case. Consider a special case when $\frac{q(1+p)}{p+q}=\frac{1}{2}$.
Then
$R^{c}(D)=\left\\{\begin{array}[]{ll}1-H(D)&\mbox{if $D\leq\frac{1}{2}$}\\\
0&\mbox{if $D>\frac{1}{2}$}\end{array}\right.$
Note that the capacity of a binary symmetric channel with error probability
$\epsilon=D<\frac{1}{2}$ is precisely $C(\epsilon)=1-H(D)$ [2]. Therefore, the
realization of the reproduction kernel is given by the cascade of encoder, the
binary symmetric channel, and decoder such that the directed information
including the encoder but not the decoder operates at the capacity
$C(\epsilon)=1-H(D)$, and it is equal to the directed information from the
source to the decoder output. Utilizing the capacity achieving encoder and
decoder for the binary symmetric channel found by Horstein in [15], the
realization is completed.
## References
* [1] T. Berger, _Rate Distortion Theory: A Mathematical Basis for Data Compression_. Englewood Cliffs, NJ: Prentice-Hall, 1971.
* [2] T. M. Cover and J. A. Thomas, _Elements of Information Theory_ , 2nd ed. John Wiley & Sons, Inc., Hoboken, New Jersey, 2006.
* [3] R. S. Liptser and A. N. Shiryaev, _Statistics of Random Processes: II. Applications_ , 2nd ed. Springer-Verlag, Berlin, Heidelberg, New York, 2001.
* [4] S. Ihara, _Information theory - for continuous systems_. World Scientific, 1993.
* [5] T. Cover and S. Pombra, “Gaussian feedback capacity,” _IEEE Transactions on Information Theory_ , vol. 35, no. 1, pp. 37–43, Jan. 1989.
* [6] C. D. Charalambous and A. Farhadi, “LQG optimality and separation principle for general discrete time partially observed stochastic systems over finite capacity communication channels,” _Automatica_ , vol. 44, no. 12, pp. 3181–3188, 2008.
* [7] M. Gastpar, B. Rimoldi, and M. Vetterli, “To code, or not to code: Lossy source-channel communication revisited,” _IEEE Transactions on Information Theory_ , vol. 49, no. 5, pp. 1147–1158, May 2003.
* [8] R. Bucy, “Distortion rate theory and filtering,” _IEEE Transactions on Information Theory_ , vol. 28, no. 2, pp. 336–340, Mar. 1982.
* [9] A. K. Gorbunov and M. S. Pinsker, “Asymptotic behavior of nonanticipative epsilon-entropy for Gaussian processes,” _Problems of Information Transmission_ , vol. 27, no. 4, pp. 361–365, 1991.
* [10] S. C. Tatikonda, “Control over communication constraints,” Ph.D. dissertation, Mass. Inst. of Tech. (M.I.T.), Cambridge, MA, 2000.
* [11] D. Neuhoff and R. Gilbert, “Causal source codes,” _IEEE Transactions on Information Theory_ , vol. 28, no. 5, pp. 701–713, Sep. 1982.
* [12] R. E. Kalman, “A new approach to linear filtering and prediction problems,” _Journal of Basic Engineering on Transactions of the ASME_ , vol. 82, no. Series D, pp. 35–45, March 1960.
* [13] P. Dupuis and R. S. Ellis, _A Weak Convergence Approach to the Theory of Large Deviations_. John Wiley & Sons, Inc., New York, 1997.
* [14] D. G. Luenberger, _Optimization by Vector Space Methods_. John Wiley & Sons, Inc., New York, 1969.
* [15] M. Horstein, “Sequential transmission using noiseless feedback,” _IEEE Transactions on Information Theory_ , vol. 9, no. 3, pp. 136–143, July 1963\.
|
arxiv-papers
| 2012-04-13T13:06:00 |
2024-09-04T02:49:29.671520
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Photios A. Stavrou, Charalambos D. Charalambous, Christos K.\n Kourtellaris",
"submitter": "Photios Stavrou",
"url": "https://arxiv.org/abs/1204.2980"
}
|
1204.3019
|
# Contractile stresses in cohesive cell layers on finite-thickness substrates
Shiladitya Banerjee Department of Physics, Syracuse University, Syracuse New
York, 13244-1130, USA M. Cristina Marchetti Department of Physics, Syracuse
University, Syracuse New York, 13244-1130, USA Syracuse Biomaterials
Institute, Syracuse University, Syracuse New York, 13244-1130, USA
###### Abstract
Using a minimal model of cells or cohesive cell layers as continuum active
elastic media, we examine the effect of substrate thickness and stiffness on
traction forces exerted by strongly adhering cells. We obtain a simple
expression for the length scale controlling the spatial variation of stresses
in terms of cell and substrate parameters that describes the crossover between
the thin and thick substrate limits. Our model is an important step towards a
unified theoretical description of the dependence of traction forces on cell
or colony size, acto-myosin contractility, substrate depth and stiffness, and
strength of focal adhesions, and makes experimentally testable predictions.
Many cell functions, such as spreading, growth, differentiation and migration,
are affected by the elastic and geometric properties of the extracellular
matrix Harris _et al._ (1980). Considerable effort has been devoted to the
study of cell adhesion to elastic substrates Discher _et al._ (2005). Cells
adhere to a substrate via focal adhesion complexes that link the substrate to
the actomyosin cytoskeleton, which in turn generates contractile forces that
deform soft substrates Balaban _et al._ (2001). The traction forces that the
cell exerts on the substrate are regulated by the cell itself in a complex
feedback loop controlled by cell activity and substrate elasticity.
Two powerful experimental techniques have been developed to measure forces by
cells on substrates: traction force microscopy, used to probe cell adhesion to
continuous substrates Dembo and Wang (1999); Butler _et al._ (2002), and the
imaging of cell-induced bending of microfabricated pillar arrays Tan _et al._
(2003). These two techniques have also been recently combined Polio _et al._
(2012). These experiments have yielded new insight on substrate rigidity
sensing and have opened up new questions on the physics of individual and
collective cell adhesion: What controls the length scale that governs the
penetration of traction forces? What is the relative role of active cellular
contractility and cell-cell-interaction in controlling the emergent response
of cell layers? In this Letter we describe minimal models of individual cells
and adhering cell colonies that reproduce qualitatively several experimental
findings. The traction stresses exerted by cells on substrates are extracted
directly from measurements of micropillar displacements or inferred from the
displacements of fiducial markers embedded in a continuum substrate. It is
found that traction stresses by isolated fibroblasts and epithelial cells on
pillar arrays are localized near the cell edge, while contractile stresses
(referred to below as cellular stresses) built-up inside the cellular material
is largest near the cell center Dembo and Wang (1999); Ghibaudo _et al._
(2008), as shown schematically in Fig. 1. This behavior, also observed in
adherent cell sheets and in migrating cell colonies Saez _et al._ (2010);
Mertz _et al._ (2012); Trepat _et al._ (2009), is predicted by our model.
Further, both substrate thickness and stiffness affect cellular and traction
stresses Lin _et al._ (2010). The magnitude of the traction stress increases
with substrate stiffness, saturating at large stiffness Ghibaudo _et al._
(2008), and it decreases sharply with substrate thickness, indicating that
cell colonies on thick substrates only probe a portion of substrate of
effective depth comparable to the lateral extent of the cell colony Sen _et
al._ (2009). Both trends are reproduced by our model (Fig. 3).
Figure 1: Schematic of a cell layer of lateral extent $L$ and thickness
$h_{c}<<L$ adhering to a substrate of thickness $h_{s}$. The build-up of
contractile stress $\sigma$ in the cell layer is indicated by the color map,
while the traction stresses in the substrate are shown as vectors (blue
online). The spatial variation of both traction and cellular stresses in the
lateral ($x$) direction are characterized by the length scale $\ell_{p}$,
referred to as the penetration length.
Our model builds on recent work Edwards and Schwarz (2011); Banerjee and
Marchetti (2011) describing the cell or cell layer as a contractile elastic
medium, with local elastic response of the substrate (as appropriate for
micropillar arrays or very thin substrates). In contrast, here we consider
substrates of finite thickness where the nonlocality of the elastic response
must be included. While previous studies have analyzed the deformations of
finite-thickness substrates due to point traction forces on their surface
Merkel _et al._ (2007); Maloney _et al._ (2008), our work considers the
inhomogeneous traction due to an extended contractile cell layer. A central
result for our work is the expression for the scaling parameter referred to as
the lateral penetration length $\ell_{p}$ (Fig. 1). This length scale
characterizes the in-plane spatial variations of both adhesion-induced
traction stresses on the substrate and cellular stresses within the cell layer
in terms of cell and substrate elastic and geometrical properties. Our model
also quantifies the experimentally-observed role of substrate thickness
$h_{s}$ in controlling the mechanical response of adhering cell layers Lin
_et al._ (2010). If $h_{s}$ is small compared to the lateral extent $L$ of the
cell sheet, the substrate elasticity plays a negligible role in determining
the mechanical response of the cell. This may explain why traction forces
exerted by cell colonies with $L\gg h_{s}$ appear insensitive to substrate
stiffness Trepat _et al._ (2009). If, in contrast, $L\ll h_{s}$, then
substrate nonlocality controls stress build-up in the cell sheet. This
crossover may be observable in large cell colonies on thick substrates.
Finally, the importance of long-range substrate elasticity has also been
emphasized in recent models of cells as active dipoles on a soft elastic
matrix, where it is crucial in controlling cell adhesion Bischofs _et al._
(2004); De _et al._ (2007). Long-range interfacial elastic stresses coupled
with gel thickness have also been shown to have a profound effect on focal
adhesion growth Nicolas and Safran (2006) and to enhance cell polarization
Bischofs and Schwarz (2005); Friedrich and Safran (2012). These important
effects are not discussed here.
## Contractile cell on a soft substrate.
To illustrate the importance of substrate nonlocality, we first analyze a
single cell, modeled as a contractile spring of stiffness $k_{c}$ and rest
length $\ell_{c0}$, adhering to a continuum substrate (described as an elastic
continuum of Young’s modulus $E_{s}$ and Poisson’s ration $\nu_{s}$) via two
focal adhesion bonds (linear springs of stiffness $k_{a}$) located at $x_{1}$
and $x_{2}$ (Fig. 2, top left) Schwarz _et al._ (2006). This is motivated by
the experimental observation that in adhering cells focal adhesions tend to be
localized near the cell periphery Wozniak _et al._ (2004).
Figure 2: Top : Schematic of a contractile cell adhering to a soft substrate
(left) and effective spring constant $k_{\text{eff}}$ versus cellular strain
$\Delta\ell$, showing strain stiffening (right). Bottom : Cell contraction
$\Delta\ell$ (solid blue line) and traction force $F_{T}$ (red dashed line) vs
substrate stiffness (left) for ${h}_{s}=10\,\mathrm{\mu m}$ and as a function
of substrate thickness (right) for ${E}_{s}=500\,\mathrm{Pa}$. Other
parameters : $F_{A}=10\,\mathrm{nN}$, $k_{c}=1\,\mathrm{nN/\mu m}$,
$k_{a}=2.5\,\mathrm{nN/\mu m}$, $E_{s}=1\,\mathrm{kPa}$,
$h_{s}=10\,\mathrm{\mu m}$, $\ell_{c0}=10\,\mathrm{\mu m}$, $\nu_{s}=0.4$.
For simplicity we consider a one dimensional model, where the cell lies on the
$x$ axis and the substrate lies in the $0\leq z\leq h_{s}$ region of the $xz$
plane. Contractile acto-myosin fibers connect the focal adhesions and exert
active forces of magnitude $F_{A}$. Once the cell has fully adhered, the cell-
substrate system is in mechanical equilibrium. Force balance at $x_{1}$ and
$x_{2}$ yields
$\displaystyle
k_{a}\left[u_{1}-u^{s}(x_{1})\right]=F_{A}-k_{c}(u_{1}-u_{2})\;,$ (1a)
$\displaystyle
k_{a}\left[u_{2}-u^{s}(x_{2})\right]=-F_{A}+k_{c}(u_{1}-u_{2})\;,$ (1b)
with $u_{i}$ the displacements of the contact points $x_{i}$ from their
unstretched positions $x_{2}^{0}-x_{1}^{0}=\ell_{c0}$, and $u^{s}(x_{i})$ the
displacement of the substrate’s surface at $x_{i}$. All displacements are
defined with respect to an initial state where the cell has length
$\ell_{c0}$. The net contraction is then
$\Delta\ell=l_{c0}-(x_{2}-x_{1})=u_{1}-u_{2}$. The traction force by the cell
on the substrate is localized at $x_{1}$ and $x_{2}$, yielding a traction
force density ${f}_{T}({x})=F_{T}\delta({x}-x_{1})-F_{T}\delta({x}-x_{2})$,
with $F_{T}=F_{A}-k_{c}\Delta\ell$. Assuming linear elasticity, the substrate
deformation is Landau _et al._ (1986),
$u^{s}({x})=\int_{-\infty}^{\infty}d{x^{\prime}}G({x}-{x}^{\prime})f_{T}({x}^{\prime})$,
where $G(x)$ is the elastic Green’s function at $z=h_{s}$. For a substrate of
thickness $h_{s}$ we use the approximate form 111To enable a direct comparison
between the penetration lengths obtained below and experimentally accessible
parameters, $E_{s}$ is the Young modulus of a three dimensional elastic
medium.
$G(x)=\frac{2}{\pi\ell_{c0}E_{s}}K_{0}\left[\frac{a+|x|}{h_{s}(1+\nu_{s})}\right]$
(2)
derived in the Supplemental Material SMp , with $a$ the size of adhesion
complexes, providing a short-distance cut-off, and $K_{0}$ denotes the
modified Bessel function of the second kind. We obtain
$F_{T}(\Delta\ell)=\frac{1}{2}k_{\text{eff}}(\Delta\ell)\Delta\ell$, with
$k_{\text{eff}}^{-1}=k_{a}^{-1}+[G(0)-G(l_{c0}-\Delta\ell)]$ the effective
stiffness of the cell-substrate adhesions. For $\Delta\ell\ll l_{c0}$,
$k_{\text{eff}}$ is independent of $\Delta\ell$ and $F_{T}$ scales linearly
with $\Delta\ell$. Stiffening sets in for
$\Delta\ell>\ell_{c0}|1-h_{s}(1+\nu_{s})/\ell_{c0}|$, as shown in Fig. 2 (top
right), with a crossover controlled by the thickness of the substrate $h_{s}$.
Using $F_{T}=F_{A}-k_{c}\Delta\ell$, we solve for both $\Delta\ell$ and
$F_{T}$, shown in Fig. 2 (bottom) as functions of the substrate thickness and
stiffness. For very thin ($h_{s}\rightarrow 0$) or infinitely rigid
substrates, where the substrate elasticity becomes local,
$\Delta\ell=F_{A}/(k_{c}+k_{a}/2)$, corresponding to a spring $k_{c}$ in
parallel with a series of two focal adhesions springs $k_{a}$. In this limit
the traction force saturates to $F_{T}=k_{a}F_{A}/(2k_{c}+k_{a})$. Conversely,
for a very soft substrate with $E_{s}\rightarrow 0$, the contraction is
maximal and given by $F_{A}/k_{c}$, and $F_{T}\rightarrow 0$. The substrate
thickness above which both cell contraction and traction force saturate is
controlled by the cell size and the substrate elasticity, in qualitative
agreement with experiments Lin _et al._ (2010).
## Contractile Cell Layer.
The continuum limit can be obtained by considering a multi-mer of
$N=\left[L/l_{c0}\right]$ contractile elemental “cells”, connected by springs
representing cell-cell interactions. The outcome is a set of coupled equations
for a contractile elastic medium. For a cell layer of thickness $h_{c}<<L$
(Fig. 1), the force balance equation, averaged over the cell thickness, is
$Y_{a}\left[u(x)-u^{s}(x)\right]=h_{c}\partial_{x}\sigma(x)\;,$ (3)
where $Y_{a}=k_{a}/(L\ell_{c0})$ describes the effective strength of the focal
adhesions, $u(x)$ is the displacement field of the cellular medium at
$z=h_{s}$, and $\sigma$ is the thickness-averaged cellular stress tensor,
$\sigma(x)={1/h_{c}}\int_{h_{s}}^{h_{s}+h_{c}}dz\ \sigma_{xx}(x,z)$, given by
$\sigma(x)=B_{c}\partial_{x}u+\sigma_{a}$, with $B_{c}$ the longitudinal
elastic modulus of the cell layer. The one dimensional model presented here
may be relevant to wound healing assays, where the cell layer is a strip with
$y$-translational invariance. Although we have neglected components of the
cellular displacements and spatial variations along $z$, the cell elastic
constants are those of a three-dimensional cellular medium. The active stress
$\sigma_{a}=F_{A}/(Lh_{c})$ arises from acto-myosin contractility Kruse _et
al._ (2005). The substrate deformation at the surface is
$u^{s}(x)=h_{c}\int
dx^{\prime}G(x-x^{\prime})\partial^{\prime}_{x}\sigma(x^{\prime})\;,$ (4)
with $G(x)$ the elastic Green’s function of a substrate of infinite extent in
$x$, occupying the region $0\leq z\leq h_{s}$, evaluated at $z=h_{s}$. Eqs.
(3)-(4) can be reduced to integro-differential equations for the cellular
stress, as
$\ell^{2}_{a}\partial_{x}^{2}\sigma+\sigma_{a}=\sigma-
B_{c}Lh_{c}\partial_{x}^{2}\int_{0}^{L}dx^{\prime}G(|x-x^{\prime}|)\sigma(x^{\prime})\;.$
(5)
The length scale $\ell_{a}=\sqrt{B_{c}h_{c}/Y_{a}}$ controls spatial
variations of cellular stresses induced by the stiffness of the focal
adhesions. It is the size of a region where the areal elastic energy density
$Y_{a}\ell_{a}^{2}$ associated with focal adhesions is of order of the areal
elastic energy density $B_{c}h_{c}$ of the cell layer. For a cell monolayer
with $B_{c}=1\,\mathrm{kPa}$, $h_{c}=0.1\,\mathrm{\mu m}$, $L=100\,\mathrm{\mu
m}$, $\ell_{c0}=10\,\mathrm{\mu m}$ and $k_{a}=2.5\,\mathrm{nN/\mu m}$ Balaban
_et al._ (2001), we get $\ell_{a}\simeq 6.3\,\mathrm{\mu m}$, comparable to
traction penetration length seen in experiments on stiff microposts Saez _et
al._ (2005, 2007). The second term on the right hand side of Eq. (5) describes
spatial variations in the cellular stress due to the (generally nonlocal)
coupling to the substrate. In the following we examine solutions to Eq. (5),
considering various limiting cases for the substrate thickness and analyze the
dependence of traction stresses on cell size, substrate stiffness and
substrate depth. The equation governing stress distribution in two dimensional
cell layers is derived in the Supplemental material SMp .
## Thin substrate.
If the substrate’s elastic response can be approximated as local, as it is the
case for $h_{s}<<L$ or for cells on micropillar arrays, the Green’s function
is given by $G(x)=\frac{2h_{s}(1+\nu_{s})}{LE_{s}}\delta(x)$. Eq. (5) can then
be written as $\ell^{2}_{p}\partial^{2}_{x}\sigma+\sigma_{a}=\sigma$, where,
$\ell_{p}=\sqrt{B_{c}h_{c}/Y_{\text{eff}}}$ and
$Y_{\text{eff}}^{-1}=Y_{a}^{-1}+2h_{s}(1+\nu_{s})/E_{s}$ describes the
combined action of the focal adhesions and the substrate, acting like two
linear elastic components in series. Assuming zero external stresses at the
boundary, i.e., $\sigma(0)=\sigma(L)=0$, the internal stress profile is
$\sigma(x)=\sigma_{a}\left(1-\cosh{\left[(L-2x)/2\ell_{p}\right]}/\cosh{[L/2\ell_{p}]}\right)$
Edwards and Schwarz (2011); Banerjee and Marchetti (2011); Mertz _et al._
(2012). The traction stress $T(x)=Y_{\text{eff}}u(x)$, is localized within a
length $\ell_{p}$ from the edge of the cell layer. The penetration length
$\ell_{p}$ can be written as $\ell_{p}=\sqrt{\ell_{a}^{2}+\ell_{s}^{2}}$, with
$\ell_{s}=\sqrt{\frac{2B_{c}h_{c}h_{s}}{E_{s}/(1+\nu_{s})}}$ the square root
of the ratio of the cell’s elastic energy to the elastic energy density of the
substrate. This form highlights the interplay of focal adhesion stiffness and
substrate stiffness in controlling spatial variation of stresses in the
lateral ($x$) direction. The two act like springs in series, where the weaker
spring controls the response. If $Y_{a}<<\frac{E_{s}}{2(1+\nu-s)h_{s}}$, then
$\ell_{p}\simeq\ell_{a}$ and the stiff substrate has no effect. Conversely, if
the focal adhesions are stiffer than the substrate, then
$\ell_{p}\simeq\ell_{s}$. For an elastic substrate with $h_{s}=10\,\mathrm{\mu
m}$, $\nu_{s}=0.4$ and $E_{s}$ in the range $0.01-100\,\mathrm{kPa}$,
$\ell_{s}$ lies in the range $0.2-17\,\mathrm{\mu m}$. This leads to typical
values of $\ell_{p}$ in the range $6.3-18\,\mathrm{\mu m}$ for a cell layer of
length $100\,\mathrm{\mu m}$, consistent with experimentally observed traction
penetration lengths on thin continuous substrates Mertz _et al._ (2012) and
on micropillar posts Saez _et al._ (2010).
## Infinitely thick substrate.
Figure 3: Internal stress $\sigma(x)/\sigma_{a}$ (top left), substrate
displacement $u^{s}(x)$ (top right) and traction stress $T(x)/\sigma_{a}$
(bottom) vs position $x$ along the cell layer, for $E_{s}=500\,\mathrm{Pa}$
(solid, blue), $50\,\mathrm{Pa}$ (dotted, green) and $10\,\mathrm{Pa}$
(dashed, red). The vertical dashed lines in the top right frame denote the
cell layer’s edges. Inset (bottom right): magnitude of contractile moment
$|{\cal P}|$ vs $E_{s}/B_{c}$. Inset (bottom left): $|{\cal P}|$ as a function
of substrate thickness for $E_{s}=10\,\mathrm{Pa}$. Other parameters:
$B_{c}=1\,\mathrm{kPa}$, $h_{c}=0.1\,\mathrm{\mu m}$,
$\ell_{a}=6.3\,\mathrm{\mu m}$, $L=100\,\mathrm{\mu m}$, $\nu_{s}=0.4$.
If $h_{s}>>L$, the substrate Green’s function can be approximated as that of
an elastic half plane, $G(x)=-\frac{2}{\pi
LE_{s}}\left[\gamma+\log{(|x|/L)}\right]$, with $\gamma$ the Euler constant
Barber (2010). The solution of Eq. (5) with boundary conditions
$\sigma(0)=\sigma(L)=0$ can be obtained by expanding $\sigma(x)$ in a Fourier
sine series as, $\sigma(x)=\sum_{n=1}^{\infty}\sigma_{n}\sin{(n\pi x/L)}$ and
solving the coupled algebraic equations for the Fourier amplitudes
$\sigma_{n}$ given in the Supplementary Material SMp . The effect of the
nonlocal elasticity of the substrate is controlled by yet another length scale
$\ell_{s\infty}=\sqrt{\frac{4B_{c}h_{c}L}{\pi E_{s}}}$ that can be obtained
from the length $\ell_{s}$ introduced in the case of thin substrate by the
replacement $h_{s}\rightarrow L$ and $(1+\nu_{s})\rightarrow 2/\pi$. This
highlights the known fact that cells or cell layers only “feel” the substrate
up to a thickness comparable to their lateral size $L$. For parameter values
quoted in the preceding paragraphs, $\ell_{s\infty}$ takes values between
$0.35-35\,\mathrm{\mu m}$ for $E_{s}$ in the range $0.01-100\,\mathrm{kPa}$,
indicating that the thin/thick substrate crossover, although not observable in
isolated cells, should be seen experimentally in cohesive cell layers where
the lateral extent can exceed $100\,\mathrm{\mu m}$. The cellular stress and
substrate displacement profiles obtained numerically by summing the Fourier
series are shown in Fig. 3 (top). The lateral variation of stresses is now
controlled by the length scale
$\ell_{p}=\sqrt{\ell_{a}^{2}+\ell_{s\infty}^{2}}$. One consequence of nonlocal
substrate elasticity is that the substrate deformation shown in the top right
frame of Fig. 3 extends outside the region occupied by the cell layer,
indicated by the two vertical dashed lines. The profile of the local traction
stress displayed in Fig. 3 (bottom frame) shows that the traction stress is
localized near the edge of the cell layer and its magnitude increases with
substrate stiffness. The inset to Fig. 3 (bottom right) shows the magnitude of
the net contractile moment defined as ${\cal
P}=\int_{-\infty}^{\infty}dxxT(x)$. This quantity is negative, as expected for
contractile systems. Its magnitude increases with $E_{s}$ at a rate consistent
with experiments, with a $25\%$ rise in $|{\cal P}|$ upon increasing the
substrate stiffness by $40\%$ Wang _et al._ (2002), and saturates for very
stiff substrates.
## Substrate of Finite Thickness.
Finally, we consider a substrate of finite thickness, $h_{s}$. The
calculations are carried out using the approximate Green’s function given in
Eq. (2), with the replacement $\ell_{c0}\rightarrow L$. The variation of the
net contractile moment with $h_{s}$ for $E_{s}=10\,\mathrm{Pa}$ is shown in
Fig. 3 (bottom left inset). As seen previously in experiments Lin _et al._
(2010), $|{\cal P}|$ drops sharply with increasing substrate thickness,
quickly reaching the asymptotic value corresponding to infinitely thick
substrates. Thinner substrates are effectively stiffer than thick ones,
inducing larger contractile moments. Our analysis suggests a general
expression for the penetration length $\ell_{p}$ that interpolates between the
thin and thick substrates limits,
$\ell_{p}=\sqrt{\frac{B_{c}h_{c}}{Y_{a}}+\frac{B_{c}h_{c}}{\pi E_{s}}h_{\rm
eff}}\;.$ (6)
Stress penetration is controlled by a substrate layer of effective thickness
$h_{\rm eff}^{-1}=\frac{1}{h_{s}2\pi(1+\nu_{s})}+\frac{1}{L}$ given by the
geometric mean of the actual substrate thickness $h_{s}$ and the lateral
dimension $L$ of the cell or cell layer. If $h_{s}<<L$, then $h_{\rm
eff}\approx 2\pi h_{s}(1+\nu_{s})$ and stress penetration is not affected by
cell layer size, as in the experiments of Mertz _et al._ (2012). On the other
hand, if $h_{s}>>L$, then cells only feel the effect of the substrate down to
an effective depth $L$.
## Discussion.
In summary, we have examined the dependence of traction stresses in adhering
cell layers on the mechanical and geometrical properties of the substrate.
Using a generic non-local model, we provide analytical results for the effect
of cell and substrate properties on the stress penetration length, that can be
tested in experiments. Although the analysis presented here is restricted to
one dimensional layers, isotropic planar cell layers with spherical symmetry
can also be considered analytically Banerjee and Marchetti (2012), with
similar predictions for the dependence of traction fields and their moments on
substrate mechanical and geometrical properties. The scaling of traction
moments on cell layer size is, however, different in two dimensions Mertz _et
al._ (2012). The model can be extended to incorporate the effects of cell
polarization, spatial variations in contractility, heterogeneities in the cell
layer or anisotropic elasticity of the substrate.
We thank Eric Dufresne and Aaron Mertz for many useful discussions and the
anonymous referees for valuable comments. This work was supported by the
National Science Foundation through awards DMR-0806511, DMR-1004789 and
DGE-1068780.
## References
* Harris _et al._ (1980) A. Harris, P. Wild, and D. Stopak, Science, 208, 177 (1980).
* Discher _et al._ (2005) D. Discher, P. Janmey, and Y. Wang, Science, 310, 1139 (2005).
* Balaban _et al._ (2001) N. Balaban, U. Schwarz, D. Riveline, P. Goichberg, G. Tzur, I. Sabanay, D. Mahalu, S. Safran, A. Bershadsky, L. Addadi, _et al._ , Nature cell biology, 3, 466 (2001).
* Dembo and Wang (1999) M. Dembo and Y. Wang, Biophysical journal, 76, 2307 (1999).
* Butler _et al._ (2002) J. Butler, I. Tolić-Nørrelykke, B. Fabry, and J. Fredberg, American Journal of Physiology-Cell Physiology, 282, C595 (2002).
* Tan _et al._ (2003) J. L. Tan, J. Tien, D. M. Pirone, D. S. Gray, K. Bhadriraju, and C. S. Chen, PMNAS, 100, 1484 (2003).
* Polio _et al._ (2012) S. R. Polio, K. E. Rothenberg, D. Stamenović, and M. L. Smith, Acta Biomaterialia, 8, 82 (2012).
* Ghibaudo _et al._ (2008) M. Ghibaudo, A. Saez, L. Trichet, A. Xayaphoummine, J. Browaeys, P. Silberzan, A. Buguin, and B. Ladoux, Soft Matter, 4, 1836 (2008).
* Saez _et al._ (2010) A. Saez, E. A. andM. Ghibaudo, O. du Roure, J.-M. D. Meglio, P. Hersen, P. Silberzan, A. Buguin, and B. Ladoux, J. Phys.: Condens. Matter, 22, 194119 (9pp) (2010).
* Mertz _et al._ (2012) A. F. Mertz, S. Banerjee, Y. Che, G. K. German, Y. Xu, C. Hyland, M. C. Marchetti, V. Horsley, and E. R. Dufresne, Phys. Rev. Lett., 108, 198101 (2012).
* Trepat _et al._ (2009) X. Trepat, M. Wasserman, T. Angelini, E. Millet, D. Weitz, J. Butler, and J. Fredberg, Nature physics, 5, 426 (2009).
* Lin _et al._ (2010) Y. Lin, D. Tambe, C. Park, M. Wasserman, X. Trepat, R. Krishnan, G. Lenormand, J. Fredberg, and J. Butler, Physical Review E, 82, 041918 (2010).
* Sen _et al._ (2009) S. Sen, A. Engler, and D. Discher, Cellular and molecular bioengineering, 2, 39 (2009).
* Edwards and Schwarz (2011) C. Edwards and U. Schwarz, Physical Review Letters, 107, 128101 (2011).
* Banerjee and Marchetti (2011) S. Banerjee and M. Marchetti, EPL (Europhysics Letters), 96, 28003 (2011).
* Merkel _et al._ (2007) R. Merkel, N. Kirchgeßner, C. Cesa, and B. Hoffmann, Biophysical journal, 93, 3314 (2007).
* Maloney _et al._ (2008) J. M. Maloney, E. B. Walton, C. M. Bruce, and K. J. van Vliet, Phys. Rev. E, 78, 041923 (2008).
* Bischofs _et al._ (2004) I. Bischofs, S. Safran, and U. Schwarz, Physical Review E, 69, 021911 (2004).
* De _et al._ (2007) R. De, A. Zemel, and S. Safran, Nature Physics, 3, 655 (2007).
* Nicolas and Safran (2006) A. Nicolas and S. Safran, Biophysical journal, 91, 61 (2006).
* Bischofs and Schwarz (2005) L. B. Bischofs and U. S. Schwarz, Phys. Rev. Lett., 95, 068102 (2005).
* Friedrich and Safran (2012) B. M. Friedrich and S. A. Safran, Soft Matter, (2012).
* Schwarz _et al._ (2006) U. Schwarz, T. Erdmann, and I. Bischofs, Biosystems, 83, 225 (2006).
* Wozniak _et al._ (2004) M. Wozniak, K. Modzelewska, L. Kwong, and P. Keely, Biochimica et Biophysica Acta (BBA)-Molecular Cell Research, 1692, 103 (2004).
* Landau _et al._ (1986) L. D. Landau, L. P. Pitaevskii, E. M. Lifshitz, and A. M. Kosevich, _Theory of Elasticity_ , 3rd ed. (Butterworth-Heinemann, 1986) ISBN 075062633X.
* Note (1) To enable a direct comparison between the penetration lengths obtained below and experimentally accessible parameters, $E_{s}$ is the Young modulus of a three dimensional elastic medium.
* (27) See Supplemental Material at [URL will be inserted by publisher] for derivation of substrate Green’s function, Integro-differential equation governing cellular stress distribution, and corresponding equations for two dimensional cell layers.
* Kruse _et al._ (2005) K. Kruse, J. Joanny, F. Jülicher, J. Prost, and K. Sekimoto, The European Physical Journal E: Soft Matter and Biological Physics, 16, 5 (2005).
* Saez _et al._ (2005) A. Saez, A. Buguin, P. Silberzan, and B. Ladoux, Biophysical journal, 89, L52 (2005).
* Saez _et al._ (2007) A. Saez, M. Ghibaudo, A. Buguin, P. Silberzan, and B. Ladoux, Proceedings of the National Academy of Sciences, 104, 8281 (2007).
* Barber (2010) J. Barber, _Elasticity_ , Vol. 172 (Springer Verlag, 2010).
* Wang _et al._ (2002) N. Wang, I. Tolić-Nørrelykke, J. Chen, S. Mijailovich, J. Butler, J. Fredberg, and D. Stamenović, American Journal of Physiology-Cell Physiology, 282, C606 (2002).
* Banerjee and Marchetti (2012) S. Banerjee and M. C. Marchetti, “On the role of substrate thickness on traction force distribution of adherent cell layers,” (2012), in preparation.
|
arxiv-papers
| 2012-04-13T14:50:38 |
2024-09-04T02:49:29.676531
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shiladitya Banerjee, M. Cristina Marchetti",
"submitter": "Shiladitya Banerjee",
"url": "https://arxiv.org/abs/1204.3019"
}
|
1204.3054
|
###### Abstract
The mechanisms of nuclear transfer reactions are described for the transfer of
two nucleons from one nucleus to another. Two-nucleon overlap functions are
defined in various coordinate systems, and their transformation coefficients
given between coordinate systems. Post and prior couplings are defined for
sequential transfer mechanisms, and it is demonstrated that the combination of
‘prior-post’ couplings avoids non-orthogonality terms, but does not avoid
couplings that do not have good zero-range approximations. The simultaneous
and sequential mechanisms are demonstrated for the 124Sn(p,t)122Sn reaction at
25 MeV using shell-model overlap functions. The interference between the
various simultaneous and sequential amplitudes is shown.
## Chapter 0 Reaction mechanisms of pair transfer
### 1 Introduction
Much of the evidence for nucleonic pairing in nuclei comes from energy
expectation values, but important further information comes from the transfer
of pairs of nucleons to or from another nucleus of known structure. In this
regard, a more fundamental understating of nuclear reactions has been, and
will continue to be (especially in the FRIB era), crucial to the nuclear
physics community. This chapter focuses on the theory, calculation and model
results for the reactions mechanisms of pair transfer.
Here we consider the reaction mechanisms for pair transfer between two nuclei,
namely reactions that we can describe as $A(B{+}2,B)A{+}2$. Here, the two
nucleons may be two neutrons, two protons, or a proton and a neutron, and are
transferred from core $B$ to core $A$. The nucleons may transfer either in one
simultaneous step, or one after the other sequentially. If a distinguishable
proton and a neutron are transferred, then both proton-then-neutron and
neutron-then-proton routes need to be considered. Furthermore, these
sequential and simultaneous routes contribute amplitudes that all add together
coherently. This feature enables us to probe the nature of coherent two-
nucleon superpositions in nuclei. Conversely, these superpositions, coupling
orders and phase conventions have all to be defined consistently in a good
calculation.
Subsequent sections will therefore consider the definition of two-nucleon
overlap functions, their coordinate transformations, the definition of
transfer matrix elements along with zero-range approximations and non-
orthogonality corrections. Finally, some results are shown to illustrate the
coherence effects in the reaction mechanisms of pair transfers.
In the last 50 years, a significant number of papers have been presented in
which absolute differential cross sections have been calculated, and compared
with experimental results [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19]. Traditionally, for example in [9], the theory predictions
have fallen well below the experimental data. This ratio has been called the
‘unhappiness factor’ [20, 21], and has sometimes been more than 100. Most
previous calculations modeled the transfer of a dineutron as a single cluster.
And only from Charlton [8] were sequential transfer contributions considered.
We find that modern calculations, such as as those of Potel et al[19], are in
considerably better agreement with experiment.
### 2 Bound states and vertex functions
The general theory of nucleon pair bound states defines the overlap function
$\phi^{J}_{I}({\bf
r},{{\mbox{\boldmath$\rho$}}})=\langle\Phi_{A}(I)|\Phi_{A+2}(J)\rangle$ in
terms of the Jacobi coordinates ${\bf r}$ between the two nucleons, and $\rho$
between their center of mass (cm) and the core $A$. The core spin is $I$ and
the spin of the $A{+}2$ composite state is $J$. When intrinsic spins
$s_{1},s_{2}$ are also included in a particular coupling order such as
$\left|\\{L,(\ell,(s_{1}s_{2})S)j\\}J_{12},I;~{}J\right\rangle$, we have the
partial wave expansion
$\displaystyle\phi^{J}_{I}({\bf
r},{\mbox{\boldmath$\rho$}})=\\!\\!\\!\sum_{L\ell
SjJ_{12}I}\\!\\!\\!\\!\\!\\!$
$\displaystyle\phi_{I\mu_{I}}(\xi_{c})\phi_{s_{1}}^{\sigma_{1}}\phi_{s_{2}}^{\sigma_{2}}~{}Y_{L}^{\Lambda}(\hat{\bf
r})Y_{\ell}^{\mu}(\hat{{\mbox{\boldmath$\rho$}}})~{}{1\over
r\rho}u_{12}(r,\rho)\langle J_{12}M_{12}I\mu_{I}|JM\rangle$ (1)
$\displaystyle\langle L\Lambda jm_{12}|J_{12}M_{12}\rangle\langle\ell\mu
S\Sigma|jm_{12}\rangle\langle s_{1}\sigma_{1}s_{2}\sigma_{2}|S\Sigma\rangle\
.$
The radial wave function $u_{12}(r,\rho)$ can be given either as a cluster
product of single-particle wave functions
$u_{12}(r,\rho)=\Phi_{L}(r)\phi_{\ell}(\rho),$ input directly as a two-
dimensional distribution e.g. from a Faddeev bound-state calculation, or
calculated from the correlated sum of products of single-particle states with
independent coordinates. These two-nucleon wave functions will in general be
the eigenstates of a three-body bound state Schrödinger equation
$\displaystyle[T_{\bf
r}+T_{\mbox{\boldmath$\rho$}}+V_{1A}+V_{2A}+V_{12}-\varepsilon]\phi^{J}_{I}({\bf
r},{\mbox{\boldmath$\rho$}})=0\ ,$ (2)
where the $V_{iA}$ are the potentials between nucleon $i$ and the core, and
$V_{12}$ is the pairing interaction between the two nucleons.
Such two-particle states that come from shell-model calculations [22] or from
Sturmian-basis calculations [23] are usually described by means of the $|{\bf
r}_{1},{\bf r}_{2}\rangle$ coordinates. This describes a pair state as
$\displaystyle\varphi_{12}({\bf r}_{1},{\bf r}_{2})$ $\displaystyle=$
$\displaystyle\sum_{i}c_{i}~{}\left|(\ell_{1}(i),s_{1})j_{1}(i),(\ell_{2}(i),s_{2})j_{2}(i);~{}J_{12}\right\rangle$
(3)
The coefficients $c_{i}$ for correlated basis states $i$ and the single-
particle wave functions $\varphi_{\ell sj}({\bf r})$ contain all the physics
information about the bound state needed to do a transfer calculation. Shell
model codes [24] can produce the coefficients $c_{i}$ needed here in terms of
previously calculated eigenstates of the $A$ and the $A+2$ systems. These
states are then transformed into the centre-of-mass coordinates $|{\bf
r},{{\mbox{\boldmath$\rho$}}}\rangle$ of Eq. (1) using ${\bf r}_{i}=x_{i}{\bf
r}+y_{i}{{\mbox{\boldmath$\rho$}}}$. For equal mass particles,
$x_{1}=x_{2}=1$, and $y_{1}=-y_{2}=\frac{1}{2}$.
The vertex functions of these bounds states are defined to be these bound
state wave functions $\phi^{J}_{I}({\bf r},{\mbox{\boldmath$\rho$}})$
multiplied by the potentials which have zero effects after the transfer step
is performed and all exit channel nuclei have completely separated. These
potentials are therefore the sum $V$ of the binding potentials $V^{\rm
sp}_{i}=V_{\ell sj}({\bf r}_{i})$, namely $V=V^{\rm sp}_{1A}+V^{\rm sp}_{2B}$.
(These are the individual potentials that should appear in the bound-state
equation $[T_{\bf r}+V_{\ell sj}({\bf r})-\varepsilon]\varphi_{\ell sj}({\bf
r})=0$.) The vertex function does not include the nucleon-nucleon pair
interaction $V_{12}({\bf r}_{1}{-}{\bf r}_{2})$, since this potential produces
binding effects in both the initial and final bound states. We denote by
$V\phi^{J}_{I}({\bf r},{\mbox{\boldmath$\rho$}})$ the vertex function after
transformation into Jacobi coordinates by the same method used to transform
the wave function itself.
### 3 Post and prior coupling forms of transfer matrix elements
We now consider the Hamiltonian $\cal H$ for the whole system of $A{+}B{+}2$
nucleons and described by system wave function $\Psi$ for the complete
transfer reaction $A(B{+}2,B)A{+}2$. Let the various $A{+}2$ and $B{+}2$ bound
states be denoted by $\Phi_{i}$ for indices $i$. Then we may expand $\Psi$ in
terms of the $\Phi_{i}$ with some coefficients $\psi_{i}({\bf R}_{i})$
depending on the two-body separation vectors ${\bf R}_{i}$. This gives the
channel expansion $\Psi=\sum_{i}\psi_{i}({\bf R}_{i})\Phi_{i}$.
The model Schrödinger’s equation $[{\cal H}-E]\Psi=0$ when projected
separately onto the different basis states $\Phi_{j},$ yields the set of
equations
$\displaystyle\left[E_{j}-H_{j}\right]\psi_{j}({\bf R}_{j})+\sum_{i\neq
j}\left\langle\Phi_{j}|{\cal H}-E|\Phi_{i}\right\rangle\psi_{i}({\bf
R}_{i})=0,$ (4)
which couple together the unknown wave functions $\psi_{i}({\bf R}_{i}).$ The
channel Hamiltonians are defined by the diagonal
$H_{j}-E_{j}=\langle\Phi_{j}|{\cal H}-E|\Phi_{j}\rangle$ such that the $E_{j}$
are the asymptotic kinetic energies in channel $j$.
The off-diagonal matrix element $\langle\Phi_{j}|{\cal H}-E|\Phi_{i}\rangle$
has two different forms, depending on whether we expand
$\displaystyle{\cal H}-E$ $\displaystyle=$ $\displaystyle
H_{j}-E_{j}+V_{j}\mbox{ (the `post' form)}$ $\displaystyle=$ $\displaystyle
H_{i}-E_{i}+V_{i}\mbox{ (the `prior' form)}.$
The name (post or prior) is determined by whether it is the initial or final
channel whose Hamiltonian is used. The above Eq. (4), as written, has $i$ as
the initial channel and $j$ as the final channel for the indicated coupling.
Thus
$\displaystyle\left\langle\Phi_{j}|{\cal H}-E|\Phi_{i}\right\rangle$
$\displaystyle=$ $\displaystyle V_{ji}^{\rm
post}+[H_{j}-E_{j}]K_{ji}{\rm~{}~{}~{}(post)}$ (5) $\displaystyle{\rm or}$
$\displaystyle=$ $\displaystyle V_{ji}^{\rm
prior}+K_{ji}[H_{i}-E_{i}]{\rm~{}~{}~{}(prior),}$
where
$\displaystyle V_{ji}^{\rm
post}\equiv\langle\Phi_{j}|V_{j}|\Phi_{i}\rangle,~{}~{}~{}V_{ji}^{\rm
prior}\equiv\langle\Phi_{j}|V_{i}|\Phi_{i}\rangle,~{}~{}~{}K_{ji}\equiv\langle\Phi_{j}|\Phi_{i}\rangle.$
(6)
The overlap function $K_{ji}=\langle\Phi_{j}|\Phi_{i}\rangle$ in Eqs. (5,6)
arises from the non-orthogonality between the basis states $\Phi_{i}$ and
$\Phi_{j}$ if these are in different mass partitions. The $K_{ji}$ are non-
local operators that go to zero asymptotically ($R_{i}$ or $R_{j}\to\infty$).
(Within the same partition, the $\Phi_{i}$ would be inelastic states, and
would form an orthogonal set.)
The first-order DWBA matrix element use entrance $\psi_{i}$ and exit
$\psi_{j}$ channel wave functions satisfying $[H_{i}-E_{i}]\psi_{i}=0$ and
$[H_{j}-E_{j}]\psi_{j}=0$ respectively. Its matrix element is
$\displaystyle T^{(1)}_{ji}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|{\cal
H}-E|\Phi_{i}\psi^{(+)}_{i}\rangle\ .$ (7)
The prior form of this is
$\displaystyle T^{(prior)}_{ji}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|H_{i}-E_{i}+V_{i}|\Phi_{i}\psi^{(+)}_{i}\rangle$
(8) $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|V_{i}|\Phi_{i}\psi^{(+)}_{i}\rangle+\langle\psi^{(-)}_{j}\Phi_{j}|\Phi_{i}[H_{i}-E_{i}]\psi^{(+)}_{i}\rangle$
$\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|V_{i}|\Phi_{i}\psi^{(+)}_{i}\rangle+0$
$\displaystyle=$ $\displaystyle\langle\psi^{(-)}_{j}|V_{ji}^{\rm
prior}|\psi^{(+)}_{i}\rangle\ .$
Similarly, the equivalent post form is
$\displaystyle T^{(post)}_{ji}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|H_{j}-E_{j}+V_{j}|\Phi_{i}\psi^{(+)}_{i}\rangle$
(9) $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|V_{i}|\Phi_{i}\psi^{(+)}_{i}\rangle+\langle\psi^{(-)}_{j}[H_{j}-E_{j}]\Phi_{j}|\Phi_{i}\psi^{(+)}_{i}\rangle$
$\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{j}\Phi_{j}|V_{j}|\Phi_{i}\psi^{(+)}_{i}\rangle+0$
$\displaystyle=$ $\displaystyle\langle\psi^{(-)}_{j}|V_{ji}^{\rm
post}|\psi^{(+)}_{i}\rangle\ .$
Thus the non-orthogonality term disappears in first-order DWBA. Post and prior
first-order DWBA matrix elements can be made to exactly agree numerically, if
sufficient care is taken to ensure convergence of the non-local form factors.
Let a second-order DWBA matrix element use entrance channel $i$, exit channel
$k$, and some intermediate channel $j$, as $i\to j\to k$. The propagation in
the intermediate channel may be described in terms of the Green’s function
$G_{j}$, or equivalently within an iterated coupled-channels set. The two-step
DWBA matrix element is
$\displaystyle T^{(2)}_{ki}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{k}|~{}\langle\Phi_{k}|{\cal
H}{-}E|\Phi_{j}\rangle G_{j}\langle\Phi_{j}|{\cal
H}{-}E|\Phi_{i}\rangle~{}|\psi^{(+)}_{i}\rangle\ .$ (10)
Now there are four matrix elements that may be calculated, according to the
first and the second Hamiltonian form chosen: post-post, post-prior, prior-
post, and prior-prior. The terms prior and post for each step are used to
refer to the initial or final channels of that step, not the overall incoming
or outgoing channels. In ‘prior-post’, the prior refers to the first step, and
the post refers to the second step.
The post-post form of this, for example, is
$\displaystyle
T^{(post,post)}_{ki}=\langle\psi^{(-)}_{k}|~{}\langle\Phi_{k}|H_{k}{-}E_{k}{+}V_{k}|\Phi_{j}\rangle
G_{j}\langle\Phi_{j}|H_{j}{-}E_{j}{+}V_{j}|\Phi_{i}\rangle~{}|\psi^{(+)}_{i}\rangle.~{}~{}~{}~{}$
(11)
Here the $[H_{k}-E_{k}]$ can operate on the final $\psi_{k}$ to give zero, but
little can simplify the $[H_{j}-E_{j}]$ since $[H_{j}-E_{j}]G_{j}\neq 0$
always. Thus
$\displaystyle T^{(post,post)}_{ki}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{k}|V_{kj}^{\rm post}G_{j}V_{ki}^{\rm
post}|\psi^{(+)}_{i}\rangle+$ (12)
$\displaystyle\langle\psi^{(-)}_{k}|V_{kj}^{\rm
post}G_{j}[H_{j}-E_{j}]K_{ji}|~{}\psi^{(+)}_{i}\rangle.$
This second term is called a ‘non-orthogonality term’ since it involves the
bound-state non-orthogonality overlaps
$K_{ji}=\langle\Phi_{j}|\Phi_{i}\rangle$, which is significant when $R_{i}$
and $R_{j}$ are both within the range of the bound states.
Similar analyses for post-prior and prior-prior two-step DWBA expression also
have non-orthogonality terms in the final form. The prior-post form, however,
is
$\displaystyle
T^{(prior,post)}_{ki}=\langle\psi^{(-)}_{k}|~{}\langle\Phi_{k}|H_{k}{-}E_{k}{+}V_{k}|\Phi_{j}\rangle
G_{j}\langle\Phi_{j}|H_{i}{-}E_{i}{+}V_{i}|\Phi_{i}\rangle~{}|\psi^{(+)}_{i}\rangle.~{}~{}~{}~{}$
(13)
Here the $[H_{i}-E_{i}]$ can also operate on the initial $\psi_{i}$ to give
zero, as well as $[H_{k}-E_{k}]$ on the $\psi_{k}$, so we have the simplest
form
$\displaystyle T^{(prior,post)}_{ki}$ $\displaystyle=$
$\displaystyle\langle\psi^{(-)}_{k}|V_{kj}^{\rm post}G_{j}V_{ji}^{\rm
prior}|\psi^{(+)}_{i}\rangle\ .$ (14)
The non-orthogonality terms can thus be made to disappear in second-order DWBA
if the first and second steps use the prior and post interactions
respectively. If the non-orthogonality terms are included as necessary, the
results should be the same whatever post or prior forms are used.
In third and higher-order transfer calculations, some non-orthogonality terms
will always be present, but most pair transfer mechanisms can be well modeled
as two-step processes.
### 4 Two-nucleon transfer interaction
We now consider the specific transfer matrix element $V_{ji}^{\rm
prior}=\langle\Phi_{j}|V_{i}|\Phi_{i}\rangle$. Given an expression for this
prior form, we may calculate the post interaction easily as $V_{ji}^{\rm
post}=(V_{ij}^{\rm prior})^{\dagger}$. Take $\Phi_{j}$ to refer to the bound
states of nucleus $A{+}2$ outside core $A$, and $\Phi_{i}$ analogously for
nucleus $B$.
The transfer interaction has therefore the non-local matrix element
$\displaystyle{\sf V}_{ji}({\bf R}_{j},{\bf
R}_{i})=\langle\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})|V^{\rm sp}_{1B}+V^{\rm
sp}_{2B}+U_{AB}-U_{i}|\phi^{J_{B}}_{I_{B}}({\bf
r},{\mbox{\boldmath$\rho$}}_{B})\rangle.$ (15)
As is usual in transfer operators, there are three kinds of potentials
appearing here. First there are the binding potentials $V^{\rm sp}_{1B}({\bf
r}_{1B})+V^{\rm sp}_{2B}({\bf r}_{2B})$. Since these binding potentials always
appear while multiplied by their bound state wave functions, we need only
store and use the vertex functions defined in section 2. Secondly, there is
the ‘core-core’ potential $U_{AB}({\bf R}_{AB})$ between the core nuclei $A$
and $B$. Finally is subtracted an optical potential. In this prior form we
subtract the optical potential in the initial channel, $U_{i}({\bf R}_{i})$.
The difference $U_{AB}-U_{i}$ of the two optical potentials is called the
remnant term, and is sometimes taken to be small.
The integrals in Eq. (15) include integrating over the two-nucleon separation
${\bf r}$ as well as over their cm distance ${\mbox{\boldmath$\rho$}}_{A}$
from the core $A$. The ${\bf r}$ coordinate appears in both the initial and
final states, and so is not labeled by $A$ or $B$. This has the important
consequence that neither the distance nor the angle of the ${\bf r}$
coordinate is changed in the transfer. Neither, therefore, is their relative
angular momentum $\ell$, and, for similar reasons, nor their spin couplings
$S$ and total angular momentum $j$. The two neutron transfer can hence be
viewed as the transfer of a ‘structured particle’
$\\{r,(\ell,(s_{1}s_{2})S)j\\}$, and then becomes similar to the more familiar
single-particle transfers. This means that when we also afterwards integrate
over the coordinate ${\mbox{\boldmath$\rho$}}_{A}$, we can use the standard
procedures already developed for one-particle transfer interactions.
### 5 Coordinate transformations
The transfer mechanism requires the pair wave function to be expressed in the
form of Eq. (1), so independent-particle forms of Eq. (3) have to be
transformed in their coordinates as
$\displaystyle\varphi_{12}({\bf r}_{1},{\bf r}_{2})=\sum_{u}{c_{i}}\sum_{L\ell
Sj}\left|L,(\ell,(s_{1}s_{2})S)j;J_{12}T\right\rangle\phi^{J_{12}T,i}_{L(\ell
S)j}(r,\rho)\ .$ (16)
A particular basis state $i$ in the $(r,\rho)$ coordinates is
$\displaystyle\phi^{J_{12},i}_{L(\ell S)j}(r,\rho)$ $\displaystyle=$
$\displaystyle\left\langle
L,(\ell,(s_{1}s_{2})S)j;J_{12}\right|\left(\ell_{1}(i),s_{1})j_{1}(i),(\ell_{2}(i),s_{2})j_{2}(i);~{}J_{12}\right\rangle$
(17) $\displaystyle\times\langle\left[Y_{L}(\hat{\bf
r})Y_{\ell}(\hat{\rho})\right]_{\lambda}|\left[\varphi_{\ell_{1}s_{1}j_{1}}({\bf
r}_{1})\varphi_{\ell_{2}s_{2}j_{2}}({\bf r}_{2})\right]_{J_{12}T}\rangle$
where (suppressing the $i$ indices for clarity), and including an isospin $T$
to define the antisymmetrization,
$\displaystyle\\!\\!\\!\langle
L,(\ell,(s_{1}s_{2})S)j;J_{12}T|(\ell_{1},s_{1})j_{1},(\ell_{2},s_{2})j_{2};~{}J_{12}T\rangle=\sum_{\lambda}\hat{\lambda}\hat{S}\hat{j_{1}}\hat{j_{2}}~{}~{}~{}~{}$
$\displaystyle\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&\lambda\\\
s_{1}&s_{2}&S\\\
j_{1}&j_{2}&J_{12}\end{array}\right){1+(-1)^{\ell+S+T}\over\sqrt{2(1+\delta_{\ell_{1},\ell_{2}}\delta_{j_{1},j_{2}})}}~{}\hat{j}\hat{\lambda}W(L\ell
J_{12}S;\lambda j)(-1)^{\ell+L-\lambda}.~{}~{}~{}$ (21)
The radial overlap integral can be derived by means of harmonic-oscillator
expansions [25], with the Bayman-Kallio expansion [26] or using the Moshinsky
solid-harmonic expansion[27]. This last method gives
$\displaystyle K^{\lambda}_{\ell
L:\ell_{1}\ell_{2}}(r,\rho)=\langle\left[Y_{L}(\hat{\bf
r})Y_{\ell}(\hat{{\mbox{\boldmath$\rho$}}})\right]_{\lambda}|\left[\varphi_{\ell_{1}}({\bf
r}_{1})\varphi_{\ell_{2}}({\bf r}_{2})\right]^{\lambda}\rangle$ (22)
$\displaystyle=$
$\displaystyle\sum_{n_{1}n_{2}}~{}\left(\begin{array}[]{c}2\ell_{1}{+}1\\\
2n_{1}\end{array}\right)^{\frac{1}{2}}\left(\begin{array}[]{c}2\ell_{2}{+}1\\\
2n_{2}\end{array}\right)^{\frac{1}{2}}(x_{1}r)^{\ell_{1}-n_{1}}(y_{1}\rho)^{n_{1}}(x_{2}r)^{n_{2}}(y_{2}\rho)^{\ell_{2}-n_{2}}$
(40) $\displaystyle\times\sum_{Q}{\bf
q}_{\ell_{1}\ell_{2}}^{Q}(r,\rho)~{}(2Q{+}1)~{}\hat{\ell_{1}}\hat{\ell_{2}}\widehat{\ell_{1}{-}n_{1}}\widehat{\ell_{2}{-}n_{2}}~{}\hat{L}\hat{\ell}$
$\displaystyle\times\sum_{\Lambda_{1}\Lambda_{2}}\left(\begin{array}[]{ccc}\ell_{1}{-}n_{1}&n_{2}&\Lambda_{1}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}\ell_{w}{-}n_{2}&n_{1}&\Lambda_{2}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}\Lambda_{1}&L&Q\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}\Lambda_{2}&\ell&Q\\\
0&0&0\end{array}\right)$
$\displaystyle\times(-1)^{\ell_{1}+\ell_{2}+L+\Lambda_{2}}(2\Lambda_{1}+1)(2\Lambda_{2}+1)W(\Lambda_{1}L\Lambda_{2}\ell;Q\lambda)$
$\displaystyle\times\left(\begin{array}[]{ccc}\ell_{1}{-}n_{1}&n_{2}&\Lambda_{1}\\\
n_{1}&\ell_{2}{-}n_{2}&\Lambda_{2}\\\
\ell_{1}&\ell_{2}&\lambda\end{array}\right),$
where $\left(\begin{array}[]{c}a\\\ b\end{array}\right)$ is a binomial
coefficient. The kernel function ${\bf q}_{\ell_{1}\ell_{2}}^{Q}(r,\rho)$ is
the Legendre expansion of the product of the two radial wave functions in
terms of $u$, the cosine of the angle between ${\bf r}$ and $\rho$:
$\displaystyle{\bf
q}^{Q}_{\ell_{1},\ell_{2}}(r,\rho)=\frac{1}{2}\int_{-1}^{+1}{\varphi_{\ell_{1}s_{1}j_{1}}(r_{1})\over
r_{1}}^{\ell_{1}+1}{\varphi_{\ell_{2}s_{2}j_{2}}(r_{2})\over
r_{2}}^{\ell_{2}+1}~{}P_{Q}(u)du$ (41)
### 6 Zero-range and other approximations
The coupling potentials ${\sf V}_{ji}({\bf R}_{j},{\bf R}_{i})$ of Eq. (15)
are non-local, in the sense that in general the initial and final radii, ${\bf
R}_{j}$ and ${\bf R}_{i}$, will be different. They will not only have
different magnitudes, but also different directions. In the early days of
transfer modeling, the calculations only became practical if a zero-range
approximation could be found, in which the coupling was restricted to ${\bf
R}_{j}=\alpha{\bf R}_{i}$ for some constant $\alpha$ (which need not be
unity).
When the projectile is a light ion such as 3H, 3He or 4He for nucleus $B+2$,
then the binding potential sum $V^{\rm sp}_{1B}+V^{\rm sp}_{2B}$ will have
short range. We may therefore consider approximating the vertex function
$\displaystyle[V^{\rm sp}_{1B}+V^{\rm sp}_{2B}]\phi^{J_{B}}_{I_{B}}({\bf
r},{\mbox{\boldmath$\rho$}}_{B})\sim
D_{0}\delta({\mbox{\boldmath$\rho$}}_{B})\phi^{B}_{nn}({\bf r})$ (42)
for some nucleon-nucleon wave function $\phi_{nn}({\bf r})$ that we are free
to choose. This a zero-range approximation. Note that it is only
${\mbox{\boldmath$\rho$}}_{B}$ which needs to have zero range, not ${\bf r}$.
The constant $D_{0}$ is called the zero-range constant.
If, furthermore, we can neglect the remnant term $U_{AB}-U_{i}$, then the
transfer coupling of Eq. (15) can be simplified as
$\displaystyle{\sf V}_{ji}({\bf R}_{j},{\bf R}_{i})$ $\displaystyle=$
$\displaystyle\langle\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})|V^{\rm sp}_{1B}+V^{\rm
sp}_{2B}|\phi^{J_{B}}_{I_{B}}({\bf r},{\mbox{\boldmath$\rho$}}_{B})\rangle$
(43) $\displaystyle=$ $\displaystyle\langle\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})|D_{0}\delta({\mbox{\boldmath$\rho$}}_{B})\phi^{B}_{nn}({\bf
r})\rangle$ $\displaystyle=$ $\displaystyle D_{0}~{}\langle\phi^{B}_{nn}({\bf
r})|\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})\rangle~{}~{}\delta({\mbox{\boldmath$\rho$}}_{B})$
$\displaystyle=$ $\displaystyle D_{0}~{}\langle\phi^{B}_{nn}({\bf
r})|\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})\rangle~{}~{}\delta(\beta\left({\bf
R}_{j}-\frac{A}{A{+}2}{\bf R}_{i}\right)),$
since
$\displaystyle{\bf R}_{j}-\frac{A}{A{+}2}{\bf
R}_{i}={\mbox{\boldmath$\rho$}}_{B}/\beta\mbox{~{}~{}~{}for~{}~{}~{}}\beta=\frac{2(A{+}B{+}2)}{(A{+}2)(B{+}2)}.$
(44)
That is, we arrive at a ‘form factor’ $\langle\phi^{B}_{nn}({\bf
r})|\phi^{J_{A}}_{I_{A}}({\bf r},-{\bf R}_{j})\rangle$ that is local in ${\bf
R}_{j}=\frac{A}{A+2}{\bf R}_{i}=-{\mbox{\boldmath$\rho$}}_{A}$ because of the
delta function $\delta({\mbox{\boldmath$\rho$}}_{B})$. To find the form
factor, we need to determine the average nucleon-nucleon relative wave
function $\phi^{B}_{nn}({\bf r})$ in the light ion, and project the heavy-
nucleus two-body wave function $\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})$ onto this relative motion. This gives a
function only of the distance $\rho_{A}=R_{j}$ and the angles. The kinematics
for this zero-range approximation are identical to those for the one-body
transfer of a mass-2 cluster from core $B$ to core $A$. A local-energy
approximation may be used to improve the treatment of the finite range of the
vertex function, just as for one-body transfers.
This is a further example the conclusion stated at the end of section 4,
namely that transfer reactions only probe in the unknown nucleus those
components of $nn$ relative motion that already exist in the known nucleus.
Since the known light nuclei 3H, 3He and 4He have predominantly $s$-wave
relative motion between the two transferred nucleons, our transfer reactions
will only probe pairing states of $s$-wave relative motion in the target. The
magnitude of the transfer cross section will be proportional to the form
factor overlap $\langle\phi^{B}_{nn}({\bf r})|\phi^{J_{A}}_{I_{A}}({\bf
r},{\mbox{\boldmath$\rho$}}_{A})\rangle$.
Zero-range approximations can be also used for some of the sequential steps
involving these light nuclei, but not for all of them if we are using ‘prior-
post’ couplings to avoid non-orthogonality corrections. For stripping
reactions such as (t,p), the first prior (t,d) step has no good zero-range
approximation, and for pickup reactions such as (p,t), the second post (d,t)
step must be treated in full finite range for the same reason.
Two-neutron overlap function for $\langle^{122}$Sn$|^{124}$Sn$\rangle$
$1g_{7/2}^{2}$ 0.62944 $2d_{5/2}^{2}$ 0.59927 $2d_{3/2}^{2}$ 0.71913
$3s_{1/2}^{2}$ 0.51892 $1h_{11/2}^{2}$ –1.24399
Figure 1: Simultaneous (short dash), sequential (dot-dash) and
simultaneous+sequential (solid line) cross sections for the reaction
124Sn(p,t)122Sn at 25 MeV, in comparison with the experimental data of
Guazzoni et al. [28]. Figure 2: Sequential cross sections for all possible
combinations of post and prior for the two steps. Figure 3: Simultaneous
(short dash), sequential (dot-dash) and simultaneous+sequential (solid line)
amplitudes at zero degrees for the reaction 124Sn(p,t)122Sn at 25 MeV. The
short lines show the individual contributions from the wave function
components of Table 6, and the longer lines with symbols are their coherent
sums.
### 7 Results
In this short paper we will focus on the reaction mechanisms for the pair
transfer 124Sn(p,t)122Sn at 25 MeV, using the overlap function shown in Table
6 we find by overlapping the shell-model wave functions for the ground states
of 122Sn and 124Sn. The structure results for 124Sn and 122Sn were obtained in
the model space of $(0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$,
$0h_{11/2})$ for neutrons with the code NuShell [24]. The model-space two-body
matrix elements are those used in Refs. [29, 30]. They were obtained starting
with a G matrix derived from the CD-Bonn [31] nucleon-nucleon interaction. The
harmonic oscillator basis was employed for the radial wave functions with an
oscillator energy $\hbar\omega$ = 7.87 MeV. The effective interaction for the
above shell-model space is obtained from the $Q$-box method and includes all
non-folded diagrams through third-order in the interaction G, to sum up the
folded diagrams to infinite order [32, 33]. The single-particle energies were
adjusted to reproduce the observed states in 131Sn.
The inputs to the reaction code are the two-nucleon spectroscopic amplitudes
(TNA) of Table 6. A center of mass correction [34] equal to
$[A/(A{-}2)]^{2n+\ell}$ for the TNA has been applied, where $A=124$. Our sign
convention is that the radial wave functions are positive at the origin. The
sequential process was calculated by a single intermediate state for each of
these orbits connected by a product of one-nucleon spectroscopic amplitudes
that are equal to the center-of-mass corrected TNA multiplied by $\sqrt{2}$
that takes into account the normalization of the two-particle amplitude.
Future calculations should also take into account the TNA obtained from the
mixing of neutron pairs for orbitals outside of the model space.
We use the triton potential of Li [35], the deuteron potential of Daehnick
[36], and the proton potential of Chapel Hill 89 [37]. All the two-neutron
wave functions are constructed within the half-separation-energy prescription.
For a triton wave function we use the pure $s^{2}$ configuration found by the
product of eigenstates at the half-separation energy (4.24 MeV) in a Woods-
Saxon potential with $V=77.83$ MeV, $R=0.95$ fm, and $a=0.65$ fm (the results
are not sensitive to these values). The Sn wave functions shown in Table 6 are
found at the half-separation energy (7.219 MeV) in a WS potential with
$r=1.17$ fm, and $a=0.75$ fm that has the fixed spin-orbit component
$V_{so}=6.2$ MeV, $r=1.01$ fm, and $a=0.75$ fm.
The complete cross section prediction is shown in Fig. 1, compared with the
experimental data of Guazzoni et al. [28]. Now we see that, with the shell-
model overlaps and proper finite-range and sequential contributions, the
unhappiness factors are much closer to unity. A better agreement between
theory and experiment has already been published[19], but in the present
calculations there are still questions about the angular oscillations which
are in not so good agreement with experiment. Note that Guazzoni et al. [28]
took the better agreement of the simultaneous transfer curve (dashed line) to
indicate small effects for sequential transfers, but this is not correct since
we do know that sequential transfers occur, and can calculate them with good
accuracy in this model (dot-dashed line).
To see the importance of the non-orthogonality terms, and hence of choosing
‘prior-post’ couplings if non-orthogonality terms are to be avoided, Fig. 2
plots the different sequential cross sections for all possible combinations of
post and prior for the two steps. The prior-post solid curve is the dot-dashed
curve in Fig. 1. The other curves are all different from this one, and cannot
be simply added to the simultaneous amplitude to get the correct result. This
also implies that no complete calculation with only zero-range couplings is
possible.
Finally, it is instructive to look at the interference effects between the
various simultaneous and sequential contributions. To display these coherence
effects, I choose to plot the scattering amplitude at zero degrees for the
non-spin-flip amplitude $m_{p}=m_{t}=1/2$ (the only non-zero amplitude at this
angle). Fig. 3 plots all the simultaneous and sequential contributions from
the different components listed in Table 6, along with their coherent sums. We
see that all the contributions to the simultaneous transfer are constructively
coherent, as are all the contributions to the total sequential amplitude. This
constructive coherence follows from the signs of the amplitudes in Table 6,
and reflects the significant pairing enhancement in 124Sn. The total
sequential and simultaneous amplitudes are not uniformly coherent with each
other, however. A uniform $90^{\circ}$ angle between the simultaneous and
sequential amplitudes in Figure 3 would indicate an incoherent summation of
the two cross sections, but that is not exactly true either. This reflects the
importance of the deuteron channel with its own specific optical potential. In
general, varying deuteron optical potentials and differing intermediate
$Q$-values require that both simultaneous and sequential terms be explicitly
calculated.
### Acknowledgements
This work was supported by the TORUS topical collaboration, and performed
under the auspices of the U.S. Department of Energy by Lawrence Livermore
National Laboratory under Contract DE-AC52-07NA27344. I thank Alex Brown (MSU)
for many discussions, and for providing shell-model overlap functions.
## References
* 1. S. Yoshida, Note on the two-nucleon stripping reaction, _Nuclear Physics_. 33, 685–692, (1962).
* 2. N. K. Glendenning, Nuclear spectroscopy with two-nucleon transfer reactions, _Phys. Rev._ 137, B102–B113 (Jan, 1965).
* 3. J. Bjerregaard, O. Hansen, O. Nathan, and S. Hinds, The (t, p) reaction with the even isotopes of sm, _Nuclear Physics_. 86(1), 145–166, (1966).
* 4. R. J. Ascuitto and N. K. Glendenning, Inelastic processes in particle transfer reactions, _Phys. Rev._ 181, 1396–1403 (May, 1969).
* 5. R. Ascuitto, N. Glendenning, and B. S√∏rensen, Confirmation of strong second order processes in (p, t) reactions on deformed nuclei, _Physics Letters B_. 34(1), 17–20, (1971).
* 6. B. Bayman, Finite-range calculation of the two-neutron transfer reaction, _Nuclear Physics A_. 168(1), 1–16, (1971).
* 7. R. Broglia, The pairing model, _Annals of Physics_. 80(1), 60–85, (1973).
* 8. L. A. Charlton, Finite-range evaluation of ($p$-$d$,$d$-$t$) with momentum space techniques, _Phys. Rev. C_. 14, 506–513 (Aug, 1976).
* 9. M. Yasue, H. Yokomizo, S. Kubono, K. Koyama, S. Takeuchi, and H. Ohnuma, 12C(p,t)10C and 12C(p,3He)10B Reactions at Ep = 51.9 MeV, _Journal of the Physical Society of Japan_. 42, 367, (1977).
* 10. N. Hashimoto and M. Kawai, The $(p,d)$ $(d,t)$ process in strong $(p,t)$ reactions, _Progress of Theoretical Physics_. 59(4), 1245–1260, (1978).
* 11. T. Takemasa, T. Tamura, and T. Udagawa, Exact finite range calculations of light-ion induced two-neutron transfer reactions, _Nuclear Physics A_. 321(2), 269–294, (1979).
* 12. B. F. Bayman and J. Chen, One-step and two-step contributions to two-nucleon transfer reactions, _Phys. Rev. C_. 26, 1509–1520 (Oct, 1982).
* 13. E. Maglione, G. Pollarolo, A. Vitturi, R. Broglia, and A. Winther, Absolute cross sections of two-nucleon transfer reactions induced by heavy ions, _Physics Letters B_. 162(1‚Äì3), 59–65, (1985).
* 14. M. Kurokawa, Y. Aoki, Y. Tagishi, and K. Yagi, One-and two-step processes in natural-and unnatural-parity 208Pb(p,t)206Pb reaction at $E_{p}=22$ MeV, _Nucl. Phys. A_. 470(2), 377–396, (1987).
* 15. M. Igarashi, K. Kubo, and K. Yagi, Two-nucleon transfer reaction mechanisms, _Physics Reports_. 199(1), 1–72, (1991).
* 16. I. Tanihata, M. Alcorta, D. Bandyopadhyay, R. Bieri, L. Buchmann, B. Davids, N. Galinski, D. Howell, W. Mills, S. Mythili, R. Openshaw, E. Padilla-Rodal, G. Ruprecht, G. Sheffer, A. C. Shotter, M. Trinczek, P. Walden, H. Savajols, T. Roger, M. Caamano, W. Mittig, P. Roussel-Chomaz, R. Kanungo, A. Gallant, M. Notani, G. Savard, and I. J. Thompson, Measurement of the two-halo neutron transfer reaction ${}^{1}\mathrm{H}(^{11}\mathrm{Li},^{9}\mathrm{Li})^{3}\mathrm{H}$ at $3A\text{\,}\text{\,}\mathrm{MeV}$, _Phys. Rev. Lett._ 100, 192502 (May, 2008).
* 17. G. Potel, A. Idini, F. Barranco, E. Vigezzi, and R. A. Broglia, Single cooper pair transfer in stable and in exotic nuclei, _arXiv/0906.4298v3_. nucl-th (April, 2011).
* 18. G. Potel, F. Barranco, E. Vigezzi, and R. A. Broglia, Evidence for phonon mediated pairing interaction in the halo of the nucleus ${}^{11}\mathrm{Li}$, _Phys. Rev. Lett._ 105, 172502 (Oct, 2010).
* 19. G. Potel, F. Barranco, F. Marini, A. Idini, E. Vigezzi, and R. A. Broglia, Calculation of the transition from pairing vibrational to pairing rotational regimes between magic nuclei ${}^{100}\mathrm{Sn}$ and ${}^{132}\mathrm{Sn}$ via two-nucleon transfer reactions, _Phys. Rev. Lett._ 107, 092501 (Aug, 2011).
* 20. M. C. Mermaz, A. Greiner, B. T. Kim, M. A. G. Fernandes, N. Lisbona, E. Müller, W. Chung, and B. H. Wildenthal, Multinucleon transfer reactions induced by ${}^{18}\mathrm{O}$ on ${}^{28}\mathrm{Si}$, _Phys. Rev. C_. 20, 2130–2142 (Dec, 1979).
* 21. J. S. Vaagen and K. Kumar, Matrix elements for scattering and two-nucleon transfer operators derived from dynamic deformation theory wavefunctions, _Journal of Physics G: Nuclear Physics_. 5(9), 1211, (1979).
* 22. S. Cohen and D. Kurath, Two-nucleon transfer in the 1p shell, _Nucl. Phys. A_. 141(1), 145–157, (1970).
* 23. J. Bang, F. Gareev, W. Pinkston, and J. S. Vaagen, One-and two-nucleon overlap functions in nuclear physics, _Phys. Rep._ 125(6), 253–399, (1985).
* 24. B. A. Brown and W. D. M. Rae. Nushell@MSU. MSU-NSCL report, (2007).
* 25. T. Brody and M. Moshinsky, _Tables of Transformation Brackets_. (Monografias del Instituto de Fisica, 1960).
* 26. B. F. Bayman and A. Kallio, Relative-angular-momentum-zero part of two-nucleon wave functions, _Phys. Rev._ 156, 1121–1128 (Apr, 1967).
* 27. M. Moshinsky, Transformation brackets for harmonic oscillator functions, _Nuclear Physics_. 13(1), 104–116, (1959).
* 28. P. Guazzoni, L. Zetta, A. Covello, A. Gargano, B. Bayman, T. Faestermann, G. Graw, R. Hertenberger, H.-F. Wirth, and M. Jaskola, High-resolution measurement of the 118,124Sn(p,t)116,122Sn reactions: Shell-model and microscopic distorted-wave born approximation calculations, _Phys. Rev. C_. 83(4), 044614 (Apr, 2011).
* 29. B. A. Brown, N. J. Stone, J. R. Stone, I. S. Towner, and M. Hjorth-Jensen, Magnetic moments of the ${2}_{1}^{+}$ states around ${}^{132}\mathrm{Sn}$, _Phys. Rev. C_. 71, 044317 (Apr, 2005).
* 30. B. A. Brown, N. J. Stone, J. R. Stone, I. S. Towner, and M. Hjorth-Jensen, Erratum: Magnetic moments of the ${2}_{1}^{+}$ states around ${}^{132}\mathrm{Sn}$, _Phys. Rev. C_. 72, 029901 (Aug, 2005).
* 31. R. Machleidt, F. Sammarruca, and Y. Song, Nonlocal nature of the nuclear force and its impact on nuclear structure, _Phys. Rev. C_. 53, R1483–R1487 (Apr, 1996).
* 32. M. Hjorth-Jensen, T. T. Kuo, and E. Osnes, Realistic effective interactions for nuclear systems, _Physics Reports_. 261(3–4), 125–270, (1995).
* 33. M. Hjorth-Jensen, H. Müther, E. Osnes, and A. Polls, Comparison of the effective interaction to various orders in different mass regions, _Journal of Physics G: Nuclear and Particle Physics_. 22(3), 321, (1996).
* 34. N. Anyas-Weiss, J. Cornell, P. Fisher, P. Hudson, A. Menchaca-Rocha, D. Millener, A. Panagiotou, D. Scott, D. Strottman, D. Brink, B. Buck, P. Ellis, and T. Engeland, Nuclear structure of light nuclei using the selectivity of high energy transfer reactions with heavy ions, _Physics Reports_. 12(3), 201–272, (1974).
* 35. X. Li, C. Liang, and C. Cai, Global triton optical model potential, _Nuclear Physics A_. 789(1‚Äì4), 103–113, (2007).
* 36. W. W. Daehnick, J. D. Childs, and Z. Vrcelj, Global optical model potential for elastic deuteron scattering from 12 to 90 MeV, _Phys. Rev. C_. 21, 2253–2274 (Jun, 1980).
* 37. R. Varner, W. Thompson, T. McAbee, E. Ludwig, and T. Clegg, A global nucleon optical model potential, _Physics Reports_. 201(2), 57–119, (1991).
|
arxiv-papers
| 2012-04-13T17:19:00 |
2024-09-04T02:49:29.684022
|
{
"license": "Public Domain",
"authors": "Ian J. Thompson",
"submitter": "Ian Thompson",
"url": "https://arxiv.org/abs/1204.3054"
}
|
1204.3072
|
# Null controllability for a parabolic-elliptic coupled system
E. Fernández-Cara, J. Limaco , S. B. de Menezes Dpto. E.D.A.N., Universidad de
Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, cara@us.es.Inst. Matemática,
Universidade Federal Fluminense, Valonguinho, 24020-140, Niterói, RJ, Brasil,
jlimaco@vm.uff.br.Corresponding author: Dpto. Matemática, Universidade Federal
do Ceará, Campus do Pici - Bloco 914, 60455-760, Fortaleza, CE, Brasil,
silvano@mat.ufc.br.
###### Abstract
In this paper, we prove the null controllability of some parabolic-elliptic
systems. The control is distributed, locally supported in space and appears
only in one PDE. The arguments rely on fixed-point reformulation and suitable
Carleman estimates for the solutions to the adjoint system. Under appropriate
assumptions, we also prove that the solution can be obtained as the asymptotic
limit of some similar parabolic systems.
Mathematics Subject Classification 2000: 35B37, 35A05, 35B40.
Keywords: Null controllability, parabolic-elliptic systems, Carleman
inequalities.
## 1 Introduction and main results
Let ${\Omega}$ be a bounded domain of ${\mathbb{R}}^{N}$ ($N\geq 1$), with
boundary $\Gamma=\partial{\Omega}$ of class $C^{2}$. We fix $T>0$ and we
denote by $Q$ the cylinder $Q={\Omega}\times(0,T)$, with lateral boundary
$\Sigma=\Gamma\times(0,T)$. We also consider a non-empty (small) open set
$\mathcal{O}\subset{\Omega}$; as usual, $1_{\mathcal{O}}$ denotes the
characteristic function of $\mathcal{O}$.
Throughout this paper, $C$ (and sometimes $C_{0}$, $K$, $K_{0}$, …) denotes
various positive constants. Frequently, we will emphasize the fact that $C$
depends on (say) $f$ by writing $C=C(f)$. The inner product and norm in
$L^{2}({\Omega})$ will be denoted, respectively, by $(\cdot\,,\cdot)$ and
$\|\cdot\|$. On the other hand, $\|\cdot\|_{\infty}$ will stand for the norm
in $L^{\infty}(Q)$.
We will consider the following semilinear parabolic-elliptic coupled systems
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}\end{array}\right.$ (1.1)
and
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (1.2)
under some hypotheses for $F$ and $f$.
In (1.1) and (1.2), we have $y=y(x,t)$ and $z=z(x,t)$; $1_{\mathcal{O}}$ is
the characteristic function of $\mathcal{O}$ and $y^{0}=y^{0}(x)$ is the
initial state. We will assume that the possibly nonlinear functions
$F:{\mathbb{R}}\times{\mathbb{R}}\mapsto{\mathbb{R}}$ and
$f:{\mathbb{R}}\mapsto{\mathbb{R}}$ satisfy:
$\left\\{\begin{array}[]{l}\displaystyle\text{$F$ and $f$ are (globally)
Lipschitz-continuous,}\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
F(0,0)=f(0,0)=0,\quad{\partial f\over\partial z}(y,z)\leq\mu<\mu_{1}\ \
\text{a.e.},\\\ \end{array}\right.$ (1.3)
where $\mu_{1}$ the first eigenvalue of the Dirichlet Laplacian in ${\Omega}$.
The analysis of systems of the kind (1.1) and (1.2) can be justified by
several applications. Let us indicate two of them:
* •
Reaction-diffusion systems with origin in physics, chemistry, biology, etc.
where two scalar “populations” interact and the natural time scale of the
growth rate is much smaller for one of them than for the other one. Precise
examples can be found in the study of prey-predator interaction, chemical
heating, tumor growth therapy, etc.
* •
Semiconductor modeling, where one of the state variables is (for example) the
density of holes and the other one is the electrical potential of the device;
see for instance [7]. Other problems with this motivation will be analyzed
with more detail by the authors in the next future.
The system (1.1) (resp. (1.2)) is well-posed in the sense that, for each
$y^{0}\in L^{2}({\Omega})$ and each $v\in L^{2}(\mathcal{O}\times(0,T))$
(resp. $w\in L^{2}(\mathcal{O}\times(0,T))$) possesses exactly one solution
$(y,z)$, with
$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;D(-\Delta)).$
This statement is justified in Appendix A.
In this paper we will analyze some controllability properties of (1.1) and
(1.2).
It will be said that (1.1) (resp. (1.2)) is null-controllable at time $T$ if
the following holds: for any given $y^{0}\in L^{2}({\Omega})$, there exist
controls $v\in L^{2}(\mathcal{O}\times(0,T))$ (resp. controls $w\in
L^{2}(\mathcal{O}\times(0,T))$) and associated solutions satisfying
$z\in C^{0}([0,T];L^{2}({\Omega}))$
and
$y(x,T)=0\ \mbox{ in }\ {\Omega},\quad\limsup_{t\to T^{-}}\|z(\cdot\,,t)\|=0,$
(1.4)
with an estimate of the form
$\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|\quad(\mbox{resp.\
}\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|).$ (1.5)
This inequality indicates that the “null controls” can be chosen depending
continuously on the initial data.
The control of PDEs equations and systems has been the subject of a lot of
papers the last years. In particular, important progress has been made
recently in the controllability analysis of semi-linear parabolic equations.
We refer to the works [1, 2, 3, 5, 6, 9, 11, 12] and the references therein.
Consequently, it is natural to try to extend the known results to systems of
the kind (1.1) and (1.2).
The main results in this paper are the following:
###### Theorem 1.1
Assume that
$F(y,z)\equiv F_{0}(y)+bz$, with $F_{0}$ Lipschitz-continuous,
$b\in{\mathbb{R}}$, (1.6) $f(y,z)\equiv cy+dz$, with $c,d\in{\mathbb{R}}$,
$c\not=0$, $d<\mu_{1}$. (1.7)
Then (1.1) is null-controllable at any time $T>0$.
###### Theorem 1.2
Let us assume that (1.6) holds and
$f(y,z)\equiv f_{0}(y)+dz$, with $f_{0}$ Lipschitz-continuous,
$d\in{\mathbb{R}}$, $d<\mu_{1}$. (1.8)
Then (1.2) is null-controllable at any time $T>0$.
The proofs of these results rely on relatively well known arguments and some
new estimates.
More precisely, in a first step, we will first consider similar linearized
systems of the form
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=a(x,t)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}\end{array}\right.$ (1.9)
and
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=a(x,t)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (1.10)
We will establish null controllability results for (1.9) and (1.10) by
previously proving appropriate Carleman estimates for the solutions to the
associated adjoint systems. Then, in a second step, we will adapt a fixed-
point argument to get the null controllability results stated in Theorems 1.1
and 1.2.
In this paper, we will also consider systems of the form
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon z_{t}-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$
(1.11)
It will be shown that, under the assumptions of Theorem 5.1, (1.11) is
uniformly null-controllable as $\varepsilon\to 0$, i.e. null-controlable with
controls $w_{\varepsilon}$ satisfying the estimates (1.5) with $C$ independent
of $\varepsilon$. We will also see that the $w_{\varepsilon}$ can be chosen in
such a way that they converge weakly to a null control of (1.2) (see Theorem
(5.2) below).
This paper is organized as follows. In Section 2, we introduce some adjoint
(backwards in time) parabolic-elliptic systems and we prove that their
solutions satisfy suitable Carleman estimates. In Section 3, we deduce from
these estimates null controllability results for (1.9) and (1.10). Section 4
deals with the proofs of Theorems 1.1 and 1.2. The uniform null
controllability property of (1.11) and the convergence of the associated null
controls are established in Section 5. Finally, we give the proofs of some
technical results in Section 6 (Appendix A).
## 2 Some Carleman estimates
We will first consider the general linear backwards in time system
$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b(x,t)\varphi+d(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi=0,\ \psi=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
(2.1)
where $\varphi^{T}\in L^{2}({\Omega})$ and we assume that
$a,b,c,d\in L^{\infty}(Q),\ \ d\leq\mu<\mu_{1}\ \text{a.e.}$ (2.2)
Also, it will be convenient to introduce a new non-empty open set
$\mathcal{O}_{0}$, with $\mathcal{O}_{0}\Subset\mathcal{O}$. We will need the
following result, due to Fursikov and Imanuvilov [6]:
###### Lemma 2.1
There exists a function $\alpha_{0}\in C^{2}(\overline{{\Omega}})$ satisfying:
$\left\\{\begin{array}[]{l}\alpha_{0}(x)>0\quad\forall
x\in{\Omega},\quad\alpha_{0}=0\quad\forall x\in\partial{\Omega},\\\
|\nabla\alpha_{0}(x)|>0\quad\forall
x\in\overline{{\Omega}}\setminus\mathcal{O}_{0}.\end{array}\right.$
Let us introduce the functions
$\beta(t):=t(T-t),\ \phi(x,t):=\frac{e^{\lambda\alpha_{0}(x)}}{\beta(t)}\,,\
\overline{\alpha}(x):=e^{k\lambda}-e^{\lambda\alpha_{0}(x)},\
\alpha(x,t):=\frac{\overline{\alpha}(x)}{\beta(t)}\,,$
where $k>\|\alpha_{0}\|_{L^{\infty}}+\log 2$ and $\lambda>0$. Also, let us set
$\begin{array}[]{l}\displaystyle\hat{\alpha}(t):=\min_{x\in\overline{\Omega}}\alpha(x,t),\quad\alpha^{*}(t):=\max_{x\in\overline{\Omega}}\alpha(x,t),\\\
\displaystyle\hat{\phi}(t):=\min_{x\in\overline{\Omega}}\phi(x,t),\quad\phi^{*}(t):=\max_{x\in\overline{\Omega}}\alpha(x,t).\end{array}$
Then the following Carleman estimates hold:
###### Proposition 2.1
Assume that (2.2) holds. There exist positive constants $\lambda_{0}$, $s_{0}$
and $C_{0}$ such that, for any $s\geq s_{0}$ and $\lambda\geq\lambda_{0}$ and
any $\varphi^{T}\in L^{2}({\Omega})$, the associated solution to (2.1)
satisfies
$\begin{array}[]{c}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}(s\phi)|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]dxdt\\\
\displaystyle\leq
C_{0}\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}|\psi|^{2}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right)dxdt\end{array}$
(2.3)
and
$\begin{array}[]{c}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!|\Delta\psi|^{2}\\!+\\!\lambda^{2}(s\phi)|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\psi|^{2}\right]dxdt\\!\\\
\displaystyle\leq
C_{0}\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}|\varphi|^{2}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}\right)dxdt\end{array}$
(2.4)
Furthermore, $C_{0}$ and $\lambda_{0}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{0}$ can be chosen of the form
$\displaystyle s_{0}=\sigma_{0}(T+T^{2}),$ (2.5)
where $\sigma_{0}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $\|b\|_{\infty}$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.
This result is proved in [6]. In fact, similar Carleman inequalities are
established there for more general linear parabolic equations. The explicit
dependence in time of the constants is not given in [6]. We refer to [4],
where the above formula for $s_{0}$ is obtained.
For further purpose, we introduce the following notation:
$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}(s\phi)|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]dxdt\end{array}$
and
$\begin{array}[]{l}\displaystyle\tilde{I}(s,\lambda;\psi)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}|\Delta\psi|^{2}\\!+\\!\lambda^{2}(s\phi)|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\psi|^{2}\right]dxdt.\end{array}$
Now, we will deduce several consequences from Proposition 2.1 under particular
hypotheses on the coefficients of (2.1). First, it will be assumed that $c$ is
a.e. equal to a non-zero constant and $b$ and $d$ do not depend of $t$:
$a\in L^{\infty}(Q),\ c\in{\mathbb{R}},\ c\not=0,\ b,d\in
L^{\infty}({\Omega}),\ d\leq\mu<\mu_{1}\ \text{a.e.}$ (2.6)
Accordingly, (2.1) reads:
$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b(x)\varphi+d(x)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi(x,t)=0,\ \psi(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$
(2.7)
###### Proposition 2.2
Assume that (2.6) holds. There exist positive constants $\lambda_{0}$, $s_{0}$
and $C_{1}$ such that, for any $s\geq s_{0}$ and $\lambda\geq\lambda_{0}$ and
any $\varphi^{T}\in L^{2}({\Omega})$, the associated solution to (2.7)
satisfies
$I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq
C_{1}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.$
(2.8)
Furthermore, $C_{1}$ and $\lambda_{0}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{0}$ can be chosen of the form
$\displaystyle s_{1}=\sigma_{1}(T+T^{2}),$ (2.9)
where $\sigma_{1}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$.
Proof: Obviously, it will be sufficient to show that there exist
$\lambda_{0}$, $s_{0}$ and $C_{1}$ such that, for any small $\varepsilon>0$,
one has:
$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq C\varepsilon
I(s,\lambda;\varphi)+C\varepsilon\tilde{I}(s,\lambda;\psi)\\\
\displaystyle\qquad+\
C_{1}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$
(2.10)
We start from (2.3) and (2.4). After addition, by taking $\sigma_{1}$
sufficiently large and $s\geq\sigma_{1}(T+T^{2})$, we obtain:
$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\\\ \displaystyle\quad\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt.\end{array}$
(2.11)
Let us now introduce a function $\xi\in\mathcal{D}(\mathcal{O})$ satisfying
$0<\xi\leq 1$ and $\xi\equiv 1$ in $\mathcal{O}_{0}$. Then
$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi|\psi|^{2}dxdt\\\
\displaystyle\quad=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi(x)\psi\left(-\frac{1}{c}(\varphi_{t}+\Delta\varphi+a(x,t)\varphi)\right)dxdt\\\
\displaystyle\quad=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,\psi\,\varphi_{t}\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,\psi\,\Delta\varphi\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\,a(x,t)\,\psi\,\varphi\;dxdt\\\
\displaystyle\quad:=M_{1}+M_{2}+M_{3}.\end{array}$ (2.12)
Let us compute and estimate the $M_{i}$.
First,
$\begin{array}[]{l}\displaystyle
M_{1}=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{2\xi(x)}{c}\lambda^{4}s^{4}\phi^{3}\alpha_{t}\psi\varphi\;dxdt\\\
\displaystyle\phantom{M_{1}=}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{3\xi(x)}{c}\lambda^{4}s^{3}\phi^{2}\phi_{t}\psi\varphi\;dxdt\\\
\displaystyle\phantom{M_{1}=}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\frac{\xi(x)}{c}\lambda^{4}(s\phi)^{3}\psi_{t}\varphi\;dxdt.\end{array}$
Using that $|\alpha_{t}|\leq C\phi^{2}$ and $|\phi_{t}|\leq C\phi^{2}$ for
some $C>0$, we get:
$\begin{array}[]{l}\displaystyle M_{1}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}s^{4}\phi^{5}|\psi|\,|\varphi|\;dxdt\\\
\displaystyle\phantom{M_{1}\leq}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\phantom{M_{1}}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}s^{5}\phi^{7}|\varphi|^{2}dxdt\\\
\displaystyle\phantom{M_{1}\leq}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt.\end{array}$
The last integral in this inequality can be bounded as follows:
$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\quad\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\hat{\alpha}}\lambda^{4}(s\phi^{*})^{3}|\psi_{t}|\,|\varphi|\;dxdt\\\
\displaystyle\quad=\displaystyle\int_{0}^{T}e^{-2s\hat{\alpha}(t)}\lambda^{4}(s\phi^{*}(t))^{3}\|\psi_{t}(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad\leq
C\displaystyle\int_{0}^{T}e^{-2s\hat{\alpha}(t)}\lambda^{4}(s\phi^{*}(t))^{3}\|\varphi_{t}(\cdot\,,t)\|\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad=C\displaystyle\int_{0}^{T}e^{-s\alpha^{*}}(s\phi^{*}(t))^{-1/2}\|\varphi_{t}(\cdot\,,t)\|\cdot
e^{-2s\hat{\alpha}+s\alpha^{*}}\lambda^{4}(s\phi^{*})^{7/2}\|\varphi(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\;dt\\\
\displaystyle\quad\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$
Here, we have used that
$\left\\{\begin{array}[]{l}\displaystyle-\Delta\psi_{t}=b\varphi_{t}+d\psi_{t}\
\text{ in }\ {\Omega},\\\ \displaystyle\psi_{t}=0\ \text{ on }\
\partial{\Omega},\end{array}\right.$
whence we obviously need $b$ and $d$ independent of $t$.
Thus, the following is found:
$\begin{array}[]{l}\displaystyle M_{1}\leq\varepsilon
I(s,\lambda;\varphi)+\varepsilon\tilde{I}(s,\lambda;\psi)\\\
\displaystyle\phantom{M_{1}}+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-4s\hat{\alpha}+2s\alpha^{*}}\lambda^{8}(s\phi^{*})^{7}|\varphi|^{2}dxdt.\end{array}$
(2.13)
Secondly, we see that
$\begin{array}[]{l}\displaystyle
M_{2}=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}\Delta\left(e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{c}\psi\right)\varphi\;dxdt\\\
\displaystyle\phantom{M_{2}\leq}C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\left[\lambda^{6}(s\phi)^{5}|\psi|+\lambda^{5}(s\phi)^{4}|\nabla\psi|+\lambda^{4}(s\phi)^{3}|\Delta\psi|\right]\varphi\;dxdt\\\
\displaystyle\phantom{M_{2}}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\varphi|^{2}dxdt.\end{array}$
(2.14)
Here, we have used the identity
$\begin{array}[]{c}\displaystyle\Delta\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\psi\right)=\Delta\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\right)\psi\\\
\displaystyle+2\nabla\left(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\right)\cdot\nabla\psi+e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c}\Delta\psi\end{array}$
and the estimates
$|\Delta(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c})|\leq
Ce^{-2s\alpha}\lambda^{2}s^{2}\phi^{5}\ \text{ and }\
|\nabla(e^{-2s\alpha}\phi^{3}\frac{\xi(x)}{c})|\leq Ce^{-2s\alpha}\lambda
s\phi^{4}.$
Finally, it is immediate that
$M_{3}\leq\varepsilon\tilde{I}(s,\lambda;\psi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}dxdt.$
(2.15)
From (2.11), (2.12) and (2.13)–(2.15), we directly obtain (2.10) for all small
$\varepsilon>0$. This ends the proof. $\Box$
An almost immediate consequence of Proposition 2.2 is the following
observability inequality:
###### Corollary 2.1
Under the assumptions of Proposition 2.2, there exists constants $M$ and $K$,
depending on ${\Omega}$, $\mathcal{O}$, $T$, $\|a\|_{\infty}$,
$\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$, such that every solution
$(\varphi,\psi)$ to (2.7) verifies:
$\begin{array}[]{l}\displaystyle\|\varphi(\cdot\,,0)\|^{2}+\|\psi(\cdot\,,0)\|^{2}\leq
M\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi|^{2}dxdt.\end{array}$
(2.16)
Proof: From the Carleman inequality in Proposition 2.2 with $s=s_{1}$ and
$\lambda=\lambda_{1}$, we see that there exists a constant $C>0$ such that
$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}\phi^{3}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}(t(T-t))^{-7}e^{-4s\hat{\alpha}+2s\alpha^{*}}|\varphi|^{2}dxdt.$
Since $2\hat{\alpha}-\alpha^{*}\equiv a^{*}/\beta(t)$ for some $a^{*}>0$ and
$e^{-2s\alpha}\phi^{3}$ is uniformly bounded from below in
${\Omega}\times[T/4,3T/4]$ we find that
$\displaystyle\int_{T/4}^{3T/4}\int_{\Omega}\left(|\varphi|^{2}+|\psi|^{2}\right)dxdt\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi|^{2}dxdt$
(2.17)
for some $C,K>0$.
On the other hand, we can easily get from (2.7) the standard (backwards)
energy inequalities
$-\frac{1}{2}\frac{d}{dt}\|\varphi\|^{2}+\|\varphi\|_{H_{0}^{1}}^{2}\leq
C\left(\|\varphi\|^{2}+\|\psi\|^{2}\right)\ \text{ and }\
\|\psi\|_{H_{0}^{1}}^{2}\leq C\|\varphi\|^{2}.$ (2.18)
This yields
$-\frac{1}{2}\frac{d}{dt}\|\varphi\|^{2}\leq C\|\varphi\|^{2},$ (2.19)
whence we deduce that
$\|\varphi(\cdot\,,0)\|^{2}\leq C\|\varphi(\cdot\,,t)\|^{2}\;\;\mbox{ for all
}t.$ (2.20)
Combining (2.17), (2.20) and the second part of (2.18), we obtain at once
(2.16). $\Box$
Now, we will assume that $b$ is a non-zero constant:
$a,c,d\in L^{\infty}(Q),\ b\in{\mathbb{R}},\ b\not=0,\ d\leq\mu<\mu_{1}\
\text{a.e.}$ (2.21)
The corresponding (2.1) becomes
$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a(x,t)\varphi+c(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle-\Delta\psi=b\varphi+d(x,t)\psi\;\;\mbox{in}\;\;Q,\\\
\displaystyle\varphi(x,t)=0,\ \psi(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\
\displaystyle\varphi(x,T)=\varphi^{T}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$
(2.22)
###### Proposition 2.3
Assume that (2.21) holds. There exist positive constants $\lambda_{2}$,
$s_{2}$ and $C_{2}$ such that, for any $s\geq s_{2}$ and
$\lambda\geq\lambda_{2}$ and any $\varphi^{T}\in L^{2}({\Omega})$, the
associated solution to (2.22) satisfies
$I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq
C_{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\psi|^{2}dxdt.$
Furthermore, $C_{2}$ and $\lambda_{2}$ only depend on ${\Omega}$ and
$\mathcal{O}$ and $s_{2}$ can be chosen of the form
$\displaystyle s_{2}=\sigma_{2}(T+T^{2}),$
where $\sigma_{2}$ only depends on ${\Omega}$, $\mathcal{O}$,
$\|a\|_{\infty}$, $|b|$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.
Proof: We start again from (2.11). Recalling that
$\xi\in\mathcal{D}(\mathcal{O})$, $0<\xi\leq 1$ and $\xi\equiv 1$ in
$\mathcal{O}_{0}$, we see that
$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}_{0}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\varphi|^{2}dxdt\leq\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi|\varphi|^{2}dxdt\\\
\displaystyle\quad=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\xi(x)\varphi\left(-\frac{1}{b}(\Delta\psi+d(x,t)\psi)\right)dxdt\\\
\displaystyle\quad=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}\varphi\Delta\psi\;dxdt\\\
\displaystyle\phantom{\quad=}-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}d(x,t)\varphi\psi\;dxdt\\\
\displaystyle\quad:=M^{\prime}_{1}+M^{\prime}_{2}.\end{array}$ (2.23)
As in the proof of Proposition 2.2, it is not difficult compute and estimate
the $M^{\prime}_{i}$. Indeed,
$\begin{array}[]{l}\displaystyle
M^{\prime}_{1}=-\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}\Delta\left(e^{-2s\alpha}\lambda^{4}(s\phi)^{3}\frac{\xi(x)}{b}\varphi\right)\psi\;dxdt\\\
\displaystyle\phantom{M^{\prime}_{1}}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\left[\lambda^{6}(s\phi)^{5}|\varphi|+\lambda^{5}(s\phi)^{4}|\nabla\varphi|+\lambda^{4}(s\phi)^{3}|\Delta\varphi|\right]|\psi|\;dxdt\\\
\displaystyle\phantom{M^{\prime}_{1}}\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{8}(s\phi)^{7}|\psi|^{2}dxdt.\end{array}$
(2.24)
On the other hand,
$M^{\prime}_{2}\leq\varepsilon
I(s,\lambda;\varphi)+C_{\varepsilon}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt.$
(2.25)
From (2.11), (2.23) and (2.24)–(2.25), we find that
$\begin{array}[]{l}\displaystyle
I(s,\lambda;\varphi)+\tilde{I}(s,\lambda;\psi)\leq C\varepsilon
I(s,\lambda;\varphi)+C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2s\alpha}\lambda^{4}(s\phi)^{3}|\psi|^{2}dxdt,\end{array}$
for all small $\varepsilon>0$.
This ends the proof. $\Box$
Arguing as in the proof of Corollary 2.1, the following can be easily
established:
###### Corollary 2.2
Under the assumptions of Proposition 2.3, there exists constants $M^{\prime}$
and $K^{\prime}$, depending on ${\Omega}$, $\mathcal{O}$, $T$,
$\|a\|_{L^{\infty}(Q)}$, $|b|$, $\|c\|_{L^{\infty}(Q)}$ and
$\|d\|_{L^{\infty}}$, such that every solution $(\varphi,\psi)$ to (2.7)
verifies:
$\|\varphi(\cdot\,,0)\|^{2}+\limsup_{t\to 0^{+}}\|\psi(\cdot\,,t)\|^{2}\leq
M^{\prime}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K^{\prime}/(T-t)}|\psi|^{2}dxdt.$
(2.26)
## 3 The null controllability of the linearized systems
In this Section, we will deduce from the observability estimates (2.16) and
(2.26) null controllability results for (1.9) and (1.10).
More precisely, we have:
###### Theorem 3.1
Assume that (2.6) holds. Then (1.9) is null-controllable at any time $T>0$.
That is to say, for any $y^{0}\in L^{2}({\Omega})$ there exist null controls
$v\in L^{2}(\mathcal{O}\times(0,T))$ for (1.9) satisfying
$\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$ (3.1)
where $C$ only depends on ${\Omega}$, $\mathcal{O}$, $T$,
$\|a\|_{L^{\infty}(Q)}$, $\|b\|_{L^{\infty}}$, $|c|$ and $\|d\|_{L^{\infty}}$.
Proof: There are several ways to prove that the observability inequality
(2.16) implies the null controllability of (1.9). One of them is the
following.
For any $v\in L^{2}(\mathcal{O}\times(0,T))$ and any $\varepsilon>0$, let us
set
$J_{\varepsilon}(v)=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|v|^{2}dxdt+\frac{1}{\varepsilon}\|y(\cdot\,,T)\|^{2}.$
(3.2)
Here, $(y,z)$ is the solution to (1.9) associated to the initial data $y^{0}$.
It is not difficult to check that $v\mapsto J_{\varepsilon}(v)$ is lower semi-
continuous, strictly convex and coercive in $L^{2}(\mathcal{O}\times(0,T))$.
Hence, it possesses a unique minimizer $v_{\varepsilon}\in L^{2}(Q)$. We will
denote by $(y_{\varepsilon},z_{\varepsilon})$ the associated state.
We will show that, at least for a subsequence, $v_{\varepsilon}$ converges
weakly in $L^{2}(\mathcal{O}\times(0,T))$ towards a control $v\in
L^{2}(\mathcal{O}\times(0,T))$ and the associated $y_{\varepsilon}(\cdot\,,T)$
converges strongly in $L^{2}({\Omega})$ to zero. Obviously, this proves that
$v$ is a null control for (1.9), i.e. that the state associated to $v$
satisfies (1.4).
Notice that the unique minimizer of (3.11) is characterized by the following
optimality system:
$\left\\{\begin{array}[]{l}\displaystyle y_{\varepsilon,t}-\Delta
y_{\varepsilon}=ay_{\varepsilon}+bz_{\varepsilon}+v_{\varepsilon}1_{\mathcal{O}}\
\ \ \mbox{ in }\ Q,\\\ \displaystyle-\Delta
z_{\varepsilon}=cy_{\varepsilon}+dz_{\varepsilon}\ \ \ \mbox{ in }Q,\\\
\displaystyle y_{\varepsilon}=z_{\varepsilon}=0\ \ \ \mbox{ on }\ \Sigma,\\\
\displaystyle y_{\varepsilon}(x,0)=y^{0}(x)\ \ \ \mbox{ in }\
\Omega,\end{array}\right.$ (3.3)
$\left\\{\begin{array}[]{l}\displaystyle-\varphi_{\varepsilon,t}-\Delta\varphi_{\varepsilon}=a\varphi_{\varepsilon}+c\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\
\displaystyle-\Delta\psi_{\varepsilon}=b\varphi_{\varepsilon}+d\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\ \displaystyle\varphi_{\varepsilon}=\psi_{\varepsilon}=0\
\ \ \;\mbox{ on }\ \Sigma\\\
\displaystyle\varphi_{\varepsilon}(x,T)=-\frac{1}{\varepsilon}y_{\varepsilon}(x,T)\
\;\;\;\mbox{ in }\;\Omega,\end{array}\right.$ (3.4)
$v_{\varepsilon}=\bigl{.}e^{-2K/(T-t)}\varphi_{\varepsilon}\bigr{|}_{\mathcal{O}\times(0,T)}\,$
(3.5)
(see for instance [8]; see also [4]).
By multiplying both sides of $(\ref{asep})_{1}$ by $y_{\varepsilon}$ and both
sides of $(\ref{asep})_{2}$ by $z_{\varepsilon}$, integrating in time and
space and adding the resulting identities, we obtain:
$\begin{array}[]{l}\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt+\frac{1}{\varepsilon}\|y_{\varepsilon}(\cdot\,,T)\|^{2}\\\\[11.0pt]
\displaystyle\leq\|y^{0}\|\;\|\varphi_{\varepsilon}(\cdot\,,0)\|\leq
M^{1/2}\|y^{0}\|\left(\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt\right)^{1/2}\\\\[11.0pt]
\displaystyle\leq
C\|y^{0}\|^{2}+\frac{1}{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt,\end{array}$
(3.6)
where we have used the observability estimate (2.16).
From (3.6), we see that
$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|v_{\varepsilon}|^{2}dxdt=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\varphi_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2}.$ (3.7)
Consequently, at least for a subsequence, one has
$\begin{array}[]{l}\displaystyle v_{\varepsilon}\to v\ \text{ weakly in }\
L^{2}(\mathcal{O}\times(0,T)),\ \text{ with }\
\|v\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|.\end{array}$ (3.8)
We also have from (3.6) that
$\begin{array}[]{l}\displaystyle y_{\varepsilon}(\cdot\,,T)\to 0\ \text{
strongly in }\ L^{2}({\Omega})\ \text{ as }\ \varepsilon\to 0.\end{array}$
(3.9)
From the usual energy method, it is clear that a subsequence can be extracted
such that
$\begin{array}[]{l}\displaystyle y_{\varepsilon}\to y\ \text{ and }\
z_{\varepsilon}\to z\;\;\mbox{strongly in }\;L^{2}(Q)\;\;\mbox{ as
}\;\varepsilon\to 0\end{array}$ (3.10)
(see for instance [10]) and, consequently, the limit $v$ is such that the
solution $(y,z)$ to (1.9) satisfies (1.4).
This ends the proof of Theorem 3.1. $\Box$
We also deduce from Corollary 2.2 the following result:
###### Theorem 3.2
Assume that (2.21) holds. Then (1.10) is null-controllable at any time $T>0$.
In other words, for any $y^{0}\in L^{2}({\Omega})$ there exist null controls
$w\in L^{2}(\mathcal{O}\times(0,T))$ for (1.10) such that
$\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$
where $C$ only depends on ${\Omega}$, $\mathcal{O}$, $T$, $\|a\|_{\infty}$,
$|b|$, $\|c\|_{\infty}$ and $\|d\|_{\infty}$.
Proof: It is very similar to the proof of Thoerem 3.1.
Indeed, we can introduce the functional $L_{\varepsilon}$, with
$L_{\varepsilon}(v):=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w|^{2}\;dxdt+\frac{1}{\varepsilon}\|y(\cdot\,,T)\|^{2}$
(3.11)
and we can again check that $L_{\varepsilon}$ possesses exactly one minimizer
$w_{\varepsilon}\in L^{2}(Q)$. The optimality system is now
$\left\\{\begin{array}[]{l}\displaystyle y_{\varepsilon,t}-\Delta
y_{\varepsilon}=ay_{\varepsilon}+bz_{\varepsilon}\ \ \ \mbox{ in }\ Q,\\\
\displaystyle-\Delta
z_{\varepsilon}=cy_{\varepsilon}+dz_{\varepsilon}+w_{\varepsilon}1_{\mathcal{O}}\
\ \ \mbox{ in }Q,\\\ \displaystyle y_{\varepsilon}=z_{\varepsilon}=0\ \ \
\mbox{ on }\ \Sigma,\\\ \displaystyle y_{\varepsilon}(x,0)=y^{0}(x)\ \ \
\mbox{ in }\ \Omega,\end{array}\right.$
$\left\\{\begin{array}[]{l}\displaystyle-\varphi_{\varepsilon,t}-\Delta\varphi_{\varepsilon}=a\varphi_{\varepsilon}+c\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\
\displaystyle-\Delta\psi_{\varepsilon}=b\varphi_{\varepsilon}+d\psi_{\varepsilon}\
\ \ \mbox{ in }\ Q\\\ \displaystyle\varphi_{\varepsilon}=\psi_{\varepsilon}=0\
\ \ \;\mbox{ on }\ \Sigma\\\
\displaystyle\varphi_{\varepsilon}(x,T)=-\frac{1}{\varepsilon}y_{\varepsilon}(x,T)\
\;\;\;\mbox{ in }\;\Omega,\end{array}\right.$
$w_{\varepsilon}=\bigl{.}e^{-2K/(T-t)}\psi_{\varepsilon}\bigr{|}_{\mathcal{O}\times(0,T)}\,.$
We can argue as before and deduce that
$\begin{array}[]{l}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w_{\varepsilon}|^{2}dxdt\\\\[5.0pt]
\displaystyle=\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{-2K/(T-t)}|\psi_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2},\end{array}$ (3.12)
whence a weakly convergent sequence of control exists and, in the limit, we
get a null control for (1.10).
Notice that, by construction,
$\limsup_{t\to T^{-}}\|w(\cdot\,,t)\|_{L^{2}(\mathcal{O})}=0.$
This, together with the energy estimates
$\|z_{\varepsilon}(\cdot\,,t)\|_{H_{0}^{1}}\leq
C\left(\|y(\cdot\,,t)\|+\|w(\cdot\,,t)\|_{L^{2}(\mathcal{O})}\right),$
ensures the second part of (1.4). $\Box$
## 4 The null controllability of the semilinear systems
In this Section, we present the proofs of the main results in this paper,
namely Theorems 1.1 and 1.2. They will be obtained by combining the linear
controllability results in the previous Section and a standard fixed-point
argument.
Proof of Theorem 1.1: Let us first assume that, in (1.6), $F_{0}$ is $C^{1}$.
In view of (1.6) and (1.7), we observe that (1.1) can be written as follows:
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(y)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
with
$A_{0}(s)=\left\\{\begin{array}[]{ll}\displaystyle{F_{0}(s)\over s}&\ \text{
if }\ s\not=0,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
F^{\prime}_{0}(0)&\ \text{ otherwise. }\end{array}\right.$ (4.1)
For any $k\in L^{2}(Q)$, let us consider the linear system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (4.2)
In view of Theorem 3.1, there exists controls $v\in
L^{2}(\mathcal{O}\times(0,T))$ such that the associated states $(y,z)$ satisfy
(1.4) and (3.1), where the constant $C$ only depends on ${\Omega}$,
$\mathcal{O}$, $T$, $\|F_{0}\|_{C^{1}({\mathbb{R}})}$, $|b|$, $|c|$ and $|d|$.
Let us introduce the mapping $\Phi:L^{2}(Q)\mapsto 2^{L^{2}(Q)}$, as follows:
for any $k\in L^{2}(Q)$, we set by definition
$\Phi_{0}(k)=\\{\,v\in L^{2}(\mathcal{O}\times(0,T)):\text{ the solution
to~{}\eqref{tk1} satisfies \eqref{eq2n} and~{}\eqref{eq3}}\,\\}$
and
$\Phi(k)=\\{\,y\in L^{2}(Q):\text{ $(y,z)$ solves \eqref{tk1} for some
$v\in\Phi_{0}(k)$}\,\\}.$
Then $\Phi$ satisfies the hypotheses of Kakutani’s Fixed-Point Theorem.
Indeed, the following holds:
* •
For any $k\in L^{2}(Q)$, $\Phi(k)\subset L^{2}(Q)$ is a non-empty compact set.
Furthermore, there exists a fixed compact set $K\subset L^{2}(Q)$ such that
$\Phi(k)\subset K$ for all $k\in L^{2}(Q)$.
This is an obvious consequence of (3.1), the energy estimates
$\|y_{t}\|_{L^{2}(0,T;H^{-1}({\Omega}))}^{2}+\|y\|_{L^{2}(0,T;H_{0}^{1}({\Omega}))}^{2}\leq
C\left(\|v\|_{L^{2}(\mathcal{O}\times(0,T))}^{2}+\|y^{0}\|^{2}\right)$
(established in the Appendix) and the compactness of the embedding
$W\hookrightarrow L^{2}(Q)$, where
$W=\\{\,\xi\in L^{2}(0,T;H_{0}^{1}({\Omega})):\xi_{t}\in
L^{2}(0,T;H^{-1}({\Omega}))\,\\}.$
* •
$\Phi$ is sequentially upper semicontinuous on $L^{2}(Q)$, i.e. if $k_{n},k\in
L^{2}(Q)$ and $k_{n}\to k$ in $L^{2}(Q)$, then
$\displaystyle\limsup_{n\to\infty}\left(\sup_{y\in\Phi(k_{n})}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}y\,\xi\;dxdt\right)\leq\sup_{y\in\Phi(k_{n})}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}y\,\xi\;dxdt\quad\forall\xi\in
L^{2}(Q).$
Therefore, $\Phi$ possesses at least one fixed-point $y$.
Obviously, $y$ solves, together with some $z$, the semilinear system (1.1) for
some $v\in L^{2}(\mathcal{O}\times(0,T))$ and (1.4) holds.
Let us now assume that the function $F_{0}$ in (1.6) is only globally
Lipschitz-continuous. Suppose that
$|F_{0}(s_{1})-F_{0}(s_{2})|\leq L|s_{1}-s_{2}|\quad\forall
s_{1},s_{2}\in{\mathbb{R}}.$
Then, we can find $C^{1}$ functions $F_{01},F_{02},\dots$ with the following
properties:
1. 1.
$F_{0n}:{\mathbb{R}}\mapsto{\mathbb{R}}$ is $C^{1}$ and globally Lipschitz-
continuous., with Lpschitz constant $L$, i.e. Suppose that
$|F_{0n}(s_{1})-F_{0n}(s_{2})|\leq L|s_{1}-s_{2}|\quad\forall
s_{1},s_{2}\in{\mathbb{R}},\quad\forall n\geq 1.$
2. 2.
$F_{0n}\to F_{0}$ uniformly on each compact interval $I\subset{\mathbb{R}}$.
For each $n\geq 1$, let us consider the system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F_{0n}(y)y+bz+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=cy+dz\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (4.3)
Let $v_{n}$ be a null control for (4.3) satisfying
$\|v_{n}\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$
with $C$ independent of $n$. In view of the previous arguments and the
properties of $F_{0n}$, such a $v_{n}$ exists. It is clear that, at least for
a subsequence, one has
$v_{n}\to v\ \text{ weakly in }\ L^{2}(\mathcal{O}\times(0,T)),$
where $v$ is a null control for (1.1) again satisfying (1.4).
This ends the proof. $\Box$
Proof of Theorem 1.2: The proof is similar to the previous one, although a
little more intrincate.
Again, let us first assume that, in (1.6) and (1.8), the functions $F_{0}$ and
$f_{0}$ are $C^{1}$. Then, (1.3) can be written in the form
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(y)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=C_{0}(y)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
where $A_{0}$ is given by (4.1) and
$C_{0}(s)=\left\\{\begin{array}[]{ll}\displaystyle{f_{0}(s)\over s}&\ \text{
if }\ s\not=0,\\\ \vskip 3.0pt plus 1.0pt minus 1.0pt\cr\displaystyle
f^{\prime}_{0}(0)&\ \text{ otherwise. }\end{array}\right.$
For each $k\in L^{2}(Q)$, we can consider the linear system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=C_{0}(k)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega}.\end{array}\right.$ (4.4)
As in the proof of Theorem 1.1, we can prove that there exist controls $w\in
L^{2}(\mathcal{O}\times(0,T))$ such that the associated solutions to (4.4)
satisfy (1.4) and
$\|w\|_{L^{2}(\mathcal{O}\times(0,T))}\leq C\|y^{0}\|,$
for some fixed $C$. We can again introduce a multi-valued mapping
$\Psi:L^{2}(Q)\mapsto 2^{L^{2}(Q)}$ (similar to $\Phi$) and we can show that
the assumptions of Kakutani’s Theorem are satisfied by $\Psi$, etc.
An easy adaptation of the remaining results leads to the desired
controllability result. We omit the details, that can be checked easily.
$\Box$
## 5 An asymptotic controllability property
In this Section, we prove that, under appropriate conditions on $F$ and $f$
(in fact the same in Theorem 1.2), the semilinear parabolic system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon\,z_{t}-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega,\end{array}\right.$
(5.1)
is uniformly null-controllable as $\varepsilon\to 0$. We also prove the
convergence of the null controls to a null control for the similar parabolic-
elliptic system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$ (5.2)
To this end, we will first consider the linear system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=ay+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon z_{t}-\Delta
z=cy+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\;\;\;z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
(5.3)
and we will establish a uniform null controllability result.
More precisely, the following holds:
###### Theorem 5.1
Assume that (2.21) holds. Then, for any $\varepsilon>0$ and any
$y^{0},z^{0}\in L^{2}({\Omega})$, there exist controls $w_{\varepsilon}\in
L^{2}(\mathcal{O}\times(0,T))$ such that the corresponding solutions
$(y_{\varepsilon},z_{\varepsilon})$ to (5.3) satisfy
$y_{\varepsilon}(x,T)=0,\;\;z_{\varepsilon}(x,T)=0\;\;\;\mbox{in}\;\;{\Omega},$
with an estimate of the form
$\|w_{\varepsilon}\|_{L^{2}(\mathcal{O}\times(0,T))}\leq
C\left(\|y^{0}\|+\varepsilon\;\|z^{0}\|\right).$ (5.4)
where $C$ is independent of $\varepsilon$.
Sketch of the proof: Let us consider the adjoint system of (5.3), that is:
$\left\\{\begin{array}[]{l}-\varphi_{t}-\Delta\varphi=a\varphi+c\psi\;\;\mbox{in}\;\;Q,\\\\[5.0pt]
\displaystyle-\varepsilon\psi_{t}-\Delta\psi=b\varphi+d\psi\;\;\mbox{in}\;\;Q,\\\\[5.0pt]
\displaystyle\varphi=0,\;\psi=0\;\;\mbox{on}\;\;\Sigma,\\\\[5.0pt]
\displaystyle\varphi(x,T)=\varphi^{0}(x),\,\,\,\psi(x,T)=\psi^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$
with $\varphi^{0},\psi^{0}\in L^{2}({\Omega})$.
As in Section 2, we will use an abridged notation for the weighted integrals
concerning $\varphi$, $\psi$ and their derivatives: for any positive $\lambda$
and $s$, we set
$\begin{array}[]{l}\displaystyle
I(s,\lambda,\varphi)=\displaystyle\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}\\!\left(|\varphi_{t}|^{2}\\!+\\!|\Delta\varphi|^{2}\right)\\!+\\!\lambda^{2}s\phi|\nabla\varphi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]\\!\,dx\,dt,\\\\[9.0pt]
\displaystyle
I_{\varepsilon}(s,\lambda,\psi)=\displaystyle\varepsilon^{2}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}e^{-2s\alpha}(s\phi)^{-1}|\psi_{t}|^{2}\,dx\,dt\\\\[9.0pt]
\displaystyle\phantom{I_{\varepsilon}(s,\lambda,\psi)}+\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{Q}\\!\\!e^{-2s\alpha}\\!\left[(s\phi)^{-1}|\Delta\varphi|^{2}\\!+\\!\lambda^{2}s\phi|\nabla\psi|^{2}\\!+\\!\lambda^{4}(s\phi)^{3}|\varphi|^{2}\right]\\!\,dx\,dt\end{array}$
Then one has
$\displaystyle I(s,\lambda,\varphi)+I_{\varepsilon}(s,\lambda,\psi)\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}(s\phi)^{7}\lambda^{8}e^{-2s{\alpha}(x,t)}|\psi|^{2}\,dx\,dt,$
(5.5)
where $C$ is independent of $\varepsilon$.
The proof of (5.5) is very similar to the proof of Proposition 2.2 and will be
omitted. An almost immediate consequence is the following observability
inequality:
$\|\varphi(\cdot\,,0)\|^{2}+\varepsilon\;\|\psi(\cdot\,,0)\|^{2}\leq
C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2s\alpha}(s\phi)^{7}{\lambda}^{8}|\psi|^{2}\,dx\,dt.$
(5.6)
Now, arguing as in Section 3, it becomes clear that the uniform null
controllability property of (5.3) is implied by the observability estimate
(5.6). $\Box$
From this result, we get the following for the semilinear systems (5.1) and
(5.2):
###### Theorem 5.2
Under the assumptions of Theorem 1.2, for any $\varepsilon>0$ and any
$y^{0},z^{0}\in L^{2}(\Omega)$, there exist null controls $w_{\varepsilon}\in
L^{2}(\mathcal{O}\times(0,T))$ for (5.1). They can be chosen such that, at
least for a subsequence,
$w_{\varepsilon}\to w\;\;\;\mbox{ weakly in
}\;\;L^{2}(\mathcal{O}\times(0,T)),$
where $w$ is a null control for (5.2).
Sketch of the proof: For the proof of the first assertion, it suffices to
argue as in the proof of Theorem 1.2.
Indeed, for any $\varepsilon>0$ and any fixed $k\in L^{2}(Q)$, we can apply
Theorem 5.1 to the linearized system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=A_{0}(k)y+bz\;\;\mbox{in}\;\;Q,\\\ \displaystyle\varepsilon\,z_{t}-\Delta
z=C_{0}(k)y+dz+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x),\,\,z(x,0)=z^{0}(x)\;\;\mbox{in}\;\;\Omega.\end{array}\right.$
(5.7)
We deduce that, for any $\varepsilon>0$, (5.7) is null-controllable, with
controls $w_{\varepsilon}$ satisfying (5.4) (where $C$ is independent of
$\varepsilon$). Observe that, in fact, we can get a stronger estimate:
$\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,T)}e^{2K/(T-t)}|w_{\varepsilon}|^{2}dxdt\leq
C\|y^{0}\|^{2}.$
We can again introduce a multi-valued mapping $\Psi_{\varepsilon}$ (similar to
$\Psi$) and we can show that the assumptions of Kakutani’s Theorem are
satisfied by $\Psi_{\varepsilon}$. Therefore, there exist controls
$w_{\varepsilon}\in L^{2}(\mathcal{O}\times(0,T))$ satisfying (5.4) such that
the associated solutions to (5.1) satisfy
$y_{\varepsilon}(x,T)=0,\;\;z_{\varepsilon}(x,T)=0\;\;\;\mbox{in}\;\;{\Omega}.$
Let us multiply both sides of $(\ref{6epsn})_{1}$ (resp. $(\ref{6epsn})_{2}$)
by $\displaystyle y_{\varepsilon}$ (resp. $\displaystyle z_{\varepsilon}$) and
let us integrate in $Q$. We easily obtain the following for all $t$:
$\begin{array}[]{l}\displaystyle\|y_{\varepsilon}(\cdot\,,t)\|^{2}+\varepsilon\|z_{\varepsilon}(\cdot\,,t)\|^{2}+\displaystyle\int_{0}^{t}\left(\|y_{\varepsilon}(\cdot\,,s)\|_{H_{0}^{1}}^{2}+\|y_{\varepsilon}(\cdot\,,s)\|_{H_{0}^{1}}^{2}\right)\,ds\\\
\displaystyle\ \
\leq\|y^{0}\|^{2}+\varepsilon\|z^{0}\|^{2}+C\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,t)}|w_{\varepsilon}|^{2}\,dx\,ds+C\displaystyle\int_{0}^{t}\|y_{\varepsilon}(\cdot\,,s)\|^{2}\,ds.\end{array}$
Using Gronwall’s inequality and extracting appropriate subsequences, we deduce
that, at least for a subsequence, $w_{\varepsilon}$, $y_{\varepsilon}$ and
$z_{\varepsilon}$ respectively converge to $w$, $y$ and $z$, where $(y,z)$
solves (5.2) and satisfies (1.4). $\Box$
## 6 Appendix A: Some technical results
In this Appendix, for completeness, we give a theoretical result for the
semilinear system
$\left\\{\begin{array}[]{l}\displaystyle y_{t}-\Delta
y=F(y,z)+v1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle-\Delta
z=f(y,z)+w1_{\mathcal{O}}\;\;\mbox{in}\;\;Q,\\\ \displaystyle
y(x,t)=0,\;z(x,t)=0\;\;\mbox{on}\;\;\Sigma,\\\ \displaystyle
y(x,0)=y^{0}(x)\;\;\mbox{in}\;\;{\Omega},\end{array}\right.$ (6.1)
where $F$ and $f$ satisfy (1.6) and (1.8), respectively.
We have the following:
###### Theorem 6.1
For any given $y^{0}\in L^{2}({\Omega})$ and $v,w\in
L^{2}(\mathcal{O}\times(0,T))$, (6.1) possesses a unique solution $(y,z)$,
with
$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;D(-\Delta)).$ (6.2)
Sketch of the proof: We will apply the Faedo-Galerkin method, see J.-L. Lions
[10].
Let $\\{h_{j}\\}$ be a special basis of $H_{0}^{1}({\Omega})$, more precisely,
the basis formed by the eigenfuncions of the Dirichlet Laplacian in
${\Omega}$. Let us introduce the finite-dimensional Galerkin approximations as
follows: find $y_{N},z_{N}$, with $y_{N}(t),z_{N}(t)\in V_{N}$ for all $t$,
such that
$\displaystyle(y_{N}^{\prime}(t),h)+(\nabla y_{N}(t),\nabla
h)=(F(y_{N},z_{N}),h)+(v1_{\mathcal{O}},h)\quad\forall h\in V_{N},$ (6.3)
$\displaystyle(\nabla z_{N}(t),\nabla
k)=(f(y_{N},z_{N}),k)+(w1_{\mathcal{O}},k)\quad\forall k\in V_{N},$ (6.4)
$\displaystyle\begin{array}[]{l}y_{N}(0)=y_{0N}.\end{array}$ (6.6)
Here, $V_{N}=[h_{1},h_{2},\dots,h_{N}]$ is the subspace of
$H_{0}^{1}({\Omega})$ spanned by the first $N$ eigenfunctions $h_{j}$ and
$y_{0N}=P_{N}y_{0}$, where $P_{N}:L^{2}({\Omega})\mapsto V_{N}$ is the
orthogonal projector.
This Cauchy problem has at least one local solution on $[0,t_{N})$. That the
maximal solution is defined in the whole interval $[0,T]$ is a consequence of
the estimates given below.
Estimates I : Let us set $h=y_{N}(t)$ in (6.3) and $k=z_{N}(t)$ in (6.4).
After some computations we obtain the following for any $t$ and for all small
$\delta>0$:
$\begin{array}[]{l}\displaystyle{1\over
2}\|y_{N}(t)\|^{2}+\displaystyle\int_{0}^{t}\|y_{N}(s)\|_{H_{0}^{1}}^{2}\,ds+\left(1-{\mu+\delta\over\mu_{1}}\right)\displaystyle\int_{0}^{t}\|z_{N}(s)\|_{H_{0}^{1}}^{2}\,ds\\\
\displaystyle\ \ \leq{1\over
2}\|y^{0}\|^{2}+C_{\delta}\displaystyle\int\\!\\!\\!\\!\displaystyle\int_{\mathcal{O}\times(0,t)}\left(|v|^{2}+|w|^{2}\right)\,dx\,ds+C_{\delta}\displaystyle\int_{0}^{t}\|y_{N}(s)\|^{2}\,ds\end{array}$
(6.7)
and
$\begin{array}[]{l}\displaystyle\left(1-{\mu+\delta\over\mu_{1}}\right)\|z_{N}(t)\|_{H_{0}^{1}}\leq
C_{\delta}\left(\|y_{N}(t)\|+\|w\|_{L^{2}(\mathcal{O})}\right).\end{array}$
(6.8)
Estimates II Since the $h_{j}$ are the eigenfunctions of $-\Delta$ in
$H_{0}^{1}({\Omega})$, one has
$\|y^{\prime}_{N}(t)\|_{H^{-1}(\Omega)}\leq\|(\Delta
y_{N}+F(y_{N},z_{N})+v1_{\mathcal{O}})(t)\|_{H^{-1}(\Omega)}$ (6.9)
for all $t$. Therefore, we get the following for some $C>0$:
$\|y^{\prime}_{N}(t)\|_{H^{-1}(\Omega)}\leq
C\left(\|y_{N}(t)\|_{H_{0}^{1}}+\|(v1_{\mathcal{O}})(\cdot\,,t)\|+\|(w1_{\mathcal{O}})(\cdot\,,t)\|\right).$
(6.10)
From (6.7)–(6.10), it is standard to deduce that, at least for a subsequence,
the $y_{N}$ and $z_{N}$ converge to a solution to (6.1). This solution must
satisfy
$y\in L^{2}(0,T;H_{0}^{1}({\Omega})),\quad y_{t}\in
L^{2}(0,T;H^{-1}({\Omega})),\quad z\in L^{2}(0,T;H_{0}^{1}({\Omega})).$
Furthermore, from the usual elliptic estimates, we also obtain that $z\in
L^{2}(0,T;D(\Delta))$. This yields (6.2).
The uniqueness of the solution is also a standard consequence of the previous
estimates (written for $y:=y^{1}-y^{2}$ and $z:=z^{1}-z^{2}$ where
$(y^{1},z^{1})$ and $(y^{2},z^{2})$ are assumed to solve the system) and the
global Lipschitz-continuity of $F$ and $f$. $\Box$
## References
* [1] J.-M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007.
* [2] A. Doubova, E. Fernández-Cara, M. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optimiz., Vol. 41, No. 3 (2002), 798-819.
* [3] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Royal Soc. Edinburgh, 125A (1995), 31–61.
* [4] E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., Vol. 45. No. 4, (2006), 1395–1446.
* [5] E. Fernández-Cara and E. Zuazua, Null and approximate controllability of weakly blowing-up semilinear heat equations, Ann. Inst. Henri Poincaré, Analyse non linéaire 17, 5 (2000), 583–616.
* [6] A. Fursikov and O. Imanuvilov, Controllability of evolution equations, Lecture Notes, Vol. 34, Seoul National University, Korea, 1996.
* [7] A. Jüngel, Transport equations for semiconductors, Lecture Notes in Physics, Vol. 773, Springer, Berlin Heidelberg, 2009.
* [8] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations, Volume 1 and Volume 2, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2000.
* [9] O. Yu. Imanuvilov, Controllability of parabolic equations (Russian), Mat. Sbornik. Novaya Seriya, 186 (1995), 109–132.
* [10] J-L. Lions - Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris (1960).
* [11] E. Zuazua, Exact boundary controllability for the semilinear wave equation, Nonlinear Partial Differential Equations and their Applications, 10 (1989), 357–391.
* [12] E. Zuazua, Controllability and observability of partial differential equations: some results and open problems, Handbook of differential equations: evolutionary equations, Vol. III, 527–621, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.
|
arxiv-papers
| 2012-04-13T18:38:16 |
2024-09-04T02:49:29.689947
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Fern\\'andez-Cara, J. Limaco and S. B. de Menezes",
"submitter": "Silvano B. de MENEZES Dias",
"url": "https://arxiv.org/abs/1204.3072"
}
|
1204.3134
|
# Transport in Graphene: Ballistic or Diffusive?
Mario F. Borunda mario.borunda@okstate.edu Department of Physics, Oklahoma
State University, Stillwater, OK 74078, USA Department of Physics, Harvard
University, Cambridge, MA 02138, USA. H. Hennig Department of Physics,
Harvard University, Cambridge, MA 02138, USA. Eric J. Heller Department of
Physics, Harvard University, Cambridge, MA 02138, USA. Department of
Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA.
###### Abstract
We investigate the transport of electrons in disordered and pristine graphene
devices. Fano shot noise, a standard metric to assess the mechanism for
electronic transport in mesoscopic devices, has been shown to produce almost
the same magnitude ($\approx 1/3$) in ballistic and diffusive graphene devices
and is therefore of limited applicability. We consider a two-terminal geometry
where the graphene flake is contacted by narrow metallic leads. We propose
that the dependence of the conductance on the position of one of the leads, a
conductance profile, can give us insight into the charge flow, which can in
turn be used to analyze the transport mechanism. Moreover, we simulate
scanning probe microscopy (SPM) measurements for the same devices, which can
visualize the flow of charge inside the device, thus complementing the
transport calculations. From our simulations, we find that both the
conductance profile and SPM measurements are excellent tools to assess the
transport mechanism differentiating ballistic and diffusive graphene systems.
###### pacs:
72.80.Vp, 73.23.Ad, 72.10.Fk
## I Introduction
As predicted by early band theory studies,Wallace (1947) charge carriers in
graphene are chiral quasiparticles that have a linear Dirac-like dispersion
relation resulting in fascinating electronic CastroNeto_2009 (2009) and
transport DasSarma_2011 (2011) properties. As a result of this linear energy-
momentum relationship, the valence and conduction bands intersect at the Dirac
points located at the Brillouin zone corners. For neutral graphene, the
chemical potential lies exactly at the Dirac points. Use of a gate potential
can shift the chemical potential away from the Dirac point and thus tune the
charge density of the graphene device over a large range (as compared to other
electronic materials). The high degree of control over the charge density and
the linear dispersion relation are features that distinguish graphene from the
two-dimensional electron gas in semiconductor heterostructures and other
metallic conductors. Understanding the mechanism for charge carrier transport
in graphene in the low carrier density regime (the linear dispersion regime)
and in the presence of disorder has received considerable
attention.CastroNeto_2009 (2009); DasSarma_2011 (2011); Geim_2009 (2009);
Rozhkov_2011 (2011)
The wide variety of fabrication techniques Novoselov_2004 (2004); Berger_2006
(2006); Meyer_2007 (2007); Bolotin_2008 (2008); Du_2008 (2008); Dean_2010
(2010) has resulted in graphene devices that operate in different disorder
regimes. The strength of the disorder charge carriers encounter can be
quantified via the electron mobility. The standard fabrication method of
mechanical exfoliation and deposition of graphene flakes on SiO2 substrates
Novoselov_2004 (2004) can result in good quality devices with mobilities
approaching a few tens of thousands $cm^{2}V^{-1}s^{-1}$, values comparable to
those found in conventional semiconductor devices. Yet, the substrate provides
a source of impurities Adam_2007 (2007) and corrugations Katsnelson_2007
(2007); Gibertini_2010 (2010) for the electrons to scatter, limiting the mean
free path $l$ to about a hundred nanometers. Etching away the substrate is
routinely done to produce suspended graphene devices, removing some sources of
scattering. Etching followed by current annealing increases the carrier
mobility in suspended samples to values exceeding 200,000
$cm^{2}V^{-1}s^{-1}$, resulting in mean free paths comparable to the size of
the device.Bolotin_2008 (2008); Du_2008 (2008) Similar sample quality (80,000
$cm^{2}V^{-1}s^{-1}$ carrier mobility) is obtained in devices where the
substrate is single-crystal hexagonal boron nitride.Dean_2010 (2010)
It is apparent that replacing oxide based substrates by high-quality crystal
substrates or suspended graphene results in devices with improved mobility and
reduced electron scattering events.Bolotin_2008 (2008); Du_2008 (2008);
Dean_2010 (2010) For transport near or at the Dirac point, where the density
of states vanishes, the conductivity has a minimum value ($4e^{2}/\pi h$),
even for ideal pristine graphene.Tworzydlo_2006 (2006) Experimentally, the
phase coherent transport in graphene devices with short-and-wide geometries
has been characterized as ballistic Miao_2007 (2007); DiCarlo_2008 (2008);
Danneau_2008 (2008) indicating evanescent wave transport. In an ideal (no
impurities or defects) graphene device, the mean free path is longer than the
size of the device resulting in ballistic conductivity. Thus, transport in
graphene qualitatively changes from metallic diffusive in SiO2 substrates to
what has been characterized as ballistic in suspended systems and when using
crystalline boron nitride substrates.
Our paper is organized as follows. In Sec. II we discuss the usual transport
measures, emphasizing the null results obtained from the Fano shot noise, and
present two alternative methods that can also estimate the transport
mechanism. In Sec. III we present the computational method used to model
transport in two-terminal graphene devices. In Sec. IV we report results of
extracting the transport mechanisms from modeled transport measurements. We
present a summary and conclusion in Sec. V.
## II Transport measures
The possibility of several transport mechanisms makes it essential to develop
measures that clearly distinguish and classify the different transport
regimes. In principle, Fano shot noise is the metric of choice to assess the
mechanism for electronic transport in mesoscopic devices. However, the Fano
shot noise is of limited applicability, with the same value for the Fano
factor found in disordered or pristine devices.
### II.1 Shot noise and Fano factor
Shot noise measurements can determine the statistics relevant for transport in
mesoscopic conductors. Blanter_2000 (2000) Shot noise is a consequence of
charge quantization and can be assessed by the Fano factor defined as the
ratio of noise power and mean current,
$\mathcal{F}=\frac{\sum_{n=1}^{N}T_{n}(1-T_{n})}{\sum_{n=1}^{N}T_{n}}.$ (1)
From the above definition, the Fano factor should be zero for conventional
ballistic systems, i.e. if perfect transmission is present. It has also been
established that for Poisson processes $\mathcal{F}=1$ and for diffusive
metallic conductors $\mathcal{F}=1/3$.Beenakker_1992 (1992)
However, the Fano factor calculated for pristine graphene,Tworzydlo_2006
(2006); Schomerus_2007 (2007); Barraza_2012 (2012) disordered
systems,Louis_2007 (2007); San-Jose_2007 (2007) samples with substrate
roughness,Lewenkopf_2008 (2008) and for transport along an n-p junction
Cheianov_2006 (2006) all have similar magnitude ($\approx 1/3$) to that found
in diffusive metallic conductors, making the Fano shot noise a cumbersome
measure of the transport mechanism in graphene devices.DasSarma_2011 (2011)
Calculations in pristine graphene predict that the Fano factor at the Dirac
point is identically 1/3 when (1) a wide device (where the aspect ratio W/L is
above 3) is contacted by doped graphitic leads, Tworzydlo_2006 (2006) (2) when
the leads are quantum wires creating an effective-contact model simulating the
metallic lead/graphene interface, Schomerus_2007 (2007) and (3) for graphene
junctions with realistic metal contacts. Barraza_2012 (2012) Away from the
Dirac point, the Fano factor also indicates diffusive metallic transport.
Schomerus_2007 (2007); Barraza_2012 (2012) A slightly smaller yet universal
value ($\mathcal{F}=0.295$) was found numerically for disordered systems.
Louis_2007 (2007); San-Jose_2007 (2007) When disorder originates mainly from
roughness in the substrate, the Fano factor lies slightly above the 1/3 value
and has been shown numerically to increase slightly with disorder.
Lewenkopf_2008 (2008) Transport along an n-p (electron rich/hole rich)
junction is selective of those quasiparticles that approach the n-p interface
almost perpendicularly and results in a Fano factor of
$\mathcal{F}=1-\sqrt{1/2}$. Cheianov_2006 (2006) Again, in all of these
different situations, the Fano factor measurement results in 1/3 or a value
numerically close to this number.
Experiments have found similar results. The Fano factor measured in clean
devices Danneau_2008 (2008) is close to the value of 1/3 found analytically
for ballistic samples. Tworzydlo_2006 (2006) Measurements in disordered
devices DiCarlo_2008 (2008) follow the numerical trends of Ref.
Lewenkopf_2008, 2008. These similar shot noise values warrant developing an
alternative quantitative understanding of the carrier dynamics crucial for
testing the transport regime of graphene devices.
### II.2 Conductance profiles
Our quest for a better measure to assess transport mechanisms begins with a
simple question: How does the conductance change with respect to the vertical
displacement of one of the leads? By calculating the conductance between two
metallic contacts, we model transport in two-terminal devices. We extract the
conductance profile from the dependence of the conductance on the position of
one of the leads, and use this to distinguish the mechanism of transport. In
particular, we study the transport properties of clean and disordered graphene
devices contacted by narrow metallic leads where one of the two leads can be
moved along the edge of the device. Moreover, we simulate scanning probe
microscopy (SPM) Topinka_2000 (2000) measurements for the same devices, which
sheds light on the charge flow _inside_ the device. Topinka_2001 (2001)
Diffusive transport in disordered systems is based on the charge carriers
scattering multiple times off impurities or boundaries as they traverse the
system. This mechanism can be described classically by a random walk. In
ballistic transport, the charge carriers traverse the system with minimal
scattering. Ballistic transport is expected in an ideal graphene strip, given
that the crystal lattice has no defects and no impurities are present. Yet for
ideal graphene, the dynamics of the electrons produce the same shot noise as
that found in classical diffusion. Tworzydlo_2006 (2006) Determining the
transport mechanism in graphene devices is important given that transport
experiments are possible in the quasi-ballistic limit, that is, where the mean
free path is of the order of the size of the system.
Here, we argue that the transition from diffusive to ballistic transport in
graphene, along with the limiting cases, can be quantified by alternative
methods based on measuring the flow of charge in the system. First, the
conductance profile, that is, the conductance as a function of the vertical
displacement of one of the leads, can give a measure of the flow inside the
device. For the case of ballistic transport, scattering does not impede the
flow of charge carriers and the conductance profiles are well described by a
linear (triangular) fit due to the convolution of the square windows created
by the two leads. In the case of diffusive transport, the charge carriers
experience several scattering events and the transport is then well described
by Brownian motion with a drift. Thus, charge flow will be a Gaussian function
characteristic of diffusion; for this reason, the conductance profile can be
fitted to a Gaussian function.
## III Computational method
We use the nonequilibrium Green’s function (NEGF) formalism Haug_2008 (2008)
to calculate the transport properties of a normal-conductor/graphene/normal-
conductor junction, given that in experiments the electronic contacts are
usually metallic. The metallic lead/graphene interface we have investigated
follows that of Robinson and Schomerus. Robinson_2007 (2007) A schematic of
the geometry of the devices investigated is shown in Fig. 1, where rectangular
(width W and length L) graphene flakes have armchair boundaries along their
longitudinal direction and narrow metallic leads connected along the zigzag
boundary. The two metallic terminals of width W${}_{\mbox{L}}$ are modeled by
semi-infinite square lattice regions that have a quadratic dispersion
relation.
Figure 1: (Color online) Schematic setup for the transport calculation. A
rectangular graphene flake, depicted by red and blue atomic sites arranged in
a honeycomb lattice, is contacted between two narrow metallic contacts,
depicted by the green atomic sites arranged in a square lattice, forming a
normal-conductor/graphene/normal-conductor junction. This schematic is not to
scale; simulated devices were considerably bigger with $\sim 700$ carbon atoms
along the zigzag edge.
The tight-binding model Hamiltonian of the device is given by
$H=\sum_{i}\epsilon_{i}c^{\dagger}_{i}c_{i}-\sum_{\langle
i,j\rangle}\gamma_{ij}c^{\dagger}_{i}c_{j},$ (2)
where $\epsilon_{i}$ is the on-site potential of the lattice site $i$,
$c_{i}(c^{\dagger}_{i})$ is the annihilation (creation) operator acting on
site $i$, the second sum is over nearest neighbors $\langle i,j\rangle$, and
$\gamma_{ij}$ denotes the hopping matrix elements. The on-site potential
changes due to contributions from impurities and the gate voltage applied to
the device, $\epsilon_{i}=\epsilon_{imp}+\epsilon_{gate}$. The graphene
section consists of atomic sites placed in the hexagonal lattice with lattice
constant $a$. As outlined in Ref. Robinson_2007, 2007, metallic contacts are
modeled as a region with a square lattice arrangement with lattice constant
$a_{L}=\sqrt{3}a$, matching the A(B)-atom in the zigzag interface at the
left(right) of the graphene flake.
Disorder is introduced in two different ways. To generate edge-disordered
samples, we randomly remove 30% of carbon atoms from sites located in the
three outer atomic layers of the device. Louis_2007 (2007) This type of
disordered edge without disorder puddles in the bulk of the system is a
reasonable model for suspended graphene devices.
Bulk disordered in graphene devices is linked to the presence of charged
impurities in the substrate. Rycerz_2007 To generate such disorder
potentials, $N_{imp}$ lattice sites are selected at random from the total
number of atomic sites ($N_{tot}$) in the device. The position $R_{i}$ of each
of the $N_{imp}$ lattice sites has an on-site potential amplitude $V_{i}$
chosen randomly from the interval $(-\delta,\delta)$ and smoothed out over a
range $\xi$ by convolution with a Gaussian,
$\epsilon_{imp}(R_{i})=\sum_{i=1}^{N_{imp}}V_{i}\exp\left(-\frac{\left|r-R_{i}\right|^{2}}{2\xi^{2}}\right).$
(3)
The range of the convolution is important given that in the extreme case of
$\xi<<a$ the atomic scale disorder potential would break the A-B symmetry
caused by having two atoms in the unit cell. For our calculations we assume
that $\xi=2a$ resulting in a short-range potential (when compared to other
length scales in the system) that varies smoothly on the atomic scale and
suppresses the effect of intervalley scattering. The parameters $N_{imp}$ and
$\delta$ determine the mean free path $l$ of the disordered system. Using the
Born approximation, the mean free path can be quantified as Rycerz_2007 ;
Suzuura_2002 (2002)
$l=\frac{4}{k_{F}K_{0}},$ (4)
where $k_{F}$ is the Fermi wave vector and $K_{0}$ is the dimensionless
correlator given by
$K_{0}=\frac{LW}{(\hbar
vN_{tot})^{2}}\sum_{i=1}^{N_{tot}}\sum_{j=1}^{N_{tot}}\langle\epsilon_{imp}(R_{i})\epsilon_{imp}(R_{j})\rangle.$
(5)
For the type of bulk disordered systems considered here ($\xi<<L,W$), Rycersz
et al. Rycerz_2007 have found that
$K_{0}=\frac{\sqrt{3}}{9}\frac{\delta^{2}}{\gamma^{2}}\frac{N_{imp}}{N_{tot}}\kappa,$
(6)
where
$\kappa=\frac{1}{N_{imp}}\sum_{i=1}^{N_{imp}}\sum_{j=1}^{N_{tot}}\exp\left(-\frac{\left|r-R_{i}\right|^{2}}{2\xi^{2}}\right).$
(7)
The NEGF formalism Haug_2008 (2008); Nicolic_2010 is a sophisticated
framework for obtaining the transmission and other quantities in realistic
devices. The retarded Green’s function in the atomic orbital basis is given by
$G^{r}(E)=\left[E-H-\Sigma(E)\right]^{-1},$ (8)
where the non-Hermitian self-energy matrix
$\Sigma(E)=\Sigma_{L}(E)+\Sigma_{R}(E),$ (9)
introduces the effect of attaching leads to the left and right ends of the
device. The energy $E$ tunes the Fermi level from the Dirac point ($E=0$) to
any charge density induced by the gate voltage in the device. The self-
energies determine the escape rates of electrons from the device into the
semi-infinite ideal leads. Using the Landauer formula Landauer_1957 , it is
possible to obtain the transmission function
$T(E,V_{ds})=Tr\left[\Gamma_{R}(E,V_{ds})G^{r}\Gamma_{L}(E,V_{ds})G^{a}\right],$
(10)
where $G^{a}(E)=(G^{r}(E))^{\dagger}$ is the advanced Green’s function. The
matrices
$\Gamma_{p}(E,V_{ds})=i\left[\Sigma_{p}\left(E-\frac{eV_{ds}}{2}\right)-\Sigma^{\dagger}_{p}\left(E+\frac{eV_{ds}}{2}\right)\right]$
(11)
introduce a level broadening due to the coupling of the leads and a source-to-
drain voltage given by $V_{ds}$. The Fano factor [Eq. (1)] for a particular
energy and $V_{ds}$ is calculated using the following expression,
$F=1-\frac{Tr\left[\Gamma_{R}G^{r}\Gamma_{L}G^{a}\Gamma_{R}G^{r}\Gamma_{L}G^{a}\right]}{T}.$
(12)
Finally, the current in two-terminal devices can be obtained from the Landauer
formula
$I(V_{sd})=\frac{2e}{h}\int_{-\infty}^{\infty}dE~{}T(E)\left[f(E-\mu_{L})-f(E-\mu_{R})\right]$
(13)
where the energy window is defined from the difference of the Fermi functions
of the macroscopic reservoirs where the leads terminate. In our calculations
we assume the linear response limit ($V_{ds}\rightarrow 0$), where the
relationship between conductance $G$ and current is given by $I=GV_{ds}$.
## IV Results and Discussion
### IV.1 Fano shot noise
We investigate rectangular graphene devices with a two-terminal geometry where
the source and drain leads are assumed to be perfect ballistic and metallic
conductors. This geometry has been investigated previously in the context of
quantum scars in graphene Huang_2009 (2009) and for the geometry-dependent
conductance fluctuations in graphene quantum dots. Huang_2011 (2011) For all
results presented here and as illustrated in Fig. 1, the edges of the graphene
along the transport direction are in the armchair configuration.
We use Eq. 12 to calculate the Fano factor for pristine, edge disordered, and
bulk disordered graphene devices. As presented in Fig. 2, our results are
similar to those found in the literature.Tworzydlo_2006 (2006); Schomerus_2007
(2007); Barraza_2012 (2012); Louis_2007 (2007); San-Jose_2007 (2007);
Lewenkopf_2008 (2008) The Fano factors found are near the theoretical value
$\mathcal{F}=1/3$ that applies to pristine graphene as $W/L\rightarrow\infty$
near the Dirac point limit. As Towrzydlo et. al. reported,Tworzydlo_2006
(2006) the $\mathcal{F}=1/3$ value is a theoretical maximum for armchair edge
graphene devices, increasing the charge density reduces the value of the Fano
factor.
For pristine devices, the Fano factor does show a trend as a function of the
system size. The longer device calculated (83 nm) presented the lowest value
of $\mathcal{F}$ while the largest value corresponds to the 44 nm graphene
sample. Likewise, the Fano factor for the graphene devices with disordered
edges do not exhibit a trend in the sizes studied. However, the fluctuations
in $\mathcal{F}$ from the mean value, illustrated with the standard deviation
(error bars), show that as the devices get longer, the deviation from the mean
increases. In contrast, for diffusive graphene, we find that the value of the
Fano factor does show a monotonic dependence as a function of the $W/L$ ratio,
approaching the $\mathcal{F}=1/3$ value in the larger devices. Yet, given that
the shot noise metric is not significantly different from the value found in
classical diffusive transport, even in the narrow-leads geometry, the Fano
factor is an inadequate metric for predicting the transport mechanism.
Figure 2: (Color online) Fano factor at a Fermi energy $E_{F}=0.5\gamma$
(with hopping matrix element $\gamma$). The limit $\mathcal{F}=1/3$ as
$W/L\rightarrow\infty$ is shown as a dotted line. The data points of the
disordered systems have error bars showing the variance of the value over
several realizations. Each of the data points corresponds to different system
lengths (L = 11, 23, 44, and 85 nm) while the widths of the graphene device
and of the leads are kept constant at W = 74 nm and W${}_{\mbox{L}}$ = 10 nm,
respectively. The mean free path is $l=27$ nm in the system with bulk
disorder.
### IV.2 Transport and the profile of the conductance
In order to visualize the charge flow at the edge of the sample, we calculate
the profile of the conductance $G(\Delta y)$, that is, the conductance as a
function of the position of the drain lead with respect to the source lead
that remains fixed. In Fig. 3, we present the conductivity as a function of
the displacement of the drain lead $\Delta y$ for several system lengths
(L=11, 21, 43 nm) and for the three models considered here. Both the width of
the device and the width of the leads are kept constant at W = 74 nm and
W${}_{\mbox{L}}$ = 10 nm, respectively. Thus, the schematic shown in Fig. 1 is
not representative of the size of the devices considered in this work as our
calculation are considerably larger with about 700 atomic sites along the
zigzag edge (width of system). Numerical calculations of these system sizes
require the use of efficient recursive Green’s function methods.
Figure 3: (Color online) Profile of the conductance for pristine, edge
disordered, and bulk disordered systems. The conductance versus displacement
of one of the leads from the center of the device $G(\Delta y)$ is calculated
at a Fermi energy $E_{F}=0.5\gamma$ (with hopping matrix element $\gamma$).
Each of the curves corresponds to different system lengths (L = 11, 21, and 43
nm) while the width of the graphene device and of the leads are kept constant
at W = 74 nm and W${}_{\mbox{L}}$ = 10 nm, respectively. (a) Clean pristine
system. (b) System with edge disorder. (c) System with bulk disorder where the
mean free path is $l=27$ nm.
While the displacement of one of the leads from the center of the device is a
theoretical construct that can be easily investigated numerically, the
experimental implementation is not trivial. Our proposal of “moving the leads”
could be realized by fabricating several samples where one of the leads is
attached at a different location in each device. Another possibility is to
replace one of the metallic leads by the tip of a scanning tunneling
microscope. When the tip is brought close enough to the edge, the tunneling
current could be measured as a function of the position of the scanning
tunneling microscope.
Figure 3(a) corresponds to pristine graphene devices, Fig. 3(b) corresponds to
realizations with edge disorder, and Fig. 3(c) to systems where the presence
of bulk disorder would form electron-rich and hole-rich puddles Martin_2008
(2008) and the mean free path is $l=27$ nm, as estimated from the Born
approximation. Figure 3(a) shows that the maximum conductance is associated
with the two leads being collinear ($\Delta y\approx 0$). For a short device
the maximum conductance occurs at $\Delta y=0$ and as the length of the system
in increased there are local maxima near $\Delta y=0$. Increasing the length
of the device reduces the conductance peak due to the increased number of
reflections at the boundary of the sample. Yet, we find an envelope of maximum
conductance when the vertical positions of the two leads overlap with each
other. Similar features are seen in the conductance profiles for the
disordered samples. We find that for all three systems the width of the
central peak is approximately $2W_{L}$, corroborating the idea that the peak
in the displacement conductance is due to the overlap of the two leads.
Although our interest is to extract the transport mechanisms from the
conductance profiles, there are certain features visible in the conductance
curves of Fig. 3 that are worth explaining such as the fluctuations near the
edges of the conductance curves. We carefully checked that these fluctuations
are not only present for individual conductance curves and remain after
averaging over an energy window. Thus, the fluctuations are consistent with
universal conductance fluctuations (UCFs), Lee_1985 (1985); Altshuler_1985
(1985) which we therefore review.
Figure 4: (Color online) The transport mechanism is classified according to
the shape of the central peak (of width $2W_{L}$ centered around $\Delta y$=0)
of the conductance profile curves. $G(\Delta y)$ for the pristine system (a,b)
is better fit by a triangular fit indicative of ballistic transport. In
systems with edge disorder (c,d) and bulk disorder (e,f), the central peak is
well described by a Gaussian envelope pointing to diffusive transport. The
dashed lines indicate the region where the two leads have an overlap.
Parameters for all calculations are $E_{F}=0.499\gamma$, $L=11\,$nm, W = 74 nm
and W${}_{\mbox{L}}$ = 10 nm.
In a disordered mesoscopic conductor where the system is of comparable size or
smaller than the phase coherence length of the charge carriers but large
compared to the average impurity spacing, the transmission of carriers is
affected by interference of many different paths through the system. As these
paths are typically long compared to the wavelength of the charge carriers,
the accumulated phase along the paths changes randomly in response to
variation in an external parameter (e.g., the magnetic field or gate voltage).
This results in a random interference pattern and reproducible fluctuations in
the conductance of a universal magnitude on the order of $2e^{2}/h$. Lee_1985
(1985); Altshuler_1985 (1985) UCFs can also be created by the displacement of
a single scatterer. Feng_1986 (1986); Altshuler_1985b (1985); Borunda_2011
(2011) But how can UCFs occur in a ballistic device? The role of disorder in
providing a distribution of random phases can as well be taken by chaos. Thus,
ballistic mesoscopic cavities like quantum dots in high mobility two-
dimensional electron gases that form chaotic billiards show the same universal
fluctuations. Marcus et al. (1992); Jalabert et al. (1990); Baranger et al.
(1993) Remarkably, experiments in graphene quantum dots found strong
indications of chaos in billiard systems. Ponomarenko et al. (2008)
In Fig. 4 we presents our conductance profile analysis for the 11 nm devices.
A striking feature of these conductance profiles is the apparent fit of the
peaks in the displacement conductance curves to either triangular or Gaussian
functions. The conductance profile for a pristine device of length $L=11\,$nm
is shown in Fig. 4(a) and the peak is presented in Fig. 4(b). The envelope of
the peak in the conductance is well described by a triangular fit (red line).
In the case of conventional ballistic transport, the classical expectation for
the charge flow at the edge is a triangular shape of width $2W_{L}$. This
triangular shape is the result of the convolution of two square windows, where
the window sizes are given by the lead width $W_{L}$. Gaussian fits for the
conductance profile (not shown) differ significantly particularly near the
cusp of the curve.
Diffusive transport is based on the multiple scattering paths taken by charge
carriers as they transverse a device. In Fig. 4(c) we present the conductance
profile for an edge disordered system of length $L=11\,$nm and a close-up of
the peak with a fit shown in Fig. 4(d). In this case, and in contrast with the
wide and ballistic device, the data for the edge-disordered device is best
fitted by a Gaussian curve, where the envelope of the conductance profile is
the result of the spatial overlap of the leads. Similarly, a Gaussian curve
describes well the conductance profile of a diffusive system, as seen in Figs.
4(e) and 4(f). We note that Barthelemy et al. Barthelemy_2008 (2008) used a
similar approach to distinguish normal diffusion (reflected by a Gaussian
profile) from the anomalous transport associated with Lévy transport.
Figure 5: (Color online) Charge flow in pristine graphene obtained from the
SPM conductance map calculations. The maps of conductance as a function of SPM
tip position, $G(x,y)$, show the charge flow in the graphene device. Each
snapshot presents the conductance map for different position of the right
lead, starting from left at the bottom portion of the device, towards the
right where the right lead is located at the same height of the left lead. We
can see that the majority of the flow is over a path connecting the left and
right lead. When the two leads are collinear, there are several bands of
enhanced conductance due to the interference caused by paths that scatter from
the tip and those that go from one lead straight to the other lead. The Fermi
energy is set to $E_{F}=0.5\gamma$.
While we have been able to account for the features of the envelope of the
central peak of width $2W_{L}$ in a quasi-classical way, the conductance
profile curves exhibit several quantum interference effects that need a
different interpretation. Along with UCFs, Fabry-Pérot conductance
interferences show up as regularly spaced fluctuations seen in the conductance
profiles of the pristine systems.
### IV.3 Charge flow
Scanning probe microscopy (SPM) Topinka_2000 (2000) has recently been used to
image mesoscopic transport effects such as universal conductance fluctuations
Berezovsky et al. (2010) and weak localization Berezovsky and Westervelt
(2010) in graphene devices. SPM has also been used to image current transport
in a graphene quantum dot connected to leads through two small constrictions,
imaging conductance resonances of the quantum dot and observing localized
states. Schnez_2010 (2010) The tip of the cryogenic SPM capacitively couples
to the graphene device inducing a movable scatterer. Conductance maps created
via SPM have been proposed as a way to probe the chiral nature of the charge
carriers in graphene. Braun_2008 (2008) The work carried out by Berezovsky et
al. Berezovsky et al. (2010); Berezovsky and Westervelt (2010) and Schnez et
al. Schnez_2010 (2010) probed the regime of coherent diffusive transport in
graphene. In light of the above, it is interesting to consider the possibility
of using SPM to gain insight into the transport mechanism by imaging the flow
of charge carriers – particularly, the ballistic to diffusive crossover regime
in graphene devices.
We simulate the effect of the capacitively-coupled tip of the SPM on the
graphene device as a point charge $q$ located a height $a$ above the
substrate. The charge induces a local charge density perturbation given by
$n(\rho)=\frac{-qa}{2\pi(\rho^{2}+a^{2})^{3/2}}$ (14)
where $\rho$ is the in-plane radial coordinate away from the position of the
point charge. The charge $q$ is chosen to yield an rms charge density
$n\approx 4\times 10^{11}~{}e~{}\mbox{cm}^{-2}$, in agreement with the
observed charge puddles in graphene. The tip height $a$ controls the width of
the induced charge density; we have chosen $a=10$ nm, a routinely used tip-to-
substrate distance in SPM experiments which results in a half-width for the
induced density puddle of about 20 nm.
During SPM measurements, a transport measurement is carried out while the tip
of the microscope scans the over the device. The resulting conductance maps
are generated by calculating the conductance as a function of SPM tip position
$G(x,y)$ and rastering the tip position in a plane above the device. The
effect of the SPM tip is included by adding to the on-site potential of each
of the lattice sites the energy due to the charge density perturbation created
by the tip. The total on-site energy is obtained from the contributions of the
perturbation from the tip, the effect from the gate voltage, and the
impurities present,
$\epsilon_{i}=\epsilon_{imp}+\epsilon_{gate}+\epsilon_{tip}$.
In the previous section we extracted conductance profiles that allowed the
classification of transport as ballistic or diffusive. To explain these
observations and to obtain an intuitive picture of the transport mechanisms,
we explore charge flow as a function of the position of one of the leads. As
in the conductance profile calculations, the injection lead remains fixed in
the left edge of the sample. The two terminals are identical. As seen in the
panels of Fig. 5, the SPM conductance maps exhibit a significant change in
conductance when the charge flow between the leads is obstructed or enhanced
by the presence of the perturbation induced by the scanning probe.
The conductance map shown in Fig. 5(a) was obtained when the right lead is
positioned near the bottom edge of the device. As the scanning probe rasters
along the top half of the device, the change in conductance is minimal as
reflected by the mostly uniform conductance in that region of the image.
However, as the tip scans the bottom half of the device, the conductance map
shows a region of lower conductance that connects the left lead to the right
lead, as indicated by the large blue feature across the bottom of the image.
In this case, the trans-conductance is reduced when the local perturbation
induced by the tip of the SPM is over a region of considerable charge flow. In
Fig. 5(b), the right lead contacts the device at a slightly higher position.
Consequently, the blue feature corresponding to the region of charge flow is
now at a higher position, “following” the position of the right lead.
Furthermore, there are two regions of higher conductance in the image,
corresponding to an increase of the conductance due to the presence of the
local perturbation. The higher conductance (red) bands are a consequence of
the redirection of the charge flow that without the tip being present would
not have contributed to the conductance. Thus, the tip redirects charge into
the drain lead that otherwise would not have exited the device.
Similarly, placing the right lead closer to the central region of the device
(Fig. 5(c)), we find that the position of conductance features are correlated
with the path starting at the left lead and ending in the right lead. In all
images, the effect of narrow lead constrictions is seen from the size of the
band of near constant conductance. Surprisingly, the feature does not become
significantly wider as the charge flows away from the source lead. All of the
panels present conductance fluctuations of order $\sim\pm 2e^{2}/h$.
The conductance map presented in Fig. 5(d) corresponds to a device where the
leads are collinear ($\Delta y=0$). Here, the conductance is higher than in
the ones to the left. Along with the central band of conductance drop, Fig.
5(d) also shows regions of constructive interference along bands parallel to
the transport. The limit to the size of the features present in the
conductance maps is proportional to the Fermi wavelength of the system
Topinka_2000 (2000); Topinka_2001 (2001), which for graphene is inversely
proportional to the Fermi energy given by
$\lambda_{F}=2\pi\frac{v_{F}}{E_{F}}.$ (15)
Figure 6: (Color online) Conductance maps for two Fermi energies, (left)
$E_{F}=0.1\gamma$ and (right) $E_{F}=0.9\gamma$, corresponding to Fermi
wavelengths of 13 nm and 1.5 nm, respectively.
The Fermi wavelength in the system presented in Fig. 5 is $\lambda_{F}=2.7$
nm. Figure 6 presents the simulated conductance maps for the same
configuration of the leads for $E_{F}=0.1~{}\gamma$ and $E_{F}=0.9~{}\gamma$,
corresponding to Fermi wavelengths of 13 nm and 1.5 nm. The resolution
resolved in the images, i.e., the size of the pixels in each of the maps, is
also 1.5 nm.
Figure 7: (Color online) SPM conductance maps simulations for disordered
systems. (Left) Graphene sample with disordered edges and (middle, right)
devices with bulk disorder. While the interference pattern survives in devices
with edge disorder, it is not present in systems with bulk disorder. At a mean
free path of $l=108$ nm, larger than the system size, $L=65$ nm, the mobility
of the disordered devices presented in the middle panel is comparable to that
of devices in crystalline substrates. The simulation presented in (c) is for a
system with $l=24$ nm. Other parameters are as in Fig. 5.
Previous analytical results of Braun et al. Braun_2008 (2008) consider
ballistic trajectories between two constrictions in the presence of the tip
scattering potential. To first order, as treated in Ref. Braun_2008, 2008,
there are only two trajectories that interfere with each other: the source-to-
drain trajectory and the source-tip potential-drain trajectory, revealing an
interference pattern in the conductance maps. In particular, the first order
calculation predicts that the interference pattern depends on the Fermi
wavelength of the system. Comparing the conductance maps for three different
energies (Fig. 5(d) and those in Fig. 6), the interference pattern are very
similar. Increasing the Fermi energy of the system does not reveal the Fermi
wavelength of the particles but rather results in better resolution of the
interference patterns. In contrast to analytical results, Braun_2008 (2008)
our numerical calculation treats all possible trajectories including
reflections from the edge of the device and multiple scattering events. We
believe that the pattern found is due to the redirection of the charge
carriers as the tip scans the device and does not dependent on the wavelength
but rather on the geometry of the system and the interference of several
trajectories the charge carriers can take as they traverse the device.
In what follows we study the effect of edge and bulk disorder on the simulated
conductance maps. In Fig. 7, we present conductance maps for graphene devices
with edge disorder (left panel) and bulk disorder (middle and right panels).
As seen in the left panel, the interference pattern is present in edge-
disordered graphene devices. The effect of disorder at the edges of the device
induces random scattering, modifying previously interfering trajectories and
thus reducing the strength of the pattern. This is more noticeable near the
edge of the device. Still, the result for edge-disordered systems is a
comparable pattern found for pristine graphene (Fig. 5(d)).
Our calculations show an important difference for bulk-disordered systems.
Surprisingly, the SPM simulations indicate diffusive rather than ballistic
transport (in the mesoscopic systems considered) even on the regime where the
mean free path (estimated from the Born approximation) is longer than the
system size. This result does not question the validity of the Born
approximation; rather, it shows the sensitivity of the SPM method for the
geometry studied. Even when the size of the system is smaller than $l$, for
this geometry, the oath traveled by the charge carrier from initially entering
the device to exiting at the drain lead can be significantly longer. Our
calculations take into account all possible trajectories that the charge
carriers can take, including the possibility of several reflections from the
leads and multiple scattering events from the walls of the device. The
presence of random disorder due to charged impurities in the substrate results
in conductance maps with no interference patterns, (Fig. 7, middle and right
panels). The middle panel of Fig. 7 is the conductance map for a system where
the mean free path ($l=107.5$ nm) is larger than the size of the device
($L=65$ nm). This value of $l$ is attainable in graphene devices with a
crystalline BN-substrate, resulting in relatively high mobility Dean_2010
(2010). The mean free path of the system pictured in the right panel is $l=24$
nm, comparable to high quality graphene devices in a silicon oxide substrate.
Ballistic transport is expected when the mean free path is larger than the
size of the system. In contrast, in graphene devices with weak bulk disorder
such that $l>L$, we find diffusive transport signatures. This indicates that
for the system and sizes considered, multiple reflections of the Dirac
electrons from the edges dominate. This mesoscopic effect needs to be
carefully considered in both theoretical and experimental studies on
mesoscopic graphene flakes. As a result, SPM imaging could be used to explore
the ballistic to diffusive transition in graphene devices. The visible
interference pattern in the conductance maps would be robust against the
presence of edge disorder, inevitable in most experimental realizations, but
is sensitive to the presence of disorder due to bulk disorder in the system.
Finally, we would like to comment on possible experimental realizations of SPM
measurements in graphene. Several SPM measurements have been carried out for
graphene devices on a SiO2 substrate Berezovsky et al. (2010); Berezovsky and
Westervelt (2010); Schnez_2010 (2010) and can be readily carried out in
suspended graphene, where the ballistic transport regime can be reached, in
graphene membranes, Wang_2012 (2012) and in devices with crystalline
substrate, where the ballistic to diffusive cross-over should be observable.
## V Conclusion
We have proposed a method to study the mechanisms of electronic transport
based on displacement conductance, that is, the conductance as a function of
the position of the drain lead in a two-terminal device. The method extracts
the conductance profile of the charge flow at the edge of the device and it
can be applied to discern the mechanism of transport from ballistic to
diffusive. It is worth noting that this technique might be capable of
uncovering signatures of anomalous transport in mesoscopic systems. Several
quantum interference effects (Fabry-Pérot resonances and UCFs) are also found
in the resulting conductance curves. The method that we applied to graphene is
general, resolves the transport mechanism in graphene which cannot be
accomplished via Fano shot noise, and can be used as well to study other
systems such as semiconductor heterostructures.
In this article, we simulated SPM measurements for the same devices considered
in the conductance profile study. Probe imaging is important as we are able to
visualize the flow of charge and do not have to rely on transport metrics. For
the graphene devices considered, the SPM simulations suggest diffusive rather
than ballistic transport even in the “clean” regime where the mean free path
is larger than the systems size. Our numerical results suggest that in the
process of escaping the device, charge carriers perform multiple reflections.
From our simulations, we expect that SPM measurements are well suited to study
the crossover between ballistic and diffusive transport in graphene devices.
## VI acknowledgments
We are grateful to S. Barraza-Lopez, S. Bhandari, J. Berezovsky, and R. M.
Westervelt for valuable discussions. M.F.B. and E.J.H. were supported by the
Department of Energy, Office of Basic Science (Grant No. DE-FG02-08ER46513),
and H.H. acknowledges funding through the German Research Foundation (Grant
No. HE 6312/1-1). While most of the calculations were run on the Odyssey
cluster supported by the FAS at Harvard University, the rest were performed on
the Vaquero cluster supported by Oklahoma State University and outfitted with
a Tesla K20 GPU donated by the NVIDIA Corporation.
## References
* Wallace [1947] P. R. Wallace, Phys. Rev. 71, 622 (1947).
* CastroNeto_2009 [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
* DasSarma_2011 [2011] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011).
* Geim_2009 [2009] A. K. Geim, Science 324, 1530 (2009).
* Rozhkov_2011 [2011] A. V. Rozhkov, G. Giavaras, Y. P. Bliokh, V. Freilikher, and F. Nori, Phys. Rep. 503, 77 (2011).
* Novoselov_2004 [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
* Berger_2006 [2006] C. Berger, et al., Science 312, 1191 (2006).
* Meyer_2007 [2007] J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature 446, 60 (2007).
* Bolotin_2008 [2008] K. I. Bolotin, K. J. Sikes, Z. Jiang, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008)
* Du_2008 [2008] X. Du, I. Skachko, A. Barker, and E. Y. Andrei, Nature Nano. 3, 491 (2008).
* Dean_2010 [2010] C. R. Dean, et al., Nature Nano. 5, 722 (2010).
* Adam_2007 [2007] S. Adam, E. H. Hwang, V. Galitski, and S. Das Sarma, Proc. Natl. Acad. Sci. 104, 18392 (2007).
* Katsnelson_2007 [2007] M. I. Katsnelson and A. K. Geim, Phil. Trans. R. Soc. A 366, 195 (2007).
* Gibertini_2010 [2010] M. Gibertini, A. Tomadin, M. Polini, A. Fasolino, and M. I. Katsnelson, Phys. Rev. B 81, 125437 (2010).
* Tworzydlo_2006 [2006] J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J. Beenakker, Phys. Rev. Lett. 96, 246802 (2006).
* Miao_2007 [2007] F. Miao, S. Wijeratne, Y. Zhang, U. C. Coskun, W. Bao, and C. N. Lau, Science 317, 1530 (2007).
* DiCarlo_2008 [2008] L. DiCarlo, J. R. Williams, Y. Zhang, D. T. McClure, and C. M. Marcus, Phys. Rev. Lett. 100, 156801 (2008).
* Danneau_2008 [2008] R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo, and P. J. Hakonen, Phys. Rev. Lett. 100, 196802 (2008).
* Blanter_2000 [2000] Y. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).
* Beenakker_1992 [1992] C. W. J. Beenakker and M. Büttiker, Phys. Rev. B 46, 1889 (1992).
* Schomerus_2007 [2007] H. Schomerus, Phys. Rev. B 76, 045433 (2007).
* Barraza_2012 [2012] S. Barraza-Lopez, M. Kindermann, and M. Y. Chou, Nano Lett. Article ASAP, (2012).
* Louis_2007 [2007] E. Louis, J. A. Verges, F. Guinea, and G. Chiappe, Phys. Rev. B 75, 085440 (2007).
* San-Jose_2007 [2007] P. San-Jose, E. Prada, and D. S. Golubev, Phys. Rev. B 76, 195445 (2007).
* Lewenkopf_2008 [2008] C. H. Lewenkopf, E. R. Mucciolo, and A. H. CastroNeto, Phys. Rev. B 77, 081410(R) (2008).
* Cheianov_2006 [2006] V. V. Cheianov and V. I. Fal’ko, Phys. Rev. B 74, 041403(R) (2006).
* Topinka_2000 [2000] M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard, Science 289, 2323 (2000).
* Topinka_2001 [2001] M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R. Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gossard, Nature 410, 183 (2001).
* Barthelemy_2008 [2008] P. Barthelemy, J. Bertolotti, and D. S. Wiersma, Nature 453, 495 (2008).
* Haug_2008 [2008] H. Haug and A.-P. Jaujo, Quantum Kinetics in Transport and Optics of Semiconductors, 2nd Ed. (Springer, Berlin, 2008).
* Robinson_2007 [2007] J. P. Robinson and H. Schomerus, Phys. Rev. B 76, 115430 (2007).
* [32] A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, EPL 79, 57003 (2007).
* Suzuura_2002 [2002] H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002).
* [34] B. K. Nicolic, L. P. Zarbo, and S. Souma, Spin currents in Semiconductor nanostructures: a nonequilibrium Green-function approach, In A. V. Narlikar and Y. Y. Fu (Eds.), The Oxford Handbook on Nanoscience and Technology Frontiers and Advances, (Oxford University Press, Oxford, 2010).
* [35] R. Landauer, IBM J. Res. Develop. 1, 233 (1957).
* Huang_2009 [2009] L. Huang, Y.-C. Lai, D. K. Ferry, S. M. Goodnick, and R. Akis, Phys. Rev. Lett., 103, 054101 (2009).
* Huang_2011 [2011] L. Huang, R. Yang, and Y.-C. Lai, EPL, 94, 58003 (2011).
* Martin_2008 [2008] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. Von Klitzing, and A. Yacoby, Nature. Phys. 4, 144 (2008).
* Lee_1985 [1985] P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
* Altshuler_1985 [1985] B. L. Altshuler and D. E. Khmelnitskii, JETP Lett. 42, 359 (1985).
* Feng_1986 [1986] S. Feng, P. A. Lee and A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986).
* Altshuler_1985b [1985] B. L. Altshuler and B. Z. Spivak, JETP Lett. 42, 447 (1985).
* Borunda_2011 [2011] M. F. Borunda, J. Berezovsky, R. M. Westervelt, and E. J. Heller, ACS Nano 5, 3622 (2011).
* Marcus et al. [1992] C. M. Marcus, A. J. Rimberg, R. M. Westervelt, P. F. Hopkins, and A. C. Gossard, Phys. Rev. Lett. 69, 506 (1992).
* Jalabert et al. [1990] R. A. Jalabert, H. U. Baranger, and A. D. Stone, Phys. Rev. Lett. 65, 2442 (1990).
* Baranger et al. [1993] H. Baranger, R. Jalabert, and A. Stone, Chaos 3, 665 (1993).
* Ponomarenko et al. [2008] L. Ponomarenko, F. Schedin, M. Katsnelson, R. Yang, E. Hill, K. Novoselov, and A. Geim, Science 320, 356 (2008).
* Berezovsky et al. [2010] J. Berezovsky, M. F. Borunda, E. J. Heller, and R. M. Westervelt, Nanotechnology 21, 274013 (2010).
* Berezovsky and Westervelt [2010] J. Berezovsky and R. M. Westervelt, Nanotechnology 21, 274014 (2010).
* Schnez_2010 [2010] S. Schnez, J. Güttinger, M. Huefner, C. Stampfer, K. Ensslin, and T. Ihn, Phys. Rev. B 82, 165445 (2010).
* Braun_2008 [2008] M. Braun, L. Chirolli, and G. Burkard, Phys. Rev. B 77, 115433 (2008).
* Wang_2012 [2012] W. L. Wang, S. Bhandari, W. Yi, D. C. Bell, R. Westervelt, and E. Kaxiras, Nano Lett. 12, 2278 (2012).
|
arxiv-papers
| 2012-04-14T02:33:32 |
2024-09-04T02:49:29.697432
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mario F. Borunda, H. Hennig, and Eric J. Heller",
"submitter": "Mario Borunda",
"url": "https://arxiv.org/abs/1204.3134"
}
|
1204.3181
|
††thanks: Present Address: Semnan Branch, Islamic Azad University, Semnan,
Iran
Electronic Address: m.valuyan@semnaniau.ac.ir; m-valuyan@sbu.ac.ir
# Casimir Energy For a Massive Dirac Field in One Spatial Dimension: A Direct
Approach
R. Saghian M.A. Valuyan A. Seyedzahedi S.S. Gousheh Department of Physics,
Shahid Beheshti University G.C., Evin, Tehran 19839, Iran
###### Abstract
In this paper we calculate the Casimir energy for a massive fermionic field
confined between two points in one spatial dimension, with the MIT Bag Model
boundary condition. We compute the Casimir energy directly by summing over the
allowed modes. The method that we use is based on the Boyer’s method, and
there will be no need to resort to any analytic continuation techniques. We
explicitly show the graph of the Casimir energy as a function of the distance
between the points and the mass of the fermionic field. We also present a
rigorous derivation of the MIT Bag Model boundary condition.
Casimir Energy; Fermion; MIT Bag Model
## I Introduction
More than $60$ years have passed from the time when H. B. G. Casimir published
his famous paper h.b.g. on what has been called the Casimir effect ever
since. It remained relatively unknown for over two decades, but from the early
$70$s this effect has attracted much attention. The Casimir energy is a pure
quantum effect with macroscopic manifestations. Generally, the Casimir energy
is defined as the difference between the vacuum energies in the presence and
the absence of any external boundary conditions or background fields. Both of
these energies are in general infinite. However the difference between the two
has almost invariably been calculated to be finite. The Casimir effects have
been calculated for a variety of fields, geometries, number of spatial
dimensions, and boundary conditions (for a review see burdag.book. ;
milton.book. ; milton.paper. ; Mostepanenko. ).
Recently the Casimir effect has been studied in connection with many physical
phenomena. For example this effect has been studied in the context of the
phenomenological chiral bag models of the nucleon chiral.casimir. ; linas. .
In such models the bag constant , $B$, is an input parameter to the theory. As
is well known, this constant is added to the Lagrangian density in order to
balance the outward pressure of the quarks by the inward vacuum pressure $B$
on the surface of the bag milton.book. . This constant can be related to the
Casimir energy bag1. . The Casimir effect for the String and the Superstring
leads to string theories having critical dimensions. The string is a finite
two-dimensional system with an infinite phonon spectrum, the sum of the zero-
point fluctuations of which leads to exactly the same calculation as in the
original Casimir effect LBrink. . Moreover, the presence of the Casimir
effects in many different phenomena in condensed matter and laser physics have
been established both theoretically and experimentally condensed. ; Krech. .
For example, the Casimir force is important in the development of
microtechnologies that routinely allow control of separation between bodies
smaller than $1\mu m$ nano.application. ; Rodrigues. ; Lambrecht. . An
interesting application of the field theoretical models with compact
dimensions recently appeared in nanophysics. The long-wavelength description
of the electronic states in graphene can be formulated in terms of a Dirac-
like theory in three dimensional space-time with the Fermi velocity playing
the role of speed of light 5\. ; Vincenzo. ; Gonzalez. ; Lee. ; Sharapov. ;
Castro. . Single-walled carbon nanotubes are generated by rolling up a
graphene sheet to form a cylinder and the background space-time for the
corresponding Dirac-like theory has topology $R^{2}\times S^{1}$ graphene.ref.
.
In order to compute the Casimir energy, physically relevant boundary
conditions must be imposed. For example the Casimir energy for an
electromagnetic field is usually calculated with the boundary conditions
appropriate for conducting boundaries as in burdag.book. ; milton.book. ;
milton.paper. ; Mostepanenko. ; EM. ; Valuyan. ; Hacgan. ; Jordan. . The
Casimir energy for scalar fields has been investigated with the Dirichlet
boundary condition burdag.book. ; milton.book. ; milton.paper. ; Mostepanenko.
; valuyanjadval. ; Moazzemi. ; Gousheh. ; valuyan2. ; Mhammadi. , the Neumann
boundary condition milton.book. ; Mostepanenko. , the mixed boundary condition
mixed. , and the Robin boundary condition Robins. ; Albuquerque. ;
someaspects. .
From this point on we concentrate on the Dirac field. Since the Dirac field
obeys a first order differential equation of motion, it is impossible to use
the aforementioned boundary conditions. Moreover, boundary conditions are in
general more disruptive to Dirac fields than boson fields because the
equations of motion are first order Jaffe. . A proper boundary condition for
the Dirac field is the MIT Bag Model boundary condition. It is usually said to
imply that there is no flux of fermions through the boundary, i.e. if
$j^{\mu}$ denotes the current of the Dirac field and $n^{\mu}$ is the normal
unit vector to the boundary, then $n_{\mu}j^{\mu}=0$. However, it implies an
even stronger condition which is the absolute confinement of the fermionic
field. One could confine the fermionic wave function by an infinite scalar
potential. This model was considered by Bogolioubov Bogolioubov. and later
developed as the MIT Bag Model by A. Chodos, et al Chodos1. ; Chodos2. for
hadrons. It is important to mention that in general the computation of the
Casimir energy for a massive Dirac field turns out to be much more difficult
than the massless case. There has been relatively few studies for the Casimir
energy inside closed surfaces, and we mentioned a few of them here. The
Casimir energy has been calculated for a spherical geometry for a massive
Elizalde. and massless fermionic fields milton.book. ; milton.paper. ;
hofmann. . It has also been recently calculated for a massless fermionic field
subject to the MIT Bag Model boundary condition confined inside a three
dimensional rectangular geometry maghale. .
As usual there are many more studies on the two parallel plates geometry, and
we shall concentrate on this problem from this point on. The first computation
of the Casimir energy for the Dirac field was done by Johnson in $1975$
Massless. . He computed this energy per unit area for a massless fermionic
field subject to the MIT Bag Model boundary condition between two parallel
plates in three spatial dimensions. Afterwards, the Casimir energy has been
calculated for massless fermions in one dimension Jaffe. , between two
parallel plates using various methods in three-dimensional space milton.paper.
; milton.book. ; Massless. ; Queiroc. and in $d+1$ dimensional space-time
Tort2. ; Setare. ; Bezerra. . The first attempt to compute the Casimir energy
for the massive case in three spatial dimensions was done by Mamayev and
Trunov who managed to find an integral form for this quantity and explicitly
computed its small and extremely large mass limits Massive. . The first
complete computation of the Casimir energy for the massive case in this
geometry in $3+1$ and $1+1$ dimensions was done by C. D. Fosco and E. L.
Losada in $2008$ Functional-approach. . In their approach, a coupling of the
bilinear form $\bar{\psi}\psi$ to a series of regularly spaced
$\delta$-function potentials with coupling constant $g$ is introduced, which
implements imperfect bag-like boundary condition. This method can produce a
fermionic propagator which satisfies the MIT Bag Model boundary condition when
$g=2$. However a direct calculation of the Casimir energy for this problem has
not been presented so far, and this will be the subject of this paper.
In order to calculate the Casimir energy from first principles one must sum
over the allowed modes. However, the vacuum energies in the presence and
absence of disturbances obtained by these direct sums are infinite. Therefore
one has to adopt regularization and analytic continuation prescriptions, in
order to compute the difference between these two divergent quantities. When
the allowed modes which appear in the summands are regular, the major
approaches used are: the zeta function analytic continuation technique
milton.paper. ; milton.book. ; someaspects. ; Elizalde2. , cut-off
regularization cut-offs. ; Miltao. ; CRHagen. ; Abel-Plana1. and box
subtraction scheme along with the Abel-Plana summation formula Moazzemi. ;
Abel-Plana1. . On the other hand, when the allowed modes which appear in the
summands are irregular, the major approaches used are: the contour integration
method countor.int. , the Green function formalism green.func. , the
functional approach Functional-approach. , and the Boyer method boyer. ;
valuyanjadval. . Unfortunately, most of these techniques do not be lead to
closed forms for the values of the Casimir energy and one has to employ
various numerical methods to obtain a value for the Casimir energy.
In this paper we calculate the Casimir energy for a massive fermionic field in
two dimensional space-time with the MIT Bag Model boundary condition, by
directly summing over the modes. In this problem the fermionic modes are
irregular and the divergences that appear are very severe. Upon using the
contour integration method, one might encounter some ambiguities, mainly due
to the severe nature of divergences inherent to this problem. Here we use a
direct approach which does not resort to any analytic continuation techniques
and is devoid of any ambiguities. Our method is based on the Boyer method,
which we shall explain in detail boyer. . Moreover, in this procedure one can
associate a physical meaning to the Casimir energy: It is equal to the work
done in forming the configuration under study from the free vacuum. In section
II, we solve the Dirac equation in one spatial dimension using the MIT Bag
Model boundary condition and we find a transcendental equation for the
discrete spectrum. The allowed modes for the massive case obtained from this
equation are not regular. In section III we compute the Casimir energy by
performing a direct sum over all modes of the field. We finally plot the
values obtained for the Casimir energy as a function of the distance between
the points for various values of the mass. We show that the results for the
small mass limit converges to the results for the massless case. The results
obtained in this paper are consistent with those obtained in Functional-
approach. . In Section IV we summarize our results. In the Appendix we present
a rigorous derivation of the MIT Bag Model boundary condition for the most
general case.
## II The solution of Dirac Equation with the MIT Bag Model boundary
condition
In this section we find the solutions to the Dirac equation with the MIT Bag
boundary condition in one spatial dimension. We consider the case where the
Dirac field is completely free inside the bag: $(-\frac{a}{2}<x<\frac{a}{2})$.
We can write the spinor $\psi(x,t)$ as:
$\displaystyle\psi(x,t)=e^{-iEt}\left(\begin{array}[]{c}f(x)\\\ g(x)\\\
\end{array}\right),$ (3)
where $E$ denotes the energy eigenvalue of the time-independent solution. We
choose the following representation for the $\gamma$-matrices:
$\gamma^{0}=\sigma_{1}$, $\gamma^{1}=i\sigma_{3}$, and
$\gamma^{5}=\gamma^{0}\gamma^{1}=\sigma_{2}$. Then the Dirac equation leads
to,
$\displaystyle\Bigg{\\{}\begin{array}[]{c}f(x)=Ce^{ikx}+De^{-ikx},\\\ \\\
g(x)=\frac{m}{E}(Ce^{ikx}+De^{-ikx})+\frac{ik}{E}(Ce^{ikx}-De^{-ikx}),\\\
\end{array}$ (7)
where $k=\sqrt{E^{2}-m^{2}}$.
The MIT Bag Model boundary condition has been derived in the Appendix A and we
show that this boundary condition can completely confine the spinor fields
between the boundaries. The MIT Bag Model boundary condition in two space-time
dimension in our convention becomes,
$\displaystyle(1\mp\sigma_{3})\psi(x)\bigg{|}_{x=\pm\frac{a}{2}}=0.$ (8)
Applying these conditions we obtain
$\displaystyle\bigg{\\{}\begin{array}[]{c}g(\frac{a}{2})=0,\\\
f(\frac{-a}{2})=0.\\\ \end{array}$ (11)
Figure 1: In this figure the real and imaginary parts of the upper and lower
components of the lowest two positive energy wavefunctions $\psi(x)$ are
plotted as a function of $x$, separately. In parts. (A,B) the real and
imaginary parts of the upper and lower components of the ground state
($k_{1}=2.0288$ and $E_{1}=+\sqrt{k_{1}^{2}+m^{2}}$ with $m=1$) are plotted.
The parity of this state is positive. In parts. (C,D) the real and imaginary
parts of the upper and lower components of the first excited state
($k_{2}=4.9132$ and $E_{2}=+\sqrt{k_{2}^{2}+m^{2}}$ with $m=1$) are plotted.
The parity of this state is negative.
Extracting $\frac{C}{D}$ from the condition $g(\frac{a}{2})=0$ and inserting
it into $f(\frac{-a}{2})=0$, we obtain an expression which determines the
quantized modes:
$\displaystyle ka\cot(ka)=-ma.$ (12)
The above expression is identical to the result obtained by Mamayev and Trunov
for the component of the momentum perpendicular to the plates for a massive
Dirac field between two parallel plates in three spatial dimensions Massive. .
In Ref. Massive. this condition has been obtained by using the MIT boundary
condition and the property of the Dirac spinors that each component also
satisfies the Klein-Gordon equation. However all of the allowed modes along
with this equation for the two parallel plate geometry in 3+1 dimension can
also be easily obtained directly, along the same lines as the derivation
presented in this section. The solutions of the above transcendental equation
(Eq. (12)), denoted by $k_{s}$, are not regular for the massive Dirac fields.
Only for the specific case of a massless Dirac field, the modes are regular
($k_{s}a=\frac{(2s-1)\pi}{2}$, $s=\\{1,2,3,\cdots\\}$) milton.paper. ;
milton.book. . Since changing the sign of the root $k_{s}$ does not lead to a
linearly independent solution, we consider only the positive roots.
As expected the MIT bag model boundary condition has transformed the spectrum
of the free Dirac field, which consisted of two continua starting at $E=\pm
m$, into two sets of discrete states with energies
$E=\pm\sqrt{k_{s}^{2}+m^{2}}$, where $s=\\{1,2,3,\cdots\\}$. This problem has
parity symmetry and in this representation the parity operation is given by
$P\psi(x,t)=\sigma_{1}\psi(-x,t)$. The energy eigenstates automatically turn
out to be parity eigenstates. The parities of the lowest lying states with
energies $E=\pm\sqrt{k_{1}^{2}+m^{2}}$ are $\pm 1$ respectively, and the
parities of the exited states alternate as the absolute value of the energy
increases. The real and imaginary parts of the upper and lower components of
the lowest two positive energy wavefunctions $\psi(x)$ are plotted in Fig.
(1). Note that the values of the wavefunctions on the boundaries are non-zero
and just outside the boundaries are exactly zero. This is another
manifestation of the imposition of the MIT bag model boundary condition, as
explained fully in the Appendix A.
In order to calculate the Casimir energy for the massive case we need to find
a relationship between the root number $s$ and the wave number $k_{s}$. For
this purpose, Eq. (12) can be written as:
$\displaystyle s=\frac{1}{\pi}\Big{[}X_{s}+\tan^{-1}(\frac{X_{s}}{M})\Big{]},$
(13)
where $s$ is the root number, $X_{s}=k_{s}a$ and $M=ma$. Note that the values
of the root indices $s$ and the corresponding wave-numbers $k_{s}$ can be
analytically continued to any real value.
## III The Casimir Energy
In this section we calculate the Casimir energy for a massive Dirac field
between two parallel plates (two points) in 1+1 space-time dimensions. In
order to obtain the Casimir energy, we should subtract the zero point energy
in the absence from the presence of the boundary conditions. Therefore, the
vacuum energies for both cases should be calculated. Since the solutions of
the Hamiltonian in the absence and presence of the boundaries are complete
mackenzie1 ; dr , the Fermi field operator can be expanded in terms of either
of these modes, as follows
$\displaystyle\displaystyle\Xi(x)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{dk}{2\pi}[b_{k}u_{k}(x)+d^{{\dagger}}_{k}\upsilon_{k}(x)]$
(14) $\displaystyle=$
$\displaystyle\sum_{s=1}^{\infty}[a_{s}\mu_{k_{s}}(x)+c^{{\dagger}}_{s}\nu_{k_{s}}(x)]$
where we have denoted the positive-energy and negative-energy modes in the
free case by $u_{k}(x)$ and $\upsilon_{k}(x)$, and in the case with the
boundary condition by $\mu_{k_{s}}(x)$ and $\nu_{k_{s}}(x)$, respectively. By
substituting the expressions for the field operator $\Xi(z)$ given in Eq. (14)
into the general definition of the Hamiltonian operator and using the usual
anticommutation relations, and evaluating the two zero point energies, we
obtain the following expression for the Casimir energy,
$\displaystyle E$ ${}_{\footnotesize\mbox{Cas.}}=\langle\Omega\mid
H\mid\Omega\rangle-\langle 0\mid H^{\footnotesize\mbox{free}}\mid 0\rangle$
(15) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\sum_{s=1}^{\infty}(-\sqrt{k_{s}^{2}+m^{2}})\nu^{{\dagger}}_{k_{s}}(x)\nu_{k_{s}}(x)$
$\displaystyle-$
$\displaystyle\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}\frac{adk}{2\pi}(-\sqrt{k^{2}+m^{2}})\upsilon^{{\dagger}}_{k}(x)\upsilon_{k}(x),$
where $\mid 0\rangle$ and $\mid\Omega\rangle$ denote the vacuum states in the
absence and the presence of the boundary condition, respectively. We have just
shown that we can obtain the vacuum energy, and therefore the Casimir energy,
by simply summing over the negative energy modes without the factor of $1/2$.
This is equivalent to the usual definition where one sums over both positive
and negative energy modes, since our problem possesses particle conjugation
symmetry along with C, P and T symmetries, separately. Integrating over $x$
the Casimir energy becomes,
$\displaystyle
E_{\footnotesize\mbox{Cas.}}=-\sum_{s=1}^{\infty}\sqrt{k_{s}^{2}+m^{2}}+\int_{-\infty}^{+\infty}\frac{adk}{2\pi}\sqrt{k^{2}+m^{2}}.$
(16)
As mentioned earlier both of these vacuum energies are infinite. However the
difference, which is the Casimir energy, is expected to be finite. In many of
the techniques for calculating the Casimir energy one starts with only the
expression for the vacuum energy for the problem at hand (the first expression
in Eq. (16) in our case) and removes the infinities that appear during the
calculation by using various methods such as analytic continuation or simply
by hand. this should precisely amount to subtracting the free vacuum energy
which was left out from the beginning (the second expression in Eq. (16) in
our case).
The dependence of the fermionic quantized momenta on the mass of the field is
one of the distinguishing features of the Fermi field as compared to the
bosonic case. In Fig. (2) we show the wave vectors for two massive and a
massless fermionic field.
Figure 2: The plot of the allowed values of the wave-number $k_{s}$, obtained
from the roots of Eq. (12), as a function of the root number $s$. We have
displayed the results for three values of the mass $m=\\{0,1,5\\}$ with $a=1$.
Note that for $m=0$ we obtain a straight line, i.e. the roots are equally
spaced. However, as is apparent from the figure, this no longer true for
$m\neq 0$.
However as mentioned earlier, for $m=0$ the wave vectors are evenly spaced and
are given by $k_{s}=\frac{(2s-1)\pi}{2a}$. Therefore the Casimir energy for a
massless fermionic field can be easily obtained from the first term in Eq.
(16) using the zeta function and its analytic continuation as follows
milton.book. ; Jaffe. ,
$\displaystyle E_{\mbox{\footnotesize{Cas.}}}^{(0)}\Big{(}M=0\Big{)}$
$\displaystyle=$
$\displaystyle\frac{-1}{a}\Bigg{[}\sum\limits_{s=1}^{\infty}(s-\frac{1}{2})\pi\Bigg{]}_{\mbox{\footnotesize{Analytic
Part}}}$ (17) $\displaystyle=$
$\displaystyle\frac{-\pi}{a}\zeta(-1,\frac{1}{2})=\frac{-\pi}{24a}.$
As shown in Fig. (2) the wave numbers obtained from Eq. (12) for a massive
Dirac field are irregular. In order to calculate the Casimir energy we use the
Boyer method boyer. , instead of using Eq. (16) directly, since the latter is
more prone to ambiguities. These two methods for calculating of the Casimir
energy are equivalent. Now we discuss the Boyer method in detail. In this
method we consider two similar configurations: two points with distance $a$
and two points with distance $b$. Then we place each of these systems in a box
with size $L>a,b$ as Fig. (3). Finally we define the Casimir energy as the
difference between the vacuum energies of these two similar configurations as
follows,
$\displaystyle
E_{\mbox{\footnotesize{Cas.}}}=\lim_{b/a\rightarrow\infty}\bigg{[}\lim_{L/b\rightarrow\infty}\big{[}E_{A}^{(0)}-E_{B}^{(0)}\big{]}\bigg{]},$
(18)
Figure 3: The geometry of the two different configurations whose energies are
to be compared. The labels $a1$, $b1$, etc. denote the appropriate sections in
each configuration separated by points. The upper configuration is denoted by
‘B’, and the lower one by ‘A’.
where $E_{A}^{(0)}=E_{a1}^{(0)}+2E_{a2}^{(0)}$ and
$E_{B}^{(0)}=E_{b1}^{(0)}+2E_{b2}^{(0)}$ are the vacuum energies of
configurations ‘A’ and ‘B’ as shown in Fig. (3). Note that upon taking the
limits indicated in Eq. (18), one recovers the original definition of the
Casimir energy given in Eq. (16). From this definition one can easily conclude
that the Casimir energy is equal to the work done on the configuration ‘B’ to
deform it to configuration ‘A’. For each of the six regions shown in the Fig.
(3), for example the region $a1$, we calculate the vacuum energy as follows,
$\displaystyle E^{\mbox{\footnotesize$(0)$}}_{a1}(M)=\lim_{\lambda\rightarrow
0}\bigg{[}-\frac{1}{a}\sum_{s=1}^{\infty}(X_{s}^{2}+M^{2})^{1/2}g(\lambda\omega_{s})\bigg{]},$
(19)
where we have introduced a convergence factor
$g(\lambda\omega_{s})=e^{-\lambda\sqrt{X_{s}^{2}+M^{2}}}$ with
$\lambda\rightarrow 0$. Now we start the calculation for the region $a1$, and
the calculations for the other regions can be done analogously. Since the
wave-vectors are not regular with respect to the root indices $s$, in order to
find an analytical form for the divergence in Eq. (19), we employ the simplest
form of the Euler-Maclaurin Summation Formula (EMSF) to obtain
$\displaystyle\displaystyle E^{\mbox{\footnotesize$(0)$}}_{a1}(M)$
$\displaystyle=$ $\displaystyle-\frac{1}{a}\lim_{\lambda\rightarrow
0}\Big{[}\int_{s=1}^{\infty}ds(X_{s}^{2}+M^{2})^{1/2}g(\lambda\omega_{s})$
$\displaystyle+\frac{1}{2}(X_{1}^{2}+M^{2})^{1/2}g(\lambda\omega_{1})$
$\displaystyle+\int_{s=1}^{\infty}ds\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{ds}[(X_{s}^{2}+M^{2})^{1/2}g(\lambda\omega_{s})]\Big{]},$
where $[s]$ is the Floor function. There is a one-to-one correspondence
between the wave vector $k_{s}$ and the root number $s$, as is manifest in Eq.
(13). Therefore we can change the variable of integration from $s$ to $X$.
Then we obtain,
$\displaystyle\displaystyle E^{\mbox{\footnotesize$(0)$}}_{a1}(M)$
$\displaystyle=$ $\displaystyle-\frac{1}{a}\lim_{\lambda\rightarrow
0}\Big{[}\int_{X=X_{1}}^{\infty}dX\frac{ds}{dX}(X^{2}+M^{2})^{1/2}g(\lambda\omega)$
$\displaystyle+\frac{1}{2}(X_{1}^{2}+M^{2})^{1/2}g(\lambda\omega_{1})$
$\displaystyle+\int_{X=X_{1}}^{\infty}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]\Big{]}.$
Now, by adding and subtracting appropriate terms, we can extend the lower
limits of all integrals in Eq. (III) to zero. We thus have
$\displaystyle\displaystyle E^{\mbox{\footnotesize$(0)$}}_{a1}(M)$
$\displaystyle=$ $\displaystyle-\frac{1}{a}\lim_{\lambda\rightarrow
0}\Big{[}\int_{X=0}^{\infty}dX\frac{ds}{dX}(X^{2}+M^{2})^{1/2}g(\lambda\omega)$
$\displaystyle-\int_{X=0}^{X_{1}}dX\frac{ds}{dX}(X^{2}+M^{2})^{1/2}g(\lambda\omega)$
$\displaystyle+\frac{1}{2}(X_{1}^{2}+M^{2})^{1/2}g(\lambda\omega_{1})$
$\displaystyle+\int_{X=0}^{\infty}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]$
$\displaystyle-\int_{X=0}^{X_{1}}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]\Big{]}.$
The last term in Eq. (III) can be simplified by noting that the Floor function
$[s]=0$ in the indicated domain, and integration by parts yields,
$\displaystyle\int_{X=0}^{X_{1}}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]$
$\displaystyle=$
$\displaystyle\int_{X=0}^{X_{1}}dX\Big{(}s-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]$
$\displaystyle=$
$\displaystyle\frac{1}{2}(X_{1}^{2}+M^{2})^{1/2}g(\lambda\omega_{1})+\frac{1}{2}Me^{-\lambda
M}$ $\displaystyle-$
$\displaystyle\int_{X=0}^{X_{1}}dX\frac{ds}{dX}(X^{2}+M^{2})^{1/2}g(\lambda\omega).$
Using Eqs. (III,III) we obtain
$\displaystyle E^{\mbox{\footnotesize$(0)$}}_{a1}(M)=$ $\displaystyle-$
$\displaystyle\frac{1}{a}\lim_{\lambda\rightarrow
0}\Bigg{[}\int_{X=0}^{\infty}dX\frac{ds}{dX}(X^{2}+M^{2})^{1/2}g(\lambda\omega)+\frac{1}{2}Me^{-\lambda
M}$ $\displaystyle+$
$\displaystyle\int_{X=0}^{\infty}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]\Bigg{]}.$
Note that only the first term on the right hand side of Eq. (III) is
divergent. Upon substituting the expression displayed in Eq. (III) into the
definition of the Casimir energy given in Eq. (18), the constant terms
automatically cancel each other in the limit $\lambda\rightarrow 0$ and by
choosing appropriate cutoffs on the upper limits of the integrals for each
region, the divergent integrals cancel each other, all due to the box
subtraction scheme. Therefore only the convergent integral terms remain. The
contributions of integrals in regions $a2$, $b1$ and $b2$ go to the zero as
$L/b\rightarrow\infty$ and $b/a\rightarrow\infty$. Therefore our final
expression for the Casimir energy is,
$\displaystyle\displaystyle E_{\mbox{\footnotesize
Cas.}}=\lim_{\lambda\rightarrow
0}\Bigg{[}-\frac{1}{a}\int_{0}^{\infty}dX\Big{(}s-[s]-\frac{1}{2}\Big{)}$
$\displaystyle\times\frac{d}{dX}[(X^{2}+M^{2})^{1/2}g(\lambda\omega)]\Bigg{]},$
(25)
where $s$ is obtained from Eq. (13). It seems that this expression does not
have a closed form solution and it should be solved numerically. As is
apparent from Eq. (III), the integrand has an infinite number of
discontinuities due to the presence of the Floor function. First the precise
positions of the jumps in the integrand have to be determined. These jumps
precisely correspond to the roots of Eq. (12), giving the values of the wave-
numbers. The integrations are done separately for all parts and then all of
the results are summed. The integration is over the continuous version of the
wave number, which extends to infinity. In order to accomplish this
numerically, we compute this integral up to a cutoff $\Lambda$ which should
eventually go to infinity. Meanwhile, we also have to take the limit
$\lambda\rightarrow 0$ as indicated in Eq. (III). We have determined that an
optimization occurs precisely when $\lambda=1/\Lambda$. In Fig. (4) the values
of the Casimir energy have been plotted as a function of the distance $a$ for
various values of $m$. This plot shows that there is a good consistency
between the results of the massless case and massive ones when $m\rightarrow
0$. This figure also shows the rapid decrease in the value of the Casimir
energy as a function of $ma$. We should mention that our results are in an
excellent agreement with the previously reported result which was obtained
indirectly through the analysis of the bilinear forms Functional-approach. .
Figure 4: The values of the Casimir energy for the massive and massless Dirac
fields with the MIT Bag Model boundary condition in one spatial dimension are
plotted as a function of the distance between the points ($a$). In this figure
we have shown a sequence of plots for $m=\\{1,0.1,0.01,10^{-6},0\\}$. It is
apparent that the sequence of the plots for the massive cases converges
rapidly to the massless case as $m$ decreases.
## IV Conclusion
In this paper we have computed the Casimir energy for a massive Dirac field
with the MIT Bag Model boundary condition in one spatial dimension. We have
used the direct mode summation method in order to compute the Casimir energy
for this field. For the massless case the modes are regular and we use the
zeta function analytic continuation technique. However, for the massive case
the modes are irregular and we use the Boyer’s subtraction scheme. Its worth
mentioning that in this technique all of the infinities are canceled
automatically and there is no need to use any analytic continuation. We have
shown that the massless limit of the massive case precisely corresponds to the
massless case. Our result for the values of the Casimir energy has been
obtained numerically, similar to the previously reported results Functional-
approach. .
## Acknowledgement
We would like to thank the Research Office of the Shahid Beheshti University
for financial support.
## Appendix A A Derivation for the MIT Bag Model Boundary condition
In this appendix we present a rigorous derivation of the MIT Bag Model
boundary condition for the Dirac field inside an arbitrary closed surface $S$.
This boundary condition ensures the complete confinement of the eigenstates of
the Dirac Hamiltonian inside an enclosed area. We show that this boundary
condition can be obtained by coupling the Dirac field to a scalar potential
$V$ and taking the limit as $V\rightarrow\infty$. However we first present the
reasons why we cannot confine fermions inside an enclosed area by the time
component of a four-vector potential $V_{0}$.
Solving the Dirac equation we obtain $p^{2}=(E-V_{0}+m)(E-V_{0}-m)$ which is
positive for $|E-V_{0}|>m$. Therefore we have oscillatory solutions out of the
barrier, i.e. we have currents of particles and antiparticles. This
contradicts the assumption of complete confinement of the fermionic field.
This is the well-known Klein’s paradox Klein. . Next we use the scalar
potential $V(x)$ which, as we shall see, does not have this problem. The Dirac
equation with the scalar potential is:
$\displaystyle\big{[}i\gamma^{\mu}\partial_{\mu}-(m+V(\vec{x}))\big{]}\psi(\vec{x},t)=0.$
(26)
Decomposing the spatial components of $\gamma^{\mu}\partial_{\mu}$ at the
surface into tangential $(t)$ and normal $(n)$ parts, we have:
$\displaystyle\gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}+\gamma^{t}\partial_{t}+\gamma^{n}\partial_{n}.$
(27)
We choose the potential $V(\vec{x})$ to be infinite outside of the enclosed
area and to vanishes inside. First we take the integral of the Dirac equation
with the scalar potential (Eq. (26)) from $\vec{a}-\epsilon\hat{n}$ to
$\vec{a}$ (a small interval inside the barrier) where $\vec{a}$ specifies a
random point on the surface and $\hat{n}$ is the normal unit vector to the
surface at point $\vec{a}$
$\displaystyle\int_{a-\epsilon}^{a}\hat{n}dn(i\gamma^{\mu}\partial_{\mu}-m)\psi(\vec{x},t)=0.$
(28)
When $\epsilon\rightarrow{0}$, all of the terms vanish except the one
containing the term $\gamma^{n}\partial_{n}$. Then:
$\displaystyle
i(\hat{n}\cdot\vec{\gamma})(\psi(\vec{a})-\psi(\vec{a}-\epsilon\hat{n}))=0\Rightarrow\psi(\vec{a})=\psi(\vec{a}-\epsilon\hat{n}).$
(29)
Second, we change the integration domain. This time we integrate from a point
just inside the volume ($\vec{a}-\epsilon\hat{n}$) to a point outside of the
volume ($\vec{a}+\epsilon\hat{n}$).
$\displaystyle\int_{a-\epsilon}^{a+\epsilon}\hat{n}dn(i\gamma^{\mu}\partial_{\mu}-(m+V(\vec{x})))\psi(\vec{x},t)=0.$
(30)
This time we have to deal with the infinite potential outside the bag and
therefore we cannot neglect the term $\epsilon V$. Now using the fact that the
Dirac equation demands $\psi(\vec{a}+\epsilon\hat{n})=0$ and Eq. (29) one
obtains:
$\displaystyle-i(\hat{n}\cdot\vec{\gamma})\psi(\vec{a})=\epsilon
V\psi(\vec{a}).$ (31)
Multiplying Eq. (31) from left by $(\hat{n}\cdot\vec{\gamma})$ yields
$\epsilon V=\pm 1$. Since $\epsilon>0$ and $V=+\infty$ we conclude $\epsilon
V=1$.
It is interesting to note that using the time component of a four-vector
potential for the confinement purpose we obtain an inconsistent result:
$\epsilon V=\pm i$.
Inserting $\epsilon V=1$ in Eq. (31) we obtain the MIT Bag Model boundary
condition:
$\displaystyle(1+i(\hat{n}\cdot\vec{\gamma}))\psi(\vec{x})\bigg{|}_{\mbox{\footnotesize
Boundary}}=0.$ (32)
## References
* (1) H. B. G. Casimir, Proc. Kon. Nederl. Akad. Wet. 51 (1948) 793.
* (2) M. Bordag, G.L. Klimchitskaya, U. Mohideen, and V.M. Mostepanenko, _Advances in the Casimir Effect_ , Oxford University Press Inc. New York (2009).
* (3) K. A. Milton, _The Casimir Effect: Physical Manifestations of Zero-Point Energy_ , (World Scientific Publishing Co. Pte. Ltd. 2001).
* (4) K. A. Milton, _The Casimir Effect: Physical Manifestations of Zero Point Energy_ , Invited Lectures at 17th Symposium on Theoretical Physics, Seoul National University, Korea, June 29-July 1, (1998); [arXiv:hep-th/9901011v1].
* (5) M. Bordag, U. Mohideen, V.M. Mostepanenko, Phys. Rep. 353 (2001) 1; [arXiv:quant-ph/0106045v1].
* (6) Hee-Jung Lee, Dong-Pil Min, Byung-Yoon Park, Mannque Rho and Vicente Vento, Nucl. Phys. A 657 (1999) 75.
* (7) Linas Vepstas, A. D. Jackson and A. S. Goldhaber, Nucl. Phys. B 140 (1984) 280.
* (8) I. O. Cherednikov, Int. J. Mod. Phys. A 17, 874 (2002).
* (9) L. Brink and H. B. Nielsen, Phys. Lett. B 45, 332 (1973).
* (10) F. De Martini, M. Marrocco and P. Mataloni, Phys. Rev. A 43, 2480 (1991).
* (11) M. Krech and S. Dietrich, Phys. Rev. Lett. 66, 345 (1991).
* (12) F. Chen, U. Mohideen, G. Klimchitskaya and V. Mostepanenko, Phys. Rev. Lett. 88 (2002) 101801.
* (13) R. Rodrigues, P. Maia Neto, A. Lambrecht and S. Reynaud, EPL 76, 822 (2006).
* (14) A. Lambrecht, I. Pirozhenko, L. Duraffourg and Ph. Andreucci, EPL 77, 44006 (2007).
* (15) G.W. Semenoff, Phys. Rev. Lett. 53, (1984) 2449.
* (16) D.P. Di Vincenzo and E.J. Mele, Phys. Rev. B 29 (1984) 1685.
* (17) J. González, F. Guinea, and M. A. H. Vozmediano, Phys. Rev. B 63 (2001) 134421.
* (18) H.-W. Lee and D.S. Novikov, Phys. Rev. B 68 (2003) 155402.
* (19) S.G. Sharapov, V.P. Gusynin, and H. Beck, Phys. Rev. B 69 (2004) 075104.
* (20) A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, Rev. Mod. Phys. 81 (2009) 109.
* (21) S. Bellucci and A. A. Saharian, Phys. Rev. D80 (2009) 105003; [arXiv:hep-th/0907.4942v1]
* (22) J. Ambjørn, and S. Wolfram, Ann. Phys. (N. Y.) 147 (1983) 1.
* (23) M. A. Valuyan, R. Moazzemi, and S. S. Gousheh, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 145502; [arXiv:hep-th/0806.1628].
* (24) S. Hacgan, R. Jáuregui, and C. Villarreal, Phys. Rev. A 47 (1993) 4204.
* (25) G. Jordan Maclay, Phys. Rev. A 61 (2000) 052110-1.
* (26) M. A. Valuyan, and S. S. Gousheh, Int. J. Mod. Phys. A 25 (2010) 1165.
* (27) R. Moazzemi, M. Namdar, and S.S. Gousheh, JHEP 09 (2007) 029; [arXiv:hep-th/0708.4127v1].
* (28) R. Moazzemi, and S. S. Gousheh, Phys. lett. B 658 (2008) 255; [arXiv:hep-th/0708.3428v2].
* (29) S. S. Gousheh, R. Moazzemi, and M. A. Valuyan, Phys. Lett. B 681 (2009) 477; [arXiv:hep-th/0911.3707].
* (30) R. Moazzemi, Abdollah Mohammadi, and S.S. Gousheh, Eur. Phys. J. C 56 (2008) 585; [arXiv:hep-th/0806.4862].
* (31) A. C. Aguiar Pinto, T. M. Britto, R. Bunchaft, F. Pascoal, and F.S.S. da Rosa, Brazilian Journal of Physics 33 (2003) 60.
* (32) A. Romeo, and A. A. Saharian, J. Phys. A: Math. Gen. 35 (2002) 1297; [arXiv:hep-th/0007242v2].
* (33) L.C. de Albuquerque, and R. M. Cavalcanti, J. Phys. A: Math. Gen. 37 (2004) 7039; [arXiv:hep-th/0311052v2].
* (34) C. Farina, Brazilian Journal of Physics 36 (2006) 1137.
* (35) P. Sundberg, and R. L. Jaffe, Ann. Phys. 309 (2004) 449; [arXiv:hep-th/0308010v1].
* (36) P. N. Bogolioubov, Ann. Inst. Henri Poincare 8 (1967) 163.
* (37) A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D 9 (1974) 3471.
* (38) A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Phys. Rev. D 10 (1974) 2599.
* (39) E. Elizalde, M. Bordag, and K. Kirsten, J. Phys. A: Math. Gen. 31 (1998) 1743; [arXiv:hep-th/9707083v1].
* (40) R. Hofmann, M. Schumann and R.D. Viollier, Eur. Phys. J. C 11 (1999) 153.
* (41) A. Seyedzahedi, R. Saghian, and S. S. Gousheh, Phys. Rev. A.82 (2010) 032517.
* (42) K. Johnson, Acta. Pol. B 6 (1975) 865.
* (43) H. Queiroc, J. C. da Silva, and F. C. Khanna, J.M.C. Malbouisson, M. Revzen and A. E. Santana, Ann. Phys. 317 (2005) 220.
* (44) E. Elizalde, F. C. Santos, and A. C. Tort, Int. J. Mod. Phys. A 18 (2003) 1761; [arXiv:hep-th/0206114v1].
* (45) A. A. Saharian and M. R. Setare, Int. J. Mod. Phys. A 19 (2004) 4301.
* (46) A. A. Saharian and E. R. Bezerra de Mello, Int. J. Mod. Phys. A 20 (2005) 2380.
* (47) S. G. Mamayev and N. N. Trunov, Sov. Phys. J 23 (1980) 551.
* (48) C. D. Fosco, and E. L. Losada, Phys. Rev. D 78 (2008) 025017; [arXiv:hep-th/0805.2922v1].
* (49) E. Elizalde, S. D. Odindsov, A. Romeo, A. A. Bitsenko and S. Zerbini, _Zeta Regularization Techniques with Appliccations_ , (World Scientific, Singapore, 1994).
* (50) N. F. Svaiter, and B. F. Svaiter, J. Phys. A: Math. Gen. 25 (1992) 979.
* (51) M. S. R. Milt$\tilde{a}$o, Phys. Rev. D 78 (2008) 065023.
* (52) C.R. Hagen, Eur. Phys. J. C 19 (2001) 677.
* (53) A. A. Saharian, _The generalized Abel-Plana formula: applications to Bessel functions and casimir effect_ , [arXiv:hep-th/0002239v1].
* (54) V. V. Nesterenko and I. G. Pirozhenko, Phys. Rev. D 57 (1998) 2.
* (55) K. A. Milton, L. L. Deraad, and J. Schwinger, Ann. Phys. (N.Y.) 115, 388 (1978).
* (56) T. H. Boyer, Phys. Rev.174 (1968) 1764.
* (57) J. J. Sakurai, _Advanced Quantum Mechanics_ , Addison Wesley; Rev sub edition (1993).
* (58) R. MacKenzie and F. Wilczek, Phys. Rev. D 30, 2194 (1984).
* (59) S. S. Gousheh and R. López-Mobilia, Nucl. Phys. B 428, 189 (1994).
|
arxiv-papers
| 2012-04-14T15:16:01 |
2024-09-04T02:49:29.704621
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Saghian, M. A. Valuyan, A. Seyedzahedi and S. S. Gousheh",
"submitter": "Madad Ali Valuyan",
"url": "https://arxiv.org/abs/1204.3181"
}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.